European Journal of Operational Research 200 (2010) 893–900

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European Journal of Operational Research

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Interfaces with Other Disciplines

Gains from diversification on convex combinations: A majorizationand stochastic dominance approach

Martin Egozcue a, Wing-Keung Wong b,*

a Department of Economics FCS, Universidad de la Republica del Uruguay, Uruguayb Department of Economics and Institute for Computational Mathematics Hong Kong Baptist University WLB, Shaw Campus, Kowloon Tong, Hong Kong, Hong Kong

a r t i c l e i n f o

Article history:Received 22 March 2008Accepted 5 January 2009Available online 16 January 2009

Keywords:MajorizationStochastic dominancePortfolio selectionExpected utilityDiversification

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.01.007

* Corresponding author. Tel.: +852 3411 7542; fax:E-mail address: awong@hkbu.edu.hk (W.-K. Wong

1 To enhance the mean–variance portfolio selectio(2008) apply the technique of the repeated measures deSharpe ratio statistic to test the hypothesis of the equawhereas Bai et al. (2009, forthcoming) develop new bofor the optimal return and its asset allocation andcorrected estimates are proportionally consistent with

2 This rule provides an excellent approximation to aunder some restrictions on the range of return; see Lemore information.

a b s t r a c t

By incorporating both majorization theory and stochastic dominance theory, this paper presents a gen-eral theory and a unifying framework for determining the diversification preferences of risk-averse inves-tors and conditions under which they would unanimously judge a particular asset to be superior. Inparticular, we develop a theory for comparing the preferences of different convex combinations of assetsthat characterize a portfolio to give higher expected utility by second-order stochastic dominance. Ourfindings also provide an additional methodology for determining the second-order stochastic dominanceefficient set.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The pioneer work of Markowitz (1952) and Tobin (1958) on themean–variance (MV) portfolio selection is a milestone in modernfinance theory for optimal portfolio construction, asset allocation,and investment diversification.1 In the procedure, investors respondto the uncertainty of an investment by selecting a portfolio thatmaximizes anticipated profit, subject to achieving a specified levelof calculated risk or, equivalently, minimizes variance, subject toobtaining a predetermined level of expected gain. However, the dis-advantage of using the MV criterion2 is that it is derived by assumingthe von-Neumann and Morgenstern (1944) quadratic utility functionand returns being examined are required to be normally distributedor elliptic distributed (Feldstein, 1969; Hanoch and Levy, 1969; Berk,1997).

To circumvent the limitations of the MV criterion, academicsrecommend adopting the stochastic dominance (SD) approach,which can be used in constructing a general framework for the

ll rights reserved.

+852 3411 5580.).n, recently Leung and Wongsign to develop a multivariatelity of multiple Sharpe ratios,otstrap-corrected estimationsprove that these bootstrap-their theoretic counterparts.

ny risk-averse utility functionvy and Markowitz (1979) for

analysis of choice and problems of diversification for risk-averseinvestors under uncertainty without any restriction on the distri-bution of the assets being analyzed and without imposing the qua-dratic utility function assumption on investors. Academics haveregarded the SD approach as one of the most useful tools for rank-ing uncertain investment prospects or portfolios because theirrankings have been theoretically justified to be equal to the rank-ings of the corresponding expected utilities. Hanoch and Levy(1969) link stochastic dominance to a class of utility functionsfor non-satiable and risk-averse investors. Hadar and Russell(1971) develop the analysis using the concept of stochastic domi-nance and its applicability to choices under conditions of uncer-tainty, whereas Tesfatsion (1976) further extends their results fordiversification using a stochastic dominance approach to maximiz-ing investors’ expected utilities. Readers may refer to Ortobelli Loz-za (2001) and Post (2008) for an exhaustive overview of otheruseful results along these lines.

By combining majorization theory with stochastic dominancetheory, we extend the theory by developing some new results forchoice in portfolio diversification. To specify, we establish somenew theorems to determine the preferences of risk-averse inves-tors among different diversified portfolios and show the conditionsunder which all risk-averse investors would prefer more diversi-fied portfolios to less diversified ones. Our findings are importantbecause they permit investors to specialize the rankings, by sec-ond-order stochastic dominance, from among a wide range of con-vex combinations of assets, and especially because they haveimplications concerning the weights of allocations. Our findingsenable investors to make choices about allocations from their

4 We note that if u 2 U2;u is Fréchet differentiable; see, for example, Machina

894 M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900

capital that result in higher expected utilities. This was one of thetopics that Levy (2006) suggested for future research.

In addition, our findings could also be used in determining thesecond-order stochastic dominance efficient set. Traditionally,there are two decision stages in determining the efficient set; seeBawa et al. (1985). In the first stage, the initial screening of pros-pects or investments is accomplished by partitioning the feasibleset into the efficient and inefficient sets using a stochastic domi-nance relation.3 At the second stage, Fishburn’s (1974) concept ofconvex stochastic dominance (CSD) is used to eliminate elementsthat are not optimal in the sense of CSD. Alternatives that are dom-inated by convex combinations of other portfolios will be eliminatedfrom the efficient set as they are classified to be inefficient. In thiscontext, our findings allow investors to rank convex combinationsof assets by majorization order, which, in turn, implies the rankingsof their preferences of second-order stochastic dominance. Thus, ourfindings assist investors in determining the second-order stochasticdominance efficient set.

Our paper is organized as follows: We begin by introducing def-initions and notations and stating some basic properties for themajorization theory and stochastic dominance theory. Section 3presents our findings on the preferences for risk-averse investorsin their choices of diversified portfolios, and Section 4 offers someconclusions.

2. Definitions and notations

In this section, we will first introduce some notations and well-known properties in stochastic dominance theory and majorizationtheory that we will use in this paper. Considering an economicagent with unitary initial capital, in this paper we study the sin-gle-period portfolio selection for risk-averse investors to allocatetheir wealth to the nðn > 1Þ risks without short selling in orderto maximize their expected utilities from the resulting final wealth.Let random variable X be an (excess) return of an asset or prospect.If there are n assets ~Xn ¼ ðX1; . . . ;XnÞ0, a portfolio of ~Xn withoutshort selling is defined by a convex combination, k

!n0X!

n, of the n as-sets ~Xn for any k

!n 2 S0

n where

S0n ¼ ðs1; s2; . . . ; snÞ0 2 Rn : 0 6 si 6 1 for any i;

Xn

i¼1

si ¼ 1

( )ð1Þ

in which R is the set of real numbers. The ith element of k!

n is theweight of the portfolio allocation on the ith asset of return Xi. Aportfolio will be equivalent to return on asset i if si ¼ 1 and sj ¼ 0for all j – i. It is diversified if there exists i such that 0 < si < 1,and is completely diversified if 0 < si < 1 for all i ¼ 1;2; . . . ;n. Aswe study the properties of majorization in this context, without lossof generality, we further assume that Sn satisfies:

Sn ¼ ðs1; s2; . . . ; snÞ0 2 Rn : 1 P s1 P s2 P � � �P sn P 0;Xn

i¼1

si ¼ 1

( ):

ð2Þ

We note that the condition ofPn

i¼1si ¼ 1 is not necessary. It could beany positive number in most of the findings in this paper. For con-venience, we set

Pni¼1si ¼ 1 so that the sum of all relative weights is

equal to one. In this paper, we will mainly study the properties ofmajorization by considering k

!n 2 Sn instead of S0

n.Suppose that an investor has utility function u, and his/her ex-

pected utility for the portfolio k!0nX!

n is E½uðk!0nX!

n�. In this context, westudy only the behavior of non-satiable and risk-averse investors

3 Readers may refer to Broll et al. (2006), Wong (2006, 2007) and Wong and Chan(2008) and the references there for more information.

whose utility functions belong to the following classes (see, e.g.,Ingersoll, 1987):

Definition 1. 4U2 is the set of the utility functions, u, defined in R

such that:

U2 ¼ fu : ð�1ÞiuðiÞ 6 0; i ¼ 1;2g;

where uðiÞ is the ith derivative of the utility function u, and the ex-tended set of utility functions is:

UE2 ¼ fu : u is increasing and concaveg:

We note that in the above definition, ‘‘increasing” means ‘‘non-decreasing”. It is known (e.g., see Theorem 11C in Roberts and Var-berg, 1973) that u in UE

2 is differentiable almost everywhere and itsderivative is continuous almost everywhere. We note that the the-ory can be easily extended to satisfy utilities defined in Definition 1to be non-differentiable.5

There are many ways to order the elements in Sn. A popular oneis to order them by majorization; see, for example, Hardy et al.(1934) and Marshall and Olkin (1979), as stated in the following:

Definition 2. Let ~an;~bn 2 Sn in which Sn is defined in (2).~bn is saidto majorize ~an, denoted by ~bn�M~an, if

Pki¼1bi P

Pki¼1ai, for all

k ¼ 1;2; . . . ;n.

Majorization is a partial order among vectors of real numbers.We illustrate it in the following example:

Example 1. 35 ;

15 ;

15

� �0�M25 ;

25 ;

15

� �0 because 35 >

25 and 3

5þ 15 P 2

5þ 25.

Vectors that can be ordered by majorization have some inter-esting properties. One of them is a Dalton Pigou transfer, as de-scribed in the following definition:

Definition 3. 6For any ~an;~bn 2 Sn;~an is said to be obtained from ~bn

by applying a single Dalton (Pigou) transfer, denoted by ~bn!d~an, if

there exist h and kð1 6 h < k 6 nÞ such that ai ¼ bi for anyi – h; k; ah ¼ bh � �; and ak ¼ bk þ � with � > 0.

For instance, consider the above example that ~a3 ¼ 25 ;

25 ;

15

� �0 and~b3 ¼ 3

5 ;15 ;

15

� �0, As a1 ¼ b1 � 15 ;a2 ¼ b2 þ 1

5, and a3 ¼ b3 ¼ 15, from Def-

inition 3, we said that~a3 can be obtained from~b3 by applying a sin-gle Dalton transfer by setting a1 ¼ b1 � 1

5 and a2 ¼ b2 þ 15. Thus, we

write ~b3!d~a3.

In this example, we also notice that ~b3 majorizes ~a3. One maywonder whether there is any relationship between majorizationand a Dalton transfer. To answer this question, we have the follow-ing theorem:

Theorem 1. Let ~an; ~bn 2 Sn;~bn�M~an if and only if ~an can be obtainedfrom ~bn by applying a finite number of Dalton transfers, denoted by~bn!

D~an.

Readers may refer to Appendix 1 for the proof of Theorem 1.This theorem states that if~bn majorizes~an, then~an can be obtainedfrom~bn by applying a finite number of single Dalton transfers, andvice versa. We illustrate the procedure in the following example:

Example 2. Consider 13 ;

13 ;

13

� �0 and 45 ;

15 ;0

� �0. As 13 ;

13 ;

13

� �0 is majorizedby 4

5 ;15 ;0

� �0, from Theorem 1, we know that 13 ;

13 ;

13

� �0 can beobtained by applying a finite number of single Dalton transfers on

45 ;

15 ;0

� �0. This could be done, for example, by setting45 ;

15 ;0

� �0 !d 23 ;

13 ;0

� �0 !d 13 ;

13 ;

13

� �0. That is, by simply first transferring

(1982) for more information.5 Readers may refer to Wong and Ma (2008) and the references there for more

information. In this paper, we will skip the discussion of non-differentiable utilities.6 Some academics suggest the reverse direction for the definition of a Dalton Pigou

transfer. In this paper, we follow Ok and Kranich (1998) for the definition.

7 ~Xn ¼ ðX1; . . . ;XnÞ0 ;~Yn ¼ ðY1; . . . ;YnÞ0 ;~Fn ¼ ðF1; . . . ; FnÞ0 a n d ~Gn ¼ ðG1; . . . ;GnÞ0

where Fi and Gi are the distribution functions of Xi and Yi , respectively.

M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900 895

215 from the first entry to the second entry of 4

5 ;15 ;0

� �0 and thentransferring 1

3from its first entry to its third entry to obtain 13 ;

13 ;

13

� �0.Thus, we write 4

5 ;15 ;0

� �0 !D 13 ;

13 ;

13

� �0.In this paper, we link a Dalton transfer and majorization order

to stochastic dominance. The theory of stochastic dominance isimportant in decision making, since the rankings of assets or port-folio preferences have been proved to be equivalent to the rankingsof their corresponding expected utilities. Before we discuss sto-chastic dominance, we define some notations as follows. Let R bethe set of extended real numbers. Suppose that X ¼ ½a; b� is a subsetof R in which a and b can be finite or infinite. Let B be the Borel r-field of X and l be a measure on ðX;BÞ. The function F of the mea-sure l is defined as:

FðxÞ ¼ lða; x� for all x 2 X: ð3Þ

The function F is called a (probability) distribution function and l iscalled a probability measure if lðXÞ ¼ 1. By the basic probabilitytheory, for any random variable X and for probability measure P,there exists a unique induced probability measure l on ðX;BÞ andthe probability distribution function F such that F satisfies (3) and

lðBÞ ¼ PðX�1ðBÞÞ ¼ PðX 2 BÞ for any B 2 B:

An integral written in the form ofR

A f ðtÞdlðtÞ orR

A f ðtÞdFðtÞ is aLebesgue–Stieltjes integral for any integrable function f ðtÞ. If theintegral has the same value for any set A that is equal toðc; d�; ½c;dÞ or ½c;d�, then we use the notation

R dc f ðtÞdlðtÞ instead.

In addition, if l is a Borel measure with lðc; d� ¼ d� c for anyc < d, then we write the integral as

R dc f ðtÞdt. The Lebesgue–Stieltjes

integralR d

c f ðtÞdt is equal to the Riemann integral if f is bounded andcontinuous almost everywhere on ½c; d�; see Theorem 1.7.1 in Ash(1972).

We consider random variables, denoted by X;Y; . . . defined onX. The probability distribution functions of X and Y are F and G,respectively. Throughout this paper, all functions are assumed tobe measurable, and all integrals and expectations are implicitly as-sumed to exist and to be finite. We next define the second-orderstochastic dominance that will be useful for risk-averse investorsin making their decision as follows:

Definition 4. Given two random variables X and Y with F and G astheir respective probability distribution functions defined on½a; b�;X dominates Y and F dominates G in the sense of SSD,denoted by X�2Y or F�2G, if and only if

R xa FðyÞdy 6

R xa GðyÞdy for

each x in ½a; b�, where SSD stands for second-order stochasticdominance.

An individual chooses between F and G in accordance with aconsistent set of preferences satisfying the von-Neumann and Mor-genstern (1944) consistency properties. Accordingly, F is preferredto G, or equivalently, X is preferred to Y for all utility functions u if

DEu � E½uðXÞ� � E½uðYÞ�P 0; ð4Þ

where E½uðXÞ� �R b

a udF and E½uðYÞ� �R b

a udG.

3. The theory

In this section, we will develop the theory of diversification forrisk-averse investors to make comparisons among different portfo-lios by incorporating both majorization theory and stochastic dom-inance theory.

We first discuss the stochastic dominance theory for randomvariables, and non-negative combinations, or equivalently convexcombinations, of random variables. Random variables X;Y ; . . . canbe regarded as returns on individual prospects, and convex combi-nations of random variables can be regarded as the returns on theportfolios for different prospects. Hence, stochastic dominance for

the random variables can be applied to examine preferences of dif-ferent prospects and the preferences of different portfolios. Thetheory of stochastic dominance is important because it is equiva-lent to the theory of utility maximization as stated in the followingtheorem:

Theorem 2. Let X and Y be random variables with probabilitydistribution functions F and G, respectively. Suppose u is a utilityfunction. Then,

X �2 Y or; equivalently; F �2 G if and only if E½uðXÞ�P E½uðYÞ�

for any u 2 UE2.

We note that Hanoch and Levy (1969) and Hadar and Russell(1969) first prove this theorem. Readers may refer to their papersfor the proof of the theorem.

Hadar and Russell (1971) first investigate the diversificationproblem for the independent and identically distributed case in abivariate setting. They verify that

E½uð a2�!0~X2Þ�P E½uð b2

�!0~X2Þ�; ð5Þ

whenever ja1 � a2j 6 jb1 � b2j where ~a2 ¼ ða1;a2Þ0 and~b2 ¼ ðb1; b2Þ

0 2 S2;u 2 U2, and ~X2 ¼ ðX1;X2Þ0 in which X1 and X2 arenon-negative independent and identically distributed randomvariables.

We note that for any pair of random variables X and Y, the state-ments X�2Y and F�2G are equivalent. But for n > 1, the statements~a0n~Xn�2

~b0n~Yn and ~a0n~Fn�2

~b0n~Gn

7 are different because the distributionfunctions of ~a0n~Xn and ~b0n~Yn are different from ~a0n~Fn and ~b0n~Gn, respec-tively. Thus, we cannot apply the convex stochastic dominance the-orems obtained in Fishburn (1974), Dekel (1989), and Wong and Li(1999) to the convex combinations of random variables. To investi-gate the properties of the convex combinations of random variables,Hadar and Russell (1971) and Tesfatsion (1976) first study theinvariance property of the stochastic dominance for the convex com-binations of random variables in a bivariate setting, whereas Li andWong (1999) further extend their work by comparing two sets ofindependent variables in a multivariate setting as shown in the fol-lowing theorem:

Theorem 3. For n > 1, let ~Xn ¼ ðX1; . . . ;XnÞ0 and Y!

n ¼ ðY1; . . . ;YnÞ0where fX1; . . . ;Xng and fY1; . . . ;Yng be two sets of independent andidentically distributed random variables. Then, Xi�2Yi for anyi ¼ 1;2; . . . ;n if and only if ~a0n~Xn�2~a0nY

!n for any ~an 2 S0

n.

One may refer to Li and Wong (1999) for the proof. We providethe following example to illustrate Theorem 3:

Example 3. Consider Yi to be independently distributed with zeromean and unit variance, and Xi ¼ aþ bYi with a P 0 and 0 < b 6 1for any i ¼ 1; . . . ;n. Applying Theorem 8 in Li and Wong (1999), onecould show that Xi�2Yi for any i ¼ 1;2; . . . ;n. Thereafter, applyingTheorem 3, one could obtain ~a0n~Xn�2~a0nY

!n for any ~an 2 S0

n.

Let 1n

!� �¼ 1

n ; . . . ; 1n

� �0. Hanoch and Levy (1969), Hadar and

Russell (1971), and Tesfatsion (1976) verify in the bivariate case

that, for any 12

!� �and a

!2 2 S0

2, if X1 and X2 are independent and

identically distributed, then

12

!0@1A0~X2�2a

!02~X2�2Xi for i ¼ 1;2:

896 M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900

Li and Wong (1999) generalize the results further to the case of ran-dom variables in a multivariate setting as shown in the followingtheorem:

Theorem 4. For n > 1, let ~Xn ¼ ðX1; . . . ;XnÞ0 where X1; . . . ;Xn areindependent and identically distributed, then, for any i ¼ 1;2; . . . ;n

and for any 1n

!� �; a!

n 2 S0n,

1n

!0@1A0~Xn�2~a0n~Xn�2Xi:

Readers may refer to Li and Wong (1999) for the proof of Theo-rem 4. We note that Samuelson (1967) has proved that 1

n

� �0~Xn at-tains maximum among ~a0n~Xn for any iid non-negative Xi. Thistheorem verifies the optimality of diversification that the maximalexpected utility will be achieved in an equally weighted portfolioof independent and identically distributed assets. The manifesta-tion of the generality of the theorem is that it places very weakrestrictions on the weights. This theorem implies that for indepen-dent and identically distributed assets, risk-averse investors willprefer the equally weighted portfolio to any convex combinationportfolio, which, in turn, is preferred to any individual asset.

Nonetheless, Theorem 4 does not permit investors to comparethe preferences of other different convex combinations of randomvariables. So far, the comparison of the preferences of differentconvex combinations of random variables has not been well stud-ied in the literature. To bridge the gap in the literature, in this pa-per we will develop a theory to compare the preferences ofdifferent convex combinations of random variables. We first exam-ine the situation in which the underlying assets are independentand identically distributed. To make our contribution clear, we firststate the situation in a bivariate setting as shown in the followingtheorem:

Theorem 5. Let ~a2;~b2 2 S2 and ~X2 ¼ ðX1;X2Þ0 where X1 and X2 areindependent and identically distributed. Then,

~b2�M~a2 if and only if ~a02~X2�2~b02~X2:

The proof of Theorem 5 is in the appendix. This theorem pro-vides a methodology for investors to make comparisons among awide range of portfolios so that they make better choices in theirinvestment decisions, especially the implications concerning theweights of allocations.

It is interesting to note from Theorem 1 that, if ~a2 is majorizedby ~b2;~a2 can be obtained from vector ~b2 by applying Dalton trans-fer(s) and vice versa. Thus, we could incorporate Theorem 1 intoTheorem 5 to obtain the following corollary:

Corollary 6. Let ~a2;~b2 2 S2 and ~X2 ¼ ðX1;X2Þ0 where X1 and X2 areindependent and identically distributed. Then,

~b2!D~a2 if and only if ~a02~X2�2

~b02~X2:

Theorem 5 and Corollary 6 extend the results developed by Ha-dar and Russell (1971), Tesfatsion (1976), and Li and Wong (1999).In this context, we further generalize the above results to a multi-variate setting as shown in the following theorem:

Theorem 7. For n > 1, let ~an;~bn 2 Sn8 and ~Xn ¼ ðX1; . . . ;XnÞ0 where

X1; . . . ;Xn are independent and identically distributed. If ~bn�M~an, then~a0n~Xn�2

~b0n~Xn.

8 We keep the conditionPn

i¼1si ¼ 1 in Sn for convenience. One could exclude thiscondition and relax it to be ~10n~an ¼ ~10n~bn.

The proof of Theorem 7 is in the appendix. The relationship be-tween stochastic dominance and majorization order characterizedby this theorem allows us to rank different convex combinations oftwo sets of independent and identically distributed assets in a mul-tivariate setting. It conveys two messages to investors: First, anyrisk-averse investor will always prefer portfolios with majorizedvectors of allocations to ones with majorizing vectors of alloca-tions. Second, if ~an and ~bn cannot be ranked by majorization, theportfolios ~a0n~Xn and ~b0n

~Xn could be SSD incomparable.9 Furtherinvestigation of their SSD determination is required.

In addition, incorporating Theorem 1 into Theorem 7, we obtainthe following corollary:

Corollary 8. For n > 1, let ~an;~bn 2 Sn and ~Xn ¼ ðX1; . . . ;XnÞ whereX1; . . . ;Xn are independent and identically distributed. If b

!n!

D a!

n,then ~a0n~Xn�2

~b0n~Xn.

We note that the necessary condition of Theorem 5 and Corol-lary 6 can only hold in a bivariate setting and not in a multivariatesetting. We illustrate by the following example that the converse ofTheorem 7 does not hold:

Example 4. Consider U ¼ 12 X1 þ 1

2 X2 and V ¼ 34 X1 þ 1

8 X2 þ 18 X3

where Xiði ¼ 1;2;3Þ is independent and identically distributed asNð0;1Þ. One can easily show that U �2 V as EðUÞ ¼ EðVÞ andVarðUÞ < VarðVÞ. Nonetheless, 1

2 ;12 ;0

� �0 and 34 ;

18 ;

18

� �0 cannot beordered by majorization.

Similarly, one could easily construct an example to show thatthe necessary condition of Corollary 8 does not hold in a multivar-iate setting. Theorem 7 and Corollary 8 further extend the resultsdeveloped by Hadar and Russell (1971), Tesfatsion (1976), and Liand Wong (1999) by providing an additional methodology forinvestors to make comparisons among a wide range of differentconvex combinations of assets in a multivariate setting to checktheir SSD preferences. These results are very useful. They permitinvestors to rank different convex combinations of independentand identically distributed random variables by second-order sto-chastic dominance and by their corresponding vectors of alloca-tions. Since the portfolios with majorizing vectors will besecond-order stochastically dominated by the ones with majorizedvectors, those with majorizing vectors should be eliminated fromthe SSD efficient set. Thus, our findings could be used in determin-ing the second-order stochastic dominance efficient set. We illus-trate how to compare different portfolios in the following example:

Example 5. Suppose that one would like to rank the followingassets by second-order stochastic dominance: ~a03~X3 ¼ 2

5 X1 þ 25 X2þ

15 X3 and ~b03

~X3 ¼ 35 X1 þ 1

5 X2 þ 15 X3. One could achieve the objective

simply by examining the majorization orders. In this example, onecould easily find that ~b3 majorizes ~a3 (see Example 1). Thereafter,applying Theorem 7, we obtain ~a03~X3�2

~b03~X3.

If one does not want to apply Theorem 7 directly, one couldconsider applying Theorems 3 and 5 instead. We illustrate theprocedure as follows: By Theorem 5, we know that since 3

5 ;15 ;0

� �0majorizes 2

5 ;25 ;0

� �010, and we have 25 X1 þ 2

5 X2�235 X1 þ 1

5 X2. ByTheorem 3, adding another independent random variable, say 1

5 X3;

on both sides of the above does not alter the stochastic dominancerelationship. Thus, we have 2

5 X1 þ 25 X2 þ 1

5 X3�235 X1 þ 1

5 X2 þ 15 X3.

9 X and Y are SSD comparable if either X�nY or Y�nX holds, and X and Y are not SSDcomparable, or are SSD incomparable, if both X†nY and Y†nX hold, where X†nYmeans X�nY does not hold.

10 Readers may refer to footnote 8 to see that it is not necessary for the sum of allweights in the vectors to be 1. As long as the summations of the weights are equal forall portfolios being compared, the results for Theorem 7 hold.

M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900 897

Can the iid assumption be dropped in the diversification prob-lem and the completely diversified portfolio still be optimal? Sam-uelson (1967) tells us that the answer is no in general. He furtherestablishes some results to relax the iid assumption. In this paper,we complement Samuelson’s work by extending the results statedin the above theorems and corollaries by relaxing the independentand identically distributed condition as stated in the followingcorollaries:

Corollary 9. For n > 1, let ~Xn ¼ ðX1; . . . ;XnÞ0 be a series of randomvariables that could be dependent. For any ~an and ~bn,

~a0n~Xn�2~b0n~Xn

if there exist ~Yn and Ann such that ~Yn ¼ ðY1; . . . ;YnÞ0 in whichfY1; . . . ;Yng are independent and identically distributed, ~Xn ¼ Ann

~Yn,and

~b0nAnn�M~a0nAnn

where ~a0nAnn;~b0nAnn 2 Sn.

Corollary 10. For n > 1, let ~Xn ¼ ðX1; . . . ;XnÞ0 and ~Yn ¼ ðY1; . . . ;YnÞ0

be two series of random variables that could be dependent. For any ~an

and ~bn,

~a0n~Xn�2~b0n~Yn

if there exist ~Un ¼ ðU1; . . . ;UnÞ0; ~Vn ¼ ðV1; . . . ;VnÞ0;Ann, and Bnn inwhich fU1; . . . ;Ung and fV1; . . . ;Vng are two series of independentand identically distributed random variables such that~Xn ¼ Ann

~Un;~Yn ¼ Bnn~Vn;Ui�2Vi for all i ¼ 1;2; . . . ;n; and

~b0nBnn�M~a0nAnn;

where ~a0nAnn;~b0nBnn 2 Sn.

One could simply apply Theorem 7 to obtain the results of Cor-ollary 9 and apply Theorem 3 to obtain the results of Corollary 10.One could then apply Theorem 1 to the above corollaries to obtainthe following results:

Corollary 11. For n > 1, let ~Xn ¼ ðX1; . . . ;XnÞ0 be a series of randomvariables that could be dependent. For any ~an and ~bn,

~a0n~Xn�2~b0n~Xn

if there exist ~Yn and Ann such that ~Yn ¼ ðY1; . . . ;YnÞ0 in whichfY1; . . . ;Yng are independent and identically distributed, ~Xn ¼ Ann

~Yn,and

~b0nAnn!D~a0nAnn;

where ~a0nAnn;~b0nAnn 2 Sn.

Corollary 12. For n > 1, let ~Xn ¼ ðX1; . . . ;XnÞ0 and ~Yn ¼ ðY1; . . . ;YnÞ0

be two series of random variables that could be dependent. For any ~an

and ~bn,

~a0n~Xn�2~b0n~Yn

if there exist ~Un ¼ ðU1; . . . ;UnÞ0; ~Vn ¼ ðV1; . . . ;VnÞ0;Ann, and Bnn inwhich fU1; . . . ;Ung and fV1; . . . ;Vng are two series of independentand identically distributed random variables such that~Xn ¼ Ann

~Un;~Yn ¼ Bnn~Vn;Ui�2Vi for all i ¼ 1;2; . . . ;n; and

~b0nBnn!D~a0nAnn;

where ~a0nAnn;~b0nBnn 2 Sn.

We provide the following examples to illustrate the usefulnessof the above corollaries.

Example 6. Let Y1;Y2 and Y3 be independent and identicallydistributed random variables. Consider X1 ¼ 3

5 Y1 þ 15 Y2 þ 1

5 Y3;

X2 ¼ 13 Y1 þ 1

3 Y2 þ 13 Y3, and X3 ¼ 3

4 Y1 þ 18 Y2 þ 1

8 Y3. Obviously,fXigði ¼ 1;2;3Þ are dependent. Corollary 9 ensures that~a03~X3�2

~b03~X3 where ~a3 ¼ 4

5 ;1

10 ;1

10

� �0;~b3 ¼ 5

6 ;1

12 ;1

12

� �0, and~X3 ¼ ðX1;X2;X3Þ0. It is because one could easily find

A33 ¼

35

15

15

13

13

13

34

18

18

264

375

such that

~b03A33�M~a03A33

and ~a03A33;~b03A33 2 S3.In addition, one could also apply Corollary 11 to ensure

~a03~X3�2~b03~X3 as one could easily observe that

~b0nAnn!D~a0nAnn:

Example 7. Let fVigði ¼ 1;2;3Þ be a series of independent andidentically distributed random variables with mean lV and vari-ance r2

V , respectively. Let Ui ¼ pþ qViði ¼ 1;2;3Þ such that0 6 q < 1 and p=ð1� qÞP lV . Then, by Theorem 8 in Li and Wong(1999), we have Ui�2Viði ¼ 1;2;3Þ.

Let X1 ¼ 12 U1 þ 1

4 U2 þ 14 U3;X2 ¼ 1

4 U1 þ 13 U2 þ 5

12 U3;X3 ¼ 16 U1þ

12 U2 þ 1

3 U3;Y1 ¼ 13 V1 þ 1

3 V2 þ 13 V3;Y2 ¼ 4

7 V1 þ 14 V2 þ 5

28 V3, andY3 ¼ 4

5 V1 þ 425 V2 þ 1

25 V3. Obviously, fXig and fYigði ¼ 1;2;3Þ aredependent.

Corollary 10 ensures that ~a03~X3�2~b03~Y3 where a3 ¼ 2

3 ;16 ;

16

� �0;

b3 ¼ 34 ;

18 ;

18

� �0;~X3 ¼ ðX1;X2;X3Þ0, and ~Y3 ¼ ðY1;Y2;Y3Þ0. It is because

one could easily find

A33 ¼

12

14

14

14

13

512

16

12

13

264

375 and B33 ¼

13

13

13

47

14

528

45

425

125

264

375

such that

b03B33�Ma03A33

and ~a03A33;~b03B33 2 S3.In addition, one could also apply Corollary 12 to ensure

~a03~X3�2~b03~Y3 as one could easily find that

~b03B33!D~a03A33:

As a consequence of the above corollaries, an additional meth-odology is provided by which investors can make comparisonsamong a wide range of different convex combinations of depen-dent assets in a multivariate setting to check their SSD preferences.The results allow investors to rank different convex combinationsof dependent assets not only by second-order stochastic domi-nance but also by their corresponding vectors of allocations. Ourresults also permit investors to eliminate non-efficient portfoliosso that our results could help investors in determining the sec-ond-order stochastic dominance efficient set.

4. Concluding remarks

By incorporating the majorization theory, this paper presentsseveral new results of interest on stochastic dominance. Specifi-cally, we establish some basic relationships in the portfolio choiceproblem by using both majorization theory and stochastic domi-nance. We also provide the foundation for applying majorizationtheory and stochastic dominance to investors’ choices underuncertainty. The results are general, but presumably they are

898 M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900

applicable to investment decision theory and comparisons ofdiversification of assets in a multivariate setting. We give new con-ditions for stochastic comparisons among different portfoliochoices and new necessary and sufficient conditions that charac-terize diversified portfolios to give higher expected utilities. Thus,risk-averse and non-satiable investors will increase their expectedutilities as the diversification of the portfolio increases. Our find-ings bring together, under a common framework, a number offairly general results about diversification that permit comparisonsamong them. Our results could also be used to demonstrate theoptimality of diversification and to obtain the preference orderingsof portfolios for different asset allocations. In addition, our findingsalso impose further restrictions on admissible portfolios on theefficient frontier, and thus, our findings could also be used in deter-mining the second-order stochastic dominance efficient set.

Nonetheless, Rothschild and Stiglitz (1970) show that for anytwo distributions with the same mean, the mean-preservingspread and SSD are equivalent, whereas Shalit and Yitzhaki(1994) verify that under some conditions, marginal conditionalstochastic dominance is equivalent to SSD. Thus, incorporatingthe theory developed in our paper, one could conclude that undersome regularity conditions, the preferences obtained from a Daltontransfer, majorization, and stochastic dominance could be equiva-lent to those obtained from the mean-preserving spread and mar-ginal conditional stochastic dominance.

Unlike the SD approach, which is consistent with utility maxi-mization, the dominance findings using the mean–variance crite-rion11 may not be consistent with utility maximization if theassets returns are not normally distributed or investors’ utilitiesare not quadratic. However, under some specific conditions, themean–variance optimality could be consistent with the SD approachwith utility maximization. For example, Meyer (1987), Wong (2006,2007) and Wong and Ma (2008) have shown that if the returns of as-sets follow the same location-scale family, then a mean–variancedomination could infer preferences by risk averters for the dominantfund over the dominated one. In addition, the Markowitz mean–var-iance optimization is equivalent to minimizing variance subject toobtaining a predetermined level of expected gain; see, for example,Bai et al. (2009a, forthcoming). Thus, by incorporating the resultsdeveloped in this paper and under some regularity conditions, theefficient set derived from a Dalton transfer, majorization, and sto-chastic dominance could belong to the same efficient set obtainedfrom the mean–variance criterion and risk minimization. Further re-search could study their relationships in detail.

Last, we note that, recently, other studies have findings that ex-tend the bivariate framework of the diversification of risky assetsto a multivariate setting. For example, Ma (2000) has removedthe assumption of independence and studied the possibility thatthe random variables are exchangeable. In addition, Pellerey andSemeraro (2005) derive some new results for the portfolio choiceproblem when risky opportunities are correlated. They give newconditions for stochastic comparison among different portfoliochoices and new necessary and sufficient conditions that charac-terize the portfolio to give the maximal expected utility. One ave-nue for further research would be to incorporate the findings in ourpaper to extend the results developed by Ma (2000), Pellerey andSemeraro (2005), and others. Since the theoretical results provedin this paper open the door for the development of real-life appli-cations, another avenue for further research would apply the the-ory developed in this paper to different real-life applications inbusiness, economics and finance. For example, one could incorpo-

11 See, for example, Markowitz (1952), Joro and Na (2006), Buckley et al. (2008),Josa-Fombellida and Rincón-Zapatero (2008), Zhao and Ziemba (2008) for moreinformation.

rate the theory in this paper to explain well-known financial phe-nomena or financial anomalies12 and to model investment risk.13

Acknowledgements

The authors are grateful to Professor Robert Graham Dyson andanonymous referees for substantive comments that have signifi-cantly improved this manuscript. The authors would also like tothank Professors Esfandiar Maasoumi, Harry M. Markowitz, andFranco Pellerey for valuable comments that have significantly im-proved this manuscript. Our deepest thanks also go to ProfessorJuan Dubra for his helpful comments and assistance with the pa-per. The second author would like to thank Professors Robert B.Miller and Howard E. Thompson for their continuous guidanceand encouragement. This research is partially supported by grantsfrom Universidad de la Republica del Uruguay and Hong Kong Bap-tist University.

Appendix 1. Proof of Theorem 1

Let Km ¼ fi1; i2; . . . ; img# Kn ¼ f1;2; . . . ;ng such that aj – bj forany j 2 Km and aj ¼ bj for any j 2 Kn n Km. Then, we have1 P ai1 P ai2 P � � �P aim P 0;1 P bi1 P bi2 P � � �P bim P 0, andPk

j¼1aij 6Pk

j¼1bijfor any k ¼ 1;2; . . . ;m.

Let dij ¼ bij� aij for j ¼ 1;2; . . . ;m and let p and q be the indices

such that jdpj ¼minfjdij j : j ¼ 1; . . . ;mg and jdqj ¼maxfjdij j :

j ¼ 1; . . . ;mg. Without loss of generality, we assume p < q. In thissituation, one can easily show that dp ¼ bp � ap > 0.

Let � ¼ dp and~cn ¼ ðc1; . . . ; cnÞ0 such that cp ¼ bp � �; cq ¼ bq þ �,

and ci ¼ bi for any i – p; q. Then, we have ~bn!d~cn by performing

one Dalton transfer and now only m� 1 items of ~cn are differentfrom those of ~an. One could simply continue this process m� 2times to obtain ~an from ~cn. Thus, we have ~bn!

D~an by performing

m� 1 times single Dalton transfers. The assertion follows. h

Appendix 2. Proof of Theorem 5

Before we prove Theorem 5, we first state the following lemma:

Lemma 13. For n > 1, suppose u 2 U2 defined in (1) andY ¼

Pni¼1kiXi with

Pni¼1ki ¼ 1 where X1; . . . ;Xn are random variables,

then

E½uðYÞ�PXn

i¼1

kiE½uðXiÞ�:

Proof. Let F~X ¼ FX1X2 ...Xn be the joint distribution function of fXigdefined on K ¼ Pn

i¼1Ki. As u 2 U2, for any~x ¼ ðx1; x2; . . . ; xnÞ0, by Jen-sen inequality, we have

uXn

i¼1

kixi

!PXn

i¼1

kiuðxiÞ: ð6Þ

Taking integration on both sides of (6) with respect to F~X on K, wehaveZ

KuXn

i¼1

kixi

!dF~X P

ZK

Xn

i¼1

kiuðxiÞdF~X ¼Xn

i¼1

ki

ZK

uðxiÞdF~X

¼Xn

i¼1

kiE½uðXiÞ�:

Thus, the assertion of Lemma 13 follows. �

12 See, for example, Wong and Bian (2000), Post (2003), Post and Levy (2005), Fonget al. (2005, 2008), Fong and Wong (2006), and Post (2008).

13 See, for example, Matsumura et al. (1990), Seppälä (1994), Wong and Chan(2004), Gasbarro et al. (2007), Wong et al. (2008), and Lozano and Gutiérrez (2008).

M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900 899

Now we prove Theorem 5. We first prove the sufficient part ofthe theorem. Let X ¼ ~a02~X2 and Y ¼~b02~X2 with distribution functionsF and G, respectively. By Theorem 2, proving X�2Y is equivalent toproving that for any u 2 U2,

DEu � EuðXÞ � EuðYÞ ¼Z b

auðxÞdFðxÞ �

Z b

auðxÞdGðxÞP 0:

First, it is easy to verify that if b!

2 majorizes ~a2, then there existsk 2 ð0;1Þ such that

X ¼ kY þ ð1� kÞ X1 þ X2

2

� �: ð7Þ

Together with the concavity property of u as shown in Lemma 13,we have

E½uðXÞ� ¼ E u kY þ ð1� kÞ X1 þ X2

2

� �� �

P kE½uðYÞ� þ ð1� kÞE uX1 þ X2

2

� �� �:

As 12

!0~X2�2 k

!0~X2 where k

!¼ ðk;1� kÞ0, we have

kE½uðYÞ� þ ð1� kÞE u12

X1 þ X2ð Þ� �� �

P kE½uðYÞ� þ ð1� kÞE½uðYÞ� ¼ E uðYÞ½ �:

Thus, we have E½uðXÞ�P E½uðYÞ�; and thereafter, DEu P 0 is ob-tained as required.

We now proceed to prove the necessary part of the theorem bycontradiction. Suppose that ~a02~X2�2

~b02~X2 but ~b2 does not majorize

~a2. Then, there are two cases for ~b2 not to majorize ~a2 as follows:

1. ~a2 majorizes ~b2, or2. ~a2 and ~b2 are not comparable by majorization.

In the first case that~a2 majorizes~b2. By the sufficient part of thistheorem, we have ~b02~X2�2~a02~X2, this creates a contradiction.

In the second case in which ~a2 and ~b2 are not comparable bymajorization. This means that a1 < b1 and a1 þ a2 > b1 þ b2. Butthis is impossible since, by assumption, we havea1 þ a2 ¼ b1 þ b2 ¼ 1. Thus,~b2 must majorize ~a2 and the necessarypart of this theorem holds. �

Appendix 3. Proof of Theorem 7

As ~bn�M~an, by Theorem 1, there exist~ai

n ¼ ðai1; . . . ;ai

nÞ0ð0 6 i 6 kÞ with k 6 n� 1 such that

~an ¼ ~a0n!

d � � �!d ~ain!

d~aiþ1

n !d � � �!d ~ak

n ¼ ~an. For any two consecutivevectors in f~ai

ng, say ~ain and ~aiþ1

n , there exist h andkð1 6 h < k 6 nÞ such that ai

j ¼ aiþ1j for any j – h; k; ai

h ¼ aiþ1h � �;

and aik ¼ aiþ1

k þ � with � > 0.Thus, to obtain the proof of this theorem, it is sufficient to prove

~ai0n~Xn�2~aiþ10

n~Xn. Nonetheless, this could be obtained by simply

applying Theorems 3 and 5 and thereby the assertion follows. �

References

Ash, R.B., 1972. Real Analysis and Probability. Academic Press, New York.Bai, Z.D., Liu, H.X., Wong, W.K., 2009. On the Markowitz mean–variance analysis of

self-financing portfolios. Risk and Decision Analysis 1 (1), 35–42.Bai, Z.D., Liu, H.X., Wong, W.K., forthcoming. Enhancement of the applicability of

Markowitz’s portfolio optimization by utilizing random matrix theory.Mathematical Finance.

Bawa, V.S., Bodurtha Jr., J.N., Rao, M.R., Suri, H.L., 1985. On determination ofstochastic dominance optimal sets. Journal of Finance 40 (2), 417–431.

Berk, J.B., 1997. Necessary conditions for the CAPM. Journal of Economic Theory 73(1), 245–257.

Broll, U., Wahl, J.E., Wong, W.K., 2006. Elasticity of risk aversion and internationaltrade. Economics Letters 92 (1), 126–130.

Buckley, I., Saunders, D., Seco, L., 2008. Portfolio optimization when asset returnshave the Gaussian mixture distribution. European Journal of OperationalResearch 185 (3), 1434–1461.

Dekel, E., 1989. Asset demands without the independence axiom. Econometrica 57(1), 163–169.

Feldstein, M.S., 1969. Mean variance analysis in the theory of liquidity preferenceand portfolio selection. Review of Economic Studies 36, 5–12.

Fishburn, P.C., 1974. Convex stochastic dominance with continuous distributionfunctions. Journal of Economic Theory 7, 143–158.

Fong, W.M., Wong, W.K., 2006. The modified mixture of distributions model: Arevisit. Annals of Finance 2 (2), 167–178.

Fong, W.M., Wong, W.K., Lean, H.H., 2005. stochastic dominance and the rationalityof the momentum effect across markets. Journal of Financial Markets 8, 89–109.

Fong, W.M., Lean, H.H., Wong, W.K., 2008. Stochastic dominance and behaviortowards risk: The market for internet stocks. Journal of Economic Behavior andOrganization 68 (1), 194–208.

Gasbarro, D., Wong, W.K., Zumwalt, J.K., 2007. Stochastic dominance analysis ofishares. European Journal of Finance 13, 89–101.

Hadar, J., Russell, W.R., 1969. Rules for ordering uncertain prospects. AmericanEconomic Review 59, 25–34.

Hadar, J., Russell, W.R., 1971. Stochastic dominance and diversification. Journal ofEconomic Theory 3, 288–305.

Hanoch, G., Levy, H., 1969. The efficiency analysis of choices involving risk. Reviewof Economic Studies 36, 335–346.

Hardy, G.H., Littlewood, J.E., Pólya, G., 1934. Inequalities. Cambridge UniversityPress, Cambridge, MA.

Ingersoll, J.E., 1987. Theory of Financial Decision Making. Rowman & Littlefield,Totowa.

Joro, T., Na, P., 2006. Portfolio performance evaluation in a mean–variance-skewness framework. European Journal of Operational Research 175 (1), 446–461.

Josa-Fombellida, R., Rincón-Zapatero, J.P., 2008. Mean–variance portfolio andcontribution selection in stochastic pension funding. European Journal ofOperational Research 187 (1), 120–137.

Leung, P.L., Wong, W.K., 2008. On testing the equality of the multiple sharpe ratios,with application on the evaluation of ishares. Journal of Risk 10 (3), 1–16.

Levy, H., 2006. Stochastic Dominance: Investment Decision Making UnderUncertainty. Springer Publishers.

Levy, H., Markowitz, H.M., 1979. Approximating expected utility by a function ofmean and variance. American Economic Review 69 (3), 308–317.

Li, C.K., Wong, W.K., 1999. Extension of stochastic dominance theory to randomvariables. RAIRO Recherche Opérationnelle 33, 509–524.

Lozano, S., Gutiérrez, E., 2008. Data envelopment analysis of mutual funds based onsecond-order stochastic dominance. European Journal of Operational Research189 (1), 230–244.

Ma, C., 2000. Convex orders for linear combinations of random variables. Journal ofStatistical Planning and Inference 84, 11–15.

Machina, M.J., 1982. Expected utility analysis without the independence axiom.Econometrica 50 (2), 277–323.

Markowitz, H.M., 1952. Portfolio selection. Journal of Finance 7, 77–91.Marshall, A.W., Olkin, I., 1979. Inequalities: Theory of Majorization and its

Applications. Academic Press, San Diego.Matsumura, E.M., Tsui, K.W., Wong, W.K., 1990. An extended multinomial-Dirichlet

model for error bounds for dollar-unit sampling. Contemporary AccountingResearch 6, 485–500.

Meyer, J., 1987. Two-moment decision models and expected utility maximization.American Economic Review 77 (3), 421–430.

Ok, E., Kranich, L., 1998. The measurement of opportunity inequality: A cardinality-based approach. Social Choice Welfare 15, 263–287.

Ortobelli Lozza, S., 2001. The classification of parametric choices under uncertainty:Analysis of the portfolio choice problem. Theory and Decision 51, 297–327.

Pellerey, F., Semeraro, P., 2005. A note on the portfolio selection problem. Theoryand Decision 59, 295–306.

Post, T., 2003. Empirical tests for stochastic dominance efficiency. Journal of Finance58, 1905–1931.

Post, T., 2008. On the dual test for SSD efficiency: With an application to momentuminvestment strategies. European Journal of Operational Research 185 (3), 1564–1573.

Post, T., Levy, H., 2005. Does risk loving drive asset prices? Review of FinancialStudies 18 (3), 925–953.

Roberts, A.W., Varberg, D.E., 1973. Convex Functions. Academic Press, New York.Rothschild, M., Stiglitz, J.E., 1970. Increasing risk: I a definition. Journal of Economic

Theory 2, 225–243.Samuelson, P.A., 1967. General proof that diversification pays. Journal of Financial

and Quantitative Analysis 2 (1), 1–13.Seppälä, J., 1994. The diversification of currency loans: A comparison between

safety-first and mean–variance criteria. European Journal of OperationalResearch 74 (2), 325–343.

Shalit, H., Yitzhaki, S., 1994. Marginal conditional stochastic dominance.Management Science 40 (5), 670–684.

Tesfatsion, L., 1976. Stochastic dominance and maximization of expected utility.Review of Economic Studies 43, 301–315.

Tobin, J., 1958. Liquidity preference and behavior towards risk. Review of EconomicStudies 25, 65–86.

von-Neumann, J., Morgenstern, O., 1944. Theory of Games and Economic Behavior.Princeton University Press, Princeton N.J.

900 M. Egozcue, W.-K. Wong / European Journal of Operational Research 200 (2010) 893–900

Wong, W.K., 2006. Stochastic dominance theory for location-scale family. Journal ofApplied Mathematics and Decision Sciences, 1–10.

Wong, W.K., 2007. Stochastic dominance and mean–variance measures of profit andloss for business planning and investment. European Journal of OperationalResearch 182, 829–843.

Wong, W.K., Bian, G., 2000. Robust Bayesian inference in asset pricing estimation.Journal of Applied Mathematics & Decision Sciences 4, 65–82.

Wong, W.K., Chan, R., 2004. The estimation of the cost of capital and its reliability.Quantitative Finance 4 (3), 365–372.

Wong, W.K., Chan, R., 2008. Markowitz and prospect stochastic dominances. Annalsof Finance 4 (1), 105–129.

Wong, W.K., Li, C.K., 1999. A note on convex stochastic dominance theory.Economics Letters 62, 293–300.

Wong, W.K., Ma, C., 2008. Preferences over Meyer’s location-scale family. EconomicTheory 37 (1), 119–146.

Wong, W.K., Phoon, K.F., Lean, H.H., 2008. Stochastic dominance analysis of Asianhedge funds. Pacific-Basin Finance Journal 16 (3), 204–223.

Zhao, Y., Ziemba, W.T., 2008. Calculating risk neutral probabilities and optimalportfolio policies in a dynamic investment model with downside risk control.European Journal of Operational Research 185 (3), 1525–1540.