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Stochastic dominance efficiency analysis of diversified portfolios: classification, comparison and refinements

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Abstract

For more than three decades, empirical analysis of stochastic dominance was restricted to settings with mutually exclusive choice alternatives. In recent years, a number of methods for testing efficiency of diversified portfolios have emerged, which can be classified into three main categories: (1) majorization, (2) revealed preference and (3) distribution-based approaches. Unfortunately, some of these schools of thought are developing independently, with little interaction or cross-referencing among them. Moreover, the methods differ in terms of their objectives, the information content of the results and their computational complexity. As a result, the relative merits of alternative approaches are difficult to compare. This paper presents the first systematic review of all three approaches in a unified methodological framework. We examine the main developments in this emerging literature, critically evaluating the advantages and disadvantages of the alternative approaches. We also point out some misleading arguments and propose corrections and improvements to some of the methods considered.

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Notes

  1. For the sake of brevity we will denote Stochastic Dominance as SD, and use FSD and SSD for First and Second order Stochastic Dominance (these concepts will be defined later in the paper), respectively.

  2. Some authors use more demanding non-linear programs (such as Linton et al. 2005, 2010) and the iterative quadratic program of Post and Versijp (2007) which, in addition to the efficiency outcome, also provide statistical significance scores under some assumptions. Since such programs do not produce a dominated portfolio and are considerably more computationally demanding, we will omit them from our analysis. As statistical significance scores can be more naturally obtained via non-parametric bootstrapping procedures in the framework of this paper, we will focus on SSD efficiency tests which are more practical in terms of the computational complexity and the information content of the result.

  3. States with different probabilities can be dealt with by a linear transformation of decision variables so that the resulting program will be equivalent to the one with equally probable states; see Dybvig and Ross (1982) for details.

  4. Although occasionally we will distinguish the weak efficiency in the sense of Definition 3, we adapt the commonly accepted SSD efficiency given by Definition 4 throughout, and unless otherwise stated, SSD efficiency will refer to this strong definition.

  5. Varian (1983) has applied Afriat’s approach to testing rationality of investor behavior in a different setting than the one considered in this paper.

  6. Kuosmanen (2004) formulates (10) with X augmented by y, as it can happen that y∉M X is SSD efficient, but is dominated by a linear combination of a marketed portfolio and itself. We omit this augmentation here for the sake of comparability with the other methods.

  7. Kuosmanen (2004) defines \(\theta_{2}^{S}(y)\) as \(\frac{m^{2}}{2} - \sum_{k=1}^{m} kd_{0k}\); however he clearly meant (13). Moreover, the summation \(\sum_{k=1}^{m} kd_{0k}\) equals m, since it counts all m elements of y precisely once.

  8. The inverse SD constraints, including those based on CVaR and used in Kopa and Chovanec (2008), were developed earlier in Dentcheva and Ruszczyński (2006a). However, the linear programming test (21) was suggested in Kopa and Chovanec (2008).

  9. In fact (8) is only valid for \(\Lambda=\{ \lambda\in\mathbb{R}^{n}: \lambda^{\mathsf{T}} e = 1,\lambda\geq0 \}\). If Λ is another polytope, X ti in (8) should be substituted by the vertices of Λ.

  10. Assuming without loss of generality that the first n rows of X are linearly independent.

  11. Note however that as n increases, the dimensionality of (29) becomes smaller, but one needs to invert a larger X 1 prior to solving (29). If X happens to be particularly ill-conditioned, one may rewrite (29) without decomposition as: find \(z \in\mathbb{R}^{m}\) such that Xz=e,z≥0. This is a linear program with m variables and 2m constraints, and therefore remains the most efficient method for the case of unbounded Λ.

  12. The portfolio budget constraint enters every method in the same form and thus was omitted from the complexity analysis for brevity’s sake.

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  1. Erasmus School of Economics and Tinbergen Institute, Erasmus University Rotterdam and BNG Bank, Koninginnegracht 2, 2514AA, The Hague, The Netherlands

    Andrey Lizyayev

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Correspondence to Andrey Lizyayev.

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This research is based on discussion paper Lizyayev (Stochastic dominance efficiency analysis of diversified portfolios: classification, comparison and refinements, Tinbergen Institute working paper series, Erasmus University Rotterdam, 10-084/2, August 2010) presented at QMF conference in Sydney in December 2010; the author is thankful to all its participants. I would also like to thank Willem Verschoor and Philippe Versijp from Erasmus School of Economics and the two anonymous referees for their constructive comments that helped me improve the paper. Financial support from Erasmus School of Economics, Trustfonds Association and Tinbergen Institute is gratefully acknowledged.

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Lizyayev, A. Stochastic dominance efficiency analysis of diversified portfolios: classification, comparison and refinements. Ann Oper Res 196, 391–410 (2012). https://doi.org/10.1007/s10479-012-1123-4

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