share prices;capitalism;financial markets

Lettau, M.

English

Research Papers in Economics

Comment on `The Spirit of Capitalism and
Stock-Market Prices' by G.S. Bakshi and
Z. Chen (AER, 1996) 
by
Martin Lettau
CentER for Economic Research and Department of Finance,
Tilburg University, and CEPR
June 2, 1997
I thank John Campbell for helpful comments. JEL code: G1, G10. ERN elds:
Macroeconomics, Capital Market.
12
1 Introduction
In a recent paper Bakshi and Chen (BC) (1996) argue that augmenting stan-
dard preferencesby allowing wealth to directlyenter the utilityfunction helps
to explain risk premia in capital markets. The motivation for this general-
ization comes from the observation that utility might not only be generated
by consumption but also by the social status implied by high wealth. The
most straightforward way to include this notion into neoclassical economics
is to modify the utility function. BC propose various functional forms of
preferences which have consumption, wealth and a social-wealth index as
arguments. They focus on the implications of these preferences on risk pre-
mia by studying the volatility bounds of the intertemporal marginal rate of
substitution (IMRS) as suggested by Hansen and Jagannathan (1991). They
conclude that their preference specication enter the volatility bounds for
more reasonable parameter values than the standard CRRA utility function.
In particular, a relative risk aversion coecient of about 10 is sucient com-
pared to about 100 for CRRA preferences.
This comment argues that this conclusion can be misleading because it
does not take into account all restrictions among variables implied by the
model. In particular, the authors treat consumption and wealth as indepen-
dent variables when they compute the Hansen-Jagannathan bounds of the
IMRS. However, one of the implicationsof the model is that the consumption-
wealth ratio is constant and innovations in consumption and wealth are per-
fectly correlated. Once these restrictions are used to substitute out wealth
from the IMRS, the same IMRS is obtained as in the standard CRRA spec-
ication (apart from a dierent discount factor) in which only consump-
tion enters the IMRS. This implies that the preferences studied in BC have
the same implications for risk premia as the standard CRRA preferences.
For example, a high risk aversion coecient is needed to enter the Hansen-
Jagannathan bounds. As shown by Kocherlakota (1990), Svensson (1989),
Campbell (1993) and Lettau and Uhlig (1997) the same issue arises in the3
related preference specication of Epstein and Zin (1989). This argument
shows that it is very important to substitute out as many variables as possi-
ble from the IMRS in order to avoid misleading conclusions concerning the
variability of the IMRS implied by the model.
The rest of this comment is organized as follows. Section 2 reproduces the
important aspects of the model studied by BC and summarizes the implica-
tions for the Hansen-Jagannathan bounds. Section 3 substitutes out wealth
from the IMRS using restrictions among consumption and wealth implied
by the model and shows that the model reduces basically to the standard
CRRA model. Section 4 shows that an extended version of the preferences
has potentially richer implications for risk premia and Section 5 concludes.
2 The Model
2.1 The Setup
This section reproduces the most important parts of BC. Let Ct denote con-
sumption of some investor, Wt her wealth and Vt some social-wealth index.
Let the period utilityfunction depend on these three arguments: U(C;W;V).
I impose the same restrictions on the derivatives of U as BC. They propose
two dierent functional forms for U. Their Model 1 is a \absolute wealth"
model:
U(C;W;V)=
C 1 − γ−1
1−γ
W
− ; (1)
while their Model 2 includes the social-wealth index:
U(C;W;V)=
C 1 − γ−1
1−γ

W
V
 − 
: (2)
See BC for motivations for these utility functions. I omit their Model 3
because it is essentially a variant of Model 2. To avoid excessive notation I
present the model which is solved in section II in BC. Investors can invest4
in two assets. First, there is a riskless asset with return r0 which is assumed
to be constant over time. The price of the risky second asset is assumed to
follow a Brownian motion:
dPt
Pt
= dt + d!t; (3)
where !t is a standard Wiener process. The investor solves the following
problem:
J(Wt;V t)=m a x
C s;s
Et
Z 1
t
e
−(s−t)U(Cs;W s;V s)ds

(4)
subject to
dWt = Wt(r0 + t( − r0)) − Ct + tWtd!t; (5)
where  denotes the discount rate.
2.2 The Solution
BC give the complete solution of the model. Here I present only the equations
which are relevant for my argument. First, the consumption-wealth ratio is
constant:
C
?
t = W
?
t ; (6)
where variables with a `?' denote the optimal choices in the above problem.
The parameter  is a complicated function of the parameters of the model and
is giveninProposition 2 of BC. Furthermore, the growth rates of consumption
equals the growth rate of wealth:
dC?
t
C?
t
=
dW ?
t
W ?
t
= wdt + wd!t; (7)
where w and w are again given in BC (p. 143). It is also worth noting that
the coecient of relative risk aversion (RRA) as measured by the curvature
of the value function with respect to wealth is given by γ + in this model.5
2.3 Implications for Asset Prices
BC use the Hansen-Jagannathan (HJ) (1991) bounds for Model 1 to show
this preference specication is more successful in explaining risk premia than
the standard CRRA specication. They do not compute HJ-bounds of Model
2 which includes the social-wealth index. They argue that it is hard to nd a
satisfying functional form for the social wealth index V and question whether
it is appropriate to use aggregate data to test such as model.
The HJ-bounds are derived from the stochastic disount factor (SDF) im-
plied by the model. The discrete time version of the SDF for for Model 1
is
Mt+1 = 

Ct+1
Ct
−γ 
Wt+1
Wt
− 
1+

γ−1
C t +1
Wt+1

: (8)
Using aggregate data, BC nd that the model satises the HJ-bounds for
parameter congurations which imply only moderately high risk aversion.
For example, when γ and  are both around 5, RRA is around 10 and the
bounds are satised. The authors stress that this is a great improvement
over the standard CRRA preferences which require a RRA of over 100 to
satisfy the bounds.
3 Substituting out Wealth
In their empiricalexercise, BC treat consumption and wealth as separate vari-
ables. However, the model has strong implications for their co-movement.
In particular, (6) implies that the consumption-wealth ratio is constant and
hence innovations in consumption growth and wealth growth are perfectly
correlated (as can be seen also from (7)). These restrictions have very impor-
tant implications for the HJ-bounds and asset prices in general. Substituting
out wealth in the SDF (8) using (6) yields
Mt+1 = 

Ct+1
Ct
−(γ+) 
1+

γ−1

6
= ~ 

Ct+1
Ct
−RRA
;
where ~  = 

1+ 
γ− 1

. Hence the SDF is the same as in the standard
CRRA preference case with a dierent discount rate. Recall the RRA is
equal to γ + here. This implies that the HJ-bounds are basically the same
as in the CRRA case (only the discount factors dier). In particular, an
extremely high risk aversion is needed to enter the bounds.
Note, that a very similar problem appears in the related preferences spec-
ication of Epstein and Zin (EZ) (1989). They propose a recursive utility
function that allows to disentangle risk aversion and intertemporal substitu-
tion. In addition to consumption, the return of the market portfolio enters
the SDF. As pointed out by Campbell (1993) this model implies a constant
consumption-wealth ratio when returns are i.i.d.. He used that fact to solve
out consumption. As shown in Kocherlakota (1990) and Lettau and Uhlig
(1997), one can alternatively solve out the return on the market portfolio.
They show that the SDF reduces to the standard CRRA case as long as
returns are i.i.d.. The important implication in the EZ framework as well as
in the BC model is that high risk aversion is needed to enter the HJ-bounds
once all the model implications are used.
4 A More General Model
BC compute the HJ-bounds only for their Model 1. They argue that it
would not be appropriate to test Model 2 with aggregate data. However, it
is instructive to compute the SDF for Model 2:
Mt+1 = 

Ct+1
Ct
−γ 
Wt+1
Wt
− 
Vt+1
Vt
− 
1+

γ−1
C t +1
Wt+1

: (9)7
Using again the fact that the consumption-wealth ratio is constant to sub-
stitute out wealth, I obtain
Mt+1 = 

Ct+1
Ct
−(γ+) 
Vt+1
Vt
− 
1+

γ−1


: (10)
Note that (10) does not reduce the the standard CRRA case as the growth of
the social-wealth index variable V remains in the SDF. Hence if V is highly
volatile the SDF becomes more volatile as well making it easier to enter th
HJ-bounds. Whether the volatility of this variable on a high frequency is
reasonable is of course open. But in principle a framework like Model 2 of
BC has an additional channel to increase the volatility of the SDF compared
to the standard CRRA model and Model 1 of BC.
5 Conclusion
This comment argues that it is important to take into account all restrictions
among variables implied by the model when computing the SDF of a model.
If this is not done, variables will be treated as independent time series often
leading to an overestimation of the volatility of the SDF. The model studied
by BC is one example of this eect, but the same argument holds for the
Epstein-Zin (1989) preferences.8
References
Bakshi, Gurdip S. and Zhiwu Chen (1996), \The Spirit of Capi-
talism and Stock-Market Prices," American Economic Review
86, 133-157.
Campbell, John Y. (1993),\Intertemporal Asset Pricing without
Consumption Data," American Economic Review 83, 487-512.
Epstein, Larry G. and Stanley E. Zin (1989), \Substitution, Risk
Aversion, and the Temporal Behavior of Consumption and
Asset Returns: A Theoretical Framework," Econometrica 57,
937-969.
Hansen, Lars Peter and Ravi Jagannathan (1991),\Implications
of security market data for models of dynamic economies,"
Journal of Political Economy 99, 225-62.
Kocherlakota (1990), \Disentangling the Coecient of Relative
Risk Aversion from the Elasticity of Intertemporal Substitu-
tion: An Irrelevance Result," Journal of Finance 45, 175-90.
Lettau, Martin and Harald Uhlig (1997), \Preferences, Consump-
tion Smoothing, and Risk Premia," working paper, Tilburg
University.
Svensson, Lars (1989), \Portfolio Choice with Non-Expected Util-
ity in Continuous Time," Economics Letters 30, 313-317.

Asset Pricing without Consumption Data,&quot;

Disentangling the Coecient of Relative Risk Aversion from the Elasticity of Intertemporal Substitution: An Irrelevance Result,&quot;

Peter and Ravi Jagannathan (1991),\Implications of security market data for models of dynamic economies,&quot;

Portfolio Choice with Non-Expected Utility in Continuous Time,&quot;

Preferences, Consumption Smoothing, and Risk Premia,&quot; working paper,

Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,&quot;

The Spirit of Capitalism and Stock-Market Prices,&quot;

Comment on "The Spirit of Capitalism and Stock Market Prices" By G.S. Bakshi and Z. Chen (AER, 1996)

Comment on "The Spirit of Capitalism and Stock Market Prices" By G.S. Bakshi and Z. Chen (AER, 1996)

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