Optimal Forecasts of Long-Term Returns: Geometric, Arithmetic, or Other Means

Abstract

It is well known that an unbiased forecast of the terminal value of a portfolio requires compounding at the arithmetic mean rate of return over the investment horizon. Yet, the procedure applied to the standard unbiased estimator of the mean return, while maximum likelihood, produces very biased forecasts. We address the forecasting of long-term returns. First, we show that the geometric estimator often proposed as alternative to the MLE is also biased. We derive the unbiased forecast of long-term returns for given estimation period and horizon. Second, we derive an efficient, minimum mean squared error estimator for the long-term. Both estimators impose a penalty that decreases the annual compounding rate as the forecasting horizon increases. The minimum MSE estimator is however lower than the alternative MLE and unbiased estimator, even than the geometric estimator as H becomes larger. Third, we show that parameter uncertainty and forecasting horizon interact to produce optimal portfolio allocations in striking contrast with conventional wisdom: Longer investment horizons require lower, not higher, allocations to risky assets. Finally, we derive a Bayes estimate of long-term expected returns. While the small sample efficiency of the Bayesian method makes it ideal for this problem, we show that standard priors have unreasonable implications and need to be modified. Finally, we show how our results may adjust to non i.i.d. returns.

Keywords: Long-term returns, parameter uncertainty, Bias, maximum likelihood, mean squared error, arithmetic and geometric mean

JEL Classification: G11, C53, C11, D91