# zbMATH — the first resource for mathematics

A second-order stochastic dominance portfolio efficiency measure. (English) Zbl 1154.91456
Summary: We introduce a new linear programming second-order stochastic dominance (SSD) portfolio efficiency test for portfolios with scenario approach for distribution of outcomes and a new SSD portfolio inefficiency measure. The test utilizes the relationship between CVaR and dual second-order stochastic dominance, and contrary to tests of Post and Kuosmanen, our test detects a dominating portfolio which is SSD efficient. We derive also a necessary condition for SSD efficiency using convexity property of CVaR to speed up the computation. The efficiency measure represents a distance between the tested portfolio and its least risky dominating SSD efficient portfolio. We show that this measure is consistent with the second-order stochastic dominance relation. We find out that this measure is convex and we use this result to describe the set of SSD efficient portfolios. Finally, we illustrate our results on a numerical example

##### MSC:
 91B28 Finance etc. (MSC2000) 91B30 Risk theory, insurance (MSC2010)
Full Text:
##### References:
 [1] Giorgi E. De: Reward-risk portfolio selection and stochastic dominance. J. Banking Finance 29 (2005), 895-926 [2] Giorgi E. De, Post T.: Second order stochastic dominance, reward-risk portfolio selection and the CAPM. J. Financial Quantitative Analysis, to appear [3] Dybvig P. H., Ross S. A.: Portfolio efficient sets. Econometrica 50 (1982), 6, 1525-1546 · Zbl 0495.90010 [4] Hadar J., Russell W. R.: Rules for ordering uncertain prospects. Amer. Econom. Rev. 59 (1969), 1, 25-34 [5] Hanoch G., Levy H.: The efficiency analysis of choices involving risk. Rev. Econom. Stud. 36 (1969), 335-346 · Zbl 0184.45202 [6] Kopa M., Post T.: A portfolio optimality test based on the first-order stochastic dominance criterion. J. Financial Quantitative Analysis, to appear [7] Kuosmanen T.: Efficient diversification according to stochastic dominance criteria. Management Sci. 50 (2004), 10, 1390-1406 [8] Levy H.: Stochastic dominance and expected utility: Survey and analysis. Management Sci. 38 (1992), 4, 555-593 · Zbl 0764.90004 [9] Levy H.: Stochastic Dominance: Investment Decision Making Under Uncertainty. Second edition. Springer Science, New York 2006 · Zbl 1109.91037 [10] Markowitz H. M.: Portfolio Selection. J. Finance 7 (1952), 1, 77-91 [11] Markowitz H. M.: Portfolio Selection: Efficient Diversification in Investments. Wiley, New York 1959 [12] Ogryczak W., Ruszczyński A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13 (2002), 60-78 · Zbl 1022.91017 [13] Pflug G. Ch.: Some remarks on the value-at-risk and the conditional value-at-risk. Probabilistic Constrained Optimization: Methodology and Applications (S. Uryasev, Kluwer Academic Publishers, Norwell MA 2000, pp. 278-287 · Zbl 0994.91031 [14] Post T.: Empirical tests for stochastic dominance efficiency. J. Finance 58 (2003), 1905-1932 [15] Rothschild M., Stiglitz J. E.: Rules for ordering uncertain prospects. J. Econom. Theory 2 (1969), 225-243 [16] Russell W. R., Seo T. K.: Representative sets for stochastic dominance rules. Studies in the Economics of Uncertainty (T. B. Fomby and T. K. Seo, Springer-Verlag, New York 1989, pp. 59-76 [17] Ruszczyński A., Vanderbei R. J.: Frontiers of stochastically nondominated portfolios. Econometrica 71 (2003), 4, 1287-1297 · Zbl 1154.91475 [18] Uryasev S., Rockafellar R. T.: Conditional value-at-risk for general loss distributions. J. Banking Finance 26 (2002), 1443-1471
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.