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

Boston College Working Paper

33 Pages Posted: 17 Dec 2002

See all articles by Eric Jacquier

Eric Jacquier

Boston University School of Management; HEC Montreal - Department of Finance

Alex Kane

University of California, San Diego (UCSD) - Graduate School of International Relations and Pacific Studies (IRPS)

Alan J. Marcus

Boston College - Department of Finance

Date Written: October 2002

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

Suggested Citation

Jacquier, Eric and Kane, Alex and Marcus, Alan J., Optimal Forecasts of Long-Term Returns: Geometric, Arithmetic, or Other Means (October 2002). Boston College Working Paper. Available at SSRN: https://ssrn.com/abstract=353242 or http://dx.doi.org/10.2139/ssrn.353242

Eric Jacquier (Contact Author)

Boston University School of Management ( email )

595 Commonwealth Avenue
Boston, MA 02215
United States

HEC Montreal - Department of Finance ( email )

3000 Chemin de la Cote-Sainte-Catherine
Montreal, QC H3T 2A7
Canada

Alex Kane

University of California, San Diego (UCSD) - Graduate School of International Relations and Pacific Studies (IRPS) ( email )

9500 Gilman Drive
La Jolla, CA 92093-0519
United States

Alan J. Marcus

Boston College - Department of Finance ( email )

Fulton Hall
Chestnut Hill, MA 02467
United States
617-552-2767 (Phone)
617-552-0431 (Fax)

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