Knowledge that Transforms

George M. Frankfurter, Herbert E. Phillips, John P. Seagle, Portfolio Selection: The Effects of Uncertain Means, Variances, and Covariances, The Journal of Financial and Quantitative Analysis, Vol. 6, No. 5 (Dec., 1971), pp. 1251-1262

Most portfolio analysis is based on the use of two parameters, the mean and variance, of the statistical distribution of returns. Exceptions to this practice can be found in an empirical work by Arditti [1] and a theoretical paper by Levy [4], both using the third moment around the mean. It is the purpose of this paper to begin a general extension of the two-parameter analysis to three or more parameters. Accordingly, some problems will be solved, but others will be suggested for further analysis.

Three main approaches to the problem of portfolio selection may be discerned in the literature. The first of these is the mean-variance approach, pioneered by Markowitz [21], [22], and Tobin [30]. The second approach is that of chance-constrained programming, apparently initiated by Naslund and Whinston [26]. The third approach, Latané [19] and Breiman [6], [7], has its origin in capital growth considerations. The purpose of this paper is to contrast the mean-variance model, by far the most well-known and most developed model of portfolio selection, with the capital growth model, undoubtedly the least known. In so doing, we shall find the mean-variance model to be severely compromised by the capital growth model in several significant respects.

Almost twenty years ago, Markowitz [4] first suggested that portfolio selection be regarded as a parametric quadratic programming problem. Risk is stated in terms of the predicted variance of portfolio return — a function that is quadratic in the decision variables (the proportions of the portfolio invested in various securities). All other functions (e.g., expected return) and constraints are assumed to be linear. The objective is to find the set of efficient feasible portfolios. A portfolio is feasible if it satisfies a set of relevant linear constraints; it is efficient if it provides (1) less variance than any other feasible portfolio with the same expected return and (2) more expected return than any other feasible portfolio with the same variance.