Geometric Mean Approximations of Individual Security and Portfolio Performance - Young, William E.; Trent, Robert H. (Journal of Financial and Quantitative Analysis, 1969)
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.