European Journal of Operational Research 215 (2011) 188–193

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Do investors like to diversify? A study of Markowitz preferences

Martín Egozcue a,b, Luis Fuentes García c, Wing-Keung Wong d,⇑, Ricardas Zitikis e

a Department of Economics, University of Montevideo, Uruguayb FCS Universidad de la Republica del Uruguay, Montevideo 11600, Uruguayc Departamento de Métodos Matemáticos e de Representación, Universidade da Coruña, 15001 A Coruña, Spaind Department of Economics and the Institute for Computational Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Konge Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada N6A 5B7

a r t i c l e i n f o

Article history:Received 7 December 2009Accepted 19 May 2011Available online 26 May 2011

Keywords:Portfolio selectionDiversified portfolioMarkowitz preferencesUtility theoryRisk aversion

0377-2217/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.ejor.2011.05.034

⇑ Corresponding author.E-mail address: awong@hkbu.edu.hk (W.-K. Wong

a b s t r a c t

We study rankings of completely and partially diversified portfolios and also of specialized assets wheninvestors follow so-called Markowitz preferences. It turns out that diversification strategies for Marko-witz investors are more complex than in the case of risk-averse and risk-inclined investors, whose invest-ment strategies have been extensively investigated in the literature. In particular, we observe that forMarkowitz investors, preferences toward risk vary depending on their sensitivities toward gains andlosses. For example, it turns out that, unlike in the case of risk-averse and risk-inclined investors, Marko-witz investors might prefer investing their entire wealth in just one asset. This finding helps us to betterunderstand some financial anomalies and puzzles, such as the well known diversification puzzle, whichnotes that some investors tend to concentrate on investing in only a few assets instead of choosing theseemingly more attractive complete diversification.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The classical von Neuman and Morgenstern (1944) expectedutility theory is based on the assumption that utility functions ofrisk-averse and risk-inclined investors are concave and convex,respectively; both are increasing. When examining the relativeattractiveness of various forms of investments, Friedman andSavage (1948) observed that concave utility functions may not bebest for explaining why investors buy lottery tickets or insurance.Markowitz (1952) addressed this concern by suggesting utilityfunctions with convex segments on both the positive and the neg-ative parts of the real line. Specifically, on the negative real half-line, the functions are initially convex and then concave, and thenon the positive real half-line, they are convex and finally concave.Thus, the utility functions have three inflection points, with one ofthem being located at 0, which we call the status quo throughoutthe paper as it serves the borderline between losses and gains.

Subsequently, a number of authors departed from the originalMarkowitz’s suggested utility function and proposed functionsthat are convex for gains and concave for losses. Sometimes suchfunctions are called reverse S-shaped, but throughout this paperwe call them Markowitz utility functions, because this type offunctions first appeared in Markowitz (1952). We call investorspossessing Markowitz utility functions ‘‘Markowitz investors.’’Such functions have only one inflection point, which is in concor-

ll rights reserved.

).

dance with a number of studies offering empirical support infavour of limiting the number of inflection points (e.g., Kraus andLitzenberger, 1976; Friend and Westerfield, 1980; Harvey andSiddique, 2000; Post and Levy, 2005, and reference therein).

Levy and Wiener (1998), Post and Levy (2005) study investorswith Markowitz utility functions. Levy and Levy (2002, 2004),and Wong and Chan (2008) extend these works by developing anew criterion – called Markowitz stochastic dominance – whichis designed for determining the dominance of one investment overanother one for all Markowitz utility functions. In the presentpaper we continue this analysis. As we shall see in the followingsections, the analysis of investor preferences is more complex forMarkowitz utility functions than for (classical) concave or convexutility functions.

It is well known that when confronted with assets whose re-turns are independent and identical distributed (i.i.d.), risk-averseinvestors – that is, those whose utility functions are concave –unanimously judge completely diversified portfolios as superiorto partially diversified portfolios, which are in turn preferred bythem to specialized assets. On the other hand, but under the samei.i.d. assumption, risk-inclined or, in other words, risk-seekinginvestors – that is, those whose utility functions are convex – pre-fer to invest in specialized assets than in partially diversified port-folios, which are in turn preferred by them to completelydiversified portfolios.

There are very few articles addressing the portfolio diversifica-tion when investors possess Markowitz utility functions. In thispaper we provide such an analysis. Specifically, in Section 2 we

M. Egozcue et al. / European Journal of Operational Research 215 (2011) 188–193 189

study rankings of diversified portfolios and specialized assets forMarkowitz investors. In Section 3 we illustrate our main resultsand their optimality using examples and graphs. Section 4 con-cludes the paper.

2. Results

Assume that investors with certain initial endowments want toknow shares of their endowments to be invested in n risky assetswhose returns denoted by Xi are random. Hence, the investorswant to maximize the expected utilities

E uXn

i¼1

aiXi

!" #

with respect to ai 2 [0,1] such thatPn

i¼1ai ¼ 1.1 This maximizationproblem is quite complex in general, and in this paper we therefore re-strict ourselves to the analysis of whether it is better to completelydiversify, partially diversify, or specialize portfolios; we call a portfolio

� completely diversified if ai = 1/n for all i = 1, . . . ,n;� partially diversified if 0 < ai < 1 for at least one i;� specialized if ai = 1 for only one i and aj = 0 for all other j – i

(note that this type of portfolio consists of only one asset).

Diversification preferences for risk-averse and risk-inclinedinvestors have been extensively studied (e.g., Li and Wong, 1999,and Wong, 2007) and the following proposition is well known.

Proposition 2.1. Let X1, . . . ,Xn be independent and identically dis-tributed (i.i.d.) random variables.

1) Risk-inclined investors prefer specialized portfolios to partiallydiversified ones, and prefer the latter ones to completely diver-sified portfolios. Namely, for every convex (and twice differen-tiable) utility function u(x), we have that

1 Wefunction

E uXn

i¼1

Xi

n

!" #6 E u

Xn

i¼1

aiXi

!" #

6 E uðXjÞ� �

for every j 2 f1; . . . ; ng: ð2:1Þ

2) Risk-averse investors prefer completely diversified portfolios topartially diversified ones, and prefer the latter ones to special-ized portfolios. Namely, for every concave (and twice differen-tiable) utility function u(x), we have that

E uXn

i¼1

Xi

n

!" #P E u

Xn

i¼1

aiXi

!" #P E uðXjÞ

� �for every j 2 f1; . . . ;ng: ð2:2Þ

We next study rankings of completely and partially diversifiedportfolios and also of specialized assets in the case of Markowitzinvestors, whose behavior turns out to be different from that ofrisk-averse and risk-inclined investors. Namely, their preferencestoward risks vary depending on the sensitivity of their utility func-tions toward gains and losses. The following theorem states ourfindings rigorously.

Theorem 2.1. Let X1, . . . ,Xn be i.i.d. and symmetric around zerorandom variables, and let the utility function u(x) be twice differen-tiable. Then we have the following two statements:

1) If u(2)(�x) P �u(2)(x) for all x 2 R, then inequalities (2.1) hold.

note that in this paper all investors are profit seeking, and thus all utilitys u: R ? R are increasing.

2) If u(2)(�x) 6 �u(2)(x) for all x 2 R, then inequalities (2.2) hold.

The right-most bounds of both inequalities (2.1) and (2.2) holdwithout the assumption of identical distributions: we only need torequire that the random variables X1, . . . ,Xn be independent, symmet-ric around zero, and have same means.

The proof of Theorem 2.1 is somewhat complex and we havetherefore relegated it to an appendix. Note that verifying the boundu(2)(�x) P �u(2)(x) for all x 2 R is equivalent to verifying the samebound for only all x P 0. The same note holds for the boundu(2)(�x) 6 �u(2)(x).

Theorem 2.1 implies that in the presence of i.i.d., symmetric,and mean zero assets, the diversification preferences of Markowitzinvestors can vary from preferring to diversify to preferring not todiversify depending on the sensitivity of their utility functions to-ward gains and losses. When, however, the sensitivities towardlosses and gains are identical, then Theorem 2.1 implies thatMarkowitz investors can be indifferent toward any diversifiedportfolio and specialized asset. We formulate this observation asthe following corollary.

Corollary 2.1. Let X1, . . . ,Xn be i.i.d. and symmetric around zerorandom variables, and let the utility function u(x) be twice differen-tiable. If the second derivative u(2)(x) is an odd function, that is,satisfies the equation u(2)(�x) = �u(2)(x) for all x 2 R, then

E uXn

i¼1

Xi

n

!" #¼ E u

Xn

i¼1

aiXi

!" #¼ E uðXjÞ

� �for every j

2 f1; . . . ;ng: ð2:3ÞThe right-most equality of (2.3) holds assuming only that the randomvariables X1, . . . ,Xn are independent, symmetric around zero, and havesame means.

Next we study Markowitz investors preferences when the statusquo, which by definition separates losses from gains, is differentfrom the mean-return of the assets. We can view this situationby setting the status quo to zero and vary the means l of Xi’s. Thisis the approach that we have adopted in the following corollary.

Corollary 2.2. Let X1, . . . ,Xn be i.i.d. and symmetric (around theirmeans l) random variables. Let the utility function u(x) be twicedifferentiable, and let its second derivative u(2)(x) be absolutelycontinuous with the third derivative (defined almost everywhere)u(3)(x) being non-negative for all x 2 R. Then we have the followingtwo statements:

(1) If u(2)(�x) P �u(2)(x) for all x 2 R, then inequalities (2.1) holdwhenever l P 0.

(2) If u(2)(�x) 6 �u(2)(x) for all x 2 R, then inequalities (2.2) holdwhenever l 6 0.

The proof of Corollary 2.2 is somewhat involved and we haverelegated it to an appendix. Note that the conditions on the secondderivative u(2)(x) in parts (1) and (2) need to be verified only for allx P 0.

Corollary 2.2 implies that when facing i.i.d. and symmetricassets, Markowitz investors prefer not to diversify in the case ofpositive mean-returns of the underlying assets, but when themean-returns are negative, then Markowitz investors prefer todiversify, assuming of course the formulated sensitivities of theirutility functions to losses and gains.

3. An illustration

This section is designed to illustrate Theorem 2.1 and Corollary2.2. In the case of Theorem 2.1 we shall use a skew-normal

Fig. 3.1. The utility function u(x) given by (3.1) with d = 0.1.

Fig. 3.2. The skew-normal density function f(x) for the skewness parameter valuesk = �5 (dotted with a negative mode), �1 (dashed), 0 (solid), and 5 (dotted with apositive mode).

190 M. Egozcue et al. / European Journal of Operational Research 215 (2011) 188–193

distribution (see Azzalini, 2005, and references therein) and varyits skewness parameter k in order to see when Theorem 2.1 failsand when it works. We shall also provide an analogous study ofCorollary 2.2 but varying the mean l of an underlying symmetricdistribution. To make the two illustrations connected, in the caseof Corollary 2.2 we shall use the same mean l as that implied bythe skew-normal distribution to be used for illustrating Theorem2.1. We note at the outset that the skewness parameter k of theskew-normal distribution differs from the mean of the distribution,as we shall see in a formula below. Details follow.

With a sensitivity to gains and losses parameter d > 0, let theutility function be given by the formula

uðxÞ ¼ x3; if x > 0;dx3

; if x 6 0;

(ð3:1Þ

which we have visualized in Fig. 3.1. The utility function is twicedifferentiable, its second derivative u(2)(x) is absolutely continuous,and we have the following expressions:

uð2ÞðxÞ ¼6x; if x > 0;6dx; if x 6 0;

�uð3ÞðxÞ ¼

6; if x > 0;6d; if x < 0:

�ð3:2Þ

We have chosen to work with the value d = 0.1 throughout this sec-tion because whenever d 2 (0,1), then the bound u(2)(�x) P �u(2)(x)holds for all x 2 R. Consequently, we are concerned with only thefirst parts of our results in the previous section, but analogous con-siderations are valid for the second parts as well.

To facilitate calculating the three expected utilities, as well asvisualizing them, we work with only two random variables, X1

and X2. Hence, n = 2 throughout this section. To explore the poten-tial influence of skewness on Theorem 2.1, we convenientlyassume that X1 and X2 are two independent random variablesfollowing the same skew-normal density (see Azzalini, 2005, andreferences therein)

f ðxÞ ¼ 2/ðxÞUðkxÞ; ð3:3Þ

where / and U are the standard normal density and distributionfunctions, respectively, and k 2 R is the skewness parameter. Whenk = 0, then the density f(x) reduces to the (symmetric) standard nor-mal density /(x). When k > 0, then the density f(x) is skewed to theright, and when k < 0, then it is skewed to the left. The density func-tion f(x) is visualized in Fig. 3.2 for the skewness parameter valuesk = �5, �1, 0, and 5, which we use in all our subsequentillustrations.2

We consider the symmetric case first, that is, we set k = 0, whichis within the framework of Theorem 2.1. For any a 2 [0,1], the con-vex combination aX1 + (1 � a)X2 of the two random variables X1

and X2 is normally distributed with the mean 0 and the variancea2 + (1 � a)2. Consequently,

E u aX1 þ ð1� aÞX2ð Þ½ � ¼ E uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ ð1� aÞ2

qXi

� ��

¼ ða2 þ ð1� aÞ2Þ3=2E uðXiÞ½ � ð3:4Þ

for any i = 1, 2. Since the function (a2 + (1 � a)2)3/2 defined on theinterval [0,1] achieves its minimum at a = 1/2 and identical maxi-mums at the end-points of the interval [0,1], we have from Eq.(3.4) that

E u12

X1 þ12

X2

� �� 6 E u aX1 þ ð1� aÞX2ð Þ½ � 6 E uðXiÞ½ �; ð3:5Þ

as predicted by Theorem 2.1.

2 We have included more negative values than positive ones because the first partsof our results in the previous section break down – as intended to illustrate – forrandom variables with negative skewness and/or negative means. For illustrating thesecond parts of the results, including more positive values would be warranted.

When k – 0, then the random variables X1 and X2 are not sym-metric, and thus Theorem 2.1 cannot be applied. To have a glimpseof what is happening in this case, in Fig. 3.3 we have visualized thefunction

HðaÞ ¼ E u aX1 þ ð1� aÞX2ð Þ½ �; 0 6 a 6 1;

in the case of the aforementioned four values of the skewnessparameter k. We see from Fig. 3.3 that bounds of Theorem 2.1 canindeed fail when the symmetry of the underlying distribution isviolated.

When the skewness changes, then the mean of the distributionalso changes. This brings us into the framework of Corollary 2.2. Toget a glipse of how the mean of a symmetric distribution and theearlier discussed skewness might influence expected utilities, wecontinue working with the same three expected utilities (corre-sponding to completely diversified, partially diversified, and spe-cialized portfolios) but now with two independent normalrandom variables X1 and X2, which are of course symmetric, withvariances equal to 1 and their (identical) means given by theformula

l ¼ffiffiffiffi2p

rkffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ k2p : ð3:6Þ

We continue using the aforementioned four values k = �5, �1, 0,and 5. Hence, X1 and X2 are two independent normal random vari-ables N(l,1) with their mean values l = �0.78239, �0.56419, 0, and

Fig. 3.4. The values of H(a) (vertical axes) for 0 6 a 6 1 (horizontal axes) when X1 and X2 are independent normal random variables N(l,1) with the means (clockwise startingwith the top-left panel) l = �0.78239, �0.56419, 0, and 0.78239.

Fig. 3.3. The values of H(a) (vertical axes) for 0 6 a 6 1 (horizontal axes) when X1 and X2 are independent skew-normal random variables with the skewness parameters(clockwise starting with the top-left panel) k = �5, �1, 0, and 5.

M. Egozcue et al. / European Journal of Operational Research 215 (2011) 188–193 191

0.78239. (Note that l in Eq. (3.6) is the mean of the skew-normaldensity (3.3).) In Fig. 3.4 we have visualized the function H(a). Wesee from the figure that the inequalities of the first statement ofCorollary 2.2 can indeed fail when the non-negativity assumptionon the mean l is violated.

4. Conclusions

We have developed a theoretical study that aids in explainingthe attitudes of Markowitz investors toward risk, including theirpreferences for specialized, partially diversified, and completely

192 M. Egozcue et al. / European Journal of Operational Research 215 (2011) 188–193

diversified portfolios. Our main results show that unlike in the caseof risk-averse and risk-inclined investors, Markowitz investorsmight prefer investing their entire wealth in just one asset. Thisfinding helps us to better understand some financial anomaliesand puzzles, including the well known diversification puzzle(e.g., Statman, 2004) which notes the tendency of some investorsnot to completely diversify investments and choose investing onlyin a few assets.

We have also provided an illustration showing that certainassumptions imposed for the validity of our results cannot in gen-eral be disposed of. Naturally, a further study exploring the validityboundaries of our results under additional distributional assump-tions on the portfolio and/or asset returns would be of interest,including a study of portfolio rankings under constrains on the var-iability of portfolio and/or asset returns.

Acknowledgements

The authors gratefully acknowledge their partial research sup-port by the Agencia Nacional de Investigación e Innovación (ANII)of Uruguay, the Research Grants Council (RGC) of Hong Kong, HongKong Baptist University, and the Natural Sciences and EngineeringResearch Council (NSERC) of Canada.

Appendix A. Appendix: Proofs

Here we prove Theorem 2.1 and Corollary 2.2, for which we firstestablish a technical result formulated as Proposition A.1.

Proposition A.1. Let X1, . . . ,Xn be independent and symmetric aroundzero random variables, and let the utility function u(x) be twicedifferentiable. Then we have the following two statements:

1) If u(2)(�x) P �u(2)(x) for all x 2 R, then

E uXn

i¼1

aiXi

!" #6 max E uðXjÞ

� �; j ¼ 1; . . . ;n

�: ðA:1Þ

2) If u(2)(�x) 6 �u(2)(x) for all x 2 R, then

E uXn

i¼1

aiXi

!" #P min E uðXjÞ

� �; j ¼ 1; . . . ;n

�: ðA:2Þ

Proof of Propositon A.1. We first prove the theorem in the casen = 2 and then apply an induction argument to prove it for any n.We have that

dda1

� �2

E u a1X1 þ ð1� a1ÞX2ð Þ½ � ¼ 12

E ðX1 � X2Þ2 uð2Þ a1X1ð�h

þð1� a1ÞX2Þ þ uð2Þ �a1X1 � ð1� a1ÞX2ð Þ i; ðA:3Þ

where the equation holds due to the independence and symmetryof X1 and X2, which imply that the random variable a1X1 +(1 � a1)X2 is symmetric around 0. Note that the right-hand side ofEq. (A.3) is non-negative because u(2)(x) P �u(2)(�x) for all x 2 Rby assumption. Hence, the expectation E[u(a1X1 + (1 � a1)X2)] is aconvex function of a1 2 [0,1] and its maximum is thereforeachieved at either a1 = 0 or a1 = 1. This proves that

E uða1X1 þ ð1� a1ÞX2½ � 6max E½uðX1Þ�;E uðX2Þ½ �f g: ðA:4Þ

To proceed, we use an induction argument. Namely, for a givenk P 2, let the bound

E uXj

i¼1

aiXi

!" #6 max E uðXiÞ½ �; i ¼ 1; . . . ; jf g: ðA:5Þ

hold for every j 2 {2, . . . ,k}, all utility functions u(x) such thatu(2)(�x) P �u(2)(x) for all x 2 R and all non-negative ai’s such thatPj

i¼1ai ¼ 1. We shall prove that bound (A.5) also holds whenj = k + 1 for all utility functions u(x) such that u(2)(�x) P �u(2)(x)for all x 2 R and all non-negative ai’s such that

Pkþ1i¼1 ai ¼ 1. To this

end, we write

E uXkþ1

i¼1

aiXi

!" #¼ E u ð1� akþ1ÞY þ akþ1Xkþ1ð Þ½ �; ðA:6Þ

where Y ¼Pk

i¼1biXi with bi = ai/(1 � ak+1) for 1 6 i 6 k. Without lossof generality we can and thus do assume that ak+1 < 1; otherwise allthe other ai’s are equal to 0, which is a trivial case. Since Xi’s areindependent and symmetric around zero, the random variable Y issymmetric around 0 and independent of Xk+1. Thus by our induc-tional assumption (case j = 2) we have that

E u ð1� akþ1ÞY þ akþ1Xkþ1ð Þ½ � 6max E½uðYÞ�;E uðXkþ1Þ½ �f g: ðA:7Þ

Furthermore, by our inductional assumption (case j = k) implies that

E½uðYÞ� ¼ E uXk

i¼1

biXi

!" #6 max E uðXjÞ

� �; j ¼ 1; . . . ; k

�: ðA:8Þ

Combining bounds (A.6), (A.7) and (A.8), we obtain that

E uXkþ1

i¼1

aiXi

!" #6 max E uðXjÞ

� �; j ¼ 1; . . . ; kþ 1

�:

This concludes the proof of Proposition A.1. h

Proof of Theorem 2.1. The right-most inequalities of both parts 1)and 2) follow directly from Proposition A.1. We next establish thebound

E uXn

i¼1

Xi

n

!" #6 E u

Xn

i¼1

aiXi

!" #; ðA:9Þ

under the conditions of part 1) of Theorem 2.1. We start with thecase n = 2. To see where the expectation E [u(a1X1 + (1 � a1)X2)]achieves its minimum with respect to a1, we equate its first deriv-ative to zero:

E ðX1 � X2Þuð1Þ a1X1 þ ð1� a1ÞX2ð Þ� �

¼ 0: ðA:10Þ

One of the solutions to this equation is a1 = 1/2, and since theexpectation is a convex function of a1(recall statement (A.3) andthe therein noted non-negativity of its right-hand side), the valuea1 = 1/2 is a global minimum of E [u(a1X1 + a2X2)]. Consequently,we have the bound

E uX1

2þ X2

2

� �� 6 E u a1X1 þ a2X2ð Þ½ �: ðA:11Þ

Assume now that, for a given k P 2, the bound

E uXj

i¼1

Xi

j

!" #6 E u

Xj

i¼1

aiXi

!" #ðA:12Þ

holds for every j 2 {2, . . . ,k}, all utility functions u(x) such thatu(2)(�x) P �u(2)(x) for all x 2 R and all non-negative ai’s such thatPj

i¼1ai ¼ 1. We shall prove that bound (A.12) also holds whenj = k + 1 for all utility functions u(x) such that u(2)(�x) P �u(2)(x)for all x 2 R and all non-negative ai’s such that

Pkþ1i¼1 ai ¼ 1. For this,

we rewrite Eq. (A.6) as follows:

E uXkþ1

i¼1

aiXi

!" #¼ E v

Xk

i¼1

biXi

!" #; ðA:13Þ

where bi = ai/(1 � ak+1) for 1 6 i 6 k, and

M. Egozcue et al. / European Journal of Operational Research 215 (2011) 188–193 193

vðyÞ ¼ E u ð1� akþ1Þyþ akþ1Xkþ1ð Þ½ �:

Without loss of generality, we can and thus do assume that ak+1 < 1.In order to apply the above inductional assumption on the right-hand side of Eq. (A.6), we need to check that v(2)(x) + v(2)(�x) P 0.This can be done as follows:

v ð2ÞðxÞ þ v ð2Þð�xÞ ¼ ð1� akþ1Þ2 E uð2Þðð1� akþ1Þyþ akþ1Xkþ1Þ� ��

þE uð2Þð�ð1� akþ1Þyþ akþ1Xkþ1Þ� �

¼ ð1� akþ1Þ2E uð2Þðð1� akþ1Þyþ akþ1Xkþ1Þ�

þuð2Þð�ð1� akþ1Þy� akþ1Xkþ1Þ�;

where the second equation holds due to the symmetry of Xk+1. Notethat the right-hand side is non-negative because u(2)(�x) P �u(2)(x)for all x 2 R by assumption. Consequently, our inductional assump-tion can be applied, and we thus have that

E vXk

i¼1

biXi

!" #P E v

Xk

i¼1

Xi

k

!" #

¼ E u ð1� akþ1ÞXk

i¼1

Xi

kþ akþ1Xkþ1

!" #: ðA:14Þ

Denote the right-hand side of (A.14) by h(ak+1). Differentiatingh(ak+1) with respect to ak+1 and equating the derivative to 0, we ob-tain that

E Xkþ1 �Xk

i¼1

Xi

k

!uð1Þ ð1� akþ1Þ

Xk

i¼1

Xi

kþ akþ1Xkþ1

!" #¼ 0:

ðA:15Þ

Since Xi’s are i.i.d. random variables, the choice ak+1 = 1/(k + 1) sat-isfies Eq. (A.15), which implies that ak+1 = 1/(k + 1) is a critical pointof the function h(ak+1). We need to verify that the critical point is aglobal minimum of the function h(ak+1) with respect to ak+1 2 [0,1].For this, we show that the function h(ak+1) is convex, which followsif its second derivative

hð2Þðakþ1Þ ¼ E Xkþ1 �Xk

i¼1

Xi

k

!2

uð2Þ ð1� akþ1ÞXk

i¼1

Xi

kþ akþ1Xkþ1

!24

35

ðA:16Þ

is non-negative. SincePk

i¼1Xi=k and Xk+1 are independent andsymmetric around 0, the non-negativity of h(2)(ak+1) follows from asimilar argument employed in the proof of Theorem 2.1 but nowusing the assumption u(2)(�x) P �u(2)(x) for all x 2 R. Consequently,we have that ak+1 = 1/(k + 1) is a global minimum of the functionh(ak+1) on the interval [0,1]. In view of this observation and sincebound (A.14) is achieved with bi = 1/k for all i = 1, . . . ,k, which isequivalent to setting ai = 1/(k + 1), we have bound (A.12) for j = k + 1.This completes the proof of part (1). Part (2) of Theorem 2.1 follows anal-ogously. This finishes the entire proof of Theorem 2.1. h

Proof of Corollary 2.2. The corollary reduces to Theorem 2.1 withthe utility function

vðxÞ ¼ uðxþ lÞ:

Hence, to prove part (1), we need to check that v(2)(�x) P �v(2)(x)for all x 2 R. The condition can be written as follows:

uð2Þð�xþ lÞP �uð2Þðxþ lÞ: ðA:17Þ

Since u(2)(x + l) P �u(2)(�x � l) due to the bound u(2)(�z) P�u(2)(z) for all z 2 R, we have that

uð2Þð�xþ lÞ þ uð2Þðxþ lÞP uð2Þð�xþ lÞ � uð2Þð�x� lÞ

¼Z l

�luð3Þð�xþ yÞdy: ðA:18Þ

Since l P 0 and u(3)(z) P 0 for all z 2 R, the right-hand side of(A.18) is non-negative, thus proving bound (A.17) and establishingpart (1) of Corollary 2.2.

To prove part (2), we need to check that v(2)(�x) 6 �v(2)(x) forall x 2 R. The condition can be rewritten as follows:

uð2Þð�x� l�Þ 6 �uð2Þðx� l�Þ; ðA:19Þ

where l⁄ = �l P 0. Since u(2)(�x � l⁄) 6 �u(2)(x + l⁄) due tou(2)(�z) 6 �u(2)(z) for all z 2 R by assumption, we have that

uð2Þð�x� l�Þ þ uð2Þðx� l�Þ 6 �uð2Þðxþ l�Þ þ uð2Þðx� l�Þ

¼ �Z l�

�l�uð3Þðxþ yÞdy: ðA:20Þ

Since l⁄P 0 and u(3)(z) P 0 for all z 2 R, the right-hand side of(A.20) is non-positive and so bound (A.19) holds. This establishespart (2) of Corollary 2.2 and completes the entire proof. h

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