Lecture 6: Recursive Preferences
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
Basics
Epstein and Zin (1989 JPE, 1991 Ecta) following work by Kreps
and Porteus introduced a class of preferences which allow to
break the link between risk aversion and intertemporal
substitution.
These preferences have proved very useful in applied work in
asset pricing, portfolio choice, and are becoming more prevalent
in macroeconomics.
Value function:
To understand the formulation, recall the standard expected
utility time-separable preferences are defined as
Vt = Et
∞
X
β s−t u(ct+s ),
s=0
We can also define them recursively as
Vt = u(ct ) + βEt Vt+1 ,
or equivalently:
Vt = (1 − β)u(ct ) + βEt (Vt+1 ) .
EZ Preferences
EZ preferences generalize this: they are defined recursively over
current (known) consumption and a certainty equivalent
Rt (Vt+1 ) of tomorrow’s utility Vt+1 :
Vt = F (ct , Rt (Vt+1 )) ,
where
Rt (Vt+1 ) = G−1 (Et G(Vt+1 )) ,
with F and G increasing and concave, and F is homogeneous of
degree one.
Note that Rt (Vt+1 ) = Vt+1 if there is no uncertainty on Vt+1 .
The more concave G is, and the more uncertain Vt+1 is, the
lower is Rt (Vt+1 ).
Functional forms
Most of the literature considers simple functional forms for F
and G:
ρ > 0 : F (c, z) = (1 − β)c1−ρ + βz 1−ρ
α > 0 : G(x) =
x1−α
1−α
1
1−ρ
.
For example:
1
1−ρ 1−ρ
1−ρ
1−α 1−α
.
Vt = (1 − β)ct + β Et Vt+1
,
Limits
Limits:
ρ = 1 : F (c, z) = c1−β z β .
α = 1 : G(x) = log x.
Hence
1−α
α > 0 : Rt (Vt+1 ) = Et Vt+1
1
1−α
,
α = 1 : Rt (Vt+1 ) = exp (Et log (Vt+1 )) .
Proof
Define f (x)
F (c, z) = cf (x)
where x = z/c and
f (x) = 1 − β + βx1−ρ
So
1
1−ρ
f 0 (x)
βx−ρ
=
f (x)
1 − β + βx1−ρ
and
f 0 (x)
β
= .
ρ→1 f (x)
X
Since f continuous this implies
lim
lim f (x) = X β
ρ→1
(note this is simply the proof that a CES function converges to a
Cobb-Douglas as ρ → 1).
Risk Aversion vs IES
In general α is the relative risk aversion coefficient for static
gambles and ρ is the inverse of the intertemporal elasticity of
substitution for deterministic variations.
Suppose consumption today is c today and consumption
tomorrow is uncertain: {cL , c, c, ....} or {cH , c, c, ....} , each has
prob 21 .
Utility today:
1
1
V = F c, G−1
G(VL ) + G(VH )
2
2
where VL = F (cL , c) and VH = F (cH , c).
Curvature of G determines how adverse you are to the
uncertainty.
If G is linear you only care about the expected value.
If not, this is the same as the definition of a certainty equivalent:
G(Vb ) = 21 G(VL ) + 12 G(VH ).
Special Case: Deterministic consumption
If consumption is deterministic: we have the usual standard
time-separable expected discounted utility with discount factor β
and IES = ρ1 , risk aversion α = ρ.
Proof: If no uncertainty, then Rt (Vt+1 ) = Vt+1 and
Vt = F (ct , Vt+1 ) . With a CES functional form for F, we
recover CRRA preferences:
1
1−ρ 1−ρ
Vt = (1 − β)c1−ρ
+
βV
t
t+1
Wt = (1 −
β)c1−ρ
t
+ βWt+1 = (1 − β)
∞
X
j=0
where Wt = Vt1−ρ .
1−ρ
β j ct+j
,
Special Case: α = ρ
Similarly, if α = ρ, then the formula
Vt =
(1 −
β)c1−ρ
t
+β
1−α
Et Vt+1
1−ρ
1−α
simplifies to
1−α
Vt1−ρ = (1 − β)c1−ρ
+ β Et Vt+1
t
Define Wt = Vt1−ρ , we have
+ βEt (Wt+1 ) ,
Wt = (1 − β)c1−ρ
t
i.e. expected utility.
1
1−ρ
Simple example with two lotteries:
Lotteries:
lottery A pays in each period t = 1, 2, ... ch or cl , the probability
is 21 and the outcome is iid across period;
lottery B pays starting at t = 1 either ch at all future dates for
sure, or cl at all future date for sure; there is a single draw at time
t = 1.
With expected utility, you are indifferent between these lotteries,
but with EZ lottery B is prefered iff α > ρ.
In general, early resolution of uncertainty is preferred if and only
1
if α > ρ i.e. risk aversion > IES
. This is another way to
motivate these preferences, since early resolution seems
intuitively preferable.
Resolution of uncertainty
For lottery A, the utility once you know your consumption is
either ch , or cl , since
1
1−ρ 1−ρ
Vh = F (ch , Vh ) = (1 − β)c1−ρ
.
+
βV
h
h
The certainty equivalent before playing the lottery is
G
−1
1
1
1 1−α 1 1−α 1−α
1
G (ch ) + G (cl ) =
c
+ cl
.
2
2
2 h
2
For lottery B, the values satisfy
Wh1−ρ
Wl1−ρ
= (1 −
β)c1−ρ
h
= (1 −
β)c1−ρ
l
+β
+β
1 1−α 1 1−α
W
+ Wl
2 h
2
1 1−α 1 1−α
W
+ Wl
2 h
2
1−ρ
1−α
,
1−ρ
1−α
,
Resolution of uncertainty
We want to compare G−1 12 G (Wh ) + 21 G (Wl ) to
G−1 21 G (ch ) + 21 G (cl ) .
1−ρ
Note that the function x → x 1−α is concave if 1 − ρ < 1 − α,
i.e. ρ > α, and convex otherwise. As a result, if ρ > α,
1 1−α 1 1−α
W
+ Wl
2 h
2
1−ρ
1−α
≥
=
1−ρ
1−ρ
1
1
Wh1−α 1−α +
Wl1−α 1−α
2
2
1 1−ρ 1 1−ρ
+ Wl
W
2 h
2
Also
Wh1−ρ
Wl1−ρ
1 1−ρ
≥ (1 −
+β
W
+
2 h
1 1−ρ
1−ρ
≥ (1 − β)cl + β
W
+
2 h
β)c1−ρ
h
1 1−ρ
W
2 l
1 1−ρ
W
2 l
Continued
These results imply that if ρ > α then
Wh1−ρ + Wl1−ρ
c1−ρ
+ c1−ρ
h
l
≥
.
2
2
in which case the certainty equivalent of lottery A is higher than
the certainty equivalent of lottery B and agents prefer late to
early resolution of uncertainty.
Technically, EZ is an extension of EU which relaxes the
independence axiom. Recall the independence axion is: if x y,
then for any z,α : αx + (1 − α)z αy + (1 − α)z. With EZ,
“Intertemporal composition of risk matters”: we cannot reduce
compound lotteries.
Euler’s Thereom:
We have
1
1−ρ
Vt = (1 − β) Ct1−ρ + βRt (Vt+1 )1−ρ
where
1−α
Rt (Vt+1 ) = Et Vt+1
1
1−α
Since Vt is homogenous of degree one, Euler’s thereom implies
Vt = M Ct Ct + Et M Vt+1 Vt+1
Euler equation:
Taking derivatives:
M Ct =
∂Vt
= (1 − β) Vtρ Ct−ρ
∂Ct
and
M Vt+1 =
where
and
∂Rt (Vt+1 )
∂Vt
∂Rt (Vt+1 ) ∂Vt+1
∂Vt
= Vtρ βRt (Vt+1 )−ρ
∂Rt (Vt+1 )
∂Rt (Vt+1 )
−α
= Rt (Vt+1 )α Vt+1
∂Vt+1
This implies
−α
M Vt+1 = βVtρ Rt (Vt+1 )α−ρ Vt+1
IES
Define the intertemporal marginal rate of substitution as
M Vt+1 M Ct+1
M Ct
ρ−α
Ct+1 −ρ
Vt+1
= β
Ct
Rt (Vt+1 )
St,t+1 =
The first term is familiar. The second term is next period’s value
relative to its certainty equivalent.
If ρ = α or there is no uncertainty so that Vt+1 = Rt (Vt+1 ) this
term equals unity.
Household wealth:
Start with the value function:
Vt = M Ct Ct + Et M Vt+1 Vt+1
Divide by M Ct :
Vt
= Ct + Et
M Ct
M Vt+1 M Ct+1
M Ct
Define
Wt =
Vt
M Ct
then
Wt = Ct + Et St,t+1 Wt+1
is the present-discounted value of wealth.
Vt+1
M Ct+1
The return on wealth
Define the cum-dividend return on wealth:
Wt+1
Rm,t+1 =
W t − Ct
Note that
V 1−ρ C ρ
Vt+1
= t+1 t+1
Wt+1 =
M Ct+1
1−β
Hence
!
1−ρ ρ
1−ρ
Vt+1
Ct+1
Vt+1
Ct+1 ρ
Rm,t+1 = 1−ρ ρ
=
Ct
V t Ct − Ct
Vt1−ρ − (1 − β)Ct1−ρ
Now use fact that
Vt1−ρ = (1 − β) Ct1−ρ + βRt (Vt+1 )1−ρ
to obtain
Rm,t+1
" #−1
Ct+1 −ρ Rt (Vt+1 ) 1−ρ
= β
Ct
Vt+1
Certainty Equivalent
Use this equation to solve for the value function relative to the
certainty equivalent:
" #
Ct+1 −ρ Rt (Vt+1 ) 1−ρ
−1
Rm,t+1 = β
Ct
Vt+1
Vt+1
=
Rt (Vt+1 )
βRm,t+1
Ct+1
Ct
−ρ !1/(1−ρ)
Comment: we can use this to directly evaluate the cost of
uncertain returns and consumption.
SDF:
From above:
St,t+1 = β
= β
θ
Ct+1
Ct
−ρ θ−1 Ct+1
Rm,t+1
Ct
Vt+1
Rt (Vt+1 )
ρ−α
θ
−ψ
where
θ =
1−α
and ψ = 1/ρ
1−ρ
Note if ρ = α we have
St,t+1 = β
Ct+1
Ct
−ρ
Now take logs
log St,t+1 = θ log β −
θ
∆ct+1 − (1 − θ) rm,t+1
ψ
Expected returns
The return on the ith asset satisfies:
i
Et St,t+1 Rt,t+1
=1
Taking logs:
log
i
Et Rt,t+1
f
Rt+1
!
i
= −cov(log St,t+1 , log Rt,t+1
)
θ
(cov (∆ct+1 , ri,t+1 ))
ψ
+ (1 − θ) cov(rm,t+1 , ri,t+1 )
=
Epstein-Zin is a linear combination of the CAPM and the
CCAPM model.
Market return:
For the market return we have
ERm
θ
2
log
= cov (∆c, rm ) + (1 − θ) σm
ψ
Rf
We can write as:
rm +
σm
θ
2
= rf + cov (∆c, rm ) + (1 − θ) σm
2
ψ
Special case: ρ = 1
In this case
Rm,t+1 =
β
Ct+1
Ct
−1 !−1
and
σ∆c = σm
This implies that:
log
ERm
Rf
2
= σm
Risk free rate:
We have:
f
log Rt+1
= log Et exp(− log St,t+1 )
In logs:
θ
f
rt+1
= −θ log β + Et ∆ct+1 + (1 − θ) rm
ψ
2 2
σ∆c
θ
σ2
θ (1 − θ)
−
− (1 − θ)2 m −
cov(∆c, rm )
ψ
2
2
ψ
Substitute in for the market return to obtain:
(1 − θ) σm
+
2
(1 − θ) θ
+ (1 − θ)2 σm +
cov (∆c, rm )
ψ
(1 − θ) rm = (1 − θ) rf −
Risk free rate:
Simplify
rtf = − log β +
θ σ2
σ2
1
Et ∆ct+1 − 2 ∆c − (1 − θ) m
ψ
ψ 2
2
Again if ρ = α so θ = 1 we have the standard risk-free rate
equation.
If α > ρ then θ < 1 and the volatility from the market return
reduces the real interest rate.
Iid Consumption
Let
∆Ct+1 = g + σc εt+1
Let vt =
Vt
Ct
and write value function as

vt = 1 − β + βEt
1−α
vt+1
Ct+1
Ct
Since consumption is iid v is constant.
1−ρ
1−α ! 1−α
1
 1−ρ

SDF with iid consumption
With vt = v
St,t+1 = β
= β
Ct+1
Ct
−ρ Ct+1
Ct
−α
Vt+1
Rt (Vt+1 )




ρ−α
1
Et
Ct+1
Ct
−(1−θ)

1−α 

Take logs:
log St,t+1 = log β − α∆ct+1 + (1 − θ) log Et exp((1 − α)∆ct+1 )
σc
= log β − α∆ct+1 + (α − ρ) g + (1 − θ) (1 − α)2
2
Risk free rate with iid consumption
Risk free rate:
σs2
= − log Et St,t+1 = − Et log St,t+1 +
2
σ2
= − log β + ρg − (1 − θ) (1 − α) + α2 c
2
rf
If ρ = α this is the standard expression
rf = − log β + ρg − ρ2
σc2
2
Price-Dividend ratio
Log dividend price ratio: Conjecture a constant q
q = Et St,t+1
Ct+1
(1 + q)
Ct
where
log St,t+1 + ∆ct+1 = log β + (1 − α) ∆ct+1
+ (1 − θ) log Et exp((1 − α)∆ct+1 )
= log β + (1 − α) ∆ct+1
+ (α − ρ) g + (1 − θ) (1 − α)2
σc
2
Gordon-Growth Formula
So the price-dividend ratio satsifies:
log
q
1+q
σ2
= log β + (1 − ρ)g − (1 − α)2 θ c
2
σc2
= −rf + g +
− ασc2
2
where the term in brackets is expected consumption growth
log(Et Ct+1 /Ct ).
Hence this is a risk-adjusted Gordon growth formula.
Risk Premium
The risk premium on a consumption claim is then
log Et Rt+1 = log Et
so that
rm +
q + 1 Ct+1
q
Ct
σm
− rf = ασc2
2
Consumption-Wealth Ratio:
Start with the identity:
Wt+1 = Rm,t+1 (Wt − Ct )
to obtain the log-linear equation:
∆wt+1
1
= rm,t+1 + k + 1 −
(ct − wt )
ρ
where ρ = 1 − exp(c − w).
Rearrange to obtain
(1 − ρ) (ct − wt ) = ρrm,t+1 − ρ∆wt+1 + ρk
= ρrm,t+1 + ρ [∆ (ct+1 − wt+1 ) − ∆ct+1 ] + ρk
Present value relationship:
ct − wt = ρ (rm,t+1 − ∆ct+1 ) + ρ (ct+1 − wt+1 ) + ρk
∞
X
ρ
=
ρs [rm,t+s − ∆ct+s ] +
k
1−ρ
s=1
Present value expression
Now combine the risk free and market rate Euler equations to
obtain:
rm,t+s − ∆ct+s = (1 − ψ) rm,t+s − µm
where µm is a constant that depends on conditional covariances
etc..
ct − wt = (1 − ψ) Et
∞
X
s=1
ρs rm,t+s +
ρ (κ − µm )
1−ρ
The consumption-wealth ratio is an increasing function of
expected future returns if the IES < 1.
Note, we started with an identity and combined it with the Euler
equation for safe vs risky returns for a given IES. Thus these
expressions are general and do not depend specifically on EZ
preferences.
Unexpected changes in consumption
Now use
ct+1 − Et ct+1 = Wt+1 − Et Wt+1
+ (1 − ψ) (Et+1 − Et )
∞
X
ρs rm,t+s+1
s=1
= rm,t+1 − Et rm,t+1
+ (1 − ψ) (Et+1 − Et )
∞
X
ρs rm,t+s+1
s=1
Unexpected returns increase consumption growth.
Unexpected future returns increase current consumption growth
if the IES < 1.
Some comments
If returns are not forecastable, the consumption-wealth ratio is a
constant.
In this case, consumption volatility equals the volatility of
wealth, or equivalently the market return.
In the data this is obviously not true – hence returns must be
predictable.
Asset pricing implications:
We can now compute
covt (ri,t+1 , ∆ct+1 ) = σic = σim + (1 − ψ) σih
where σih is the covariance of ri,t+1 with the surprise in future
market returns:
σih = cov(ri,t+1 , (Et+1 − Et )
∞
X
s=1
ρs rm,t+s+1 )
Epstein-Zin preferences and Risk Premiums:
Using EZ preferences, the risk premium is:
Et ri,t+1 − rf,t+1 +
σic
σi2
=θ
+ (1 − θ) σim
2
ψ
The risk premium for asset i depends on its covariance between
current returns and its covariance with news about future market
returns:
Et ri,t+1 − rf,t+1 +
σi2
= ασim + (α − 1) σih
2
where α is the coefficient of relative risk aversion.
Note we don’t need to know the IES or consumption growth to
price risk in this framework.
EZ preferences and the Equity premium puzzle:
For EZ preferences we can now write the risk premium on the
market return as: .
Et rm,t+1 − rf,t+1 +
2
σm
2
= ασm
+ (α − 1) σmh
2
If returns are unforecastable, σih = 0. Given σim = 0.17 we
need α = 2 to obtain a risk premium of 6% So we succeed in
matching the risk premium with low relative risk aversion but fail
on the fact that the consumption-wealth ratio will be a constant,
and consumption volatility should equal wealth volatility.
If there is mean reversion and future returns are negatively
correlated with current returns then σmh < 0 we would need a
higher α. Since mean-reversion is difficult to determine, the
estimate could be substantially higher.
Predictable consumption growth:
Consumption growth:
∆ct+1 = g + xt + ut
xt = φxt−1 + vt
Again use Campbell-Shiller decomposition:
rt+1 = ρqt+1 − qt + ∆ct+1
where qt = Pt /Ct the price of a consumption claim and
ρ=
q
<1
1+q
Solving forward
qt = E t
∞
X
s=1
ρs [∆ct+s+1 − rt+s+1 ]
Price-dividend ratio
Euler equation
∆ct+s = ψrt+s
where ψ is the IES so that
1
[∆ct+s+1 − rt+s+1 ] = 1 −
∆ct+s+1
ψ
The price-dividend ratio satisifies
ρφ
1
qt =
1−
xt
1 − ρφ
ψ
Implications
If the IES > 1 then an increase in current consumption growth
causes an increase in the price-dividend ratio.
Intuition: A persistent increase in consumption growth provides
news about future cash flows and discount rates that go in
opposite directions.
If the IES is high, interest rates don’t need to move very much in
response to the change in consumption growth. The cash flow
effect dominates.
Comments
We need a high φ to get large volatility in the price-dividend
ratio.
But this comes from predictable dividend growth not from
time-varying returns (the risk free rate is moving but the risk
premium is not).
Time-varying volatility:
Now add time-varying volatility:
xt = φxt−1 + σt ut
σt = (1 − γ)σ + γσt−1 + vt
We then have a solution of the form
1
ρφ
1−
xt + aσt2
qt =
1 − ρφ
ψ
where a > 0 if IES > 0 and risk-aversion > 0.
We will also get time-varying risk premia – “discount rate news”
that offsets the “cash flow” news of the consumption growth
shock.
In other words, we need time-varying volatility to match the
equity premium combined with persistent movements in
consumption growth to match the volatility of the price dividend
ratio.
Calibration and empirical implementation
Calibration:
IES = 1.5, α = 10
Very high persistence and volatility for shocks to volatility and
peristent consumption-growth process.
Issues to think about:
IES > 1 is controversial.
Difficult to estimate long-run risk.