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Portfolio Selection Under Exponential and Quadratic Utility

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Portfolio Selection Under Exponential and Quadratic Utility Portfolio Selection Under Exponential and Quadratic Utility Steven T. Buccola The production or marketing portfolio that is optimal under the assumption of quadratic utility may or may not be optimal under the assumption of exponential utility. In certain cases, the necessary and sufficient condition for an identical solution is that absolute risk aversion coefficients associated with the two utility functions be the same. In other cases, equality of risk aversion coefficients is a sufficient condition only. A comparison is made between use of exponential and quadratic utility in the analysis of a California farmer's marketing problem. A large number of utility functional forms difference. Under other conditions, the have been proposed for use in selecting opti- choice has a significant impact on optimal mal portfolios of risky prospects [Tsaing]. For behavior. purposes of both theoretical and applied eco- Suppose an analyst is studying a continu- nomic research, attention has primarily been ously divisible set of alternative production directed to the quadratic and exponential or marketing options offering approximately forms. After enjoying widespread use in the normally-distributed returns, and that he 1950s and 1960s, the quadratic form came wishes to select the portfolio of these options under criticism by Pratt and others and lost with highest expected utility. The analyst is much of its reputability. In contrast the expo- considering using an exponential utility func- nential form has become increasingly popu- tion U = -exp(-Xx), X > 0, or a quadratic lar, due in large measure to its mathematical utility function U = x -vx,v, > 0, x < /2v, tractability [Attanasi and Karlinger; O'Con- where U is utility and x is wealth. Corre- nor]. However, Anderson, Dillon, and Har- sponding to each of these functions is a daker (pp. 94-95) have recently defended the unique absolute risk aversion function r(x) = quadratic, and it continues in use [Lin, - U"(x)/U'(x) from which a risk aversion coef- Dean, and Moore; Hanoch and Levy; Kall- ficient can be determined at any level of x. berg and Ziemba]. The objective of the pre- For exponential utility, risk aversion r is in sent paper is to explore conditions under fact invariant with respect to x, whereas for which use of quadratic utility results in opti- quadratic utility it rises monotonically with x mal choices identical to, or different from, [Pratt, p. 132]. Thus at one, and only one, those resulting from use of exponential utili- wealth level the two risk aversion functions ty. The analysis makes clear that under cer- will be equated. The wealth level at which a tain conditions, the choice between quadrat- risk aversion coefficient is properly evaluated ic and exponential utility makes very little is the decision maker's initial wealth plus his expected return from the portfolio con- sidered; hence the portfolio considered gen- Steven Buccola is Assistant Professor in the Department erally affects the risk aversion coefficient it- of Agricultural and Resource Economics at Oregon State self. University. The helpful remarks of Ben French, Bruce McCarl, John Pratt, and three anonymous reviewers are The thesis of the present paper may be gratefully acknowledged. Oregon State Agricultural Ex- stated as follows: if the optimal portfolios periment Station Technical Paper No. 5724. include each production or marketing option 43 July 1982 Western Journal of Agricultural Economics at a nonzero level and if the decision maker is stituting the above mean and variance into assumed to operate under at most one linear the latter formula gives resource constraint (such as an acreage or 2 capital constraint), then necessary and suffi- (1) CEe = x + E Pii - X(. Pi2'i + i 1 cient conditions for selecting the same port- E PiPjaij)/2. folio using either quadratic or exponential j:i utility is that the absolute risk aversion coeffi- The optimal quantity Pi*e allocated by the cients be equal at the optimal point. If some decision maker to the ith prospect is found by options are optimally operated at zero level satisfying the Kuhn-Tucker conditions in or if more than one resource constraint is which resource constraints are not specified, employed, then equality of risk aversion co- that is by satisfying aCEe/aPi < 0, efficients is sufficient but not necessary for (aCEe/aPi)Pi = 0, Pi > 0 (Intrilligator, p. 50). selecting the same portfolio. An implication It is assumed at the outset that Pi > 0, so it is of this argument is that in many cases the only required that aCEe/aPi = 0, all i. That composition of optimal portfolios depends is, upon the utility functional form selected. Only in special circumstances can identical 2 dCEe/aPi = - X(Pioi + 2 Pijoij) = 0. behavior in the face of risk be used to infer i identical risk aversion, although the reverse Solving for the optimal Pi*e gives2 inference can more frequently be made. (Ai/) - Pjaij Portfolio Problem Under (2) Pi:e = 2 Unlimited Resources Oi2 The situation first considered here is that Expected utility reaches a maximum at this in which an individual contemplates n alter- point because the matrix of second deriva- native risky prospects Ri, i = 1, 2, ... , n, tives I| a2 CEe/aPiPj ||, which equals 2 where Ri ~ N([Li, ai ). The individual has 1I- oUij 1 for all i, j, has a negative semi- access to an unlimited nonnegative number definite form. Pi of units of each prospect. Under these Corresponding first order conditions for conditions, portfolio return is z = E PiRi. If x the quadratic utility function are derived by is the individual's pre-risk wealth, total maximizing the expected utility function EUq wealth after realization of the random return = jW - v(Jw 2 - vocw 2 with respect to port- is w = x + z. Hence U and r may be folio quantity Pi. If Pi is positive, in the evaluated in terms of expected post-risk or optimum it is required that terminal wealth Rjw. Terminal wealth has mean w = x + E PiLi and variance 2 =2 2 aw OEUq/aPi = Ri(l-2vx)- 2v(Pipi + Li .j i I Pi2ai 2 + E Z PiPjo'ij. Pj j + Pioi 2 + E Pjrij) = 0, j~-i j~i Optimal Portfolio Selection If the decision maker has an exponential 2 utility function with parameter X and returns Conditions (2) would be solved simultaneously for all Pi, i = 1, 2, ... , n. It is not necessarily true in (2) that are normally distributed, Freund has shown 2 aPi*e/aX = -pi/ < 0 because quantity Pjzi in the that the certainty equivalent of expected ter- numerator of Pi*e is, at the optimum, also a negative 2 minal wealth is CEe = (w- (X/2)r .'1 Sub- function of X. Hence in the simultaneous solution, aPi*e/aX\ 0 for any particular risky prospect Ri. For the 1 All e subscripts refer to the exponential utility function two-variable case, it may be shown that increases in risk and q subscripts to the quadratic utility function. Sub- aversion coefficient \ decrease the proportion optimally scripts w refer to terminal wealth. allocated to the prospect with higher mean. 44 Buccola Exponential and Quadratic Utility so that optimal quantity Pi q is (7) rq* = 1/[ (/2v) - (x + z*) ]. pLi[('/2V) - X] -(1LiI Pg.LR+ I Pi aj) j :Aji j :Ai Substituting from %L*in (6) and from Pi*q in (3) Pi*q 2 (Ri + o.i2) (3): Second order conditions for a maximum are (7)' r* = 1 again satisfied because the matrix of second (W/2V) - (x + Lo,)- 2 derivatives IIa EUq/aPiaPj I| is equivalent to R[ 2'[(/2V)- X]- j(i(oi+ o + io)] II- 2v(LiJj + oij) 11, a negative semi-definite 2 2 ) form. (7i + fi Conditions for Identical Multiplying numerator and denominator of 2 Optimal Portfolios (7)' by (xi 2 + ri ) gives To derive the necessary conditions under 2 (p.i2 + (ai ) which the composition of the optimal port- rq = 2 2 folio would not depend upon utility function- (1i2 + C(T )/2v-(x + pJ,,)([i2 + (T) 2 al form, optimal quantity Pi*e in (2) is equated - Vi [(1/2V)- X] + Li(piJo + (7io) with optimal quantity Pi*q in (3). To simplify notation, let orio = j Pjcij and [o = E j i jLi: Cancelling and collecting terms in the de- PjLj, where o refers to alternatives other than nominator, the ith. Then 2 (fi,2 + oi ) ( + (4) (i/x) - cri. .Li[(l/2V)- X] - (LiL, + oJio) (7)" r* 2 2 2 q [(,/2V) - X] Oi - (i2, - io cri2 (p1 + Ci ) ) Solving for the exponential utility's constant which is identical to the risk aversion coeffi- risk aversion coefficient gives cient (5) for the exponential utility function. Thus, if resources are unlimited and each portfolio option is optimally held at a nonzero (T 2 (5) =+ ) level, a necessary condition for the optimal [('/2V) - X](Ti - (0ai2o,- i(io) quantities Pi*, i = 1, 2, . ., n, to be the same regardless of functional form is that the It now remains to be shown that the quadrat- absolute risk aversion coefficients be the ic utility's risk aversion coefficient rq equals A same at expected in (5). terminal wealth. To prove that equality of risk aversion coefficients is To show that this is so, recall from the risk sufficient for agreement on Pi*, it is only aversion coefficient formula r = -U"(lLw)/ required that X be set equal to (7) or equiva- U'(Lw) that rq = 2v/(1 - 2vw) = 1/[ (2v) - lently to (7)"; this is the same as equation (5), (x + piz) ]. Coefficient rq varies with Rz and which is a rearranged form of (4) where the thus with the particular portfolio selected.
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