the persistence of expected returns: a state space approach

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The Persistence of Expected Returns:

a State Space ApproachCarlo Ambrogio Favero Marco Giacoletti Andrea Tamoni

Universita Bocconi

December 6, 2010

Abstract

In this paper we focus on the link between stock returns predictability and low persis-

tence in the stochastic process of expected returns. Moreover, we show that the Middle

Young Ratio and Realized Volatilities are relevant variables which determine the price div-

idends mean level. We implement different specifications of Present Value Models in state

space form; under these settings we can filter out the unobservable state variable describing

expected returns and define it as a Markov Chain. The lower is the size of the autoregres-

sive coefficient of this series, the higher is the forecasting power of the model. We test the

predictive performance of the different specifications both in-sample and out-of-sample.

Keywords: Asset Allocation, Kalman Filter, Long Run Regressions, Maximum Likelihood

Estimate, No Arbitrage, Present Value Models, Stock Returns Predictability


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1 Introduction

When looking at the stock markets performance, the information set is represented by realizedreturns and historical values of state variables. In an ideal world, researchers should be able toselect all the relevant variables, include them in a model and build accurate forecasts of future

returns. However, a key problem in modern financial econometrics is model uncertainty, whichis also captured by the concept of imperfect predictors. Observable state variables might beable to explain even a large part of stock returns behavior, but nobody can say whether or notone or more explanatory variables are still missing in the model. Expected real returns cantherefore be represented as a stochastic process, which is however unobservable in real life. Wecan expect the process to have memory and a follow a mean reverting path. It would thereforebe very convenient to use a Markov Chain representation:

t+1 = 0 + 1(t 0) + t

Where t+1 is the expected return in t + 1 (t+1 = E[rt+1]) and t is a random disturbance.In the literature, we can find several attempts to filter out this unobservable process from ob-

served real returns. What researchers usually find is that 1 1; expected returns follow arandom walk. This is the same as saying that investors have nearly infinite memory and thatthe effects of past events should bear a relevant weight on future investment decisions for longtime periods. This also means that long run predictability is impossible; if the process in theequation above has a unit root, its volatility is not stationary and has an explosive behavior.In our opinion, this results are not realistic, and they might just be due to misspecification of thefiltering algorithm. In other words, our idea is that it is possible to obtain a more meaningfulspecification for expected returns by identifying an appropriate set of state variables and usingthem to improve the quality of the filtering process. A more precise filtered series could then beused in a forecasting exercise in order to make asset allocation decisions. Our point can also bepresented from a decision theoretic point of view: if you improve your information set, forecastsbecome more accurate, predictability becomes possible and uncertainty fades away.The most basic returns predictability framework is the Present Value Model for the equity mar-ket, proposed by Campbell and Shiller (1988); the model relies on the price dividend as the onlyrelevant state variable. Koijen and Van Binsbergen (2009) propose an implementation of thissimple model in state space form. In this way they can filter out the series of unobservable statevariables for expected returns and expected dividend growth. The autoregressive coefficient ofexpected returns estimated using a Kalman Filter with annual data is smaller than one, butstill above 0.9. Thus, the filtered process of expected returns is very persistent and not veryuseful in a forecasting exercise.An issue with the basic Present Value Model is that the price dividend is locally mean reverting,but globally affected by level shifts (see figure 1).In order to address this problem financial econometricians have introduced additional variablesin the cointegration vector of prices and dividends. These variables should account for shifts inthe mean level of the ratio, and eliminate the persistent component of the vectors residuals.An intuitive way to represent expected returns for the extended model is to write:

t+1 = 0,t+1 + 1(t 0,t) + t+1

Where 0,t is a function of the variables that explain the time varying mean of the price divi-dend. As a matter of fact, the persistent component in expected returns should now load onthis coefficient.

1


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Figure 1: Price dividend ratio for the US stock market (S&P 500 Index); annual observations, sample 1948:2009.

In this work we propose several alternative specifications for the price dividends mean, and

we analyze how the persistence of expected returns (the size of 1) changes under each differentspecification. Our conclusion is that the better the model for the price dividends mean, thelower is 1 and the stronger is the forecasting power on future returns.The paper is structured in the following way: section 2 gives a general description of thetheoretical framework, of our main assumptions and of the state space model; section 3 to 6show different specifications of the model and analyze empirical results; section 7 contains a testof the predictive performance of the different specifications on multiperiod horizons; section 9shows the results of an asset allocation strategy based on different specifications of our model.In section 8 we make a robustness check by comparing models estimate obtained with andwithout including mean levels. Conclusions are in section 10.

2 The Model

Our model is a state space form representation of the Present Value (PV) Framework for theequity market. The Present Value Model describes a general equilibrium condition for equityreal returns. For any risky assets, gross returns can be written as:

Rt+1 =Pt+1 + Dt+1

Pt(2.1)

While net returns are:

rt+1 = Pt+1 + Dt+1

Pt 1 (2.2)

Where Pt and Dt are respectively the price and cashflow at time t; since the financial asset weare considering is a stock or a stock index, the cashflow will coincide with the dividend paid attime t. Starting from this simple relation, the PV model links current prices, future expectedshifts in cashflows (dividends) and future expected real returns1:

pdt = pd +

j=1

jEt

dt+j d (rt+j r)

(2.3)

Where pd, d and r are the means of price dividend, returns and dividend growth. Of course,

sample means can be considered as steady state levels, or equilibrium expected values for the

1Full derivation of the model is in Appendix A

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series. In the basic formulation of the PV model, the steady state price dividend, dividendgrowth rate and expected real return are constant. However, in order to suppress the persistentcomponent in the price dividend residuals, we want to allow the long term mean of the ratio tobe time varying. Since this value is a function of the long term mean of dividend growth and ofthe one of returns, this is the same as saying that all the mean levels in equation (2.3) should

change along with time. At time t, the equilibrium growth rate of dividends is Dt and steadystate expected (gross) returns are Rt; the equilibrium level of the price-dividend ratio is:

P Dt =Dt

Rt Dt(2.4)

If we take logs:pdt log(P Dt) = dt log(exp (rt) exp

dt

) (2.5)

By log linearizing around the steady state level in t + 1, and then expressing the variables interms of deviations from equilibrium we get:

pdt pdt = (dt+1 dt) (rt+1 rt) + t+1(pdt+1 pdt+1) + pdt+1 + rt+1 dt+1

We solve by taking conditional expectations:

pdt pdt =

j=1

j1t Et

dt+j dt Et [rt+j rt]

The variable is a discount factor, defined as = P /D1+P /D

. Note that according to our

assumptions on the mean of the price dividend, should be a time varying stochastic discountfactor. However, changes in the value of would affect our results only as a second order effect;in addition the variability of the discount factor is small. Rytchkov (2008) shows that in a

Present Value framework for the equity market, a relatively small shift in the value of doesnot significantly affect parameters values. We therefore take as a constant computed usingthe sample means of dividends and prices.We now face a tricky identification problem. The mean of the price dividend is a function ofthe mean of returns and of the one of dividend growth rates; thus, in order to calculate at timet the steady state of two variables, we need to take as given (fixed) the value of the third one.Our assumption is that dt is constant, i.e. dt = d t. The intuition is that cashflowsgrowth rates tend to be stable and constant on the long run; shifts in the mean of the pricedividend will therefore coincide with shifts in the steady state of returns. This intuition issupported both by the literature and by empirical evidence. First, the time series of dividendgrowth rates, even though highly volatile, does not show any relevant level shift (see figure 2).

Second, there is empirical evidence on breaks in the equity premium, presented in Pastor andStambaugh (2001) and by the literature discussing the decline in the expected equity premiumoccurred during the Seventies. Our empirical findings are highly consistent with this secondtheoretical argument. Figure 3 shows the time plot of steady state expected returns derivedusing the specification of the model proposed in section 6. The decline in the steady state levelhas the correct timing and a relevant magnitude.

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Figure 2: Dividend growth rates for the US stock market (S&P 500 Index); annual observations, sample

1948:2009.

Figure 3: Long run expected real equity returns for the US stock market (S&P 500 Index) derived with the

model in section 6; annual observations, sample 1948:2009.

A key problem in financial econometrics literature on returns predictability is whether theinvestor is able to identify relevant observable state variable; in a state space framework wecan encompass this problem by specifying the expected levels of dividend growth and expectedreal returns in terms of unobservable state variables. This is a way to deal with the imperfectregressors problem. When the means are constant, the only observable predictive variable isthe price dividend. When not constant, the intercept of the expected returns equation and themean of the price dividend will be a function of some observables.The behavior of expected returns (t) and of expected dividend growth rates (gt) can be de-scribed as a Markov Chain:

t+1 = 0,t + 1(t 0,t) + t+1

gt+1 = 0 + 1(gt 0) + gt+1

If we omit intercepts (and take demeaned series):

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t+1 = 1t + t+1

gt+1 = 1gt + gt+1

This is the same as writing:

t = Et [rt+1 rt]

gt = Et

dt+1 dt

The equilibrium equation for the price dividend can then be derived as:

pdt pdt =gt

1 1

t1 1

= B2gt B1t

While:

B1 1

1 1

B2 1

1 1

The system is composed by two observation equations (one for the price dividend and one forobserved dividend growth rates) and a measurement equation (expected dividend growth rate):

gt+1 = 1gt + gt+1

dt+1 dt = gt + dt+1

pdt+1 pdt+1 = B2(1 1)gt + 1(pdt pdt) B1t+1 + B2

gt+1

Expected returns can be reverse engeneered from the price dividend equation:

pdt = B1t + B2gt

B1t = pdt + B2gt

t = (pdt + B2gt)/B1

The variance covariance matrix of errors does not vary with time:

=

gt+1

t+1Dt+1

=

2g g gD

g 2 D

Dg D 2D

(2.6)

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3 Constant Mean Specification

As a first step we specify our model by imposing constant means for dividend price, returnsand dividend growth rates2; in formulas:

P D =

D

R D (3.1)

We collect observables in a vector {yt}Tt=1 = (dt, pdt).

Parameters are estimated with maximum likelihood. In order to avoid identification issuesdue to the large number of dimensions in the problem, we set two restrictions on the variancecovariance matrix of errors. The first one imposes gD = 0; this means that variations individends are not correlated with their current level. The second restriction follows just as aconsequence of the first one; since dividend levels and growth rates are uncorrelated, we needthat:

2D +2g

(1 21

)< V ar(d1:T)

The right-hand side of the equation is the sample variance of observed growth rates of divi-dends. Table 1 reports values of the model parameters estimated with annual observations forthe sample 1946:2009. Standard errors are calculated using a bootstrapping technique with1000 iterations.

Estimate S.E. Estimate S.E

1 0.930 (0.0703) 1 0.347 (0.0842)

0.016 (0.0134) g 0.061 (0.0056)

d -0.203 (0.1724) d 0.023 (0.0014)

g 0.417 (0.1658)

R2rets 8.1%

R2divs 7.0%

Table 1: Estimation results for the model with constant means; annual observations, sample: 1946-2009.

Bootstrapped SE.

The autoregressive coefficient of the unobservable variable describing expected real returnsis higher than 0.9; this means high persistence and perhaps non-stationarity. Moreover, the fitof expected returns to actual realizations is quite weak (the R-square is only 8.1%). In Figure4 we plot the unobservable series against observations; expected returns have long memory andtend to move slowly; t is almost flat as compared to rt.On the other hand, expected dividend growth rates have a smaller autoregressive coefficient;they tend to be more reactive, even if the R-square with respect to actual realizations is againquite small (see Figure 5).

2This model is actually identical to the one with cash-invested dividends in Koijen Van Binsbergen (2009).

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Figure 4: Expected versus realized annual returns; sample 1949:2009.

Figure 5: Expected versus realized annual dividend growth rates; sample 1949:2009.

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4 A No Arbitrage Rule: pdt proxied by Yield and Realized

Volatilities Ratio

We now remove the assumption of constant means and try to model the time varying behavior of

the price dividend level. Financial econometrics literature makes reference to the No Arbitragerelationship between the price dividend (or price earnings) and long-term (usually 10 years)bond yields. The basic relationship should state that (Yt is the yield of 10 years bonds):

pdt = 0 + 1Yt (4.1)

However, real world investors are risk adverse, and tend to price in a different way the riskinessof equities and the one of bonds. Thus, as proposed by Asness (2000), the rule in (4.1) shouldbe adjusted, in order to take into account the relative riskiness of the two asset classes. Thissecond effect can be captured using the relationship between historical volatilities of long termbonds and equity returns. The equilibrium model for the mean becomes:

pdt = 0 + 1Yt + 2etbt

(4.2)

We compute historical volatilities by taking sums of monthly squared returns over a 10 yearsrolling window and annualizing figures. We then compute the ratio between the historicalvolatility of equity returns and the one of bonds. This variable can also be interpreted as ameasure of investors relative risk aversion for the equity market.We now test the equilibrium framework by regressing the price dividend over yields and volatili-ties ratio. Table 3 collects our results; the reference sample is 1946:2008. Coefficients are highlysignificant, and the R-square is close to 50%. However, the time plot of realizations againstfitted values (Figure 4) shows a high frequency behavior in the mean of the price dividend

ratio, which is not consistent with the idea of steady state level. Moreover, this model faces adramatic theoretical problem, since while the price dividend is expressed in real terms, yieldsare a nominal variable.

Estimate tstat Estimate tstat

0 4.2714 (31.1004) 1 -0.0915 (-5.1508)

2 -0.0383 (-7.5799)

R2 48.62%

Table 2: Regression of price dividend over 10 years bond Yield and Volatility Ratio; annual observations, sample1946:2008.

What we think is that the explanatory power of yields is due to spurious correlation. Inother words, yields and price dividend do actually co-move, but this happens because there is athird, exogenous (and hopefully smoother) variable influencing both of the financial time series.

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Figure 6: Fitted versus realized annual values of the price dividend; model with yields and volatilities, sample

1946:2008.

5 An Exogenous Variable: pdt approximated by MY

We now analyze more carefully the No-Arbitrage relationship stated at the beginning of theprevious section. Even thought it makes sense to think that dividend yields and yields onlong-term bonds should be linked, an interesting issue to be addressed is whether there is anyexogenousvariable driving the behavior of both the markets. An ideal candidate, as suggestedby Favero Gozluklu and Tamoni (2010), is the demographic variable MY, or Middle Young ratio.

This series is a Support Ratio of the population; to be precise, it is the ratio of the number ofindividuals aged between 40 and 49 to the ones between 20 and 29. Empirical evidence showsthat MY is correlated both with shifts in the price dividend ratio and in the level of the yieldcurve. We can therefore specify a model of the kind:

pdt = 0 + 1M Yt (5.1)

When regressing the price dividend over MY, we get an R-square (see Table 3) even higher thanthe one we have seen before. In addition, the fitted series follows now a smooth and persistentpattern.


0 2.0390 (11.5106) 1 1.635 (8.1289)

R2 51.19%

Table 3: Regression of price dividend over MY; annual observations, sample 1946:2008.

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Figure 7: Fitted versus realized annual values of the price dividend; model with MY, sample 1946:2008.

Table 4 reports coefficients estimated with maximum likelihood for the Kalman Filter. Notehow the coefficient for MY is very close to the one estimated in the regression model above.The autoregressive coefficient of expected returns is now smaller than the one observed in theprevious section. In this specification of the model, the presence of time varying means, stronglyreduces the persistence of the expected returns filtered series. We can therefore consider MYas a relevant state variable for price dividend and returns. In addition, consistently with ourhypothesis, also the predictive power increases. In Figures 8 and 9 we plot unobservables againstrealizations. The series of expected returns is now more reactive and shows a better fit to actualobservations.


1 0.828 (0.1441) 1 0.136 (0.0313)

0.023 (0.0168) g 0.030 (0.0126)

d -0.298 (0.0344) d 0.047 (0.0153)

g 0.289 (0.1571) MY 1.662 (0.0825)

R2rets 22.6%

R2divs 6.9 %

Table 4: Estimation results for the model with MY; annual observations, sample: 1946-2009. Bootstrapped SE.

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Figure 9: Expected versus realized annual dividend growth rates; sample 1949:2009.

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6 The Complete Model: pdt approximated by MY and Realized

Volatilities Ratio

In order to complete the specification presented in the previous section we need to include

investors Risk Aversion. We therefore specify a new model for the mean of the price dividend:

pdt = 0 + 1M Yt + 2etbt

(6.1)

Table 5 reports coefficients for the regression of price dividend over MY and Volatilities Ratio.The R-square increases impressively with respect to what we have seen in the previous sections.If we plot fitted against realized values (Figure 9) we can note that, due to the inclusion of thevolatilities ratio, there has been a counter-clockwise rotation of the profile of expected values.In this way the fit becomes more coherent with actual realizations.


0 2.2379 (17.1821) 1 1.593 (10.9371)

2 -0.0213 (-7.7490)

R2 75.10%

Table 5: Regression of price dividend over MY and Volatility Ratio; annual observations, sample 1946:2008.

Figure 10: Fitted versus realized annual values of the price dividend; model with MY and volatilities, sample

1946:2008.

Table 6 reports the model parameters; coefficients for the equation of the price dividendmean are again very close to the ones estimated in table 5. The value of 1 is now slightlyhigher than 0.7. In Figures 11 and 12 we plot unobservables for expected returns and dividendgrowth rates against actual realizations. The fit for the expected returns series has improvedsensibly with respect to the two previous specifications. However, also the one for dividendgrowth is better than what observed previously. These results are consistent with the idea that

the better and the more consistent the specification of the observable state variables, the loweris the autoregressive coefficient of expected returns and the higher is the predictive power.

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1 0.719 (0.0810) 1 0.073 (0.0986)

0.075 (0.0019) g 0.045 (0.0079)

d -0.095 (0.0861) d 0.043 (0.0091)g 0.390 (0.1894)

MY 1.555 (0.0314) V ol -0.022 (0.0003)

R2rets 32.3 %

R2divs 11.6 %

Table 6: Estimation results: cash-invested dividends, demographics and volatility ratio. Sample 1946:2008


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Figure 12: Expected versus realized dividend growth rates; sample 1949:2008.

7 Multiperiod Returns - 10 Years Horizon

As a further step in our analysis we compare the different specifications of the state spacemodel using the performance of forecasts for multiperiod returns. The predictive horizon is setto 10 years; we compute cumulated returns and then annualize figures. We then measure the

goodness of fit for each specification by determining the R-square of a regression of expectedreturns over actual realizations.For what concerns the constant mean specification, the forecasting power is weak, as it is shownin Figure 12. The R-square is just 3.83%.

Figure 13: Expected versus realized returns constant means model; 10 years horizon, sample 1949:2009.

The performance of the model including MY alone is impressively better, as it can be seen in

figure 13; the R-square is 71.17%. When adding also Risk Aversion the R-square raises slightlyto 74.13% (see Figure 14).

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Figure 14: Expected versus realized returns MY model; 10 years horizon, sample 1949:2009

Figure 15: Expected versus realized returns MY and realized volatilities model; 10 years horizon, sample

1949:2008

This evidence is again consistent with the idea that when the observable information setimproves, forecasting power raises. We have also to note that while the R-square for the lasttwo specifications is higher for the longer horizon, the one of the first model is smaller. Thisresult is consistent with the point we made in section 1. If expected returns are persistent,long run predictability is impossible. On the other hand, empirical evidence suggests meanreversion on longer horizons. A model which includes good observable predictors and has amean reverting feature of expected returns delivers a better performance on longer rather than

on shorter horizons. The point is very interesting and is linked to the general framework ofstock returns predictability.Equity returns are characterized by a noise and an information component. While on the shortrun noise is the relevant element, its power fades away as the horizon increases. On the longrun, information becomes dominant and returns become predictable.

8 Robustness Check: De-meaning

As a robustness check we propose an alternative specification startegy for the models in sections5 and 6. In fact, what we have done in so far has been to estimate the model in one singlestep, by calculating the parameters of the equations in the Kalman Filter at the same time as

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estimating the slope coefficients for MY and Realized Volatilities in the specification of the pricedividend mean.We now show the results of a two step approach. In other words we now first estimate themodels by running the regressions based on equilibrium relationships in (5.1) and (6.1). Wethen use fitted values from the regressions to de-mean the price dividend, and run the Kalman

Filter on de-meaned observable series.This strategy is in general inefficient and prone to many specification and robustness issues; onestep procedures are in general preferred. However, as we can see in tables 7 and 8, results arevery similar to the ones obtained with the one step estimation. This is true in particular forautoregressive roots and slope coefficients of MY and Realized Volatilities.This is supporting evidence of our frameworks robustness.


1 0.798 (0.1306) 1 0.153 (0.0909)

0.041 (0.0252) g 0.055 (0.0188)

d -0.316 (0.1448) d 0.041 (0.0055)

g 0.294 (0.1467)

Table 7: Estimation results: demographic specification (sample 1946:2009).


1 0.707 (0.0314) 1 0.067 (0.0426)

0.080 (0.0176) g 0.034 (0.0216)

d -0.093 (0.1021) d 0.052 (0.0096)

g 0.374 (0.2284)

Table 8: Estimation results: demographics and volatilities. Sample 1946:2008

9 Asset Allocation Strategy and Parameters Stability

The aim of this section is to propose an asset allocation strategy based on the different spec-ifications of the state space model. As a first step, we estimate parameters over the sample1936:1980. We can then derive a forecast for the expected real return t+1 for 1981. We com-pare this value with the real yield of one year bonds (rft+1), which is a proxy for the risk free

rate. When t+1 > rft+1 we invest 100% of our portfolio in the equity market; on the other

hand, if t+1 < rft+1 we invest the whole portfolio in one year bonds. For the following year,

we extend the estimation sample, which is now 1936:1981 and repeat our exercise for the in-vestment strategy in 1982. In this way we determine our annual asset allocation for the period

1981:2008 (28 years). Note that this is an out-of-sample test for the one-step-ahead forecastingaccuracy of the model.The investment strategy is tested for all the three different specifications: the one with constant

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means, the one including MY and the one including both MY and Risk Aversion. Table 9summarizes results and statistics for these three frameworks and for a buy-and-hold strategywith 100% of the portfolio invested in equity.The first column collects Sharpe Ratios computed ex-post as the difference between the returnof the investment strategy minus the return of the risk free bonds divided by the portfolio re-

turns volatility. Columns InvBond and InvEquity show respectively the number of years whenthe portfolio was respectively invested in bonds or in equity. Correct Choices is the percentageof correct investment choices over the whole sample. Correct investment choice means investingin the asset class carrying the higher return for the year being. The last column shows theannualized total return of each investment strategy.

Sharpe InvBond InvEq Correct Choices TotReturn (Annual)

Equity 2.8776 0 28 67.86% 7.93%

ConstMean 12.5233 7 21 71.43% 9.72%

M Y 15.8178 10 18 67.86% 10.44%

M Y V ol 28.3922 9 19 100% 12.95%

Table 9: Asset allocation exercise; on-line strategies, sample: 1981-2008.

All the investment strategies based on the model outperform the buy-and-hold equity port-folio. Strategies which allow for a time varying mean level of the price dividend perform betterthan the one based on the constant mean assumption. In addition, the strategy exploiting the

largest information set (MY and Risk Aversion) achieves the best results. The percentage ofcorrect choices for this last strategy is equal to 1 (this means that the asset manager made thecorrect allocation choice for every year in the sample); the Sharpe Ratio is almost twice as largeas the one for the MY model. The equity-only portfolio is neatly dominated both in terms ofannualized returns and of risk adjusted performance (the Sharpe of the allocation determinedaccording to the constant means model is more than 4 times larger than the one of equity).All these results confirm our conclusions on the relationship between expected returns persis-tence, price dividend level shifts and predictive performance. The most reliable out-of-sampleforecasts are the ones determined with the model specification discussed in section 6.

We would like now to conduct a robustness check on parameters stability using the results ofthe asset allocation exercise. As a matter of fact, we are now able to plot estimates of relevant

parameters at different dates conditional on different information sets.Figures 16 to 19 show the values of the autoregressive coefficients 1 and 1 for all the modelspecifications, estimated over samples from 1936:1988 to 1936:2008. The constant means modelalways delivers high values for 1; the autoregressive coefficient is several times close to the floorof the unit value. In addition, its volatility does not decrease as the sample size increases. Alsothe autoregressive coefficient of expected dividend growth shows a certain degree of volatility.The specification with MY has the more stable results; while the sample size increases, pa-rameters values remain constant. The model including the Risk Aversion adjustment shows asexpected the lowest values for 1. However, with this model autoregressive coefficients seem tobe volatile, at least for the shortest samples. As a matter of fact, while MY just represents a veryslow moving component of returns, the Risk Aversion factor captures a higher frequency effect

which makes parameters more sensible to market information. As the sample size increases, thevolatility of autoregressive coefficients goes down.

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Figure 16: Autoregressive coefficients; constant means model; samples 1936:1988 to 1936:2008.

Figure 17: Autoregressive coefficients; MY model; samples 1936:1988 to 1936:2008.

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Figure 18: Autoregressive coefficients; MY and Realized Volatilities model; samples 1936:1988 to 1936:2008.

10 Conclusions

Agents who are making an investment decision can observe past realizations of equity returns;the historical relationship between this series and some observable state variables can be used inorder to derive a predictive framework for future returns. However, it is impossible to identifywhich is the correct information set to be used in the forecasting exercise. This problem canbe encompassed using models allowing for imperfect predictors, able to filter the true unob-servable stochastic process of expected returns. Evidence presented in the literature shows how

expected returns can be described as an autoregressive process with a root very close to one.This specification is in line with the assumption that investors have long memory; the empiricalresult is de facto no predictability.What we suggest is that the high absolute value of the autoregressive coefficient might just bedue to bad specification of the model and failure to include relevant regressors in the observableinformation set. Koijen and Van Binsbergen (2009) specify a Present Value Model for the equitymarket in state space form, and get for the unobservable state variable of expected returns anautoregressive coefficient above 0.9. However, the accuracy of this model is hampered by theassumption that the mean of the price dividend ratio is constant in time. This paper showshow, by modeling consistently level shifts of the price dividend, it is possible to obtain specifica-tions with smaller roots for the process of expected returns and better forecasting performance.

The stronger the predictive power, the lower the persistence of expected returns and thereforethe value of the autoregressive coefficient. In order to build a model for the mean of the pricedividend, we start from the No Arbitrage considerations of Asness (2000); according to this firstapproach, the mean of the price dividend should be captured by the yield on 10 years Treasuriesadjusted for a Risk Aversion factor. However, we think that bond yields are too volatile to de-scribe the behavior of the persistent component of the price dividend ratio. In addition, theyare a nominal variable. As an alternative to bond yields, recent financial econometrics literaturehas proposed exogenous variables, such as the Middle Young Ratio. Empirical evidence showsthat MY explains both bond yields levels and long horizon equity returns. We therefore proposea specification with MY alone and one including both MY and the Risk Aversion Factor. Thestate space formulation of this last model delivers the better fit to data and the better predictive

performance. Furthermore, it also shows the smaller value of the autoregressive coefficient forexpected returns. Eventually, in order to test the out-of-sample forecasting performance of our

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models and to perform a robustness check on coefficients stability, we run an asset allocationexercise, based on one-step-ahead out-of-sample forecasts. Basically, we use the different spec-ifications of the model in order to decide whether to invest in equity or in one year bonds forthe following year. If the expected real return for the equity market is above the real yield ofone year bonds, we buy stocks, otherwise bonds. The model with constant means is the worst

performer, while the one with MY and Risk Aversion is the best. However, all these modelsoutperform a long-only equity position.

This paper can be considered as part of the wider literature investigating stock returnspredictability and possible extensions of the Present Value framework. However, its contributionis relevant and original. The link between predictability and low persistence of expected returnsis a powerful intuition, which can become a useful tool also when studying other markets andsecurities. In addition, this paper offers a nice theoretical background for predictive regressionsof stock returns, and highlights how Risk Aversion (together with demographics) is one of thekey predictors for the future behavior of the equity market.

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References

[1] Favero C. Gozluklu A. and A. Tamoni. Fluctuations in the dividend/price ratio and de-mographic factors: Does age composition predict risk premia? mimeo, 2009.

[2] Goyal A. and I. Welch. A comprehensive look at the empirical performance of equitypremium prediction. Review of Financial Studies, 2007.

[3] Lander J. Orphanides A. and M. Douvogiannis. Earning forecasts and the predictability ofstock returns: evidence from trading the s&p. Board of Governors of the Federal ReserveSystem, 1997.

[4] C. S. Asness. Stocks versus bonds: Explaining the equity risk premium. Association forInvestment Management and Research, 2000.

[5] Favero C. and A. Tamoni. Demographics and the term structure of stock market risk.2010.

[6] J. H. Cochrane. The dog that did not bark: A defense of return predictability. OxfordUniversity Press, 2007.

[7] Fama E. and K. R. French. Dividend yields and expected stock returns. Journal ofFinancial Economics, 1988.

[8] E. F. Fama. Efficient capital markets: A review of theory and empirical work. Journal ofFinance, 1970.

[9] Koop G. and D. Korobilis. Bayesian multivariate time series methods for empirical macroe-conomics. Working Paper, 2009.

[10] Campbell J.Y. and R.J. Shiller. Interpreting cointegrated models. NBER, Working Paper2568, 1988.

[11] Campbell J.Y. and L. Viceira. Long-horizon mean-variance analysis. a primer. 2004.

[12] Campbell J.Y. and L. Viceira. The term structure of risk-return trade-off. 2005.

[13] Pastor L. and R. F. Stambaugh. Liquidity risk and expected stock returns. Working PaperNBER, 2001.

[14] Pastor L. and R. F. Stambaugh. Are stock less volatile in the long-run? CEPR DiscussionPaper No. DP7199, 2009.

[15] Bandi M. and B. Perron. Long-run risk-returns tradeoffs. Journal of Econometrics, 2008.

[16] R. Priestly. Time varying persistence in expected returns. Journal of Banking and Finance,2000.

[17] O. Rytchkov. Filtering out expected dividends and expected returns. 2008.

[18] R. J. Shiller. Do stock prices move too much to be justified by subsequent changes individends? American Economic Review, 1981.

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A Deriving the Present Value Model

We define stock (net) returns as:

rt,T =PT + DT

Pt 1

By multiplying both sides by PtDt and dividing by (1 + rt+1) I get:

PtDt

= (1 + rt+1)1Dt+1

Dt(1 +

Pt+1Dt+1

)

We then take logs and get to the following (small case letters are natural logarithms):

pt dt = rt,T + dT + log(1 + epTdT)

By taking the Taylor-McLaurin expansion in a point PD

= epd and omitting the infinitesimalterm (overlines are sample means):

pt dt = rt,T + dT + log(1 + P /D) +P /D

1 + P /D(pT dT (p d))

I then set = P /D1+P /D

:

pt dt = rt,T + dT + log(1 + P /D) + (pT dT (p d))

Assuming that the process of the price dividend is stationary, and therefore its mean is constant,we get by forward iteration to:

pdt = pd +

j=1

jEt

dt+j d (rt+j r)

Where pd , d , r are the means of log price dividend, log dividend growth and log returns.We now rewrite:

pt dt = rt,T + dT + (pT dT (p d))

We then get to the Campbell and Shiller definition of log stock returns:

rt,T = dt pt + dT + (pT dT (p d))

And then:

k = (d p)

rt,T = k + dt pt + dT + (pT dT)

rt,T = k + dt pt + pT + (1 )(pT dT)

We omit the constant in the final representation:

rt,T = pT + (1 )(dT pT)

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B Kalman Filter

In this section we present the Kalman Filter for our model. We do not include intercepts inthese equations. Define an expanded state vector:

Xt =

gt1

dtgtt

which satisfies:Xt+1 = F Xt +

Xt+1

Matrixes are:

F =

1 0 1 0

0 0 0 00 0 0 00 0 0 0

=

0 0 01 0 00 1 00 0 1

The measurement equation is:

Yt = M1Yt1 + M2XtWith:

M1 =

0 00 1

M2 =

1 1 0 0 1 1

B2(1 1) 0 B2 B1 1 1

For a detailed description of the filtering recursion see van Binsbergen - Koijen (2009).

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C Data Sources and Descriptions

Series of prices and dividends for the American equity market have been downloaded fromCRSP, and refer to the S&P 500 Index for the period 1936:2009.

The Middle Young ratio for the United States is calculated as a support ratio of the popu-lation. Its value is equal to the ratio of the number of individuals aged between 40 and 49 tothe ones between 20 and 29.

Realized Volatilities are calculated as the sum of monthly squared returns, aggregated overa 10 years horizon and then annualized. In order to calculate these figures we download fromCRSP monthly total returns for the S&P Index and a 10 years constant maturity benchmarkprice index for US Government Bonds.

Yields for 10 Years bonds are again downloaded from CRSP and refer to the 10 Years USTreasury, while one year yields refer to the one year T-Bill.

the persistence of expected returns: a state space approach

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