Pitfalls of Downside Performance Measures with Arbitrary Targets

• Published date: 01 December 2017
• Author: Peter Reichling,Benedikt Hoechner,Gordon Schulze
• Date: 01 December 2017
• DOI: http://doi.org/10.1111/irfi.12137

Pitfalls of Downside Performance

Measures with Arbitrary Targets*

BENEDIKT HOECHNER,PETER REICHLING AND GORDON SCHULZE

Department of Banking and Finance, Otto-von-Guericke University Magdeburg,

Faculty of Economics and Management, Magdeburg, Germany

ABSTRACT

Downside performance measures relate above target returns with lower partial

moments. They were developed to resolve restrictive assumptions of the

classical Sharpe ratio. While the Sharpe ratio evaluates whether portfolios of

a mutual fund and the risk-free asset dominate passive portfolios of the

benchmark and the risk-free asset, this characteristic cannot be transferred to

downside performance measures with arbitrary targets. We show that

downside performance measures assign different values to passive benchmark

strategies if the target differs from the risk-free rate. This effect can lead to

reverse rankings of financial assets. Therefore, downside performance

measures are only applicable in asset management if the target is set equal to

the risk-free rate.

JEL Codes: D81; G11

I. INTRODUCTION

The classical Sharpe (1966) ratio, i.e., expected rate of return of a financial asset

above the risk-free rate divided by its volatility, has been criticized with regard

to the assumptions of mean–variance based investment decisions. Some authors

link the Sharpe ratio with quadratic utility that exhibits increasing risk aversion.

Actually, quadratic utility is only required for consistency of mean–variance

decisions with expected utility theory if arbitrary distribution functions are

allowed, in particular, if continuously as well as discretely distributed returns

are evaluated by the investor (Johnstone and Lindley 2011). However, in many

cases of performance measurement and other applications of the mean–variance

criterion in capital markets, we can assume that assets’returns belong to the

same class of distribution functions. It is well known that no further assumption

about the utility function is needed for a normal distribution because it can be

* We gratefully acknowledge detailed and helpful comments from an anonymous referee.

© 2017 International Review of Finance Ltd. 2017

International Review of Finance, 2017

DOI: 10.1111/irfi.12137

International Review of Finance, 17:4, 2017: pp. 597–610

DOI: 10.1111/irfi .12137

© 2017 International Review of Finance Ltd. 2017