Journal of Financial Economics 2 (1975) 9S-121. (QNorth-Holland Publishing Company

OPTIMAL, RULES FOR ORDERING UNCERTAIN PROSPECTS+

Vijay S. BAWA

Bell Laboratories, Holmdel, N.J. 07733, U.S.A.

Received October 1974

In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of altema- tives. those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm lo obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in Enance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule lo others previously advocated in the literature. including the more popular mean-variance rule as well as the semi-variance rule.

1. Introduction

Decision-making under uncertainty may be viewed as choices between alternative probability distributions of returns, and the individual chooses between them in accordance to a consistent set of preferences. Von Neumann and Morgenstern (1967) have shown that under reasonable assumptions about individual preferences, the individual chooses an alternative which maximizes the expected utility of returns, where the utility function is determined uniquely, up to a positive linear transformation, by individual preferences. However, in most situations. such a selection is not possible since complete information about an individual’s preference set, and hence his utility function, is not available. Thus, with only partial information that an individual’s utility function belongs to a certain restricted class of admissible functions, one is interested in the optimal selection rules which minimizes the admissible set of alternatives by discarding, from among the given set of alternatives, those that are inferior (for each utility function in the restricted class) to a member of the given set.

*We thank Nils Hakansson for stimulating discussions and acknowledge helpful comments by Nils Hakansson, Karl Borch, Peter Fishbum, and Alex Whitmore on an earlier draft.

96 V.S. Bawa. Oplimal rules for or&ring uncertain prospects

The more restrictive the class of utility functions, the smaller will be the admissible set and thus the more useful will it be in practical situations. However, more restrictions on the utility functions imply that the admissible set is relevant for a smaller group of individuals and may involve a severe loss in generality. Thus, one is interested in determining the admissible set of alternatives for the most restrictive class of utility functions that is consistent with observed economic phenomena. Arrow (1971) and Pratt (1964) have pointed out that the observa- tion of certain economic phenomena indicate that individual utility functions exhibit decreasing absolute risk aversion and to a lesser extent increasing relative risk aversion. Stiglitz (1970) has raised doubts whether increasing relative risk aversion is a plausible assumption. Thus, it appears that decreasing absolute risk aversion is the most restrictive class of utility functions acceptable to most economists and we are interested in the optimal selection rule for this class of utility functions.

For the portfolio selection problem, which may be viewed as a canonical representation for certain types of economic problems involving decision- making under uncertainty, Markowitz (1952, 1970) and Tobin (1958, 1965) proposed for risk averse individuals, a mean-variance selection rule in which, from among a given set of investment alternatives, the admissible set is obtained by discarding those investments with a lower mean and a higher variance than a member of the given set. But even though the mean-variance approach has spawned a considerable body of literature, including most notably the Sharpe- Lintner-Mossin Capital Asset Pricing model [Sharpe (1964), Lintner (1965), Mossin (1966)], it has been known for some time [see, for example. Borch (1969). Feldstein (1969) and Hakansson (1972)] that the approach is of limited generality since it is the optimal selection rule only if the utility function is quadratic or the probability distributions of returns are normal. Arrow (1971) and Hicks (1962) have pointed out that the assumption of quadratic utility is highly implausible in that it implies increasing absolute risk aversion. Also, the assumption of normal distribution of return on risky investments is not realistic as it rules out asymmetry or skewness in the probability distribution of returns. Cootner (1964) has shown that the returns on financial investments are more likely to be lognormal than normal. In a recent empirical study, Lintner (1972) has shown that even returns on portfolios of risky assets are more likely to be lognormal than normal. Furthermore, progressive taxation and limited liability of corporations imply that the distribution of net returns is quite likely to be skewed. Thus a selection rule based on mean and variance alone is indeed not justifiable on theoretical grounds but is an approximate and computationally feasible selection rule. Samuelson (1970) and Tsiang (1972) have shown that the approximation is reasonable only when the ‘riskiness’ of returns is limited.

The use of variance as a measure of risk for non-symmetric distributions has been questioned by financial theorists [see for example, Hirshleifer (1970, pp. 278-284). Mao (1970) and Markowitz (1970, pp. 188-201)]. Semivariance,

V.S. Bawa. Optimal rules for ordering uncertain prospccrs 97

rather than variance, has been proposed [Mao (1970) Markowitz (1970)] as a measure of risk on the grounds that semivariance concentrates on reducing losses as opposed to variance which considers extreme gains, as well as extreme losses, as undesirable. Semivariance is less tractable mathematically than variance and an algorithm for obtaining the mean-semivariance admissible set has been provided by Hogan and Warren (1972). We note, however, that the mean-semivariance selection rule, just as the mean-variance rule, is the optima1 selection rule for an increasing absolute risk averse utility function and hence this approach also appears to be of limited generality.

On recognizing the restrictiveness of the mean-variance rule and its variants, Quirk and Saposnik (1962) and later Fishbum (1964), Hadar and Russell (1969, 1971). and Hanoch and Levy (1969) obtained the optimal selection rule for the entire class of increasing utility functions (i.e., the utility function is an increasing function of returns). The selection rule, called First Order Stochastic Dominance (FSD) rule, is that a probability distribution F dominates a proba- bility distribution G if and only if F never lies above and somewhere lies below G; the FSD admissible set contains distributions that are not dominated by the FSD rule. Thus, among the class of all distribution functions, the subclass that can be ordered by the FSD rule will indeed be small. Hence, as is to be theoreti- cally expected, and has been empirically verified by Levy and Hanoch (1970) Levy and Sarnat (1970) and Porter and Gaumnitz (1972) a large proportion of the given set of alternatives will still be members of the FSD admissible set; this restricts the practical applicability of the selection rule.

Perhaps so motivated, and since individual behavior exhibits risk aversion, Hadar and Russell (1969, 1971), and Hanoch and Levy (1969) considered the restricted class of increasing utility functions that are risk averse, i.e., utility functions are increasing and have decreasing marginal utility everywhere, and obtained the optimal selection rule. This selection rule, called the Second Order Stochastic Dominance (SSD) rule, is that a probability distribution Fdominates a probability distribution G if and only if the integral of F never lies above and somewhere lies below the integral of G; the SSD admissible set contains distribu- tions that are not dominated by the SSD rule. Thus, a larger subclass of dis- tributions will be ordered by the SSD rule and the SSD admissible set will be smaller than that under the FSD criterion. [This is empirically varified in Levy and Hanoch (1970), Levy and Samat (1970). and Porter and Gaumnitz (1972).] Rothschild and Stiglitz (1970, 1971) also obtained SSD rule for the special case of distributions with equal means and finite range and provided applications of these to several economic problems.

Whitmore (1970) considered the class of increasing and risk averse utility functions with the additional restrictions that the third derivative of the utility functions be positive and for the case when the random variables are deEned on a finite range obtained the optimal selection rule, called the Third Order Stochastic Dominance (TSD) rule. The economic grounds for assuming that the

98 VS. &wa, Opliml n&s for or&ring wtccrtain prospects

utility functions have positive third derivative is that it is implied by decreasing absolute risk aversion; it is more desirable to obtain the optimal selection rule directly tied to decreasing absolute risk aversion.

The optimal selection rules (FSD and SSD) involve pairwise comparisons and require complete knowledge of the entire distribution function, as opposed to the knowledge of only the mean and the variance. With no restrictions on the probability distribution functions, the selection rule is complex and for practical applications, one would need an efficient algorithm to obtain the admissible set. Algorithms to obtain the admissible sets are outlined in Levy and Hanoch (1970) and Levy and Samat (1970) and an algorithm which is efficient even for large numbers of alternatives is provided in Porter, Wart and Ferguson (1973). However, with certain restrictions on the class of distribution functions, the optimal selection rules reduce to manageable and simple forms; results for certain special cases are given in Hanoch and Levy (1969) and Hadar and Russell (1971). As noted earlier, for the case of normal distributions, the SSD rule reduces to the mean-variance selection rule, but otherwise even for the restricted class of two parameter distributions, the SSD and mean-variance rules result in different admissible sets. It is pointed out in Quirk and Saposaik (1962), Hadar and Russell (1969). Hakansson (1971), and Hanoch and Levy (1969) that it may be the case that some distribution on the mean-variance admissible set may be dominated under the SSD (or even the FSD) rule and hence do not belong on the SSD admissible set. Conversely, a member of the SSD admissible set may not be on the mean-variance admissible set. Thus, a major deficiency of the mean-variance rule is that it fails to obtain the SSD admissible set; it uses neither a necessary nor a sufficient condition for dominance in eliminating inferior alternatives.

In section 2, we show that for the entire class of distribution functions and for the class of decreasing absolute risk averse utility functions, the Third Order Stochastic Dominance (TSD) rule is the optimal selection rule when the dis- tributions have equal means. We also show that for distributions with unequal means, TSD rule is a sufficient condition for dominance, a large mean is also a necessary condition for dominance but only a part of the integral condition underlying TSD rule is necessary for dominance. However, with no restrictions and the class of distribution functions, there is no selection rule which is both necessary and sufficient for dominance. In view of these results TSD rule may be used as a reasonable approximation to the optimal selection rule for the class of decreasing absolute risk averse utility functions and the entire class of distribution functions. The TSD rule requires complete knowledge of the entire distribution function, is complex and involves pairwise comparisons of the alternatives. In section 3, we provide an efficient algorithm to obtain the TSD admissible set using the TSD rule. In section 4, we show that Third Order Stochastic Dominance implies dominance under the mean-lower partial variance rule: the mean-lower partial variance rule uses a necessary condition for

V.S. Bawa, Optimal rules for or&ring uncertain prospects 99

dominance for the entire class of distribution functions. In view of the complexity of the TSD rule, this provides a strong rationale for using the mean-lower partial variance rule (as opposed to the popular mean-variance rule) as a reasonable approximation to TSD and hence the optimal selection rule for the entire class of distribution functions. In section 5, we consider certain restricted classes of distribution functions considered extensively in the literature (and which cover most distributions of practical interest in economics and finance) and show that the TSD rule is indeed the optimal selection rule and that it reduces to simple manageable forms. Some concluding remarks are provided in section 6.

2. Optimal selection rules

We are interested in rules for ordering a pair of uncertain prospects charac- terized by random variables X and Y with known probability distribution functions F( *) and G(s), respectively. The random variables X and Y, represent- ing the outcomes of uncertain prospects, may be discrete, continuous or mixed with range, represented by a closed interval [u, b], u < b, where either one or both end points may be infinite. The distribution functions F(e) and G( *) are

non-decreasing, continuous on the right with F(u) = G(u) = 0 and F(b) =

G(b) = 1. The analysis is not affected by whether these probabilities are ‘objective’ or ‘subjective’ as long as the distribution functions are completely specified. The uncertain prospects are thus equivalently characterized by dis- tributions Fand G and we are interested in rules for ordering them.

An individual chooses between F and G in accordance with a consistent set of preferences satisfying the Von Neumann-Morgenstern (1967) consistency properties. Accordingly, F is preferred to G if’ *’

AEu = E+(X)-E&X) > 0, (0

(i.e., expected utility under F is greater than that under G) where u(x) is deter- mined uniquely, up to a. positive linear transformation, by individual preferences. Since complete information about individual preference may not be available, we are interested in ordering rules for certain restricted classes of utility functions. We define the following sets of utility functions that are commonly used in economic problems, with R denoting [a, b):

‘The decision-maker is interested in maximizing the expected utility of the end of period wealth. Hence, Xand Y denote the end of period wealth under the two alternatives being con- sidered. The end of period wealth could equivalently be defined as (W+X) and (W+ Y), where Xand Yare now defined as additions (or reductions) to the initial wealth level W. Hadar and Russell (1971) and Levy and Sarnat (1971) have shown that if W is either a constant or a random variable distributed independently of X and Y. then the ordering rule is invariant to the initial wealth position W. Hence, in the ordering rule (1). X and Y may equivalently be viewed as money payoffs on uncertain ventures that are additions (or reductions) to individual’s initial wealth position W.

‘It is assumed throughout this paper that the expected utility exists.

100 V.S. Bawo, Optimal rules for ordering uncerroin prospects

Dejnition 1:

CJ, = {u(x) 1 U(X) is finite for every finite x, u’(x) > 0 Vx E R) ;

Definition 2:

U, = {u(x)~~(x)~U,, - co < u”(x)~<Otlx~R};

Definition 3:

U, = (u(x)]u(x)~U~,u-(x) > OVXER};

Definition 4:

U, = {u(x) 1 u(x) E U, , r’(x) s (-u”(x)/u’(x))’ < 0 Vx E R} .

WI is the set of all increasing and continuously differentiable utility functions u(x) assumed to have finite values for finite values of x. U, is the subset of CJ, with risk averse utility functions; U, is the subset of U, with utility functions having positive third derivative, while U, is the subset of II, with decreasing absolute risk averse utility functions. We are primarily interested in considering utility functions which vary only in a range bounded from below. This is so if, for example, an individual cannot be worse off than loosing everything; then U(X) needs to be defined only for x z 0. Hence, it is assumed throughout (with no loss in generality) that the interval R( = [a, b)) is a subset of the non-negative half line (i.e., u 2 0).

The classes (I,, U, and U, of utility functions have been considered extensively in the literature [Quirk and Sapasnik (1962). Fishbura (1964). Hadar and Russell (1969, 1971), Hanoch and Levy (1969), Whitmore (1970)], and the optimal selection rules for these classes (i.e., the FSD, SSD and TSD rules) are summarized by the following theorems,‘**

Theorem I. For any two distributions F and G, F is preferred to G for all utility functions in II, if and only i/

F(x) s G(x) Vx E R and < for some x.

Theorem 2. For any two distributions F and G, F is preferred to G for all utility functions in U2 if and only if

‘The integrals used throughout are Stieltjcs-Lebcsgues integrals, which are assumed to be bounded.

.It should be noted that in proof of Theorem 3, it is immaterial whether b is finite or infinite. Thus our Theorem 3 extends Whitmore’s TSD rule to cover infinite range as well.

V.S. Bawa, Optimal rules for or&ring uncertain prospects 101

j;F(r)dt s j: G(r)dr V x E R and c for some x.

Theorem 3. For any two distributions F and G, F is preferred to G /or all utility functions in U, ifand only if

cc, 2 c1G

and

j: j: F(r)dr dy < j: j: G(f)dr dy Vx E R and < for some x.

We note that the proof of Theorem 3 [in Whitmore (1970)] and Theorems 1 and 2 [in Hadar and Russell (1969, 1971)] is incomplete. We now provide a simple proof of Theorems l-3.

ProofofTheorems 1-3. We note that

AEu = E&X)- E&Y)

= j: u(x) d( F(x) - G(x)).

Carrying out integration by parts several times with H,(x) = F(x)- G(x),

H,(x) = j: %-,Qdy,

for n 2 2, we obtain the following equivalent expressions for AEu:

AEu = -j: u’(x)H,(x)dx,

AEu = -u’(bW,(b) + j: u”(x)Hz(xW,

AEu = -u’(b)H,(b) + u”(b)H,(b) - j.” u”(x)H,(x)dx.

(2)

(3)

(4)

(5)

(6)

Using (4)-(6) and noting that H,(b) z pc -pF proves the sufficiency of selection rules in Theorems l-3. To prove necessity, it needs to be shown that if the conclusion of Theorem i (i = 1, 2, 3) fails to hold then there exists a utility function in U, for which the hypothesis of Theorem i is contradicted (i.e., if for some arbitrary x0, with a < x0 < x,, +b d 6, H,(x) > 0 for x E [x,, x0 +6], then there is a utility function u, E U, such that AEu, < 0; in addition, for

102 VS. Bawa, Optimal rules for ordering uncertain prospects

Theorem 3, it also needs to be shown that A/J = ~+-c(o >, 0 is also necessary). The contradiction is provided by means of a utility function defined through the following function, with E = y6,

i

E, a < x < x0,

4(x> = E-Y(X--X0), x0 < x < x,+6,

0, x0 +6 cx.’ To prove Theorem 1, we consider utility function ur(x) defined by u;(x) = k- f(x), where k > 0; then as u;(x) > 0, ur(x) E WI ,’ and we obtain sub- stituting in (4) and noting that the integrals are bounded,

AEu, = -y If:+” H,(x)dx+M,.

Thus, by choosing y large enough, one can make AEu, < 0; this completes proof of Theorem 1.

To prove Theorem 2, we consider utility function uI(x) defined by u’;(x) E -k + 4’(x), where k > 0. Then as u’;(x) < 0, u;(x) I k, - k(x-a) + b(x) > 0 by appropriate choice of k, , u2(x) E U,, and we again obtain for sufficiently

large Y,

AEu, = M,-Y ,;+, Hz(x) dx < 0.

To prove Theorem 3, we consider utility function u,(x) defined by u;(x) E eI - 4’(x), where .sr > 0. We note that uj (x) > 0, u’;(x) = -(k, -Q(x-u)) - 4(x) c 0 by appropriate choice of k, . Similarly,

u;(x) 3 k,-(x-u) k, -2 (x-u) >

-j; d(y)dy,

where

I 4x -a), a < x < x0,

j-10 Ody = 1 4x-++~,,l, x0 f x G x0+6,

‘We note that the differentiability requirements can be satisfied by rounding the edges which does not change the analysis.

V.S. B4wa, Optimal rules for or&ring uncertain prospecrs 103

and k2 is an arbitrary constant chosen such that u;(x) > 0 Vx E R; hcna u,(x) fz U,. We let

k, = c,(b-a)(1 +k;),

k, = e,(b-a)‘(k;+k;+#+E(x,-o+b) -;6’ +k,yb2;

also let

M3 = k;(b-a)2-k;(b-a)H,(b)-Jf H,(x)dx,

and noting that -H,(b) = Ap(=j+-po),

AEu, = k,yS2Ap+eIM,-y I:;+” H,(x)dx

< vW,W--,l+c,M,,

where for x E [x0, x0 +S], H,(x) 2 M4 > 0. Thus, if we let E~ -+ 0, 6 + 0, y + co such that E = yS + C > 0, then

AEU, < 0 for sufficient small value of 6. This proves that H,(x) < 0 Vx, < 0 for some x is a necessary condition for dominance.

We consider utility function u.,(x) defined by

464 = exp (eSPX),

wherep > 0. Then we note that as uk > 0, u; < 0 and u; > 0, u,&x) E U, and we obtain, after some algebraic simplifications,

AEu, = -exp(e-pb)[H2(b)+pe-pbH,(b)]

+J: exp (e-b”)A(x)Hs(x)dx,

where ,4(x) z -p2e-P”(1 aemPX) < 0.

It thus follows that since A(x) +Oasp-,O,ifH,(b)>O, AEu,<Ofor sufficiently small value of p. This completes proof of Theorem 3.

We are interested in the class U, of decreasing absolute risk averse utility functions and the optimal selection rule for this class is given by the following:

Theorem 4. For any two distributions F and G with pF = po, F is prefered (0 G for all utility functions in U4 if and only if

I: JI F(l)dfdy < J: J: G(t)dldv Vx E R and < for somex.

104 VS. Bawa, Optimal rules for or&ring uncertain prospects

Proof of Theorem 4. We obtain from (a), remembering that r(x) E -u’(x)/ u’(x) and H,(b) = 0,

AEu = -u’(ww)+~ u’(x)[“(x)-(4x))2]H3(x)dx. (7)

For U(X) E Uq, r’(x) < 0, r(x) > 0 and thus H,(x) < 0 Vx and < 0 for some x is a sufficient condition for dominance. To prove necessity, we consider a utility function us(x) defined by r;(x) = $‘(x),‘j and note that us(x) E I/,, and we obtain

AEu, = -E2 j:” &(x)H,(x)dx-y j:;‘” u;(x)H,(x)d_X

_ j:;+* 4(x)(&-Y(x-xo))2wx)dx.

We note that since for x E [x,,, x,+6], H,(x) Z MI > 0 and since IH3(x)I < M2 VXER, AEu, < Oif

YMI j;+* u’(x)& > &2M2 j:” u’(x)dx. (8)

We obtain after some algebraic simplifications, remembering that u’(x) = exp (-j: r(r)df),

I

x0 e-” _e-exo u’(x)dx = e ,

0

and

j “,I+” u’(x)dx = e-‘Xo-(N2) Id,_0 e(r/2)(~-*)a dy

> ~e-exo-W/2). I

hence to prove necessity of the selection rule, it suffices to show that

e -m

yM,(Je -txo-W/2)) , E2~2 _,-cxo

.

E ’ that is,

Mle-‘xo-h’/2) > M~(~-u__~-IJo). (9)

We note that with 6 + 0, y + 03 such that E = y6 + 0, inequality (9) will hold for sufficiently small b. This completes the proof of Theorem 4.

*To insure that r,‘(x) < 0 Vx, we need to define r,‘(x) = d’(x) --e, , where cI > 0. In the proof, we will take c, arbitrarily close to zero and since it does not effect the analysis, we assume hereafter for typographical simplicity that c1 = 0.

V.S. Bawa, Optimal rules for or&ring wtccrtain prospects 105

For the more general case of distributions with unequal means, our results on the optimal selection rule are summarized by the following, with H,(x) as given by (3):

Theorem 5. For any two distributions F and G, F is preferred to G for all utility functions in U,,

(i) if pr > ~1~ and H,(x) < 0 Vx E R, >O for some x, (ii) only if c(r k ~1~ and for some x0 E R, H,(x) Q 0 for (I d x < x0,

<Oforsome x < x0.

Proof of Theorem 5. We note that since U, c U,, it follows that TSD, as given in (i), is sufficient condition for dominance. To prove (ii), we obtain from (6), noting that r(x) E -I/(X)/U’(X),

AEu = - u’(b)( H,(b) + r(b)H,(b))

To prove (ii), it needs to be shown that if for some x0 E R, H,(x) Z 0, a G x < x0 with strict inequality for some x < x0, then there exists a utility function in u4 such that AEu -C 0. The contradiction is provided by means of the utility function Q(X) with

U:(x) = exp (e-P(x-xO)), (11)

wherep is a freely chosen positive parameter. We note that u’(x) > 0, Vx E R,

and u;(x) = -pe-P’=-%;(x) < 0,

r6(x) = pe-p(“-*o) > 0,

r:(x) = -p2e-P(X-X0) < 0.

Thus I+,(X) E U, and we obtain using (10) that for the utility function given

by (11). AEug = - cxp (e -P’b-Xo))[H2(b)+pe-p’b-r~)H)(6)] (12)

+ j:” exp (e -P(“-“0$4(x)H,(x)d_X

+ j:, exp (e -p’“-‘o’)A(x)H,(x)dx,

where A(x) E r:(x)-r:(x) < 0. Since exp (e-p(X-*o)) is an increasing and

106 VS. Bawa, Optimal rules for ordering uncertain prospects

unbounded function of p for x < x0, and a decreasing function of p for x > x,, , it follows that AEu, can be made negative for a sufficiently large value of p; similarly, it follows that it is also necessary to have H,(b)(=p,-~1~) < 0 as otherwise dEu6 can be made negative for a sufficiently small value of p. This completes the proof of Theorem 5.

We have shown that the Third Order Stochastic Dominance (TSD) rule that is optimal for the class U, of utility functions is also the optimal selection rule for the class U, of decreasing absolute risk averse utility functions when the distributions have equal means. For the general case of unequal means, TSD rule is a sufficient condition for dominance, a larger mean is also a necessary con- dition for dominance, but only a part of the integral condition underlying TSD are necessary for dominance. Indeed H,(x) Q 0 Vx E R is not necessary for dominance [see Vickson (forthcoming a, b) for several counter examples], and, unfortunately, there is no known selection rule which is both necessary and sufficient for dominance. In view of these results, TSD rule may be used as a reasonable approximation to the optimal selection rule for the entire class of distribution functions.

3. Algorithm to obtain the TSD admissible set

The Third Order Stochastic Dominance Rule, involves pairwise comparison of the given set of alternative probability distributions. It requires complete knowledge of the entire distribution function, not just means and variances as needed for the mean-variance selection rule. Thus for a given set of n alternative distributions, (;) = n(n- I)/2 paired comparisons are needed; in addition, if each distribution is discrete with M possible values then for each paired com- parison, H,(x) has to be computed and checked to be non-positive at 2m values. Thus, a total of n(n - I)m comparisons would be needed to obtain the admissible set from a given set of n alternatives. (The same number of comparisons would be needed for the FSD and SSD rules as well.) Even for moderate values of n and m, the number of comparisons can be prohibitively large; for example for n = 10 and m = 10, 900 comparisons would be needed, while for n = 100 and m = 10, 99,ooO comparisons would be needed to obtain the admissible set. Thus, in order to obtain the admissible set of alternatives in any practical situation, one needs an algorithm which makes efficient use of all the available information about properties of the distribution functions and the ordering rule.

For the case of discrete distributions, Porter, Wart and Ferguson (1973) have developed an algorithm to obtain the admissible set for the TSD rule,’ which is

‘This algorithm also obtains FSD and SSD admissible sets; Levy and Hanoch (1970) and Levy and Sarnat (1970) have also provided algorithms to obtain FSD and SSD admissible sets. Our algorithm is more efficient than these other algorithms as well in obtaining the admissible sets.

VS. Bawa, Optimal rules for ordering uncertain prospects 107

claimed to be efficient even for large numbers of alternatives. The efficiency of this algorithm basically results from the following use of three properties of the ordering rule :

(1) List alternatives in descending order of mean values. Since larger mean is a necessary condition for dominance, an alternative cannot dominate the alternatives listed above it. Thus the alternative with the largest mean belongs to the efficient set.

(2) An alternative F cannot dominate an alternative G if the smallest possible value of the random variable under F is smaller than that under G; this is so as in this case F(x) - G(x) > 0 and hence H,(x) > 0 at the smallest value of x.

(3) The ordering rule is transitive; hence eliminate from further pairwise

comparisons an alternative dominated by another.

The algorithm in Porter-Wart-Ferguson (1973) takes advantage of these properties to reduce the number of pairwise comparisons to a minimum. In addition, when comparing an alternative F to an alternative G (with pF 2 po), the efficiency is increased by computing H,(x) sequentially for increasing values of x: If H,(x) < 0 for all x, then it is noted that Fdominates G, but, on the other hand, as soon as H,(x) > 0 is recorded, further comparisons are stopped and it is noted that F does not dominate G.

The algorithm in Porter, Wart and Ferguson (1973) applies only to the case of discrete distributions. In addition, since in comparing F to G, the algorithm computes and checks for N,(x) to be non-positive for all possible values of x before concluding that Fdominatcs G, the efficiency of the algorithm is inversely proportional to the number of possible values (m) of the random variables. Thus, for large values of m (which may occur, for example, when approximating a continuous distribution by a discrete distribution), the algorithm in Porter, Wart and Ferguson (1973) may not be practical.

We now propose an algorithm which is applicable to continuous as well as discrete distributions. The algorithm uses all the clever features used in Porter, Wart and Ferguson (1973) to reduce the number of pairwise comparisons to a minimum, and, in addition, improves the efficiency by making use of available information about the distribution functions. This is done by avoiding, when comparing F to G, the computation of H,(x) for all possible values of x; instead the algorithm concentrates on the zero crossings of the function H,(x) E F(x)-G(x) and uses the minimum possible number of computation necessary to check if F dominates G. For discrete distributions (with m possible values for each distribution), the number of zero crossings is at most 2m but will usually be much smaller; hence the algorithm will be, in most cases substantially more efficient than the one proposed in Porter, Wart and Ferguson (1973). Also the algorithm can handle continuous distributions as well since the number of zero crossings, except for pathological cases, will be finite for continuous distributions.

We now provide the basic steps of the algorithm used in comparing two un-

108 V.S. Bawa, Optimal rules for or&ring wucrtain prospect3

certain prospects F and G,s noting that the random variables are defined on [u, 61 and for the case of continuous distributions the following equivalent expressions for H,(x) and H,(x) may be used recursively to improve the c5ciency of computation,

H,(x) = &(9+(x-Wz(Z)+~(x-y)HJ_y)dy, f < x,

Algorithm to order Fund G (with pp > c(,J

Step I:

If H&z+) > 0 GO TO STEP 7; otherwise GO TO STEP 2.

Step 2:

(Initialization)

Let N = number of zero crossings from below of H,(x). (Let X*,X2...., X, denote the N zero crossings.)

If N = 0 GO TO STEP 6; otherwise let Y, , Y, , . . ., Yn_i denote the (N- 1) zero crossings from above of H,(x), x E [a, 6).

Let

last zero crossing from YN = above of H,(x), if H,(b) < 0,

b, otherwise.

Step 3:

Let n = 1. Compute H2( Y,).

If Hz( Yi) < 0 GO TO STEP 4; otherwise compute H,( Y,).

If HJ( Y,) < 0 GO TO STEP 5; otherwise GO TO STEP 7.

Step 4: Let n = n+l.

If n = N+ 1 GO TO STEP 6; otherwise compute H2( Y,).

If H2( Y.) B 0 GO TO STEP 4; otherwise compute H,( Y,).

If H,( Y,) f 0 GO TO STEP 5; otherwise GO TO STEP 7.

‘The other parts of the algorithm, which essentially reduce the number of pairwix compari- sons (between distributions) to a minimum, arc the same as that in Porter, Wart and Ferguson (1973).

V.S. Bawa, Optimal rules for ordering uncertain prospects 109

Step 5:

Letn = n+l.

Ifn = N+ 1 GO TO STEP 6; otherwise compute H2(X,,).

If HJXJ > 0 compute H,(X,): if H,(X,) < 0 GO TO STEP 5; otherwise GO TO STEP 7.

If H2(Xn) < 0, find Y*, Y,,_, < Y* < X, such that H2( Y*) = 0

and compute H,( Y*).

If H,( Y*) > GO TO STEP 7; otherwise let n = n- 1 and GO TO STEP 4.

Step 6:

Terminate: F dominates G.

Step 7:

Terminate: F does not dominate G.

The above algorithm terminates in a finite number of steps and, by noting that in between successive zero crossings of H,(x), the algorithm checks that the maximum of H,(x) is non-positive, it can be easily verified that when the algorithm terminates and concludes that F dominates G, it is indeed true that H,(x) < 0 for all x E [a, 61. This is summarized in the following:

Theorem 6. For discrete or continuous distributions (with finitely many zero crossings in [a, b]) F and G, the algorithm orders F and G under TSD rule in a finite number ofsteps.

4. Rationale for mean-lower partial variance selection rule

The admissible set of alternatives for the class U, of utility functions (as well as the class U, of decreasing absolute risk averse utility functions when the distributions have equal means) is provided by the Third Order Stochastic Dominance (TSD) rule. With no restrictions on the class of distribution func- tions, the rule involves pairwise comparisons, is complex and requires complete knowledge of the entire distribution function. The efficient algorithm developed in the last section may be used for practical implementation of the TSD rule. This requires that the set of alternatives being considered is prespecified and finite. Thus, for problems like capital budgeting under uncertainty, where the number of alternatives is finite, the algorithm can be used to obtain the admis- sible set. For certain other types of problems, like the portfolio selection

110 V.S. &Iwo, Optimal rules for or&ring uncertain prospects

problem, not only the basic alternatives but all linear combinations of these alternatives are possible alternative choices available to the individual. Thus, even with a finite number of basic alternatives (e.g., n basic securities in the stock market) the number of possible alternatives to be considered is infinite.9 The following theonm provides the result needed to overcome this problem:

Theorem 7. TSD implies dominance under mean-lower partial variance rule.

Proof. By definition, the lower partial variance LPVJx) for distribution function F(x) is given as

LPV,(x) = j: (JJ - x)~ dF@) ,

and hence,

dLPV(x) E LPV,(x) - LPV,(x)

= j: (u-x)2dKti).

Integrating by parts, several times, we obtain

LYLPWX) = (Y-x)~H~(Y) I:--2 j: OI-xW,(y)

= -2 j (v-x)dH2b)

= -2(y-x)H,(y) I:+2 j; dH,Q

= 2H,(x).

In comparing distributions F and G under the mean-LPV(r), rule F dominates G if and only if pF-pG 2 0 and dLPV(r) < 0 (where r E R is a prespecified value), whereas under the TSD rule, F dominates G if and only if pF-pc b 0 and H,(x) < 0 for Vx E R. Since dLPV(x) = 2H,(x), it follows that dominance under the TSD rule implies dominance under the mean-semivariance rule. This completes the proof of Theorem 7.

Theorem 7 provides the rationale for using the mean-lower partial variance selection rule as an approximation to TSD rule and hence the optimal selection rule for the class of decreasing absolute risk averse utility functions. Since it uses a necessary condition for dominance, the mean-LPV(r) admissible set will be

9The literature on stochastic dominance rules make the ad hoc assumption that the number of alternatives is finite, thus limiting the usefulness of the approach. Theorem 6 overcomes this and shows the full generality of this approach.

V.S. Bowo, Optimol rules for ordering wcertoin prospects 111

contained in the TSD admissible set; indeed TSD admissible set contains the inclusion of all the mean-LPV(r) rules. By combining the admissible sets for the mean-LPV(r) rules for several values of r, one gets sufficiently close to the TSD admissible set (as the algorithm of last section indicates, only finitely many r values are needed to obtain the TSD admissible set). Thus, the mean- LPV(r) rule for one prespecified value of r can be directly used as an approxima- tion for the TSD rule. Since it can be easily shown that for the class U, (as we!! as class U4) of utility functions, mean-variance is neither a necessary nor a sufficient condition for dominance,‘O it appears that on theoretical grounds mean- lower partial variance rather than mean-variance should be used as an approxima- tion to the optimalselection rule. Quite remarkably, we have placed no restrictions on the class of distribution functions, and hence mean-lower partial variance can be used as an approximation for the entire class of distributions.

We note that the mean-lower partial variance rule, just as the popular meao- variance rule, allows one to consider a!! possible linear combinations of basic alternatives and can be used to generate the mean-lower partial variance admissible boundary. [See, for example, Hogan and Warren (1972) for com- putational feasibility of this approach.] This overcomes the difficulty of infinitely many alternatives and hence the mean-lower partial variance can be used for the portfolio selection problem as a suitable approximation to the TSD rule and hence the optima! selection rule.

It should also be noted that since the lower partial variance function provides for explicit consideration of asymmetry or skewness of the probability dis- tribution, it is to be preferred to selection rules based on mean, variance and third moment of the distribution function. Indeed, it can be easily shown that selection rules based on the first n moments (n > 3) use neither a necessary nor a sufficient condition for dominance for the class Cl., of utility functions and instead the mean-lower partial variance rule should be used. Thus, at least on theoretical grounds, the approaches recently advocated in Jean (1971) should be abandoned in favor of the mean-lower partial variance rule.

5. Optimal rule for restricted classes of distribution functions

With no restrictions on the class of distribution functions, the optima! selection rule is not known and even the TSD rule used as a reasonable approxi- mation is complex and it would be necessary to use the algorithm to obtain the TSD admissible set. Thus, one might be tempted to consider a certain restricted family of distribution functions, obtain the optima! selection rule for that family of distributions and check if the optima! rule reduces to a simple form involving only certain parameters of the distribution function (e.g., mean and variance).

“‘We leave it to the reader to verify this proposition.

112 VS. Bawa, Optimal rules for ordering uncertain prospecfs

We consider a special case of the general problem of decision making under uncertainty with the number of alternatives considered being finite and pre- specified and with the probability distribution for these alternatives belonging to a restricted class of distributions. rl*l z We consider two classes of distributions Se and FI which include most distributions that have been extensively coo- sidered in the literature and are defined as follows:

Definition

A distribution function F(x), x E [a, 61. belongs to class P,, if F(x) = ${(x - IF)/+} for all x E [a, b] and s, > 0.

Definition

A distribution function F(x), x E [a, b]. belongs to class 6t if F(x) = +(( 4(x) - I,)/+} with 4’(x) > 0 for all x E [u, 61 and sr > 0.

4r,, is the class of distributions characterized by a location parameter (I) and a scale parameter (s), whereas 9, is the class of distributions in which a mooo- tonic transformation (q5(*)) of the random variables belongs to the location and scale parameter family 9,. Distributions like normal, r-distribution, exponen- tial, uniform and double exponential all belong to the class Fe while the log- normal distribution, which [in view of empirical results of Cootner (1964) and Lintner (1972)] has a special importance in optimal portfolio selection problem, belongs to the class 9,.

We are interested in comparing distributions Fand G that belong to class .aC,,. Thus, by definition

G(x) = t)

and hence

= I,+s,A ;

“The results of this section would thus be useful for problems like capital budgeting under uncertainty. For portfolio selection problem, the results serve as useful approximations in that if one is willing to assume that the distribution of returns on the linear combinations of the basic securities ‘approximately’ belongs to the same restricted class of distributions as the basic securities, then the optima1 rule obtained herein can be used as a reasonable approxima- tion to the true optimal.

“The results obtained in this section are also useful in deriving the optimal rule for the port- folio selection problem wherein all linear combinations of basic alternatives are also considered. These results will be presented in a forthcoming paper ‘On Optimal Portfolio Selection Rules’.

V.S. Bawa. Oprinral rules for ordering uncertain prospccfs 113

similarly EGX = t,+s,A.

where A =jyWCv)dp d P e en s u on the function $ bit is independent of the

location and scale parameters. We also note that for F, G c 5,,, if we let

and

x = max {.v 1 $(x) = 0),

2 = min{xj I(/(x) = 1).

then in comparing distributions F and G, the range [u, 61 is quite naturally defined as the values of the random variables for which either F(x) or G(x) (or both) are between zero and one, i.e.,

and 0 = min {IF+s,:,,I,+sG.\-},

6 = max {I, + s& I, + s,.C) ,

[as beyond these values H,(X) E F(x)-G(x) is identically zero and hence does not effect the analysis]. It should be noted that a may be -co or finite and similarly 6(> u) may be finite or co. It appears to be implicitly assumed in the literature [see for example Hanoch and Levy (1969, Theorem 4)] that [a, b] s [-co, co] although it need not be necessarily true. We allow for all the possi- bilities and with x, given (for the case sr # sc) as the solution to (xr -IF)/rF =

(x1 - Ml%, i.e., x1 E (I,s, -ICrF)/(rc-rF), the optimal selection rule is given

by the following:

Theorem 8. For any two distributions F and G belonging to class 9,, F is preferred to G for all utility functions in 11, if and only i/:

(A) For sF = sG,

(B) For sF # sG,

I. for u -z x, c b

(9 l,+s,A z I,+s,A

and

(ii) SF < SG;

114 Y.S. Bawa, Optimd rules for or&ring uncertain prospects

II. for x1 2 b

Proof. For F, G c .9,, ,

H,(x) = F(x)-G(x)

Thus, H,(x) 2 0 according as

hence, for s, < sG, H,(x) $ 0 as x g xi , for s, > sG, H,(x) f 0 as x $g x1 , andfors, = sG, H,(x) Z$ OVxasl, s I,.

Given s, = so, I, > I, satisfies necessary and sufficient conditions for dominance of Theorem 5. This completes proof of Case A.

Case B: sF # sG

Fora<x, cb,ifs,>s,,H,(x)2Ofora<x<x,andhenceH,(x)~O for a < x < xi. Thus F cannot dominate G for s, > sG. However, ifs, < sG, H,(x) < 0 and hence H,(x) < 0 for u 6 x < x, while for x 2 x,, Hz(x) is an increasing function of x.

Also if (i) holds, then EFX 2 EGX, i.e., H,(b) z E,X-E,X < 0. This implies that Hz(x) < 0 for all x, xi ,< x ,< b; hence H,(x) 6 0 for all x E [a, b]. As (i) and (ii) satisfy the condition of Theorem 5, this completes the proof of Case B-I.

For x1 2 b, H,(x) > 0 for all x E [a, b) if sF > sc, whereas H,(x) < 0 for all x E [u, 6) if sF < sG. Thus sF < sG satisfies the necessary and sufficient condition for dominance of Theorem 5. This completes proof of Case B-II.

For xi < a, H,(x) > 0 for all x E (a, b] if sF < sc, whereas H,(x) < 0 for all x E (a, b] if sF > So. Thus sF > So satisfies necessary and sufficient conditions for dominance of Theorem 5. This completes proof of Case B-III.

Similarly, if F, G c 9,, i.e.,

VS. Bawa, Opfinud rules for or&ring uncertain prospecls 11s

F(x) = $ fP(-+~F ( 1 - for all x E [u, b], SF

forallxE[u,b],

and if we let, for the case sp # sG, x2 denote the solution to

+(x2blF = ddx2)-k , SF SG

i.e.,

x2 = 6 , then the optimal selection rule is given by the following:

Theorem 9. For any IWO distributions F and G rhar belong lo class 9,) F is preferred to G for all utility functions in U, if and on Iy i/:

(A) For SF = SG,

1, > I,.

(B) For SF # SG,

I. for a < x2 < b

W ErX 2 EoX,

and

(ii) SF < s,;

I.I. for x2 z b

sp < s,;

III. for x2 < u

sr > so.

(The proof of Theorem 9 i’s similar to that of Theorem 8 and is omitted.)

116 VS. Bawa, Optimal rules for ordering uncertain prospects

Similarly, if we consider a subclass F2 of secontaining symmetric distribu- tions, i.e.,

.%s = {FE 9,I F is symmetric},

thensinceF,Gc.%,,A= 0, ErX = l,, EoX = I, and for distributions with finite variances, s: = B*VarrX and si = B.Var,X (where B > 0), Theorem 8 reduces to the following:

Theorem 10. For any two distributions Fand G withjinite variances that belong to class 9 2 , F is preferred to G for aN utility functions in U4 i/and only lfz

(A) For Var,X = Var,X,

E,X > EoX.

(B) For Var,X # Var,X,

I. fora-zx, <b

(0 ErX 2 E,X,

and

(ii) Var,X c Var,X;

II. for x1 2 b

Var,X < Var,X;

III. for xi G a

Var,X > Var,X.

It is interesting to note that for the restricted classes 9,, 9, and 9, of two parameter distributions, the optimal selection rule naturally reduces to com- parison of two parameters but these parameters are neither the mean and variance nor the location (I) and scale (s) parameters that characterize the restricted classes of distributions. Instead, the selection rule involves comparison of the mean and the scale parameter. A larger mean is a necessary condirion for dominance for the entire class of distributions (Theorem 5);’ ’ thus, for a restricted

“The stochastic dominance rules (FSD, SSD and TSD -as well as higher order rules that may be obtained by placing further restrictions on fourth and higher moments of the utility functions) all involve a comparison of the mean. Thus larger mean may be taken to be a universal condition for dominance.

VS. Bawa, Optimal rules for or&ring uncertain prospects 117

class of distributions (.%,,, Fl and F,) the mean will obviously be one of the two parameters being compared. However, as the scale parameter (and not the variance) is the natural measure of dispersion, it is quite appropriate that it be the other parameter that is compared in the optimal selection rule. Even for the class 9z of symmetric distributions and with finite variances, when variance is indeed proportional to the scale parameter and hence may be used as the second parameter, the optimal selection rule reduces to the mean-variance rule only if a < x1 < b. This condition (Case B-I of Theorem 10) does not hold necessarily for class 4Fz (for example, it always holds for normal distributions but need not for uniform distributions); hence even for the (unrealistic) case of symmetric distributions characterized by location and scale parameter that have been used in the literature to rationalize the mean-variance rule, the mean-variance rule is not necessarily the optimal selection rule.

For lognormal distributions (that belong to class F1), we note that a < x, < b obtains and since the scale parameter is the logarithmic variance, Theorem 9 reduces to the following:

Theorem II. For any two lognormal distributions, one is preferred to another for all utility functions in U, if and only if it has at least as large a mean (in natural units) and a smaller logarithmic variance.

In view of the empirical study of Lintner (1972) (that the portfolio of log- normal securities is approximately lognormal) this result is of special significance in portfolio theory. It implies that in order to generate the admissible set of portfolios, instead of the popular mean-variance rule, the mean-logarithmic variance rule should be used.

Finally, we consider the class of gamma distributions which is a two-parameter distribution distinctly different from .%O and f, . This class is of importance as in certain situations, the gamma distribution can be used, with appropriate choice of parameters, to reasonably approximate empirical distributions. The density function for a gamma distribution is

e-‘“~“~-’

f(n) * x 2 0,

and thus the parameters (A, n) (usually called scale and shape parameters respectively in the statistics literature) characterize the distribution. For this class of distributions, the optimal selection rule reduces to the following:

Theorem 12. For any two gamma distributions F and G, with (A,, nF) and

(Ai. no) as the respective parameters, F is preferred to G for all utility functions in U, ifand only if

118 VS. Bawa, Optimal rules for or&ring uncertain prospects

with at least one inequality holding strictly.

The proof of this theorem follows simply by noting that n,, 2 no and E,,X( E n&J z EoX(= n&o) with at least one inequality holding strictly satisfies necessary and sufficient conditions of Theorem 5.

The different classes of distribution functions considered in this paper have the common property that any two distributions F, G belonging to the restricted class cross at most once. Thus we have indeed proved the following:

Theorem 13. For any two distributions F and G that cross at most once F is preferred to G for all utility functions in U, if and only ifpp 2 po and F initially lies below G.

We note that in general it will be hard to check if F initially lies below G. However, for the important classes of distributions considered earlier in this section, this is easily identifiable by looking at the appropriate parameter that characterizes the dispersion of the distribution.

6. Conclodlng remarks

We have shown that the Third Order Stochastic Dominance (TSD) rule is the optimal selection rule for ordering uncertain prospects with equal means which minimizes the admissible set of alternatives by discarding from among a given set of alternatives, those that are inferior (for every decreasing absolute risk averse utility function) to a member of the given set of alternatives. It was also shown that in ordering uncertain prospects with unequal means, no known selection rule uses both necessary and sufficient conditions for dominance and TSD rule is a reasonable approximation to the optimal rule. In view of the common acceptance among economists of decreasing absolute risk aversion and almost universal reluctance to place restrictions on higher derivatives of utility functions, the smallest acceptable admissible set will be given by the TSD rule.

The TSD rule is complex as it involves pairwise comparison of the means and the lower partial variance function of the probability distributions (which has to be determined over the entire range of the distribution). We developed an efficient algorithm which can be used to obtain the admissible set for discrete as well as continuous probability distributions (of returns on each alternative). We showed that the two parameter mean-lower partial variance rule (with the lower partial variance computed at any one point in the range), which uses a

VS. Bawa, Oprimal rules for ordering unccrrain prospeers 119

necessary condition for dominance may be used as a reasonable approximation to the TSD and hence optimal selection rule for the entire class of distribution functions. For the class of decreasing absolute risk averse utility functions, and with no restrictions on the distribution functions, the mean-lower partial variance rule is thus to be preferred (as an approximation to the optimal selection rule), at least on theoretical grounds, to the popular mean-variance rule or the recently proposed mean-variance-skewness preference rules [see, e.g., Jean (1971)] which use neither necessary nor sufficient conditions for dominance. We hope that this will provide a strong impetus for further research in the development of efficient algorithms for the mean-lower partial variance rule.

We considered certain restricted but important classes of two parameter distributions that include most distributions of practical interest in economics and finance and obtained the optima1 rule for decreasing absolute risk averse utility functions. We also showed that the optimal rule reduces to simple manageable forms, which involves naturally only two parameters, one of which is always the mean but the other parameter is the appropriate measure of dispersion. This second parameter is usually the scale parameter which, except for the unrealistic case of symmetric distributions, is not the variance. For the important case of lognormal distributions, the optimal rule reduces to one where at least as large a mean and smaller logarithmic variance are the necessary and sufficient conditions for dominance. This result along with the empirical linding of Lintner (1972) (that linear combination of jointly lognormally distributed returns is approximately lognormal) allows us to analytically obtain the efficient set for the portfolio problem and the results will be presented in a later paper. Our results for the restricted classes of distribution functions considered also show that the second parameter characterize uncertainty and is the mean preserving spread parameter of Rothschild and Stiglitz (1970). Hence, for a large and important class of distribution functions, one can easily obtain unambiguous economic effects of increasing risk and these results will be presented in a later paper.

References

Arrow, K.J.. 1971, Theory of risk aversion, in: Essays in the theory of risk bearing, ch. 3 (Markham, Chicago, Ill.).

Borch. K., 1969. A note on uncertainty and indifference curves, Review of Economic Studies 36, I-4.

Cootner, P.H.. 1964, The random character of stock market prices (M.I.T. Press, Cambridge, Mass.).

Feldstein. MS, 1969, Mean-variance analysis in the theory of liquidity preference and port- folio selection, Review of Economic Studies 36, S-12.

Fishburn, P.C.. 1964. Decision and value theory (Wiley, New York). Hadar. J. and W.R. Russell, 1969, Rules for ordering uncertain prospects, American Economic

Review 59.25-34. Hadar, J. and W.R. Russell, 1971, Stochastic dominance and diversification, Journal of

EconomicTheory 3,288-30X

120 V.S. Bawa, Optlmal rides for eraking uncertain prospects

Hakansson, N.. 1971. Capital growth and the mean-variance approach to portfolio selection, Journal of Fiiancial and Quantitative Analysis 6.517-557.

Hakansson. N., 1972, Mean&iance analysis in a finite world, Journal of Financial and Quantitative Analysis 5.1873-1880.

Hanoch, G. and H. Levy, 1969, The eflicicncy analysis of choices involving risk, Review of Economic Studies 36.335-346.

Hicks, J.R.. 1962. Liquidity, Economic Journal 72.787-802. Hirshleifer, J., 1970. Investment, interest and capital (Prentice Hall, E&wood cliffs, N.J.). Hogan, W.W. and J.M. Warren. 1972. Computation of efIicicnt boundary in the E-S portfolio

selection model, Journal of Financial and Quantitative Analysis 7.1881-1896. Jean, W.H.. 1971, The extension of portfolio analysis to three or more parameters, Journal

of Financial and Quantitative Analysis 5.505-515. Levy, H. and G. Hanoch, 1970, Relative effectiveness of efficiency criteria for portfolio

selection. Journal of Financial and Quantitative Analysis 5.63-76. Levy. H. and M. Sarnat, 1970, Alternative efficiency criteria: An empirical analysis, Journal of

Finance 25.1153-I 158. Levy. H. and M. Samat. 1971. A note on portfolio selection and investors wealth, Journal of

Financial and Quantitative Analysis 6.639-642. Lintner, J., 1965, &urity prices, risk and maximal gain from diversification, Journal of

Finance 30.587-615. Lintner, J., 1972. Equilibrium in a random walk and lognormal securities market, Harvard

Institute of Economic Research, Discussion Paper no. 235 (Harvard University, Cambridge, Mass.).

Mao, J.C.T., 1970, Models of capital budgeting. E-V vs. E-S, Journal of Financial and Quanti- tative Analysis 4.657-675.

Markowitz, H., 1952, Portfolio selection. Journal of Finance 7,77-91. Markowitz, H., 1970, Portfolio selection: Efficient diversification of investments (Wiley,

New York). Mossin, J,, 1966, Equilibrium in a capital market, Econometrica 34.768-783. Potter. R.B. and J.E. Gaumnitz, 1972, Stochastic dominance vs. mean-variance oortfolio

analysis: An empirical evaluation, American Economic Review 62,438-446. - Porter. RB.. J.R. Wart and D.L. Ferauson. 1973. Efficient algorithms for conductinn stochastic

dominance tests on large numb&s of.portfolios, Journal of Financial and Quantitative Analysis 8,71-81.

Pratt, J.W., 1964. Risk aversion in the small and in the large. Econometrica 32.122-136. Quirk, J.P. and R. Sapasnik. 1962, Admissibility and measurable utility functions, Review of

Economic Studies 29.140-146. Rothschild, M. and J.E. Stiglitz. 1970, Increasing risk I: A definition, Journal of Economic

Theory 2, 225-243. Rothschild, M. and J.E. Stigiitz. 1971, Increasing risk II: Its economic consequences, Journal

of Economic Theory 3.6684. Samuelson, P.A., 1970, The fundamental approximation theorem of portfolio analysis in

terms of means, variances and higher moments, Review of Economic Studies 37, 537-542. Sharps, W.F., 1964, Capital asset prices: A theory of market equilibrium under conditions of

risk, Journal of Finance 29,425-442. Stigiitz, J.E., 1970, Review of some aspects of theory of risk bearing by K.J. Arrow,

Econometrica 38. Tobin, J., 1958. Liquidity preference as behavior towards risk, Review of Economic Studies

25.65-86. Tobin, J., 1965, The theory of portfolio selection. in: F.H. Hahn and F.P.R. Brcchling, eds..

The theory of interest rates (MacMillan, London). Tsiang, S.C., 1972. The rationale of the mean-standard deviation analysis, skewness

preference and the demand for money, American Economic Review 62,354-371. Vickson, R.G., forthcoming a, Stochastic dominance for decreasing absolute risk aversion,

Journal of Financial and Quantitative Analysis. Vickson, R.G., forthcoming b, Stochastic dominance tests for decreasing absolute risk

aversion I: Discrete random variables, Management Science.

VS. Bawa, Optimal rules /or ordrrhg uncertain prospects I21

Von Neumann, 1. and 0. Morgcnstem, 1967, Theory of games and economic behavior (Wiley, New York).

Whitmore, G.A., 1970, Third order stochastic dominana, American Economic Review 50, 457459.