ALTERNATIVE INVESTMENT RESEARCH CENTRE WORKING PAPER SERIES Working Paper # 0008 Diversification and Yield Enhancement with Hedge Funds Gaurav S. Amin Manager Schroder Hedge Funds, London Harry M. Kat Professor of Risk Management, Cass Business School, City University, London Alternative Investment Research Centre Cass Business School, City University 106 Bunhill Row, London, EC2Y 8TZ United Kingdom Tel. +44.(0)20.70408677 E-mail: harry@airc.info Website: www.cass.city.ac.uk/airc
DIVERSIFICATION AND YIELD ENHANCEMENT WITH HEDGE FUNDS Gaurav S. Amin * Harry M. Kat # Working Paper This version: October 7, 2002 Please address all correspondence to: Harry M. Kat Professor of Risk Management and Director Alternative Investment Research Centre Cass Business School, City University 106 Bunhill Row, London, EC2Y 8TZ United Kingdom Tel. +44.(0)20.70408677 E-mail: harry@harrykat.com *Ph.D student, ISMA Centre, University of Reading, #Professor of Risk Management, Cass School of Business, City University, London. The authors like to thank Hans de Ruiter and ABP Investments for generous support and Tremont TASS (Europe) Limited for supplying the hedge fund data. 2
DIVERSIFICATION AND YIELD ENHANCEMENT WITH HEDGE FUNDS ABSTRACT In this paper we study the diversification effects from introducing hedge funds into a traditional portfolio of stocks and bonds. We find that although the inclusion of hedge funds may significantly improve a portfolio s mean-variance characteristics, it can also be expected to lead to significantly lower skewness as well as higher kurtosis. This means that the case for hedge funds is less straightforward than often suggested and includes a definite trade-off between profit and loss potential. Our results also emphasize that to have at least some impact on the overall portfolio, investors will have to make an allocation to hedge funds which by far exceeds the typical 1-5% that many institutions are currently considering.. 3
1. INTRODUCTION Due to low interest rates and falling equity markets as well as substantial marketing and media hype, hedge funds have become the favourites of many private as well as institutional investors. Recently, two of the largest pension funds in the world, CalPERS and ABP, both announced plans to invest several billions of dollars in this asset class. Hedge funds are often said to provide investors with the best of both worlds: an expected return similar to equity with a risk similar to that of bonds. As we will see, when risk is defined as the standard deviation of the fund return, this is indeed true. Recently, however, several studies 1 have shown that the risk and dependence characteristics of hedge funds are substantially more complex than those of stocks and bonds. This means that when hedge funds are involved it is no longer appropriate to use the standard deviation as the sole measure of risk. Investors will have to give weight to the return distribution s higher moments as well. 2 Most investors (in)tend to hold hedge funds as part of a balanced portfolio containing stocks, bonds and possibly real estate as well. In this paper we therefore investigate what exactly happens to the portfolio return distribution when combining hedge funds with stocks and bonds. We not only look at the means and standard deviations of the resulting portfolios return distributions, but also at their symmetry (as measured by skewness) and the probability of extreme outcomes (as measured by kurtosis). Our results make it clear that hedge funds do not mix very well with equity and that the case for hedge funds is less straightforward than often suggested. Although including hedge funds in a traditional investment portfolio may significantly improve that portfolio s mean-variance characteristics, it can also be expected to lead to significantly lower 4
skewness as well as higher kurtosis. The case for hedge funds therefore includes a definite trade-off between profit and loss potential. 2. THE DATA The hedge fund data used in this study were obtained from Tremont TASS, which is one of the largest hedge fund databases currently available. After eliminating funds with incomplete and ambiguous data as well as funds of funds, per May 2001 the database at our disposal contains monthly net of fee returns on 1195 live and 526 dead funds. As shown in Amin and Kat (2001) for example, concentrating on live funds only will on average overestimate the mean return on individual funds by around 2% as well as introduce a significant downward bias in estimates of the standard deviation, an upward bias in the skewness and a downward bias in the kurtosis estimates of individual fund returns. To correct for this in our analysis we decided not to work with the raw return series of the 264 funds that survived the period 1994-2001. Instead we created 455 7-year monthly return series by, starting off with the 455 funds that were alive in June 1994, replacing every fund that closed down during the sample period by a fund randomly selected from the set of funds alive at the time of closure, following the same type of strategy and of similar age and size. For simplicity, we will still refer to the data series thus obtained as fund returns. Implicitly we assume that in case of a fund closure investors are able to roll from one fund into the other at the reported end-of-month net asset values and at zero additional costs. This will underestimate the true costs of fund closure to the investor for two 5
reasons. First, when a fund closes shop its investors will have to look for a replacement. This search takes time and is not without costs. Second, investors may get out of the old and into the new fund at values that are less favourable than the endof-month net asset values contained in the database. Unfortunately, it is impossible to incorporate this into the analysis in a satisfactory way without further detailed information. To represent stocks we use the S&P 500 index, while bonds are represented by the 10- year Salomon Brothers Government Bond index. Over the sample period monthly S&P returns have a mean of 1.46%, a standard deviation of 4.39%, a skewness of 0.80 and a kurtosis of 3.92. Monthly bond index returns have a mean of 0.43%, a standard deviation of 1.77%, a skewness of 0.56 and a kurtosis of 4.29. 3. DIVERSIFICATION WITH HEDGE FUNDS One reason why investors allocate to hedge funds is to reduce risk without loss of expected return. Based on monthly return data over the period 1994-2001, a portfolio of 50% stocks and 50% bonds has an expected return of almost 1% per month. The same is true for a diversified hedge fund portfolio. With hedge funds only loosely correlated with stocks and bonds, this means that by replacing stocks and bonds with hedge funds investors can reduce the standard deviation of the portfolio return while maintaining the expected return at around 1%. To study this diversification process in more detail, we created 500 different portfolios containing 20 hedge funds each by random sampling without replacement from the above 455 funds. Subsequently, we combined every one of these hedge fund portfolios with stocks and bonds in proportions ranging from 0% to 6
100% invested in hedge funds. Doing so, it is assumed that the proportions of wealth invested in stocks and bonds are always equal. This gives rise to portfolios like 40% stocks, 40% bonds and 20% hedge funds, 30% stocks, 30% bonds and 40% hedge funds, etc. From the monthly returns on the resulting portfolios we calculated four different sample statistics: the mean, standard deviation, skewness, and kurtosis. For hedge fund allocations ranging from 0% to 100%, the 5 th, 10 th, etc. percentiles of the frequency distributions of these four statistics are shown in Exhibit 1-4. Many will argue that investors (including fund of funds managers) do not select portfolios by random sampling. This is certainly true. However, although many investors spend a lot of time and effort selecting hedge funds, this does not necessarily mean that in many cases a randomly sampled portfolio is not a good proxy for the portfolio ultimately selected. So far, there is no evidence that some investors are consistently able to select future out-performers 3 nor of the existence of specific patterns or anomalies. When corrected for possible biases, there is no significant persistence in hedge fund performance nor is there any significant difference in performance between older and younger funds, large and small funds, etc. In addition, older funds may be (more or less) closed for new investments. Investors that are relatively new to hedge fund investing are therefore often forced to invest in funds with little or no track record. If so, selecting funds based on (the statistical properties of) their track record is not an option. The fund prospectus and interviews with managers may provide some information, but in most cases this information will only be sketchy at best and may add more confusion than actual value. 7
<< Insert Exhibit 1-2 >> Exhibit 1 shows the frequency distributions of the mean portfolio return for varying hedge fund allocations. A portfolio of 50% stocks and 50% bonds has a mean return of 0.95%. Since the mean return of a portfolio is simply the weighted average of the means of its components, the introduction of hedge funds makes the median mean return change linearly from 0.95% when no hedge funds are included to 0.99% (the mean return on the median basket of 20 hedge funds) when 100% is invested in hedge funds. Exhibit 2 shows the frequency distributions of the standard deviation of the portfolio return. Here a more interesting picture emerges. Starting at 2.49% for the case of no hedge funds, the median standard deviation drops first but rises later to end at a standard deviation of 2.44% when 100% is invested in hedge funds. The drop represents the relatively low correlation of hedge funds with stocks and bonds. The median standard deviation reaches its minimum at a hedge fund allocation of 50%, which emphasizes that to obtain at least some diversification benefits investors will have to allocate a very substantial part of their wealth to hedge funds. << Insert Exhibit 3-4 >> The frequency distributions of the skewness of the portfolio return are shown in Exhibit 3. The graph shows a remarkable similarity with that of the standard deviation. Starting at 0.32, the median skewness drops first and rises later to end at 0.52 when 100% is invested in hedge funds. The median reaches a minimum of 0.86 at a hedge fund allocation of 55%. Finally, Exhibit 4 shows the frequency distributions 8
of the kurtosis of the portfolio return. Starting at 2.90, the median kurtosis rises gradually towards the kurtosis level of the median portfolio of hedge funds (5.39). The graph is somewhat S-shaped though, meaning that most of the rise takes place for hedge fund allocations between 25% and 65%. For allocations smaller than 25% the effect from the inclusion of hedge funds on the kurtosis of the overall portfolio return is relatively limited. Overall, it seems that hedge funds can indeed be expected to reduce a portfolio s standard deviation, but only at the cost of lower skewness and increased kurtosis. In addition, but not completely unexpected, Exhibit 1-4 show that to realize at least some of the diversification effect investors will have to invest much more in hedge funds than most of them are currently contemplating. 4. YIELD ENHANCEMENT WITH HEDGE FUNDS Another application of hedge funds that is often suggested is to use hedge funds to replace bonds. A good example can be found in McFall Lamm (1999). The idea is that since hedge funds have a relatively high mean and low standard deviation and are only loosely correlated with equity, replacing bonds by hedge funds will substantially raise the expected return without an accompanying rise in standard deviation. To investigate the exact workings of this, we again created 500 different portfolios containing 20 randomly selected hedge funds and combined every one of these hedge fund portfolios with stocks and bonds. Doing so, the equity allocation was kept constant at 50%. In other words, starting with 50% stocks and 50% bonds, the hedge fund allocation is assumed to come fully out of the bond allocation. This gives rise to portfolios like 50% 9
stocks, 40% bonds and 10% hedge funds, 50% stocks, 30% bonds and 20% hedge funds, etc. As before, from the monthly returns on the resulting portfolios we calculated the mean, standard deviation, skewness, and kurtosis. For hedge fund allocations ranging from 0% to 50%, the 5 th, 10 th, etc. percentiles of the frequency distributions of these four statistics are shown in Exhibit 5-8. << Insert Exhibit 5 6 >> From Exhibit 5 we see that, as intended, when the hedge fund allocation increases the median expected return rises in a linear fashion from 0.95% on a portfolio without hedge funds to 1.24% on a portfolio of 50% equity and 50% hedge funds. From Exhibit 2, which shows the frequency distributions of the standard deviation of the portfolio return, we see that replacing bonds by hedge funds in the way we did does not leave the portfolio standard deviation completely untouched. Over the range studied, it rises from 2.4% with no hedge funds to 3.1% with 50% hedge funds. A somewhat different allocation rule could easily correct this though. << Insert Exhibit 7 8 >> So far things are not too different from what investors (are told to) expect when replacing bonds by hedge funds. However, this is no longer the case if we look at the skewness of the portfolio return distribution. Exhibit 7 shows very clearly that replacing bonds by hedge funds will lead to a very substantial reduction in skewness. In addition, as shown by Exhibit 8, it also causes a substantial rise in the return distribution s 10
kurtosis. The drop in skewness is very interesting. With the median hedge fund portfolio exhibiting a skewness of only 0.52, it is clear that in terms of skewness hedge funds and equity do not mix very well. In economic terms, the data suggest that when things go wrong in the stock market, they also tend to go wrong for hedge funds. In a way, this makes sense. A significant drop in stock prices will often be accompanied by a widening of a multitude of spreads, a drop in market liquidity, etc. As a result, many hedge funds will show relatively bad performance as well. A similar reasoning explains why the median portfolio of 50% hedge funds and 50% equity already has a kurtosis that is almost as high as that of 100% hedge funds (5.39). We reach a similar conclusion as before. The improvement in expected return that is observed when bonds are replaced by hedge funds is not a free lunch. The higher expected return is obtained at the cost of substantially lower skewness as well as substantially higher kurtosis. Exhibit 5-8 also confirm our previous conclusion with respect to the required size of the hedge fund allocation. 5. BRINGING IT ALL TOGETHER From the previous discussion it is clear that the beneficial effect of hedge funds on the mean or standard deviation of the portfolio return tends to go hand in hand with an opposite effect on the return distribution s skewness and kurtosis. As a result, the overall shape of the portfolio return distribution can be expected to change substantially as a result of the inclusion of hedge funds. Exhibit 9 shows the return distribution of a portfolio of 50% stocks and 50% bonds as well as the return distribution of the median portfolio of 30% stocks, 30% bonds and 40% hedge funds. 11
<< Insert Exhibit 9 >> Comparing both distributions we see that they intersect several times. Reading the graph from left to right, the net effect of the inclusion of hedge funds consists of: (1) a higher probability of a very large loss, (2) a lower probability of a smaller loss, (3) a higher probability of a low positive return, and (4) a lower probability of a high positive return. Most investors that use hedge funds for diversification will expect to trade in profit potential for reduced loss potential on a more or less equal basis. However, as shown clearly by Exhibit 9, because of the increase in negative skewness the trade-off is not symmetrical. Investors can expect to give up more on the upside than on the downside. << Insert Exhibit 10 >> Exhibit 10 shows the return distribution of a portfolio of 50% stocks and 50% bonds as well as the return distribution of the median portfolio of 50% stocks and 50% hedge funds. Comparing both graphs we see that the net effect of replacing bonds by hedge funds consists of (1) a higher probability of a large loss, (2) a lower probability of a smaller (positive or negative) return, and (3) a higher probability of a higher positive return. Most investors who replace bonds by hedge funds will expect the return distribution to simply shift to the right without changing shape. Exhibit 10, however, makes it clear that this shift will be accompanied by an extension of the left tail, i.e. a higher probability of a large loss. 12
6. CONCLUSION Our study shows clearly that the case for hedge funds is less straightforward than often suggested and requires investors to make a trade-off between profit and loss potential. In essence, hedge funds offer investors a way to modify the risk-return characteristics of their portfolio. Whether the resulting portfolio makes for a more attractive investment than the original is primarily a matter of taste though, not a general rule. The next question is of course what type of investor would be interested in trading in skewness for a higher mean return and/or a lower standard deviation. Since in general institutional investors will be better equipped to deal with a relatively large loss (after all they can raise premiums) than private investors, one could argue that hedge funds are more suitable for institutional investors than for retail investors. So far, however, private investors have been the main investors in hedge funds. Although institutional investors are showing interest, not many have made a significant allocation to hedge funds yet. 13
FOOTNOTES 1. See for example Agarwal and Naik (2001), Fung and Hsieh (2001), Lo (2001), Brooks and Kat (2002), or Favre and Galeano (2002). 2. Scott and Horvath (1980) show that under fairly weak assumptions concerning investors utility functions investors will desire high odd moments and low even moments. Hedge funds offer relatively high means and low variances, but as we will see they also tend to give investors skewness and kurtosis attributes that are exactly opposite to what investors desire. 3. Although funds of hedge funds often claim to possess superior fund selection skills, it is shown in Kat and Lu (2002) that over the period 1994 2001 the average fund of funds underperformed an equally-weighted portfolio of randomly selected hedge funds by almost 3% per annum. 14
REFERENCES Agarwal, V. and N. Naik (2001), Characterizing Systematic Risk of Hedge Funds with Buy-and-Hold and Option-Based Strategies, Working Paper London Business School. Amin, G. and H. Kat (2001), Welcome to the Dark Side: Hedge Fund Attrition and Survivorship Bias 1994-2001, Working Paper ISMA Centre, University of Reading. Brooks, C. and H. Kat (2002), The Statistical Properties of Hedge Fund Index Returns and Their Implications for Investors, Journal of Alternative Investments, Fall, pp. 26-44. Favre, L. and J. Galeano (2002), An Analysis of Hedge Fund Performance Using Loess Fit Regression, Journal of Alternative Investments, Spring, pp. 8-24. Fung, W. and D. Hsieh (2001), The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers, Review of Financial Studies, Vol. 14, pp. 313-341. Kat, H. and S. Lu (2002), An Excursion into the Statistical Properties of Individual Hedge Fund Returns, Working Paper ISMA Centre, University of Reading. Lo, A. (2001), Risk Management for Hedge Funds: Introduction and Overview, Financial Analysts Journal, November/December, pp. 16-33. McFall Lamm, R. (1999), Portfolios of Alternative Assets: Why Not 100% Hedge Funds?, Journal of Alternative Investments, Winter, pp. 87-97. Scott, R. and P. Horvath (1980), On the Direction of Preference for Moments of Higher Order Than the Variance, Journal of Finance, Vol. 35, pp. 915-919. 15
Exhibit 1: Mean of Portfolios of Stocks, Bonds and 20 Hedge Funds (A). 1.4 1.2 Mean Return 1 0.8 0.6 0 10 20 30 40 50 60 70 80 90 100 % in Hedge Funds Exhibit 2: Std Deviation of Portfolios of Stocks, Bonds and 20 Hedge Funds (A). 3.5 Standard Deviation 3 2.5 2 1.5 0 10 20 30 40 50 60 70 80 90 100 % in Hedge Funds 16
Exhibit 3: Skewness of Portfolios of Stocks, Bonds and 20 Hedge Funds (A). 0.5 0 Skewness -0.5-1 -1.5 0 10 20 30 40 50 60 70 80 90 100 % in Hedge Funds Exhibit 4: Kurtosis of Portfolios of Stocks, Bonds and 20 Hedge Funds (A). 10 8 Kurtosis 6 4 2 0 10 20 30 40 50 60 70 80 90 100 % in Hedge Funds 17
Exhibit 5: Mean of Portfolios of Stocks, Bonds and 20 Hedge Funds (B). 1.4 1.3 Mean Return 1.2 1.1 1 0.9 0 5 10 15 20 25 30 35 40 45 50 % in Hedge Funds Exhibit 6: Std Deviation of Portfolios of Stocks, Bonds and 20 Hedge Funds (B). 4 Standard Deviation 3.5 3 2.5 2 0 5 10 15 20 25 30 35 40 45 50 % in Hedge Funds 18
Exhibit 7: Skewness of Portfolios of Stocks, Bonds and 20 Hedge Funds (B). -0.2-0.4 Skewness -0.6-0.8-1 -1.2-1.4 0 5 10 15 20 25 30 35 40 45 50 % in Hedge Funds Exhibit 8: Kurtosis of Portfolios of Stocks, Bonds and 20 Hedge Funds (B). 7 6 Kurtosis 5 4 3 2 0 5 10 15 20 25 30 35 40 45 50 % in Hedge Funds 19
Exhibit 9: Diversification with Hedge Funds. 0.2 0.16 0.12 0.08 No HF Yes HF 0.04 0-12 -9-6 -3 0 3 6 9 Exhibit 10: Yield Enhancement with Hedge Funds. 0.18 0.15 0.12 0.09 No HF Yes HF 0.06 0.03 0-12 -9-6 -3 0 3 6 9 20