This paper examines an economy in which aggregate shocks are not dispersed equally throughout the population. Instead, while these shocks affect all individuals ex ante, they are concentrated among a few ex post.The equity premium in general depends on the concentration of these aggregate shocks; it follows that one cannot estimate the degree of risk aversion from aggregate data alone. These findings suggest that the empirical usefulness of aggregation theorems for capital asset pricing models is limited.

N. Gregory Mankiw

English

Research Papers in Economics

NBER WORKING PAPER SERIES
THE EQUITY PREMIUM AND THE
CONCENTRATION OF AGGREGATE SHOCKS
N. Gregory Mankiw
Working Paper No. 1788
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
January 1986
I am grateful to Fischer Black, Olivier Blanchard, Stanley Fischer,
John Long, and David Romer for helpful comments. The research
reported here is part of the NBER's research programs in Economic
Fluctuations and Financial Markets and Monetary Economics. Any
opinions expressed are those of the author and not those of the
National Bureau of Economic Research.ER Working Paper #1788
January1986
TheEquity Premium and
the Concentration of Aggregate Shocks
ABSTPACT
Thispaper examines an economy in which aggregate shocks are not
dispersed equally throughout the population. Instead, while these shocks
affect all individuals ex ante, they are concentratedamong a few ex post.
The equity premium in general depends on the concentration of these
aggregate shocks; it follows that one cannot estimate the degree of risk
aversion from aggregate data alone. These findings suggest that the
empirical usefulness of aggregation theorems for capital asset pricing
models is limited.
Professor Gregory Ivlankiw
Harvard University
123 Littauer Center
Canbridge, NA 021381. Introduction
Several recent and important studies have attempted to exp)ain the
joint behavior of asset returns and aggregate consumption using
representative consumer models.1 This empirical work raises the obvious
question of whether it is valid to aggregate across consumers. In this
paper I present a simple model economy in which aggregation is not valid
and, in particular, obscures the economic forces underlying relative asset
returns. I assume that aggregate shocks to consumption are not dispersed
equally across all consumers. Instead, while all consumers are subject to
adverse aggregate shocks ex ante, these shocks affect only some consumers ex
post.I show that the concentration of aggregate shocks is a potentially
important determinant of relative asset returns.
The model illustrates how the absence of certain contingent claims
markets can render representative consumer models largely ineffective as
approximations to a complex economy with ex post heterogeneous consumers.
Rubinstein (1974) and Grossman and Shiller (1982) prove aggregation
theorems that do not require complete markets. The results I present here
suggest that these theorems cannot be greatly extended. More important,
they suggest that these theorems do not fully justify the use of
representative consumer models in empirical studies of asset pricing.
The general principle is that the absence of complete markets implies
that individual consumption is more variable than per capita consumption,
even if individuals are identical ex ante. Unless individuals have
quadratic utility, so that the marginal utility schedule -is linear, this-2—
extra variability generally affects both the mean of marginal utility and
its covariance with asset returns. It is generally not possible to
aggregate individuals' first-order conditions relating consumption and asset
returns to a relation holding with per capita consumption data.
The model also suggests a possible solution to the equity premium
"puzzle" discussed by Mehra and Prescott (1985) among others. The nature of
this puzzle can be seen using the consumption-beta relation Grossman and
Shiller (1982) derive. They show that
E Rt =ACov(Rt, 1nCt), (1)
where Rt is the difference in return between any two assets, A is the
harmonic mean of individuals' Arrow-Pratt coefficient of relative risk
aversion, and C is aggregate consumption. This relation implies that
A E Rt /a(Rt)a(AlnCt). (2)
In United States data, the equity premium is about six percent, the standard
deviation of the realized equity premium is about twenty percent, and the
standard deviation of the growth in consumption of non-durables and
services is about three precent. The inequality in (2) therefore implies
that the coefficient of relative risk aversion exceeds ten. Using (1) and
noting that the correlation of the market return and consumption growth is
about one-third, we find that the implied coefficient of relative risk
aversion is about thirty, which is generally considered implausible.
The model presented here suggests that the level of the equity
premium is -in part attributable to the role of incomplete markets in—3—
determining the equilibrium return on marketable assets. In particular,
for any set of aggregate variables, the equity premium may be made
arbitrarily large or small by changing the concentration of the
aggregate shock among the population. This finding implies that one cannot
judge the appropriateness of the equity premium from aggregate data alone.
2. A Simple Illustrative Model
I illustrate the importance of the concentration of aggregate shocks
using the simplest possible model. I first describe the aggregate economy
and how an observer might attempt to infer the degree of risk aversion from
aggregate data. I then consider the disaggregate distribution of the
aggregate shocks and the implications for relative asset returns.
2.1 The Aggregate Economy
There are two points of time in the model. At time zero, while the
endowment of the consumption good is uncertain, portfolio choices are made.
At time one, the endowment is realized and consumption takes place.
Per capita consumption in the economy takes on two values: a good value
of j.z,anda bad value of (1 -q)p,where 0 <q<1.Each state occurs
with probability 3.
I examine a portfolio that pays —1 in the bad state and pays 1 +it in
the good state, where itisthe "premium." One can think of this portfolio
as consisting of two assets: a short position in an asset that pays off in
both states (Treasury bills) together with a long position in an asset that
pays off only in the good state (equity).
Consider a representative consumer with utility function U(•) deciding
how much of the security to purchase. His goal is to maximize—4—
E U(C) (3)
where C is consumption. If R is the payoff of the portfolio, then the
standard first—order condition is
E [R U'(C)] =0. (4)
The marginal utility weighted mean return is zero.
Given the distribution of per capita consumption, this first-order
condition can be written as
(1 + it) U'(jt)—u'((l— = 0 (5)
If it is valid to describe the economy as generated by this representative
consumer, equation (5) must hold at the equilibrium level of it.Equation
therefore (5) produces the following value of the premium:
it =— [U'(ji)— — q)t)]/ u'(4. (6)
For small values of 4, the premium is approximately
11 =— [iU''(&)/U'(.z)] 4. (7)
=
whereA is the coefficient of relative risk aversion.
An economist observing the size of the aggregate shock (4) and the
premium on the portfolio (it) might wish to estimate the degree of risk
aversion. Using the approximation in (7) he would obtain—5-
A =7t/4. (8)
Alternatively, he might explicitly parameterize the utility function as
U(C) =C1/1—A. (9)
In this case, equation (6) implies
A =- log(].+ir)/log(1-q), (10)
which isapproximatelythe same as (8) for small itand.
2.2.Individuals and Equilibrium
Suppose there are an infinite number of individuals that are identical
ex ante. That is, as of time zero, the distribution of consumption is the
same for all individuals.I assume, however, that their consumption is not
the same ex post. In particular, I assume that in the bad state, the fall
in aggregate consumption of qjiisconcentrated among a fraction A of the
population.
The stochastic environment facing any given individual is therefore as
follows. With probability 3, a good state occurs: his consumption is .tand
the portfolio pays 1 +it. Withprobability 3, a bad state occurs. In the
bad state, the portfolio pays -1; his consumption is jiwithprobability 1. -A
and is (1 -4/A)gi withprobability A.I assume there do not exist
contingent claims markets through which individuals can diversify away this
latter risk.2
The parameter A measures the concentration of the aggregate shock. If
A =1,then all individuals have the same consumption ex post.As A-6—
approaches q, the aggregate shock becomes more highly concentrated. At
A =, theaggregate shock is fully concentrated on a few individuals whose
consumption falls to zero.
The first-order condition (4) holds for each individual, which implies
(1 +ii) U'(ji)—(1—A)U'(ji) —Au'((i —q/X)p)=o. (11)
The premium is therefore
it= x -4/X)i)-W[gi])/U'[j.t]}. (12)
The premium (it)ingeneral depends not only on the size of the aggregate
shock (4) but also on its distribution within the population (A).
2.3. The Implications of Concentration
I now consider how the concentration of the aggregate shock affects
the size of the equity premium and the apparent degree of risk aversion
that an observer might infer from aggregate data.I assume that the
observer knows the size of the aggregate shock 4 and the size of the
premium itanduses the results from the representative consumer model--that
is, equations (9) and (1O)—-to estimate the coefficient of relative risk
aversion.
The first result Is:
Proposition 1: If the utility function U(S) is quadratic, then the
premium is independent of the concentration of the
aggregate shock. That is, it does not depend on A.—7—
This result follows directly from equation (12). It implies that if
utility is quadratic, then the concentration of the aggregate shock does
not affect the apparent degree of risk aversion. Hence, our observer isnot
led astray by his representative consumer model.
This result does not generalize, however, as the nextproposition
makes clear:
Proposition 2:If the third derivative of the utility function is
positive, then an increase in the concentration of the
aggregate shock increases the premium. That is,
if U''' > 0, then
an
Proof:By differentiating equation (12), we obtain
—U'[(l- - U'[ji)+ (qi/x) U''[(l - ax — u'J
This can be rewritten as
J.1
air {U''[(l —q/A)i]—U''[Z]}dZ
=(1-q/x).
U' Eu]
If U''' > 0, then the expresssion in the integral is negativeover
the range of integration. This completes the proof.
The condition of a positive third derivative isvery plausible; indeed, it
is even weaker than the condition of non-increasing absolute risk-8-
aversion.3 The implication of Proposition 2 is that one cannot determine
the size of the equity premium from aggregate data alone. It further
suggests that our observer could be badly mistaken using a representative
consumer model, that is, equation (10). In particular, since our observer
estimates the degree of risk aversion correctly if A =1,Proposition 2
implies that if A <1,our observer overestimates the degree of risk
aversion.
The assumption that the concentrated shock is an adverse one is
crucial to the direction of this bias. If, instead, we considered a model
with a concentrated windfall, greater concentration would imply a smaller
premium. The general case is discussed in Section 3.
The next proposition shows that the error from using the representative
consumer model in fact can be great.
Proposition 3: Suppose the utility function satisfies the Inada
condition: urn U'(C) =
c—,0
Then limir =
Proposition3 follows directly from equation (12). It shows that
regardless of the size of the aggregate shock, the equity premium can be
made arbitrarily large by making the shock more and more concentrated.
Thus, if the Inada condition is satisfied, one cannot place an upper bound
on the equity premium from only the degree of risk aversion and the
aggregate shock. Conversely, one cannot place a lower bound on the degree
of risk aversion from the aggregate shock and the equity premium alone.-9-
It may be instructive to apply some numbers to the model. Suppose
=0.05,so that the aggregate endowment falls by five percent in the bad
state. Table 1 presents the ratio of the true premium (equation (12)) to
the premium one would expect from the representative consumer model
(equation 7). Suppose A =0.2,so that twenty percent of the population
experiences a fall in endowment of twenty-five percent in the bad state.If
utility is logarithmic, then the true equity premium is 1.3 times what one
would expect from a representative consumer model. If the constant relative
risk aversion is six, then the equity premium is 2.6 times what one would
expect. While the model is clearly too stylized to draw firm empirical
conclusions, the numbers in Table 1 do suggest that the concentration of
aggregate shocks is a potentially important determinant of the equity
premium.
3. Discussion
This section provides a less formal and perhaps more intuitive
discussion of the effects highlighted in the model of Section 2. As above,
consider an economy in which all individuals are honiogenous ex ante but
heterogeneous ex post. Let R be the difference in return between two tra-
dable assets and C. be the consumption of individual i. The first-order
condition each individual satisfies is
E [R tJ'(C1)] =0. (13)
Letbe the expectation of consumption. Since individuals are identical ex
ante, this mean is the same for all individuals, and therefore does not-10-
require a subscript i. The second-order Taylor approximation of marginal
utility around w is
U'(C) =U'(w)+U''()(C —w)+iU'''(w) (Ci —w)2. (14)
Substituting (14) into (13) yields
E(R)=- ---- E[R(C
- -1'''E[R(C
-w)2], (15)
where the derivatives are evaluated at w. Now sum (15) over the
individuals in the economy. Letting C denote per capita consumption and N
the number of individuals in the population, we obtain the following
expression for the expected excess return:
— ,,, — I,,
E(R) =- jj--—E[R(C-w)] -tE{R(C-w)2] — E[R{E(C.- )2/N}]. (16)
The three terms in equation (16) provide some insight into the determinants
of relative asset yields.
If utility is quadratic, the second and third terms in equation (16)
disappear. Expected return then depends only on the covariance of per
capita consumption with return. If the third derivative is positive,
then expected return depends on the third cross-moment of per capita
consumption with return, as represented in the second term of equation (16).
The third term shows how the deviations of individual consumption
from per capita consumption affect expected return. In particular, if
U' '' >0,then expected return depends on the cross-moment of return with—11—
ex post heterogeneity. In the model of Section 2, heterogeneity is great
when return is low; this cross-moment is therefore negative, which exerts a
positive influence on expected return. In general, however,
non-diversifiable individual risk can exert either a positive or negative
influence on the equity premium.
4. Conclusion
The simple model presented here illustrates how one might be misled
using a representative consumer model to estimate the degree of risk
aversion from the size of the equity premium. Unless aggregate shocks to
income affect all investors equally ex post, relative asset returns in
general depend on the distribution of aggregate shocks among the population.
It is therefore not possible to infer investors' risk aversion from aggregate
data alone.
It seems plausible that the concentration of aggregate shocks is an
important determinant of the equity premium. It is well-known that
recessions do not affect all individuals equally; rather, they fall on a
small fraction of the population that experiences very large losses in
income. From 1929 to 1933, consumption of nondurables and services
per capita fell only 20 percent. One suspects that certain investors
experienced much larger drops in their standard of living.
The results obtained here require the absence of contingent claims
markets through which individuals can agree ex ante to spread this aggregate
risk among themselves ex post. This assumption appears a reasonable
approximation to observed behavior. Undoubtedly, the reason such markets—12—
do not exist is a combination of moral hazard and adverse selection
considerations.
In light of these results, one might wonder whether representative
consumer models remain a useful paradigm in empirical work. It is probably
impossible to justify rigorously these models once we admit that many
contingent claims markets do not exist. Vet representative consumer models
may nonetheless remain a useful approximation for applications in which the
failure of Arrow-Debreu assumptions is not critical. Moreover, models using
a "surrogate" consumer with a hypothetical utility function may be useful for
some purposes even if this surrogate cannot be interpreted as representative
of actual individuals in the economy. Delineating the boundary between the
(approximately) valid and invalid uses of representative consumer models is
an important topic for future research.—13—
Table 1
Ratio of the True Premium to the Premium Inferred from Aggregate Model
=0.05
A=1 4=3 A=6
X =0.1 1.9 4.2 17.5
A =0.2 1.3 1.6 2.6
Note: A =Coefficientof relative risk aversion,
=Sizeof adverse aggregate shock,
A =Fractionof population affected by aggregate shock.—14—
Notes
1.
See, for example, Sh-iller (1982), Hansen and Singleton (1983),
Mehra and Prescott (1985), and Dunn and Singleton (1984).
2.
It is this assumption that makes Rubinstein's (1974) aggregation
theorem inapplicable. Rubinste-in assumes that all risky assets are traded,
so that the portfolio of risky assets is the same for all individuals.
3.
This condition is related to the precautionary demand for saving; see
Leland (1968) and Sandmo (1970). It is also related to skewness preference
in asset demand; see Kraus and Litzenberger (1976) for some empirical
support for the assumption of a positive third derivative.—15—
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Hansen, Lars P., and Kenneth J. Singleton, 1983, Stochastic consumption,
risk aversion, and the temporal behavior of asset returns,
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Kraus, Alan, and Robert H. Litzenberger, 1976, Skewness preference and the
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An aggregation theorem for securities markets,

Consumption correlatedness and risk measurement in economies with non-traded assets and heterogeneous information,

Consumption, asset markets, and macroeconomic fluctuations,

Modeling the term structure of interest rates under nonseparable utility and durability of goods,

Saving and uncertainty: the precautionary demand for saving,

Skewness preference and the valuation of risk assets,

Stochastic consumption, risk aversion, and the temporal behavior of asset returns,

The effect of uncertainty on saving decisions,

The equity premium: a puzzle,

The Equity Premium and the Concentration of Aggregate Shocks

The Equity Premium and the Concentration of Aggregate Shocks

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