Don’t use quotients to calculate performance

Abstract

Quotients, ratios, are among the most applied tools for measuring performance of mutual funds and investment portfolios, the Jensen index being an exception to the general rule. In this paper, we show some problems that arise when quotients are applied, closely related to their statistical meaning, which is too often forgotten. We also raise some advantages of the use of linear penalization, introducing a little known methodology for performance measurement. With this purpose, this paper’s approach is comprehensive: we conceptually analyze performance indexes’ geometric and statistical meaning, complementing this with a numerical example and empirical testing that confirm our view. This paper’s main contribution is to demonstrate and empirically test how the use of quotients to measure performance may create problems due to their denominators, which may be solved by applying linear penalization.

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© 2015 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license

Keywords:

• performance
• ratio
• risk
• linear penalization

JEL Classifications:

• G11
• G23

Public Interest Statement

Academics, practitioners, and general public very often apply quotients, like the ratio between return and risk, to measure the performance of their investments (e.g. the Sharpe and Treynor indexes or the Information Ratio). In this paper, we show how the application of these quotients may have problems and could lead analysts to take wrong investment decisions, partially due to the statistical implications of these quotients’ use. We propose linear penalization by risk as an interesting alternative and introduce a little known methodology for performance measurement. With this purpose, this paper’s approach is comprehensive including conceptual, geometric, and statistical analyses that we complement with a numerical example and empirical testing. We think that the explained ideas are especially interesting under the current economic environment, in which optimal prioritization of investments and accurate analysis of balance between return and risk are a must.

Acknowledgments

The authors want to thank the editor, referees, and other colleagues for their comments and suggestions, which have undoubtedly contributed to the improvement of the present paper.

Notes

1. Taking the tracking error definition applied by Roll (1992): difference between portfolio’s return and benchmark’s return for the same period of time, and applying the IR definition proposed by Modigliani and Modigliani (1997).

2. In this approach, the capital asset pricing model (CAPM) logic is underlying. A recent test of this model in the Spanish market can be seen in Gómez-Bezares, Ferruz, and Vargas (2012).

3. We can also give a statistical meaning to the t value. From Formula 4, we get to (PIRR−µ)/σ = −t; so, if return follows a normal distribution, the −t value would leave at its left a probability in the standard normal distribution equal to the probability for the actually obtained return of the mutual fund or portfolio in a certain period to be less than the PIRR. In this sense, an investor may look for a different t value from the one offered by the market (we have seen that this is the market Sharpe ratio). For instance, a t value of 1 would leave a probability (in the standard normal distribution tables) equal to 0.1587, which means that the PIRR would be a “guaranteed value” with a probability of (1−0.1587) = 0.8413. We could also interpret the straight lines, parallel to the CML, as indifference straight lines, giving a greater value to the portfolio or mutual fund that reaches the highest straight line; but, this would lead us to another type of reasoning that would take us away from this paper’s goal.

4. Taking Formula 1 (Sharpe ratio), (µ−r₀)/σ is also, changed its sign, the standardized value of r₀, which we call t. This t, searching in the tables of the standard normal distribution, shows us the probability of the return to fall below the risk-free rate in a certain period.

5. A t = 3.5 implies, in the tables of the standard normal distribution, a 0.00023 probability of the return falling below r₀, and a t = 4 would result in a probability of 0.00003; the latter is smaller, but both are insignificant. On the other hand, the probability of C mutual fund return falling below the average return of D would be obtained this way: t = (5−1.9)/1 = 3.1, with a corresponding probability of 0.00097, which makes mutual fund C clearly more attractive.

6. Its statistical meaning is not so obvious, but assuming that diversifiable risk will be eliminated by diversification, the β will be proportional to the resulting σ, and we can maintain the same reasoning.

7. There is a lack of logic in the negative value obtained for Treynor when we have a mutual fund with a positive risk premium and a negative beta. A more detailed explanation can be seen in Gómez-Bezares and Gómez-Bezares (2012).

8. We want to show our gratitude to Norbolsa (Broker), Raquel Arechabala, and Manu Martín-Muñío for their help to obtain the required data. We also want to specially thank Alba Díaz Gómez for her support with the information processing.

9. Based on variations of net asset value.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

Fernando Gómez-Bezares

Fernando Gómez-Bezares, as professor of Finance at the Deusto Business School, has written 25 books and many papers in academic and professional journals related to the topic faced in this paper: risk treatment and value creation.

Fernando R. Gómez-Bezares

Fernando R. Gómez-Bezares, as a practitioner, currently at The Boston Consulting Group and previously at Morgan Stanley, knows the importance given daily to the performance indexes analyzed in this paper. Apart from the works mentioned below, he has published in the “Revista Española de Capital Riesgo” (Spanish Journal of Venture Capital).

Both authors have published together the following papers: “On the use of hypothesis testing and other problems in financial research” in “The Journal of Investing,” “Classic performance indexes revisited: axiomatic and applications” in “Applied Economics Letters,” “An analysis of risk treatment in the field of finance” as a chapter in the “Encyclopedia of Finance, 2nd edition,” edited by Springer, and “The efficiency-CAPM paradigm” in the Spanish journal “Análisis Financiero”.