Geometric or Arithmetic Mean: A Reconsideration

Posted: 26 Jan 2004

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

Abstract

An unbiased forecast of the terminal value of a portfolio requires compounding of its initial value at its arithmetic mean return for the length of the investment period. Compounding at the arithmetic average historical return, however, results in an upwardly biased forecast. This bias does not necessarily disappear even if the sample average return is itself an unbiased estimator of the true mean, the average is computed from a long data series, and returns are generated according to a stable distribution. In contrast, forecasts obtained by compounding at the geometric average will generally be biased downward. The biases are empirically significant. For investment horizons of 40 years, the difference in forecasts of cumulative performance can easily exceed a factor of 2. And the percentage difference in forecasts grows with the investment horizon, as well as with the imprecision in the estimate of the mean return. For typical investment horizons, the proper compounding rate is in between the arithmetic and geometric values.

Keywords: Portfolio management, asset allocation, investment theory, portfolio theory, quantitative tools, econometric and statistical methods

Suggested Citation

Jacquier, Eric and Kane, Alex and Marcus, Alan J., Geometric or Arithmetic Mean: A Reconsideration. Financial Analysts Journal, Vol. 59, No. 6, pp. 46-53, November/December 2003. Available at SSRN: https://ssrn.com/abstract=489522

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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