Lecture 5: Asset Pricing Model with Habit
Formation
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
Habit model:
Assume:
U =E
∞
X
β t u (ct , ht )
t=0
,with u given, for instance, by the formula
u(c, h) =
(c − θh)1−σ
,
1−σ
where θ > 0 is a parameter and ht is the habit level.
The habit level ht satisfies a law of motion, e.g. it is a function of
past consumption choices:
ht = (1 − δ)ht−1 + ct−1 =
∞
X
j=1
.
(1 − δ)j−1 ct−j
Comments:
External habits (keeping up with the Joneses) – agents take h as
exogenous. Internal habits – agents recognize the effect of
current consumption on future habits.
In addition to asset pricing, habits are used in DSGE models to
deliver hump-shaped business cycle dynamics.
Microfoundations are weak: very little empirical evidence and
unclear how such a model aggregates.
Volatility
If agent’s utility is v(c, h) = u(c − h) instead of u(c), and h
grows over time so that its distance to c is always rather small,
then a given percentage change in c generates a larger
percentage change in c − h :
∆(c − h)
∆c c
∆c
=
>
.
c−h
c c−h
c
This is just a “leverage” effect coming from the “subsistence
level” h. Hence for a given volatility of c, we certainly get more
volatility of marginal utility of consumption ∂u
∂c . This will allow
us to become closer to the Hansen-Jagannathan bounds: marginal
utility of consumption is volatile, which is what you need for the
equity premium puzzle.
Time-varying risk aversion:
When agents’ consumption becomes closer to the habit level h,
they fear further negative shocks since their utility is concave.
The curvature of the utility function, i.e. the “index of relative
risk aversion”, is
IR (c) =
−cu00 (c − h)
−cv 00 (c)
=
,
v 0 (c)
u0 (c − h)
which is time-varying.
When u is CRRA, u(c) =
c1−γ
1−γ .
Direct calculation yields
IR (c) = γ
c
,
c−h
As c → h, IR (c) → ∞. Hence “time-varying risk aversion”, and
hence time-varying risk premia.
Multiplicative habits
Utility:
E0
∞
X
t=0
βt
(Ct /Xt )1−γ − 1
,
1−γ
Assume habit is external then
Ct+1 −γ
Xt γ−1
Mt+1 =
Ct
Xt+1
Assume one lag for simplicity:
κ
Xt = Ct−1
then
Mt+1 =
Ct+1
Ct
−γ Ct−1
Ct
κ(γ−1)
Implications:
Assume joint-log normality of consumption growth and asset
returns then the risk free rate is:
σc2
+ γEt ∆ct+1 − κ (γ − 1) ∆ct
2
The equity premium is unchanged relative to the CRRA model:
σr
Et (rt+1 − rf,t+1 ) +
= γσc
2
On average:
rf,t+1 = − log β − γ 2
σc2
+ (γ (1 − κ) + κ) g
2
so increasing γ will not have as large an impact on reducing rf
when 0 < k < 0
Increasing γ for 0 < κ < 1 will also raise the variability of the
interest rate in the short-run – this will tend to be counterfactual.
Overall difficult to argue this model solves the risk-free rate
puzzle for very high γ.
rf,t+1 = − log β − γ 2
Campbell-Cochrane:
Utility:
E0
∞
X
t=0
βt
(Ct − Xt )1−γ − 1
,
1−γ
where γ denotes the risk-aversion coefficient, Xt the external
habit level and Ct consumption.
Assume that consumption growth is i.i.d and log-normal:
∆ct+1 = g + ut+1 , where ut+1 ∼ i.i.d. N (0, σ 2 ).
Surplus consumption ratio:
Define the surplus consumption ratio St ≡ (Ct − Xt )/Ct . Note
that 0 < St < 1.
Assume that st = log(St ) is related to consumption through the
following heteroskedastic AR(1) process:
st+1 = (1 − φ)s + φst + λ(st )(∆ct+1 − g).
This is a generalization of a standard AR(1), i.e.
Xt = (1 − δ)Xt−1 + Ct−1 . The function λ(.) introduces a
nonlinearity, which will prove important.
Pricing kernel:
External habits: The habit is assumed here to depend only on
aggregate, not on individual, consumption. Thus, the
inter-temporal marginal rate of substitution is here:
Uc (Ct+1, Xt+1 )
Uc (Ct, Xt )
St+1 Ct+1 −γ
= β
S t Ct
= βe−γ [g+(φ−1)(st −s)+(1+λ(st ))(∆ct+1 −g)] .
Mt+1 = β
In contrast, with internal habits, the consumer is forward-looking
and realizes that increasing C today will result in a higher habit
in the future. In this case the SDF is more complicated:
Mt+1 = β
Uc (t + 1) +
∂Xt+1+j
j
j=1 β Ux (t + 1 + j) ∂Ct+1
P∞ j
∂Xt+j
j=1 β Ux (t + j) ∂Ct
P∞
Uc (t) +
.
Sensitivity function:
Campbell and Cochrane use the following sensitivity function:
λ(st ) =
1p
1 − 2(st − s) − 1, when s ≤ smax , 0 elsewhere,
S
where S and smax are respectively the steady-state and upper
bound of the surplus-consumption ratio, which we set as:
r
γ
,
S=σ
1−φ
and
2
smax = s +
1−S
.
2
These two values for S and smax are one possible choice, which
they justify to make the habit locally predetermined.
Risk free rate:
This sensitivity function allows them to have a constant risk-free
interest rate. To see this, note that the risk-free rate is
f
rt+1
= − log β + γg − γ(1 − φ)(st − s) −
γ 2σ2
(1 + λ(st ))2 .
2
Two effects where st appears: intertemporal substitution and
precautionary savings. When st is low, households have a low
IES which drives the risk free rate up. High risk aversion induces
more precautionary savings which drives the risk free rate down.
CC offset these two effects by picking λ such that
γ(1 − φ)(st − s) +
γ 2σ2
(1 + λ(st ))2 = constant.
2
Alternative specifications
By picking a different λ, you can have a risk-free rate which
depends, say, linearly on the state variable st , i.e.
f
rt+1
= A − Bst .
Extensions of the CC model to study the yield curve (Wachter,
JFE) or the forward premium puzzle (Verdelhan, JF) consider the
case B < 0 and B > 0, modifying slightly the CC model.
Depending on the value of the structural parameters, the model
implies constant, pro- or counter-cyclical interest rates.
Key mechanism
Time-varying local risk-aversion coefficient:
γt = −
CUCC
γ
= .
UC
St
Counter-cyclical market price of risk. Start from:
e ) Et (Rt+1
≤ σt (Mt+1 ) = M P Rt ,
SRt = e
σt (Rt+1 ) Et (Mt+1 )
with an equality for assets that perfectly correlated with the SDF.
In this model the market price of risk is:
M P Rt = γσ (1 + λ(st )) =
γσ
S
q
1 − 2(st − s).
At the steady-state, SR = γσ/S, but the market price of risk is
countercyclical, and hence so is the Sharpe ratio.
Proof of result on MPR:
In this model, the SDF is conditionally log-normally distributed,
hence we can apply the general formula:
p
SRt = eV art (log Mt+1 ) − 1 ' σt (log Mt+1 )
Proof: use the log-normal formula
E(exp(X)) = exp E(X) + 12 V ar(X) and compute
1
Et (Mt+1 ) = eEt (log Mt+1 )+ 2 V art (log Mt+1 ) ,
and
2
V art (Mt+1 ) = Et (Mt+1
) − [Et (Mt+1 )]2 ,
= e2Et (log Mt+1 )+2V art (log Mt+1 )
− e2Et (log Mt+1 )+V art (log Mt+1 ) ,
hence the result.
Solving the model
Solving numerically this model is complicated, because of the
important nonlinearities.
The aggregate market is represented as a claim to the future
consumption stream. Let Pt denote the ex-dividend price of this
claim. Then, Et [Mt+1 Rt+1 ] = 1 implies that in equilibrium Pt
satisfies:
Pt+1 + Ct+1
Et Mt+1
=1
Pt
Pt
Ct
Pt+1
Ct+1
= Et Mt+1
+1
Ct+1
Ct
−γ !
St+1
Ct+1 1−γ Pt+1
= Et β
+1
St
Ct
Ct+1
Solution:
The state variable is st .
We solve for a fixed point, i.e.
h(st ) = Eut+1
β
St+1
St
Pt
Ct
= h(st ) with
−γ Ct+1
Ct
!
1−γ
(h(st+1 ) + 1) ,
with
∆ct+1 = g + ut+1 ,
st+1 = (1 − φ)s + φst + λ(st )ut+1 ,
ut+1 ∼ i.i.d.N (0, σ 2 ).
Fixed point equation
This implies the following fixed point equation in h(s)):
Z ∞
h(s) = β
exp(f (s, u))dΦ(u).
−∞
where
f (u, s) = −γ ((1 − φ) (s − s) + λ(s)u) + (1 − γ)g
+ (1 − γ)u (h ((1 − φ)s + φs + λ(s)u) + 1)
To compute this fixed point we need to evaluate the integral on
the right hand side.
Recipe:
Set a grid for s, {s1 , s2 , ..., sN } ;
Take a guess for h, i.e. {h (s1 ) , h (s2 ) , ..., h (sN )} ;
For each value of s in the grid, compute the RHS of the
fixed-point equation numerically. We need to interpolate h to
find its value at the point (1 − φ)s + φs + λ(s)u since it lies
outside the grid.
To compute the integral
numerically,
one way is to use a
R
P
quadrature, e.g. k(u)du = i ωi k(ui ) for some points ui .
This gives a new guess for h using the fixed-point equation. We
keep iterating on this equation until convergence.
Once we know h, we have the P-D ratio and the rest can be
computed simply, as in Mehra-Prescott. (See Wachter, Finance
research letters, for a note detailing the computation of that
model.)
Basic results:
The model matches the level of the riskless rate and the equity
return, and their volatilities.
no variation in expected cash flows, no variation in the risk-free
rate, but a lot of variation in risk premia.
Intuition: Risk premia are higher after a sequence of some bad
shocks, which push C close to X, hence a very low s.
Suggests that risk premia should be highest in the depth of a
recession.
Covariance of returns with st is the key source of risk here.
Additional results:
Volatility of returns is higher in bad times.
The model also matches the time-series predictability evidence:
dividend growth is not predictable, but returns are predictable,
and the volatility of the P-D ratio is accounted for by this latter
term (“discount rate news”).
Long-run equity premium: because of mean-reversion in stock
prices, excess returns on stocks at long horizons are even more
puzzling than the standard one-period ahead puzzle. CC note
that if the state variable is stationary, the long-run standard
deviation of the SDF will not depend on the current state. Key
point: in their model, S −γ is not stationary – variance is growing
with horizon!