Almost Stochastic Dominance and Efficient Investment Sets

ABSTRACT

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.

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.

Cite this paper

M. Levy, "Almost Stochastic Dominance and Efficient Investment Sets,"*American Journal of Operations Research*, Vol. 2 No. 3, 2012, pp. 313-321. doi: 10.4236/ajor.2012.23038.

M. Levy, "Almost Stochastic Dominance and Efficient Investment Sets,"

References

[1] M. Leshno and H. Levy, “Preferred by ‘All’ and Preferred by ‘Most’ Decision Makers: Almost Stochastic Dominance,” Management Science, Vol. 48, No. 8, 2002, pp. 1074-1085. doi:10.1287/mnsc.48.8.1074.169

[2] G. Chamberlain, “A Characterization of the Distributions that Imply Mean-Variance Utility Functions,” Journal of Economic Theory, Vol. 29, No. 1, 1983, pp. 185-201. doi:10.1016/0022-0531(83)90129-1

[3] J. Owen and R. Rabinovitch, “On the Class of Elliptical Distributions and their Applications to Portfolio Choice,” Journal of Finance, Vol. 38, No. 3, 1983, pp. 745-752. doi:10.1111/j.1540-6261.1983.tb02499.x

[4] J. Berk, “Necessary Conditions for the CAPM,” Journal of Economic Theory, Vol. 73, No. 1, 1997, 245-257. doi:10.1006/jeth.1996.2218

[5] H. Levy and H. Markowitz, “Approximating Expected Utility by a Function of Mean and Variance,” American Economic Review, Vol. 69, No. 3, 1979, pp. 308-317.

[6] H. Markowitz, “Foundations of Portfolio Theory,” Journal of Finance, Vol. 46, No. 2, 1991, pp. 469-477. doi:10.1111/j.1540-6261.1991.tb02669.x

[7] J. Hadar and W. Russell, “Rules for Ordering Uncertain Prospects,” American Economic Review, Vol. 59, No. 1, 1969, pp. 25-34.

[8] G. Hanoch and H. Levy, “The Efficiency Analysis of Choices Involving Risk,” Review of Economic Studies, Vol. 36, 1969, pp. 335-346. doi:10.2307/2296431

[9] M. Rothschild and J. Stiglitz, “Increasing Risk: I. A Definition,” Journal of Economic Theory, Vol. 2, No. 3, 1970, pp. 225-243. doi:10.1016/0022-0531(70)90038-4

[10] E. De Giorgi, “Reward-Risk Portfolio Selection and Stochastic Dominance,” Journal of Banking and Finance, Vol. 29, No. 4, 2005, pp. 895-926. doi:10.1016/j.jbankfin.2004.05.027

[11] D. Dentcheva and A. Ruszczynski, “Portfolio Optimization with Stochastic Dominance Constraints,” Journal of Banking and Finance, Vol. 30, No. 2, 2006, pp. 433-451. doi:10.1016/j.jbankfin.2005.04.024

[12] G. Constantinides and S. Perrakis, “Stochastic Dominance Bounds on American Option Prices in Markets with Frictions,” Review of Finance, Vol. 11, No. 1, 2007, pp. 71-116. doi:10.1093/rof/rfl001

[13] D. Gasbarro, W. Wong and J.K. Zumwalt, “Stochastic Dominance Analysis of iShares,” European Journal of Finance, Vol. 13, No. 1, 2007, pp. 89-101. doi:10.1080/13518470601025243

[14] W. K. Wong, “Stochastic Dominance and Mean-Variance Measures of Profit and Loss for Business Planning and Investment,” European Journal of Operational Research, Vol. 182, No. 2, 2007, pp. 829-843. doi:10.1016/j.ejor.2006.09.032

[15] A. Abhyankar, K. Y. Ho and H. Zhao, “Value versus Growth: Stochastic Dominance Criteria,” Quantitative Finance, Vol. 8, No. 7, 2008, pp. 693-704. doi:10.1080/14697680701668426

[16] J. Annaert, S. Van Osselaer and B. Verstraete, “Performance Evaluation of Portfolio Insurance Strategies Using Stochastic Dominance Criteria,” Journal of Banking and Finance, Vol. 33, No. 2, 2009, pp. 272-280. doi:10.1016/j.jbankfin.2008.08.002

[17] M. Kopa and T. Post, “A Portfolio Optimality Test Based on the First-Order Stochastic Dominance Criterion,” Journal of Financial and Quantitative Analysis, Vol. 44, No. 5, 2009, pp. 1103-1124. doi:10.1017/S0022109009990251

[18] H. Levy, Stochastic Dominance: Investment Decision Making Under Uncertainty, Springer, New York, 2006.

[19] H. Zhang, W. Song, X. Peng and X. Song, “Evaluate the Investment Efficiency by Using Data Envelopment Analysis: The Case of China,” American Journal of Operations Research, Vol. 2 No. 2, 2012, pp. 174-182.

[20] R. B. Porter and J. E. Gaumnitz, “Stochastic Dominance vs. Mean Variance Portfolio Analysis: An Empirical Evaluation,” American Economic Review, Vol. 62, No. 3, 1972, pp. 438-446.

[21] H. Tehranian, “Empirical Studies in Portfolio Performance Using Higher Degrees of Stochastic Dominance,” Journal of Finance, Vol. 35, No. 1, 1980, pp. 159-171. doi:10.1111/j.1540-6261.1980.tb03478.x

[22] V. S. Bawa, J. Bondurtha, M. R. Rao and H. L. Suri, “On Determination of the Stochastic Dominance Optimal Set,” Journal of Finance, Vol. 40, No. 2, 1985, pp. 417-431. doi:10.1111/j.1540-6261.1985.tb04965.x

[23] M. Levy, “Are Rich People Smarter?” Journal of Economic Theory, Vol. 110, No. 1, 2003, pp. 42-64. doi:10.1016/S0022-0531(03)00024-3

[24] P. C. Benitez, T. Kuosmanen, R. Olschewski, and G. C. van Kooten, “Conservation Payments Under Risk: A Stochastic Dominance Approach,” American Journal of Agricultural Economics, Vol. 88, No. 1, 2006, pp. 1-15. doi:10.1111/j.1467-8276.2006.00835.x

[25] J. Huang, “Almost First Stochastic Dominance: What Do We Know from the Options Market?” SSRN Working Paper, 2007.

[26] H. Levy, M. Leshno and B. Leibovich, “Economically Relevant Preferences for All Observed Epsilon,” Annals of Operations Research, Vol. 176, No. 1, 2010, pp. 153- 178. doi:10.1007/s10479-008-0470-7

[27] M. Levy, “Almost Stochastic Dominance and Stocks for the Long Run,” European Journal of Operations Research, Vol. 194, No. 1, 2009, pp. 250-257. doi:10.1016/j.ejor.2007.12.017

[28] P. C. Fishburn, “Convex Stochastic Dominance with Continuous Distributions,” Journal of Economic Theory, Vol. 7, No. 2, 1974, pp. 143-158. doi:10.1016/0022-0531(74)90103-3

[29] I. Friend and M. Blume, “The Demand for Risky Assets,” American Economic Review, Vol. 65, No. 5, 1975, pp. 900-922.

[30] A. Tversky and D. Kahneman, “Advances in Prospect Theory,” Journal of Risk and Uncertainty, Vol. 5, No. 4, 1992, pp. 297-323. doi:10.1007/BF00122574

[1] M. Leshno and H. Levy, “Preferred by ‘All’ and Preferred by ‘Most’ Decision Makers: Almost Stochastic Dominance,” Management Science, Vol. 48, No. 8, 2002, pp. 1074-1085. doi:10.1287/mnsc.48.8.1074.169

[2] G. Chamberlain, “A Characterization of the Distributions that Imply Mean-Variance Utility Functions,” Journal of Economic Theory, Vol. 29, No. 1, 1983, pp. 185-201. doi:10.1016/0022-0531(83)90129-1

[3] J. Owen and R. Rabinovitch, “On the Class of Elliptical Distributions and their Applications to Portfolio Choice,” Journal of Finance, Vol. 38, No. 3, 1983, pp. 745-752. doi:10.1111/j.1540-6261.1983.tb02499.x

[4] J. Berk, “Necessary Conditions for the CAPM,” Journal of Economic Theory, Vol. 73, No. 1, 1997, 245-257. doi:10.1006/jeth.1996.2218

[5] H. Levy and H. Markowitz, “Approximating Expected Utility by a Function of Mean and Variance,” American Economic Review, Vol. 69, No. 3, 1979, pp. 308-317.

[6] H. Markowitz, “Foundations of Portfolio Theory,” Journal of Finance, Vol. 46, No. 2, 1991, pp. 469-477. doi:10.1111/j.1540-6261.1991.tb02669.x

[7] J. Hadar and W. Russell, “Rules for Ordering Uncertain Prospects,” American Economic Review, Vol. 59, No. 1, 1969, pp. 25-34.

[8] G. Hanoch and H. Levy, “The Efficiency Analysis of Choices Involving Risk,” Review of Economic Studies, Vol. 36, 1969, pp. 335-346. doi:10.2307/2296431

[9] M. Rothschild and J. Stiglitz, “Increasing Risk: I. A Definition,” Journal of Economic Theory, Vol. 2, No. 3, 1970, pp. 225-243. doi:10.1016/0022-0531(70)90038-4

[10] E. De Giorgi, “Reward-Risk Portfolio Selection and Stochastic Dominance,” Journal of Banking and Finance, Vol. 29, No. 4, 2005, pp. 895-926. doi:10.1016/j.jbankfin.2004.05.027

[11] D. Dentcheva and A. Ruszczynski, “Portfolio Optimization with Stochastic Dominance Constraints,” Journal of Banking and Finance, Vol. 30, No. 2, 2006, pp. 433-451. doi:10.1016/j.jbankfin.2005.04.024

[12] G. Constantinides and S. Perrakis, “Stochastic Dominance Bounds on American Option Prices in Markets with Frictions,” Review of Finance, Vol. 11, No. 1, 2007, pp. 71-116. doi:10.1093/rof/rfl001

[13] D. Gasbarro, W. Wong and J.K. Zumwalt, “Stochastic Dominance Analysis of iShares,” European Journal of Finance, Vol. 13, No. 1, 2007, pp. 89-101. doi:10.1080/13518470601025243

[14] W. K. Wong, “Stochastic Dominance and Mean-Variance Measures of Profit and Loss for Business Planning and Investment,” European Journal of Operational Research, Vol. 182, No. 2, 2007, pp. 829-843. doi:10.1016/j.ejor.2006.09.032

[15] A. Abhyankar, K. Y. Ho and H. Zhao, “Value versus Growth: Stochastic Dominance Criteria,” Quantitative Finance, Vol. 8, No. 7, 2008, pp. 693-704. doi:10.1080/14697680701668426

[16] J. Annaert, S. Van Osselaer and B. Verstraete, “Performance Evaluation of Portfolio Insurance Strategies Using Stochastic Dominance Criteria,” Journal of Banking and Finance, Vol. 33, No. 2, 2009, pp. 272-280. doi:10.1016/j.jbankfin.2008.08.002

[17] M. Kopa and T. Post, “A Portfolio Optimality Test Based on the First-Order Stochastic Dominance Criterion,” Journal of Financial and Quantitative Analysis, Vol. 44, No. 5, 2009, pp. 1103-1124. doi:10.1017/S0022109009990251

[18] H. Levy, Stochastic Dominance: Investment Decision Making Under Uncertainty, Springer, New York, 2006.

[19] H. Zhang, W. Song, X. Peng and X. Song, “Evaluate the Investment Efficiency by Using Data Envelopment Analysis: The Case of China,” American Journal of Operations Research, Vol. 2 No. 2, 2012, pp. 174-182.

[20] R. B. Porter and J. E. Gaumnitz, “Stochastic Dominance vs. Mean Variance Portfolio Analysis: An Empirical Evaluation,” American Economic Review, Vol. 62, No. 3, 1972, pp. 438-446.

[21] H. Tehranian, “Empirical Studies in Portfolio Performance Using Higher Degrees of Stochastic Dominance,” Journal of Finance, Vol. 35, No. 1, 1980, pp. 159-171. doi:10.1111/j.1540-6261.1980.tb03478.x

[22] V. S. Bawa, J. Bondurtha, M. R. Rao and H. L. Suri, “On Determination of the Stochastic Dominance Optimal Set,” Journal of Finance, Vol. 40, No. 2, 1985, pp. 417-431. doi:10.1111/j.1540-6261.1985.tb04965.x

[23] M. Levy, “Are Rich People Smarter?” Journal of Economic Theory, Vol. 110, No. 1, 2003, pp. 42-64. doi:10.1016/S0022-0531(03)00024-3

[24] P. C. Benitez, T. Kuosmanen, R. Olschewski, and G. C. van Kooten, “Conservation Payments Under Risk: A Stochastic Dominance Approach,” American Journal of Agricultural Economics, Vol. 88, No. 1, 2006, pp. 1-15. doi:10.1111/j.1467-8276.2006.00835.x

[25] J. Huang, “Almost First Stochastic Dominance: What Do We Know from the Options Market?” SSRN Working Paper, 2007.

[26] H. Levy, M. Leshno and B. Leibovich, “Economically Relevant Preferences for All Observed Epsilon,” Annals of Operations Research, Vol. 176, No. 1, 2010, pp. 153- 178. doi:10.1007/s10479-008-0470-7

[27] M. Levy, “Almost Stochastic Dominance and Stocks for the Long Run,” European Journal of Operations Research, Vol. 194, No. 1, 2009, pp. 250-257. doi:10.1016/j.ejor.2007.12.017

[28] P. C. Fishburn, “Convex Stochastic Dominance with Continuous Distributions,” Journal of Economic Theory, Vol. 7, No. 2, 1974, pp. 143-158. doi:10.1016/0022-0531(74)90103-3

[29] I. Friend and M. Blume, “The Demand for Risky Assets,” American Economic Review, Vol. 65, No. 5, 1975, pp. 900-922.

[30] A. Tversky and D. Kahneman, “Advances in Prospect Theory,” Journal of Risk and Uncertainty, Vol. 5, No. 4, 1992, pp. 297-323. doi:10.1007/BF00122574