sources of entropy in dynamic representative agent...

Sources of Entropy inDynamic Representative Agent Models

David Backus (NYU), Mikhail Chernov (LBS and LSE),and Stanley Zin (NYU)

University of Geneva | December 2010

This version: December 5, 2010

1 / 39

Understanding dynamic models

How does one discern critical features of modern dynamicmodels?

The size of equity premium is no longer an overidentifyingrestriction

1 / 39

Excess returns, log r j − log r 1

Standard ExcessVariable Mean Deviation Skewness Kurtosis

EquityS&P 500 0.40 5.56 -0.40 7.90Fama-French (small, high) 0.90 8.94 1.00 12.80Fama-French (large, high) 0.60 7.75 -0.64 11.57Nominal bonds1 year 0.11 1.00 0.57 9.924 years 0.15 2.00 0.10 4.87CurrenciesAUD -0.15 3.32 -0.90 2.50GBP 0.35 3.16 -0.50 1.50OptionsS&P 500 ATM straddles -62.15 119.40 -1.61 6.52

2 / 39




3 / 39




The models are built up from different state variables

Evidence points to an important role of extreme events

Which pieces are most important quantitatively?

We start by thinking about how risk is priced in these models

We find the concept of entropy and entropy bounds useful for thistask

3 / 39

Outline

Preliminaries: entropy, entropy bound, cumulants

Example: Additive preferences, i.i.d. consumption growth

Generalizing to the non-i.i.d. case

Example: External habit, i.i.d. consumption growth

Example: Recursive preferences, non-i.i.d. consumption growth

4 / 39

Entropy bound

Pricing relation

Et

(

mt+1r jt+1

)

= 1

Entropy: for any x > 0

L(x) ≡ log Ex −E logx ≥ 0

Entropy bound, i.i.d. case (Bansal and Lehmann, 1997; Alvarezand Jermann, 2005; Backus, Chernov, and Martin, 2009)

L(m) ≥ E(log r j − log r1

)

5 / 39

Entropy bound

Pricing relation

Et

(

mt+1r jt+1

)

= 1

Entropy: for any x > 0

L(x) ≡ log Ex −E logx ≥ 0

Entropy bound, i.i.d. case (Bansal and Lehmann, 1997; Alvarezand Jermann, 2005; Backus, Chernov, and Martin, 2009)


)

Name: it is a relative entropy of Q w.r.t. P

L(m) = L(q/p) = logE(q/p)−E log(q/p) = −E log(q/p)

5 / 39

Connection to HJ bound

Net return, rnet,j = r j −1

HJ bound

HJ(m) ≡σ(m)

E(m)≥

E(rnet ,j − rnet,1

)

σ(rnet ,j − rnet,1)

6 / 39



HJ bound

HJ(m) ≡σ(m)

E(m)≥


)


Entropy bound


)

≈ E(rnet ,j − rnet,1

)−E

((rnet ,j)2 − (rnet ,1)2

)/2

6 / 39



HJ bound

HJ(m) ≡σ(m)

E(m)≥


)


Entropy bound


)

≈ E(rnet ,j − rnet,1

)−E

((rnet ,j)2 − (rnet ,1)2

)/2

The two coincide when m is normal

6 / 39

Cumulants

Cumulant generating function

k(s;x) = log Eesx =∞

∑j=1

κj(x)sj/j!

Cumulants are almost moments

mean = κ1(x)

variance = κ2(x)

skewness = κ3(x)/κ3/22 (x)

excess kurtosis = κ4(x)/κ22(x)

If x is normal, κj(x) = 0 for j > 2

7 / 39

Entropy and cumulants

Computation

L(m) = logEelogm −E logm = k(1, log m)−κ1(logm)

8 / 39

Entropy and cumulants

Computation

L(m) = logEelogm −E logm = k(1, log m)−κ1(logm)

Intuition

L(m) = κ2(logm)/2!︸︷︷︸

(log)normal term

+κ3(log m)/3!+ κ4(log m)/4!+ · · ·︸︷︷︸

high-order cumulants

8 / 39

Barro’s disasters

Consumption growth iid

log gt+1 = wt+1 + zt+1

wt+1 ∼ N (µ,v)

zt+1|j ∼ N (jθ, jδ2)

j ≥ 0 has probability e−hhj/j!

Pricing kernel and entropy

logmt+1 = log β+(α−1) loggt+1

L(m) = log Eelogm −E logm

= k(α−1; log g)︸︷︷︸

k(1;log β+(α−1) log gt+1)

− (α−1)κ1(log g)︸︷︷︸

κ1(log β+(α−1) log gt+1)

9 / 39

Barro’s disasters

Consumption growth iid

log gt+1 = wt+1 + zt+1

wt+1 ∼ N (µ,v)

zt+1|j ∼ N (jθ, jδ2)

j ≥ 0 has probability e−hhj/j!

Pricing kernel and entropy

logmt+1 = log β+(α−1) loggt+1

L(m) = log Eelogm −E logm

= k(α−1; log g)︸︷︷︸

k(1;log β+(α−1) log gt+1)

− (α−1)κ1(log g)︸︷︷︸

κ1(log β+(α−1) log gt+1)

= (α−1)2v/2+ h(e(α−1)θ+(α−1)2δ2/2 − (α−1)θ−1)

9 / 39

Entropy

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Risk Aversion α

Ent

ropy

of P

ricin

g K

erne

l L(m

)

Alvarez−Jermann lower boundnormal

10 / 39

Entropy

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Risk Aversion α

Ent

ropy

of P

ricin

g K

erne

l L(m

)

Alvarez−Jermann lower boundnormal

disasters

10 / 39

Entropy bound, non-i.i.d. case

Entropy bound


)+ L(q1)

︸︷︷︸

non-i.i.d. piece

q1 − price of a one-period riskless bond

Conditional entropy:

Lt(mt+1) = logEtmt+1 −Et logmt+1

Average conditional entropy (ACE)

L(m) = ELt(mt+1)+ L(Et(mt+1)) = ELt(mt+1)+ L(q1)

ELt(mt+1) ≥ E(log r j − log r1

)

11 / 39

Entropy and the Yield Curve

There is a tight connection between the dynamic properties of thepricing kernel and the yield curve

One should be able to use the yield curve as an additionalinformation source about the entropy of the prcing kernel

We illustrate both points using the Vasicek model and thenextend to the representative agent models

12 / 39

Multi-period ACE

One period:

Lt(mt+1) = logEtmt+1 −Et logmt+1 = −y1t −Et logmt+1

ELt(mt+1) = −Ey1 −E logm.

Two periods:

Lt(mt+1mt+2) = log Et(mt+1mt+2)−Et log(mt+1mt+2)

ELt(mt+1mt+2) = −2Ey2 −2E logm

2ELt(mt+1)−ELt(mt+1mt+2) = 2E(y2 − y1)

n periods:

ELt(mt+1)−ELt(mt,t+n)/n = E(yn − y1),

where mt,t+n = Πnj=1mt+j .

13 / 39

Loglinear pricing kernel

Let the pricing kernel be

logmt = δ+∞

∑j=0

ajwt−j + but = δ+ a(L)wt + but

with {wt} ∼ NID(0,1) and {ut} ∼ NID(0,1).

The serial covariance of order k is

cov(logmt , log mt−k) =∞

∑j=0

ajaj+k .

ACE (An = ∑nj=0 aj )

ELt(mt+1) = Lt(mt+1) = (a20 + b2)/2 = (A2

0 + b2)/2,

ELt(mt,t+n) = Lt(mt,t+n) =1

2

(n−1

∑j=0

A2j +(n−1)b2

)

.

14 / 39

Vasicek

ARMA(1,1) kernel

logmt+1 = δ(1−ϕ)+ ϕ logmt + v1/2wt+1 + θv1/2wt ,

a0 = v1/2,a1 = (θ+ ϕ)v1/2

aj+1 = ϕaj , j > 1,

An = a0 + a1(1−ϕn)/(1−ϕ)

Vasicek interest rate

rt = −Lt(mt+1)−Et(log mt+1)

= −(δ+ v/2)(1−ϕ)+ ϕrt−1 +(θ+ ϕ)v1/2wt

15 / 39

Multi-Period Entropy

0 20 40 60 80 100 120−0.02

0

0.02

0.04

0.06

0.08

0.1

aj, j=0, 1, ...

0 20 40 60 80 100 120−7

−6

−5

−4

−3

−2

−1

0

1x 10

−4 aj, j=1, 2, ...

0 20 40 60 80 100 1200.065

0.07

0.075

0.08

0.085

0.09

Maturity, months

Aj

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

Maturity, months

annu

al %

E(yn−y1)

16 / 39

Advantages ofaverage conditional entropy (ACE)

Transparent lower bound: expected excess return (in logs) –helps with quantitative assesment

Alternatively, ACE measures the highest risk premium in theeconomy – helps with qualitative assesment

A tight link between the one-period and n−period ACEs: the termspread

Conditional entropy is easy to compute; to compute ACE evaluateconditional entropy at steady-state values (for log - linear modelsor approximations)

ACE is comparable across different models with different statevariables, preferences, etc.

17 / 39

Key models

External habit

Recursive preferences

Heterogeneous preferences

....

18 / 39

External habit

Equations (Abel/Campbell-Cochrane/Chan-Kogan/Heaton)

Ut =∞

∑j=0

βju(ct+j ,xt+j),

u(ct ,xt) = (f (ct ,xt)α −1)/α.

Habit is a function of past consumption: xt = h(ct−1),e.g., Abel: xt = ct−1.

Dependence on habit

Abel: f (ct ,xt) = ct/xt

Campbell-Cochrane: f (ct ,xt) = ct − xt

Pricing kernel:

mt+1 = βuc(ct+1,xt+1)

uc(ct ,xt)= β

(f (ct+1,xt+1)

f (ct ,xt)

)α−1( fc(ct+1,xt+1)

fc(ct ,xt)

)

19 / 39

Example 1:Abel (1990) + Chan and Kogan (2002)

Preferences: f (ct ,xt) = ct/xt

Chan and Kogan have extended the habit formulation:

logxt+1 = (1−φ)∞

∑i=0

φi logct−i = φ log xt +(1−φ) logct

Relative (log) consumption

logst ≡ log(ct/xt) = φ log st−1 + loggt

Pricing kernel:

logmt+1 = logβ+(α−1) loggt+1 −α log(xt+1/xt)

= logβ+(α−1) loggt+1 −α(1−φ) logst

20 / 39

ACE: Abel-Chan-Kogan

Pricing kernel:

logmt+1 = logβ+(α−1) loggt+1 −α(1−φ) logst

Conditional entropy: Lt(mt+1) = logEtelog mt+1 −Et logmt+1

log Etelog mt+1 = logβ+ k(α−1; logg)−α(1−φ) log st

Et logmt+1 = logβ+(α−1)κ1(log g)−α(1−φ) log st

Lt(mt+1) = k(α−1; log g)− (α−1)κ1(logg)

21 / 39

ACE: Abel-Chan-Kogan

Pricing kernel:

logmt+1 = logβ+(α−1) loggt+1 −α(1−φ) logst

Conditional entropy: Lt(mt+1) = logEtelog mt+1 −Et logmt+1

log Etelog mt+1 = logβ+ k(α−1; logg)−α(1−φ) log st

Et logmt+1 = logβ+(α−1)κ1(log g)−α(1−φ) log st

Lt(mt+1) = k(α−1; log g)− (α−1)κ1(logg)

ACE: ELt(mt+1) = k(α−1; log g)− (α−1)κ1(log g)

It is exactly the same as in the CRRA case

21 / 39

Example 2: Campbell and Cochrane (1999)

Preferences: f (ct ,xt) = ct − xt

Campbell and Cochrane specify (log) surplus consumption ratiodirectly:

logst = log[(ct − xt)/ct ]

logst = φ(log st−1 − log s̄)+ λ(logst−1)(log gt −κ1(log g)).

22 / 39






Compare to relative (log) consumption in Chan and Kogan

logst ≡ log(ct/xt) = φ log st−1 + loggt

22 / 39






Pricing kernel:

logmt+1 = logβ+(α−1) loggt+1 +(α−1) log(st+1/st)

= logβ− (α−1)λ(logst)κ1(log g)

+ (α−1)(1+ λ(logst)) log gt+1

+ (α−1)(φ−1)(log st − log s̄)

23 / 39



Pricing kernel:

logmt+1 = logβ+(α−1) loggt+1 +(α−1) log(st+1/st)

= logβ− (α−1)λ(logst)κ1(log g)

+ (α−1)(1+ λ(logst)) log gt+1

+ (α−1)(φ−1)(log st − log s̄)


Lt(mt+1) = k((α−1)(1+ λ(log st)); log g)

− (α−1)(1+ λ(logst))κ1(log g)

24 / 39

Additional assumptions

To compute ACE, we have to specify λ and log g

Conditional volatility of the consumption surplus ratio

λ(logst) =1

σ

√

1−φ−b/(1−α)

1−α√

1−2(logst − log s̄)−1

In discrete time, there is an upper bound on logst to ensurepositivity of λIn continuous time, this bound never binds so we will ignore itIn Campbell and Cochrane, b = 0 to ensure a constant log r1

Consumption growth is i.i.d.Case 1. loggt+1 = wt+1 ∼ N (µ,σ2)Case 2. loggt+1 = wt+1 − zt+1, zt+1|j ∼ Gamma(j,θ−1), j̄ = h

25 / 39

ACE: Campbell and Cochrane, Case 1


Lt(mt+1) = ((α−1)(φ−1)−b)/2+ b(log st − log s̄)

ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2

26 / 39




ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2

All authors use α = −1

ACE for different calibrations (quarterly)

26 / 39




ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2


ACE for different calibrations (quarterly)Campbell and Cochrane (1999): φ = 0.97, b = 0;ELt(mt+1) = 0.0300 (0.120 annual)

26 / 39




ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2


ACE for different calibrations (quarterly)Campbell and Cochrane (1999): φ = 0.97, b = 0;ELt(mt+1) = 0.0300 (0.120 annual)Wachter (2006): φ = 0.97, b = 0.011;ELt(mt+1) = 0.0245 (0.098 annual)

26 / 39




ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2


ACE for different calibrations (quarterly)Campbell and Cochrane (1999): φ = 0.97, b = 0;ELt(mt+1) = 0.0300 (0.120 annual)Wachter (2006): φ = 0.97, b = 0.011;ELt(mt+1) = 0.0245 (0.098 annual)Verdelhan (2009): φ = 0.99, b = −0.011;ELt(mt+1) = 0.0155 (0.062 annual)

26 / 39



Lt(mt+1) = (α−1)(1+ λ(log st))hθ+ ((1+(α−1)(1+ λ(log st))θ)−1 −1)h

+ ((α−1)(φ−1)−b)/2+ b(log st − log s̄)

ACE: use log-linearization around log s̄

ELt(mt+1) = hd2/(1+ d)+ ((α−1)(φ−1)−b)/2︸︷︷︸

no-jump case

d =θσ√

(α−1)(φ−1)−b

27 / 39



Lt(mt+1) = (α−1)(1+ λ(log st))hθ+ ((1+(α−1)(1+ λ(log st))θ)−1 −1)h

+ ((α−1)(φ−1)−b)/2+ b(log st − log s̄)

ACE: use log-linearization around log s̄

ELt(mt+1) = hd2/(1+ d)+ ((α−1)(φ−1)−b)/2︸︷︷︸

no-jump case

d =θσ√

(α−1)(φ−1)−b

Calibration as above + vol of logg + jump parameters:σ2 = (0.035)2/4−hθ2

Barro-Nakamura-Steinsson-Ursua: h = 0.01/4, θ = 0.15Backus-Chernov-Martin: h = 1.3864/4, θ = 0.0229

27 / 39


Calibration ACE ACE (case 1) ACE jumps

CC + BNSU 0.0341 0.0300 0.0041W + BNSU 0.0281 0.0245 0.0036V + BNSU 0.0181 0.0155 0.0026

CC + BCM 0.0883 0.0300 0.0583W + BCM 0.0737 0.0245 0.0492V + BCM 0.0487 0.0155 0.0332

28 / 39

Recursive preferences

Equations (Kreps-Porteus/Epstein-Zin/Weil)

Ut =[(1−β)c

ρt + βµt(Ut+1)

ρ]1/ρ

µt(Ut+1) =(EtU

αt+1

)1/α

EIS = 1/(1−ρ)

CRRA = 1−αα = ρ ⇒ additive preferences

29 / 39

Recursive preferences: pricing kernel

Scale problem by ct (ut = Ut/ct , gt+1 = ct+1/ct )

ut = [(1−β)+ βµt(gt+1ut+1)ρ]1/ρ

Pricing kernel (mrs)

mt+1 = β(

ct+1

ct

)ρ−1( Ut+1

µt(Ut+1)

)α−ρ

= β gρ−1t+1︸︷︷︸

short-run risk

(gt+1ut+1

µt(gt+1ut+1)

)α−ρ

︸︷︷︸

long-run risk

Note the role of recursive preferences: if α = ρSecond term disappearsNo role for predictable consumption growth or volatility (coming)

30 / 39

Loglinear approximation

Loglinear approximation

log ut = ρ−1 log [(1−β)+ βµt(gt+1ut+1)ρ]

= ρ−1 log[

(1−β)+ βeρ logµt (gt+1ut+1)]

≈ b0 + b1 logµt(gt+1ut+1).

Exact if ρ = 0 : b0 = 0, b1 = β

Otherwise, b1 = βeρ logµ/[(1−β)+ βeρ logµ]

Important: in general, b1 reflects preferences

Solve by guess and verify

31 / 39

Data-generating processes

Use MA representation for the variables, e.g.,

xt = x + γ(L)wt .

An important term that shows up in many expressions

γ(b1) ≡∞

∑i=0

bi1γi .

The term captures interaction of variable’s persistence γ andpreferences b1.

32 / 39

Example 1: Long-Run Risk and SV

Consumption growth

loggt = g + γw(L)v1/2t−1w1t

vt = v + νw(L)w2t

(w1t ,w2t) ∼ NID(0, I)

Guess value function

logut = u + ωg(L)v1/2t−1w1t + ωv(L)w2t

Solution includes

ωg,0 + γw ,0 = γw(b1)

ωv ,0 = b1(α/2)γw (b1)2νw (b1)

33 / 39

ACE: Long-Run Risk and SV

Conditional entropy

Lt(mt+1) = [(ρ−1)γw ,0 +(α−ρ)γw(b1)]2vt/2

+ (α−ρ)2(b1(α/2)γw (b1)2νw(b1))

2/2

Two calibrations (Bansal and Yaron, 2004; Bansal, Kiku, andYaron, 2009)

νw ,j = σv νj , νBY = 0.985, νBKY = 0.999

ACE

BY: 0.0715+ 0.0432 = 0.1147 (1.38 annual)

34 / 39

ACE: Long-Run Risk and SV

Conditional entropy

Lt(mt+1) = [(ρ−1)γw ,0 +(α−ρ)γw(b1)]2vt/2

+ (α−ρ)2(b1(α/2)γw (b1)2νw(b1))

2/2

Two calibrations (Bansal and Yaron, 2004; Bansal, Kiku, andYaron, 2009)

νw ,j = σv νj , νBY = 0.985, νBKY = 0.999

ACE

BY: 0.0715+ 0.0432 = 0.1147 (1.38 annual)

BKY: 0.0592+ 0.1912 = 0.2504 (3.00 annual)

34 / 39

Vasicek vs Bansal - Yaron

Both models have a loglinear pricing kernel (BY with constant vol)

logmt+1 = δ(1−ϕ)+ ϕ logmt + v1/2wt+1 + θv1/2wt ,

a0 = v1/2,a1 = (θ+ ϕ)v1/2

aj+1 = ϕaj , j > 1,

An = a0 + a1(1−ϕn)/(1−ϕ)

Bansal - Yaron (constant volatility)

a0 = [(α−ρ)γw (b1)+ (ρ−1)γω,0]v1/2,

aj+1 = (ρ−1)γw ,j+1 = (ρ−1)δγj = γaj ,

An = a0 +(ρ−1)δ(1− γn)/(1− γ)

35 / 39

Multi-Period Entropy

0 20 40 60 80 100 120−0.4

−0.3

−0.2

−0.1

0

0.1

aj, j=0, 1, ...

0 20 40 60 80 100 120−7

−6

−5

−4

−3

−2

−1

0

1x 10

−4 aj, j=1, 2, ...

0 20 40 60 80 100 120−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Maturity, months

Aj

0 20 40 60 80 100 120−20

−15

−10

−5

0

5

Maturity, months

annu

al %

E(yn−y1)

36 / 39

Example 2: Disasters

Consumption growth

loggt = g + v1/2w1t + γz,0zt

ht = h + ωw(L)w3t

(w1t ,w3t) ∼ NID(0, I)

zt |j ∼ N (jθ, j)

j ≥ 0 has jump intensity ht

Guess value function

logut = u + ωh(L)w3t

Solution includes

ωh,0 = b1ωw (b1)(eαγz,0θ+α2γ2

z,0/2 −αγz,0θ−1)/α

37 / 39

ACE: Disasters

Conditional entropy

Lt(mt+1) = (α−1)2v/2

+ (e(α−1)γz,0θ+(α−1)2γ2z,0/2 − (α−1)γz,0θ−1)ht

+ (α−ρ)2[b1ωw (b1)(eαγz,0θ+α2γ2

z,0/2 −1)/α]2/2

Two calibrations (Bansal and Yaron, 2004; Wachter, 2009)

αBY = −9,ρBY = 1/3;αW = −2,ρW = 0;

ACE

W: 0.0002+ 0.0012+ 0.0019 = 0.0033 (0.04 annual)

38 / 39

ACE: Disasters

Conditional entropy

Lt(mt+1) = (α−1)2v/2

+ (e(α−1)γz,0θ+(α−1)2γ2z,0/2 − (α−1)γz,0θ−1)ht

+ (α−ρ)2[b1ωw (b1)(eαγz,0θ+α2γ2

z,0/2 −1)/α]2/2

Two calibrations (Bansal and Yaron, 2004; Wachter, 2009)

αBY = −9,ρBY = 1/3;αW = −2,ρW = 0;

ACE

W: 0.0002+ 0.0012+ 0.0019 = 0.0033 (0.04 annual)

BY: 0.0026+ 0.0810+ 2.6509 = 2.7345 (32.81 annual)

38 / 39

Conclusions

Time dependence via external habitNo time-dependence in consumption growthNevertheless: habit with varying volatility (Campbell andCochrane, 1999) may have a substantial impact on the entropy ofthe pricing kernel

39 / 39

Conclusions


Time dependence via recursive preferences

Little time-dependence in pricing kernelNevertheless: interaction of (modest) dynamics in consumptiongrowth and recursive preferences can have a substantial impacton the entropy of the pricing kernel

39 / 39

Conclusions


Time dependence via recursive preferences

Little time-dependence in pricing kernelNevertheless: interaction of (modest) dynamics in consumptiongrowth and recursive preferences can have a substantial impacton the entropy of the pricing kernel

More broadly, entropy offers an intuitive way to asses complicatedmodels with various forms of non-normalities

39 / 39

sources of entropy in dynamic representative agent...

Documents