risk, return, and utility

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Risk, Return, and Utility Author(s): David E. Bell Source: Management Science, Vol. 41, No. 1 (Jan., 1995), pp. 23-30Published by: INFORMSStable URL: http://www.jstor.org/stable/2632897Accessed: 18-06-2015 14:55 UTC

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Risk, Return, and Utility

David E. Bell Harvard Business School, Boston, Massachusetts 02163

E xpected utility theory is widely acknowledged to be a rational approach to making decisions involving risk. Yet the methodology gives no explicit role to measures of risk and return.

In this paper we identify those families of utility functions that are compatible with a risk- return interpretation. From these families we deduce utility-compatible measures of risk. (Risk; Return; Utility; Investments)

Introduction In this paper we consider a person deciding among a set of financial alternatives with uncertain payoffs. Each alternative is summarized by a probability distribution over cashflows that are incremental to a known initial wealth. The results that follow may be adapted fairly easily to cases where financial payoffs are multiplicative rather than incremental, where initial wealth is itself uncertain, or where the outcomes are not financial.

We will denote uncertain alternatives generically by the notation x, y, z - *, and the base wealth by w. The decision maker selecting x would then have uncertain wealth w + x which, upon resolution, would lead to some particular wealth w + x. For the case in which an alternative x has two possible outcomes xi and x2, with xl occurring with probability p, we will write

= (Xl, p, X2)- Von Neumann and Morgenstern (1947) showed that

if a decision maker accepts a certain set of assumptions concerning "rational choice," then the decision maker should compare alternatives by use of an expected utility calculation. For some function u, which varies by decision maker, an alternative x should be preferred to an alternative g if and only if Eu(x) > Eu(y), and be indifferent between them if and only if Eu(x) = Eu(y)). The symbol E represents expectation over the underlying probability distribution. For example, (xl, p, x2) should be preferred to (yi, q, Y2) if and only if

pu(x1) + (1 - p)u(x2) > qu(yl) + (1 - q)U(y2). Because we wish to give an explicit role to the decision

maker's initial wealth, w, we will describe outcomes in

terms of final total wealth, so that x is to be preferred to g if and only if Eu(w + x) > Eu(w + y). The beauty of the expected utility approach lies in the elegance and compelling nature of von Neumann and Morgenstern's axioms. Though criticized and challenged over the last 50 years, their theory remains the benchmark economic approach to such problems.

Their theory gives no explicit role, however, to the concepts of risk and return. Informal discussion of alternatives by decision makers often includes statements such as "alternative A is more attractive than alternative B, but is too risky," suggesting that decisions are thought of, at an intuitive level at least, as a tradeoff between the risk inherent in the alternatives, and their levels of "return" (their attractiveness were it not for the "risk"). The notion of risk as a distinct entity has been examined extensively, although in ways at best loosely connected with von Neumann-Morgenstern utility; see, for example, Markowitz (1959) and, for an axiomatic treat- ment of risk as a primitive, Fishburn (1984, 1982).

We suggest that a more intuitively satisfying model than expected utility would recognize a measure of the return r(x) and risk R(x) of an alternative. The tradeoff between risk and return might well depend on the decision maker's wealth; we therefore propose an evaluation function of the form f (r(x), R(x), w), for some risk-return function f.

The purpose of this paper is to consider solutions to the equation Eu(w + x) = f(r(x), R(x), w). We will show that there is a surprisingly wide range of utility functions that have a risk-return interpretation. We give special attention to the nontrivial subset of solutions

0025-1909/95/4101/0023$01.25 Copyright ? 1995, Institute for Operations Research and the Management Sciences MANAGEMENT SCIENCE/VOl. 41, No. 1, January 1995 23


BELL Risk, Return, and Utility

satisfying desirable properties of u, r, R, and f. More motivation for this inquiry can be found in ?7 of this paper.

1. Illustrations (i) Suppose that a decision maker has a utility function u(x) = -ecx, where c is a constant. We make the role of initial wealth explicit by writing

u (w + x) =-e -c(w+x) = _e -cw e -cx

The evaluation of an alternative x may be written as:

Eu(w + x) = E(-exp(-cw) exp(-cx)) =-exp (-cw) E exp (-dcx) =-exp(-cw) exp(-cx) E exp(-c(x- x)).

Let

r(x) = x and R() =-log E exp(-c(x-x)).

Then

f(r, R, w) =-e-cw exp[-c(r - R)]. Thus the relation Eu(w + x) > Eu(w + y) may be

written equivalently, and suggestively, as r(x) - R(x) > r(y) - R(g). My choice of definitions of r and R in this example are not unique; however, they do result from arguments presented later in this paper.

(ii) Suppose that a decision maker has a quadratic- utility function

u(w + x) = a(w + X)2 + b(w + x) + c = aw2 + bw + c + 2awx + bx + ax2

= u(w) + (2aw + b)x + ax2. Let r(x, - x-and R(x) = E(x- X)2. Then f(r, R, w) = aw2 + bw + c + (2aw + b)r + ar2 + aR.

In this case, though the definitions of r and R seem plausible, a consequence of the quadratic utility function is that f is either increasing in R (if a > 0) or decreasing in r (a < 0, w large).

2. Background The axioms of utility impose no conditions on the shape of the utility function, but for financial applications it

is reasonable to suppose that u (w) is continuous and strictly increasing. These conditions imply that every alternative x has a certainty equivalent c(x, w), uniquely defined by the equation

Eu(w + x) = u(w + c(x, w)). A person is termed risk averse if c(x, w) < Ex = x for all nonconstant x, and decreasingly risk averse if c(x, w) increases with w.

The following result is well known. However, the proof is new and forms the basis of other proofs that follow.

RESULT 1. Suppose that if two alternatives are judged to be indifferent at any one level of wealth, then they will be judged to be indifferent at any other level of wealth. If the utility function is continuous, then it is either linear or exponential. That is either u(w) = aw + b or u(w) = a + becw for some constants a, b, c.

PROOF. We will first show that if the condition stated in the result is true, then u is either constant or strictly monotonic. Suppose that u has a turning point at wealth w *. If u is continuous, there must be nearby wealth levels w* - 61 and w* + 62 such that u(w* - 61) = U(w* + 62). But then

U(w* + 62) = U(w* + 62 + n(62 + 61))

for all integral n. Since arbitrarily small 61, 62 exist, u must be constant.

If u is strictly monotonic, let p be the unique solution to the indifference statement 1 - (2, p, 0 ); that is, u ( 1 ) = pu(2) + (1 - p)u(0). Then for all integers n we have n + 1 - (n + 2, p, n), so that

u(n + 1) = pu(n + 2) + (1 - p)u(n) or u(n + 2) = (1/p)u(n + 1) - ((1 - p)/p)u(n) or

u(n + 2) - u(n + 1) = ((1 - p)/p)(u(n+l)-u(n)). This simple recurrence relation has general solutions

u(n) = a + bn (if p = I ) and u(n) = a + b((1 p)/p)n (otherwise) for some constants a, b, p. (Gray 1967, Chapter 5, provides a review of solution techniques for recurrence relations). Note that general solutions such as these must hold for any choice of origin (w = 0) and increments (what is chosen to constitute the unit interval). Continuity dictates that these solutions converge to a + bw and a + becw, respectively.

24 MANAGEMENT SCIENCE/Vol. 41, No. 1, January 1995



The following related result reviews and extends results from Bell (1988).

RESULT 2. The following four conditions are equivalent if u is continuous:

I. If two alternatives are judged to be indifferent at any two wealth levels, then either they will be judged indifferent at all wealth levels or u is of the form aecw cos bw for some constants a, b, c.

II. u(w + x) = a(x)b(w) + c(x)d(w) + e(w) for some functions a, b, c, d, and e.

III. u is differentiable and satisfies the differential equation u"' = au" + bu' for some constants a, b.

IV. u belongs to one of the following five families of functions:

(i) Linear plus exponential: aw + bec"', (ii) Quadratic: aw2 + bw, (iii) Sumex: aebw + cedw (iv) Linear times exponential: (aw + b)ecw, (v) Exponential cosine: aeczv cos bw,

where a, b, c, and d are arbitrary constants. PROOF. Consider the equations:

pu(3) + (1 - p)u(l) = qu(2) + (1 - q)u(O) (1) and

pu(4) + (1 - p)u(2) = qu(3) + (1 - q)u(1). (2) If these equations have a solution with p and q

between 0 and 1, then we have found two gambles (2, q, 0) and (3, p, 1) that are indifferent at both w = 0 and w = 1. (Such solutions will exist if u is increasing.) Even p, q solutions outside the 0 - 1 range rep- resent equivalent gambles. For example, if a solution is p =

-2 and q = 2, then we have

-2u(3) + l-u(l) = 2u(2) - lu(O) or lu(O) + 1lu(1) = 2u(2) + lu(3)

or (1, 3, 0) (3, 1, 2). No unique solution to (1) and (2) exists if and only

if

(u(3) - U(1))2 = (u(4) - u(2))(u(2) - u(0)) or

(u(3) - u(1))/(u(2) - u(0)) = (u(4) - u(2))/(u(3) - u(1)) = k

say. In the case where k is positive, we have

(3, 1/(1 + k), 2) - (2, k/(l + k), 1). If k is negative, then

(3, 1/(1 + k), 2) - (1, 1/(1 + k), 0). If k is zero, then 2 - 0.

In summary, we have shown that there will always be two gambles involving payoffs of 0, 1, 2, 3 that are indifferent at the two wealth levels w = 0 and w = 1.

If the first part of condition I holds, they are then indifferent for all w. Thus, generalizing (1) and (2) we have for all n the recurrence relation

u(n + 3) = (q/p)u(n + 2) - ((1 - p)/p)u(n + 1) + ((1 - q)/p)u(n). (3)

(If p = 0 the recurrence relation is one order lower and u is linear. Equation (3) also holds for the exponential cosine.)

The general recurrence relation

u(n + 3) = alu(n + 2) + a2u(n + 1) + a3u(n) (4) has one of the following solutions (Gray 1967, p. 126), where z1, Z2, and Z3 are roots of the equation z3 = a1z2 + a2z + a3:

(i) b1z" + b2zn + b3z3 if all roots are distinct, (ii) b1zl + (b2n + b3)z2 if z2 = z3, or (iii) (b1n2 + b2n + b3)z if z1 Z2 = z3 for all n. Since a1 + a2 + a3 = 1 (compare (3) and (4)) at least

one of the zi is equal to 1. As with Result 1, a recurrence relation such as (4) may be deduced for arbitrary origins (where w = 0), and using arbitrarily small intervals (not just integers). As the intervals tend to zero the recurrence relations converge, respectively, to

(i) aebw + cedw or aecw cos bw, (ii) (aw + b)ecw or aw + becw, or (iii) aw2 + bw, or (aw + b)ecw. For example, in (ii) the first case arises if z1 = 1, the

second if z2 = 1. The cosine solution arises from the possibility of imaginary roots of the cubic equation. We have shown that I implies IV.

It is straightforward to check that the functions in IV satisfy conditions I, II and III. That III implies IV is rou- tine (see, for example, Piaggio 1965, p. 25). To see that II implies an equation such as (4) (which then implies IV), note that II may be written as (substituting n for w and k for x)

MANAGEMENT SCIENCE/VOL. 41, No. 1, January 1995 25



u(n + k) = a(k)b(n) + c(k)d(n) + e(n). Let k equal, in turn, 0, 1, 2, and 3. The first 3 of these equations may be solved for b(n), d(n) and e(n) and substituted into the fourth to obtain a recurrence relation such as (4). (If the first 3 equations are collinear this implies a lower order recurrence relation.) But, as argued at the beginning of this proof, an equation such as (4) must, in fact, be of the form (3). 0

3. Risk-return Functions Implications for risk-return functions are fairly imme- diate from Result 2. In this section we describe the main conclusions regarding such functions. In the next section we examine the implications for measures of risk and return. Finally, in ?5 we present some variations on the theorem in this section.

To review, we seek functions u, f, r, and R such that

Eu(w + x) f(r(x), R(x), w). What are some reasonable properties of these functions? In this section we will consider properties of u and f. In the next section we will consider reasonable properties of r and R. The following assumptions seem to be the most analogous to those commonly assumed about u:

ASSUMPTION 1. More money is better than less. We assume that u strictly increases with w and that f strictly increases with r(x). That is, more return is better.

ASSUMPTION 2. Risk aversion. We assume that u is risk averse and that f strictly decreases with R(x). That is, more risk is worse.

ASSUMPTION 3. Decreasing risk aversion. We assume that u is decreasingly risk averse. Forf, we assume that if x - y at w then x > y at higher values of w if and only if R(x) > R(y). That is, risk becomes less of a liability as w increases

THEOREM 1. Suppose u is continuous. Then if Eu(w + x) = f(r(x), R(x), w),

and if u and f reflect increasing appreciation for wealth, risk aversion and decreasing risk aversion, then either u(w) = w - becw or u(w) = -e-aw - be-cw where a, b, c are arbitrary positive constants.

PROOF. The decreasing risk aversion assumption implies that two alternatives x, y can be judged indif-

ferent at two wealth levels w1 and w2 only if R(x) = R(g). But since f is strictly increasing in r, this also implies that r(x) = r(g). But then x must be indifferent to y for all w. This means that condition I of Result 2 holds. Hence u must belong to one of the five families cited in IV. The only functions in IV that satisfy the conditions on u for all w are those stated in the theorem. Indeed, families (ii), (iv), and (v) of IV are increasingly risk averse for all w for all choices of constants.

4. Measures of Risk and Return If u(w) = w - becw then

Eu(w + x) = w + x- be-zE exp(-cx). Certainly there would be some charm to assuming r(x) = xand R(x) = E exp(-cx), for then

Eu(w + x) = w + r(x) - becwR(Y). However, though r (x) = x seems reasonable enough,

R(x) = E exp(-cx), for me at least, is not. It suggests, for example, that the risk of receiving $5 is less than the risk of receiving $4. I think that the riskiness of a sure thing should be zero.

If we let r(x) = x-and R(x) = E exp(-c(x - x)) - 1, then in the case u(w) = w - be-"' we may write

Eu(w + x) = w + x- becw exp(-cx) - becw exp(-cx)R(x) = w + r(x) - becw exp(-cr(x)) - be-(w+r)R(f) = u(w + r(x)) - be-(w+r)R(x).

This last representation is not without charm either and involves (to my mind) a more reasonable definition of risk. The purpose of the current and next sections is to state explicit conditions on the forms of r(x) and R(x) and derive their implications. We will consider, in turn, the implications of the following restrictions on the risk and return functions.

ASSUMPTION 1 (RETURN). The return of the alternative (x, p, g) is continuous in p.

ASSUMPTION 2 (RETURN). The return function satisfies the axioms of utility.

It seems to me that if we expect our overall preferences to satisfy the axioms of utility, the concept of return,

26 MANAGEMENT SCIENCE/Vol. 41, No. 1, January 1995



which is in some sense a simpler concept (in that "risk" has been removed as an issue), should do so also. Both these assumptions, however, would rule out the use of the median, for example, as the measure of return.

ASSUMPTION 3 (RETURN). The return function is the mean.

This assumption could be stated more subtly by say- ing that in addition to Assumption 2, the return function is risk neutral. Note that Assumption 3 is stronger than Assumption 2 which is stronger than Assumption 1.

ASSUMPTION 4 (RISK). The riskiness of an alternative depends only on the decision maker's final distribution of wealth. That is, R(x) = R(w + x).

We have already assumed that the risk of an alternative is independent of initial wealth; it is the reason we may write R(x). We also know that the riskiness of a final distribution of wealth, w + x, is independent of the particular division of w + x between initial wealth and the alternative (because R(w + x) = R(w + k + x - k)). Assumption 4 adds the idea that all alternatives that lead to the same distribution of final wealth (assuming constant initial wealth) have the same riskiness. An initial wealth of w - k and alternative x + k leads to the same distribution of final wealth as an initial wealth of w and an alternative Y. Assumption 4 says R(Y + k) = R(Y). A corollary of this assumption is that all constant payoffs have the same risk. Assumption 5 is a little more specific.

ASSUMPTION 5 (RISK). The risk of an alternative is positive if and only if its outcome is uncertain. Otherwise it is zero.

Note that this assumption is implicit in our usual un- derstanding of risk aversion. Our definitions for decreasing risk aversion for u and f (see previous section) are consistent only in the context of Assumption 5.

THEOREM 2. Given the conditions of Theorem 1, con- sistency with assumptions 2 and 4 requires that either

(i) r(x) =x, R(x) = (llc) log Eexp(-c(x-x)), and u (w) = w - be-"' for positive constants b and c, or

(ii) r(x) = -E exp(-ax), R(x) = (1/c) log E exp(-c(x-x))

- (1 /a) log E exp(-a(x-x ))

and u(w) = -e -a - be-w for positive constants a, b, and c (with c > a).

Each definition of r is unique up to positive linear trans- formations. Each definition of R is unique up to an arbitrary strictly increasing transformation.

PROOF. The proof is similar for the two cases. We will demonstrate the more difficult case in which u (w) =

-e-aw - becw and

Eu(w + x) = -e-awE exp(-ax) - becwE exp(-cx). For some functions h, and h2 we must have

r(x) = hi(E exp(-ax), E exp(-cx)) and R(x) = h2(E exp(-ax), E exp(-cx)).

Since r is a utility function we know that

hi(E exp(-ax), E exp(-cx)) equals the expectation of hi (exp (-ax), exp (- cx)), and hence h, must be linear in its arguments. That is,

r(x) = aE exp(-ax) + fE exp(-cx) for some constants a, 0.

Now consider a case in which x is a constant, say x -k. We have r(k) = aeak + fe ck. By assumption 4, R(k) is independent of k. Also u (w + k) is monotonically increasing both in k and in r. So it must be that r is monotonically increasing in k. (We might reasonably have assumed that directly!) This implies that a and: are each nonpositive.

But now consider an uncertain alternative x for which r(x) = r(k). Thus,

aE exp(-ax) + f3E exp(-cx) = aeak + 1e-ck. (5) Since r(i1) = r(k), the preference ordering of x and k

must be the same for all w. That is,

-e-aw[E exp(-ax)-e -ak] -becw[E exp(-cx) - e-ck] has constant sign for all w. By consideration of extreme values of w we see that this means that the signs of

E exp(-ax)-e -ak and E exp(-cx) - e-ck must be equal. But this is inconsistent with (5), unless one of a and : is zero. Thus either

r(x) = -E exp(-ax) or r(x) = -E exp(-cx).

MANAGEMENT SCIENCE/VOL 41, No. 1, January 1995 27



We argue that if c > a then r(x) = -E exp(-ax). The assumption of decreasing risk aversion for / requires that if two alternatives x, y are indifferent at w, then for levels of wealth above w, x is preferred if and only if R(x) > R(y) or, equivalently, if and only if r(x) > r (y).

Suppose x - y at w. Then

-e-azw(E exp(-ax) - E exp(-ay)) - becw(E exp(-cx) - E exp(-cg)) = 0.

If c > a then x will be preferred to y for wealth levels above w if

-E exp(-a) > -E exp(-ay) and -E exp(-cx) < -E exp(-cg).

If r(x) = -E exp(-ax) then the first of these two in- equalities is consistent with decreasing risk aversion. If r(x) = - E exp (- cx) then the second inequality is not. Hence r(x) = -E exp(-ax).

Now let us consider

R(x) = h2(E exp(-ax), E exp(-cx)). This may be written, equivalently, as

R(x) = h3(y- (1/a) log E exp(-a(x-x-), x- (1/c) log E exp(-c(x-x)

Assumption 4 implies that h3(x, y) = h3(x - y, 0). Hence we may write, for some function h, R (x) = h(S(x)) where

S(x) = (1/ c) log E exp(-c(x-x)) - (1/a) log E exp(-a(x- x)).

We will show that h is strictly increasing, and thus we may assume R (x) = S (). As a matter of identity we have

E exp(-cx) = exp(cS(x))(E exp(-aY))c/a. We may therefore write

Eu(w + ) = re -aw - be-CU(-r) c/a e cs From this equation we can see that, for fixed r, Eu

strictly declines with S, and by assumption, Eu strictly declines with R. Hence R increases strictly with S. Since

R is arbitrary up to positive transformation we may as well assume R = S. Thus

Eu(w + x) = re-"' - be -cw( r)C/aecR In the case where u(w) = w - be-cw, the argument

is very similar, but leads to the conclusions that r(x) = x,

R(x) = (/c) log E exp(-c(x-x)) and Eu(w + ) = w + rbe-ce- -crecR

= w + r - be-C(u?+r-R). m

5. Discussion The definitions of R(x) that emerge from Theorem 2 also satisfy Assumption 5. For example, if x is normal, N(,u, '2), then if u(w) = w - be-Cu' we have r(x) = and R(x) = ' cou2. If u(w) = e-aw - be-Cw then

r(x) =-exp(-a(,t-I au2) and R() = (c-a)o-2. Because the normal distribution has such prominence,

and because the variance is widely used as a measure of risk, there is some merit to adapting our definitions of risk, in a minor way, to

R(x) = (2 / c2) log E exp(-c(x- f)) and R(x) = (2/(c - a))[(1/c) log E exp(-c(x- x))

-(1/a) log E exp(-a(x-x) respectively.

For nonnormal distributions, these definitions are seemingly more complex than the variance. In this age of spreadsheets, however, this is not really an impedi- ment to their use. One major difference is that they do depend on parameters (c and a) that vary by individual. This means that we cannot talk about the risk of an alternative for all people. In the simpler definition of R(x), the parameter c controls the degree to which our concern about risk is due to extreme downside outcomes rather than simply spread. If c is close to zero, then the risk measure approximates the variance which gives equal weight to upside and downside spread. As c gets large, the risk measure simply reflects "worst case" outcomes.

28 MANAGEMENT SCIENCE/VOl. 41, No. 1, January 1995



6. Related Results The proof of Theorem 1 made substantial use of the decreasing risk aversion concept as applied to f. On the other hand, we made no use of conditions about the nature of r(x) and R(x). In this section we relax the decreasing risk aversion condition on f and instead use some of the assumptions about risk and return from the last section.

In this first result, we make explicit the desire to have r(x) be the mean.

THEOREM 3. If u is continuous and has a continuous derivative, then Eu(w + x) is consistent with a risk-return function f(r(x), R(x), w) that is strictly increasing in r(x) and strictly decreasing in R(x) and where r(x) = x, if and only if u belongs to the linear plus exponential family or the quadratic family. If, for all w, u(w) is to be strictly increasing and risk averse, then u(w) = aw - be-cv for some nonnegative constants a, b, c.

PROOF. Since r(x) = x, any two alternatives having the same mean will have the same preference ordering for all w, since preference will then depend only on relative riskiness.

If x and y have the same mean, then consider derivative gambles Z1 = (x + 6, , 2y) and Z2 = (X, 2', Y + 6) where 6 is a small amount. Note that z1 and Z2 also have the same mean. Thus the quantity

[2Eu(w + x + h ) + 2 Eu(w +)] _-[21 Eu(w + x) + 2 Eu(w + y + 6)]

has the same sign for all w. Rewriting twice this quantity as

[Eu(w + x + 6) - Eu(w + x)] -[Eu(w + y + ? )-Eu(w +)]

we note that this too has constant sign for all w. Now consider the function v (w) = u (w + 6) -u (w). Viewing v as a utility function in its own right, we see that the quantity Ev (w + x) - Ev (w + y) has constant sign for all w.

By Result 1 this means that v (w) must be linear or exponential in w. By choosing 6 small enough and in- voking continuity we see that the derivative of u must be linear or exponential in w. Integrating the linear case

gives us a quadratic utility function; integrating the exponential gives us a linear plus exponential utility function.

Though the quadratic can be increasing and risk averse on an interval of w, it never has these properties for all w. O

THEOREM 4. Suppose that u is continuous, increasing, and risk averse for all w. If Eu(w + x) is consistent with a risk-return function f(r(x), R(x), w), strictly increasing in r(x), strictly decreasing in R(x) and where r is itself a utility function, then either u (w) = aw - be-cz or u (w) = -e-azv - be-c" for some nonnegative constants a, b, c.

PROOF. Consider solutions in p, q to the equations u(3, p, 1) = u(2, q, 0) and r(3, p, 1) = r(2, q, 0). If such p and q exist then it must also be that R(3, p, 1 ) = R(2, q, 0). Hence (3, p, 1) (2, q, 0) for all w. But then we obtain the recurrence relation (4) used in Result 2. But since r(x) is assumed to be a utility function, solutions in p and q must exist, for we have

pu(3) + (1 - p)u(l) = qu(2) + (1 - q)u(O) and pr(3) + (1 - p)r(l) = qr(2) + (1 - q)r(O).

The only dilemma occurs if there is no unique solution to these equations for any choice of w. But this implies that u(w + x) = a + br(x) for some constants a, b, which implies that u is linear or exponential. [H

In Theorem 3 we used Assumption 3 about r(x); in Theorem 4 we used Assumption 2. Now we assume only Assumption 1 but add Assumption 5 about R(x).

THEOREM 5. Suppose that u is continuous, increasing, and risk averse for all w, and that Eu(w + x) is consistent with a risk-return function f (r(x), R(x), w) which is increasing in r(x) and decreasing in R(x) for all w. Suppose that r(x, p, y) is continuous in p and that R(x) is positive if and only if x is uncertain, and zero if and only if x is constant. Then either u (w) = aw - bec-cv or u (w) =-e -aw - becw for some nonnegative constants a, b, c.

PROOF. As in Theorem 4 we try to establish the existence of solutions p, q to the equations

u(3, p, 1) = u(2, q, 0) and r(3, p, 1) = r(2, q, 0). We know that there must exist values p*, q* between zero and one such that

MANAGEMENT SCIENCE/VOL. 41, No. 1, January 1995 29



(3,p*, 1) (2, 1,0) and (3,0, 1) - (2,q*,0). By Assumption 5 we know that

R(3, p*, 1) >R(2, 1, 0), and therefore

r(3, p*, 1) < r(2, 1, 0). Also by Assumption 5, we know that

R(3, 0, 1) < R(2, q*, 0) and therefore that

r(3, 0, 1) > r(2, q*, 0). Now, for a range of probabilities, X, consider the two

compound gambles

((3,p*,1),X,(3,0,1)) and ((2,1,0),X,(2,q*,0)). For all X, these two have the same utility. For X = 1 the first has the higher return; for X = 0 the second has the higher return. By Assumption 1, continuity of r, there exists a value of X, X*, at which they have the same return. Set p = p*X* and q = X* + q*(1 - X*). Now proceed as in the proof of Theorem 4. O

7. Discussion One may think of the linear utility function and its associated statistic, the mean, as first-order approximations to the "truth" of nonlinear utility. Historically the second-order approximation to utility has been the quadratic, along with its associated statistics of mean and variance. But the quadratic utility function is dis- liked by those who believe in sensible global properties, such as increasing appreciation for wealth, risk aversion, and decreasing risk aversion.

This paper shows that there is another way to gain a second-order approximation. Moreover, it is one that meets the rationality concerns of utility theorists on the one hand, and our intuitive concepts of risk and return on the other.

Consider the virtues of the utility function u (w) = w - be -cw, where b and c are positive constants. It is increasing in w, risk averse, and decreasingly risk averse. Alternatives may be compared by use of a risk-return function with return measured by the mean. The parameters b and c leave considerable flexibility to match personal attitudes to what constitutes relative riskiness (the parameter c) and of aversiveness to that riskiness (the parameter b).

It is my belief that these properties of u (w) = w - be-cw (and others explained in Bell 1988 and 1993) are sufficiently compelling that it could be adopted as the generic utility for wealth in assessments and economic analyses.

References Bell, D. E., "One-Switch Utility Functions and a Measure of Risk,"

Management Sci., 34 (1988), 1416-1424. "Contextual Uncertainty Conditions for Utility Functions,"

Harvard Business School Working Paper No. 94-044, 1993. Forthcoming in Management Sci.

Fishbum, P. C., "Foundations of Risk Measurement. I. Risk as Probable Loss," Management Sci., 30 (1984), 396-406. ,"Foundations of Risk Measurement. II. Effects of Gains on Risk," J. Mathematical Psychology, 25 (1982), 226-242.

Gray, J. R., Probability, Oliver and Boyd, Edinburgh, Scotland, 1967. Markowitz, H., Portfolio Selection, John Wiley and Sons, New York,

1959. Piaggio, H. T. H., Differential Equations, G. Bell and Sons, London,

England, 1965. Von Neumann, J., and 0. Morgenstern, Theory of Games and Economic

Behavior (second edition) Princeton University Press, Princeton, NJ, 1947.

Accepted by Irving H. LaValle,former Departmental Editor; received September 11, 1991. This paper has been with the author 5 months for 2 revisions.

30 MANAGEMENT SCIENCE/VOL 41, No. 1, January 1995


Article Contentsp. 23p. 24p. 25p. 26p. 27p. 28p. 29p. 30

Issue Table of ContentsManagement Science, Vol. 41, No. 1, Jan., 1995Front MatterDecision Analytic Networks in Artificial Intelligence [pp. 1 - 22]Risk, Return, and Utility [pp. 23 - 30]Noise and Learning in Semiconductor Manufacturing [pp. 31 - 42]Quoting Customer Lead Times [pp. 43 - 57]Efficient Estimation of Arc Criticalities in Stochastic Activity Networks [pp. 58 - 67]Forecaster Diversity and the Benefits of Combining Forecasts [pp. 68 - 75]A Maximum Decisional Efficiency Estimation Principle [pp. 76 - 82]A Min Cost Flow Solution for Dynamic Assignment Problems in Networks with Storage Devices [pp. 83 - 93]The One-Machine Problem with Delayed Precedence Constraints and Its Use in Job Shop Scheduling [pp. 94 - 109]Optimal Mean-Squared-Error Batch Sizes [pp. 110 - 123]Measuring Systematic Risk Using Implicit Beta [pp. 124 - 128]The Demand for Parimutuel Horse Race Wagering and Attendance [pp. 129 - 143]Optimal Monopolist Pricing under Demand Uncertainty in Dynamic Markets [pp. 144 - 162]Decentralized Regulation of a Queue [pp. 163 - 173]Holt-Winters Method with Missing Observations [pp. 174 - 178]NoteOn the Efficiency of Imbalance in Multi-Facility Multi-Server Service Systems [pp. 179 - 187]

Back Matter

risk, return, and utility

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