Almost Stochastic Dominance and Efficient Investment Sets

A major drawback of Mean-Variance and Stochastic Dominance investment criteria is that they may fail to determine dominance even in situations when all “reasonable” decision-makers would clearly prefer one alternative over another. Leshno and Levy [1] suggest Almost Stochastic Dominance (ASD) as a remedy. This paper develops algorithms for deriving the ASD efficient sets. Empirical application reveals that the improvement to the efficient sets implied by ASD is substantial (64% reduction for FSD). Direct expected utility maximization shows that investment portfolios excluded from the ASD efficient set would not have been chosen by any investors with reasonable preferences.

• Stochastic Dominance; Efficient Investment Set; Investment Choice

1. Introduction

The most popular tool for portfolio optimization and selection is the Mean-Variance (MV) analysis. The MV criterion conforms to Expected Utility (EU) maximization in the special case of normal return distributions¹. Moreover, if rates of return are relatively small in absolute value, MV provides a good approximation for EU maximization, even when the return distributions are not normal (see Levy and Markowitz [5], and Markowitz [6]). In all other cases, the MV criterion may lead to choices contradicting EU maximization, and the selection criteria corresponding to EU maximization are the Stochastic Dominance (SD) criteria².

The MV and SD efficient sets are important tools which allow the investor or investment company to narrow down the menu of relevant investments under consideration: inefficient investments should not be selected by any investor (in a given class of preferences), and therefore one can safely focus only on investments in the efficient set. Thus, it is not surprising that the MV and SD efficient investment sets have been the focus of many studies³. It is clear that the smaller the efficient set, the more helpful the investment criterion⁴.

While MV and SD are the most widely used investment criteria, they both suffer from the following drawback. Both the MV and SD criteria may fail to determine dominance even in situations where all “reasonable” investors would clearly prefer one investment to another. For instance, consider the following rather extreme example: suppose prospect G yields a return of 2% with a probability of and a return of 3% with a probability of. Prospect F yields a return of 1% with a probability of and a return of 100% with a probability of. There is no MV or SD dominance of one prospect over the other in this case, i.e. both F and G are in the efficient MV and SD sets⁵. However, it is obvious that any “reasonable” investor would prefer investment F. Thus, a serious drawback of the standard investment criteria is that they do not exclude from the efficient set investments which “common sense” tells us that no real-world investor would ever select⁶.

The Almost Stochastic Dominance (ASD) and Almost Mean-Variance (AMV) criteria, developed by Leshno and Levy [1], provide a solution to such paradoxical situations⁷. The idea at the heart of ASD can be graphically presented by Figure 1, which describes the cumulative return distributions of prospects F and G in the example above. The cumulative distribution of F is almost entirely below of the cumulative distribution of G. Thus, F “almost” dominates G by First-order Stochastic Dominance (FSD). However, there is a small “violation” region where F is above G, and therefore there is no FSD dominance of F. Another way to present the nondominance of F over G is to state that there are various “pathological” non-decreasing utility functions that yield higher expected utility under G. The idea of ASD is that if the area between the two cumulative distributions that causes the violation of FSD (area A₁ in Figure 1) is very small relative to the total area enclosed between the two distributions (A₁ + A₂), then a dominance relationship holds for all “reasonable” investors. Thus, ASD describes dominance for the set of all “reasonable” preferences, excluding extreme preferences that are considered “pathological”. Although the ASD criterion is fairly recent, it has been already employed in several studies⁸.

This paper investigates two questions: 1) How significant is the reduction of the efficient sets obtained by employing ASD? And 2) What are the characteristics of the “pathological” preferences excluded? The paper is organized as follows. In the next section we review the Almost FSD (AFSD) and Almost SSD (ASSD) criteria. In Section 3 we develop algorithms for determining AFSD and ASSD dominance for empirical return distributions (with n return observations). In Section 4, we construct the AFSD and ASSD efficient sets for the 100 Fama-French portfolios, and we compare these sets with the standard FSD and SSD efficient sets. This section also investigates the degree to which preferences have to be “pathological” to be excluded from the ASD preference set. Section 5 concludes.

2. The Almost Stochastic Dominance Criteria

We discuss and employ two Almost SD criteria: AFSD (Almost First degree SD) and ASSD (Almost Second degree SD).

2.1. AFSD

Consider two alternative investments, F and G with cumulative distributions F(t) and G(t), respectively. Denote

the range of possible outcomes (for both prospects) by S. F dominates G by FSD if and only if F(t) ≤ G(t) for all (with at least one strict inequality). Almost FSD dominance of F over G means that F(t) ≤ G(t) for most of the range S, except for a relatively small segment that “violates” the dominance. Define the range over which FSD is violated by S₁:

Denote the ratio between the area of FSD violation and the total area between the cumulative distributions by ε₁:

For ε₁ < 0.5 F is said to “ε₁-Almost FSD dominate” G. The smaller ε₁, the stronger this dominance. Leshno and Levy [1] prove that if F ε₁-Almost FSD dominates G then E_Fu ≥ E_Gu for all preferences, where is the set of all “well-behaved” or “reasonable” non-decreasing utility functions, given by:

Note that for a given range S, the smaller the relative area violation, ε₁, the larger the set of preferences. When ε₁ = 0, i.e. there is no violation area at all, coincides with the usual set of all non-decreasing utility functions, U₁, and AFSD reduces to the standard FSD criterion.

Moreover, note that if the actual violation area is ε₁, then F ε-Almost FSD dominates G for any ε > ε₁. In other words, if the allowed area violation, ε, is larger than the actual area violation, ε₁, F ε-Almost FSD dominates G in the sense that for any F is preferred over G.

Finally, note that all the preferences that are included in U₁ (the set of all non-decreasing preferences) but are not included in are those “pathological” preferences excluded in order to avoid paradoxical choices. We investigate the characteristic of these preferences in Section 4.

2.2. ASSD

F dominates G by SSD if and only if

for all. Assume that this inequality holds for most of the range S, but not for all of it. Denote the area of SSD violation by:

and denote the complement area of S₂ by (i.e. ). Define ε₂ as the ratio:

For ε₂ < 0.5 F is said to “ε₂-Almost SSD dominate” G. If F ε₂-Almost SSD dominates G then E_Fu ≥ E_Gu for all preferences, where is the set of all “reasonable” risk-averse utility functions given by:

For ε₂ = 0 is the set of all non-decreasing concave preferences and ASSD reduces to the standard SSD criterion. For proof of the ASD criteria and more detail, see [1,27]⁹.

3. ASD Algorithms

For FSD, SSD, and MV there are well-known algorithms for constructing the efficient sets. However, for AFSD and ASSD such algorithms have not yet been developed, and therefore, the effectiveness of these novel investment criteria has not yet been tested. Below we construct algorithms for obtaining the AFSD and ASSD efficient sets. In the next section we employ these algorithms to derive the empirical efficient ASD sets, and to compare them with the standard SD efficient sets.

3.1. AFSD Algorithm

The algorithms for constructing the empirical ASD efficient sets are based on pair-wise comparisons of all the different investments. If an investment is ASD dominated by some other investment, it is excluded from the efficient set. If it is not dominated by any other investment, it is included in the efficient set¹⁰. Generally, empirical studies rely on historical return observations, e.g. a set of n annual or monthly returns on various assets. Suppose that we want to compare two investments, F and G, each with n return observations. Denote the observations by x_i and y_i, corresponding to F and G, respectively (i = 1, 2, ···, n). Assume, without loss of generality, that the observations are ordered such that: x₁ ≤ x₂ ≤ ··· ≤ x_n and y₁ ≤ y₂ ≤ ··· ≤ y_n.

The total area between the cumulative distributions F and G is given by:      .

The area of violation of the FSD dominance of F over G is given by         

(see Figure 2(a) and Leshno and Levy [1]). Thus, the relative area violation is given by:

Recall that ε₁ is the actual area violation for F and G.

Thus, if we are considering the AFSD efficient set with some ε, then if ε₁ ≤ ε, F will exclude G from the efficient set, but if ε₁ > ε, it will not.

3.2. ASSD Algorithm

Note that ε₂ can be written as follows:

where the last equality stems from the fact that

(because), and the fact that

.

Recall that

where E_F and E_G denote the mean returns of F and G (see [8]). Finally, let us denote the SSD violation area in short by VA,

to obtain:

.¹¹(8)

The ASSD algorithm is composed of the following steps:

1) Calculate the mean returns:

,.

2) For all i: 1, 2, ···, n define

.

Every pair of return observations (x_i, y_i) adds a “block” of height to the area enclosed between the two cumulative distributions. δ_i is the area between the two cumulative distributions up to the i’th observation, where the range in which G > F makes a positive contribution to the area, and the range in which F > G makes a negative contribution, see Figure 2(b). Define δ₀ = 0.

3) Define by VA_i the total area violation up to the ith observation. Define VA₀ = 0. The total area violation for the entire distributions, VA in Equation (8), is given by VA = VA_n.

4) For every i: 1, 2, ··· n:

o     if δ_i ≥ 0: VA_i = VA_i–1 (there is no SSD violation and therefore there is no contribution to the SSD violation area. See, for example, the cases i = 1 and i = 4 in Figure 2(b)).

o     if δ_i < 0 and δ_i–1 ≥ 0 (i.e., this is an area of violation, following a block of no violation): VA_i = VA_i–1 +. (See, for example, the case i = 2 in Figure 2(b)).

o     if δ_i < 0 and δ_i–1 < 0 (i.e., this is an area of violation, following a previous block of violation): (See, for example, the case i = 3 in Figure 2(b)).

Figure 2(b) illustrates these three different cases (note that the second and third cases are different—if we had applied VA_i = VA_i–1 + in the third case we would have double-counted the previous violation area δ_i–1, see Figure 2(b)).

5) VA = VA_n. Plug VA, E_F, and E_G into Equation (8) to obtain the relative are violation, ε₂.

A Matlab program for deriving the AFSD and ASSD efficient sets for any number of investments and any desired level of ε is available from the author upon request. For 100 assets and 50 return observations the program constructs the ASD efficient sets within a matter of seconds.

4. Empirical Results

4.1. The ASD Efficient Sets

To investigate the magnitude of the reduction of the efficient sets achieved by employing ASD instead of the standard SD, we perform the following analysis. As our investment universe we take the Fama-French 100 portfolios formed on size and book-to-market, with annual returns in the period 1956-2005 (for more details on the construction of these portfolios, and for the data itself, see Ken French’s website: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library-.html). We first construct the standard MV, FSD, and SSD efficient sets, employing the standard algorithms in the literature (see, for example, Levy 2006). Next, we employ the algorithms developed in Section 3 to construct the AFSD and ASSD efficient sets for various levels of ε. We should stress that the results are not sensitive to the sample employed: similar results are obtained for other assets and sample periods.

Figure 3 illustrates one empirical comparison. The figure shows the cumulative return distributions of two portfolios in the FSD efficient set: portfolio 96, which is the portfolio with the lowest variance of all of the 100 Fama-French portfolios, and portfolio 29 (portfolio 96 is in the largest stock decile, with medium book-to-market; portfolio 29 is in the third smallest stock decile with high book-to-market). Note that portfolio 29 does not dominate portfolio 96 by FSD, because of a small FSD area violation around a return value of –20%, where P29 > P96 and P29 and P96 denote the respective cumulative return distributions (see Figure 3). However, the relative area violation is very small, and P29 ε-AFSD dominates P96 for ε = 0.01. Thus, P96 is in the FSD efficient set, however, it is not in the AFSD efficient set for ε ≥ 0.01. Indeed, looking at the cumulative return distributions it is evident that the preferences that assign higher expected utility to P96 than to P29 have to be quite extreme.

The FSD, MV, and SSD efficient sets contain 74, 13and 9 of the 100 portfolios, respectively. These sizes of the efficient sets (as a proportion of the total number of investments) are quite typical of the results in the literature obtained for other sets of assets. The sizes of the AFSD and ASSD efficient sets as a function of ε are shown in Figure 4. Note that the standard FSD and SSD sets correspond to the AFSD and ASSD for the special case of ε = 0 (i.e. no violation whatsoever is allowed). The AFSD and ASSD efficient sets decrease rather dramatically as ε increases.

An important question is what is a reasonable value of ε to employ? Levy, Leshno and Leibovich [26] experimentally estimate the relevant value as ε = 0.06. They find that if the area violation is smaller than this value, all subjects in their experiments choose the ASD dominating investment. With this experimental value, the reduction in the efficient sets is significant: The AFSD efficient set contains only 27 portfolios (a reduction of 64% relative to the 74 portfolios in the FSD efficient set), and the ASSD efficient set contains 7 portfolios (a reduction of 22% relative to the 9 portfolios in the SSD efficient set, see Figure 4).

Levy, Leshno and Leibovich experimentally find ε = 0.06. Hence, they exclude some preferences considered “unreasonable” as they do not characterize any of the subjects in their experiments. It is interesting to find the range of preference parameters corresponding to this area violation value. In other words, how “crazy” does the utility function have to be for portfolios outside of the ASD efficient set to be selected? To this question we turn next.

4.2. Direct Expected Utility Analysis

The portfolios excluded from the ASD efficient sets

could, in principle, be the optimal portfolios for some EU maximizers. The experimental results of [26] suggest that in practice individuals with these preferences are very hard to find. In order to shed more light on this issue, we ask the following question: we look at the AFSD efficient set with ε = 0.06, and we ask what preferences will imply a choice of investments outside of this AFSD efficient set. We focus on the AFSD efficient set because we will consider different preference classes, some of which are not risk averse (Prospect Theory preferences).

We first consider investors with CRRA preferences:

.

The relative risk aversion coefficient, γ, is typically estimated in the range 1 - 3 (see, for example, [29]). In our analysis we consider a much wider range: we take γ in the huge range 1 ≤ γ ≤ 100. For each value of γ (with increments of 0.01) we numerically find the optimal investment (by direct EU calculation for all the 100 portfolios), and we check whether this portfolio is in or out of the AFSD efficient set. We find that the optimal portfolios for all preferences in the range 1≤ γ ≤ 100 are included in the AFSD efficient set. Thus, we find that no individual with conceivable CRRA preferences chooses any investment outside the AFSD efficient set. We repeat this analysis for other standard preferences. For CARA preferences of the form

We find that no investor with parameters in the range 0 ≤ bW₀ ≤ 500, where W₀ is the initial wealth, chooses any investment outside the AFSD efficient set.

We also examine the case of the non-risk-averse Prospect Theory preferences suggested by Tversky and Kahneman [30]:

where x is the change in wealth relative to the current wealth, and α, β and λ are constants experimentally estimated as: α = 0.88, β = 0.88, λ = 2.25. The investor’s optimal investment is determined by the maximization of E[V(x)]¹². We find the optimal investment for preferences with loss aversion in the very wide range of 1 ≤ λ ≤ 10. Again, we find that all choices fall within the AFSD efficient set with ε = 0.06. Thus, it seems that the reduction of the efficient set comes at very little cost in terms of excluding investments that would have been chosen by any reasonable investors.

5. Conclusion

Empirical and experimental results suggest that the standard SD and MV efficient sets include a large number of investments that are very unlikely to be chosen by any real-world investor. Almost FSD (AFSD) and Almost SSD (ASSD) are new criteria that allow us to eliminate such investments from the efficient sets, by excluding very unrealistic preferences that are considered “pathological”.

In this study we employ MV, SD, and ASD criteria with two purposes:

1) To find the reduction of the efficient sets obtained with the ASD criteria.

2) To examine whether the preferences excluded by ASD are indeed unrealistic.

The algorithms developed in this paper are used to derive the AFSD and ASSD efficient sets. The reduction of the efficient sets obtained by ASD is significant: for the set of 100 Fama-French portfolios formed on size and book-to-market we find that the AFSD efficient set with ε = 0.06 includes only 27 portfolios (relative to the 74 portfolios in the standard FSD efficient set), and the ASSD efficient set includes only 7 portfolios (relative to the 9 portfolios in the standard SSD efficient set).

Experimental evidence and direct EU maximization suggest that this improvement comes at very little cost in the sense of excluding portfolios that would have been chosen by any real-world investors. To be more specific, we examine three commonly-employed preference families and we find that the excluded preferences are characterized by very extreme parameter values. Thus, it seems justified to exclude these preferences from the set of preferences relevant for economic decision-making.

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