Optimal Estimation of the Risk Premium for the Long Run and Asset Allocation: A Case of Compounded Estimation Risk

Posted: 29 Feb 2008

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: 2005

Abstract

It is well known that an unbiased forecast of the terminal value of a portfolio requires compounding at the arithmetic mean return over the investment horizon. However, the maximum-likelihood practice, common with academics, of compounding at the estimator of mean return results in upward biased and highly inefficient estimates of long-term expected returns. We derive analytically both an unbiased and a small-sample efficient estimator of long-term expected returns for a given sample size and horizon. Both estimators entail penalties that reduce the annual compounding rate as the investment horizon increases. The unbiased estimator, which is far lower than the compounded arithmetic average, is still very inefficient, often more so than a simple geometric estimator known to practitioners. Our small-sample efficient estimator is even lower. These results compound the sobering evidence in recent work that the equity risk premium is lower than suggested by post-1926 data. Our methodology and results are robust to extensions such as predictable returns. We also confirm analytically that parameter uncertainty, properly incorporated, produces optimal asset allocations, in stark contrast to conventional wisdom. Longer investment horizons require lower, not higher, allocations to risky assets.

Keywords: arithmetic mean, asset allocation, estimation risk, geometric mean, long-term returns, maximum likelihood, mean-squared error, risk premium, small sample

Suggested Citation

Jacquier, Eric and Kane, Alex and Marcus, Alan J., Optimal Estimation of the Risk Premium for the Long Run and Asset Allocation: A Case of Compounded Estimation Risk ( 2005). Journal of Financial Econometrics, Vol. 3, Issue 1, pp. 37-55, 2005. Available at SSRN: https://ssrn.com/abstract=821736

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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