Algorithms for Stochastic Dominance

Abstract

To obtain the efficient sets corresponding to the various stochastic dominance (SD) rules, we need to know the precise shape of the various distributions under comparison. For example, for SSD, in order to check whether dominance exists or not, we need to calculate the area enclosed between the two distributions under consideration, F and G up to any outcome x; and in order to be able to carry such a calculation, we need to know the precise distribution of the rates of return. Thus, we need to know the distributions (theoretical or empirical distributions) in applying the various dominance rules. As generally the theoretical distributions are unknown, in practice, stochastic dominance rules are commonly applied to empirical distributions (e.g., rates of returns of mutual funds or of other available portfolios). In such cases, if we have n observations, say, n annual rates of return, or n monthly rates of return, then each observation is commonly assigned an equal probability of 1/n. Thus, we have empirical distributions which, for investment decision making, generally serve also as estimates of future distributions. Relatively simple SD and SDR algorithms exist for this framework of uniform discrete distributions. The first algorithm for FSD and SSD was developed by Levy and Hanoch in 1969, and the algorithms for TSD as well as FSDR, SSDR and TSDR were developed by Levy and Kroll. Porter, Wart and Ferguson also developed algorithms that employ necessary rules to help cut down the number of necessary comparisons. This is particularly beneficial when a large number of portfolios are compared. This chapter is devoted to the various SD and SDR algorithms. However, we would like to stress at the outset that it is assumed that there is a set of portfolios, e.g. mutual funds and we employ the algorithm in order to divide this set of portfolios to the efficient and inefficient sets, but the algorithms discussed in this chapter are unable to construct an efficient portfolio composed from the individual available assets, e.g., stocks and bonds traded in the market.