1

Characterizing Efficient Portfolios*

Shlomo Yitzhaki and Joram Mayshar The Hebrew University of Jerusalem

Revised: December 2001 ABSTRACT We provide here a necessary and sufficient operational condition for

determining whether a given portfolio is efficient in the sense of second-degree stochastic Dominance (SSD). This condition also enables one to find a direction for improving on an inefficient portfolio, in the sense that all risk averse investors would weakly prefer that change in the portfolio composition. This condition can be applied among others to resolve questions that have long been posed in the literature, concerning on whether the portfolios that are promoted by portfolio managers are in fact efficient in the above sense.

Key Words: Stochastic Dominance, Optimal Portfolio, Efficient Portfolio * We thank Jim Bodurtha, Josef Hadar and Haim Levy for their comments on an earlier draft. Mailing Address: Joram Mayshar Dept. of Economics Hebrew University Jerusalem, Israel, 91905 E-Mail: msjoram@mscc.huji.ac.il Phone:972-2-5883138 Fax: 972-2-5816071

2

Characterizing Efficient Portfolios The need to diversify is among the key factors contributing to the complexity of financial investment. Given the complexity of this task, many investors rely on professional agents — public mutual funds or private portfolio managers — to select and update portfolios on their behalf. Portfolio managers try to meet the requirements of their clients, even though these clients' preferences are not fully known to them. Under such circumstances, how can one tell (ex-ante) whether a portfolio manager is acting appropriately? Following the literature, our answer rests on two standard assumptions: firstly that the investor and the portfolio manager share the same expectations about the joint distribution of assets' returns; and secondly that the portfolio manager knows that the client is a risk-averse, expected-utility maximizer, but nothing more than that. Given these assumptions, a standard approach to examine the appropriateness of a portfolio that is selected by a portfolio manager is to examine whether the selected portfolio is efficient according to second-degree stochastic dominance (SSD) criterion. 1 By this criterion the manager has failed to serve his client's interests if she chose an inefficient portfolio, in the sense that there exists an alternative portfolio that all risk-averse investors prefer over the manager's selected portfolio. The vast literature on stochastic dominance criteria for characterizing the efficient set of portfolios was recently surveyed by Levy (1998). In concluding that book, Levy notes that one of the key disadvantages of the SSD approach, in comparison with the mean variance approach, is that it does not provide an applicable criterion for determining efficient portfolios: “A relatively simple algorithm exists for M-V portfolio selection but not for SSD selection. We need to establish a simple method that would tell us which asset to add and which to exclude in constructing the SSD efficient portfolio.” (p. 340) . This paper, we believe, answers this need. The main obstacle to a general characterization of the efficient set of portfolios seems to be that the common SSD procedure applies to the comparison of two wealth 1 The literature often characterizes a portfolio as efficient according to a particular criterion (mean-variance, SSD, etc.), if there is no alternative portfolio that outperforms it according to that selection criterion. By this terminology, we thus characterize a portfolio as "efficient" if it is SSD-efficient. It may be noted that Bawa and his associates defined an SSD-inefficient portfolio as inadmissible [see Bawa and Goroff (1982) and Bawa et al. (1985)].

3

distributions. In a portfolio context, in order to check whether a given portfolio is efficient according to this procedure, one has to make pair-wise SSD comparisons with the infinite number of alternative portfolio-derived wealth distributions. In this paper we do not seek to provide a method for computing the efficient set of portfolios. Our more modest goal is to provide a procedure for establishing whether any given portfolio belongs to the efficient set. That procedure also implies a method for identifying how to improve on an inefficient portfolio. Among others, we also prove here that if a portfolio is efficient, then it is also “optimal”, in the sense that there exists a risk-averse utility function for which this portfolio is optimal. The method suggested here can shed light on the long-going debate regarding the efficiency of the "market portfolio". The ability to practically check the efficiency of portfolios selected by a portfolio managers on clients’ behalf, can also resolve related problems in the literature. In a recent paper, Canner, Mankiw and Weil (1997) for example raised what they identified as the “asset allocation puzzle.” This puzzle amounts to their finding that portfolios that are based on popular recommendations to the public, were found to be inefficient according to the mean-variance criterion. Shalit and Yitzhaki (2000) in fact apply the method that we suggest here, for the data considered by Canner, Mankiw and Weil, and find out that the recommended portfolios, were indeed inefficient by the mean-variance criterion, but were not inefficient according to the SSD criterion. Another issue for which our approach can be applied concerns the merit of international diversification: in an international setting, is it possible to construct efficient portfolios that are based on domestic assets only? [For a recent answer that is based on the idea of mean-variance spanning see Errunza, Hogan and Hung, (1999)]. The method employed here extends the approach of Marginal Conditional Second-Degree Stochastic Dominance (MCSSD) suggested by Shalit and Yitzhaki (1994).2 The question that Shalit and Yitzhaki ask is whether a given portfolio can be improved upon (according to the SSD criterion) by a marginal increase in the share of one asset and a corresponding decrease in the share of another asset. According to that approach, the absence of any such pair of assets provides a necessary condition for efficiency. Nonetheless, a portfolio that passes this pair-wise necessary criterion for efficiency may

2 The approach adopted here is also related to the literature on the distributional impact of taxation, see Mayshar and Yitzhaki (1995).

4

prove inefficient if a change in the holding of more than two assets can generate an improvement for all risk-averse expected-utility maximizers. Our present formulation generalizes the approach of Shalit and Yitzhaki (1994) in several ways: we consider the local (marginal) revision of any combination of assets; we provide necessary and sufficient conditions for the efficiency of a portfolio; we demonstrate that if a portfolio is locally efficient it is also globally efficient; we also prove that, in the present context, any efficient portfolio is, in fact, optimal; and finally, the approach is generalized to accommodate the possibility of risky wealth from non-financial sources, such as human capital.3 Section I introduces the formal definitions of efficient and optimal portfolios in the framework of a finite set of states of nature and provides some preliminary results. In section II we prove the exact nature of the relations between efficiency and optimality and specify necessary and sufficient conditions for efficiency. Section III briefly considers some applications, and section IV presents an extension to continuous random variables. Here we suggest necessary conditions for efficient portfolios that can be interpreted as conditions on the expected returns and the betas of the assets, and also prove that the SSD efficiency conditions that we proposed here reduces to the mean-variance efficiency condition, given normal distributions. The paper concludes with suggestions for further research. I. Preliminary Definitions and Propositions The purpose of this section is to present the notation and some standard definitions and propositions. These are provided here for completeness, even though they are covered in one way or another in the literature. Let there be n assets indexed by i, and denote by ri(s) the return to asset i in state

3 The present approach is somewhat related to the literature on recovering an investor's utility function from his portfolio choices. Thus, in Varian (1983), for example, it is assumed that several portfolio selections have been made, under different asset prices. The aim is to recover properties of the underlying utility function, or to test (by revealed preference theory) for the consistency of these selections with the existence of such a function. Our concern here, is to characterize the efficiency of a given, single portfolio.

5

of nature s, where the space of states of nature is assumed to be finite, with S elements. A portfolio is characterized by a vector α = (α1, α2, ..., αn) ∈ Rn, with Σi αi ≤ 1. The

component αi represents the share of financial wealth invested in asset i.4 We will consider an investor who starts off with initial wealth w > 0, and with y(s) units of future wealth in state s that stem from non-financial sources. We assume that the joint distribution of r1, ..., rn and y, or the probability πs > 0 of each state s, is known to everyone. The future wealth of the person who holds portfolio α is a random variable: x(α) = y + wαr, that takes the value x(αs) = y(s) + wΣi αi ri(s) in state s .

We shall denote the feasible consumption space by X and assume that it is of the form (x, ∞) ⊆ R, for some lower bound x. We then denote by A the set of portfolios α that generate in every state a feasible consumption level x(αs)∈X. That is, A = α∈Rn

Σi αi ≤ 1, x(αs) ≥ x, for all s. Given that it is defined by a set of weak linear

inequalities, it is clear that the set A is closed and convex. We also define an open convex subset of A: A* = α∈Rn

Σi αi = 1, x(αs) > x, for all s

Assumption 1. The distribution of asset returns satisfies: (i) there exists a portfolio α° with Σi α°i = 1 for which Σi α°i ri(s) ≥ 0 for all s,

with a strict inequality for at least some s; (ii) there exists a portfolio α' with Σi α'i = 1, for which x(α's) = y(s) + wΣi α'i ri(s) > x for all s; (iii) there exists no (non-zero) portfolio α" such that Σi α"i ri(s) ≡ 0 for all s.

The first assumption guarantees that one can invest a dollar and obtain a non-negative payout in every state. The second assumption guarantees the non-emptiness of the subset of portfolios A* (our key theorem below will be applied to portfolios in this set). The third assumption rules out linear dependence among the assets. In particular this guarantees that two different portfolios will always generate two distinct wealth vectors across states. At this point it may help to point out that our assumptions do not rule out the possibility of an arbitrage portfolio: a non-zero portfolio with Σi αi = 0, for which Σi αi ri(s) 4 The feasibility of Σi αi < 1, where some initial wealth is freely discarded, is included for technical reasons (see Proposition 7 below). For economic reasons, such a portfolio will not be efficient (see Proposition 2).

6

≥ 0 for all s. Neither do we impose any restrictions on short sales. This is simply a result

of the fact that we do not aim to characterize the optimal portfolio for a given utility function, or to characterize equilibrium prices. As stated in the introduction, our goal here is to identify whether any given portfolio is efficient. If an arbitrage portfolio exits, then, quite simply, no portfolio can be efficient. 5 We will be interested in risk-averse investors. Accordingly, we denote by Ω the set of utility functions u: X→R, that are continuously differentiable, increasing and strictly concave.6 Given w and y, for any (u, α) where u∈Ω and α∈A we can define U(u, α)

by: U(u, α) ≡ Eu(y + wαr) = Σs πs u[y(s) + wΣi αi ri (s)] .

Given w, y, two portfolios α′∈A and α∈A, and utility u∈Ω we will say that α' improves on α relative to u, if U(u, α′) ≥ U(u, α); we will further say that α' improves on α if U(u, α′) ≥ U(u, α) for all u∈Ω.

Proposition 1.7 Given (w, y), if u∈Ω and α∈A, the set of portfolios that improve on

α relative to u is convex. We are now in a position to define the concepts of efficiency and optimality. In extending the standard definition of SSD-efficient portfolios to the case where there exists a non-market source of wealth, y, we define: Definition 1: Given (w, y), a portfolio α∈A is inefficient if there exists an alternative portfolio, α′∈A (α′ ≠ α) that improves on α. A portfolio will be referred to as efficient if

5 Another way to realize the difference between characterizing optimal portfolios and our current goal

to characterize efficient and inefficient portfolios is to note that, unlike the condition of no arbitrage,

the first two components of Assumption 1, can be stated in a manner that makes them invariant to a

change in asset prices. That is, if one looks at return as the ratio of future payout to current price, then

these assumptions concern only the payout pattern, and are independent of the current pricing.

(Assumption 1-(iii) can be interpreted as a condition on initial wealth, rather on asset return). 6 Strict concavity and weak monotonicity imply that utility is strictly increasing.

7 All the proofs are presented in the appendix.

7

it is not inefficient. Definition 2. Given (w, y), a portfolio α∈A is optimal if there is a utility function, u∈Ω such that α improves on every alternative portfolio, α′∈A, relative to u.

Before going any further, we want to briefly illustrate these two definitions, and to show that they are rather distinct concepts in a context were only a discrete set of portfolios (or wealth distributions) exists, see Bawa et. al. (1985). Consider the case in the enclosed table, where only three investors (utility functions) and five alternative portfolios are possible, and where the figures refer to the expected utility of the given investor, if he holds the respective portfolio.

portfolio investor

α1 α 2 α 3 α 4 α5

u 1 2 2 4 3 2

u 2 3 2 1 2 2

u 3 2 2 1 1 1 The problem that we address here is that faced by a portfolio manager who has to select one of the five portfolios on behalf of a client, knowing only that the client might be one of these three investors. By the above definitions, portfolio α5 is inefficient since all investors (weakly) prefer portfolio α4 over it. By the same token, also portfolio α2 is inefficient as well (see α1). All the other three portfolios would thus be characterized as efficient (since they are not inefficient). Portfolios α1 and α3 are optimal, since they are the preferred ones by the second and first investors, respectively. Since the third investor is indifferent between choosing either α1 or α2, by our definition, also α2 is optimal. In this illustrative example then, α2 is optimal but inefficient, and α4 is efficient but not optimal. (As we will show later, our assumptions on the dense space of available portfolios, rule out this possibility). Returning to the formal presentation, it is worth noting that strict concavity of the utility function insures that a portfolio is either efficient or inefficient. Strict concavity insures the following too:

8

Proposition 2. If α∈A is inefficient, for any u∈Ω there is some α'∈A such that U(u,

α') > U(u, α). Proposition 3. A portfolio α∈A is efficient only if Σi αi = 1.

Proposition 4. Given w, y, if a portfolio α∈A is optimal, then it is efficient.

Strict concavity of the utility function, and Assumption 1-(iii), insure that for each utility function u* there is only one optimal portfolio. According to these simple propositions, the set of optimal portfolios is a subset of the set of efficient portfolios, and in the latter set no initial wealth is disposed of. II. Conditions for Efficiency and Optimality To formally present the main results of this paper, we now state: Theorem. Given w, y, and α∈A* (so that Σi αi = 1 and x(αs) > x for all s),

(i) The existence of a solution for a set of linear inequalities provides a necessary and sufficient condition for the inefficiency of α, and for identifying a direction of improving on α if it is inefficient. (ii) The portfolio α is efficient if and only if it is optimal. The rest of this section is devoted to prove this Theorem. Our approach to provide such a complete characterization of whether a portfolio is efficient is based on a local analysis. Definition 3. Given w, y, a portfolio α∈A is locally inefficient if there exists a differential portfolio δ (a non-zero vector of differentials in Rn), where (α+δ)∈A , and

where α+δ improves on α. We refer to such a differential portfolio δ as a marginal, α-conditional arbitrage portfolio.8 The differential portfolio δ in the above definition is an arbitrage portfolio in a special sense. First, it provides an improvement for any increasing and concave utility function in Ω (rather than for any increasing utility function, as an arbitrage portfolio is

8 Clearly, every arbitrage portfolio is an α-conditional arbitrage portfolio, but not vice versa (see section 4 below).

9

typically defined). Second, it provides an improvement only over the particular initial portfolio α, therefore it is termed α-conditional. Third, it provides an improvement only for a local departure from the wealth distribution x(α), hence it is termed marginal. Proposition 5. Given (w, y), a portfolio α∈A is inefficient if and only if it is locally

inefficient. Proposition 3 proved that for a portfolio α to be efficient it is necessary that Σi αi = 1. Proposition 5 allows us to restrict attention to local improvement in a portfolio. Therefore, we shall henceforth restrict the discussion to local conditions that enable to check the efficiency and optimality of any given portfolio α where Σi αi = 1. In order to be unrestricted in our ability to locally perturb the portfolio, we shall also limit our attention to a given portfolio where x(αs) > x, for all s. That is, we will be interested only in portfolios α∈A*. In addition to these minor restrictions we now assume:

Assumption 2. For the combination of w, y and α∈A* that concerns us, there are no two states of nature s and s' with the same wealth level [i.e., where x(αs) = x(αs')]. As a

result, the states of nature s = 1, 2, ..., S will be taken to be ranked so that if s < s' then x(αs) < x(αs').

Assumption 2 may seem to be surprisingly restrictive. The following proposition demonstrates that this is not the case. Proposition 6. If Assumption 2 fails to hold, then we can consolidate states of nature so that this assumption will hold and so that if δ is a differential portfolio, α+δ improves on α in the consolidated, reduced state-space, if and only if it does so also in the original state-space. According to Proposition 6 it may seem then that we did not need to list Assumption 2 altogether as a restriction. This however is not the case, since the process of consolidating states may cause Assumption 1-(iii) to fail. Thus, henceforth, we will assume that both assumptions hold. Given the ranking of states by the specific portfolio α, in accordance with

10

Assumption 2, for each asset i we now define: (1) Ci (sα) = Σs′=1

s ri (s′)πs′ = Σs′, x(αs′)≤x(αs) ri (s′)πs′ s = 1, 2, ..., S.

Ci(sα) is the cumulative first moment of the probability-weighted rates of return ri(s) over those states s′ whose x(αs′) is not higher than x(αs). Or, in another way, Ci(sα) is the cumulative conditional expected return on asset i, provided that the return on the portfolio does not imply wealth in excess of x(αs). 9 Hence for the top state, Ci(Sα) is simply the expected value of the return ri .

Proposition 7. Given w, y, α∈A* and Assumptions 1-2, the portfolio α is inefficient if

and only if there exists a non-zero differential portfolio δ, such that: (2) Σi δi ≤ 0 and Q(sδ, α) ≡ Σi δi Ci (sα) ≥ 0 for s = 1, 2, ..., S.

Proposition 7 proves the first claim of our Theorem and provides the main contribution of this paper. For completeness, and for the purpose of proving the second part of our Theorem, we now also provide a complete characterization of optimal portfolios.10 Proposition 8. Given w, y, α∈A* and Assumptions 1-2, a necessary and sufficient condition for portfolio α to be optimal is that there exists a positive vector φ ∈RS that

satisfies the conditions (3a) and (3b): (3a) Σs φs πs ri (s) = 1, for i = 1, 2, ..., n. (3b) φ1 > φ2 > ... > φS > 0. Using the local approach, we were thus able to obtain necessary and sufficient conditions to check whether any portfolio α∈A* is efficient, or optimal. In Proposition 4

we have also established that every optimal portfolio is efficient. Using the last two

9 Ci (sα) is referred to in statistics as the absolute concentration curve; for a discussion of its properties see Shalit and Yitzhaki (1994) and Yitzhaki (2001). 10 The relation of the following proposition to the theorems by Peleg and Yaari (1975) is discussed below.

11

propositions we now aim to prove also that every efficient portfolio is optimal. For that purpose we now restate conditions (2) and (3) in matrix form. Assume that the states are ranked in accordance with assumption 2, given the portfolio α. Let R represent the n x S matrix, whose i-s element is the return to asset i in state s: Ris = ri(s), and let D represent the n x S matrix, whose i-s element is the probability-weighted return: Dsj = πsrj(s). It is clear that D can be obtained by post-multiplying R by a diagonal matrix Π with the probabilities πs along the diagonal. Thus since R has full row rank (by assumption 1-iii), so will D. Now let C represent the n x S matrix, whose i-s element is the cumulative return: Cis = Ci(sα). Further define also the S × S matrix B, whose h-th column is the column vector (1, 1, 1, ..., 0, 0)', starting with

h times 1. It can be then recognized from the definition in (1) that C = DB. Thus, since B is a regular matrix, also D will be of full row rank. Finally, denote by e∈Rn the

column vector e = (1, 1, ..., 1, 1)'. In view of the fact that we assume that the vector of differentials δ∈Rn is non-zero,

and the full rank of C, we can rewrite condition (2) as stating that δ has to satisfy:11 (2') δ'e ≤ 0, δ'C ≥ 0S,

where 0S∈RS. Condition (3a) states that the vector φ∈RS satisfies Dφ = e. It is convenient to rewrite this condition as DBB-1φ = CB-1φ = e, and then to define ψ∈RS by ψ = B-1φ.

From the definition of B it can be recognized that its inverse B-1 has all 1's in the main diagonal, has all -1's in its one-up off diagonal, and has zeros every where else. Thus ψ = B-1φ = (φ1 - φ2, φ2 - φ3, ..., φS-1 - φS, φS)'. Conditions (3a) and (3b) can now be rewritten as requiring the existence of a vector ψ∈RS such that:

(3') ψ > 0, Cψ = e. Lemma (Farkas). Either there exists a vector δ that solves (2') or there exists a vector ψ that solves (3').

11 We use the symbol ≥ on vectors to denote strict inequality in at least one component, and the symbol > to denote strict inequality in all components.

12

This Lemma is in fact only one of the variants of Farkas' Theorem. This specific variant is an application of Propositions 12.3 and 12.4 in Yaari (1971).12 It is easy to prove that if there exists a solution for (3') then there is no solution for (2'). Given, our propositions above, this is a restatement of our Proposition 4 that an optimal portfolio cannot be inefficient. The difficult part of this Lemma is to prove that if there is no solution to (2'), then there must exist a solution for (3'). The meaning of this implication of the above version of Farkas’ Lemma is that it proves: Proposition 9. Given w, y, α∈A* and assumptions 1-2, the portfolio α is efficient if

and only if it is optimal. With this proposition, we have finished proving the Theorem that was stated at the beginning of this section. III. Further Interpretations and Applications By now we have provided an essentially complete characterization of when a given portfolio is efficient or inefficient. We also demonstrated that given our assumptions, a portfolio is either optimal or inefficient. Furthermore, we have also shown how to improve upon an inefficient portfolio. In this section we seek to illustrate the applications of the above sets of linear conditions, and relate our results to the literature. As noted above, Proposition 7 translates the problem on whether a portfolio is efficient into a set of linear inequalities. Given w , y in order to find out whether a given portfolio α∈A* is efficient we have to follow a three stage algorithm: 12 The second part Proposition 12.3 in Yaari (1971), states that if A is any matrix, then exactly one of the following two assertions is true: (i) the equation Ax = 0 has a (strictly) positive solution; (ii) the inequality y'A ≥ 0 has a solution. (Yaari, uses a different notation for a semi-positive vector.) As in his Proposition 12.4, the above version of the Lemma follows from adjoining the vector -e to the matrix C, as the last column of A = [C, -e], so that A is n x (S+1). If x = (ψ, ψ0) then the first assertion translates to the existence of a strictly positive ψ and ψ0 such that Cψ -ψ0e = 0. By normalizing ψ0 = 1, this is condition (3'). By replacing y by δ, the second assertion involves the existence of δ such that δ'C ≥ 0, and such that δ'(-e) ≥ 0, with at least one of the S+1 inequalities being strict. This is equivalent to condition (2'), where the condition of semi-positiveness is guaranteed by the full row rank of matrix C.

13

(i) Rank the states according to the magnitude of x(αs) = y(s) + wΣαi ri (s), (and check that x(s) ≥ x for all s);

(ii) For each asset i and state s, construct Ci (sα) according to (1);

(iii) Check whether there exists an MCSSD arbitrage portfolio δ that solves the set of linear inequalities in (2). To illustrate this algorithm, we now apply it to an almost trivial portfolio problem. Consider the case presented in the attached table, of three equally probable states of nature and two assets, whose return ri(s) is as specified in the table. Assume that x = y(s)

≡ 0 and w =1. We now seek to assess whether the portfolio α = (1.5, -0.5) is efficient or

not. This portfolio generates the future wealth distribution x(α) stated in the table, where the ranking of states is in accordance with Assumption 2. The fifth and sixth columns present the two cumulative returns (absolute concentration): Ci(s| α).

S r1 r2 X(α) C1(s| α) C2(s| α)

1 1 2 0.5 1/3 2/3 2 1 1 1 2/3 1 3 1 0.5 1.25 1 7/6

According to Proposition 7, in order to find out whether portfolio α is inefficient, in stage

(iii) of our algorithm we should seek a non-zero solution to the following set of 4 inequalities in the two unknowns, δ1 and δ2:

δ1 + δ2 ≤ 0 ; (1/3)δ1+ (2/3)δ2 ≥ 0 ; (2/3)δ1 + δ2 ≥ 0 ; δ1 + (7/6)δ2 ≥ 0. It is easy to see here that δ = (-1, 1) solves these inequalities. According to Proposition 7,

this finding has two implications: first, the portfolio α is inefficient; second, a small enough decrease in the first asset, with a parallel increase in the second asset is guaranteed to provide an improved portfolio that would be preferred by all risk-averse investors. Needless to say, this example was indeed almost trivial, and one could have verified directly that the portfolio α was inefficient.13 The merit of our proposed approach is that it applies generally, to much more complicated situations, where until now there was no

13 A less trivial example, with three assets and eight states is presented in Shalit and Yitzhaki (1994). As explained in the introduction, they nevertheless do not consider the possibility of altering the entire portfolio, and examine only the pair-wise substitution of two assets.

14

algorithm to assess whether a given portfolio is efficient or not; or to identify a direction for improvement, when a portfolio turns out to be inefficient [see the quote from Levy (1998) in the introduction]. In practical applications, the third stage of our proposed algorithm can be done with a linear programming model, or with a numerical optimization program [see Shalit and Yitzhaki (2000)]. In Relating our main Theorem to the literature, it should be emphasized that when Bawa et al. (1985) considered the possibility of efficient portfolios that were non-optimal, they were concerned with the efficiency and optimality of wealth distributions from a given discrete set of distributions. That is, they did not explicitly considered the case that concerned us here, where all the feasible wealth distributions are derived from an initial wealth, through the selection of a portfolio of assets. The thrust of our ability to provide a full characterization in that framework through linear conditions is due to the use of a local approach, which given the concavity of preferences, easily extends also globally. This local approach has also an important additional advantage in allowing for the, practically important, existence of a non-financial source of future wealth (such as human capital). The existence of a marginal α-conditional (MCSSD) arbitrage portfolio signals that all risk-averse investors may benefit from revising the composition of the portfolio α. Our definition of an arbitrage portfolios that is conditional on a distribution x(α), contrasts in several respects the standard definition of an (unconditional) arbitrage portfolio. As usually defined, a non-zero vector δ is an absolute arbitrage portfolio if it satisfies: (4) Σi δi ≤ 0 and Σi δi ri (s) ≥ 0 for all s .

If δ is an absolute arbitrage portfolio, the wealth distribution x(α+δ) = y + w(α + δ)r dominates the wealth distribution x(α) = y + wαr by first-degree stochastic dominance. That is, x(α+δ) would be preferred to x(α) by all (not necessarily risk-averse) investors, and for every combination (y, α, w), where w > 0 and α∈A. Obviously, the current

concept of an MCSSD arbitrage portfolio is much weaker (and thus admits a larger set of possible solutions), since it conditional on a distribution x(α), and since it requires only second, rather than first, degree stochastic dominance. Finally, we would like to relate what we did here to the important article by Peleg

15

and Yaari (1975). In that article, Peleg and Yaari provide a necessary and sufficient condition for a wealth distribution to be efficient, in terms of the existence of appropriate state prices. Their condition for efficiency is in fact equivalent to our condition for optimality, as stated in Proposition 8. To see the connection one should look at the parameters φs in our Proposition 8 as the ratios of state prices Ps and state probabilities πs.

It should be emphasized however, that Peleg and Yaari (1975) do not mention assets and portfolios and their primitive is the space X of available wealth distributions. As such, they do not provide a proposition like our proposition 7 that enables one not only to check whether a given portfolio of assets is efficient, but also to identify a direction for improvement when it is inefficient. As stated, this is the main contribution of this paper. IV. Some Applications to Continuous Distributions A. The case of joint normality In this section we apply the preceding results, which were derived for a finite set of states of nature, to the case of continuous random variables (without verifying that the expected utility integral converges). Accordingly, y, ri and x = x(α) will now be treated as continuous random variables. To simplify the notation, we will denote here by indices α and δ the wealth distribution and return that are generated by the given portfolio α∈A, and by the conditional arbitrage portfolio δ. Let Fα(⋅) denote the cumulative distribution

function of xα. The cumulative conditional first moment distributions Ci(X) in (1) can now be defined as: (5) Ci (X α) = ∫X′≤X E(ri X′)dFα(X′) ,

where E(riX′) = E(ri xα = X′) is the expected return on asset i, conditional on that the wealth level generated by α is equal to X′. Ci (X α) represents the expected return on

asset i, conditioned on that the wealth level generated by α is not higher than X. To apply the condition for inefficiency in (2), we redefine: (6) Q(Xδ, α) = Σi δi Ci (X α) .

Let us now investigate the implications of applying the MCSSD condition for inefficiency to the case where the distribution of assets ri is multivariate normal and with

16

no non-financial wealth (y = 0.). In this case, xα = wΣi αi ri , is normal as well, and (7) E(riX) = µi + βi(X - µα) ,

where, µi = E(ri), µα = E(xα), and, as usually defined, βi = Cov(ri, xα)/Var(xα) is the beta coefficient of asset i relative to the distribution xα. Further denote µδ = Σi δi µi and βδ = Σiδiβi = Cov(Σiδiri , xα)/Var(xα). By the definition of Q(Xδ, α) in (6), condition (2) then becomes:

(8) Q(Xδ, α) = ∫X’≤X [µδ + βδ (X’ - µα)]dFα(X’) ≥ 0 for all X.

Now define gα(X) = -∫X’≤X (X’ - µα )dFα(X’) / ∫X’≤X dFα(X’). Condition (8) can then be

presented as requiring that (9) µδ ≥ βδ gα(X) , for all X .

It can be established that gα(X) is non-negative, and monotonically declining from infinity to zero. For X at infinity condition (9) means that µδ ≥ 0; and for X at minus infinity it requires βδ ≤ 0. Thus, as an application of our theorem above,

Proposition 10: Given the joint normality of all assets, xα = wαr is efficient if and only if there exists no non-zero solution δ for the system of three linear conditions: (10) Σi δi ≤ 0 , µδ = Σi δi µi ≥ 0 , βδ = Σi δi βi ≤ 0 ,

with at least one strict inequality. The intuitive meaning of proposition 10 is well known. In the mean-variance framework implied by joint normality, SSD efficiency is equivalent to mean-variance efficiency [see also Bodurtha and Shen (2001)]. Alternatively, where xα+δ = xα + wxδ , in the mean-variance framework, an MCSSD arbitrage portfolio has to marginally increase the mean [i.e. wE(xδ) = E(xα+δ) - E(xα) ≥ 0], or to marginally decrease the variance of the wealth distribution [wβδ = Var(xα+δ) – Var(xα) ≤ 0].

17

B. A necessary condition for efficiency It is reassuring that in the case of joint-normality the MCSSD condition in (2) reduces to the known condition for mean-variance efficiency. This, though, serves to point out that in the general case the necessary and sufficient conditions in our theorem are not stated in terms of parameters customarily used in finance. We aim now to use standard finance terminology, such as the expected returns and the betas of assets, to present a set of general necessary (though not sufficient) conditions for the efficiency of a portfolio. Proposition 11. If α is an efficient portfolio then it is impossible to find a non-zero vector δ such that: (11) Σi δi ≤ 0 , Σi δi µi ≥ 0 and Σi δi (µi - β°i Γα ) ≥ 0 .

Here, β°i is the (Gini) beta coefficient of asset i with respect to the wealth distribution

x, and Γα is the Gini mean-difference of the portfolio (for properties of the Mean-Gini CAPM see Shalit and Yitzhaki, 1984). The analogy between Propositions 7 and 11 is clear. The condition that E(xδ) = Σi δi

µi ≥ 0 is similar to Samuelson's (1967) condition for characterizing an efficient portfolio

when assets are independently distributed. A necessary condition for a wealth distribution x′ to SSD-dominate another distribution x is that the expected return of x′ be higher. The interpretation of the third condition in (13), that Σi δi (µi - β°i Γα ) ≥ 0, is that the risk-adjusted return of the dominating portfolio x′ has to be higher than the risk-adjusted

return of the dominated portfolio x.14 V. Conclusions In this paper we provided a complete characterization of when a given portfolio is efficient in the sense of second-degree stochastic dominance (SSD). Our procedure for checking whether a portfolio is efficient identifies a direction for improvement, when it turns out to be inefficient. Our general necessary and sufficient condition for efficiency was shown to reduce to the standard case of mean-variance efficiency in the case of joint-

14 The necessary condition for stochastic dominance can be stated in the mean-Gini framework because the Gini can be used to construct necessary conditions for SSD.

18

normality. Our analysis also demonstrated that in the context of portfolio-generated wealth distributions, the distinction that was made in the literature between optimal and efficient portfolios is unnecessary. The above characterization of efficient portfolios enables one to answer the question we started with. A portfolio manager can be said not to act according to the best interest of her clients, if she offers an inefficient portfolio; provided that the probability distribution is agreed upon, our paper offered a feasible way of examining whether this is the case. The present analysis can be extended in many directions. By way of generalizing the results, one could examine the implications of binding constraints on the feasible portfolios, and also check the conditions for convergence in the case of continuous random variables. To obtain more specific results, one might check how the MCSSD criterion applies to various specifications of the joint distributions of assets, and in particular to the cases considered in the literature thus far, were the portfolios were restricted in various ways. Finally, the difficult task of specifying the probability distribution notwithstanding, one can follow Shalit and Yitzhaki (2000) in applying the criterion that was offered here in order to empirically check for the efficiency of particular portfolios, such as "the" market portfolio, or the portfolios offered by various financial agencies.

19

Appendix Proof of Proposition 1: What we have to prove is the quasi-concavity of U in α. This follows from the concavity of u. Formally, assume that for two portfolios α′ and α″ in A, Eu[x(α′)] ≥ Eu[x(α)] and Eu[x(α″)] ≥ Eu[x(α)]. We then have to prove that for any 0 < t < 1 Eu[x(tα′+(1-t)α″)] ≥ Eu[x(α)] . By linearity, x(tα′+(1-t)α″s) = tx(α′s) + (1-t)x(α″s) for every s. Thus, due to the concavity of the utility function, for every state s, u[x(tα′+(1-t)α″s)] ≥ tu[x(α′s)] + (1-t)u[x(α″s)]. By taking expectations, Eu[x(tα′+(1-t)α″s)] ≥ tEu[x(α′s)] + (1-t)u[x(α″s)] ≥ Eu[x(αs)].

Proof of proposition 2: consider any u∈Ω. Since α is inefficient, there is α'∈A such that U(u, α') ≥ U(u, α). If the inequality is strict we are done. If not, consider an

intermediate portfolio α" = tα + (1-t)α'. As in the proof of Proposition 1, we must have u[x(α″s)] ≥ tu[x(αs)] + (1-t)u[x(α′s)] for all s. Because of assumption 1-(iii) there must be a state s' for which x(αs') ≠ x(α's'). Because u∈Ω it is strictly concave, so

that the last inequality must be strict for s'. By taking expectations we obtain the desired result. Proof of Proposition 3: By assumption 1-(i), there exists a portfolio α° such that Σi α°i = 1, and Σi α°iri (s) ≥ 0 for all s, with strict inequality for some s. If Σi αi < 1, then, by looking at the portfolio α+tα°, for t > 0 that is small enough so that Σi (αi+tα°i) ≤ 1, we

obtain a portfolio in A that improves on α, by first order stochastic dominance. Proof of Proposition 4: If α is optimal, there is u* ∈Ω for which U(u*, α) ≥ U(u*, α') for every α'∈A, distinct from α. Suppose α is inefficient then by Proposition 2, there

is α' such that U(u*, α') > U(u*, α). This is a contradiction. Proof of Proposition 5: Given that, by definition, local inefficiency implies global inefficiency, all we have to prove is the converse. Suppose that α∈A is globally inefficient. Then there exists an alternative portfolio, α′∈A, such that for every u∈Ω, α′

improves on α relative to u. For any such u, and for any scalar t, 0 < t < 1, from Proposition 1, α″ = tα′ + (1 - t)α = α + t(α′ - α) improves on α relative to u. For a small enough t, δ = t(α′ - α) is then a marginal α-conditional arbitrage portfolio, which proves

20

that α is locally inefficient. Proof of Proposition 6: What has to be shown is that it does not limit the generality to assume that all the wealth levels x(αs) for the various states are distinct. Suppose then that there are two states, s′ and s″, for which x(αs′) = x(αs″) = x°. We will

rearrange the space of states of nature by consolidating the two states into a single combined state, s°, that occurs with probability πs° = πs′ + πs″. For that purpose we also redefine the return to each asset i in state s° : ] + [ /)]"(sr + )'(r[ = )s( r s"'si"si's

oi ππππ s .

Consider now a local differential departures, δ, from the initial portfolio α. We need that α+δ improves on α in the consolidated space if and only if it does so in the original space. Yet, due to the differentiability of any u∈Ω, πs′u(x(α+δs′)) + πs″u(x(α+δs″)) = πs′[u(x°) + u'(x°)Σi δiri(s′)] + πs″[u(x°) + u'(x°)Σi δiri(s″)] = πs°[u(x°) + u'(x°)Σi δiri(s°)] = πs°u(x(α+δs°)). Thus the comparison of α+δ with α, in the relevant states is not

affected by the consolidation of the two states into one. Proof of Proposition 7: By Proposition 5, it suffices to consider the case of local inefficiency. Sufficiency. From Definition 1 and by our assumption of differentiability of u∈Ω, α∈A

is locally inefficient if there is a non-zero marginal α-conditional arbitrage portfolio δ, such that (α+δ)∈A and such that for every u∈Ω, α+δ improves on α relative to u

(that is, such that α+δ dominates α by SSD). Given differentiability of u, this requires that for every utility function u∈Ω, dU ≡ wΣs πs u′[x(αs)]Σi ri (s)δi be positive. First note that given that by (2) Σi δi ≤ 0, it follows that Σi (αi+δi) ≤ 1. Furthermore, since for α∈A* it is the case that x(αs) > x for all s, we can, if necessary, scale δ proportionally down to guarantee that also x(α+δs) ≥ x for all s, and thus guarantee that α+δ∈A. Further note that Assumption 1-iii on the linear independence of the returns ri(s) also implies the linear independence of the cumulative returns Ci(s).15 Thus, if δ≠0, then at least one of the S weak inequalities Q(sδ, α) ≥ 0, in (2) has to be strict. Given that w > 0, all that we have to prove is that if δ satisfies (2) then for every u∈Ω,

15 This will be clarified below when we introduce matrix notation.

21

Yet if (2) holds, then all the Q(sδ, α) components in equation (A1) are non-negative and at least one is positive. Since every u∈Ω, is increasing and strictly concave, each of

the components enclosed in curled brackets in equation (A1) is positive. Thus the entire expression (A1) is positive, proving that δ is a marginal α-conditional arbitrage portfolio and that portfolio α is locally (and globally) inefficient. Necessity. Suppose that α is locally inefficient. Then by Definition 3 and the assumed differentiability of u∈Ω, there has to be a differential portfolio δ such that α+δ∈A and such that dU/w = Σs πs u′[x(αs)]Σi ri (s)δi ≥ 0 for all s, and for all u∈Ω. Given that α∈A* and α+δ∈A, we have Σi αi = 1 and Σi (αi+δi) ≤ 1, so that Σi δi ≤ 0. What is left to prove is that for this δ, Q(sδ, α) ≥ 0 for all s. Suppose that for some s, Q(sδ, α) < 0, then we will construct a utility function u∈Ω, for which dU/w is negative, meaning

that α+δ does not improve on α relative to u, in contradiction to the assumption that it does so for every u∈Ω. Assume then that Q(s'δ, α) < 0 for some s' < S.16 For scalars, η > 0 and ε > 0, we then define: u'[x(αs'-1)] - u'[x(αs')] = η; while for all s≠s', u'[x(αs-1)] - u'[x(αs)] = u'[x(αS)] = ε. This defines u'[x(αs)], for all s, so that the marginal utilities are positive and strictly declining in wealth x. In the range of wealth x∈[x(αs), x(αs+1)], for s, 1, 2, ... S-1, one can extend the definition of the marginal utility by: u′(x) = u'[x(αs)] + u'[x(αs+1)] - u'[x(αs)][x - x(αs)] / [x(αs+1) - x(αs)]. The resulting

u'(x) thus defined is continuous, positive and strictly decreasing. For this constructed utility function, we then obtain that dU/w in (A1) equals ηQ(s'δ, α) + εΣs≠s'Q(sδ, α).

16 The construction for the case where Q(sδ, α) < 0 only for s=S is practically identical.

(A1)

. 0 > ... + ),| 1Q(S S)]|[x(u 1)]S|[x(u + ),| Q(S S)]|[x(u

= ... + )| 1(S R S)]|[x( u )] 1S|[x( u + )| (S R )]| [x(u

= ... + (s)] r [ S)]|[x(u 1)]S|[x(u +

+ (s)] r [ S)]|[x(u = (s)r s)]|[x(u =w /dU

///

iii

//ii

i

/

is

1S

1=si

i

//

is

S

1=si

i

/ii

n

1=i

/s

S

1=s

αδαααδα

αδαααδα

πδαα

πδαδαπ

ΣΣ

ΣΣ

ΣΣΣΣ

S

22

Now, if the second sum is negative, then dU/w is also negative. And if that sum is positive then define: η = 2ε[Σs≠s'Q(sδ, α)]/[-Q(s'δ, α)] > 0. This will also make dU/w

negative, obtaining the desired contradiction. Proof of Proposition 8: Necessity. Suppose that α∈A* is optimal, then there exists u∈Ω for which U(u, α) is maximal in A and thus also in A*. Given that u is differentiable we can define φs = u′[x(αs)]. Given that α is optimal and interior and

given that u is strictly increasing and concave, the first order conditions (3a) have to meet. Given Assumption 2, (3b) has to be fulfilled as well. Sufficiency. We have to prove the existence of u∈Ω for which α is optimal. We construct such a utility function by first defining u′[x(αs)] = φs for all s, and then

extending the definition for all intermediate wealth levels, as in the proof of the preceding proposition.17 Condition (3b) guarantees that this constructed utility function will be differentiable increasing and strictly concave. To prove that α is optimal for the constructed u∈Ω, note that the set of equations in (3b) presents the first-order conditions

for maximizing the constructed utility function, given the budget constraint Σi αi = 1. Given the concavity of the utility function, and given our assumption that the portfolio α generates consumption in the interior of X, the first-order conditions are sufficient to guarantee the optimality of the portfolio α.

17 In the range of wealth x∈[x(αs), x(αs+1)], for s, 1, 2, ... S-1, we then define the marginal utility by: u′(x) = φs + (φs+1 - φs)[x - x(αs)] / [x(αs+1) - x(αs)]. Given that by (3b), the φs are in strictly decreasing order, we have thus defined u'(x) as a continuous, and strictly positive and decreasing function.

23

References Bawa, Vijay S., and Daniel L. Goroff, 1982, Admissible Efficient, and Best Choices

Under Uncertainty, working paper 81/82 –2-8, Department of finance, University of Texas-Austin.

Bawa, Vijay. S., James M. Bodurtha, Jr., M. R. Roa, and Hira L. Suri, 1985, On Determination of Stochastic Dominance Optimal Sets, Journal of Finance 40, (2, June), 417–431.

Bodurtha, James N., and Qi Shen, 2001, Normally Distributed Admissible Choices are Optimal, working paper, McDonough School of Business, Georgetown University.

Canner, Niko, N. Gregory Mankiw, and David N. Weil, 1997, An Asset Allocation Puzzle, American Economic Review 87, (1, March), 181-191.

Errunza, Vihang, Ked Hogan and Mau-Wei Hung, 1999, Can the Gains from International Diversification Be Achieved without Trading Abroad?, Journal of Finance 54, 2075-2107.

Levy, Haim, 1998, Stochastic Dominance: Investment Decision Making under Uncertainty (Kluwer Academic publishers).

Mayshar, Joram, and Shlomo Yitzhaki, 1995, Dalton-Improving Indirect Tax Reform, American Economic Review 85, (September), 793-807.

Peleg, Betzalel and Menahem E. Yaari, 1975, A Price characterization of efficient Random variables, Econometrica 43, (2, March), 283-292.

Samuelson, Paul A., 1967, General Proof that Diversification Pays, Journal of Financial and Qualitative Analysis 2, 1–13.

Shalit, Haim and Shlomo Yitzhaki, 1984, Mean-Gini, Portfolio Theory and the Pricing of Risky Assets, Journal of Finance 39, 1449–1468.

____ and ____ ,1994, Marginal Conditional Stochastic Dominance, Management Science 40, (5, May), 670-684.

_____and _____ , 2000, Solving the Portfolio Allocation Puzzle, working paper. Varian, Hal R., 1983, Nonparametric Tests of Models of Investor Behavior, Journal of

Financial and Quantitative Analysis 18, (3, September), 269–278. Yaari, Menahem E., 1971, Linear Algebra for social Sciences (Prentice-Hall). Yitzhaki, Shlomo, 2001, The GMD: A Superior Measure of Variability for Non-Normal Distributions, working paper, The Hebrew University.