Geometric or Arithmetic Mean: A Reevaluation

Abstract

This study evaluates two competing forecasting models of rates of returns and recommends the preferable model for academicians and practitioners. In the first model, which was developed by Jacquier, Kane, and Marcus (2002), the forecast is a weighted mean between the geometric mean and the sum of the geometric mean and half the variance, where the weights are determined by the relative importance of the estimation period and the forecasting period. The second model, which is an adaption by Bodie, Kane, and Marcus (2008) of the first model, where the arithmetic mean is substituted for the sum of the geometric mean and half the variance. This substitution is not explained or justified in any way. The purpose of this paper is to explore the statistical significance and impact on forecasts of this substitution. In theory, these two models could be the same in large samples generated from normally distributed returns. However, the relative ability of these two competing models to forecast for small samples of actual returns is unknown. We find statistically significant differences in the inputs and the forecasts, but no meaningful difference in the models’ performance of forecasts as indicated by forecasting errors. In light of these results, despite the statistical differences, we find no economic difference between the forecasting errors of the two models and recommend the simpler of the two models which uses the arithmetic mean.