The Case Against Constant Relative Risk Aversion and a Suggested Alternative
Sami Alpanda
Amherst College
Geoffrey Woglom
Amherst College
November 2006
Abstract
This paper first reviews some of the paradoxes and implausible, strong results that follow
from the assumption that preferences exhibit constant relative risk aversion. We then
propose an alternative where preferences exhibit constant absolute risk aversion in relative
wealth, where relative wealth is defined as wealth relative to a habit level of consumption.
We show how these preferences are consistent with the empirical evidence on risk aversion,
eliminate the implausible results, and can resolve the risk-free, equity premium asset pricing
puzzles.
D810: Criteria for Decision-Making under Uncertainty
G110: Portfolio Choice
2
I. Paradoxes and Problems of Constant Relative Risk Aversion
The assumption of constant relative risk aversion preferences (CRRA) is commonly
used in the theoretical literature both due to its analytical convenience and also because it is
supported on an empirical basis. For example, the assumption of CRRA preferences leads
to closed form solutions to lifetime portfolio selections models (Merton [1992]). In addition,
starting with Friend and Blume (1975), the empirical evidence suggests that risk aversion
among a cross section of individuals with different levels of wealth appears to be roughly
constant.
Despite its attractiveness, however, the assumption of CRRA preferences does lead
to problems that we believe are serious enough to warrant a reconsideration of its
widespread use. These problems are all caused by the fact that with CRRA preferences
events with vanishingly small probabilities of occurrence can have a finite effect on behavior.
This section explains and gives examples of one well-known paradox and two related
problems (also well known) that arise with the assumption of CRRA preferences. None of
this is new, but the significance of these issues will help motivate our proposed alternative to
CRRA preferences.
A. St. Petersburg Paradoxes with Unbounded Utility
As noted in Meyer and Meyer (2005), there is a long-standing and continuing
tradition of utilizing “introspection” to draw inference on the degree of risk aversion. For
example, economists have long argued that there must be an upper bound to utility, in order
to avoid a St. Petersburg-style paradox whereby agents will choose to take on gambles with a
large probability of considerable losses, but a tiny probability of enormous gains.
In the context of a CRRA utility function, an upper bound requires the degree of
relative risk aversion to be greater than one. It will prove useful to review the mathematics
of this result, in a variation of the St. Petersburg paradox game.
Consider the “game” where your wealth (after paying an “entrance fee”, F) will
multiply by a factor X, for every successive time you toss a head on a fair coin. The first
time a tail appears the game stops. If a tail comes up on the first toss, your terminal wealth,
WT, is W0 – F, where W0 is the initial wealth. If a tail comes up on the second toss, your
terminal wealth is X(W0 - F); if a tail comes up on the third toss, your terminal wealth is
X2(W0 - F), and so on. The expected utility for playing the game when preferences are
CRRA is
[(W  F )( X ) j 1 ]1
E0 [V (WT )]   0
1 
j 1

(I.1)
(W  F )1
 0
2(1   )
1
 
2
 ( X )1 



2 
j 1 

j
j 1
For any  < 1 and X  21/(1 ) , E0 [V (WT )] is infinite regardless of the size of W0  F  0.
For example, if   0.5, then the infinite series of expected utility is divergent for X  4.
Suppose we have a prospective player whose initial wealth is $100,000 and X = 4.1
with a  = 0.5. According to expected utility maximization, this individual would play this
game even for an entrance fee of $99,999. Notice, however, that our player would have to
toss at least 9 consecutive heads in order for terminal wealth to exceed $100,000, an event
with a probability of less than two-tenths of one percent. The idea that vanishingly small
probabilities of enormous gains could have such a large effect on the willingness to take on
gambles leads economists to argue that utility must be bounded from above.
2
Arrow (1971) also recognized that similar problems arise if utility was not also
bounded from below.1 In this case, vanishingly small probabilities of enormous losses may
preclude agents from taking on a gamble. This can be easily illustrated with the “inverse
game” to the one above.
In this game, you are paid an entrance fee, F, to play, but every time you toss a head
on a fair coin, you have to return a part of your remaining wealth (including the entrance
fee). The game stops when the first tail appears. In particular, if a tail comes up on the
1

second toss, you return (W0  F ) 1   and your terminal wealth is also (W0 + F)/X. If a
 X
1 

tail comes up on the third toss, you return (W0  F ) 1  2  , and your remaining wealth is
 X 
(W0 + F)/X2, and so on.
The expected value for playing this game with CRRA preferences is given by
1
(I.2)
 (W0  F ) 
 
( X ) j 1 
E0 [V (WT )]   
1 
j 1
(W  F )1
 0
2(1   )
1j
 
2 
 ( X ) 1 



2 
j 1 

j 1
For any  >1 and X  21/  1 , E0 [V (WT )] is negative infinity regardless of the size of W0  F .
For example, if  =1.5, then the infinite series is divergent for X  4.
Now, again suppose that we have a prospective player whose initial wealth is also, $100,000,
but whose relative risk aversion is 1.5 and we set X =4.1. Unfortunately, our potential player
It is not clear why Arrow’s concerns with unbounded utility from below as well as from above did not have
more influence. For example, Samuelson wrote [1969, p.887]: “To apply our results, let us consider the
interesting Bernoulii case where U = log C. This does not have the bounded utility that Arrow [1] and many
writers have convinced themselves is desirable for an axiom system. Since I do not believe that Karl Menger
paradoxes of the generalized St. Petersburg type hold any terrors for the economist, I have no particular
interest in boundedness of utility…”
1
3
will refuse the gamble for all finite values of the entrance fee, F. So for example, our player
would find the game to be “too risky,” even if after the entrance fee, her wealth was $1
billion (=$100,0002). Again, in this case she has less than a 0.2% chance of being worse off
after playing the game, but those very small probabilities of a disaster are sufficient for her to
forego the $1 billion.
B. Indifference to “Compound Gambles”
A related problem with CRRA utility is that an individual’s preferences for a gamble
do not depend on the number of times the gamble is performed. 2 Another simple example
is useful to illustrate this point. Consider the gamble where wealth will increase by the factor
X >1, with probability p > ½ and will decrease by 1/X, with probability (1-p). Assume that
the values of p and X have been selected so that our CRRA subject is indifferent to the
gamble. In this case, the values of X and p are related as follows:
E0 V (W1 )   p
(I.3)

(W0 X )1
(W / X )1
 (1  p) 0
1 
1 
W01
pX 1  (1  p) X  1
1 


Note that V (W0 )  E0 V (W1 ) if pX 1  (1  p) X  1  1
Now consider an alternative gamble where the outcomes are based on two binomial
trials with the same values for X and p. In this case, the possible outcomes are:
WX 2 with probability p 2 , W with probability 2(1- p), and W / X 2 , with probability (1  p)2 .
The agents’ expected value from this “compounded” gamble is given by
Samuelson’s (1963) in an article entitled the “Risk and Uncertainty: A Fallacy of Large Numbers” has a sharp
criticism of a faculty lunchroom colleague who believes the number of times the gamble is performed matters
by the “Law of Large Numbers.” While the example in the text is not quite the same, we believe that it shows
that the colleague’s argument has substantial merit.
2
4
E0 V (W2 )  p 2
(I.4)

(W0 X 2 )1
W 1
(W / X 2 )1
 2 p(1  p) 0  (1  p) 2 0
1 
1 
1 
2
W01
 pX 1 y  (1  p) X  1   V W0 
1 
We are not imagining the second gamble as being sequential. Instead, we are asking
our constant relative risk aversion subject to evaluate two one-time gambles with different
outcomes and different probabilities. Our CRRA subject, however, will be indifferent to the
outcomes based on 1 or 2 binomial trials, as long as the payoffs are compounded. Of
course, an exactly similar argument can be made to a gamble based on 3 or more binomial
trials where the outcomes are adjusted accordingly.
But herein lies the paradox: with p > 0.5, the probability of a loss gets smaller and
smaller as the number of trials increases. As p increases without limit, the chance of a loss
approaches zero, but our CRRA subject will value this gamble as equal to the first gamble.
In the case of , this is because even though the probability of losses are going to zero,
the maximum size of those losses is increasing without limit and the utility consequences of
those losses have finite consequences.2
It is worth showing this last result more explicitly. Consider the one-trial gamble with
the following properties:
Pick an X  1, Y  1, and p  0.5 such that:
pX 1  (1  p)Y 1  1
Now consider an alternative gamble with a larger value for p (call this pY) and smaller value
for Y (call this YP) such that the individual is still indifferent to the gamble:
pY X 1  (1  pY )YP1  1
For the utility consequences of the large losses are finite, but the extra benefits of the large gains is
going to zero.
2
5
which implies
1
1
Yp  1  pY X 1 1 1  pY  1
Since   1 and 0  1  pY X 1  1 , there always exists a YP.> 0 that will meet the indifference
condition as long as pY < 1.
Now consider a (pY , YP ) couple that satisfies the indifference condition and pY is
arbitrarily close to (but less than) 1. Also consider a Y that is arbitrarily close to (but less
than) YP . A person with constant relative risk aversion will not accept this gamble even if
winning is a nearly “sure thing.” The size of the disaster in the bad state of nature is large
enough that the agent would rather not take the gamble even if the probability of a disaster
is tiny. This strikes us as implausible as people do seem to take chances such as those in
many everyday situations (e.g. driving, flying on an airplane).
The example above illustrates why the Law of Large Numbers doesn’t work for
individuals with CRRA preferences (with > 1). Think of the payoff X in our simple
gamble as the expected return from a gamble. The Law of Large Numbers implies that as
the sample size increases, the sample mean will become arbitrarily close to the population
mean. In our simple gamble, this is captured by pY approaching 1. Thus with CRRA, even
though the sample return is approaching the expected return, expected utility is not
changing. This is a consequence of unbounded utility, where events with vanishingly small
probabilities can have finite effects on expected utility.
C. Diversification over Time.
The last of our related problem with CRRA was stated most clearly by Samuelson
(1969, pp.883-4):
6
Third, being still in the prime of life, the business man can “recoup” any present
losses in the future. The widow or the retired man nearing life’s end has no
such “second or nth chance.”
Fourth, (and apparently related to the last point), since the businessman will be
investing for so many periods, “the law of averages will even out for him,” and
he can afford to act almost as if he were not subject to diminishing marginal
utility.
However, before writing this paper, I had thought that points three and four
could be reformulated to give a valid demonstration of businessman’s risk, my
thought being that investing for each period is akin to agreeing to take a 1/nth
interest in insuring n independent ships.
The present model does not itself (emphasis in the original) introduce extra
tolerance for riskiness at early, or any, stage of life.
The “present model” was the discrete time version of the lifetime portfolio selection
model where a closed-form solution was derived for the CRRA case. When theory (based
on CRRA preferences) conflicted with introspection, Samuelson chose theory.3 Samuelson
had modeled the instantaneous utility from consumption, U(C ), as exhibiting CRRA and
then had shown that the indirect utility function for wealth, V(W) was also CRRA with the
same degree of relative risk aversion.
To illustrate this point, consider the following lifetime portfolio selection problem:
(I.5)
V Wt  = max U (Ct )   Et V Wt 1  
Ct ,t
s.t. Wt 1  Wt  Ct  [( t ,t 1  R)t  R]
where  is the discount factor,  is the share of assets allocated to the risky-asset, and  and
R are the gross rate of return on the risky and risk-free assets respectively.  only enters
the problem through the second term of (I.5). If V(W) is CRRA, then the first order
condition for can be written as:
In fact, in the conclusion of this article, Samuelson specifically criticizes a school of investing that focuses on
maximizing the expected geometric return for investors with long-time horizons. This criticism set off a longrunning argument between Samuelson and Harry Markowitz, among others. In a subsequent article,
Samuelson (1971) recants a bit by recognizing that without CRRA (which follows from bounded utility),
portfolio choice will depend on wealth. For a summary of this debate see Brainard, et. al. (1991).
3
7
(I.6)
Et V ' Wt 1  ( t ,t 1  R)   Wt  Ct 
1

Et  ( t ,t 1  R)t  R    0


so that the optimal  does not depend on the level of wealth but only on the individual’s
relative risk aversion and the distribution of asset returns.
Note that as long as the risk-free rate and the distribution of the risk premium are
time invariant, optimal portfolio choice will not depend on time. Therefore, there are no
life-cycle or time-horizon effects on optimal portfolio choice. This very important and
powerful result is based on CRRA preferences, and it strikes many (including Samuelson
pre-1969) as implausible.
II. Bounded Utility Where the Degree of Relative Risk Aversion Is Constant Across
Individuals.
As was pointed out in the beginning, the assumption of CRRA preferences is often
justified by the empirical fact that people with different levels of wealth on average exhibit
similar relative risk aversion. But there is an important distinction between the assumption
of CRRA for individual preferences and the observation that average relative risk aversion
among groups of individuals appears to be constant across groups with different average
levels of wealth. The strong theoretical results discussed in the last section depend on
preferences where at a moment in time the individual’s elasticity of marginal utility of wealth
is constant for all possible levels of wealth. In this section, we show that it is possible to
develop a utility function for wealth where the elasticity of marginal utility is an increasing
function of wealth, so that utility is bounded, but the average degree of relative risk aversion
across groups of individuals with different average levels of wealth is constant.
8
Our utility function is based on a variation of “habit formation” with constant
absolute risk aversion, where the specification of the habit level of consumption has been
chosen for analytical convenience.4
W
V t
 Zt
(II.1)


  e

AWt
Zt
where Z t is the "habit level" of consumption given by:
Z t  Ct 1
(II.2)
The relative risk aversion (RRA) for these preferences is given by:
(II.3)
RRA  
2V / Wt 2
W
WA t
V / Wt
Zt
Hence in this CARA utility function, the RRA depends on the absolute risk aversion times
the relative wealth, W/Z.5
We have not been able to find a closed-form solution for the direct utility function
of consumption that gives rise to this indirect utility function for wealth, but we do know
that this unknown function is homogeneous of degree zero in C/Z. While this function
exhibits CARA for any given level of habit consumption, it can also be consistent with the
Friend and Blume (1975) evidence of the relative constancy of relative risk aversion across
groups of individuals. The key point is that when we observe individuals with different
wealth levels, they will also have different habit levels of consumption. If wealth and habit
consumption are roughly proportional across individuals, so too will be the average value of
Sundaresan (1989) solves the life-time portfolio choice problem with habit formation for both the CRRA and
the CARA cases. Constantinides (1990) analyzes the CRRA case in detail and how it can be used to solve the
“equity-premium puzzle”. In both cases, the argument of the utility function is the difference between current
consumption and the habit level, as opposed to our specification of consumption relative to the habit level.
5 We are using the simplest functional form of a bounded utility function in relative wealth. In the Appendix
we illustrate a more general specification that does not imply, as (II.3) does, that for a given Z relative risk
aversion is increasing in W.
4
9
RRA for different wealth categories. In some ways, our solution to the cross-section
evidence is similar to Friedman’s solution to the consumption puzzle, where the average
value of the Average Propensity to Consume (APC) is roughly constant across groups of the
poor and the affluent, but for an individual the marginal propensity to consume is less than
the APC. Similarly, the degree of RRA at the initial level of wealth can be constant across
groups of individuals while decreasing in wealth for any individual.
With CARA, the utility from wealth is bounded both from above and below and
thereby St. Petersburg Paradoxes are eliminated. For example, in our “inverse” gamble from
section I.A., where RRA =1.5 and X is equal to 4, a reverse entrance fee of about $450,000
would induce our CARA subject to accept the gamble. This individual would take the
gamble, in spite of the fact that there is a 25% probability that terminal wealth will be below
initial wealth.
Similarly, the CARA utility function implies that the individual will not be indifferent
to the number of trials. We were somewhat surprised that there is not a general statement
for whether the 2-trial gamble will be preferred to the one-trial gamble when the expected
utility of the one-trial gamble is equal to the utility of initial wealth. We have been able to
show through numerical simulations that for gambles of more than 2% of wealth and where
RRA is greater than 3, the two-trial gamble is preferred to the one-trial gamble.
More importantly, however, the CARA function implies that events with vanishingly
small probabilities of occurrence can have no finite effect on expected utility. To see this,
pick an X  1, Y  1, and p  0.5 such that:
pe
A
XW
Z
A
( X 1)W
Z
 (1  p )e
A
YW
Z
e
A
W
Z
which implies
pe
 (1  p )e
A
(1Y )W
Z
1
10
Now consider the pairs of pY and YP that will provide indifferent gambles:
pY e
A
( X 1)W
Z
 (1  pY )e
A
(1YP )W
Z
1
Note that
(1  pY )e
A
(1YP )W
Z
 1  pY e
As pY  1, (1  pY )e
A
A
(1YP )W
Z
( X 1)W
Z
 1 e
A
( X 1)W
Z
0
 0, for all 0  YP  1
Therefore, with CARA, there exist a pY sufficiently close to 1, where the individual will prefer
the gamble, even when the bad state of nature involves the total loss of wealth. Thus the
Law of Large Numbers will, in fact, work to the advantage of CARA individuals.
Time diversification is a more complicated issue, but CARA preferences do not lead
to the strong, but implausible results of CRRA preferences. The lifetime portfolio selection
problem and the first-order condition for  (the analogs of (I.5) and (I.6)) can be written as:
(II.4)
C
V Wt  = max U  t
Ct ,t
 Zt
  Wt 1  

   Et V 


  Zt 1  
s.t. Wt 1  Wt  Ct  [( t ,t 1  R)t  R]; Z t 1  Ct
(II.5)
Wt Ct [(t ,t 1  R )t  R ] 

A
  Wt 1 

Zt 1
0
Et V ' 
 ( t ,t 1  R)   Et ( t ,t 1  R)e


  Z t 1 

and in this case the optimal value of  will vary with the relative level of wealth.
11
By taking a second-order Taylor-series expansion around the expected value of
relative wealth in period t+1, it can be shown that (see Appendix) the optimal portfolio
allocation, , would also maximize:
Et ( t ,t 1  R)t  R  1 
(II.6)
2
A (Wt  Ct ) 
Et  ( t ,t 1  R)t  R  1 


2 Z t 1
Note that
RRAV
A (Wt  Ct ) Wt 1

2 Zt 1 (Wt  Ct )
2
and thus the optimal portfolio allocation depends on the moments of the distribution of the
returns on the risky asset and on preferences as measured by relative risk aversion. 6
The difference is that relative risk aversion can vary over time with a CARA utility function.
In our case, it will vary positively with relative wealth, and it is certainly possible that lifecycle effects will lead to increasing relative wealth and relative risk aversion as the planning
horizon shortens with age. Time-varying portfolio allocation in this case is more about the
tolerance for risk varying with the horizon, than it is about taking advantage of
diversification over time. Time diversification is taking advantage of the Law of Large
Numbers and our CARA individual does not avail herself of this opportunity, even though it
could work to her advantage.
We can state this result in a slightly different way. If you force an individual to make
one portfolio allocation and then stick to that allocation for the rest of his life, the CRRA
individual will not find this constraint to be a problem and different individuals with
different planning horizons will pick the same allocation if they have the same degree of
If we had set the problem up in continuous time, as in Merton (1992), all that would matter in our case would
be the degree of relative risk aversion and the mean and variance on the risky asset, which is exactly the same as
the CRRA case with no habit formation. We are using the notation of Meyer and Meyer (2005) to distinguish
between the curvature of the indirect utility function, RRAV and the curvature of the direct utility function,
U(.), RRAU.
6
12
relative risk aversion. This is not the case for an individual whose preferences are given by
the CARA utility function in relative wealth. Individuals with the same relative wealth and
the same degree of relative risk aversion but with different planning horizons will not select
the same portfolio allocation. In general, the longer the horizon the riskier the allocation
selected. This is just a generalization of the “Law of Large Numbers” working for our
CARA person.
The reason that individuals with CARA preferences do not time-diversify is that they
can do even better. Consider the CARA individual who is indifferent to the one-trial gamble
but prefers the 2-trial gamble. Time diversification means that the “trials” are sequential and
because of liquid financial markets one is not forced to accept the second gamble. For
example, consider the one-trial gamble for a CARA individual (where we have replaced X
with 1+x):7
(II.7)
W
A 0
Z
xW0
A
  A xW0

e
(1 x ) Z
Z
E0 V W1   
 1  p  e
 pe

 A 

Let p and x be such that the individual is indifferent between taking the gamble or not:
V W0   E0 V W1   when pe
A
xW0
Z
 1  p  e
A
xW0
(1 x ) Z
1
Our CARA subject will prefer the two-trial gamble when the following condition holds:
E0 V W2    E0 V W1    V (W0 ) if and only if
 2 x  x2 W0 

 2 x  x2 W0
A
A
2
2
Z
H   p 2e
 2 p 1  p   1  p  e (1 x ) Z   1




As already noted this condition does not always hold for small gambles and low risk
aversion. But consider the case where H < 1 does hold. Because these trials occur over
7
See the Appendix for details of the calculations.
13
time (i.e., the first trial in period 1 and second in period 2) a better gamble is the following:
Accept the first- period gamble and then if one loses (in which case relative risk aversion will
be lower) take the second gamble, which is now preferred to not gambling. If on the first
trial one wins, call off the second gamble because the increase in relative risk aversion
implies that you prefer certain wealth of W0(1+x) to the expected utility of the second
gamble.8
Individuals with CARA preferences choose not to “take advantage of the Law of
Large Numbers” because a better strategy is to adapt one’s portfolio allocation to evolving
relative risk aversion due to past successes or failures or to life-cycle effects. This timevarying portfolio allocation is due to changing risk tolerance and not to the desire to have
winners offset losers over time. Perhaps the folk wisdom that people in the prime of life
should take on a riskier portfolio than widows and orphans is due to the combination of lifecycle effects of risk tolerance along with the recognition that most people will solve the
portfolio allocation very infrequently. Given the fact that people voluntarily tie their hands,
thereby forcing themselves into multi-trial gambles, the longer the planning horizon the
riskier the portfolio should be.9
III. CARA Utility for Relative Wealth and Asset Pricing Puzzles
The original models of habit formation were developed to explain the equitypremium puzzle [Mehra and Prescott (1985)] and the related risk-free rate puzzle [Weil
(1989)]. While our specification of “relative” habit formation is different from that of the
This is a simple example of a more general proof in Samuelson (1971) that shows that in the case of bounded
utility functions “uniform” strategies are non-optimal, where uniform amounts to committing one’s self to a
multi-period gamble.
9 Samuelson’s “proof” of his lunchroom colleagues folly of accepting a 100 trial gamble when he would not
accept a 1 trial gamble (see footnote 2) is based on analyzing the “sequence” of each of the 100 gambles. We
read the lunchroom colleague as asking for a one-time gamble where the payoffs are based on the outcome of a
100 trial binomial experiment.
8
14
original models, it is still capable of explaining the puzzles. The key idea in habit formation
models is that trend growth in consumption leads to less of a decrease in marginal utility
than does a one-time change in consumption. We can illustrate this idea with a first-order
Taylor series expansion of marginal utility:
U '(Ct 1 )  U '(Ct )  U "(Ct )(Ct 1  Ct )  U '(Ct )  U "(Ct )Ct gt 1 ,
(III.1)
where
gt 1 
Ct 1
 1 is a random variable.
Ct
The Euler equation for the risk-free asset relates the change in utility from saving more
today and investing in the risk-free asset.
(III.2)
U '(Ct 1 ) 
U '(Ct )   REt [U '(Ct 1 )]  0, or  REt 
 1  0
 U '(Ct ) 
Using (III.1), the risk-free rate puzzle (assuming there were, in fact, a completely risk-free
asset) can be written as:
U '(Ct )   RU '(Ct 1 )   RU "(Ct 1 )Ct 1Et ( gt 1 )  0
(III.3)
(1   R)  RRAU Et ( gt 1 )  0, where
RRAU  
U "(Ct 1 )Ct 1
U '(Ct )
Therefore, if R is less than one, then any growth in consumption is inconsistent with the
Euler equation. If R is greater than one, then growth in consumption can be consistent
with the Euler equation so long as the rate of growth in consumption and/or the degree of
relative risk aversion are not too large. Thus, in general, the Euler equation for the risk-free
rate requires a small value for relative risk aversion in order to accommodate trend growth in
consumption.
15
The two related asset pricing puzzles arise because while the degree of relative risk
aversion has to be low to accommodate trend growth in consumption, it has to be high to
accommodate the volatility in the risk premium. The Euler equation relating to the risk
premium, is derived from the condition for the optimal share of the portfolio in the risky
asset and can be written in either one of two equivalent ways:
(III.4)
E[U '(Ct 1 )( t ,t 1  R)]  0, or
(III.4’)
E[V '(Wt 1 )( t ,t 1  R)]  0
where (III.4) shows the expected consequences of the excess return on the risky asset for
consumption in period t+1, and (III.4’) shows the consequences for wealth. Using (III.1)
with (III.4) yields:
U '(Ct ) E ( t ,t 1  R)  U "(Ct )Ct E ( t ,t 1  R, gt 1 )  0
(III.5)
E ( t ,t 1  R)  RRAU [cov( t ,t 1  R, gt 1 )  E ( t ,t 1  R) E ( gt 1 )]
Notice that the last term in parentheses will be (relatively) small because it is the product of a
growth rate and interest rate. Therefore, trend growth in consumption is not terribly
important in this Euler equation. Volatility in consumption growth is more important in
helping to determine the covariance term. But if consumption is not very volatile and the
average risk premium is high, relative risk aversion will have to be high to accommodate the
Euler equation.
In our model of habit formation we cannot directly test the Euler equation related to
the risk-free rate, (III.2) because we do not know the functional form of the direct utility
function U  Ct / Zt  . While we don’t know the exact functional form, we can write the
analogous equations to (III.1) and (III.2) and derive similar qualitative results: 1) If R is
less than one, no growth in the argument of the utility function is consistent with the Euler
16
equation; 2) If R is greater than one, then the Euler equation can be fulfilled if either
growth in the argument of the utility function and/or relative risk aversion is small. The
important difference with the habit utility function is that not consumption, but
consumption relative to the habit level is the argument of the utility function; and in a steady
state there is no growth in this relative consumption. With little or no growth in the
argument of utility, the Euler equation for the risk-free rate can be fulfilled for higher
degrees of relative risk aversion.
Similarly, we cannot use equation (III.4) to test the equity premium, Euler equation,
but in this case we can use the first-order condition from the indirect utility function. As we
showed in (II.5) the first-order condition for  leads to a condition very similar to (III.3),
where we can write:
 W 

W
Et V '  t 1  ( t ,t 1  R)   0, so that a Taylor series expansion around t implies:
Z t 1
  Z t 1 

W 
W W
V '  t  Et  t ,t 1  R   V "  t  t Et  t ,t 1  R  g W t 1   0,
 Z t 1 
 Z t 1  Z t 1
(III.6)
where g W t 1 
Wt 1
1
Wt
E ( t ,t 1  R)  RRAV cov( t ,t 1  R, g W t 1 )  Et  t ,t 1  R  Et  g W t 1  
While (III.6) looks similar to (III.3), it is important to note that the degree of relative risk
aversion in the indirect utility function is, in general, not the same as in the direct utility
function [c.f. Meyer and Meyer (2005)], and the distribution of the growth in wealth is not
the same as the distribution for the growth in consumption. But we can, and in the next
17
section do, look at historical data to see whether the data are consistent with the first-order
condition on optimal portfolio choice.
It is important to note that our version of habit formation “solves” the two asset
pricing puzzles the same way that the traditional models do. Habit formation allows relative
risk aversion in the indirect utility function to be large enough to solve the equity premium
puzzle, while lessening (and in our case eliminate for the steady state) the effect of trend
growth in consumption on the risk-free puzzle.
IV. Can CARA Preferences over Relative Wealth Explain the Asset Pricing Puzzle?
To investigate whether CARA preferences over relative wealth can explain the asset pricing
puzzles, we will follow the methodology laid out in Kocherlakota (1996) and Meyer and
Meyer (2005). We see if the Euler conditions implied by the lifetime portfolio allocation
problem are satisfied given data and reasonable parameter values.
Before turning to CARA preferences, we first show how the equity premium and
risk-free rate puzzles emerge under CRRA utility function. Consider the following 3 statistics
(where  is the constant degree of relative risk aversion): 10

(IV.1)
C 
u   Rt ,t 1  t 1 
 Ct 
(IV.2)
C 
v   t ,t 1  Rt ,t 1   t 1 
 Ct 
C
t
1

C
t
Note that the “risk-free” asset is not totally risk-free given uncertainty about inflation. We introduce timesubscript for the risk-free asset to be able match it with the data as in Kocherlakota (1996).
10
18
(IV.3)
W 
v   t ,t 1  Rt ,t 1   t 1 
 Wt 

W
t
For a particular value of , utC is the statistics that corresponds to the Euler condition related
to the risk-free rate, equation (III.3). If that Euler condition holds this statistic has an
expected value of zero. If the sample mean of the statistics is statistically significantly
different from zero, however, we can reject the hypothesis that this Euler condition is
fulfilled for that particular value for the degree of relative risk aversion.
Similarly, vtC is the statistic that corresponds to the Euler equation relating the risk
premium and the direct utility function U(C), i.e. (III.4), and vtW is the statistic that
corresponds to the Euler equation relating the risk premium and the indirect utility function
V(W), i.e. (III.4’). 11 The related asset pricing puzzles are the values of A that are needed to
avoid rejecting the risk-free Euler equation (viz., low values) lead to the rejection of the riskpremium puzzles (viz., high values) and vice versa.
The variable C is matched to Real, per Capita Consumption Expenditures data from
the NIPA tables of the Bureau of Economic Analysis (BEA). W, per capita wealth, is
matched to the (beginning-of-period) net financial assets of households (i.e. Net worth –
Tangible assets) deflated by the Personal Consumption Expenditure (PCE) deflator. The
household wealth data were obtained from the Sectoral Balance Sheets provided in the Flow
of Funds (FOF) Accounts of the Federal Reserve Board and the deflator is from NIPA of
the BEA. The real return on risky asset, , was matched to the real rate of return of the S&P
500 and the real return on the risk-free asset, R, was matched to the real rate of return of 3month U.S. Treasury-bills (the inflation numbers used were derived from the PCE deflator,
however the resultant series are very similar to those derived from the GDP deflator). Both
We follow Kocherlakota in dividing the Euler equation by marginal utility in period t. This allows us to
measure percentage deviations from the respective Euler equations.
11
19
series were obtained from Ibbotson (2006). The data used was for the years 1955-2005. Data
are available for earlier post-war years, but the Fed’s practice of pegging the T-bill rate in the
pre-accord period led us to discard these data [c.f. McGrattan and Prescott (2003)]. Table 1
reports the summary statistics of the data used.
Table 1: Summary Statistics of the Data
Rt ,t 1
 t ,t 1  Rt ,t 1 
g tC1
g tW1
Mean
0.0122
0.0684
0.0240
0.0213
Variance
0.0005
0.0291
0.0003
0.0046
Covariance
0.0005
0.0007
0.0001
0.0005
0.0291
0.0001
0.0006
0.0003
0.0006
0.0046
The statistics in Table 1 suggest that CRRA preferences will have a hard time
fulfilling the two Euler equations: 1. There is significant average growth in consumption
and wealth per capita. 2. The average risk premium is high, but the covariance of the risk
premium with growth in consumption per capita or wealth per capita is relatively low. The
first fact implies that a high degree of risk aversion will make it more difficult to fulfill the
risk-free Euler equation and the second that a low degree of risk aversion will have trouble
fulfilling the risk-premium Euler equations.
We test these results statistically, using the data described above. Table 2 presents the
sample mean and the t-statistics of the 3 test statistics for different values of the relative risk
aversion parameter.12
12
We used a discount factor = 0.99.
20
Table 2: Evaluating Euler Equations (CRRA preferences)
v C x104
uC
v W x104

mean
t-stat
mean
t-stat
mean
t-stat
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
-0.02119
-0.03252
-0.04364
-0.05456
-0.06530
-0.07584
-0.08620
-0.09638
-0.10638
-0.11620
-0.12585
-0.13534
-0.14465
-0.15380
-0.16279
-0.17163
-0.18031
-0.18883
-0.19721
-6.25683
-7.94274
-8.80311
-9.25390
-9.50445
-9.65225
-9.74405
-9.80355
-9.84346
-9.87093
-9.89019
-9.90384
-9.91351
-9.92025
-9.92477
-9.92757
-9.92896
-9.92917
-9.92838
0.06670
0.06588
0.06508
0.06430
0.06353
0.06277
0.06203
0.06130
0.06059
0.05989
0.05921
0.05853
0.05787
0.05723
0.05659
0.05597
0.05535
0.05475
0.05416
2.77791
2.76397
2.74973
2.73518
2.72034
2.70521
2.68981
2.67414
2.65821
2.64203
2.62561
2.60896
2.59208
2.57499
2.55770
2.54022
2.52255
2.50472
2.48671
0.06653
0.06569
0.06491
0.06417
0.06348
0.06284
0.06224
0.06168
0.06116
0.06067
0.06023
0.05981
0.05944
0.05909
0.05878
0.05850
0.05825
0.05803
0.05785
2.67405
2.59931
2.51906
2.43397
2.34482
2.25243
2.15766
2.06139
1.96447
1.86768
1.77177
1.67738
1.58510
1.49538
1.40863
1.32514
1.24514
1.16878
1.09616
The results above confirm the intuition reached in Section III. The first column
shows that there is no value of risk aversion that makes the data consistent with the risk-free
Euler equation. It is worth noting that not only is the average growth in consumption
during this period high, but it is uniformly positive (in 46 out of the 51 years), thus the first
term in (IV.1) is uniformly below 1 (in 45 out of the 51 years) even when the degree of
relative risk aversion is equal to 1.
The equity-premium Euler equation, however, can be reconciled with the data using
higher values of relative risk aversion (in particular  greater than 3).13 Note the results are
qualitatively invariant to the choice of the Euler equation specified in terms of the direct
The qualitative results here are very similar to results reached by other researchers, the quantitative
differences presumably are due to the different sample.
13
21
utility or indirect utility functions. In each case, the t-statistic falls with higher degrees of
relative risk aversion. The t-statistics are uniformly lower for the indirect utility function,
which probably reflects the greater volatility of wealth (see Table 1).
We now repeat the above exercise using our version of habit formation with CARA
preferences over relative wealth. The Euler condition for the risk premium is given by
equation (II.5) and can be tested by testing whether the following statistic has a sample mean
different from 0:
W
Z
t
v   t ,t 1  Rt ,t 1  e
(IV.4)
W
W 
 A t 1  t  
 Zt 1 Zt 
  t ,t 1  Rt ,t 1  e
W 
A W
  t 1  t 
  Ct Ct 1 
We first calculate our summary statistics for the data using the growth of
consumption and wealth relative to the habit level of consumption.14
Table 3: Summary Statistics of Data using Relative Variables
Rt ,t 1
 t ,t 1  Rt ,t 1 
gtC1/ Z
g tW1/ Z
Mean
0.01219
0.06838
0.00066
-0.00190
Variance
0.00046
0.02912
0.00046
0.00472
Covariance
0.00046
0.00074
0.00010
0.00050
0.02912
0.00091
0.00139
0.00046
0.00081
0.00472
The most important thing to note in Table 3 is that there is no positive trend growth
in relative wealth or relative consumption. So the major problem of fulfilling the risk-free
puzzle is gone.
We again calculate the mean and t-statistic of the statistic given in (IV.4) for different
values of .15 We pick the value of A, so that the degree of relative risk version at the
14
We used a parameter value for =0.7.
22
sample mean of W/Z varies from 1 to 10. Note from (IV.1) that only the value of A/
matters. The results are summarized below in Table 4:
Table 4
v
W
Z
AW
Z
mean
t-stat
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
0.06696
0.06626
0.06555
0.06482
0.06406
0.06328
0.06245
0.06157
0.06063
0.05962
0.05851
0.05731
0.05599
0.05453
0.05293
0.05116
0.04919
0.04701
0.04459
2.64036
2.54786
2.44935
2.34555
2.23729
2.12544
2.01091
1.89459
1.77738
1.66011
1.54357
1.42846
1.31540
1.20494
1.09751
0.99348
0.89313
0.79665
0.70419

As shown above, the equity premium puzzle again can be solved for relative risk
aversion greater or equal to 3 using CARA preferences over relative wealth. Because we
cannot find the functional form for the direct utility function, we cannot test formally the
risk-free puzzle. But we believe that the data on relative consumption growth in Table 3 are
suggestive that our preferences are consistent with the risk-free puzzle. In a steady state,
relative consumption would have no trend growth. In our sample, average growth in relative
consumption is negative (see Table 3). With this low relative growth there are many periods
15
Note that now the degree of relative risk aversion is given by AW/Z .
23
where relative consumption is falling as well as rising. For example, in 25 of the 51 years in
our sample, relative consumption growth is negative. Therefore, while we don’t know the
functional form of marginal utility from relative consumption, it less likely that there is a
trend in marginal utility that is causing the risk-free premium.
V. Conclusions and Further Thoughts.
We have shown that the assumption of constant relative risk aversion and
unbounded utility leads to problems, paradoxes and strong but counterintuitive implications
about portfolio choice. All of these problems can be resolved by assuming CARA
preferences over relative wealth. In addition, because the degree of relative risk aversion
depends on relative wealth, our model is not inconsistent with the evidence of Friend and
Blume (1975).
But do these preferences meet the “introspection test?” For further insights we can
appeal to Daniel Bernoulli himself, as quoted by the psychologist Daniel Gilbert (2006):
Although a poor man generally obtains more utility than does a rich man
from an equal gain, it is nevertheless conceivable, for example, that a rich
prisoner who possesses two thousand ducats but needs two thousand ducats
more to repurchase his freedom, will place a higher value on a gain of two
thousand ducats than another man who has less money than he. Though
innumerable examples of this kind may be constructed, they represent
exceedingly rare exceptions.
Our model doesn’t directly involve imprisonment, but we can easily construct an
example where the rich man gains more utility from an extra 2000 ducats than does a poorer
man. Consider a rich man who has suffered some recent losses (perhaps he was just
imprisoned), so that his relative wealth is low. The gain of 2000 ducats will raise his relative
wealth by less than that of a poor man who has a lower level of habit consumption. But if
the poor man started with relative wealth sufficiently greater than the rich man, the poor man
24
will gain less of an increase in utility.16 We therefore share Gilbert’s conclusion (although for
somewhat different reasons) that,”…the ‘innumerable exceptions’ that Bernoulli swept
under the rug are not exceedingly rare.
For example, suppose the rich man’s wealth is 8,000 ducats while his habit level of consumption is 2,000
ducats. Our poorer man has only 5,000 ducats in wealth, but a habit level of consumption of 1,000 ducats.
Therefore the relative wealth of the rich man is less than that of the poorer man. As long as the common
preferences are such that the rich man’s degree of relative risk aversion is greater than or equal to 2 (i.e., A < .5), the rich man will enjoy more utility from the 2000 ducat gain.
16
25
Appendix
A. The General Form of Bounded Utility in Relative Wealth
In the text we have used the simplest possible specification of bounded utility in relative
wealth. A more general specification of utility is the following
( A) V (W / Z)  e f (W / Z ) ; where f '  0; f "  0, or
(B) V (W / Z)  e f (W / Z ) ; where f '  0; f "  ( f ')2
Since relative wealth must be non-negative, the negative exponential guarantees
boundedness. In case (A):
f '(W / Z)  f (W / Z )
e
 0;
Z
  f '(W / Z )  2 f "(W / Z) 
 e f (W / Z )  0;
V "(W / Z)    

2

Z

Z
 

WV " 
f "(W / Z)  W

  f '(W / Z ) 

V'
f '(W / Z)  Z

V '(W / Z) 
In case (B):
V '(W / Z )  
f '(W / Z )  f (W / Z )
e
 0;
Z
 f '(W / Z )  2 f "(W / Z ) 
 e f (W / Z )  0;
V "(W / Z )  

2

Z

Z


WV " 
f "(W / Z )  W

   f '(W / Z ) 

V'
f '(W / Z )  Z

In both cases, relative risk aversion depends on relative wealth, and therefore on average can
be constant among groups of people with different average levels of wealth. The simplest
example of case (A) is the preferences assumed in the text where f (W / Z )  A
W
. We use
Z
these preferences in the text because it keeps the algebra simple.
26
Some, however, might be concerned that for a given value of Z, case (A) implies
increasing relative risk aversion in W. This implication is not necessary in the more general
specification of preferences. For example, consider the case (B) example where
f (W / Z )  A
Z
Z
Z2
W 1
, so that f '   A 2 , and f "  2 A 3 , where
 . In this case, for a
W
Z 2
W
W
constant Z relative risk aversion is A
Z
 2 , which for a given value of Z is decreasing in W
W
B. Compound Gambles with CARA preferences.
Consider the one-trial gamble for a CARA individual (where we have replaced X with
(1+x)):
 W 
E0 V W1    pV 1  x W0  + 1  p V  0 
 (1  x) 
p
p
e
A
1 x W0
Z
 1  p 
A
e
A
W
A 0
Z
A
xW
A 0
Z
e
A
e
 1  p 
W0
(1 x ) Z
A
e
A
W0
Z
A
xW0
(1 x ) Z
e
A
W0
Z
xW0
A
  A xW0

(1 x ) Z
Z

 1  p  e
 pe

 A 

e
Let p and x be such that the individual is indifferent between taking the gamble or not:
V W0   E0 V W1   when pe
A
xW0
Z
 1  p  e
A
xW0
(1 x ) Z
1
The expected utility of the two trial gamble is given by:
27
 W0 
 (1  x) 2 


2
2
E0 V W2    p 2V 1  x  W0  +2 p 1  p V W0   1  p  V


 p2
 p2
e
A
1 x 2 W0
Z
 2 p 1  p 
A
e
A
W0
Z
e
A
x
2
e
A
W0
Z
A
A
 1  p 

 2 x W0
Z
 2 p 1  p 
A
e
A
W0
Z
A
2
e
W0
(1 x )2 Z
A
 1  p 
2
e
A
W0
Z
A
e
x
2

 2 x W0
(1 x )2 Z
A
 2 x  x W0 

 2 x  x2 W0
A
A
2
2
2

Z

pe
 2 p 1  p   1  p  e (1 x ) Z 

A 


e
W
A 0
Z
2
 2 x  x2 W0 

 2 x  x2 W0
A
A
2
2
Z
 V W0   p 2 e
 2 p 1  p   1  p  e (1 x ) Z   0




Our CARA subject will prefer the two-trial gamble when the following condition holds:
E0 V W2    E0 V W1    V (W0 ) iff:
 2 x  x2 W0 

 2 x  x2 W0
A
A
2
2
Z
H   p 2e
 2 p 1  p   1  p  e (1 x ) Z   1




This condition does not hold for in general (e.g., it is violated for p=0.50745, x =.01,
AW/Z=-4), but it generally holds for larger gambles (e.g., it holds for p=0.51485, x =.02,
AW/Z=-4). Now consider the case where H < 1, but the gambles are sequential and the
individual turns down the second gamble if the good state of nature occurs in the first
period.
 2 x  x2 W0 

xW0
A

A
e
 pe Z  p 1  p   1  p 2 e (1 x )2 Z  ,
E (V (W2 ) 

A 


A
W0
Z
The alternative gamble is preferred to the two-trial gamble if the expected utility of the
second gamble after a good state of nature on the first trial is less than the utility of the
wealth after the first trial (i.e., V(W0(1+x))):
28
  A (1 x ) W0

Z
 1  p  e  AW0 
 pe


EV [W0 (1  x)]  
A
( x 2  x )W0
 A (1 x )W0 

A
e
Z
EV [W0 (1  x)] 
 1  p  e AxW0 
 pe
 A 

2
  A( x
EV [W0 (1  x)]  V (W0 (1  x))  pe

2
 x )W0
Z

 1  p  e AxW0   0

But because V (W0 )  E[V (W1 )]:
pe
e
A
A
pe
xW0
Z
 1  p  e
( x  x )W0
Z
2
A
e
( x 2  x )W0
Z
A
xW0
Z
A
xW0
(1 x ) Z
 1, and
; e AxW0  e
 1  p  e
AxW0
A
xW0
(1 x ) Z
 pe
A
, so
xW0
Z
 1  p  e
A
xW0
(1 x ) Z
 1, and
EV [W0 (1  x)]  V (W0 (1  x)) and E[V (W2 )]  E[V (W2 )]
C. The Homogeneity of U(C, Z) when V (W , Z )  e
W
A
Z
 V (W / Z )
Consider two individuals with different levels of absolute wealth but the same levels of
expected relative wealth at the start of the terminal period.
ET 1[T ]
T

ET 1[WT ]
, and T  WT ,   0.
ZT
ET 1[WT ] WT 1  CT 1
  T 1

ET 1[(T 1,T  R)T *  R]  T 1
ET 1[(T 1,T  R)T *  R]
ZT
ZT
T
 W
T 1  T 1 WT 1  CT 1

, and E T-1  V  T
T
ZT
  ZT

e
  

A
WT 1 CT 1
(T 1,T  R )T * R
ZT
A
 
 E T-1  V  T
  T



Now consider the first-order condition at time T-1:
29
  W  [(
 R)  R]
W 
- E T-2  V'  T  T 1,T
  T2   0
CT 1
ZT
ZT 
  ZT 
 W 
  
W 
 
But  E T-2  V'  T  [(T 1,T  R)  R]   T    E T-2  V'  T  [(  R)  R]   T  ,
ZT 
T 
  ZT 
  T 
so that:
U(CT-1, Z t 1 )
U( T-1, t 1 )

Z t 1  
 t 1
CT 1
T 1
U(CT-1, Z t 1 )

Because this last relationship must hold true for all  U (CT 1, Z t 1 )  U (CT 1 / Z t 1 ) 
D. Optimal Portfolio Choice with CAR preferences Over Relative Wealth
W 
 W  Ct 
The second-order Taylor series approximation of V  t 1  around V  t
 can
 Zt 1 
 Zt 1 
be written as:
A
Wt Ct [(t ,t 1  R )t  R ]
Zt 1
e
A

A
e
Wt Ct 
Zt 1
A
e
A
Wt Ct 
Zt 1
Wt 1  (Wt  Ct )   A e A  Z
Wt Ct 
Z t 1
t 1
Wt 1  (Wt  Ct ) 
Z t 12
2
2
,
where Wt 1  (Wt  Ct )  (Wt  Ct ) ( t ,t 1  R)t  R  1
A

e
Wt Ct 
Zt 1
A
e
A
Wt Ct 
Z t 1
(Wt  Ct )
A A
( t ,t 1  R)t  R  1  e

Z t 1
2
Wt Ct 
Zt 1
(Wt  Ct )
Z t 12
2
 (
t ,t 1
 R)t  R 
2
  A Wt Ct [(t ,t 1  R )t  R ] 
Zt 1
e

Et 

A




A

e
Wt Ct 
Zt 1
A
e
A
Wt Ct 
Z t 1
(Wt  Ct )
A A
Et ( t ,t 1  R )t  R  1  e
Z t 1
2
Wt Ct 
Z t 1
2
(Wt  Ct )
Et  ( t ,t 1  R )t  R  1 


Z t 1
2
W C 
  A Wt Ct [(t ,t 1  R )t  R ] 
A t t
Zt 1
e
 Z t 1e Zt 1
Z t 1
A (Wt  Ct ) 
Et 

 Et ( t ,t 1  R )t  R  1 
Et  ( t ,t 1  R)t



A
(
W

C
)

A
(
W

C
)
2
Z
t
t
t
t
t

1




W C 
  A Wt Ct [(t ,t 1  R )t  R ] 
A t t
Zt 1
e
 Z t 1e Zt 1
Z t 1
RRA (Wt  Ct ) 
Et 

 Et ( t ,t 1  R )t  R  1 
Et  ( t ,t 1  R)

A
 A(Wt  Ct )
2 Et (Wt 1 ) 

 (Wt  Ct )


30
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32