Vol. 63 (1996), No. 2, pp. 139-150
Journal of
Zeitschrift
Economics
fur Nationatokonomie
© Spiinjer-Verlag 1996 - Printed in Austiia
Risk Preference and Indirect Utility
in Portfolio-choice Problems
Santanu Roy and Rien Wagenvoort
Received May 24, 1995; revised version received March 5, 1996
We consider a portfolio-choice problem with one risky and one safe asset,
where the utility function exhibits decreasing absolute risk aversion (DARA).
We show that the indirect utility function of the portfolio-choice problem need
not exhibit DARA. However, if the (optimal) marginal propensity to invest is
positive for both assets, which is true when the utility function exhibits nondecreasing relative risk aversion, then the DARA property is carried over from
the direct to the indirect utility function.
Keywords: portfolio choice, absolute risk aversion, relative risk aversion, indirect utility.
JEL classification: D81, D92.
1 Introduction
One of the basic models used to study the relationship between risk
preference of agents and allocation of resources is the model of risk
portfolio choice. The simplest static version of this problem, analyzed
fairly extensively in the literature, is one where a risk-averse agent decides on how to allocate his total wealth between investment in an asset
with stochastic return (the risky asset) and an asset with deterministic
return (the safe asset), so as to maximize the expected utility of return.
If the meein return on the risky asset is less than that on the safe asset,
'lie agent concentrates all his investment in the safe asset. On the other
hand, if the mean return on the risky asset is greater than the safe return,
the agent invests a positive fraction of his wealth in the risky asset.
Given the asset-return structure, the amount of investment in the
risky asset depends on the degree of risk aversion of the agent embodied in the utility function. The Arrow-Pratt measures of absolute and
relative risk aversion {see Pratt, 1964; Arrow, 1970) provide a precise
140
S. Roy and R. Wagenvoort
characterization ofthe risk-taking behavior of agents in such situations.
These measures depend on the curvature and slope of the utility function. In particular, if the von Neumann-Morgenstem utility function
exhibits decreasing (increasing) absolute risk aversion, then the optimal investment in the risky asset is increasing (decreasing) in the level
of initial wealth.
Consider a simple two-period model of successive risk taking. In
each period, the agent chooses a portfolio allocation of his current
wealth between a risky and a risk-less asset. The wealth retum from
the first period's portfolio choice determines the current wealth in the
second period. At the end of period two, total wealth is consumed. For
the portfolio decision in the first period, the risk preference as embodied
in the (direct) utility function of the agent is not very relevant. Rather,
it is the risk-aversion properties of tbe indirect utility, generated by the
problem of expected-utility maximization through risk portfolio choice
in the second period, which determines the qualitative properties of
optimal risk choice in the first period. The purpose of this paper is
to determine whether the risk-aversion properties of the (direct) utility
function are inherited by the indirect utility function.
While it is important to study the issue of such "inheritance" in relation to several types of risk-aversion properties, in this paper we focus
on one particular property namely, decreasing absolute risk-aversion
(DARA). In the existing literature, there appears to be a broad measure of support for DARA property as being the intuitively reasonable.
The prediction about risky investment being a nonnal good, which is a
consequence of the DARA property, has been generally confirmed by
quite a few empirical and experimental studies (see, e.g.. Levy, 1994).
though there are well-known exceptions.
In a multi-period model with time-separable additive utility, where
the agent in each period decides on the allocation of current wealth between consumption and investment in a single risky asset, Neave (1971)
showed that the value function (the indirect utility for multi-period decision problems) exhibited DARA if the one-period utility function wa.s
assumed to do so. In the Neave modei, the agent does not face any
problem of investment portfolio choice between alternative assets.
A multi-period model of consumption, investment, and portfolio
choice where the agent not only decides on allocation between consumption and investment but also on how to allocate the investment
between alternative assets with stochastic retum, is analyzed by Hakansson (1970). His analysis confines attention to the class of one-period
utility function which exhibit constant absolute or relative risk aversion (CARA/CRRA) and concludes that the indirect utility or value
function inherits the CARA/CRRA property. Though widely used, this
Risk Preference in Portfolio-choice Problems
141
class of utility functions yields optimal portfolio-allocation rules which
are wealth independent and, from that standpoint, not so interesting. It
may be tempting to inquire (especially, in the context of Neave's result
for models with single investment asset) whether Hakansson's result
also extends to the case where the one-period utility function exhibits
DARA. If this is true, then the optimal dynamic decision rule can be
shown to be one where investment in the risky asset in each period
increases with current wealth.' While our analysis does not explicitly
incorporate the possibility of consumption over fime, it can be seen as
a step towards answering this question.
Our results indicate that the DARA property is not necessarily carried over from the (direct) utility function to the indirect utility generated by a one-period portfolio-choice problem. In Sect. 4, we provide
an example to demonstrate this point. In the context of multi-period decision making this indicates that even if the (one-period) utility function
of an agent satisfies the DARA property, the agent's optimal portfolio
policy (in ail but the last period) can be such that he invests less in
the risky asset when his total wealth increases. The latter behavior is
usually associated with increasing absolute risk aversion.
In Sect. 3, we show that if the optimal portfolio choice is such that
the marginal propensity to invest is positive for both assets (that is, both
risky and risk-less investments are normal goods) then the DARA property is carried over to the indirect utility function. A sufficient condition
for this to occur is that the utility function satisfies increasing relative
risk aversion (IRRA), in addifion to DARA. In the literature, there appears to be some support for the hypothesis of increasing relative risk
aversion.^
1 This would be true if the agent maximizes the expected discounted sum
of one-period utility over finite horizon with i.i.d. risky return over time. Basically, one can decompose the maximization problem in each period into two
stages; first, a portfolio-choice problem for any given level of total investment
so as to maximize the expected value (for the remaining time periods) of the
return from investment; next, take the indirect utility from this static problem
and decide on the consumption-investment trade-off. Suppose that the indirect
utility from portfolio choice inherits the DARA property of the value function
for the remaining time periods, then by following the recursive techniques
from Neave (1971), one can show that the value function for all finite horizon
problems satisfies DARA and, thus, the optimal policy is one where investment
•n the risky asset is always increasing in current wealth. This could also be
extended to infinite horizons. Our negative result in Sect. 4 shows that this
approach does not work.
2 Arrow (1970) remarks that the hypothesis of IRRA gives a wealth elas-
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S. Roy and R. Wagenvoort
In the following Sect. 2, we outline the problem formally and state
the preliminary results.
2 The One-period Portfolio-choice Problem
To begin, the agent's preferences on the space of lotteries over wealth
are assumed to satisfy the expected utility property. Let L denote the
real line augmented by the point )—oo) and let H: R+ - * L be the
(von Neumann-Morgenstem) utility function of the agent.^ We make
the following assumptions on u;
(U.I) u is continuous on R+ and thrice continuously differentiable on
(U.2) u'iy) > 0 for all y > 0;
(U.3) u"iy) < O f o r a i r y > 0;
(U.4)
Assumptions (U.I) through (U.3) are fairly standard. Assumption (U.4)
is required to ensure an interior solution to the portfolio-choice problem.
There are two assets, one risky and the other safe. Both assets
mature in one period. The risky asset yields stochastic return p and the
safe asset has return r. The following assumptions are imposed on the
return structure of the assets:
(T.l) The support of p is a finite set contained in R+ and Prob|p = 0 }
>0;
(T.2) M = E(p) > r > 0.
Let w > 0 be the level of initial wealth and q denote the amount of
wealth invested in the risky asset. We shall assume that 0 < ^ < w,
that is, only a nonnegative amount can be invested and total investment
cannot exceed total wealth; short sales and borrowing are not allowed.
ticity of demand for cash balances of at least one, which is supported by
empirical evidence.
3 The utility function is allowed to take values in the extended real lme
in order to potentially allow for utility functions for which M(0) = -OO, e.g,
uiy) = lny, u(y) = -y~^, P > 0, etc. For such functions, the assumption
of right-hand continuity at zero, implied by (U.I), is to be interpreted as
I i ( ) = «(0) = -00.
Risk Preference in Portfolio-choice Problems
143
Assumption (T.2) implies that the risky asset is not mean-variance
dominated. Assumption (T. i) is a simplifying assumption whose only
role is to ensure, along with (U.4), that a positive quantity of wealth is
always invested in the safe asset. This allows us to write the first-order
necessary conditions as equalities and use the method of comparative
statics.
Let V(w) denote the indirect utility function. Then Viw) and the
maximization problem faced by the agent are given by:
Viw) = max Eluipq-i-riw
- q))\ .
(1)
The existence of an optimal solution is easily verified, while strict
concavity of the utility function implies uniqueness of the optimal solution. Assumption (T.2) ensures the optimal investment in the risky
asset is strictly positive, while Assumptions (T.I) and (U.4) ensure that
optimal investmetit in the risk-less asset is also strictly positive (see
also Arrow, 1970). Thus we have:
Lemma 1: There exists a unique interior optimal solution qiw) to the
portfolio-choice problem (1).
In the rest of the paper, we shall use the notation qiw) to denote the
opfimal solution to (1). Using Lemma 1 and Assumptions (U.1)-(U.3)
and successively applying the implicit function theorem (to the first
condition for an interior maximum), one can show that
Lemma 2: Viw) is thrice continuously differentiable on R++ and qiw)
is at least twice continuously differentiable on M++.
The next result characterizes the slope and curvature of V and, among
other things, states that for a risk-averse agent, the indirect utility function will also exhibit risk aversion.
Lemma 3: V is strictly increasing on R+; V iw) is strictly concave on
K++; V'iw) > 0 on E+ and V"iw) < 0 on K++.
Proof: Using (U.2), it is easy to check that V(u)) is strictly increasing
on K.,^. Using the envelope theorem, it can be verified that for u) > 0,
y'iw) — E[u'ipqiw) -h r{w — qivu))r] which is strictly positive as
»'(>>) > 0 on K++. Strict concavity of V(u!) on iR++ can be shown as
144
S. Roy and R. Wagenvoort
follows: Let 9, := qiwi), i = 1,2, W] ^ W2, and 0 < a < 1. Using
(U.3) we have
-I- (i - «)u'2) > E[u{piaq\ -^(1
-f (1 - a)w2 - aqj - (1 - a)q2))j
pq -I- r(W] - qi))
-f (1 - a)ipq2 + riw2 - ?2)))J
E|_au(p^i -\-riwi - q\))
-I- (1 - a)u{pq2 4-
il-a)Viw2).
•
We now proceed to state a set of definitions. The Arrow-Pratt measure of absolute risk aversion Ru is defined by /?„(>') = -u"iy)/u'iy)
and the Arrow-Pratt measure of relative risk aversion r^ is defined by
ruiy) = -yu"iy)/u'iy). A function / : E ^ M is said to be increasing
(decreasitig) if x > y implies fix) > (<) /(y); / is said to be strictly
increasing (strictly decreasing) if x > v implies fix) > (<) /(>). « is
said to exhibit decreasing or increasing absolute risk aversion (DARA
or IARA) if Ruiy) is, respectively, decreasing or increasing in y. Similarly, u exhibits decreasing or increasing relative risk aversion (DRRA
or IRRA) if r^iy) is, respectively, decreasing or increasing in y. If,
in particular, ruiy) is constant in y, then u is said to exhibit constant
relative risk aversion (CRRA).
Lastly, from Arrow (1970) and Pratt (1964), we have the following
implications of DARA and IRRA (as defined above):
Lemma 4: If u exhibits DARA on IR+, then qiw) is increasing on IR+
and q'iw) > 0 on R++. If « exhibits IRRA on R+, then qiw)/w is
decreasing on M.+ and q'iw) < 1 on R++.
3 Positive Results on Inheritance of DARA Property
by Indirect Utility
In this section we investigate the conditions under which the indirect
utility funcfion inherits the DARA property of the utility function. As
stated in Lemma 4, the DARA property of u implies that investment in
the risky asset increases with initial wealth. Proposition 1 shows that if,
in addition, the optimal investment rule is such that investment in the
Risk Preference in Portfolio-choice Problems
145
risk-less asset is also increasing in initial wealth, that is, q'iw) < I, then
the indirect utility function exhibits DARA. The latter restriction is by
no means necessary for V to exhibit DARA; it is, however, a mathematical requirement iti our approach needed in order to apply the CauchySchwarz inequality. It ensures that even when the risky asset yields its
lowest possible rate of retum, the final wealth resulting from the optimal portfolio choice increases whenever there is an increase in initial
wealth. In Sect. 4, we demonstrate an example where u is such that the
marginal propensity to invest in the risky asset is greater than one and
the indirect utility does not inherit the DARA property satisfied by M.
Proposition 1: If uiy) exhibits DARA on R++ and q'iw) < 1, that is,
optimal investment in the risk-less asset is increasing in initial wealth,
then V(w;) exhibits DARA on R++.
Proof: We shall show that Rviw) = —V'iw)/V'iw)
is decreasing
in w on M++. Note that from Lemma 3, Rviw) is well-defined. The
tirst-order necessary condition for an interior maximum in optimization
problem (1) can be written as:
Yi.[u{pqiw) + riw - qiw))){p
- r)J = 0 .
(2)
Using Lemma 2 and differentiating (2) with respect to it' leads to:
E[u"{pqiw) + riw - qiw))){p - r){r + (p - r)q'iw))\ = 0 . (3)
Differentiating (3) with respect to w:
E\u'"{pqiw) + riw - qiw))){p - r){r + ip ~ r)q'iw)f\
+ E[u"{pqiw) + riw - qiw))){p - r)'\q"iw) - 0 .
Now, Viw) = E[u{pqiw) + riw - qiw)))\ so
V'(iL') = E[u'{pqiw) + riw - qiw)))[r + (p - r)q'iw))\
.
(5)
Using (2) in (5) we have
V'(u;) = E\u'[pqiw) -t- riw - qiw)))r\
,
(6)
so that
y'ixti) = E[u"{pqiw) -t- riw - qiw))){r -Yip- r)q'iw))r\
(7)
146
S. Roy and R. Wagenvoort
and
y"'(«)) = E[u"'{pqiw) + riw - q{w))){r + ip -
r)q'iw)fr\
+ E[u"{pqiw) -j-riw - qiw))){p -r)rq"iw)\
.
(o)
Substituting for q"{w) from (4) in (8):
"'{w) = E[u"'{pqiw) + riw - qiw))){r + ip + hiw)Elu'"{pqiw) + riw - qiw))) x
r)q'iw)fr\
(9)
X {p-r)(r + ip-r)
where
E[u"l.pq+r{w~q)){p-r)-\
From (3) we have hiw) = q'iw). So, V"'iw) can be written as:
V"'{w) = E[u"'{pqiw) + r{w - qiw))){r + (p - r)q'iw)f \ . (10)
Further, V"iw) can be written as:
V'iw) = E[u"{pqiw) + riw - qiw))){r + (p - r)q'iw))r\
+ E[u"{pq{w) + riw - qiw))) x
X (p - r)q'iw){r + (p - r)q'{w))\
[using (3)]
= E\u"{pqiw) + riw - qiw))){r + (p - r)q'iw)f \ .
Using (5), (10), the Cauchy-Schwarz inequality, 0 < q'iw) < 1 [so
that r -\- {p — r)q'iw) assumes only nonnegative values] and the fact
that Ruiy) is decreasing [or equivalently u"'iy)u'iy) > (H"(V))^], we
have:
V"'iw)V'iw) > E[{u"'{pqiw) + riw - qiw))) x
X u'{pqiw) + riw - g(w))})'^^ x
> E[u"{pqiw) + riw -qiw))){r + (p - r)q'iw)f f
= [V"iw)f
[using (11)].
Thus, V"'{w)V'iw) > [V"iw)f, that is, the measure of absolute risk
Risk Preference in Portfolio-choice Problems
147
Table 1. Examples of utility functions which exhibit decreasing absolute
(DARA) and increasing (IRRA) or constant relative risk aversion (CRRA)
1
= v'
0<c < 1
M(V)
2
3
4
r >0
«(y)=log(y)
DARA
CRRA
DARA
CRRA
DARA
CRRA
DARA
IRRA
DARA
IRRA
DARA
IRRA
fl > 0, 0 < c < 1
5
6
a > 0, c > 0
u(y) = log(y -J- a)
a >0
aversion /?v(w) associated with the indirect utility function V, is decreasing on R++.
D
As noted in Lemma 4, if the utility function u satisfies IRRA, then
an increase in wealth leads to an increase in the fraction of wealth
invested in the risk-less asset which, in particular, implies that the
absolute amount invested in the risk-less asset increases with current
wealth, that is, q'iw) S L This gives us the following useful corollary
to Proposition 1.
Corollary 1: If uiy) exhibits DARA and IRRA on R++, then
exhibits DARA on M4-+.
Viw)
There is a fairly large class of widely used utility functions which
satisfy both DARA and IRRA (which according to our definition include
CRRA utility functions). Table 1 lists such functions.
4 An Example to Show that the DARA Property
Need not Be Inherited by the Indirect Utility Function
Suppose the utility function is given by uiy) = —\y~^ + y. It is easy
to check that u exhibits DARA on M+.^-. However, contrarily to the
assumption of Corollary 1, u does not exhibit IRRA. We choose the following parameter values: r = 0.1, Prob[p = 0] = 0.04, Prob[p = 0.01]
= 0.86, and Prob[(0 = 1 ] = 0.1. One can check that Assumptions (U.I)-
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S. Roy and R. Wagenvoort
(U.4) and (T.I) and (T.2) hold. The first-order condition of the maximization problem (1) can be written as:
(12)
Suppose w = 100. The derivative of the maximand with respect to q
is given by the expression on the left-hand side of (12). If we evaluate
this expression ai q = 86.36 we can see that this derivative is strictly
positive; on the other hand, at ^ = 86.38, it is strictly negative. From
strict concavity of the maximand, it follows that the optimal choice of
g at w = 100, denoted by ^(100), satisfies
86.36 < ^(100) < 86.38 .
Differentiafing (12) with respect to w and solving for q'iw), we obtain:
q (W)
12
,
0.02322
12
_|
0.020898
0.027
_^
0.243
(13)
SetUng w = 100 in (13), and using the bounds on ^(100) stated above,
we can derive the following upper and lower bounds on ^'(100):
1.0740218 < ^'(100) < 1.0837062 .
Note that this implies that the hypothesis of Proposition 1 is violated as g'(lOO) > I. Consider the expression for V'"iw) in (10).
Using the bounds on ^(100) and ^'(100), it can be shown that at
w = 100, the right-hand side of (10) is negative and that, in fact,
V"'iw) < -3.422036 x 10^^ As stated in Lemma 3, V'iw) > 0 on
R++. Therefore, V"'iw)V'iw) < [V"(u))]2 at lu = 100. The ArrowPratt measure of absolute risk aversion Ryiw) associated with the indirect ufility Vim) is, therefore, strictly increasing on a subset of R++.
Therefore, V violates DARA property.
Risk Preference in Portfolio-choice Problems
149
5 Conclusion
We conclude that in multi-period portfolio-choice problems, the nature
of optimal portfolio choice is not necessarily determined by the riskaversion measures associated with the one-period utility function. In
particular, if the one-period utility function satisfies DARA, this property may not be carried over to the indirect-utility or value function so
that the optimal portfolio policy in early periods can exhibit features
only associated with increasing absolute risk aversion in static models.
In multi-period consumption, investment, and portfolio-choice models
similar issues arise, so in general one cannot expect that the value
function inherits all the risk-preference features of the utility function.
However, for simple two-period models, if the one-period utility function exhibits DARA and, in addition, investment in the risk-less asset
is a normal good, then the indirect utility function satisfies the DARA
property and optimal investment in the risky asset is increasing in the
level of current wealth for both time periods. A sufficient condition for
this to occur is that u satisfies both DARA and IRRA.
From the standpoint of dynamic portfolio choice, our result naturally leads to the question whether assuming that u satisfies the conditions in the hypothesis of Proposition 1 or Corollary 1 implies that V
also satisfies all these conditions (and not just DARA). For example, if
u satisfies both DARA and IRRA, does V also satisfy both DARA and
IRRA. If this were true, then using the recursive methods of dynamic
programming, one could extend the DARA property to value functions
of multi-period portfolio-choice problems of more than two periods.
We have not been able to show this and leave it as an open question
for future research.
A cknowledgemen ts
Helpful comments and suggestions from two anonymous referees are gratefully
acknowledged.
References
Arrow, K. J. (1970): "Aspects of the Theor>' of Risk-Bearing." In Essays in
the Theory of Risk-Bearing. Amsterdam: North-Holland.
Hakansson, N.H. (1970); "Optimal Investment and Consumption Strategies
under Risk for a Class of Utility Functions." Econometrica 38: 587-607.
H. (1994): "Absolute and Relative Risk Aversion: an Experimental
Study." Joumal of Risk and Uncertainty 8: 289-307.
150
Roy and Wagenvoort: Portfolio-choice Problems
Neave, E. H. (1971): "Multiperiod Consumption-Investment Decisions and
Risk Preference." Joumal of Economic Theory 3: 40-53.
Pratt, J. W. (1964): "Risk Aversion in the Small and in the Large." Econometrica 32: 122-136.
Addresses of authors: Santanu Roy, Department of Economics, H9-20,
Erasmus University, 3000 DR Rotterdam, The Netherlands; - Rien Wagenvoort, European University Institute, Badia Fiesolana, 1-50016 San Domenico
di Fiesole (F!), Italy.