Stochastic dominance and diversification

In Section 1 we presented several theorems on stochastic dominance. The first two of these establish the basic relationship between stochastic dominance and preference, and as such they provide the foundation for the applicability of stochastic dominance to problems of behavior under uncertainty. The other theorems in that section deal with some properties of distributions which satisfy the stochastic dominance conditions. These results are quite general, and presumably are applicable to problems outside the domain of decision theory; in the present paper, however, they provide the basis for some of the formal results on diversification. In Section 2 we presented a number of theorems on diversification in the order of decreasing generality. One purpose of doing so was to bring together under a common framework a number of fairly general results in the area of diversification so as to permit a comparison between them. It was shown that by placing various restrictions on the distributions in question one can, in addition to demonstrating optimality of diversification, obtain preference orderings of portfolios for different values of the mixture coefficient. In two special cases—identical distributions and portfolios involving a sure prospect—one can in fact predict the particular mixture coefficient that each risk averter will actually choose. Our analysis was confined to portfolios consisting of two independently distributed random variables. It seems that the exclusion of portfolios with more than two distributions is not too serious since the critical factors relevant to diversification are easily identified in the analysis of the two-distribution case. If anything, the resulting simplification is probably advantageous from the pedagogical point of view. In any event, the results can presumably be generalized to n-distribution portfolios by mathematical induction. As for the independence assumption, since the case of interdependent prospects is quite a bit more complicated, it will be taken up in a separate study.