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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
Understanding dynamic models
How does one discern critical features of modern dynamicmodels?
The size of equity premium is no longer an overidentifyingrestriction
3 / 39
Understanding dynamic models
How does one discern critical features of modern dynamicmodels?
The size of equity premium is no longer an overidentifyingrestriction
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)
L(m) ≥ E(log r j − log r1
)
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
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)
Entropy bound
L(m) ≥ E(log r j − log r1
)
≈ E(rnet ,j − rnet,1
)−E
((rnet ,j)2 − (rnet ,1)2
)/2
6 / 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)
Entropy bound
L(m) ≥ E(log r j − log r1
)
≈ 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(m) ≥ E(log r j − log r1
)+ 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
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)).
Compare to relative (log) consumption in Chan and Kogan
logst ≡ log(ct/xt) = φ log st−1 + loggt
22 / 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)).
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
Example 2: Campbell and Cochrane (1999)
Preferences: f (ct ,xt) = ct − xt
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)
Conditional entropy:
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
Conditional entropy:
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: Campbell and Cochrane, Case 1
Conditional entropy:
Lt(mt+1) = ((α−1)(φ−1)−b)/2+ b(log st − log s̄)
ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2
All authors use α = −1
ACE for different calibrations (quarterly)
26 / 39
ACE: Campbell and Cochrane, Case 1
Conditional entropy:
Lt(mt+1) = ((α−1)(φ−1)−b)/2+ b(log st − log s̄)
ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2
All authors use α = −1
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: Campbell and Cochrane, Case 1
Conditional entropy:
Lt(mt+1) = ((α−1)(φ−1)−b)/2+ b(log st − log s̄)
ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2
All authors use α = −1
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: Campbell and Cochrane, Case 1
Conditional entropy:
Lt(mt+1) = ((α−1)(φ−1)−b)/2+ b(log st − log s̄)
ACE: ELt(mt+1) = ((α−1)(φ−1)−b)/2
All authors use α = −1
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
ACE: Campbell and Cochrane, Case 2
Conditional entropy:
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
ACE: Campbell and Cochrane, Case 2
Conditional entropy:
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
ACE: Campbell and Cochrane, Case 2
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)
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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)
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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
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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
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
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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
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
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