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Volume 2006 |Article ID 082049 | https://doi.org/10.1155/JAMDS/2006/82049

Wing-Keung Wong, "Stochastic dominance theory for location-scale family", Advances in Decision Sciences, vol. 2006, Article ID 082049, 10 pages, 2006. https://doi.org/10.1155/JAMDS/2006/82049

Stochastic dominance theory for location-scale family

Received17 Jan 2006
Revised01 Aug 2006
Accepted02 Aug 2006
Published07 Nov 2006

Abstract

Meyer (1987) extended the theory of mean-variance criterion to include the comparison among distributions that differ only by location and scale parameters and to include general utility functions with only convexity or concavity restrictions. In this paper, we make some comments on Meyer's paper and extend the results from Tobin (1958) that the indifference curve is convex upwards for risk averters, concave downwards for risk lovers, and horizontal for risk neutral investors to include the general conditions stated by Meyer (1987). We also provide an alternative proof for the theorem. Levy (1989) extended Meyer's results by introducing some inequality relationships between the stochastic-dominance and the mean-variance efficient sets. In this paper, we comment on Levy's findings and show that these relationships do not hold in certain situations. We further develop some properties among the first- and second-degree stochastic dominance efficient sets and the mean-variance efficient set.

References

  1. G. Anderson, “Toward an empirical analysis of polarization,” Journal of Econometrics, vol. 122, no. 1, pp. 1–26, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  2. D. P. Baron, “On the utility theoretic foundations of mean-variance analysis,” Journal of Finance, vol. 32, no. 5, pp. 1683–1697, 1977. View at: Publisher Site | Google Scholar
  3. U. Broll, J. E. Wahl, and W.-K. Wong, “Elasticity of risk aversion and international trade,” Economics Letters, vol. 91, no. 1, pp. 126–130, 2006. View at: Publisher Site | Google Scholar
  4. K. V. Chow, “Marginal conditional stochastic dominance, statistical inference, and measuring portfolio performance,” Journal of Financial Research, vol. 24, no. 2, pp. 289–307, 2001. View at: Google Scholar
  5. A. M. Dillinger, W. E. Stein, and P. J. Mizzi, “Risk averse decisions in business planning,” Decision Sciences, vol. 23, no. 4, pp. 1003–1008, 1992. View at: Google Scholar
  6. M. Doumpos, S. Zanakis, and C. Zopounidis, “Multicriteria preference disaggregation for classification problems with an application to global investing risk,” Decision Sciences, vol. 32, no. 2, pp. 333–385, 2001. View at: Google Scholar
  7. M. S. Feldstein, “Mean-variance analysis in the theory of liquidity preference and portfolio selection,” Review of Economics Studies, vol. 36, no. 1, pp. 5–12, 1969. View at: Publisher Site | Google Scholar
  8. W. M. Fong and W.-K. Wong, “The modified mixture of distributions model: a revisit,” Annals of Finance, vol. 2, no. 2, pp. 167–178, 2006. View at: Publisher Site | Google Scholar
  9. W. M. Fong, W.-K. Wong, and H. H. Lean, “International momentum strategies: a stochastic dominance approach,” Journal of Financial Markets, vol. 8, no. 1, pp. 89–109, 2005. View at: Publisher Site | Google Scholar
  10. J. Hadar and W. R. Russell, “Stochastic dominance and diversification,” Journal of Economic Theory, vol. 3, no. 3, pp. 288–305, 1971. View at: Publisher Site | Google Scholar | MathSciNet
  11. J. S. Hammond, “Simplifying the choice between uncertain prospects where preference is nonlinear,” Management Science, vol. 20, no. 7, pp. 1047–1072, 1974. View at: Google Scholar | Zentralblatt MATH
  12. G. Hanoch and H. Levy, “Efficiency analysis of choices involving risk,” Review of Economic Studies, vol. 36, no. 3, pp. 335–346, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. J. C. Hershey and P. J. H. Schoemaker, “Risk taking and problem context in the domain of losses: an expected utility analysis,” Journal of Risk and Insurance, vol. 47, no. 1, pp. 111–132, 1980. View at: Publisher Site | Google Scholar
  14. D. Kahneman and A. Tversky, “Prospect theory: an analysis of decision under risk,” Econometrica, vol. 47, no. 2, pp. 263–291, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. T. Kuosmanen, “Efficient diversification according to stochastic dominance criteria,” Management Science, vol. 50, no. 10, pp. 1390–1406, 2004. View at: Publisher Site | Google Scholar
  16. H. Levy, “Two-moment decision models and expected utility maximization: comment,” American Economic Review, vol. 79, no. 3, pp. 597–600, 1989. View at: Google Scholar
  17. M. Levy and H. Levy, “Prospect theory: much ado about nothing?,” Management Science, vol. 48, no. 10, pp. 1334–1349, 2002. View at: Publisher Site | Google Scholar
  18. H. Levy and M. Levy, “Prospect theory and mean-variance analysis,” Review of Financial Studies, vol. 17, no. 4, pp. 1015–1041, 2004. View at: Publisher Site | Google Scholar
  19. H. Levy and Z. Wiener, “Stochastic dominance and prospect dominance with subjective weighting functions,” Journal of Risk and Uncertainty, vol. 16, no. 2, pp. 147–163, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. C.-K. Li and W.-K. Wong, “Extension of stochastic dominance theory to random variables,” RO Recherche Opérationnelle, vol. 33, no. 4, pp. 509–524, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  21. H. M. Markowitz, “Portfolio selection,” Journal of Finance, vol. 7, no. 1, pp. 77–91, 1952. View at: Publisher Site | Google Scholar
  22. E. M. Matsumura, K. W. Tsui, and W.-K. Wong, “An extended multinomial-Dirichlet model for error bounds for dollar-unit sampling,” Contemporary Accounting Research, vol. 6, no. 2, pp. 485–500, 1990. View at: Google Scholar
  23. J. R. McNamara, “Portfolio selection using stochastic dominance criteria,” Decision Sciences, vol. 29, no. 4, pp. 785–801, 1998. View at: Google Scholar
  24. J. Meyer, “Second degree stochastic dominance with respect to a function,” International Economic Review, vol. 18, no. 2, pp. 477–487, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  25. J. Meyer, “Two-moment decision models and expected utility maximization,” American Economic Review, vol. 77, no. 3, pp. 421–430, 1987. View at: Google Scholar
  26. J. Meyer, “Two-moment decision models and expected utility maximization: reply,” American Economic Review, vol. 79, no. 3, p. 603, 1989. View at: Google Scholar
  27. M. Myagkov and C. R. Plott, “Exchange economies and loss exposure: experiments exploring prospect theory and competitive equilibria in market environments,” American Economic Review, vol. 87, no. 5, pp. 801–828, 1997. View at: Google Scholar
  28. T. Post, “Empirical tests for stochastic dominance efficiency,” The Journal of Finance, vol. 58, no. 5, pp. 1905–1931, 2003. View at: Publisher Site | Google Scholar
  29. G. V. Post and J. D. A. Diltz, “A stochastic dominance approach to risk analysis of computer systems,” MIS Quarterly, vol. 10, no. 4, pp. 362–375, 1986. View at: Publisher Site | Google Scholar
  30. T. Post and H. Levy, “Does risk loving drive asset prices? a stochastic dominance analysis of aggregate investor preferences and beliefs,” Review of Financial Studies, vol. 18, no. 3, pp. 925–953, 2005. View at: Publisher Site | Google Scholar
  31. M. Rothschild and J. E. Stiglitz, “Increasing risk. I. A definition,” Journal of Economic Theory, vol. 2, no. 3, pp. 225–243, 1970. View at: Publisher Site | Google Scholar | MathSciNet
  32. M. Rothschild and J. E. Stiglitz, “Increasing risk. II. Its economic consequences,” Journal of Economic Theory, vol. 3, no. 1, pp. 66–84, 1971. View at: Publisher Site | Google Scholar | MathSciNet
  33. W. F. Sharpe, “A simplified model for portfolio analysis,” Management Science, vol. 9, no. 2, pp. 277–293, 1963. View at: Google Scholar
  34. H.-W. Sinn, Economic Decisions under Uncertainty, vol. 32 of Studies in Mathematical and Managerial Economics, North-Holland, Amsterdam, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  35. D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Chichester, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  36. L. Tesfatsion, “Stochastic dominance and the maximization of expected utility,” Review of Economic Studies, vol. 43, no. 2, pp. 301–315, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  37. J. Tobin, “Liquidity preference as behavior towards risk,” Review of Economics Studies, vol. 25, no. 2, pp. 65–86, 1958. View at: Publisher Site | Google Scholar
  38. A. Tversky and D. Kahneman, “Advances in prospect theory: cumulative representation of uncertainty,” Journal of Risk and Uncertainty, vol. 5, no. 4, pp. 297–323, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  39. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, New Jersey, 1944. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  40. J. K. Weeks, “Stochastic dominance: a methodological approach to enhancing the conceptual foundations of operations management theory,” Academy of Management Review, vol. 10, no. 1, pp. 31–38, 1985. View at: Publisher Site | Google Scholar
  41. J. K. Weeks and T. R. Wingler, “A stochastic dominance ordering of scheduling rules,” Decision Sciences, vol. 10, no. 2, pp. 245–257, 1979. View at: Google Scholar
  42. W.-K. Wong and G. Bian, “Robust estimation in capital asset pricing model,” Journal of Applied Mathematics and Decision Sciences, vol. 4, no. 1, pp. 65–82, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  43. W.-K. Wong and R. H. Chan, “On the estimation of cost of capital and its reliability,” Quantitative Finance, vol. 4, no. 3, pp. 365–372, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  44. W.-K. Wong and C.-K. Li, “A note on convex stochastic dominance,” Economics Letters, vol. 62, no. 3, pp. 293–300, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  45. Y. Zhao and W. Ziemba, “A dynamic asset allocation model with downside risk control,” Journal of Risk, vol. 3, no. 1, pp. 91–113, 2000. View at: Google Scholar

Copyright © 2006 Wing-Keung Wong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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