Billy P. Helms

Geometric Mean Approximations

In 1959, Henry LatanAi [2] proposed an approximation to the geometric mean that was a simple function of the arithmetic mean and variance, thereby indicating a mathematical relationship between the risky investment choice model of Bernoulli and the Markowitz mean-variance model. In 1969, Young and Trent [4] presented empirical test results of the LatanA© approximation, as well as a set of other approximations to the geometric mean based on moments, and concluded that the Latane formula yielded a quite accurate approximation to the geometric mean. In Jeans 1980 paper [1] relating the geometric mean model to stochastic dominance models, the infinite series representation of the geometric mean used suggests a more accurate approximation with moments of the geometric mean than that contained in the earlier papers may be possible. Various forms of that series expressed in alternate-origin moments are tested empirically below, and the results confirm that this later series does yield the greatest accuracy of the three approaches.

The identification of stochastic dominance efficient sets by moment combination orderings

Abstract A new procedure using algebraic combinations of central moments is derived to identify stochastic dominance efficient sets of security portfolios. The number of computations involved with the moment combinations are significantly smaller than those with regular stochastic dominance tests. The effectiveness of the moment combination procedure is demonstrated with a large-scale empirical study.

An Empirical Comparison of Stochastic Dominance among Lognormal Prospects

l. Introduct ion The theory of portfolio selection and diversification developed by Markowitz [22] and Tobin [33] was based primarily on the criterion of meanvariance (MV) efficiency. The objective was to select an efficient set of portfolios from which every risk averter will choose the optimal portfolio which maximizes his expected utility. The MV criterion is the appropriate rule either for the case in which the utility function is quadratic or if the returns are normally distributed and risk aversion is assumed. Two approaches to the choice among risky alternatives that have been developed independently over the past two decades are the geometric mean criterion (GM) and the stochastic dominance (SD) decision model. Both can be justified by the expected utility hypothesis, with the geometric mean cri? terion following as a result of the assumption that the decision-maker has a logarithmic utility function, and the stochastic dominance models requiring the less restrictive assumption of signs of the first few derivatives of the decision-makers utility function. This paper compares the concept of ordinary stochastic dominance to stochastic dominance of the lognormally distributed prospects from an empirical point of view. Further 1ight is also shed on the issue of applying the Markowitz-Tobin mean-variance rule