For Normal Returns, Mean-Variance Admissible Choices are Optimal

For mutually exclusive investments or portfolio choices with normally distributed returns, we show that all elements of the second-order stochastic dominance admissible choice set are elements of the optimal or minimally efficient set. (That is, these elements will be chosen by some investor manifesting non-satiation and risk aversion.) This result helps clarify the relation between mean-variance theory and stochastic dominance; and, in the absence of mean and variance parameter estimation risk, implies the optimality of Sharpe's (1966) classic risk measure. Our results also highlight a caveat regarding investment strategy evaluation with discrete distribution approximations to unbounded distributions. Please address correspondence to the first author at School of Business, Georgetown University, Old North 313, 37 & O Streets, NW, Washington, DC 20057, (202) 687-6351, fax: (202) 687-4031, and e-mail: [email protected] For Normal Returns, Mean-Variance Admissible Choices are Optimal Abstract For mutually exclusive investments or portfolio choices with normally distributed returns, we show that all elements of the second-order stochastic dominance admissible choice set are elements of the optimal or minimally efficient set. (That is, these elements will be chosen by some investor manifesting non-satiation and risk aversion.) This result helps clarify the relation between mean-variance theory and stochastic dominance; and, in the absence of mean and variance parameter estimation risk, implies the optimality of Sharpe's (1966) classic risk measure. Our results also highlight a caveat regarding investment strategy evaluation with discrete distribution approximations to unbounded distributions.For mutually exclusive investments or portfolio choices with normally distributed returns, we show that all elements of the second-order stochastic dominance admissible choice set are elements of the optimal or minimally efficient set. (That is, these elements will be chosen by some investor manifesting non-satiation and risk aversion.) This result helps clarify the relation between mean-variance theory and stochastic dominance; and, in the absence of mean and variance parameter estimation risk, implies the optimality of Sharpe's (1966) classic risk measure. Our results also highlight a caveat regarding investment strategy evaluation with discrete distribution approximations to unbounded distributions. Stochastic dominance and mean-variance (normal distribution) utility theories emphasize the reduction of an investment choice set to a more efficient subset. When investors manifest both non-satiation and risk-aversion, and investment returns are normally distributed, it is well-known that a simple mean-variance rule identifies inefficient or dominated choice set elements. Any alternative distribution with equal or lower mean and higher variance than another choice distribution will not be chosen, e.g. Bawa (1975). The remaining undominated choices make up the associated admissible set. The question of whether or not normally distributed admissible choice set elements will be chosen by some investor manifesting non-satiation and risk-aversion was addressed by Meyers (1979) and Bawa et. al. (1985), but not resolved. When choices are mutually exclusive or are portfolios of choice set elements (which are the two cases of interest), we show that all admissible choices will be chosen. Therefore, the admissible choice set is optimal (minimally efficient). In developing our proofs, we treat the portfolio choice set and the mutually exclusive choice set cases separately. In each case, we first develop the stochastic dominance-related aspects of our arguments. Subsequently, we identify choice set element-specific utility functions that assign higher expected utility to the associated choice than to all other admissible choice set elements.