On the dual test for SSD efficiency: With an application to momentum investment strategies☆
Introduction
Theoretically, stochastic dominance (SD) is an attractive approach to analyzing decision making under uncertainty. In nonparametric fashion, SD allows the data ‘to speak for themselves’, rather than being forced to speak the idiom of prior assumptions about the preferences of the decision maker and the statistical distribution of the choice alternatives. For example, the popular criterion of second order SD (SSD) assumes only non-satiation and risk aversion for the preferences and it effectively considers the entire return distribution rather than a finite set of moments.
The traditional approach to apply SSD uses crossing algorithms that check if the cumulated empirical distribution functions of a pair of choice alternatives ‘cross’ at some point (see, for example, Levy (1992, Appendix A)). Unfortunately, the number of pairs increases rapidly as the number of choice alternatives increases, and the computational burden explodes if many alternatives are considered.1 This problem arises naturally in the context of investment problems where the investor can diversify fully across financial assets and hence infinitely many choice alternatives exist. To circumvent this problem, Kuosmanen, 2004, Post, 2003 develop linear programming tests for testing if a given portfolio is SSD efficient relative to all possible portfolios formed from a set of assets.
The Post test basically asks if we can find an increasing and concave utility function for which the evaluated portfolio is optimal. Post also derived a dual formulation of this test. The dual test asks if we can find a portfolio that has more favorable conditional means (conditional upon the return of the evaluated portfolio) than the evaluated portfolio. While the primal formulation allows for directly imposing restrictions on the utility function (for example, to impose decreasing absolute risk aversion), the dual formulation is useful for directly altering the restrictions placed on the portfolio weights (for example, to restrict short-selling or riskless borrowing).
The dual test can be seen as a simplification of the more general Kuosmanen (2004) test. While the dual solution portfolio has more favorable conditional means than the evaluated portfolio, it generally does not SSD dominate the evaluated portfolio, because the ranking of the solution portfolio is not taken into account. By contrast, the Kuosmanen (2004) test does account for the ranking and identifies a dominating benchmark portfolio, albeit at the cost of substantial additional computational burden. We may ask if the lack of a dominating benchmark portfolio is a limitation. After all, the objective is to test for efficiency of the evaluated portfolio and the efficiency classification is identical for the two tests. The solution portfolio—even if it dominates the evaluated portfolio—comes as a ‘side-product’ that is not likely to be robust to sampling variation. Still, a dominating benchmark portfolio could be useful for, for example, graphically illustrating an (sample) inefficiency classification.
This paper further elaborates on the dual SSD test:
- 1.
In Post (2003), the primal formulation was derived from the first principles of expected utility theory. The dual formulation was derived indirectly by using the duality theorem of linear programming. In this paper, we provide a direct proof of the dual formulation that applies also outside the context of expected utility theory. This is useful because the SSD rule is also economically meaningful in the context of some non-expected utility theories, including ones that allow for subjective distortion of probabilities (see for instance Levy and Wiener, 1998).
- 2.
According to Post’s (2003) definition, a portfolio is inefficient only if another portfolio exists with a strictly higher (unconditional) mean. By contrast, the original definition requires a weakly higher mean only. From an empirical perspective, the two definitions generally are indistinguishable, because arbitrary small data perturbations suffice to change weak inequalities into strict inequalities and vice versa. Still, in this paper, we adhere to the original definition, and we demonstrate that a simple variation to the test suffices to implement this definition.
- 3.
We phrase in terms of a general polyhedral portfolio possibilities set rather than the simplex used by Post (2003). This allows for altering the restrictions placed on the portfolio weights, for example to allowing for (or exclude) short-selling and riskless borrowing.
- 4.
We show that a simple weighted average of the evaluated portfolio and the solution portfolio yields a benchmark portfolio that has a perfect positive rank correlation with the evaluated portfolio. This portfolio dominates the evaluated portfolio not only in terms of conditional means but also in terms of SSD.
The remainder of this paper is structured as follows. Section 1 provides preliminaries on the notation, assumptions and definitions that will be used throughout the paper. Section 2 introduces the dual test statistic and provides a direct proof from first principles rather than from duality with the original primal test statistic. Section 3 illustrates the dual SSD test by means of an application that analyzes if short selling restrictions affect the profitability of momentum strategies. Finally, Section 4 presents some concluding remarks.
Section snippets
Preliminaries
The investment universe consists of N assets, with returns .2 Investors may diversify between the assets, and we will use the vector λ ∈ K for the portfolio weights and the polytope for
Dual LP test for SSD efficiency
In the spirit of Post (2003), we propose the following test statisticwhere , t = 1, … ,T, is a ‘conditional mean’, that is, the mean return of xTλ, conditional upon xTτ ⩽ Qτ(t/T), and s ≡ (s1 … sT)T is a vector of slacks.
To repeat, Post (2003) deviates from the traditional definition of SSD by requiring a strictly higher (unconditional) mean. Related to this, he used the test statistic
Application to momentum investment strategies
Many empirical studies have investigated the profitability of investment strategies based on price momentum; see for example Fama and French, 1996, Jegadeesh and Titman, 2001, Korajczyk and Sadka, 2004 for some recent studies. Since the returns to momentum strategies typically are highly skewed, there appears to be a need to look beyond mean and variance alone. In this respect, the nonparametric SSD efficiency criterion seems appropriate here. Further, momentum strategies generally require
Concluding remarks
- 1.
Post (2003) derived a linear programming test for SSD efficiency from the first principles of expected utility theory. In this paper, we have derived a direct proof of the dual formulation of this test that applies also outside the context of expected utility theory. This direct proof is useful because the SSD rule is also economically meaningful in the context of some non-expected utility theories, including ones that allow for subjective distortion of probabilities.
- 2.
We have shown that a simple
Acknowledgements
Financial support by Tinbergen Institute, Erasmus Research Institute of Management and Erasmus Center of Financial Research is gratefully acknowledged. We thank Philippe Versijp and Pim van Vliet for useful comments and suggestions. Any remaining errors are our own.
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2015, Omega (United Kingdom)Citation Excerpt :We say that an investment opportunity is efficient with respect to SSD if there is no other investment opportunity which strictly dominates it. The tests, which state necessary and sufficient conditions for SSD efficiency, were derived by Grechuk [33], Kopa and Chovanec [41], Kuosmanen [43], and Post [61]. Relations between mean-risk (mean-deviation) models and SSD were investigated by [59,60].
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2013, European Journal of Operational ResearchCitation Excerpt :The concept is used in numerous empirical studies and practical applications, ranging from agriculture and health care to financial management and public policy making; see, for example, the extensive survey in the text book of Levy (2006). A selection of recent studies in OR/MS journals includes Post (2008), Lozano and Gutiérrez (2008), Blavatskyy (2010), Dupačová and Kopa (2012), Lizyayev and Ruszczyński (2012), Lizyayev (2012) and Brown et al. (2012). SD imposes general preference restrictions without assuming a functional form for the decision maker’s utility function.
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This study forms part of a research program on stochastic dominance and asset pricing. Details on the program are available at the program homepage: http://www.few.eur.nl/few/people/gtpost/stochastic_dominance.htm.