Alternative Investment Management Association Representing the global hedge fund industry
KCS Fund Strategies Inc and Simon Fraser University, respectively
This article is the winner of the 2005 AIMA Canada Research Award, which was sponsored by JCClark and RBC Capital Markets.
Hedge funds have existed since the 1960s but remained under the radar screen of the general public and most investors until 1998 when Long Term Capital Management lost 92% of investor capital. Since that time, sophisticated individual investors and institutions have continued to earn consistent and sometimes substantial returns from hedge funds, but the general perception among most investors is that hedge funds are very risky with a high potential for substantial losses. In recent years a number of researchers, including Brooks and Kat (2002), Lamm (2003), Getmansky, Lo and Makarov (2004), Agarwal and Naik (2004) and Malkiel and Saha (2004) have reported that while hedge funds look superior from a mean-variance perspective, the potential for large losses (or extreme event risk) is captured in the skew and kurtosis of hedge fund distributions.
In a recent working paper (2005) the authors of this article also looked at the extreme event risk of hedge funds and found that the skew and kurtosis in hedge fund returns do not necessarily imply that investors are exposed to undue risks. To make our argument, a good place to start is to look at pictures of hedge fund and traditional asset distributions. Figure 1 presents histograms for three common hedge fund indices and compares them with two well known equity indices. It is immediately apparent that none of the hedge fund indices has ever had monthly returns outside of 10% (either positive or negative) while both of the equity indices have. The Nasdaq has even had monthly returns exceeding 20% on both the positive and negative sides. It is also apparent that the hedge fund indices return 1% or 2% a month much more consistently than do the equity indices. Despite the concerns over extreme event risks in hedge funds, there does not appear to be any evidence of it in the distributions, at least relative to equity indices.
We also looked for evidence of extreme event risk in the maximum drawdowns for the indices. Table 1 shows the maximum drawdowns and recovery times for the same indices as shown in the histograms. The worst drawdown for any of the hedge fund indices was a 13.8% drawdown for the CSFB/Tremont Hedge Fund Index during the fall of 1998 at the time of the Long Term Capital meltdown. The drawdown lasted three months and the index took 13 months to recover the loss. Again this is minor in comparison to the two equity indices which still have not recovered from losses incurred since 1999 and which may take several more years to recover. Once again there is no evidence of substantial extreme event risk in the hedge fund indices.
Interpreting the data
Do skew and kurtosis provide a clue to some subtle risk that is not evident in other types of analysis or are skew and kurtosis problematic? Kaplansky (1945) in a brief article entitled “A Common Error Concerning Kurtosis”, written long before the first hedge fund existed, voiced concern about the way kurtosis is typically interpreted. Using examples, Kaplansky points out that the peakedness of different distributions does not match the common interpretations of kurtosis. Kaplansky focused on the peakedness of the distributions but his work can be extended to demonstrate the same inconsistencies when considering the tails, or extreme events, of distributions.
Scott and Horvath (1980) further developed the use of higher moments in return analysis when they demonstrated that each term in a Taylor series expansion of an investor’s utility function represented each subsequent statistical moment. They determined that investors prefer higher first and third moments but lower second and fourth moments. These preferences are now often applied to skew and kurtosis to claim that investors prefer higher skew and lower kurtosis. However, it should be noted that skew and kurtosis are not the same thing as the third and fourth statistical moments that Scott and Horvath worked with. Skew and kurtosis are ‘normalised’ by dividing the third and fourth statistical moment by the standard deviation raised to the third and fourth power respectively. Do Scott and Horvath’s conclusions also apply to skew and kurtosis or does the standard deviation in the denominator affect the interpretations?
Table 2 shows the standard deviation, skew and kurtosis, and third and fourth moments for the same indices as shown in the histograms. The table shows that the skew and kurtosis of the hedge funds are indeed inferior to those of the equities, as is often claimed. However the third and fourth statistical moments are substantially better for the hedge funds. This suggests that the lower standard deviation of hedge funds is hiding what are actually superior higher moments within the poorer skew and kurtosis measures.
While Scott and Horvath used utility functions to get their results, a more general result relying on fewer assumptions can be seen in the work of Ingersoll (1987) who used leverage to equalize the mean or variance of two different investments and then used a stochastic dominance argument to show that investors would prefer one investment over the other. In our paper, we extend this approach to include the unscaled statistical third and fourth moments. We use leverage to equalize the fourth statistical moment and then use Scott and Horvath’s conclusions on investor preference, that investors prefer higher first and third moments and lower second and fourth moments, to see if one investment dominates another. Applying this approach to the hedge fund and equity indices show that the hedge fund indices dominate, as shown in Table 3. They tend to have higher returns, lower standard deviation, and higher third moments after the fourth moments have been equalised.
However, if leverage is used to equalise the second moments without regards for the higher moments as is often done by mean-variance analysis, hedge funds do not dominate. Table 4 shows that the hedge funds tend to have higher first and third moments but also higher fourth moments. Mean-variance analysis on hedge funds is exposing investors to increased risk in the fourth moment. This leads us to the conclusion that it is the analysis tools that investors are using which are causing exposure to higher risks and not the hedge funds themselves.
In conclusion, our results highlight several implications for investors. First, the large allocations that some investors have made to hedge funds have been justified. Based on the historical data, these investors have enjoyed higher returns without taking on undue risk. Second, the use of leverage on a portfolio of hedge funds, as is often applied by funds of hedge funds, may also be appropriate. The lower standard deviation and fourth statistical moments of hedge funds allows leverage to be applied without resulting in risk greater than that of equities.
However, investors should be careful with the tools they use in analysing hedge funds. Mean-variance tools can expose investors to risks in the higher moments that they are not aware of. Also, beware of using the standard measures of skew and kurtosis. The use of these tools is not consistent with academic theory and can lead to erroneous results.
Agarwal, V. and N.Y. Naik, (2004) “Risks and Portfolio Decisions Involving Hedge Funds”, Review of Financial Studies, 17, 63-98.
Brooks, C. and H. M. Kat, (2002) “The Statistical Properties of Hedge Fund Index Returns and Their Implications for Investors”, The Journal of Alternative Investments, 5, 26-44.
Getmansky, M., A. W. Lo and I. Makarov, (2004) “An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns”, Journal of Financial Economics 74, 529-609.
Ingersoll, J. E., (1987) Theory of financial decision making, Rowan and Littlefield: Savage, Maryland.
Kaplansky, I., (1945) “A Common Error Concerning Kurtosis”, American Statistical Association Journal, 40, 259.
Lamm Jr., R. M., (2003) “Asymmetric Returns and Optimal Hedge Fund Portfolios”, The Journal of Alternative Investments, 6, 9-21.
Malkiel, B. G. and A. Saha, (2004) “Hedge Funds: Risk and Return”, working paper.
Scott, R. C. and P. A. Horvath, (1980) “On the Direction of Preference for Moments of Higher Order Than the Variance”, Journal of Finance 35, 915-919.
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