While Stochastic Dominance has been employed in various forms as early as 1932, it has only been since 1969--1970 that the notion has been developed and extensively employed in the area of economics, finance, agriculture, statistics, marketing and operations research. In this survey, the first-, second- and third-order stochastic dominance rules are discussed with an emphasis on the development in the area since the 1980s.stochastic dominance application, risky investment choice, stochastic dominance, theory, decision analysis, choice under uncertainty, decision-making, valuation theory: utility/preference theory, risk

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W A hile "most" decision makers may prefer one uncertain prospect over another, stochastic dominance rules as well as other investment criteria, will not reveal this preference due to some extreme utility functions in the case of even a very small violation of these rules. Such strict rules relate to "all" utility functions in a given class including extreme ones which presumably rarely represents investors' preference. In this paper we establish almost stochastic dominance (ASD) rules which formally reveal a preference for "most" decision makers, but not for "all" of them. The ASD rules reveal that choices which probably conform with "most" decision makers also solve some debates, e.g., showing, as practitioners claim, an ASD preference for a higher proportion of stocks in the portfolio as the investment horizon increases, a conclusion which is not implied by the well-known stochastic dominance rules. (Stochastic Dominance; Almost Stochastic Dominance; Mean-Variance; Risk Aversion) The detailed development of SD paradigm is given in the survey of Levy (1992, 1998).

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This paper examines the intertemporal relation between downside risk and expected stock returns. Value at Risk (VaR), expected shortfall, and tail risk are used as measures of downside risk to determine the existence and significance of a risk-return tradeoff. We find a positive and significant relation between downside risk and the portfolio returns on NYSE/AMEX/Nasdaq stocks. VaR remains a superior measure of risk when compared with the traditional risk measures. These results are robust across different stock market indices, different measures of downside risk, loss probability levels, and after controlling for macroeconomic variables and volatility over different holding periods as originally proposed by Harrison and Zhang (1999).

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Levy and Markowitz showed, for various utility functions and empirical returns distributions, that the expected utility maximizer could typically do very well if he acted knowing only the mean and variance of each distribution. Levy and Markowitz considered only situations in which the expected utility maximizer chose among a finite number of alternate probability distributions. The present paper examines the same questions for a case with an infinite number of alternate distributions, namely those available from the standard portfolio constraint set.