RECENT empirical evidence on the relation between risk and return on common stocks has suggested that diversification beyond eight to ten securities may be superfluous [1, 2]. This evidence is consistent with the observed performance record of open-end mutual funds, many of which hold substantially more than one hundred securities. Two well-known empirical studies, for example, have suggested that the net risk-adjusted returns obtained by holders of mutual fund shares during the 1950's and 60's were somewhat worse than those achieved by the popular market indices [4, 12]. If there are consistent rewards to be gained by holding a large number of securities, and if these rewards exceed transactions costs and management expenses, they have successfully evaded discovery by much empirical research to date. Meanwhile, the small investor seems forced to choose between investment in mutual fund shares and direct investment in a very few securities. In many cases, depending on the amount of funds to be invested, the choice may be one of over-diversification vs. inadequate diversification. Little evidence is available on the relative desirability of the alternatives, other than that based on hypothetical transactions cost comparisons [16]. Existing evidence on the benefits of diversification as a function of the number of securities held has been based on random selection.' Clearly, such evidence would tend to bias the comparison of actual alternatives in favor of mutual fund selection. Since the early work of Markowitz [9], much attention has been devoted to the development of mean-variance portfolio selection models. Most such models [9, 13, 14], however, are more suited to the problem of institutional portfolio choice than to that of the individual. Some, such as the full-covariance model [9] and the Sharpe Single-Index Model [13] tend to generate portfolios having a large number of securities in which some securities are held in very small proportions. On the other hand, the Sharpe linear model [14], which allows the investor some control over the proportions to be invested in each of the securities, is a valid approximation of the quadratic problem only if the number of securities held is sufficient to render unsystematic risk virtually negligible. The portfolio diversification strategy recently suggested by Mao [8] is also of this type. While unsystematic risk may be as undesirable for the individual as it is for the institution, it is total variability of return that counts. It may be optimal for the individual to accept some unsystematic risk