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A.I.R.A.P. - Alternative RAPMs for Alternative Investments Authors: Milind Sharma Source: SSRN Date: November 2003 Size: 205290 Bytes |
This study introduces an innovative risk-adjusted performance measure that is especially designed to be applied to hedge funds. The new measure is called Alternative Investments Risk Adjusted Performance (AIRAP). The author justifies the proposition of a new risk-adjusted performance measure by the fact that the specific characteristics of hedge funds cause the current performance measures, namely the Treynor ratio, the Sortino ratio, Jensen’s alpha and the Sharpe ratio, to be inappropriate.
AIRAP is constructed on the basis of the Expected Utility theory. The selected form of utility is a Constant Relative Risk Aversion (CRRA). AIRAP is formulated as follows :
when c (Arrow-Pratt coefficient) is different to 1 and greater than or equal to 0:
and when c is equal to 1 :
Sharma recommends an Arrow-Pratt coefficient (represented by c) from 1 to 10. Because a geometric mean is used to measure the average performance, c=1 corresponds to risk neutrality (in this case the risk premium is nil)*. Cases with c comprised between 0 and 1 assume that rational investors accept the risk of insolvency, and according to the author this is implausible. In a cautious view, the author assumes c=4. It corresponds to a case where investors accept a risk of a maximum loss of 20.7% of their wealth.
An approach that only involves using the ratio of gross and net assets is inadequate to take into account the impact of leverage on the performance of hedge funds, because of the presence of derivatives. This justifies a risk-based approach. AIRAP captures the impact of leverage through a credit for the higher mean and a penalty for the higher volatility as a function of the CRR parameter. The optimal leverage, which maximizes AIRAP for a range of CRRA, can be defined by standard optimization techniques.
The data covers the period from January 1997 to December 2001. At the index level, the data is provided by EACM. At the individual fund level, the data is provided by HFR.
Rank reversals between Sharpe and AIRAP and between Jensen’s alpha and AIRAP are presented, for 19 different levels of Constant Relative Risk Aversion, for the HFR universe. The percentage of Sharpe ratio rank reversals is between 99% and 100%, while the percentage of Jensen’s alpha rank reversals is between 98% and 100%. The Spearman rank correlation confirms the lack of correlation between standard RAPMs and the AIRAP. At the intra-strategy level, even if the rank reversal is somewhat lower, it also indicates discrepancies between the Sharpe ratio and AIRAP.
According to the author, AIRAP presents several advantages. It takes into account leverage, investor preferences, the non-normality of the return distribution, negative mean excess returns and higher moments. Unlike traditional RAPMs, AIRAP penalizes negative skew and positive kurtosis. Moreover, it is scale invariant and can be used for non-directional strategies, unlike the Treynor ratio. Another advantage is the intuitive interpretation of this performance measure.
Sharma highlights the importance of the optimization of style weights, which is the first step of the two-step top-down procedure for fund of hedge fund construction (the second step is the optimization of the individual manager weights). In the context of the FoHF optimization, Expected Utility maximization leads to maximization of the AIRAP.
*When a geometrically compounded arithmetic mean is used, c>0 always represents risk-aversion.
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