Performance measures: advantages of linear risk penalization

Abstract

This work corrects the risk quotient penalization carried out by the Sharpe and Treynor ratios, which, assuming normality in returns distribution, are equivalent to classifying the funds according to the probability of their returns being below that of the risk-free asset, without considering the entire distribution of the funds’ returns. Different performance measures are applied to a sample of Spanish investment funds, and it is shown that all the measures lead to similar fund classifications, enabling us to conclude that the measures proposed, though methodologically improving the Sharpe and Treynor ratios, maintain their economic meaning and their financial nature.

Acknowledgements

The authors would like to express their thanks to the Spanish Directorate General for Higher Education for the award of Project PB97-1003, to the Regional Government of Aragon for the award of Project P06/97, to Ibercaja for the award of Project 268-96 and to the University of Zaragoza for the award of funding through Research Projects 268-77, 268-84, 268-93 and 268-96. Any possible errors are, of course, the exclusive responsibility of the authors.

Notes

¹ Copeland and Weston (1988) can be seen.

² In this work, we applied the Sharpe index to all the portfolios, although it would not be correct to apply it to specialized portfolios, we have no evidence of having a considerable number of specialized portfolios in our database.

³ We could also criticize the CAPM here, as extensive literature has done over the last few years, in which case the use of Beta would not be correct, at least on its own, but we are not going to enter into that problem now.

⁴ See Appendix A on analysis of normality and symmetry of the sample analysed.

⁵ For example, monthly mean return over 5 years.

⁶ The value of k would be negative in this case.

⁷ Understand multivariate normal; if the portfolio returns in the period were normal and independent this would also be valid.

⁸ Net Present Value (NPV).

⁹ Penalized Present Value (PPV).

¹⁰ This interpretation is equivalent to using ‘straight lines’ instead of indifference curves, which is a simplification that can be alleviated by using increasing values of t for increasing risks. On the other hand, if the NPV follows the normal distribution it is easy to interpret t in accordance with the probability that exists of an NPV being less than the PPV. Further information can be found on all of this in Gómez-Bezares (1993).

¹¹ Internal Rate of Return (IRR).

¹² Penalized Internal Rate of Return (PIRR).

¹³ Maintaining the statistic interpretation of parameter t in normal distributions of IRR and the exception of the implicit use of indifference straight lines.

¹⁴ Following Gómez-Bezares et al. (2004).

¹⁵ In a perfectly efficient market, well-diversified portfolios will be situated on the capital market line, and there would be no sense in talking about different performances. What we do here is to calculate t assuming that the market is efficient.

¹⁶ Investing or borrowing.

¹⁷ Which would be the capital market line in an efficient market.

¹⁸ It can be sensed that the majority of what has been said related to the Sharpe index can also be applied to the Treynor index, simply considering that if we dispense with the diversifiable risk (supposition assumed by Treynor) the beta could replace the SD as a variability measure. Understood in this way, Treynor would also classify the funds in accordance with their ‘probability’ of falling below the risk-free rate.

¹⁹ Following Gómez-Bezares et al. (2004).

²⁰ See Appendix B on composition of portfolios analysed. To abbreviate we have shown, for each agency, the mean composition of the portfolio of all the funds considered that are managed by the same. The data have been taken at the end of the year 2000.

²¹The percentage of funds not considered in the period analysed amounts to approximately 16%. This percentage has not been considered as these are the funds for which no data exist for more than a 2-year period, which is the minimum period required to correctly apply the methodology applied in this work.

²² See Appendix C on analysis of outliers. The method used for detecting outliers has been the univariate method of z-scores (if |z-score|>3 outlier). We can see that the maximum percentage of outliers that a fund has obtained is 3.12%. These outliers have been eliminated from the sample.

²³ Although it is true that whenever a fund beats the market in Jensen, it must beat it in Treynor. However, our database shows us that this is not the case in the first and second years of analysis due to the special features of fund ‘294’, which has a negative beta over these 2 years.