Bias and Consistency of the Maximum Sharpe Ratio

Posted: 8 Nov 2005

See all articles by Ross Maller

Ross Maller

Australian National University (ANU) - School of Finance and Applied Statistics

Robert B. B. Durand

Curtin University of Technology - School of Economics and Finance - Department of Finance and Banking

Peter Lee

affiliation not provided to SSRN

Abstract

We show that the maximum Sharpe ratio obtained via the Markowitz optimization procedure from a sample of returns on a number of risky assets is, under commonly satisfied assumptions, biased upwards for the population value. Thus investment advice, decisions and assessments based on the estimated Sharpe ratio will be overly optimistic. The bias in the estimator is shown theoretically and illustrated using a data set of Spiders and iShares. We obtain bounds on the difference between the sample maximum Sharpe ratio and its population counterpart and show that the sample estimator is consistent for the population value; thus the bias disappears asymptotically under some reasonable assumptions. However, the bias can be significant infinite samples and can persist even in very large samples. We demonstrate this with simulations based on portfolios formed from normally and t-distributed returns. As expected, the over-optimistic risk-return tradeoff predicted by the procedure is not reflected in corresponding good out-of-sample portfolio performance of the Spiders and iShares.

Keywords: Sharpe ratio, Markowitz optimization procedure, population value, Spiders and iShares

Suggested Citation

Maller, Ross and Durand, Robert B. B. and Lee, Peter T., Bias and Consistency of the Maximum Sharpe Ratio. Journal of Risk, Vol. 7, No. 4, Summer 2005. Available at SSRN: https://ssrn.com/abstract=839745

Ross Maller (Contact Author)

Australian National University (ANU) - School of Finance and Applied Statistics ( email )

Canberra, Australian Capital Territory 0200
Australia

Robert B. B. Durand

Curtin University of Technology - School of Economics and Finance - Department of Finance and Banking ( email )

Bentley 6102 WA
Australia

HOME PAGE: http://tiny.cc/y85rh

Peter T. Lee

affiliation not provided to SSRN

No Address Available

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