Risk Preference and Indirect Utility in Portfolio-choice...

Vol. 63 (1996), No. 2, pp. 139-150 Journal of EconomicsZeitschrift fur Nationatokonomie© Spiinjer-Verlag 1996 - Printed in Austiia

Risk Preference and Indirect Utilityin 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 neednot exhibit DARA. However, if the (optimal) marginal propensity to invest ispositive for both assets, which is true when the utility function exhibits non-decreasing relative risk aversion, then the DARA property is carried over fromthe direct to the indirect utility function.

Keywords: portfolio choice, absolute risk aversion, relative risk aversion, in-direct utility.

JEL classification: D81, D92.

1 Introduction

One of the basic models used to study the relationship between riskpreference of agents and allocation of resources is the model of riskportfolio choice. The simplest static version of this problem, analyzedfairly extensively in the literature, is one where a risk-averse agent de-cides on how to allocate his total wealth between investment in an assetwith stochastic return (the risky asset) and an asset with deterministicreturn (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 otherhand, 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 therisky asset depends on the degree of risk aversion of the agent embod-ied in the utility function. The Arrow-Pratt measures of absolute andrelative 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 func-tion. In particular, if the von Neumann-Morgenstem utility functionexhibits decreasing (increasing) absolute risk aversion, then the opti-mal investment in the risky asset is increasing (decreasing) in the levelof initial wealth.

Consider a simple two-period model of successive risk taking. Ineach period, the agent chooses a portfolio allocation of his currentwealth between a risky and a risk-less asset. The wealth retum fromthe first period's portfolio choice determines the current wealth in thesecond period. At the end of period two, total wealth is consumed. Forthe portfolio decision in the first period, the risk preference as embodiedin the (direct) utility function of the agent is not very relevant. Rather,it is the risk-aversion properties of tbe indirect utility, generated by theproblem of expected-utility maximization through risk portfolio choicein the second period, which determines the qualitative properties ofoptimal risk choice in the first period. The purpose of this paper isto determine whether the risk-aversion properties of the (direct) utilityfunction are inherited by the indirect utility function.

While it is important to study the issue of such "inheritance" in rela-tion to several types of risk-aversion properties, in this paper we focuson one particular property namely, decreasing absolute risk-aversion(DARA). In the existing literature, there appears to be a broad mea-sure of support for DARA property as being the intuitively reasonable.The prediction about risky investment being a nonnal good, which is aconsequence of the DARA property, has been generally confirmed byquite 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, wherethe agent in each period decides on the allocation of current wealth be-tween consumption and investment in a single risky asset, Neave (1971)showed that the value function (the indirect utility for multi-period de-cision problems) exhibited DARA if the one-period utility function wa.sassumed to do so. In the Neave modei, the agent does not face anyproblem of investment portfolio choice between alternative assets.

A multi-period model of consumption, investment, and portfoliochoice where the agent not only decides on allocation between con-sumption and investment but also on how to allocate the investmentbetween alternative assets with stochastic retum, is analyzed by Hakans-son (1970). His analysis confines attention to the class of one-periodutility function which exhibit constant absolute or relative risk aver-sion (CARA/CRRA) and concludes that the indirect utility or valuefunction 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 whichare wealth independent and, from that standpoint, not so interesting. Itmay be tempting to inquire (especially, in the context of Neave's resultfor models with single investment asset) whether Hakansson's resultalso extends to the case where the one-period utility function exhibitsDARA. If this is true, then the optimal dynamic decision rule can beshown to be one where investment in the risky asset in each periodincreases with current wealth.' While our analysis does not explicitlyincorporate the possibility of consumption over fime, it can be seen asa step towards answering this question.

Our results indicate that the DARA property is not necessarily car-ried over from the (direct) utility function to the indirect utility gener-ated by a one-period portfolio-choice problem. In Sect. 4, we providean example to demonstrate this point. In the context of multi-period de-cision making this indicates that even if the (one-period) utility functionof an agent satisfies the DARA property, the agent's optimal portfoliopolicy (in ail but the last period) can be such that he invests less inthe risky asset when his total wealth increases. The latter behavior isusually associated with increasing absolute risk aversion.

In Sect. 3, we show that if the optimal portfolio choice is such thatthe marginal propensity to invest is positive for both assets (that is, bothrisky and risk-less investments are normal goods) then the DARA prop-erty is carried over to the indirect utility function. A sufficient conditionfor this to occur is that the utility function satisfies increasing relativerisk aversion (IRRA), in addifion to DARA. In the literature, there ap-pears to be some support for the hypothesis of increasing relative riskaversion.^

1 This would be true if the agent maximizes the expected discounted sumof one-period utility over finite horizon with i.i.d. risky return over time. Ba-sically, one can decompose the maximization problem in each period into twostages; first, a portfolio-choice problem for any given level of total investmentso as to maximize the expected value (for the remaining time periods) of thereturn from investment; next, take the indirect utility from this static problemand decide on the consumption-investment trade-off. Suppose that the indirectutility from portfolio choice inherits the DARA property of the value functionfor the remaining time periods, then by following the recursive techniquesfrom Neave (1971), one can show that the value function for all finite horizonproblems satisfies DARA and, thus, the optimal policy is one where investment•n the risky asset is always increasing in current wealth. This could also beextended to infinite horizons. Our negative result in Sect. 4 shows that thisapproach does not work.

2 Arrow (1970) remarks that the hypothesis of IRRA gives a wealth elas-


In the following Sect. 2, we outline the problem formally and statethe preliminary results.

2 The One-period Portfolio-choice Problem

To begin, the agent's preferences on the space of lotteries over wealthare assumed to satisfy the expected utility property. Let L denote thereal line augmented by the point )—oo) and let H: R+ -* L be the(von Neumann-Morgenstem) utility function of the agent.^ We makethe 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 assetsmature in one period. The risky asset yields stochastic return p and thesafe asset has return r. The following assumptions are imposed on thereturn 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 ofwealth invested in the risky asset. We shall assume that 0 < ^ < w,that is, only a nonnegative amount can be invested and total investmentcannot exceed total wealth; short sales and borrowing are not allowed.

ticity of demand for cash balances of at least one, which is supported byempirical evidence.

3 The utility function is allowed to take values in the extended real lmein 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 assumptionof right-hand continuity at zero, implied by (U.I), is to be interpreted asI i ( ) = «(0) = -00.


Assumption (T.2) implies that the risky asset is not mean-variancedominated. Assumption (T. i) is a simplifying assumption whose onlyrole is to ensure, along with (U.4), that a positive quantity of wealth isalways invested in the safe asset. This allows us to write the first-ordernecessary conditions as equalities and use the method of comparativestatics.

Let V(w) denote the indirect utility function. Then Viw) and themaximization 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 strictconcavity of the utility function implies uniqueness of the optimal so-lution. Assumption (T.2) ensures the optimal investment in the riskyasset is strictly positive, while Assumptions (T.I) and (U.4) ensure thatoptimal investmetit in the risk-less asset is also strictly positive (seealso Arrow, 1970). Thus we have:

Lemma 1: There exists a unique interior optimal solution qiw) to theportfolio-choice problem (1).

In the rest of the paper, we shall use the notation qiw) to denote theopfimal solution to (1). Using Lemma 1 and Assumptions (U.1)-(U.3)and successively applying the implicit function theorem (to the firstcondition 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, amongother things, states that for a risk-averse agent, the indirect utility func-tion will also exhibit risk aversion.

Lemma 3: V is strictly increasing on R+; V iw) is strictly concave onK++; 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 increasingon 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


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 mea-sure of absolute risk aversion Ru is defined by /?„(>') = -u"iy)/u'iy)and the Arrow-Pratt measure of relative risk aversion r^ is defined byruiy) = -yu"iy)/u'iy). A function / : E ^ M is said to be increasing(decreasitig) if x > y implies fix) > (<) /(y); / is said to be strictlyincreasing (strictly decreasing) if x > v implies fix) > (<) /(>). « issaid to exhibit decreasing or increasing absolute risk aversion (DARAor IARA) if Ruiy) is, respectively, decreasing or increasing in y. Sim-ilarly, u exhibits decreasing or increasing relative risk aversion (DRRAor 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 constantrelative risk aversion (CRRA).

Lastly, from Arrow (1970) and Pratt (1964), we have the followingimplications 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 isdecreasing on M.+ and q'iw) < 1 on R++.

3 Positive Results on Inheritance of DARA Propertyby Indirect Utility

In this section we investigate the conditions under which the indirectutility funcfion inherits the DARA property of the utility function. Asstated in Lemma 4, the DARA property of u implies that investment inthe risky asset increases with initial wealth. Proposition 1 shows that if,in addition, the optimal investment rule is such that investment in the


risk-less asset is also increasing in initial wealth, that is, q'iw) < I, thenthe indirect utility function exhibits DARA. The latter restriction is byno means necessary for V to exhibit DARA; it is, however, a mathemat-ical requirement iti our approach needed in order to apply the Cauchy-Schwarz inequality. It ensures that even when the risky asset yields itslowest possible rate of retum, the final wealth resulting from the opti-mal portfolio choice increases whenever there is an increase in initialwealth. In Sect. 4, we demonstrate an example where u is such that themarginal propensity to invest in the risky asset is greater than one andthe 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 decreasingin w on M++. Note that from Lemma 3, Rviw) is well-defined. Thetirst-order necessary condition for an interior maximum in optimizationproblem (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)


and

y"'(«)) = E[u"'{pqiw) + riw - q{w))){r + ip - r)q'iw)fr\(o)

+ E[u"{pqiw) -j-riw - qiw))){p -r)rq"iw)\ .

Substituting for q"{w) from (4) in (8):

"'{w) = E[u"'{pqiw) + riw - qiw))){r + ip - r)q'iw)fr\

+ hiw)Elu'"{pqiw) + riw - qiw))) x (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 [sothat r -\- {p — r)q'iw) assumes only nonnegative values] and the factthat Ruiy) is decreasing [or equivalently u"'iy)u'iy) > (H"(V))^], wehave:

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


Table 1. Examples of utility functions which exhibit decreasing absolute(DARA) and increasing (IRRA) or constant relative risk aversion (CRRA)

1

2

34

5

6

M(V) = v'0 < c < 1

r > 0

«(y)=log(y)

fl > 0, 0 < c < 1

a > 0, c > 0

u(y) = log(y -J- a)a > 0

DARA

DARA

DARA

DARA

DARA

DARA

CRRA

CRRA

CRRA

IRRA

IRRA

IRRA

aversion /?v(w) associated with the indirect utility function V, is de-creasing on R++. D

As noted in Lemma 4, if the utility function u satisfies IRRA, thenan increase in wealth leads to an increase in the fraction of wealthinvested in the risk-less asset which, in particular, implies that theabsolute amount invested in the risk-less asset increases with currentwealth, that is, q'iw) S L This gives us the following useful corollaryto Proposition 1.

Corollary 1: If uiy) exhibits DARA and IRRA on R++, then Viw)exhibits DARA on M4-+.

There is a fairly large class of widely used utility functions whichsatisfy both DARA and IRRA (which according to our definition includeCRRA utility functions). Table 1 lists such functions.

4 An Example to Show that the DARA PropertyNeed not Be Inherited by the Indirect Utility Function

Suppose the utility function is given by uiy) = —\y~^ + y. It is easyto check that u exhibits DARA on M+.̂ -. However, contrarily to theassumption of Corollary 1, u does not exhibit IRRA. We choose the fol-lowing 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)-


(U.4) and (T.I) and (T.2) hold. The first-order condition of the maxi-mization problem (1) can be written as:

(12)

Suppose w = 100. The derivative of the maximand with respect to qis given by the expression on the left-hand side of (12). If we evaluatethis expression ai q = 86.36 we can see that this derivative is strictlypositive; on the other hand, at ^ = 86.38, it is strictly negative. Fromstrict concavity of the maximand, it follows that the optimal choice ofg 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:

12 , 0.02322 0.027

q (W) 12 _| 0.020898 _̂ 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 vio-lated as g'(lOO) > I. Consider the expression for V'"iw) in (10).Using the bounds on ^(100) and ^'(100), it can be shown that atw = 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 onR++. Therefore, V"'iw)V'iw) < [V"(u))]2 at lu = 100. The Arrow-Pratt measure of absolute risk aversion Ryiw) associated with the in-direct ufility Vim) is, therefore, strictly increasing on a subset of R++.Therefore, V violates DARA property.


5 Conclusion

We conclude that in multi-period portfolio-choice problems, the natureof optimal portfolio choice is not necessarily determined by the risk-aversion measures associated with the one-period utility function. Inparticular, if the one-period utility function satisfies DARA, this prop-erty may not be carried over to the indirect-utility or value function sothat the optimal portfolio policy in early periods can exhibit featuresonly associated with increasing absolute risk aversion in static models.In multi-period consumption, investment, and portfolio-choice modelssimilar issues arise, so in general one cannot expect that the valuefunction inherits all the risk-preference features of the utility function.However, for simple two-period models, if the one-period utility func-tion exhibits DARA and, in addition, investment in the risk-less assetis a normal good, then the indirect utility function satisfies the DARAproperty and optimal investment in the risky asset is increasing in thelevel of current wealth for both time periods. A sufficient condition forthis to occur is that u satisfies both DARA and IRRA.

From the standpoint of dynamic portfolio choice, our result natu-rally leads to the question whether assuming that u satisfies the condi-tions in the hypothesis of Proposition 1 or Corollary 1 implies that Valso satisfies all these conditions (and not just DARA). For example, ifu satisfies both DARA and IRRA, does V also satisfy both DARA andIRRA. If this were true, then using the recursive methods of dynamicprogramming, one could extend the DARA property to value functionsof multi-period portfolio-choice problems of more than two periods.We have not been able to show this and leave it as an open questionfor future research.

A cknowledgemen ts

Helpful comments and suggestions from two anonymous referees are gratefullyacknowledged.

References

Arrow, K. J. (1970): "Aspects of the Theor>' of Risk-Bearing." In Essays inthe Theory of Risk-Bearing. Amsterdam: North-Holland.

Hakansson, N.H. (1970); "Optimal Investment and Consumption Strategiesunder Risk for a Class of Utility Functions." Econometrica 38: 587-607.

H. (1994): "Absolute and Relative Risk Aversion: an ExperimentalStudy." Joumal of Risk and Uncertainty 8: 289-307.

150 Roy and Wagenvoort: Portfolio-choice Problems

Neave, E. H. (1971): "Multiperiod Consumption-Investment Decisions andRisk Preference." Joumal of Economic Theory 3: 40-53.

Pratt, J. W. (1964): "Risk Aversion in the Small and in the Large." Economet-rica 32: 122-136.

Addresses of authors: Santanu Roy, Department of Economics, H9-20,Erasmus University, 3000 DR Rotterdam, The Netherlands; - Rien Wagen-voort, European University Institute, Badia Fiesolana, 1-50016 San Domenicodi Fiesole (F!), Italy.

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