J. M. Keynes' theory of portfolio management (modified and refined by Tobin)occupied an important role in his analysis of the demand for money. According to this theory, financial investors were thought to vary the composition of their portfolios between money and securities on the basis of expected yields on securities. When yields were expected to rise, investors would shift out of securities and into money. Conversely, when yields were expected to fall, investors would shift out of money and into securities. Hence, the asset, or portfolio, demand for money was argued to be negatively related to the expected yields on securities.

The objectives of this paper are to derive the relationship of the geometric mean of a distribution of positive values to the conventional first four moments — arithmetic mean, variance, absolute skewness, and absolute kurtosis — and to empirically evaluate certain approximations involving these four moments for estimating the geometric means of monthly and annual holding period returns for individual stocks and for portfolios. The geometric mean is shown to be positively related to the arithmetic mean and absolute skewness and negatively related to variance and absolute kurtosis. In the case of a normal distribution a very good approximation to the geometric mean is revealed to be a function of just the arithmetic mean and variance. Additionally, empirical evidence indicates that even though a number of the monthly and annual distributions deviate significantly from normality, the approximation involving only the mean and variance produces quite accurate estimates of the geometric means of these distributions.

In this article, we shall discuss several of the alternative definitions of risk that have been proposed from time to time. We shall show that one definition — risk is the probability of loss — leads to a formulation of the investment decision problem as a chance constrained problem. Three different strategies are then proposed by which an investor can reduce risk. It is our belief that professional investors utilize all three strategies and that risk, in many such cases, is not a substantial constraint on investor behavior.