learningaboutraredisasters:implicationsfor ... 02, 2014 · the key intuition for our results ......

Review of Finance (2014) pp 1ndash52doi101093rofrfu016

Learning about Rare Disasters Implications For

Consumption and Asset Prices

MAX GILLMAN1 MICHAL KEJAK2 and MICHAL PAKOS21University of MissourindashSt Louis and 2CERGE-EI Prague

Abstract Rietz (1988) and Barro (2006) subject consumption and dividends to rare disas-

ters in the growth rate We extend their framework and subject consumption and dividendsto rare disasters in the growth persistence We model growth persistence by means of twohidden types of economic slowdowns recessions and lost decades We estimate the model

based on the postwar US data using maximum likelihood and find that it can simultan-eously match a wide array of dynamic pricing phenomena in the equity and bond marketsThe key intuition for our results stems from the inability to discriminate between the short

and the long recessions ex ante

JEL Classification E13 E21 E32 E43 E44 G12

1 Introduction

Rietz (1988) proposes to model rare disasters as sudden cataclysms shortbut deep declines in the standards of living Using the large economicdeclines in the USA associated with World War I the Great Depressionand World War II Barro (2006) calibrates the probability of the disastersand argues that it is possible to account for the level of the equity premium1

Rietz and Barro consider a constant probability of disaster Farhi andGabaix (2011) Gabaix (2008 2012) Gourio (2008 2012 2013) GourioSiemer and Verdelhan (2013) Seo and Wachter (2013) Tsai and Wachter

The financial support of the Czech Science Foundation project No P40212G097(DYME Dynamic Models in Economics) is gratefully acknowledged Helpful commentsfrom Marek Kapicka Christopher Sims Sergey Slobodyan Laura Veldkamp the Editorand the anonymous referee are gratefully acknowledged CERGE-EI a joint workplace of

Charles University in Prague and the Economics Institute of the Academy of Sciences of theCzech Republic Prague Czech Republic1 In addition Brown Goetzmann and Ross (1995) study the long-term survival of finan-

cial markets while Barro and Jin (2011) and Barro and Ursua (2012) analyze the largeeconomic declines in international macroeconomic data

The Authors 2014 Published by Oxford University Press on behalf of the European Finance AssociationAll rights reserved For Permissions please email journalspermissionsoupcom

Review of Finance Advance Access published June 2 2014 at C

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(2013) and Wachter (2013) extend their work by making the probabilityvariable while Martin (2013) exploits cumulant-generating functionsIn accordance with Timmermann (1993) Gourio (2012) suggests learningas a fruitful way to endogenize the disaster probability Neverthelesslearning in this BarrondashRietz framework inevitably plays marginal rolebecause the deep declines in consumption are learned almostinstantaneouslyThis article proposes an alternative model of rare disasters as protracted

stagnation in the standards of living so-called ldquolost decadesrdquo in the macro-economics literature on depressions (Hayashi and Prescott 2002 Kydlandand Zarazaga 2002 Bergoeing Kehoe and Soto 2002) Interpreting disas-ters as protracted stagnation makes learning rather slow and generates asizable increase in the magnitude as well as in the variation of economicuncertainty thus dramatically enhancing the match of a broad range ofmacroeconomic and finance phenomenaIn the language of macroeconomics the uncertainty shocks of Bloom

(2009) arise endogenously as the consumption volatility fluctuates due tolearning contrary to the exogenous specification in the long-run riskmodels of Bansal and Yaron (2004) and Bansal Gallant and Tauchen(2007) Bansal Kiku and Yaron (2010) Bansal et al (2012) In a relatedpaper Orlik and Veldkamp (2013) propose a different way to endogenizethese uncertainty shocksIn the language of finance learning induces a procyclical variation in

consumption and dividend forecasts and a countercyclical variation in theEpstein and Zin (1989 1991) discount rates in response to changes in theaverage time to the (partial) resolution of uncertainty and thus our modelcan simultaneously match a wide array of dynamic pricing phenomena in theequity and bond marketsWe followMehra and Prescott (1985) and consider a version of Lucas (1978)

representative-agent model of asset pricing with exogenous stochastic andperishable dividends as extended to a continuous-time incomplete-informationsetting by Veronesi (2004) and David and Veronesi (2013) Similarly to Pakos(2013) we extend the regime-switching models of Cecchetti Lam and Mark(1990) David (1997) David and Veronesi (2013) Hamilton (1989) andVeronesi (1999 2000 2004) by subjecting consumption and dividends tohidden shifts in the growth rate and growth persistence as well The variabilityin the growth persistence is modeled by considering two types of recessions withidentical repressed growth rates but different mean duration the former cor-responds to a regular business-cycle recession while the latter is a rare lostdecade which happens on average once a century

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From the perspective of Mehra and Prescott (1985) and Weil (1989) ourunderlying hidden chain is not Markov with exponentially distributedsojourn times but rather semi-Markov2 with the sojourn times followingany distribution in our case a time-varying mixture of two exponential dis-tributions one for each recession type Modeling multiple recessions withdifferent mean duration inculcates a tail uncertainty about the sojourn timesas in Weitzman (2013) interpreted as long-run risk in Pakos (2013) Inrelated studies Branger Rodrigues and Schlag (2012) and quite recentlyJin (2014) emphasize the interplay between rare events and long-run riskIn comparison to the model of Rietz (1988) semi-Markov chains can be

reformulated as Markov ones by augmenting their state space Such refor-mulation in our setting leads to a Markov chain with three states expansionshort recession and long recession subject to the restriction that the reces-sions share exactly the same growth rate We think of ldquoa low-probabilitydepression-like third staterdquo of Rietz (1988) as a decade-long stagnation inconsumption with the disaster probability (the subjective belief about thethird state) fluctuating in response to changing economic conditionsOur model of hidden growth persistence is closely related to Cogley and

Sargent (2008) who study learning about the mean duration of recessions3

In their setting the duration distribution of expansions as well as recessionsis governed by fixed but unknown parameters while their representative agentis endowed with pessimistic priors based on the negative experience of theGreat Depression Such a calibrated model matches well to many pricingpuzzles in the equity market In a related study Collin-Dufresne Johannesand Lochstoer (2013) point out that the best unbiased estimate of a fixed buthidden parameter is a martingale that induces permanent shocks They extendCogley and Sargent (2008) by using the recursive preferences of Epstein andZin (1989 1991) so as to inculcate long-run risk into asset price dynamicsOur analysis differs from Cogley and Sargent (2008) and Collin-Dufresne

Johannes and Lochstoer (2013) in the following ways First rather thanusing pessimistic priors from the Great Depression we instead estimatethe consumption and dividend parameters by maximum likelihood fromthe postwar US data from 1952 to 2011 Second the persistence in ourmodel follows a hidden two-state Markov chain rather than being a fixedparameter which has the advantage that the risk premiums are stationaryThird each slowdown in economic activity confronts the investor with a

2 See Howard (1971 Chapter 10)3 Weitzman (2007) and Johannes Lochstoer and Mou (2012) also emphasize the import-ance of Bayesian updating about unknown structural parameters

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Peso-type problem about the mean duration of the recession generating atail uncertainty as in Weitzman (2013)The additional related literature includes Backus Chernov and

Martin (2011) Bates (2000) Branger Rodrigues and Schlag (2012)Santa-Clara and Yan (2010) and quite recently Schreindorfer (2014)These studies suggest to measure the frequency and size of such disastersusing the option price data and other derivatives on US equity indexesFurthermore while working on our article we have come across a studyof Lu and Siemer (2013) who study learning about rare events in a frame-work without a tail uncertainty about the recession durationThe article is organized as follows In Section 2 we present the formal

model and derive the theoretical implications of variable growth persistencefor consumption and asset prices In Section 3 we present the results ofestimation Section 4 describes the quantitative implications of learning forconsumption and asset prices while in Section 5 we present sensitivityanalysis and discuss our preliminary results for option pricing Weconclude in Section 6 Detailed mathematical proofs are found in theSupplementary Appendix

2 Model

We start by briefly describing the representative investorrsquos preferences andspecifying the hidden semi-Markov model of the cash-flow growth rates Wethen go on to solve the investorrsquos optimal consumption-portfolio problemIn order to do this we first solve the inference problem by introducing theposterior distribution over the discrete number of hidden states and derive arecurrent relationship for its law of motion by applying the Bayes rule Wethen discuss how the variation in the posterior distribution generates time-varying endogenous economic uncertainty in terms of changing forecasts aswell as changing forecast error variances of the T-period cash-flow growthrates Second we take the posterior distribution and make it a part of thestate vector in the dynamic programming problem This is relevant as itmakes the optimization problem Markovian leading to the standardHamiltonndashJacobindashBellman (HJB) equation Using the derived first-orderconditions and the guess-and-verify method we then derive the pricingequations for unlevered and levered equity real zero-coupon bonds as wellas European options The section additionally relates the real yield curvesand the bond risk premiums to the term structure of the T-period forecastsas well as the T-period forecast error variances of the consumptiongrowth rate

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21 PREFERENCE SPECIFICATION

The representative agent maximizes the recursive utility function of Epsteinand Zin (1989 1991) over his consumption stream ct and the continuationutility Jt defined by the recursion

Jt frac14 Et

Z 1t

U c Jeth THORNd

eth1THORN

with the CES utility aggregator

U c Jeth THORN frac14

1 1

c11 1 eth THORNJeth THORN

1

1 eth THORNJeth THORN11

eth2THORN

In these expressions Et denotes the conditional expectation operator 0is the rate of time preference 0 is the coefficient of the relative riskaversion 0 is the magnitude of the elasticity of the intertemporal sub-stitution and frac14 1 eth THORN= 1 1

is a measure of the nonindifference to

the timing of the resolution of uncertainty as we relax the independenceaxiom of the expected utilityThe investor prefers early resolution of the uncertainty when the current

marginal utility Uc falls as relatively more of the consumption occurs in the

future measured by higher continuation utility J In this case the cross-derivative

JUc

is negative which happens for gt 1 The expected

utility is nested for J

Uc

frac14 0 which happens for frac14 1

22 ASSET MARKETS

We endow the investor with a single Lucas tree called unlevered equity (orconsumption claim) and denote it with the superscript u The asset yields acontinuous flow of dividends at the rate Du

t In addition we distinguishbetween the total wealth which is unobservable and the aggregate equitymarket We thus introduce levered equity denoted with the superscript l Thelevered equity yields a continuous flow of dividends at the rate Dl

t which isdifferent from Du

t We refer to the unlevered and levered dividends jointly asthe cash flows and distinguish them using the superscript e 2 E frac14 u lf g Wefurthermore introduce real zero-coupon bonds with maturities of up to 30years (superscript b) and European call options (superscript c) For the sim-plicity of notation we denote the class of the securities A frac14 E [ b cf g Weassume that all assets in A except the unlevered equity are in zero net supplyOur bivariate time series model for the cash-flow growth rates generalizes

the standard Markov-trend model in logs introduced by Hamilton (1989) to


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a semi-Markov setting by subjecting the instantaneous cash-flow growthrates dget frac14 d logDe

t for e 2 E to hidden semi-Markov shifts

dget frac14 estdtthorn edzet eth3THORN

The predictable component est2 e e

with e lt e is driven by a two-state semi-Markov chain st which is hidden in the standard Brownian noisezt frac14 zut z

lt

0 We assume for tractability that zt and st are statistically inde-

pendent processesOur cash flow model in Equation (3) implies that the forecast error

variance of the cash-flow growth rate over the next instant

eeth THORN2dt frac14 vart dget Et dget

eth4THORN

is constant Nonetheless our learning model with hidden shifts generates apredictable variation in the T-period forecast error variances when the in-stantaneous cash-flow growth rates dget are time-aggregated from the infini-tesimal decision intervals to their T-period counterparts

R tthornTt dge as shown in

detail in Section 24 Our setting thus differs significantly from the extensivelong-run risk literature where the predictable variation in the forecast errorvariance et

2is exogenous rather than endogenous due to learning4

22a Semi-Markov Chain

Current literature models business-cycle fluctuations in terms of two-statehidden Markov chains with the state space

S frac14 fs1 frac14 expansion s2 frac14 recessiong eth5THORN

In a continuous-time setting it is natural to express the transitionprobabilities in terms of the hazard rates of transition

i frac14X

sj2Sn sif g

ij eth6THORN

where ij denotes the nonnegative transition intensity for any si sj 2 S andi 6frac14 j If the hazard rates are constant the density of the sojourn time i forifrac14 1 2 is given by the exponential distribution

fi teth THORN frac14 i exp iteth THORN eth7THORN

4 See in particular Bansal and Yaron (2004) and Bansal Gallant and Tauchen (2007)Bansal Kiku and Yaron (2010) Bansal et al (2012)

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for nonnegative t Exponential distribution tends to be a common choice formodeling sojourn times due to the mathematical tractability allowed by thememoryless property

P i gt xthorn yi gt x

frac14 P i gt y

eth8THORN

However exponential distributions have the drawback that they feature lightright tails if the hazard rate is inferred from the macroeconomic data whichin other words means that long recessions are extremely rare In fact thefollowing back-of-the-envelope calculation suggests that the probability ofobserving an economic recession with a duration of more than 10 years2 10f g equals

P 2 gt 10js frac14 s2f g frac14

Z 110

f2 eth THORNd frac14 exp 102eth THORN frac14 exp 10eth THORN frac14 000005 eth9THORN

when the mean duration of the recession state 12 is four quarters In thatcase the number of slowdowns until the first appearance of a lost decadefollows the geometric distribution with the mean 000005eth THORN

1frac14 22000 In

other words it takes on average about 22000 transitions in order to drawat least a decade-long recession which is arguably implausible due to theextreme rareness of the event56

In order to model the long-lasting recessions in a more plausible way wepropose to generalize the standard two-state Markov chain setting with thestate space S to a two-state semi-Markov chain setting where the probabilitylaw governing the recession sojourn time is a mixture of two exponentialdistributions Although semi-Markov chains can be arguably less tractablethere are special instances when they can be easily represented in terms ofrestricted Markov chains by augmenting their state space As shown inMurphy (2012 Section 176) the two-state semi-Markov chain can be ex-pressed in terms of a three-state augmented Markov chain in our case withtwo substates for the downturn which differ in the mean duration The firstsubstate corresponds to the common business-cycle recession and has themean duration 12 The second substate corresponds to the rare but

5 As a piece of anecdotal evidence consider the lost decade experienced by Japan at theend of the 20th century6 Diebold Lee and Weinbach (1994) suggest modeling transition intensities as logisticfunctions of certain exogenous variables The drawback however is that in a generalequilibrium setting one needs to specify the dynamics of those exogenous variables infine detail


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protracted recession where we set the mean duration 13 equal to fortyquarters The augmented state space is

~S frac14 f~s1 ~s2 ~s3g eth10THORN

where ~s1 frac14 s1 1eth THORN ~s2 frac14 s2 2eth THORN and ~s3 frac14 s2 3eth THORN The semi-Markov propertyimplies equality of the growth rates across the recession types that ise1 frac14 e and e

2 frac14 e frac14 e

3 for each e 2 E As a result of two recessiontypes the sojourn times of a low-growth epoch follow a mixture of twoexponential densities with different means7

Furthermore we assume that upon leaving the expansion state naturetosses a biased coin according to the Bernoulli probability distributionq 1 qeth THORN for some q 2 0 1eth THORN where the outcome of the toss decides thetype of the downturn As a result the transition probability matrixbetween times t and tthornT equals the matrix exponential exp Tf g wherethe transition intensity matrix8 is given by

frac141 q1 1 qeth THORN12 2 03 0 3

0

1A eth11THORN

We note that the two-state model without long recessions is nested for qfrac14 1because then the hazard rate of entering the long recession 1 qeth THORN1 is zeroThe invariant distribution frac14 1 2 3eth THORN

0 is given as the left eigenvectorthat corresponds to the zero eigenvalue of the transition intensity matrixsubject to the restriction that

P3ifrac141 i frac14 1 In particular the invariant

probability

3 frac141 qeth THORN 13

11 thorn q12 thorn 1 qeth THORN 13

eth12THORN

equals the average time spent in the long recession 1 qeth THORN13 divided by theaverage length of one whole cycle 11 thorn q12 thorn 1 qeth THORN13 This result isimportant in the empirical section where we propose to model the rarelong recession as a lost decade that occurs on average once a century thussetting 13 frac14 10 years and 3 frac14 01 exactly in line with Equation (12)

7 Such a mixture density is called hyperexponential distribution Hyperexponential densityis thus the probability distribution that governs the sojourn time spent in recession when thetype is hidden8 Identification requires that we rule out instantaneous transitions between short and longrecessions by setting 23 frac14 32 frac14 0 This assumption however is not particularly restrictivebecause the transition probabilities P stthornT frac14 ~sj

st frac14 ~si

are positive for any finite intervalTgt 0 and i j frac14 1 2 3

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23 INFERENCE PROBLEM

The investorrsquos inference problem is to extract the current but hidden state stfrom the history of the cash-flow signals F t frac14 gu g

l

for t

For that

purpose we define the belief

i t frac14 P st frac14 ~sijF tf g for i frac14 1 2 3 eth13THORN

and introduce the so-called ldquoinnovation processrdquo ~zet the increment of which isthe normalized forecast error of the cash-flow growth rates over the nextinstant

d ~zet frac141

edget Et dget

eth14THORN

First Liptser and Shiryaev (1977) show that ~zet is a Brownian motion in theinvestorrsquos filtration which makes the investorrsquos intertemporal optimizationproblem Markovian by allowing us to treat the beliefs t frac14 1 t2 t3 t

0as part of the state vector that reflects the variation in the investment op-portunity set perceived by the investor Second it is straightforward to seethat the innovation process is correlated with the hidden semi-Markovchain st Third the innovation process enables us to express the cash-flowdynamics in Equation (3) as the sum of the predictable part me

tdt frac14 Et dget

and the cash-flow news dget Et dget

by using Equation (14)

dget frac14 metdtthorn

ed ~zet for e 2 E eth15THORN

We can then apply the Bayes rule and obtain the following law of motion forthe beliefs t

di t frac14 i tdtthornXe2E

ei td ~zet eth16THORN

23a Intuitive explanation

Although the reference to the formal proof is provided in the SupplementaryAppendix B the outline of the intuition behind Equation (16) is rela-tively straightforward First the predictable part given by the drifti t frac14

P3jfrac141 j tji reflects the dynamics of the perfectly observable semi-

Markov chain when augmented to the three-state Markov chain Secondthe volatility

e1 frac14 1 t 1 1 t

e e

e

eth17THORN


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and

ei t frac14 1 ti te e

e

for i frac14 2 3 eth18THORN

measures the weight that the investor puts in his sequential updating on thenormalized news d ~zet in Equation (16) Indeed the weight e1 t is proportionalto the economic uncertainty about the underlying instantaneous growth ratemeasured by the prior variance

vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN

of the Bernoulli distribution over the growth rates e and e at the beginningof the instant t tthorn dteth THORNSpeaking more formally the Bayes rule says that the posterior odds equal

the prior odds times the likelihood ratio Thus the increment in the log ofthe odds

O1 23 frac141 t

1 1 teth20THORN

in favor of the expansion equals the log-likelihood ratio9 The log-likelihoodratio in favor of the hypothesis H0 e

stfrac14 e against the alternative

H1 estfrac14 e conditional on the new data dget and no regime shifts in the

interval t tthorn dteth THORN equals

1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN

As a result of Equation (15) the increment in the log odds due to the arrivalof the new information is given by

d logO1 23 frac14 O dteth THORN thorne e

e

d ~zet eth22THORN

and we recover the diffusion term e1 t in Equation (16) by applyingIto lemma to Equation (20) As can be seen in Equation (22) good cash-flow news d ~zet always raises the posterior odds O1 23 in favor of theexpansion

9 In the sequential Bayesian updating the posterior for the previous instant t dt teth THORN

becomes the prior for the next instant t tthorn dteth THORN

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Furthermore the total weight e2 t thorn e3 t frac14

e1 t lt 0 is split10 across the

beliefs 2 t and 3 t according to the prior odds at the beginning of theinstant t tthorn dteth THORN in favor of the short recession

O23 frac142 t

3 teth23THORN

as e2 t=e3 t As a result good cash-flow news d ~zet during the short recession

lowers not only the beliefs 2 t and 3 t but also brings down the relativelyhigh odds in favor of the shorter recession Indeed suppose we know thatst 6frac14 ~s1 and we try to discriminate between short and long recessions O1 23

11

Speaking more formally let us denote T the random time spent in the low-growth state st 2 ~s2 ~s3f g and recall that it follows an exponential distributionwith mean 12 in the short recession and 13 in the long recession The Bayesrule implies that the increment in the log of the odds O23 again equals thelog-likelihood ratio

d logO23 frac14 2 3eth THORNdt eth24THORN

which is basically a special case of Equation (16) for 1 t frac14 0 As a result theposterior odds in favor of the short recession O23 have tendency to decreasewith the amount of time spent in the low-growth state and the learning aboutthe growth persistence is time-consuming in proportion to the differencebetween the hazard rates for the short recession and for the long recession2 3

23b Endogenous disaster probability

Rare consumption disasters in our semi-Markov model manifest themselvesas unfavorable draws of the recession duration When we represent the two-state semi-Markov chain in terms of a restricted three-state Markov chainsubject to the equality constraint e

2 frac14 e frac14 e

3 for each e 2 E frac14 u lf g weidentify the rare consumption disasters as the long recessions st frac14 ~s3 and thedisaster probability as the belief

3 t frac14 P st frac14 ~s3jF tf g eth25THORN

The endogenous variation in the disaster probability 3 t comes fromthe fluctuations in the posterior odds in favor of the expansion O1 23 inEquation (20) and the posterior odds about the type of the recession O23

10 Note that the restriction that the beliefs sum to oneP3

ifrac141 i t frac14 1 implies that the in-crement d

P3ifrac141 i t

frac14 d

1is zero and hence the drifts as well as volatility must sum to

zero as wellP3

ifrac141 i t frac14 0 frac14P3

ifrac141 ei t

11 We can equivalently think of the analysis as being conditional on st 6frac14 ~s1


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in Equation (23) In fact each recession st frac14 ~s2 carries with it the subjectiverisk that it may correspond to the lost decade regime ~s3 due to theunobservability of the recession type Such a novel model of consumptiondisasters is an example of a Peso problem which refers to a situation inwhich the possibility of some infrequent event (such as a long recession) hasan effect on asset prices12

24 FLUCTUATING ECONOMIC UNCERTAINTY

It is well-known in the literature that the variation in economic uncertainty isthe key to successfully explaining the variation in asset prices13 Economicuncertainty can be measured by the degree of difficulty in making preciseforecasts of future cash-flow growth rates measured by the term structure ofthe forecast error variances of the T-period-ahead cash-flow growth ratesWe show that the introduction of hidden regime shifts generates endogen-

ous variation in the forecast error variance of the cash-flow growth rates andthus in economic uncertainty The key to showing this consists in time-aggregating the growth rates from the infinitesimal decision intervals totheir T-period intervalsIn order to time-aggregate the instantaneous cash-flow growth rates dget

for each e 2 E let us denote the T-period growth rate

getT frac14

Z tthornT

t

dge eth26THORN

the mean T-period growth rate

etT frac14

Z tthornT

t

esd eth27THORN

and the T-period innovation

zetT frac14

Z tthornT

t

edze eth28THORN

As a result of the time aggregation the cash-flow model leads to

getT frac14 etT thorn zetT eth29THORN

12 See Evans (1996) for a review of the Peso literature13 See in particular Bansal and Yaron (2004) Bansal Gallant and Tauchen (2007) BansalKiku and Yaron (2010) Bansal et al (2012)

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with the following variance decomposition of the T-period cash-flow growthrate

vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN

Expressed in words the variance of the T-period cash-flow growthrate vart getT

n ois given by the sum of the forecast error variance of the

mean T-period growth rate (first term) and the forecast error variance ofthe T-period innovations (second term) In particular the volatility of theannual consumption growth rate corresponds to ut 1 The commonly usedautoregressive processes imply that the forecast error variance of the meanT-period growth rate (first term) vart

etT

n o is constant14

We show in the following two sections that learning about hidden regimeshifts can generate countercyclical fluctuations in the forecast error varianceof the T-period cash-flow growth rates with constant e and hence constant

vart zetT

n ofrac14 eeth THORN2T eth31THORN

24a Time-varying forecast error variance

Let us denote the T-period forecast conditional on the hidden state

metTji frac14 E getT

F t st frac14 ~si

n oeth32THORN

and the T-period forecast error variances conditional on the hidden state

etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN

The conditional moments given the hidden state st vary due to the possibilityof a regime change with the more persistent state of lower hazard rate oftransitioning i displaying lower volatility etTji Furthermore variance

14 For example Bansal and Yaron (2004) in their Model I specify the expected consump-

tion growth rate as an AR(1) process subject to homoscedastic innovations Therefore theirmodel generates constant forecast error variance of the T-period consumption growth ratewhich they relax in their Model II by introducing exogenous variation in the consumptionvariance et

2as an AR(1) process


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decomposition conditional on the hidden state analogous to Equation (30)can be expressed as

etTji

2frac14 etTji

2thorn eeth THORN2T eth35THORN

We then condition down to the investorrsquos information set F t which does notcontain the hidden state st The mean T-period cash-flow growth rateme

tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141

metTjii t eth36THORN

Furthermore using the decomposition that the variance equals the varianceof the conditional mean plus the mean of the conditional variance15

vart xf g frac14 vart E xjF t stf gf g thorn Et var xjF t stf gf g eth37THORN

yields the following decomposition of the corresponding T-period cash-flowvariance vart getT

n oin Equation (30)

etT

2frac14X3ifrac141

etTji

2thorn eeth THORN2T|fflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflffl

Variance under Complete Info

0BB

1CCAi t

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflMean Variance under Incomplete Info

thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflVariance of Mean Growth under Incomplete Info

eth38THORN

As we can see from Equation (36) and Equation (38) the variation in boththe mean T-period growth rate forecast me

tT as well as the volatility etTdepends on the evolution of the beliefs 1 t2 t3 t

016 In addition the

T-period forecast metT attains its maximum when the confidence in favor of

the expansion state is the highest and its minimum when the confidence infavor of the lost decade is the highest whereas the corresponding forecast

15 Note that the conditional moments Et and vart are conditional only on F t which doesnot include the hidden state st16 This is consistent with Veronesi (1999) who shows in Proposition 6 that if expectedconsumption growth rate follows a hidden Markov chain then shocks to the instantaneousexpected dividend growth rate are necessarily heteroscedastic Although he does not time

aggregate the cash-flow growth rates from infinitesimal decision intervals to the finite oneshis two-state Markov chain setting is able to generate time-varying consumption volatilityafter the aggregation However such variation would be quantitatively smaller in compari-son to our two-state semi-Markov setting

14 M GILLMAN ETAL

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error varianceetT

2attains values based on the magnitude of the economic

uncertainty measured by the dispersion of the beliefs

24b Countercyclical consumption and dividend volatility

A two-state continuous-time Markov chain can be expressed as a linearcombination of two independent compensated Poisson processes leadingto a continuous-time AR(1) process with innovations that are non-Gaussian and heteroscedastic having the instantaneous variance propor-tional to the persistence of the state e e

2stdt for each st 2 S17 It

can be shown that the analogous result carries over to our two-state semi-Markov settingOur empirical estimates in Section 321 confirm that the transition hazard

rates satisfy 2 gt 1 and we thus obtain the ordering on the forecast errorvolatility of the mean T-period cash-flow growth rate conditional on thehidden state st as

etTj2 gt

etTj1 Note that we omit the discussion related

to the rare state by assuming 3 t 0 In view of Equation (35) the volatilityof the T-period cash-flow growth rate under complete information is thuscountercyclical because the mean duration of the recessions 1st

is empiric-ally shorter in comparison to the expansionsFurthermore in case of incomplete information about the underlying state

the variance of the T-period cash-flow growth rate decomposed in Equation(38) has two terms the mean of the conditional variance under completeinformation plus the variance of the conditional mean hidden due to incom-plete information The first term etTj11t thorn

etTj2 1 1t

is a decreasing

function of the belief 1 t due to the ordering of the conditional volatilityetTji displayed above and it is thus countercyclical The second term

metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN

is a quadratic function of the belief p1t and it is an increasing function inthe belief 1 t for 1 t lt

12 but decreasing for 1 t gt

12 due to the ordering of

the T-period-ahead forecasts metTj1 gt me

tTj2 implied by e1 gt e

2 As we seelater in our parametrization using maximum likelihood estimates the secondterm is usually dominated by the first one and thus the total cash-flowvolatility etT remains countercyclical even after accounting for incompleteinformation

17 See Hamilton (1989) for discrete time treatment


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24c Cash-flow dynamics over the phases of the business cycle

The forecast as well as the forecast error variance of the T-period cash-flowgrowth rate vary monotonically over the separate phases of the businesscycle First transitioning to the high-growth state st frac14 ~s1 is associatedwith a gradual improvement in the T-period forecast me

tT frac14 Et

getT

as

well as the T-period forecast error varianceetT

2frac14 vart

getT

These

gradual changes are driven by the rise in the posterior odds O1 23 inEquation (20) as the high-growth state is being recognized Second transi-tioning to the low-growth state st frac14 ~s2 is associated with a gradual deterior-ation in the forecast and the forecast error variance of the T-periodcash-flow growth rate Again these gradual changes are driven not onlyby falling posterior odds O1 23 as the low-growth regime is beingrecognized but also rising posterior odds in favor of the long recessionO123 in Equation (23) as the likelihood of a protracted slowdown isincreasing18

25 INVESTORrsquoS PROBLEM

The investorrsquos financial wealth Wt comprises the unlevered and the leveredequity as well as the real zero-coupon bond with a given maturity T and theriskless cash account offering the continuously compounded rate of return rtWe denote the share of each asset a 2 A in the wealth portfolio Wt as

at and

let the investor decide continuously how much to consume and how much tosave out of his current wealth Wt The dynamic budget constraint takes thestandard form as in Merton (1971)

dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN

where we still need to specify the law of motion for the asset return dRat

According to Equation (14) the increment in the innovation process d ~zet isthe normalized instantaneous forecast error of the cash-flow growth rate dgetfor each e 2 E frac14 u lf g and it is to be thought of as the news about the currenthidden state st 2 ~S In informationally efficient asset markets news arrivalleads to an instant revision in the price of each asset a 2 A generating a

18 Of course the economic uncertainty in s 2 ~s3f g will eventually decline once the unob-servable state is recognized but it takes a long time in comparison to the mean duration ofthe short recession ~s2

16 M GILLMAN ETAL

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surprise return (also called news innovation or forecast error) in proportionto the asset volatility a e

t

dRat Et dRa

t

frac14Xe2E

a et d ~zet eth41THORN

where the net return is defined as usual

dRat frac14

dPat thornDa

tdt

Pat

eth42THORN

The realized return dRat is composed of the predictable part given by the

expected return Et dRat

frac14 13at dt and the unpredictable part given by the

surprise return in Equation (41)

dRat frac14 13

at dtthorn

Xe2E


In our model the expected return 13at and the return volatility a et for each

e 2 E and each asset a 2 A are determined jointly by market clearing ingeneral equilibriumThe investorrsquos consumption-portfolio problem is to maximize his lifetime

utility defined recursively in Equation (1) subject to the dynamic budgetconstraint in Equation (40) leading to the standard HJB equation19

0 frac14 maxculbf g

U c Jeth THORNdtthorn Et dJ W12eth THORN

eth44THORN

where the posterior distribution becomes a part of the state vector inaddition to the wealth W20 Ito lemma applied to the continuation utilityJ frac14 J W12eth THORN along with the budget constraint in Equation (40) and thedynamics of the return dRa

t in Equation (43) then leads to a nonlinearpartial differential equation of the second order for J

25a First-order conditions and equilibrium

The first-order condition for the consumption rate c states that the marginalutility of consumption equals the marginal utility of wealth U

c frac14JW The

first-order condition for the portfolio weight a for the asset a 2 A statesthat the total demand for asset a equals the myopic demand plus the

19 See Duffie and Epstein (1992a 1992b)20 Note that the belief p3 is given implicitly due to the restriction that the probabilities sumto one


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intertemporal hedging demand that arises from the fluctuations in the in-vestorrsquos own uncertainty about the state of the macroeconomy (Merton1973 Veronesi 1999)In equilibrium the conditions ct frac14 Du

t ut frac14 1 l

t frac14 0 and bt frac14 0 must

hold for the asset and the goods markets to clear

25b Value function and wealth-consumption ratio

The first-order condition for the consumption rate implicitly defines theoptimal policy function c frac14 c W12eth THORN Invoking the homotheticity of therecursive preferences log c

logW frac14 1 implies that the policy function c W12eth THORN isseparable across the financial wealth W and the beliefs 12eth THORN The separ-ability of the policy function in turns implies through the first-order condi-tion that the value function is also separable across W and 12eth THORN

J W12eth THORN frac14 u 12eth THORNfrac12 W1

1 eth45THORN

where we choose to parametrize it in terms of the equilibrium wealth-consumption ratio

u 1 t2 t

frac14Wt=ct eth46THORN

The conjecture in Equation (45) reduces the nonlinear PDE coming fromthe HJB equation for the continuation utility J in Section 25 to thenonlinear degenerate-elliptic partial differential equation of the secondorder for u presented in Proposition D1 in the SupplementaryAppendix DWhen the investor prefers early resolution of uncertainty (ie lt 0) the

cross-derivative of the marginal utility i

JW

is negative The intuition for

these results is simple A positive short-run news d ~zet always raises the pos-terior odds in favor of the expansion O1 23 and thus the beliefs p1 and p2go up This in turn leads to an improvement in the T-period forecasts offuture consumption growth rate mu

tT raising the duration of the consump-tion stream and so delaying the mean time to the (partial) resolution ofuncertainty about the consumption stream This is disliked by the investorwith a preference for early uncertainty resolution and the marginal utilityfalls The fall in the marginal utility is larger when the increase in theduration is bigger allowing us to order the cross-derivatives as1

JW

lt

2JW

lt 0

As a corollary the wealth-consumption ratio is procyclical being anincreasing function of the beliefs

u

1gt u

2gt 0

18 MGILLMAN ETAL

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26 STATE-PRICE DENSITY

The absence of arbitrage implies the existence of a positive state-price densityprocess Mt which in case of the EpsteinndashZin preferences in Equation (1) isgiven by the formula21

Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN

The following proposition presents the law of motion for the state-pricedensity and decomposes the corresponding risk prices into the Lucas-Breeden component reflecting the covariance with the consumptiongrowth and the variable timing component reflecting the changing forecastsof the time to the (partial) resolution of the consumption uncertainty interms of the posterior odds in Equation (20) and in Equation (23) In ourparametrization of the preferences late resolution of the uncertainty isdisliked by the investor and the cross-derivative

JUc

is negative as

argued in Sections 21 and 252 In case of the expected utility we recoverthe standard consumption-based capital asset pricing model with zero timingcomponents because the independence axiom implies that the marginalutility of consumption does not depend on the continuation utility andhence

JUc

is zero which happens for frac14 1

Proposition 21

Let the equilibrium state-price density Mt be given by Equation (47) Then

Mt satisfies

logMt frac14 t 1 eth THORN

Z t

0

u 1 2

1d xt eth48THORN

with

xt frac14 logDut thorn 1 eth THORN logu 1 t2 t

eth49THORN

Mt evolves according to the stochastic differential equation

dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN

21 See Duffie and Epstein (1992a 1992b) and Schroder and Skiadas (1999)


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where

the instantaneous riskless interest rate rt frac14 r 1 t2 t

is given by

Equation (86)

the risk price functions et frac14 e 1 t2 t

for each e 2 E are given by

et frac14 ue u|fflfflfflfflzfflfflfflffl

LucasBreedenComponent

thorn 1 eth THORNX2ifrac141

ei t1

ut

u

i1 t2 t

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl

TimeVaryingTimingComponent

eth51THORN

where the symbol e u is the Kronecker delta22

Proof

See the Supplementary Appendix C laquoProposition 21 together with the equilibrium conditions allows us to

express the first-order condition for the portfolio weights at in Problem

(44) for each asset a 2 A as the restriction that the risk premium equalsthe negative of the covariance with the state-price density growth rateEt dR

at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN

As can be seen the risk prices et measure the increase in the asset risk

premiums in response to the marginal increase in the exposure to theBrownian shock d ~zet for each e 2 E

26a Risk prices

The risk prices et for e 2 E in Equation (51) measure the sensitivity of the

growth rate of the marginal utility of wealth to the news carried by theBrownian shocks23 d ~zet

M1t dMt Et dMtf geth THORN|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflMarginalUtilityGrowthRate Surprise

frac14 utd ~zut|fflfflzfflffl

Response toConsumption Surprise

thorn ltd ~zlt|fflfflzfflffl

Response toDividend Surprise

eth53THORN

22 Recall that the Kronecker delta satisfies e e frac14 1 and e u frac14 0 for u 6frac14 e23 The levered dividend shock d ~zlt enters because it is correlated with the hidden state s andis thus also a source of news as shown in (16)

20 M GILLMAN ETAL

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The innovations in the cash-flow growth rate d ~zet are iid and correspondto the short-run cash-flow news Good short-run cash-flow news alwaysincreases the posterior odds O1 23 in Equation (20) in favor of the expansionstate and generates good long-run cash-flow news in terms of the improvedforecasts of the T-period cash-flow growth rates me

tT24 The effect of short-

run cash-flow news on the long-run growth prospects is called the ldquocash-floweffectrdquo in the literature Furthermore the good short-run news also changesthe duration of the consumption stream as well as the forecast error varianceand tends to lengthen the mean time to the (partial) resolution of the con-sumption uncertainty leading to a rise in the discount rates The effect of theshort-run news on the discount rates is called the ldquodiscount-rate effectrdquo inthe literature According to the analysis in Section 252 the cash-flow effectdominates the discount rate effect and the wealth-consumption ratiou 1 t2 t

is pro-cyclical rising unexpectedly on good short-run news

d ~zet We call the innovation in the wealth-consumption ratio coming fromthe long-run cash-flow as well as discount rate news simply the long-runnewsA positive piece of news in both the short-run and the long-run generates a

positive surprise in the return on the wealth portfolio

dRat Et dRa

t

frac14 dgat Et dgat

|fflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflshortrunnews

thorn at

1da

t Et dat

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffllongrunnews

for a frac14 u

eth54THORN

and a negative surprise in the marginal utility of wealth When we invokeEquations (45) and (54) and apply Ito lemma to u

t frac14 u 1 t2 t

we

easily recover the formulas for the risk prices et in Proposition 21

27 LEVERED EQUITY PRICES

The absence of arbitrage implies that unlevered and levered equity pricesequal the expected discounted value of the future dividend stream

MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g

24 Recall that the shocks to the beliefs eth12THORN are persistent in proportion to the meanduration of the states 1j for jfrac14 1 2 3 as well as the conditional probability of transi-tioning to the short recession out of the expansion q


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The equity price Pat trends upward making it more tractable to solve for the

equilibrium price function in terms of the corresponding pricendashdividendratio25

at frac14

Pat

Dat

eth55THORN

Proposition 2 in the Supplementary Appendix D exploits the martingaleproperty of the gain process Mt

at D

at thorn

R t0 MD

ad and derives the

Fichera boundary value problems to be solved numerically as described inthe Supplementary Appendix D for the ratio

at frac14 a 1 t2 t

eth56THORN

27a Procyclical pricendashdividend ratio

A preference for early resolution of uncertainty (ie lt 0) implies thatequity prices rise on good news This happens because a positive innovationin the cash-flow growth rate d ~zet (short-run news) raises the posterior odds infavor of the expansion state in Equation (20) improving the T-period fore-casts in the cash-flow growth rate (cash-flow effect) Although the increase inthe posterior odds tends to lower future mean discount factors Mtthorn

(discount rate effect) the cash-flow effect dominates the discount rateeffect in our parametrization The dominance of the cash-flow effect thenimplies that increasing the belief i t for each ifrac14 1 2 necessarily lowers thebelief 3 t frac14 1 1 t 2 t ceteris paribus which improves the growth pro-spects and leads to an increase in the pricendashdividend ratio a

t Hence thederivative

a

iis positive In fact the belief 1 t corresponds to the expansion

state and its increase improves growth prospects more than the correspond-ing increase in 2 t allowing us to order the derivatives a

1gt a

2gt 0

However short recessions in our parametrization last on average about 1year which indicates that the long-run improvement in the growth prospectsis about the same hence the derivatives a

1and a

2are of comparable

magnitudes

27b Conditional return moments

The procyclical variation in equity prices in turn generates a correspondingcountercyclical variation in the conditional moments of the equity returns

25 Observe that the pricendashdividend ratio for the unlevered equity in equilibrium must equalthe wealth-consumption ratio

Put

Dutfrac14 u

t frac14Wt

ct and its pricing equation was partially analyzed

already in Section 251

22 M GILLMAN ETAL

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To see this let us look first at the conditional equity return volatility a et for

e 2 E which measures the sensitivity of the equity return to the cash-flownews d ~zet as shown in Equation (41) According to the analog of Equation(54) for afrac14 u we can also decompose the news for afrac14 l into short-run newsand long-run news The long-run news can be further decomposed using Itolemma as da

t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total

sensitivity to news a et frac14

a e 12eth THORN equals the sum of the sensitivity tothe short-run news which is constant by assumption and the sensitivity tothe long-run news which depends on the beliefs

a e|zTotal News Sensitivity

frac14 ea e|fflfflzfflfflShortRun News Sensitivity

thornX2ifrac141

ei1

a

a

i|fflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflLongRun News Sensitivity

eth57THORN

where the symbol a e is the Kronecker deltaThe volatility a e

t in Equation (57) is positive and countercyclical Positiveshort-run news d ~zet leads to a rise in the odds in favor of the expansion statein Equation (20) increasing 1 t but decreasing the sum 2 t thorn 3 t by exactlythe same amount But the belief 3 t is relatively small due to the inherentrareness of long recessions and thus the news di t Et di t

for ifrac14 1 2 are

of comparable magnitude but opposite sign The inequality a

1gt a

2from

the previous section then implies that the long-run news is always positivelycorrelated with the short-run news In addition the sensitivity of beliefs toshort-run news e1 t in Equation (16) is proportional to the prior variancevart

est

in Equation (19) which tends to be large during times of

heightened economic uncertainty measured by the dispersion of the beliefsand leads to countercyclical variation in the magnitude of the long-run newsand hence in the equilibrium equity volatility a e

t for each e 2 ESecond the first-order condition in Equation (52) says that the equity risk

premium can be decomposed into the Lucas-Breeden component (short-runrisk premium) plus the timing premium due to the nonindifference to thetiming of the resolution of uncertainty about future consumption growthinherent in the EpsteinndashZin preferences (the long-run risk premium)

Et dRat rtdt

|fflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflEquity Risk Premium

frac14 covt dRat Et dRa

t

dgut Et dgut

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflShortRun Equity Risk Premium

thorn 1 eth THORNcovt dRat Et dRa

t

1

udu

t Et dut

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl

LongRun Equity Risk Premium

eth58THORN


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The equity risk premium inherits the property of countercyclical variationfrom the volatility a e

t as well as the risk prices et We note again that

although positive short-run news does generate positive long-run news themagnitude of the long-run news is countercyclical as explained above whichtends to lower both covariances through ei t in Equation (16) in times of highconfidence when 1 t 0 or 1 t 1The return variance vart dRa

t

frac14

Pe2E a e

teth THORN2

dt as well as the expected

return Et dRat

frac14 13at dt are instantaneous moments corresponding to infini-

tesimal decision intervals and thus must be time-aggregated to finite intervalsas described in Proposition E1 in the Supplementary Appendix E in order tobe comparable to the data

28 REAL ZERO-COUPON BOND PRICES

Absence of arbitrage implies that the price of the real zero-coupon bond Pbt

with a given maturity T equals the expected discounted value of the principalpayment

MtPbt frac14 Et MTP

bT

eth59THORN

where we normalize the principal PbT 1 Proposition D2 in the

Supplementary Appendix D exploits the martingale property of thedeflated price MtP

bt and derives the partial differential equation for

the bond price Pb as a function of the beliefs 12eth THORN and time t

Pbt frac14 Pb 1 t2 t tT

eth60THORN

As discussed in Section 261 and applied to pricendashdividend ratios later inSection 271 the effect of uncertainty on asset prices can be decomposedinto the procyclical cash-flow effect and the countercyclical discount-rateeffect As the cash-flow effect is not present in case of nondefaultablezero-coupon bonds their prices are driven solely by the countercyclical vari-ation in the discount rates The countercyclical variation in the bond pricesgenerates a corresponding procyclical variation in the bond risk premiumbut countercyclical variation in the bond return volatility and the bondyields Such countercyclical fluctuations in the bond prices imply negativesurprise in the bond returns during the good times and positive surprise inthe bond return during the bad times The real bonds thus carry negative riskpremiums exactly because they help to smooth consumption26

26 As before the expected bond return Et dRbt

frac14 13bt dt as well as the bond return variance

vart dRbt

frac14P

e2E

a bt

2dt correspond to infinitesimal decision intervals In order to be

24 MGILLMAN ETAL

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28a Real yield curves

The intertemporal price between consumption today and consumption in Tperiods ahead equals the gross yield-to-maturity 1thorn Y

ethTTHORNt frac14 exp

yethTTHORNt

on a

real zero-coupon bond that matures in T periods The functional dependenceof the annualized yield on the beliefs 1 t2 t3 t

0presented in the

previous proposition is driven by three distinct effects First the subjectivediscount rate measures the investorrsquos desire for immediate consumptionwith more impatient investors demanding higher yields in order to willinglyaccept lower consumption today relative to the one in T periods ahead Thesecond effect reflects the desire for a smooth consumption growth profileThe increased desire to borrow against improved economic prospectsmeasured by mu

tT shifts the demand for consumption to the right whichhowever cannot be met in an endowment economy without a correspondingchange in the equilibrium yield YethTTHORNt The strength of such an effectmoreover is measured by the elasticity of intertemporal substitution cThe consumption smoothing effect explains why yields are high in times ofgood economic prospects and low in times of bad economic prospects Thethird and last motive is related to the desire to save Such precautionarysaving is inherently related to the degree of economic uncertainty measuredby the forecast error volatility of the T-period consumption growth rate utTAs discussed in Section 24 the model with hidden regime shifts endogen-ously generates the variation in the forecast error variance in response tofluctuations in the posterior distribution 1 t2 t3 t

0

The following proposition links the real yield curve to the optimal fore-casts and the forecast error variances of the T-period consumption growthrate

Proposition 22

Denote yethTTHORNt the continuously-compounded yield-to-maturity on a T-period real

zero-coupon bond and rbtthorn1T the corresponding continuously-compoundedholding period return Then the annualized yield-to-maturity on the T-periodbond is given by

yethTTHORNt thorn 1 eth THORN u eth THORNeth THORN

1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN

where frac14 1 2 3eth THORN denotes the invariant distribution of the Markov chain

comparable to their discrete-time counterparts they must be time aggregated to Et

Rb

tthorn1T

and vart

Rb

tthorn1T

as explained in Proposition E1 in the Supplementary Appendix E


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Proof

See the Supplementary Appendix F laquoOur quantitative results discussed later show that the real yield curve is

driven predominantly by the intertemporal substitution effect in response tochanging forecasts of future mean consumption growth rates mu

tT=T Theterm structures of the T-period mean forecasts mu

tT=T and hence of theT-period real log yields y

ethTTHORNt slope down during the expansions but up

during the recessions because of the mean-reverting nature of the instantan-eous consumption growth rate u

st The slope is moreover steeper during the

long recessions because of the dramatically more inferior short- andmedium-term forecasts of the consumption growth rate averaged over theT periodsIn addition our model has implications for the volatility of the real yield

curve The volatility curve of the real yields y Teth THORNt

for Tgt 0 depends on the

variability of the mean T-period consumption growth rate forecastsmu

tT=T Such forecast variability necessarily declines with the forecast

horizon T and hence the model generates a downward-sloping volatilitycurve for the real yield curve

28b Bond risk premiums

The annual bond risk premium depends crucially on the time seriesproperties of consumption as the following proposition shows

Proposition 23

The annual geometric risk premium on the T-period real zero-coupon bond isgiven by

Et rbtthorn1T yeth1THORNt

n o

2

2vart Etthorn1 Eteth THORNgutT

n o vart Etthorn1 Eteth THORNgut 1

n o

eth62THORN

Proof

See the Supplementary Appendix F laquoThe above proposition is consistent with the findings in Campbell (1986)

First bond prices carry a zero risk premium when the consumption growthrate gut 1 is IID because then Etthorn1 Eteth THORNgutT frac14 Etthorn1 Eteth THORNgut 1 Secondbond prices carry a negative risk premium when the expected consumptiongrowth rate is positively autocorrelated

vart Etthorn1 Eteth THORNgutT

n ogt vart Etthorn1 Eteth THORNgut 1

n o eth63THORN

26 MGILLMAN ETAL

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Moreover the magnitude of the bond risk premium is an increasing functionof the consumption growth rate persistence

29 EUROPEAN OPTIONS

Absence of arbitrage implies that the price of the European call option Pct

with the given maturity time T and the strike price Pl equals the expecteddiscounted value of the option payoff at the maturity

MtPct frac14 Et MTP

cT

eth64THORN

where the option price at the maturity equals the final payoffPcT frac14 max Pl

T Pl 0

Proposition 4 in the Supplementary Appendix D exploits the martingaleproperty of the discounted option price MtP

ct and derives the partial differ-

ential equation for the no-arbitrage price

Pct frac14 Pc 1 t2 t xt tT Pl

eth65THORN

as a function of the beliefs 1 t2 t

the log of the levered equity price to

the strike price xt frac14 log Plt=

Pl

and time t

3 Empirical Section

We follow the empirical strategy used by Cecchetti Lam and Mark (2000)and estimate our two-state semi-Markov model for consumption and divi-dends hidden in IID Gaussian shocks in Equation (3) by the maximumlikelihood method We discuss the point estimates and their standard errorsWe then demonstrate the plausibility of these parameter estimates bycalculating the variance ratios as well as the long-run forecasts of the con-sumption and dividend growth rates The fact that the decade-long optimalforecast of the consumption growth rate during the long recession comes outclose to zero motivates our interpretation of the long recession as the lostdecade Finally we show that learning about hidden growth persistenceendogenizes the variation in the probability of the consumption disasterspecified exogenously in Gourio (2012 2013) Seo and Wachter (2013) andWachter (2013)

31 SUMMARY STATISTICS

Our data construction of US time series is similar to Bansal Gallant andTauchen (2007) and it is described in full in the Supplementary Appendix A1


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Table I presents the summary statistics for consumption and aggregate equitymarket dividends in the USA The geometric growth rates of the series hovermost of the time around their unconditional means of 187 and 206 butthe dividend series are more volatile with the annualized standard deviation of1038 compared to 126 for the consumption series The first-orderannualized autocorrelations in both series are negligible The skewness coef-ficient is negative 044 and 045 due to the marked tendency to experiencedeclines during economic downturns while the excess kurtosis of about 141and 378 along with the quantile-to-quantile plots and the KolmogorovndashSmirnov tests against the null of Gaussian distribution (not reported) favora leptokurtic distribution such as a Gaussian mixture density Table I alsopresents the summary statistics for the aggregate equity market The averageequity risk premium (simple compounding) is about 726 with a volatility ofabout 1729 per year The null hypothesis of zero equity risk premium canbe rejected at 5 significance level The average pricendashdividend ratio is about3124 per year with annual volatility of about 3359 and the first-orderannualized autocorrelation coefficient of about 082The long-term annual data for real per-capita consumer expenditures for

forty-two countries is from Barro and Ursua (2012) and it is described in theSupplementary Appendix A2 The confidence interval for the relativefrequencies of the periods with negative T-year growth rate are plotted inFigure 1 for T frac14 1 30 We classify periods of negative 10-year growthrate as lost decades and plot in Figure 2 the histogram of their relativefrequencies considering each country separately The relative frequency of

Table I Summary statistics US dataa

aThe sample means and the sample standard deviations are in percentages The cash-flow

growth rates are geometric averages while returns are reported with simple compoundingAC1 denotes first-order autocorrelation Standard errors obtained by performing a blockbootstrap with each block having geometric distribution The sample period is quarterly

1952Indash2011IV

Mean Standard deviation AC1

Time series Estimate (SE) Estimate (SE) Estimate (SE)

Cash-flow growth rates

Consumption 187 (010) 126 (011) 000 (000)

Levered dividends 206 (036) 1038 (074) 001 (001)

Securities

Risk-free rate 106 (045) 215 (034) 065 (004)

Equity risk premium 726 (159) 1729 (110) 013 (000)

Pricendashdividend ratio 3324 (345) 3359 (475) 082 (009)

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lost decades displays a large cross-sectional variation negligible in Indonesiaand Philippines but more than 30 in Venezuela The summary statistics inTable II reveals that the lost decades in the USA occur in the 1790ndash2009sample with a relative frequency of about 12 with a standard error of 5and average a mean growth rate of about 068 with a standard error of009 This is significantly below a mean growth rate in the full BarrondashUrsua cross-section of about 141 with a standard error of 015 butnonetheless it is surprisingly close to our maximum likelihood estimatee frac14 079 in the USA 1952Indash2011IV sample

32 MAXIMUM LIKELIHOOD

The cash-flow model in Equation (3) depends on the parameter vector

frac14 uu ll u l 1 2 3 q 0

eth66THORN

subject to the restriction that the long-run geometric means of the consump-tion and the dividend growth rates E e

st

n ofor e 2 E are equal

Rare long recessions are not observed in the US postwar data and thuswe cannot estimate their mean duration 13 as well as their relative

0 5 10 15 20 25 30

005

01

015

020

025

030

Mean Duration of Long Recession (no of years)

Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons

Figure 1 Relative frequency of long recessions international evidence Notes The figureuses the international consumption data from Barro and Ursua (2012) as described in moredetail in the Supplementary Appendix A2 The confidence bounds correspond to percentileintervals from bootstrap with a random block length having geometric distribution


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ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance

GermanyGreeceIceland

IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia

NetherlandsNewZealand

NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden

SwitzerlandTaiwanTurkey

UnitedKingdomUruguay

UnitedStatesVenezuela

Relative Frequency of Lost D

ecades

000

005

010

015

020

025

030

Figure 2 Relative frequency of lost decades country-level evidence Notes The interna-tional data are from Barro and Ursua (2012) as described in more detail in theSupplementary Appendix A2

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frequency 3 For this reason we calibrate these parameters based on theBarrondashUrsua sample for the USA (1790ndash2009) Table II reveals that longrecessions with mean duration of 10 years occur in this sample with therelative frequency of about 12 with a standard error of 5 We set13 frac14 10 years and 3 frac14 10 which implies exactly one lost decade percentury on average Our choice is also consistent with the broaderevidence in the cross-section of forty-two countries in the BarrondashUrsuasample where the relative frequency of lost decades is 13We demonstrate in Section 321 that our choice of 3 and 3 together

with the maximum likelihood estimates of the remaining parametersuu l u l 1 q 0

implies empirically plausible magnitudes of theT-period-ahead forecasts of the dividend growth rate me

tT for each e 2 E

32a Parameter estimates

Table III reports the parameter values and the asymptotic standard errors ofthe maximum likelihood estimates for the transition intensities i consump-tion and the dividend growth rates e

i as well as the volatility e for eache 2 E and ifrac14 1 2 3First consumption is estimated to grow instantaneously at the annualized

rate of about u frac14 265 in expansions and about u frac14 079 in reces-sions which is consistent with the sample mean growth rate of about 068with a standard error of 009 experienced by the USA during its lostdecades between 1790 and 2009 but not used in the estimation (Table II)27

Table II Summary statistics international dataa

aThe international data are from Barro and Ursua (2012) as described in more detail in the

Supplementary Appendix A2 The growth rates are decade-long geometric means Standarderrors are obtained by performing a block bootstrap with each block having geometricdistribution

Mean Standard deviation Relative frequency


International sample

Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)

US sample 1790ndash2009

Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)

27 Table II shows that the average is about 141 per year with a standard error 015in the Barro-Ursua cross-section of forty-two countries


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Next the aggregate dividend is estimated to grow instantaneously at theannualized rate of about l frac14 428 in expansions and about l frac14 633in recessions while the annualized estimate of the consumption volatilitycomes out around u frac14 109 whereas it is about l frac14 1016 for theaggregate dividends28

In addition Table IV reports the long-run forecasts of consumption anddividend growth rates conditional on all the hidden states The annual con-sumption growth rate forecasts are guTfrac141j1 frac14 244 in expansions whereasguTfrac141j2 frac14 037 in downturns and guTfrac141j3 frac14 063 in lost decades Inaddition the annualized decade-long forecasts are guTfrac1410j1 frac14 209

Table III Cash-flow model maximum likelihood estimatesa

aWe estimate the parameters of a two-state continuous-time hidden semi-Markov model

that is discretely observed from the bivariate time series of consumption and dividendgrowth rates We map the semi-Markov chain into a restricted three-state Markov chainby imposing the restriction e

2 frac14 e frac14 e3 for each e 2 E The instantaneous volatility e is

constant across the hidden regimes for each e 2 E The maximum likelihood estimationallows for transitions of the continuous-time chain only at the quarter ends The initialprobability p0 is set equal to the invariant distribution of the chain denoted by the symbol frac14 1 2 3eth THORN The reported parameter estimates are annualized The sample period isquarterly 1952Indash2011IV

bThe parameter 2 is restricted by Equation (12)

cWe set the mean duration of the long recession equal to 10 years and so the hazard rate13 frac14 0100

dWe posit that the long recessions occur on average once a century and so the invariantprobability 3 frac14 0100

Parameter Estimate (SE) Parameter Estimate (SE) Parameter Estimate (SE)

Instantaneous Instantaneous

Transition hazard rates Consumption growth rate Levered dividend growth rate

1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)

Invariant distribution of the chain Probability of the short recession

1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d

28 For comparison David and Veronesi (2013) calibrate consumption volatility six timeshigher at 634 per year

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expansions guTfrac1410j2 frac14 179 in downturns and guTfrac1410j3 frac14 029 in lostdecades The lost decade ~st frac14 3 may in fact be thought of as a protracteddecade-long period of anemic growth during which the consumption level isforecast to stagnate29

As regards dividends the decade-long forecasts come out glTfrac1440j1 frac14 254in expansions glTfrac1440j2 frac14 164 in downturns and glTfrac1440j3 frac14 300 in lostdecades For example the cumulative drop in dividends over the whole 10-year duration of the lost decade which happens once a century is300 10 frac14 300 This quantitative exercise demonstrates thatmuch less consumption and dividend risk measured in terms of the differ-ence in the hidden growth rates u u and l l per unit of the volatilityu and l is needed when the growth persistence itself is subject to change asopposed to the common two-state models of asset prices that featureconstant persistenceSecond the mean duration of the expansion 11 comes out almost

6 years and according to Equation (12) the mean duration of the shortrecession 12 comes out slightly above 1 year The transition probabilityto the short recession conditional on leaving the expansion stateq frac14 q1= q1 thorn 1 qeth THORN1eth THORN is estimated around 092 These estimates implythat the invariant distribution is frac14 0773 0127 0100eth THORN with each centuryexperiencing on average 77 years of good times interrupted by about thirteen

Table IV Cash-flow model implied T-period forecastsa

aThe optimal T-period-ahead forecast from Equation (36) is conditional on the hidden state

mutTji frac14 E

gutT j st frac14 i

ForecastT-period forecast mtTji

HorizonConsumption growth rate Dividend growth rate

T Expansion Recession Lost Decade Expansion Recession Lost Decade

Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300

29 We note that the estimation procedure restricts only the mean duration of the lostdecades 13 and their relative frequency in a century-long series (ie 3 frac14 01) The factthat the decade-long consumption-growth forecast comes out close to zero is dictated by therealized consumption and dividends series


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brief business-cycle recessions and about one lost decade The GreatRecession of 2008 is exactly the 13th recession after the Great Depression

32b Hidden state estimates

We use the maximum likelihood estimates from Table III and followHamilton (1989) in order to obtain the time series of the filtered beliefs

i t frac14 P st frac14 ~sijF t n o

eth67THORN

for each ifrac14 1 2 3 as well as the smoothed beliefs Pst frac14 ~sijFT

which are

conditional on the whole sample period The filtered belief 3 t correspondsto the probability of the consumption disasterFigure 3 shows that the filtered beliefs nicely track the NBER recessions

when 1 t falls and 2 t and 3 t both rise due to the Peso problem discussedin Section 23 Speaking quantitatively the estimated disaster probability3 t reaches magnitudes of almost 30 in the recessions while the corres-ponding smoothed probability is estimated almost always below 10 Thisfact suggests that the US economy did not experience significantly pro-tracted recessions ex post in the postwar sample but investors nonethelessworried about such a possibility on an ongoing basis

4 Implications for Consumption and Asset Prices

We assess the performance of the model by comparing the model-impliedunconditional and conditional asset-pricing moments to their sample coun-terparts The estimates of the dividend growth rate model in Equation (3) arefrom Table III while the utility aggregator in Equation (2) is configured withthe relative risk aversion frac14 100 the elasticity of intertemporal substitution frac14 150 and the subjective discount rate frac14 0015 Our assumed level of therelative risk aversion is 10 a value considered plausible by Mehra andPrescott (1985) and used also by Bansal and Yaron (2004)In order to obtain the conditional asset-pricing moments we need to solve

the partial differential equations (87) and (88) for the unlevered and leveredpricendashdividend ratios u 12eth THORN and l 12eth THORN and further (98) for thewhole term structure of the zero-coupon real bond prices Pb 12 tTeth THORNand finally (123) and (124) for the first two moments of the time-aggregatedannual levered equity return Ml

ieth12 tTTHORN for ifrac14 1 2 We then calculatethe (gross) bond yields YethTTHORNeth12 tTHORN as

YethTTHORNeth12 tTHORN frac14 Pbeth12 tTTHORN 1

T eth68THORN

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Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes

PosteriorprobabilitiesP S

tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


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and the annual holding-period return RbTeth12 tTHORN on the T-period zero-coupon real bond as

RbTeth12 tTHORN frac14Pbeth12 tT 1THORN

Pbeth12 tTTHORN eth69THORN

We additionally calculate the levered equity risk premium El 12 teth THORN as theexpected levered equity return in excess of the 1-year yield

Ml 12 teth THORN frac14Ml1eth12 t 1THORN Yeth1THORN 12 teth THORN eth70THORN

the levered equity volatility Vleth12 tTHORN as the second moment minus the firstmoment squared

Vleth12 tTHORN frac14 Ml2eth12 t 1THORN Ml

1eth12 t 1THORN 2 1

2

eth71THORN

The unconditional moments are obtained by Monte Carlo integration withrespect to the invariant distribution of the beliefs First we use the EulerndashMaruyama scheme in order to solve the stochastic differential equation inEquation (3) in the finer filtration Gt frac14 F t [ s 0 tf geth THORN for tgt 0 andobtain the time series of the cash flows De

t for e 2 E the shocks zet and thehidden states st We then invoke Equation (14) in order to construct the timeseries of the instantaneous cash-flow forecast errors d ~zet as well as the beliefst frac14 1t2t3teth THORN

0 The time series obtained Dut D

lt st1 t2 t3 t

0contains both the hidden state as well as the beliefs about that hiddenstate given the coarse information set F t available to the investorSecond we construct the time series of the T-period expected consumption

growth rate metT and the T-period consumption growth rate volatility etT by

means of Equations (36) and (38) Furthermore we use Equations (68) and(69) in order to construct the real bond yields Y

ethTTHORNt and the annual holding

period returns RbTt for maturities up to 30 years In addition we construct

the equity risk premium Mlt equity return volatility Vl

t equity Sharpe ratioMl

t=Vlt and the equity pricendashdividend ratios e

t for e 2 E by means ofEquations (55) (70) and (71) Finally we calculate the unconditionalmoments such as the mean the standard deviation or the first-order auto-correlation as the corresponding sample statisticsIn addition we evaluate the performance of the model based on the beliefs

estimated from the actual postwar US consumption and dividend data Sucha stringent test is often absent in the literature because generating plausiblehistorical posterior beliefs based on the actual consumption data ischallenging see the recent paper of Ju and Miao (2012 Figure 3) for anotable exception As we document below our model generates plausible

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and comparable dynamics for the levered equity prices using both the his-torical and the simulated posterior beliefs

41 CONSUMPTION

In Table V we report the means the standard deviations and the first-orderautocorrelations of the realized and the simulated data as well as the cor-responding annual variance ratios for the horizons up to 5 years As can beseen the simulated cash-flow model in Equation (3) nicely matches thesalient features of the consumption and dividend dataFurthermore the analysis in Section 243 predicts that the annual con-

sumption growth rate forecast mut 1 is procyclical and the consumption

growth rate volatility ut 1 is countercyclical Table VII confirms these pre-dictions the mean forecast mu

t 1 is about 104 in recessions and 208during expansions and the consumption volatility ut 1 is about 174 inrecessions but only 136 in expansions In addition Section 243 predicts arising pattern for the annual forecast and a falling one for the annual vola-tility over the expansion and vice versa for the recession This prediction isconfirmed in Table VII as well

Table V Consumption and dividend momentsa

aThe reported entries for the mean standard deviation and the first-order autocorrelationare quarterly moments annualized by multiplying by 4 2 and taking to power of 4 re-

spectively The population moments are obtained from a Monte Carlo simulation of thelength 4 million quarters and then time-aggregated from the infinitesimal quantities to theirquarterly counterparts using the parameter estimates from Table III Annual variance ratios

are computed in the same way as in Cecchetti Lam and Mark (1990 2000)

Annualized Moments

Variance ratios

One Two Three Four Five

Mean SD AC1 Year Years Years Years Years

Sample

Panel A 1952Indash2011IV

Consumption growth 189 126 001 100 142 168 106 164

(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population

Panel B With lost decades


Dividend growth 195 1038 000 100 109 116 122 127


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Our model of consumption given by Equation (3) is thus able to endogen-ously generate countercylical uncertainty shocks in Bloom (2009) Bloomet al (2012) and Baker and Bloom (2012)

42 ASSET PRICES

Table VI in Panel A presents the model-implied levered-equity and real-bondpricing moments and compares them to their sample counterparts Table VIin Panel B repeats the same analysis using the inferred beliefs from the actualpostwar US consumption and dividend data Table VII presents the vari-ation in the moments of the conditional distributions of the levered-equityand real-bond prices over the various phases of the business cycle andcompares them to their empirical evidence in Lustig and Verdelhan (2013)

42a Levered equity

In this Section we refer to Tables VI and VII jointlyFirst the unconditional risk premium of about E Ml

t

frac14 629 per year

from Panel A in Table VI compares well to the sample estimate of 726with astandard error of 159 in Table I The conditional risk premium varies sig-nificantly over time having a standard deviation of about Ml

t

frac14 411 In

addition Table VII reveals that such variation in the risk premium occurs atthe business cycle frequency the mean equity risk premium in short recessionscomes out about 888 significantly above the mean risk premium of 552during expansions These numbers compare surprisingly well to the point esti-mates of 1131 with the standard error 220 and 528 with standard error187 obtained by Lustig and Verdelhan (2013)Second the mean return volatility from Panel A in Table VI is about

E Vlt

frac14 1578 per year and also displays a large variation over time

having the standard deviation of about Vlt

frac14 359 The total volatility

of the realized excess return is given by the square root of the mean of theconditional variance E

Vl

t

2plus the variance of the conditional mean

2 Mlt

It comes out about 1631 per year close to the point estimate

of 1729 with a standard error 110 in Table I As before Table VIIreveals the strong business-cycle variation the mean volatility in short re-cessions comes out about 1943 which is significantly higher than themean volatility of 1471 during expansionsThird the mean Sharpe ratio from Panel A in Table VI is about

E Mlt=V

lt

frac14 037 with a standard deviation of Ml

t=Vlt

frac14 015 As

before Table VII reveals that the large variation in the Sharpe ratioMl

t=Vlt is tightly linked to the business cycle the mean Sharpe ratio during

short recessions is about 043 and during expansions about 035 These

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Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare

theasset-pricingmoments

obtained

asfollows

FirstwesolveEquation(87)fortheprice-consumptionratio

ueth

1

2THORN

SecondwesolveEquation(88)forthepricendashdividendratio

l eth1

2THORN

Thirdwesolveforthewhole

term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up

to30yearsFourthwesolveEquation

(123)and

Equation

(124)forthefirsttw

omoments

ofthetime-aggregated

annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the

(gross)bondyieldsasYethTTHORN eth1

2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20

1THORNYeth1THORN eth1

20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1

2THORNandthelong-term

yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn

onT-periodzero-couponrealbondasHPRethTTHORN eth1

2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down

totheirunconditionalcounterpartswith

Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs

(4millionquartersofartificialdata)

Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011

Pricendashdividendratio

11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


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Table

VIIBusiness-cyclevariationin

consumptionandlevered

equitypricesa

aTheasset-pricingmoments

are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one

~ s2f~ s 1gandthelow-growth

one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments

conditionaljointlyontheepoch

typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)

Business-cyclerecession

Business-cycleexpansion

Conditionalmoment

Startingin

n-thquarter

after

theregim

e

shiftandending1yearlater

Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552

Conditionalvolatility

1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

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numbers again compare surprisingly well to the point estimates of 066 withstandard error 014 and 038 with standard error 014 in Lustig andVerdelhan (2013)Fourth the mean pricendashdividend ratio comes out as E l

t

frac14 2248 and

displays volatility of nearly loglt

frac14 1220 per year The pricendash

dividend ratio volatility is below the sample counterpart of 3359 with astandard error of 475 reported in Table I It is arguably difficult to matchthe volatility of prices in a model where the mean consumption growth rateswitches between high and low values only Nonetheless the model canmatch the persistence of the pricendashdividend ratio measured by the first-order autocorrelation AC1 which comes out about 075 in annual data inPanel A of Table VI and compares favorably with the point estimate of 082with a standard error of 009 in Table IFifth Panel B in Table VI reveals that the model generates comparable

values for unconditional moments for levered equity prices using the histor-ical beliefs as well

42b Real bonds

Speaking quantitatively the average short-term yield 178 is above theaverage long-term yield of about 080 The short-term yield is also morevolatile with the standard deviation of about 090 than the long-termyield with the standard deviation of about 010 As a result the averageyield curve is mildly downward sloping which is consistent with the empir-ical findings in Ang Bekaert and Wei (2008) Campbell Shiller and Viceira(2009) and Piazzesi and Schneider (2007)Furthermore our model with learning features significantly lower persistence

of the consumption growth rate which according to Proposition 23 generatesa negligible bond risk premium on a 30-year zero-coupon bond of about108 The negative sign for the bond risk premium comes from thecountercyclical variation in the real bond prices which fall in good times andrise in bad times as explained in Section 28 In contrast the related long-runrisk literature can generate a sizable equity risk premium only if the expectedconsumption growth rate is highly persistent in which case the risk premium onreal zero-coupon bonds is highly negative as discussed in Beeler and Campbell(2012)

43 CONSUMPTION AND ASSET PRICES OVER THE PHASES

OF THE BUSINESS CYCLE

As explained in Section 243 a regime shift to the high-growth state s frac14 ~s1 isassociated with rising T-period forecasts of the cash-flow growth rate


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mutT frac14 Et

utT

as well as falling economic uncertainty measured by the

corresponding T-period cash-flow volatility utT frac14 tgutT

These predict-

able changes occur gradually while the regime shift is being recognized interms of the posterior odds in favor of the high-growth state O1 23 inEquation (20) From the perspective of the equity pricing the shift isassociated with falling levered-equity risk premiums return volatility andSharpe ratios as well as a rising levered-equity pricendashdividend ratio as shownin Table VII From the perspective of the real-bond pricing the shift isassociated with rising bond yields and holding-period excess returns inTable VIII From the perspective of macroeconomics the shift isassociated with rising annual forecasts of the consumption growth ratemu

t 1 and falling annual consumption growth rate volatility ut 1 as alsoshown in Table VIIIn contrast a regime shift to the low-growth state s 2 ~s2 ~s3f g is associated

with falling T-period consumption growth-rate forecasts mutT as well as

rising economic uncertainty measured by the corresponding T-period con-sumption volatility utT These predictable changes also occur gradually overtime as the regime shift is being recognized in terms of the posterior oddsO1 23 and O23 The posterior odds O1 23 are informative about the growthrate (ie high growth rate versus low growth rate) whereas the posteriorodds O23 in Equation (23) are informative about the growth rate persistence(ie high persistence versus low persistence) In fact the uncertainty aboutthe persistence does not arise in a standard two-state Markov chain settingFrom the perspective of the equity pricing the shift is associated with risinglevered-equity risk premiums return volatility and Sharpe ratios as well as afalling levered-equity pricendashdividend ratio as shown in Table VII From theperspective of the real-bond pricing the shift is associated with falling bondyields and holding-period excess returns in Table VIII From the perspectiveof macroeconomics the shift is associated with falling annual forecasts of theconsumption growth rate mu

t 1 and rising annual consumption growth-ratevolatility ut 1 as also shown in Table VIIThe predictable variation in the cash-flow forecasts and the discount rates

depends crucially on the fact that the economic uncertainty is declining overtime after the regime shift to the high-growth state but rising after the shiftto the low-growth state This single learning mechanism has the power togenerate (i) the procyclical variation in the pricendashdividend ratio (ii) thecountercyclical variation in mean risk premium return volatility andSharpe ratio (iii) the rising pattern of the risk premiums the return volatil-ity and the Sharpe ratios during recessions and falling pattern during theexpansions (iv) the leverage effect (v) the mean reversion of excess returnsand (vi) the predictability of consumption volatility from the pricendashdividend

42 MGILLMAN ETAL

at Central E



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Table

VIII

Business-cyclevariationin

realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e

shiftandendingayearlater

Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from

ratio In particular Table VII reveals that the variation in the conditionalmoments of the levered-equity prices over the various phases of the businesscycle compares quantitatively to the empirical evidence in Lustig andVerdelhan (2013)

5 Robustness of Results

In this section we perform the following robustness analysis First we assessthe performance of the two-state Markov chain model which is nested inour semi-Markov setting for qfrac14 1 We then examine the sensitivity of ourresults to the choice of the two parameters 3 3eth THORN which cannot beestimated from the short sample We end by briefly assessing the implica-tions for the European options

51 TWO-STATE MARKOV CHAIN AS A NESTED MODEL

The standard two-state Markov chain is nested in our framework for qfrac14 1Table VI reveals that the model without lost decades performs marginallybetter than Mehra and Prescott (1985) because the states si for ifrac14 1 2 arehidden which generates a small uncertainty premium due to the preferencefor early resolution of uncertainty coming from the EpsteinndashZin preferencesDespite that the semi-Markov model with lost decades dramatically outper-forms the Markov model in every dimension reported

52 SENSITIVITY TO NONESTIMATED PARAMETERS

The cash-flow model parameters 3 3eth THORN are difficult to estimate given ourshort sample In our empirical approach which is discussed in Section 32we choose thoughtfully the mean duration of the long recession 13 frac14 10years and the invariant distribution of the lost decade 3 frac14 01 We theninvert the constraint in Equation (12) for the mean duration of the shortrecession 12 as a function of q 11 13 and 3 Such choice of the param-eters 3 3eth THORN implies that each century features on average about one lostdecadeIn order to examine the sensitivity of our results to the above choice of 3

and 3 it is more natural to consider the pair 2 3eth THORN and then obtain 3 fromthe constraint in Equation (12) We consider the hazard rate of the shortrecession 2 2 05 10 15f g and the hazard rate of the long recession3 2 008 010 012f g obtaining 3 3 frac14 9 candidate cases to considerTable IX presents the asset-pricing implications for the levered equity and

44 MGILLMAN ETAL

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the real bonds in the form of a 2Dmatrix30We additionally report the impliedinvariant distribution frac14 1 2 3eth THORN

0 As can be observed the asset-pricingmoments for the candidate calibrations are of comparable magnitudes to thebenchmark results in Table VI Interestingly higher mean duration of theshort recessions 12 lowers the equity premium because it weakens the Pesoproblem by slowing down the learning about the recession type

Table IX Asset-pricing moments sensitivity analysisa

aThe reported entries are constructed by following the steps as in Table VI In each case we

modify the calibration in Table III by varying the durations of the short recession2 2 f05 10 15g and the long recession 3 2 f008 010 012g The stationary probability is specific to each case

Transition intensity of the lost decade duration of the lost decade in years

Asset-pricing

3 frac14 012 833 years 3 frac14 01 10 years 3 frac14 008 125 years

Moments Mean SD AC1 Mean SD AC1 Mean SD AC1

frac14 eth083 008 009THORN frac14 eth081 008 010) frac14 eth079 008 013THORN

Transitionintensity

oftherecessiondurationoftherecessionin

years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028

Equity volatility 1584 401 041 1576 378 038 1562 358 035

Equity Sharpe ratio 040 019 021 042 020 022 043 021 026

Pricendashdividend ratio 2409 1251 072 2282 1279 074 2174 1312 077

One-year bond yield 184 094 044 173 096 042 161 098 040

30-Year bond yield 082 011 053 079 009 050 078 008 046

30-Year term premium 116 273 000 104 246 001 090 220 002

frac14 eth080 012 008THORN frac14 eth078 012 010THORN frac14 eth076 012 012THORN

2=101year Equity premium 653 446 035 675 447 032 690 451 031


Equity Sharpe RAtio 037 016 025 039 017 024 040 018 026



30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520

years Equity premium 538 287 030 570 302 025 593 317 022





30-Year bond yield 094 013 069 087 012 067 083 010 064


30 The calibration in the middle of the matrix 12 frac14 1 and 13 frac14 10 is closest to the choicein Section 32 with 12 frac14 09441 and 13 frac14 10


at Central E



ownloaded from

53 DISCUSSION OF EUROPEAN OPTIONS

Modeling consumption disasters as large negative declines in the realizedconsumption growth rate results in option prices less in line with the dataIn the words of Backus Chernov and Martin (2011) on p 1994

The consumption-based calibration has a steeper smile greater curva-ture and lower at-the-money volatility This follows in part from itsgreater risk-neutral skewness and excess kurtosis They suggesthigher risk-neutral probabilities of large disasters and lowerprobabilities of less extreme outcomes

Our results suggest in contrast that modeling consumption disasters asprotracted periods of anemic consumption growth rate results in optionprices consistent with the data Because the gist of our article is not optionpricing our analysis is necessarily briefWe first solve numerically the partial differential equation in Equation

(106) in Proposition D3 for the equilibrium price Pc 12 teth THORN of theEuropean call option with 3-months to maturity as described in theSupplementary Appendix D We then calculate the implied volatilityfunction I

12

Pl

Pl

by equating the equilibrium option price Pc to the

BlackndashScholes formula The average implied volatility curve as a functionof the moneyness log Pl=Pl

is then constructed as the sample mean

1T

PTtfrac141 I

1 t 2 t

Pl

Pl

where the beliefs 1 t and 2 t are estimated from

the consumption and dividend data We consider nine candidate calibrationsfor the two nonestimated parameters 2 2 05 10 15f g and3 2 008 010 012f gThe results are plotted in Figure 4 a counterpart to Figure 5 in Backus

Chernov and Martin (2011) As can be observed our model seems to be ableto resolve the discrepancy stated in the quote by Backus et al the impliedvolatility curves in our model are mildly downward sloping and displaynegligible curvature

6 Conclusion

Our model is a minimal extension of the MehrandashPrescottndashRietz asset-pricingframework to an incomplete information setting that can explain a broadrange of dynamic phenomena in macroeconomics and finance We show thatthe success of the model is attributable to the interplay of two key factorsFirst we extend the standard two-state hidden Markov model to a two-statehidden semi-Markov setting which introduces variable growth persistenceLearning about growth persistence dramatically magnifies the level as well as

46 MGILLMAN ETAL

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minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month

levered-equitycalloptionNotes

Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered

equity(theunderlyingasset)


at Central E



ownloaded from

the variation in economic uncertainty Second we relax the independenceaxiom of the expected utility by using the recursive EpsteinndashZin preferencesconfigured so that early resolution of uncertainty is preferred This makesassets with uncertain future payoffs comparably less valuable increasingtheir equilibrium risk premiumsIt is the interplay of the fluctuations in the economic uncertainty due to

learning and higher risk premiums due to the preference for the early reso-lution of uncertainty which makes the asset desirability not only lower butalso fluctuating over time in response to the variation in the average timeneeded to resolve the payoff uncertaintyWe estimate the model using maximum likelihood on the US postwar sam-

ple of consumption and dividend series The model can generate endogen-ously the following array of consumption and asset-pricing phenomena

consumption growth rate procyclical variation in the T-period forecasts including

g their rising pattern during expansions

g their falling pattern during recessions

countercyclical variation in the T-period forecast error volatilityincludingg their falling pattern during expansions

g their rising pattern during recessions

for any forecast horizon T

equity prices procyclical variation in the pricendashdividend ratios including



countercyclical variation in risk premiums return volatility andSharpe ratios includingg their rising pattern during recessions

g their falling pattern during expansions

These effects naturally induce the leverage effect the mean reversion ofexcess returns as well as the predictability of consumption volatility frompricendashdividend ratio

real bond prices average real yield curve

level variability and persistence of real yields

mean bond risk premiums

48 MGILLMAN ETAL

at Central E



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Additionally our preliminary results for option prices suggest that ourmodel could be useful for understanding the behavior of implied volatilityof SampP 500 index options observed in the dataThe recent study of David and Veronesi (2013) uses expected-utility pref-

erences and shows that learning is important for understanding the comove-ment of stocks and nominal bonds We show that additionally relaxing theindependence axiom by employing the recursive utility is key for understand-ing the comovement of consumption and asset prices in general Suchcomovement depends crucially on the fact that the economic uncertaintyin our semi-Markov model is declining over time after the shift to thehigh-growth state but rising over time after the shift to the low-growth stateTo sum up modeling rare consumption disasters in terms of protracted

but mild recessions rather than deep short declines in the realized consump-tion growth rate helps to understand consumption and asset prices throughthe lens of rare disasters The fact that the probability of the rare event isendogenous being a result of Bayesian updating rather than an exogenouslyspecified stochastic process makes our results more credible

Supplementary Material

Supplementary appendices A B C D E and F are available at Review ofFinance online

References

Ang A Bekaert G and Wei M (2008) The term structure of real rates and expected

inflation Journal of Finance 63 797ndash849

Backus D Chernov M and Martin I (2011) Disasters implied by equity index options

Journal of Finance 66 1969ndash2012Baker S and Bloom N (2012) Does uncertainty drive business cycles Using disasters as a

natural experiment Working Paper Stanford UniversityBansal R Gallant A R and Tauchen G (2007) Rational pessimism rational exuber-

ance and asset pricing models Review of Economic Studies 74 1005ndash1033Bansal R Kiku D Shaliastovich I and Yaron A (2012) Volatility macroeconomy and

asset prices NBER Working Paper No 18104

Bansal R Kiku D and Yaron A (2010) Long-run risks the macroeconomy and asset

prices American Economic Review Papers and Proceedings 100 542ndash546Bansal R and Yaron A (2004) Risks for the long run a potential resolution of asset

pricing puzzles Journal of Finance 59 1481ndash1509Barro R J (2006) Rare disasters and asset markets in the twentieth century Quarterly

Journal of Economics 121 823ndash866Barro R J and Jin T (2011 09) On the size distribution of macroeconomic disasters

Econometrica 79 1567ndash1589


at Central E



ownloaded from

Barro R J and Ursua J F (2012) Rare macroeconomic disasters Annual Review of

Economics 4 83ndash109Bates D S (2000) Post-rsquo87 crash fears in the SampP 500 futures option market Journal of

Econometrics 94 181ndash238Beeler J and Campbell J Y (2012) The long-run risks model and aggregate asset prices

an empirical assessment Critical Finance Review 1 141ndash182Bergoeing R Kehoe T J and Soto R (2002) A decade lost and found Mexico and

Chile in the 1980s Review of Economic Dynamics 5 166ndash205Bloom N (2009) The impact of uncertainty shocks Econometrica 77 623ndash685Bloom N Floetotto M Jaimovich N Saporta-Eksten I and Terry S J (2012) Really

uncertain business cycles NBER Working paper No 18245Branger N Rodrigues P and Schlag C (2012) The role of volatility shocks and rare

events in long-run risk models Manuscript Goethe University FrankfurtBrown S J Goetzmann W N and Ross S A (1995) Survival Journal of Finance 50

853ndash873Campbell J Y (1986) Bond and stock returns in a simple exchange model Quarterly

Journal of Economics 101 185ndash804Campbell J Y Shiller R J and Viceira L M (2009) Understanding inflation-indexed

bond markets Brookings Papers on Economic Activity 79ndash120Cecchetti S G Lam P-S and Mark N C (1990) Mean reversion in equilibrium asset

prices American Economic Review 80 398ndash418Cecchetti S G Lam P-S and Mark NC (2000) Asset pricing with distorted beliefs are

equity returns too good to be true American Economic Review 90 787ndash805Cogley T and Sargent T J (2008) The market price of risk and the equity premium a

legacy of the great depression Journal of Monetary Economics 55 454ndash476Collin-Dufresne P Johannes M and Lochstoer L A (2013) Parameter learning in

general equilibrium the asset pricing implications Working paper Columbia Business

SchoolDavid A (1997) Fluctuating confidence in stock markets Implications for returns and

volatilities Journal of Financial and Quantitative Analysis 32 427ndash462David A and Veronesi P (2013) What ties return volatilities to price valuations and

fundamentals Journal of Political Economy 121 682ndash746Diebold F X Lee J-H and Weinbach G C (1994) Nonstationary Time-Series Analysis

and Cointegration (Advanced Texts in Econometrics) Oxford University Press OxfordDuffie D and Epstein L G (1992a) Asset pricing with stochastic differential utility

Review of Financial Studies 5 411ndash436Duffie D and Epstein L G (1992b) Stochastic differential utility Econometrica 60

353ndash394Duffie D and Lions P-L (1992) PDE solutions of stochastic differential utility Journal of

Mathematical Economics 21 577ndash606Epstein L G and Zin S E (1989) Substitution risk-aversion and the temporal behavior

of consumption and asset returns a theoretical framework Econometrica 57 937ndash969Epstein L G and Zin S E (1991) Substitution risk aversion and the temporal behavior

of consumption and asset returns an empirical analysis Journal of Political Economy 99

263ndash286Evans M D (1996) Peso problems their theoretical and empirical implications in G

S Maddala and C R Rao (eds) Handbook of Statistics Vol 14 Elsevier Science BV

New York pp 613ndash646

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at Central E



ownloaded from

Farhi E and Gabaix X (2011) Rare disasters and exchange rates Working paper

Harvard UniversityFriedman A (2006a) Stochastic Differential Equations and Applications Vol I Dover

Mineola New YorkFriedman A (2006b) Stochastic Differential Equations and Applications Vol II Dover

Mineola New YorkGabaix X (2008) Variable rare disasters a tractable theory of ten puzzles in macro-finance

American Economic Review Papers and Proceedings of the One Hundred Twentieth Annual

Meeting of the American Economic Association 98 64ndash67Gabaix X (2012) Variable rare disasters an exactly solved framework for ten puzzles in

macro-finance The Quarterly Journal of Economics 127 645ndash700Gourio F (2008) Time-series predictability in the disaster model Finance Research Letters

5 191ndash203Gourio F (2012) Disaster risk and business cycles The American Economic Review 102

2734ndash2766Gourio F (2013) Credit risk and disaster risk AEJ Macroeconomics 5 1ndash34Gourio F Siemer M and Verdelhan A (2013) International risk cycles Journal of

International Economics 89 471ndash484Hamilton J (1989) A new approach to the economic analysis of nonstationary time series

and the business cycle Econometrica 57 357ndash384Hayashi F and Prescott E C (2002) The 1990s in Japan a lost decade Review of

Economic Dynamics 5 206ndash235Howard R A (1971) Dynamic Probabilistic Systems Vol II Semi-Markov and Decision

Processes John Wiley and Sons Inc New YorkJin T (2014) Rare events and long-run risks for asset prices empirical identification and

evaluation Manuscript Harvard UniversityJohannes M Lochstoer L and Mou Y (2012) Learning about consumption dynamics

Manuscript Columbia Business SchoolJu N and Miao J (2012) Ambiguity learning and asset returns Econometrica 80 559ndash592

Kydland F E and Zarazaga C E J M (2002) Argentinarsquos lost decade Review of

Economic Dynamics 5 152ndash165

Liptser M and Shiryaev A (1977) Statistics of Random Processes General Theory

Springer New York

Lu Y K and Siemer M (2013) Learning rare disasters and asset prices Working Paper

Federal Reserve Board No 2013-85

Lucas R E (1978) Asset prices in an exchange economy Econometrica 46 1429ndash1445Lustig H N and Verdelhan A (2013) Business Cycle Variation in the Risk-Return Trade-

Off Journal of Monetary Economics Supplement Issue October 15ndash16 2010 Research

Conference on lsquoDirections for Macroeconomics What Did We Learn from the Economic

Crisesrsquo Sponsored by the Swiss National Bank 59Ma T and Yu Y Q (1989) The Keldys-Fichera boundary value problem for degenerate

quasilinear elliptic equations of second order Differential and Integral Equations 2 379ndash388Martin I (2013) Consumption-based asset pricing with higher cumulants Review of

Economic Studies 80 745ndash773Mehra R and Prescott E C (1985) The equity premium a puzzle Journal of Monetary

Economics 15 145ndash161Merton R C (1971) Optimum consumption and portfolio rules in a continuous-time

model Journal of Economic Theory 3 373ndash413


at Central E



ownloaded from

Merton R C (1973) An intertemporal capital asset pricing model Econometrica 41

867ndash887Murphy K P (2012) Machine Learning A Probabilistic Perspective MIT Press

Cambridge MassachusettsOrlik A and Veldkamp L (2013) Understanding uncertainty shocks and the role of the

black swan Working paper New York University Stern School of BusinessPakos M (2013) Long-run risk and hidden growth persistence Journal of Economic

Dynamics and Control 37 1911ndash1928Piazzesi M and Schneider M (2007) Equilibrium Yield Curves NBER Macroeconomics

Annual 2006 MIT Press 389ndash442Rietz T A (1988) The equity risk premium a solution Journal of Monetary Economics 22

117ndash131Santa-Clara P and Yan S (2010) Crashes volatility and the equity premium lessons

from SampP 500 options Review of Economics and Statistics 92 435ndash451Schreindorfer D (2014) Tails fears and equilibrium option prices Job Market Paper

Tepper School of Business Carnegie Mellon University pp 1ndash57Schroder M and Skiadas C (1999) Optimal consumption and portfolio selection with

stochastic differential utility Journal of Economic Theory 89 68ndash126Seo S B and Wachter J A (2013) Option prices in a model with stochastic disaster risk

Working paper University of Pennsylvania (Wharton)Timmermann A G (1993) How learning in financial markets generates excess volatility

and predictability of stock returns Quarterly Journal of Economics 108 1135ndash1145Tsai J and Wachter J A (2013) Rare booms and disasters in a multi-sector endowment

economy Working paper University of Pennsylvania (Wharton)Veronesi P (1999) Stock market overreaction to bad news in good times a rational ex-

pectations equilibrium model Review of Financial Studies 12 975ndash1007Veronesi P (2000) How does information quality affect stock returns Journal of Finance

55 807ndash837Veronesi P (2004) The Peso problem hypothesis and stock market returns Journal of

Economic Dynamics and Control 28 707ndash725Wachter J A (2013) Can time-varying risk of rare disasters explain aggregate stock market

volatility Journal of Finance 68 987ndash1035Weil P (1989) The equity premium puzzle and the risk-free rate puzzle Journal of

Monetary Economics 24 401ndash421Weitzman M (2013) A precautionary tale of uncertain tail fattening Environmental and

Resource Economics 55 159ndash173Weitzman M L (2007) Subjective expectations and asset-return puzzles American

Economic Review 97 1102ndash1130Wonham W M (1964) Some applications of stochastic differential equations to optimal

nonlinear filtering SIAM Journal on Control 2 347ndash369

52 M GILLMAN ETAL

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(2013) and Wachter (2013) extend their work by making the probabilityvariable while Martin (2013) exploits cumulant-generating functionsIn accordance with Timmermann (1993) Gourio (2012) suggests learningas a fruitful way to endogenize the disaster probability Neverthelesslearning in this BarrondashRietz framework inevitably plays marginal rolebecause the deep declines in consumption are learned almostinstantaneouslyThis article proposes an alternative model of rare disasters as protracted

stagnation in the standards of living so-called ldquolost decadesrdquo in the macro-economics literature on depressions (Hayashi and Prescott 2002 Kydlandand Zarazaga 2002 Bergoeing Kehoe and Soto 2002) Interpreting disas-ters as protracted stagnation makes learning rather slow and generates asizable increase in the magnitude as well as in the variation of economicuncertainty thus dramatically enhancing the match of a broad range ofmacroeconomic and finance phenomenaIn the language of macroeconomics the uncertainty shocks of Bloom

(2009) arise endogenously as the consumption volatility fluctuates due tolearning contrary to the exogenous specification in the long-run riskmodels of Bansal and Yaron (2004) and Bansal Gallant and Tauchen(2007) Bansal Kiku and Yaron (2010) Bansal et al (2012) In a relatedpaper Orlik and Veldkamp (2013) propose a different way to endogenizethese uncertainty shocksIn the language of finance learning induces a procyclical variation in

consumption and dividend forecasts and a countercyclical variation in theEpstein and Zin (1989 1991) discount rates in response to changes in theaverage time to the (partial) resolution of uncertainty and thus our modelcan simultaneously match a wide array of dynamic pricing phenomena in theequity and bond marketsWe followMehra and Prescott (1985) and consider a version of Lucas (1978)

representative-agent model of asset pricing with exogenous stochastic andperishable dividends as extended to a continuous-time incomplete-informationsetting by Veronesi (2004) and David and Veronesi (2013) Similarly to Pakos(2013) we extend the regime-switching models of Cecchetti Lam and Mark(1990) David (1997) David and Veronesi (2013) Hamilton (1989) andVeronesi (1999 2000 2004) by subjecting consumption and dividends tohidden shifts in the growth rate and growth persistence as well The variabilityin the growth persistence is modeled by considering two types of recessions withidentical repressed growth rates but different mean duration the former cor-responds to a regular business-cycle recession while the latter is a rare lostdecade which happens on average once a century

2 MGILLMAN ETAL

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ownloaded from








at Central E



ownloaded from




2 Model


4 M GILLMAN ETAL

at Central E



ownloaded from



Jt frac14 Et

Z 1t

U c Jeth THORNd

eth1THORN



1 1


1


eth2THORN






JUc



Uc


22 ASSET MARKETS







at Central E



ownloaded from






lt





eth4THORN









i frac14X

sj2Sn sif g

ij eth6THORN




6 M GILLMAN ETAL

at Central E



ownloaded from



frac14 P i gt y

eth8THORN



Z 110



1frac14 22000 In





at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

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ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

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ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

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ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

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ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from








at Central E



ownloaded from




2 Model


4 M GILLMAN ETAL

at Central E



ownloaded from



Jt frac14 Et

Z 1t

U c Jeth THORNd

eth1THORN



1 1


1


eth2THORN






JUc



Uc


22 ASSET MARKETS







at Central E



ownloaded from






lt





eth4THORN









i frac14X

sj2Sn sif g

ij eth6THORN




6 M GILLMAN ETAL

at Central E



ownloaded from



frac14 P i gt y

eth8THORN



Z 110



1frac14 22000 In





at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from




2 Model


4 M GILLMAN ETAL

at Central E



ownloaded from



Jt frac14 Et

Z 1t

U c Jeth THORNd

eth1THORN



1 1


1


eth2THORN






JUc



Uc


22 ASSET MARKETS







at Central E



ownloaded from






lt





eth4THORN









i frac14X

sj2Sn sif g

ij eth6THORN




6 M GILLMAN ETAL

at Central E



ownloaded from



frac14 P i gt y

eth8THORN



Z 110



1frac14 22000 In





at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



Jt frac14 Et

Z 1t

U c Jeth THORNd

eth1THORN



1 1


1


eth2THORN






JUc



Uc


22 ASSET MARKETS







at Central E



ownloaded from






lt





eth4THORN









i frac14X

sj2Sn sif g

ij eth6THORN




6 M GILLMAN ETAL

at Central E



ownloaded from



frac14 P i gt y

eth8THORN



Z 110



1frac14 22000 In





at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






lt





eth4THORN









i frac14X

sj2Sn sif g

ij eth6THORN




6 M GILLMAN ETAL

at Central E



ownloaded from



frac14 P i gt y

eth8THORN



Z 110



1frac14 22000 In





at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



frac14 P i gt y

eth8THORN



Z 110



1frac14 22000 In





at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from




2 frac14 e frac14 e




0

1A eth11THORN




probability



eth12THORN



st frac14 ~si


8 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



l

for t

For that




d ~zet frac141

edget Et dget

eth14THORN



tdt frac14 Et dget












e1 frac14 1 t 1 1 t

e e

e

eth17THORN


at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

and


e



vart est

n ofrac14 1 t 1 1 t

e e

eth19THORN



O1 23 frac141 t

1 1 teth20THORN





1

2

dget e dt 2

eeth THORN2 dt

dget e dt

2eeth THORN2 dt

( ) eth21THORN



e

d ~zet eth22THORN




10 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from




O23 frac142 t

3 teth23THORN



11






2 frac14 e frac14 e






P3ifrac141 i t

frac14 d


zero as wellP3


ifrac141 ei t



at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






getT frac14

Z tthornT

t

dge eth26THORN


etT frac14

Z tthornT

t

esd eth27THORN


zetT frac14

Z tthornT

t

edze eth28THORN




12 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


vart getT

n ofrac14 vart

etT

n othorn vart zetT

n o eth30THORN




etT

n o is constant14


vart zetT





F t st frac14 ~si

n oeth32THORN


etTji

2frac14 var getT

F t st frac14 ~si

n oeth33THORN

and

etTji

2frac14 var e

tT

F t st frac14 ~si

n o eth34THORN






at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


etTji

2frac14 etTji



tT frac14 Et getT

n ois given by

metT frac14

X3ifrac141





n oin Equation (30)

etT

2frac14X3ifrac141

etTji



0BB

1CCAi t


thornX3ifrac141

metTji

2i t

X3ifrac141

metTjii t

2


eth38THORN



016 In addition the





14 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

error varianceetT








etTj2 gt





etTj2 1 1t

is a decreasing


metTj1

21t thorn me

tTj2

21 1t

metTj11t thornme

tTj2 1 1t 2

eth39THORN









at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



tT frac14 Et

getT

as


2frac14 vart

getT

These




at and


dWt frac14Xa2A

at dRa

t rtdt

thorn rtdt

Wt ctdt eth40THORN




16 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


t

dRat Et dRa

t

frac14Xe2E



dRat frac14

dPat thornDa

tdt

Pat

eth42THORN





dRat frac14 13

at dtthorn

Xe2E





0 frac14 maxculbf g


eth44THORN





c frac14JW The




at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


t ut frac14 1 l







1 eth45THORN


u 1 t2 t




JW




JW

lt

2JW

lt 0


u

1gt u

2gt 0

18 MGILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



Mt frac14 exp

Z t

0

U

Jd

U

c eth47THORN


JUc

is negative as


JUc


Proposition 21


Mt satisfies


Z t

0

u 1 2

1d xt eth48THORN

with


eth49THORN


dMt

Mtfrac14 rtdt

Xe2E

etd ~zet eth50THORN



at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

where


is given by

Equation (86)






ei t1

ut

u

i1 t2 t



eth51THORN


Proof




at rtdt

frac14 covt

dMt

Mt dRa

t rtdt that is

13at rt frac14Xe2E

et

a et eth52THORN



26a Risk prices








eth53THORN


20 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from







dRat Et dRa

t

frac14 dgat Et dgat


thorn at

1da

t Et dat


for a frac14 u

eth54THORN


t frac14 u 1 t2 t

we




MtPat frac14 Et

Z 1t

MDad

for a 2 u lf g



at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



at frac14

Pat

Dat

eth55THORN


at D

at thorn

R t0 MD

ad and derives the


at frac14 a 1 t2 t

eth56THORN





a



1gt a

2gt 0


1and a

2are of comparable

magnitudes




Put

Dutfrac14 u

t frac14Wt



22 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



t Et dat

frac14P2

ifrac141a

idi t Et di t

and thus the total





thornX2ifrac141

ei1

a

a


eth57THORN



for ifrac14 1 2 are


1gt a

2from


est





Et dRat rtdt



t

dgut Et dgut



t

1

udu

t Et dut



eth58THORN


at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from




t

frac14

Pe2E a e

teth THORN2


return Et dRat






MtPbt frac14 Et MTP

bT

eth59THORN






eth60THORN




vart dRbt

frac14P

e2E

a bt


24 MGILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from




yethTTHORNt

on a


0presented in the



0


Proposition 22




1thorn

1

T

mu

tT 1

2T

2 utT

2 eth61THORN



Rb

tthorn1T

and vart

Rb

tthorn1T



at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

Proof
















Proposition 23



n o

2



n o

eth62THORN

Proof





n o eth63THORN

26 MGILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


29 EUROPEAN OPTIONS



MtPct frac14 Et MTP

cT

eth64THORN


T Pl 0





eth65THORN




Pl

and time t

3 Empirical Section





at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






1952Indash2011IV




Consumption 187 (010) 126 (011) 000 (000)


Securities

Risk-free rate 106 (045) 215 (034) 065 (004)



28 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from





eth66THORN


st



0 5 10 15 20 25 30

005

01

015

020

025

030


Rel

ativ

e Fr

eque

ncy

of L

ong

Rec

essi

ons



at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

ArgentinaAustralia

AustriaBelgium

BrazilCanada

ChileChina

ColombiaDenmark

EgyptFinlandFrance


IndiaIndonesia

ItalyJapanKorea

MexicoMalaysia


NorwayPeru

PhilippinesPortugal

RussiaSAfrica

SingaporeSpain

SriLankaSweden





ecades

000

005

010

015

020

025

030


30 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from




tT for each e 2 E











Full 180 (010) 211 (011)

Only declines 141 (015) 175 (021) 013 (001)


Full 158 (025) 119 (013)

Only declines 068 (009) 044 (005) 012 (005)



at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from














1 0168 (0038) u 2650 (0092) l 4278 (0716)

2 0944b (0314) u 0786 (0358) l 6330 (2088)

3 0100c u 1086 (0041) l 10162 (0332)


1 0773 (0088) q 0923 (0034)

2 0127 (0088)

3 0100d


32 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






mutTji frac14 E

gutT j st frac14 i




Quarter 258 042 074 407 519 620

Year 244 037 063 363 276 583

Decade 209 179 029 254 164 300



at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from





eth67THORN


which are








T eth68THORN

34 MGILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

Boo

m

00

02

04

06

08

10

Bus

ines

sminusC

ycle

Rec

essi

on

00

02

04

06

08

10

Lost

Dec

ade

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

1949

1959

1969

1979

1989

1999

2009

00

01

02

03

04

Figure

3Tim

eseries

ofbeliefsNotes


tfrac14

~ s ijF

t M

L

plotted

assolid

lines

and

smoothed

probabilities

P S

tfrac14

~ s ijF

T

ML

asdot-dashed

linesShaded

bars

representtheNBER

recessions

Thesample

period

isquarterly

from

1952I

to2011IV


at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from








1eth12 t 1THORN 2 1

2

eth71THORN




lt st1 t2 t3 t











36 MGILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


41 CONSUMPTION









Annualized Moments

Variance ratios



Sample



(010) (012) (000) (017) (028) (040) (044)

Dividend growth 206 1038 001 100 133 094 110 125

(036) (072) (001) (013) (031) (034) (038)

Population



Dividend growth 195 1038 000 100 109 116 122 127


at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


42 ASSET PRICES


42a Levered equity


t

frac14 629 per year


t

frac14 411 In


E Vlt





Vl

t


2 Mlt



E Mlt=V

lt


t=Vlt

frac14 015 As




38 MGILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

Table

VI

Asset

pricingmomentstw

o-state

Markovchain

versustw

o-state

semi-Markovchain

a

aThereported

entriesare


obtained

asfollows


ueth

1

2THORN


l eth1

2THORN


term

structure

ofthe

zero-coupon

realbond

pricesPbeth

1

20

TTHORNby

solvingEquation

(98)forT

up


(123)and

Equation

(124)forthefirsttw

omoments


annuallevered

equityreturn

Ml ieth1

20

TTHORN

Wethen

calculate

the


2THORNfrac14

1=Pbeth

1

20

TTHORN

1=T

thestock

risk

premium

asM

l 1eth

1

20


20

1THORNthestock

vola-

tility

asM

l 2eth

1

20

1THORN

Ml 1eth

1

20

1THORN

2 t

heshort-term

yield

asYeth1THORN eth1


yield

asYeth30THORN eth1

2THORNandtheannual

holdingperiodreturn


2THORNfrac14

Pbeth

1

20

T1THORN=Pbeth

1

20

TTHORN

Theseconditional

moments

being

dependenton

thebeliefseth

1

2THORNare

then

conditioned

down


Monte

Carlo

integration

by

simulating

asample

path

ofthebeliefs

from

Equation

(72)ofthelength

1million

yearsThecash-flow

parameter

estimatesare

from

Table

IIIandthepreference

parameter

values

are

settofrac14

100 frac14

150andfrac14

0015

Panel

Asimulatedbeliefs


Panel

Bhistoricalbeliefs

(239quartersfrom

1952IIto

2011IV)

Two-state

Markov

model

Two-state

semi-Markov

model

Two-state

Markov

model

Two-state

semi-Markov

model

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Mean

SD

AC1

Levered

equity

Riskpremium

095

044

027

629

411

035

090

044

026

558

352

023

Volatility

1165

057

029

1578

359

047

1159

058

028

1499

299

040

Sharperatio

008

003

027

037

015

029

008

003

026

035

014

011


11181

002

033

2248

1220

075

11188

002

035

2326

808

043

Indexed

bondprices

Short-term

yield

251

040

033

178

090

054

252

044

035

203

079

045

Long-term

yield

239

002

033

080

010

058

239

002

035

083

008

045

30-Y

earterm

premium

013

063

007

108

234

001

013

072

003

130

246

009


at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

Table



equitypricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g

bSource

Panel

IIin

Table

IIin

Lustig

andVerdelhan(2013)



Conditionalmoment

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Consumptiongrowth

Mean

194

143

093

065

051

104

134

184

201

206

207

208

Volatility

147

172

178

179

179

174

172

154

142

138

136

136

Levered

equity

Data

b

Meanreturn

753

1413

1145

1296

1049

1131

745

277

189

559

867

528

(527)

(520)

(538)

(466)

(470)

(220)

(505)

(530)

(492)

(436)

(459)

(187)

Sharperatio

043

078

062

082

067

066

051

019

014

042

067

038

(031)

(031)

(031)

(031)

(031)

(014)

(030)

(032)

(029)

(029)

(030)

(014)

Two-state

semi-Markovmodel

Meanrisk

premium

686

901

1015

1094

1154

888

1096

1055

898

753

650

552


1587

1811

1954

2052

2124

1943

1989

1845

1692

1584

1516

1471

Sharperatio

041

049

051

053

054

043

053

053

049

043

039

035


2288

2160

2069

2006

1961

1911

2042

2151

2235

2292

2327

2347

40 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from


t

frac14 2248 and





42b Real bonds







at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

mutT frac14 Et

utT



These predict-







42 MGILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

Table

VIII


realbondpricesa


are

constructed

analogouslyto

Table

VIWeletndenote

thenumber

ofquarterssince

thebeginningofeach

epoch

(iethehigh-growth

one


one

~ s2f~ s 2

~ s 3g)Thereported

entriesin

thetable

are

themeansoftheasset-

pricingmoments


typef~ s 1gf~ s 2

~ s 3g

fgandn2f1

23

45g



Asset

Startingin

n-thquarter

after

theregim

e


Mean

Startingin

n-thquarter

after

regim

e


Mean

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

nfrac141

nfrac142

nfrac143

nfrac144

nfrac145

Yield-to-m

aturity

1-year

178

120

086

064

050

071

081

113

148

176

194

209

3-year

149

108

081

064

052

071

082

109

133

151

163

171

5-year

130

098

078

065

057

071

080

101

119

132

140

146

10-year

106

088

076

068

064

071

077

089

100

107

112

115

30-year

088

078

072

068

066

069

073

079

085

089

091

093

Riskpremium

3-year

062

017

011

026

033

000

163

133

107

090

079

071

5-year

094

026

018

040

051

002

241

195

156

130

114

104

10-year

116

031

023

051

065

005

301

242

194

161

140

127

30-year

123

032

025

056

071

007

327

263

210

174

151

136


at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from









44 MGILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from







Asset-pricing




Transitionintensity


years

2=15067years

Equity premium 692 518 029 712 518 028 722 519 028





30-Year bond yield 082 011 053 079 009 050 078 008 046








30-Year bond yield 084 012 058 081 010 055 079 009 051



2=0520






30-Year bond yield 094 013 069 087 012 067 083 010 064




at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






12

Pl

Pl




1T

PTtfrac141 I

1 t 2 t

Pl

Pl




6 Conclusion


46 MGILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

minus01

minus00

50

005

01

015

013

5

014

014

5

015

015

5

016

Pan

el A

λ 2 =

05

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

Pan

el B

λ 2 =

10

Mon

eyne

ss

Mean Implied Vol

minus01

minus00

50

005

01

015

014

5

015

015

5

016

016

5

017

Pab

el C

λ 2 =

15

Mon

eyne

ss

Mean Implied Vol

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

λ 3=01

2

λ 3=01

λ 3=00

8

Figure

4Im

plied

volatility

curves

onthree-month


Themoneynessis

measuredin

term

soflog

Pl =Pl

where

Plis

thestrikeprice

andPlis

theprice

ofthelevered



at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



















48 MGILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






References














at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



























50 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from



















Springer New York












at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from






















52 M GILLMAN ETAL

at Central E



ownloaded from

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