On the Third Order Stochastic Dominance for Risk-Averse and Risk-Seeking Investors

Chan, Raymond H. and Clark, Ephraim and Wong, Wing-Keung (2012): On the Third Order Stochastic Dominance for Risk-Averse and Risk-Seeking Investors.

Abstract

This paper studies some properties of stochastic dominance (SD) for risk-averse and risk-seeking investors, especially for the third order SD (TSD). We call the former ascending stochastic dominance (ASD) and the latter descending stochastic dominance(DSD). We first discuss the basic property of ASD and DSD linking the ASD and DSD of the first three orders to expected-utility maximization for risk-averse and risk-seeking investors. Thereafter, we prove that a hierarchy exists in both ASD and DSD relationships and that the higher orders of ASD and DSD cannot be replaced by the lower orders of ASD and DSD. Furthermore, we study conditions in which third order ASD preferences will be 'the opposite of' or 'the same as' their counterpart third order DSD preferences. In addition, we construct examples to illustrate all the properties developed in this paper. The theory developed in this paper provides investors with tools to identify first, second, and third order ASD and DSD prospects and thus they could make wiser choices on their investment decision.

• Item Type: MPRA Paper
• Language: English
• Keywords: Third order stochastic dominance, ascending stochastic dominance, descending stochastic dominance, expected-utility maximization, risk averters, risk seekers
• Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods

References

Anderson, G.J. (2004). Toward an Empirical Analysis of Polarization. Journal of Econometrics, 122, 1-26.

Bawa, V.S. (1975), Optimma Rules for Ordering Uncertain Prospects, Journal of Financial Economics, 2, 95-121.

Bawa, V.S., E.B., Lindenberg, L.C., Rafsky, (1979), An efficient algorithm to determine stochastic dominance admissible sets, Management Science, 25(7), 609-622.

Davies, J., Hoy, M., 1994. The normative significance of using 3rd-degree stochastic-dominance in comparing income distributions, Journal of economic theory 64(2), 520-530.

Dillinger, A.M., Stein, W.E., Mizzi, P.J., 1992. Risk averse decisions in business planning, Decision Sciences 23, 1003-1008.

Eeckhoudt, L., and M., Kimball, (1992), Background risk, prudence, and the demand for insurance, Contributions to insurance economics, ed. by G. Dionne. Boston: Kluwer, 239-254.

Fishburn, P.C. (1964), Decision and Value Theory, (New York: Wiley).

Fishburn, P.C. (1974). Convex stochastic dominance with continuous distribution functions. Journal of Economic Theory, 7, 143-158.

Fishburn, P., Vickson, R. (1978). Theoretical foundations of stochastic dominance. In G. Whitmore M. Findlay (Eds.), Stochastic dominance: An approach to decision-making under risk. Lexington: Lexington Books, D.C. Heath and Company.

Fong, W.M., H.H. Lean, and W.K. Wong, 2008, Stochastic Dominance and Behavior towards Risk: The Market for Internet Stocks, Journal of Economic Behavior and Organization, 68(1), 194-208.

Fong, W.M., W.K. Wong, and H.H. Lean, 2005, International Momentum Strategies: A Stochastic Dominance Approach, Journal of Financial Markets, 8, 89-109.

Gasbarro, D., W.K. Wong, and J.K. Zumwalt, 2007, Stochastic dominance analysis of iShares, European Journal of Finance 13, 89-101.

J Gotoh, H Konno, 2000, Third degree stochastic dominance and mean-risk analysis, Management Science, 46, 2, 289-301.

Hadar J., and Russell W.R. (1971), Stochastic Dominance and Diversification, Journal of Economic Theory 3, 288-305.

Hammond, J.S. 1974. Simplifying the choice between uncertain prospects where preference is nonlinear. Management Science, 20(7), 1047-1072.

Hanoch, G, H. Levy. 1969. The efficiency analysis of choices involving risk. Review of Economic Studies, 36(3) 335-346.

Michel Le Breton, Eugenio Peluso 2009. Third-degree stochastic dominance and inequality measurement. Journal of Economic Inequality, 7(3), 249-268.

Levy, M., and H. Levy, 2002, Prospect Theory: Much Ado About Nothing? Management Science, 48(10), 1334-1349.

Li, C.K., and W.K. Wong, (1999). Extension of Stochastic Dominance Theory to Random Variables, RAIRO Recherche Operationnelle, 33, 509-524.

Menezes, C., Geiss, C., Tressler, J. (1980). Increasing downside risk. American Economic Review, 70, 921–932.

Meyer, J.(1977), Second Degree Stochastic Dominance with Respect to a Function, International Economic Review, 18, 476-487.

Ng, M.C. (2000). A Remark on Third Degree Stochastic Dominance, Management Science, 46(6), 870-873.

T Post, H Levy, 2005, Does risk seeking drive stock prices? A stochastic dominance analysis of aggregate investor preferences and beliefs, Review of Financial Studies 18(3), 925-953.

T Post, P Versijp, 2007, Multivariate tests for stochastic dominance efficiency of a given portfolio, Journal of Financial and Quantitative Analysis, 42(2), 489-515.

Quirk J.P.,and Saposnik R.(1962), Admissibility and Measurable Utility Functions, Review of Economic Studies, 29, 140-146.

Rothschild, M. and Stiglitz, J.E. (1970), Increasing risk: I. A definition, Journal of Economic Theory 2, 225-243.

Rothschild, M. and Stiglitz, J.E. (1971),Increasing risk: II. Its economic consequences, Journal of Economic Theory 3, 66-84.

Schmid, F. 2005. A note on third degree stochastic dominance. OR SPECTRUM 27(4), 653-655.

Stoyan, D. (1983), Comparison Methods for Queues and Other Stochastic Models, (New York: Wiley).

Tesfatsion, L.(1976), Stochastic Dominance and Maximization of Expected Utility, Review of Economic Studies 43, 301-315.

Thorlund-Petersen, L., 2001. Third-degree stochastic dominance and axioms for a convex marginal utility function, Mathematical Social Sciences 41(2), 167-199.

von Neumann, John, and Oskar Morgenstern (1944), Theory of Games and Economic Behavior, Princeton University Press, Princeton N.J.

Weeks, J.K., 1985. Stochastic dominance: a methodological approach to enhancing the conceptual foundations of operations management theory. Academy of Management Review 10(1), 31-38.

Whitmore, G. A. (1970), Third-Degree Stochastic Dominance, American Economic Review 60(3), 457 - 59.

Wong, W.K. (2007): Stochastic Dominance and Mean-Variance Measures of Profit and Loss for Business Planning and Investment, European Journal of Operational Research, 182, 829-843.

Wong, W.K., and C.K. Li (1999): A Note on Convex Stochastic Dominance Theory, Economics Letters, 62, 293-300.

Wong, W.K., and C. Ma, 2008, Preferences over Meyer's location-scale family, Economic Theory 37(1), 119-146.

R Zagst, J Kraus, 2011, Stochastic dominance of portfolio insurance strategies, Annals of Operations Research 185, 75-103.