Abstract

In this summary paper we show how investors can neutralise the unwanted skewness and kurtosis effects from investing in hedge funds by (1) purchasing out-of-the-money equity puts, (2) investing in managed futures, and/or by (3) overweighting equity market neutral and global macro and avoiding distressed securities and emerging market funds. We show that all three alternatives are up to the job but also come with their own specific price tag.

1. Introduction

Due to their relatively weak correlation with other asset classes, hedge funds can play an important role in risk reduction and yield enhancement strategies. Recent research, however, has also shown that this diversification service does not come for free. Amin and Kat (2003), for example, show that although the inclusion of hedge funds in a portfolio may significantly improve that portfolio's mean-variance characteristics, it can also be expected to lead to significantly lower skewness and higher kurtosis. This means that the case for hedge funds is not as straightforward as is often suggested and includes a definite trade-off between profit and loss potential.

The sting of hedge funds is literally in the tail as in terms of skewness, hedge funds and equity do not mix very well. When things go wrong in the stock market, they also tend to go wrong for hedge funds as a significant drop in stock prices is typically accompanied by a drop in market liquidity, a widening of a multitude of spreads, etc. Equity market neutral and long/short funds have a tendency to be long in smaller stocks and short in larger stocks and need liquidity to maintain market neutrality. As a result, when the stock market comes down this type of funds can be expected to have a hard time. Likewise, when the stock market comes down mergers and acquisitions will be postponed which will have a negative impact on the performance of risk arbitrage funds. Problems are not limited to funds that invest in equity though. A drop in stock prices will often also lead to a widening of credit spreads, which in turn will seriously damage the performance of fixed income and convertible arbitrage funds. As they all share it, diversification among different funds will not mitigate this.

In this article we discuss a number of ways to solve the above skewness problem and the associated costs. We will look at the use of out-of-the-money stock index puts, managed futures and sophisticated strategy selection. Before we do so, however, we briefly discuss the exact nature of hedge fund returns and the associated skewness problem.

2. The effects of introducing hedge funds in a portfolio

Generally speaking, risk is one word, but not one number. The returns on portfolios of stocks and bonds risk are more or less normally distributed. Because normal distributions are fully described by their mean and standard deviation, the risk of such portfolios can be measured with one number: the standard deviation. Confronted with non-normal distributions, however, it is no longer appropriate to use the standard deviation as the sole measure of risk. In that case investors should also look at the degree of symmetry of the distribution, as measured by its so-called 'skewness', and the probability of extreme positive or negative outcomes, as measured by the distribution's 'kurtosis'. A symmetrical distribution will have a skewness equal to zero, while a distribution that implies a relatively high probability of a large loss (gain) is said to exhibit negative (positive) skewness. A normal distribution has a kurtosis of 3, while a kurtosis higher than 3 indicates a relatively high probability of a large loss or gain. Since most investors are in it for the longer run, they strongly rely on compounding effects. This means that negative skewness and high kurtosis are extremely undesirable features as one big loss may destroy years of careful compounding.

Table 1: Average Skewness and Kurtosis Individual Hedge Fund Returns

Table 1 shows the average skewness and kurtosis found in the returns of individual hedge funds from various strategy groups. The average hedge fund's returns tend to be non-normally distributed and may exhibit significant negative skewness as well as substantial kurtosis. Put another way, hedge fund returns may exhibit low standard deviations but they also tend to provide skewness and kurtosis attributes that are exactly opposite to what investors desire. It is this whole package that constitutes hedge fund risk, not just the standard deviation. Actually, this is not the whole story yet, as strictly speaking we should also include the relationship between the hedge fund return and the returns on other assets and asset classes in the definition of risk. We will look at this shortly.

The skewness and kurtosis properties of hedge funds do not come as a complete surprise. If we delve deeper into the return generating process it becomes obvious that most spread trading and pseudo-arbitrage strategies will generate these features by their very nature as the profit potential of trades will typically be a lot smaller that their loss potential. Consider a merger arbitrage fund for example. When a takeover bid is announced the share price of the target will jump towards the bid. It is at this price that the fund will buy the stock. When the takeover proceeds as planned the fund will make a limited profit equal to the difference between the relatively high price at which it bought the stock and the bid price. When the takeover fails, however, the stock price falls back to its initial level, generating a loss that may be many times bigger than the highest possible profit. Spread traders are confronted with a similar payoff profile. When the spread moves back to its perceived equilibrium value they make a limited profit, but when the market moves against them they could be confronted with a much larger loss. This is why strategies like this are sometimes referred to as "picking up nickels in front of a steamroller". Of course, there is no reason why a trader could not get lucky and avoid getting hit by the steamroller for a long period of time. This does not mean that the risk was never there, however. It always was. It just never materialised so it does not show from the trader's track record.

Since individual hedge funds carry quite some idiosyncratic risk, combining hedge funds into a basket, as is standard practice nowadays, will substantially reduce the standard deviation of the return on that portfolio. However, it can also be expected to lower the skewness and raise the correlation with the stock market.

Table 2: Individual Hedge Fund and Hedge Fund Portfolio Risks

Table 2 shows the standard deviation, skewness and correlation with the S&P 500 of the average individual hedge fund in the various strategy groups as well as an equally-weighted portfolio of all funds in each group. From the table we see that forming portfolios indeed leads to a very substantial reduction in standard deviation. With the exception of emerging market funds, the portfolio standard deviations are approximately half the standard deviations of the average individual fund. Apparently, there are many different ways in which the same general strategy can be executed. Contrary to standard deviation, skewness is not diversified away and drops as portfolios are formed. With the exception of equity market neutral funds, the portfolio skewness figures are quite a bit lower than for the average individual fund, with especially merger arbitrage and distressed securities funds standing out. Despite the lack of overall correlation, it appears that when markets are bad for one fund, they tend to be bad for other funds as well. Finally, comparing the correlation with the S&P 500 of individual funds and portfolios we clearly see that the returns on portfolios of hedge funds tend to be much more correlated with the stock market than the returns on individual funds. Although individual hedge funds may be more or less market neutral, the portfolios of hedge funds that most investors actually invest in definitely are not.

Table 3: Effects of Combining Hedge Funds with Stocks and Bonds

So far we have seen that hedge fund returns tend to exhibit a number of undesirable features, which cannot be diversified away. Skewness, kurtosis and correlation with stocks worsen significantly when portfolios are formed. But we are not there yet, as we haven't looked at what happens when hedge funds are combined with stocks and bonds. Although the inclusion of hedge funds in a portfolio may significantly improve that portfolio's mean-variance characteristics, it can also be expected to lead to significantly lower skewness as well as higher kurtosis. Table 3 shows what happens to the standard deviation, skewness and kurtosis of the portfolio return distribution if, starting with 50% stocks and 50% bonds, we introduce hedge funds (modelled by the average equally-weighted random portfolio of 20 funds) in a traditional stock-bond portfolio. As expected, when hedge funds are introduced the standard deviation drops significantly. This represents the still relatively low correlation of hedge funds with stocks and bonds. This is the good news. The bad news, however, is that a similar drop is observed in the skewness of the portfolio return. In addition, we also observe a rise in kurtosis.

Especially the skewness effect goes far beyond what one might expect given the hedge fund skewness results in Table 2. When things go wrong in the stock market, they also tend to go wrong for hedge funds. Not necessarily because of what happens to stock prices (after all, many hedge funds do not invest in equity), but because a significant drop in stock prices will often be accompanied by a widening of credit spreads, a significant drop in market liquidity, higher volatility, etc. Since hedge funds are highly sensitive to such factors, when the stock market drops, hedge funds can be expected to show relatively bad performance as well. Recent experience provides a good example. Over the year 2002, the S&P 500 dropped by more than 20% with relatively high volatility and substantially widening credit spreads. Distressed debt funds, at the start of 2002 seen by many investors as one of the most promising sectors, suffered substantially from the widening of credit spreads. Credit spreads also had a negative impact on convertible arbitrage funds. Stock market volatility worked in their favour, however. Managers focusing on volatility trading generally fared best, while managers actively taking credit exposure did worst. Equity market neutral funds suffered greatly from a lack of liquidity, while long/short equity funds with low net exposure outperformed managers that remained net long throughout the year. As a result, overall hedge fund performance in 2002 as measured by the main hedge fund indices was more or less flat.

So here is the main problem. Individual hedge fund returns tend to exhibit some negative skewness. When combined into portfolios, however, this negative skewness becomes worse. When subsequently those portfolios are combined with equity, skewness drops even further. The increase in negative skewness will tend to offset the lower standard deviation that results from the inclusion of hedge funds. In other words, when adding hedge funds, the investor's downside risk will largely remain unchanged while at the same time part of his upside potential is diversified away. Exactly the opposite of what we want a good diversifier to do.

In the next sections we discuss three possible ways to reduce the above skewness effect, as well as the associated costs, while maintaining the benefits of the lower standard deviation.

3. Purchasing Out-of-the-Money Puts

Since the increase in negative skewness that tends to come with hedge fund investing is highly undesirable, it is important to look for ways to neutralise this effect. One solution is to buy hedge funds in 'guaranteed' form only. In essence, this means buying a put on one's hedge fund portfolio so that in down markets the link between the hedge fund portfolio and the stock market is severed. Unfortunately, the market for put options on (baskets of) hedge funds is still in an early stage. As a result, counterparties for the required contracts are likely to be hard to find as well as expensive. With hedge funds so closely related to the ups and especially downs of the stock market, there is a very simple alternative though: the purchase of out-of-the-money puts on a stock index. As discussed in Kat (2003), over the last 10 years a strategy of buying and rolling over out-of-the-money S&P 500 puts would have generated returns with very high positive skewness. It therefore makes sense to use this option strategy to neutralise the negative skewness in hedge funds.

Suppose we added stock index put options to a portfolio of stocks, bonds and hedge funds with the aim to bring the skewness of the overall portfolio return back to what it was before the addition of hedge funds. Obviously, there is a price tag attached to doing so. Since we are taking away something bad (negative skewness), we will have to give up something good. If we used leverage to keep overall portfolio volatility at the same level as before the addition of the puts, i.e. if we aimed to preserve the volatility benefit of the addition of hedge funds, this means we will have to accept a lower expected return. Economically, this of course makes perfect sense as the puts that we add won't come for free and, since they are out-of-the-money, are unlikely to pay off (which of course is just another way of saying that the option strategy by itself has a highly negative expected return).

Table 4: Effects of Combining Portfolios of Stocks, Bonds and
Hedge Funds with Puts and Leverage

Assuming investors can leverage their portfolio at a rate of 4% and the expected returns on stocks, bonds and hedge funds are equal to their historical 10-year means, Table 4 shows the effect of using puts and leverage in a portfolio of stocks, bonds and hedge funds (with always equal allocations to stocks and bonds). Starting with the situation shown in Table 3, adding puts to bring the skewness of the overall portfolio back to what it was before the addition of hedge funds (-0.33), while maintaining the volatility benefit, requires only a small allocation to options. As is also clear from the change in portfolio kurtosis, this small allocation, however, goes a long way in restoring the (near-) normality of the return distribution. Unfortunately, the costs in terms of expected return (4th column) are quite significant. For example, with a 25% hedge fund allocation, investors can expect to loose 61 basis points in expected return. This drop in expected return can be interpreted as the option market's price of the additional skewness introduced by hedge funds.

Of course, the above conclusion depends heavily on the assumption that investors can leverage (either directly or through the futures market) their portfolios at 4%, which in the current interest rate environment does not seem unrealistic. Obviously, if the interest rate were higher, the costs of the skewness reduction strategy would be higher as well because the difference between the expected return on the unlevered portfolio and the interest rate, i.e. the pick-up in expected return due to the leverage, would be smaller. A similar reasoning applies in case of a lower expected return on stocks, bonds and/or hedge funds.

Another important element of the above analysis concerns the assumption that the allocations to stocks and bonds are always equal. If we assumed that investors always divided their money in such a way that 1/3 was invested in stocks and 2/3 in bonds (as opposed to the '50/50 portfolio' discussed earlier we will refer to such a portfolio as a '33/66 portfolio'), our results would of course change. Under the assumptions made, a portfolio made up of 1/3 stocks and 2/3 bonds has a skewness of 0.03. With 25% hedge funds, the portfolio's skewness will come to down to -0.43, while with 50% hedge funds it will drop to -0.75. Because when hedge funds are introduced skewness for a 33/66 portfolio drops faster than for a 50/50 portfolio, we will have to buy more puts and apply more leverage. Because the mean of the 33/66 portfolio is substantially lower than that of the 50/50 portfolio, however, the increased leverage will not be sufficient to rescue the expected return. As can be seen in the last column of Table 4, the costs of the skewness reduction strategy for a 33/66 portfolio are very substantial. With 25% hedge funds, the costs of skewness reduction will amount to 3.20%, as opposed to 'only' 0.61% for the 50/50 portfolio.

In sum, after introducing hedge funds, purchasing out-of-the money puts can restore the (near-) normality of the portfolio return distribution fairly easily. However, this may come at a substantial cost to the portfolio's expected return, especially for investors that are overweighted in bonds.

4. Investing in Managed Futures

In principle, any asset or asset class that has suitable (co-)skewness characteristics can be used to hedge the additional skewness from incorporating hedge funds. One obvious candidate is managed futures. Managed futures programmes are often trendfollowing in nature. What these programmes do is somewhat similar to what option traders do to hedge a short call position. When the market moves up, they increase exposure and vice versa. By moving out of the market when it comes down, managed futures programmes avoid being pulled in. As a result, the (co-)skewness characteristics of managed futures programmes are more or less opposite to those of many hedge funds.

The term 'managed futures' refers to professional money managers known as commodity trading advisors or CTAs who manage assets using the global futures and options markets as their investment universe. Managed futures have been available for investment since 1948 when the first public futures fund started trading. The industry did not take off until the late 1970s though. Since then the sector has seen a fair amount of growth with currently an estimated $50 billion under management.

There are 3 ways in which investors can get into managed futures. First, investors can buy shares in a public commodity (or futures) fund, in much the same way as they would invest in a stock or bond mutual fund. Second, investors can place funds privately with a commodity pool operator (CPO) who pools investors' money and employs one or more CTAs to manage the pooled funds. Third, investors can retain one or more CTAs directly to manage their money on an individual basis or hire a manager of managers (MOM) to select CTAs for them. The minimum investment required by funds, pools and CTAs varies considerably, with the direct CTA route open only to investors that want to make a substantial investment. CTAs charge management and incentive fees comparable to those charged by hedge funds, i.e. 2% management fee plus 20% incentive fee. Similar to funds of hedge funds, funds and pools charge an additional fee on top of that.

Initially, CTAs were limited to trading commodity futures (which explains terms such as public commodity fund, CTA and CPO). With the introduction of futures on currencies, interest rates, bonds and stock indices in the 1980s, however, the trading spectrum widened substantially. Nowadays many CTAs trade both commodity and financial futures. Many take a very technical, systematic approach to trading, but others opt for a more fundamental, discretionary approach. Some concentrate on particular futures markets, such as agricultural, currencies, or metals, but most diversify over different types of markets.

In this study, the asset class 'managed futures' is represented by the Stark 300 index. This asset-weighted index is compiled using the top 300 trading programmes from the Daniel B. Stark & Co. database. All 300 of the CTAs in the index are classified by their trading approach and market category. Currently, the index contains 248 systematic and 52 discretionary traders, which split up in 169 diversified, 111 financial only, 9 financial & metals and 11 non-financial trading programmes.

As shown in Kat (2004a), historically managed futures returns have exhibited a lower mean and a higher standard deviation than hedge fund returns. However, managed futures exhibit positive instead of negative skewness and much lower kurtosis. In addition, the correlation of managed futures with stocks and hedge funds is extremely low, which means that managed futures make very good diversifiers. Table 5 shows the effect of incorporating either hedge funds or managed futures in a traditional 50/50 stock-bond portfolio.

Table 5: Return Statistics Portfolios of Stocks, Bonds and
Either Hedge Funds or Managed Futures

From the table we, again, see that if the hedge fund allocation increases, both the standard deviation and the skewness of the portfolio return drop substantially, while at the same time the return distribution's kurtosis increases. With managed futures the picture is significantly different, however. If the managed futures allocation increases, the standard deviation drops faster than with hedge funds. More remarkably, skewness rises instead of drops while the reverse is true for kurtosis. Although (assuming future performance will resemble the past) hedge funds offer a somewhat higher expected return, from an overall risk perspective managed futures appear much better diversifiers than hedge funds.

Table 6: Allocations and Annualised Change in Expected Return Portfolios of Stocks, Bonds, Hedge Funds, and Managed Futures

Now suppose we did the same thing as before: choose the managed futures allocation such as to bring the skewness of the portfolio return back to what it was before the addition of hedge funds (-0.33), while at the same time preserving the volatility benefit of the addition of hedge funds by the use of some leverage. The results are shown in Table 6, which shows that for smaller hedge fund allocations of up to 15% the optimal managed futures allocation will be more or less equal to the hedge fund allocation. Looking at the change in expected return, we see that as a result of the introduction of managed futures the expected portfolio return increases significantly. With a 25% hedge fund allocation for example, the investor stands to gain 205 basis points in annualised expected return. This of course compares very favourably with the results on out-of-the-money puts. One should, however, always keep in mind that the outcomes of analyses like this heavily depend on the inputs used. A lower expected return for managed futures and/or a higher borrowing rate (used to leverage the portfolio volatility back to its initial level) could easily turn these gains into losses. In addition, we have to keep in mind that although the expected return does not seem to suffer from the use of managed futures to neutralise the unwanted skewness effect from hedge funds, this does not mean it comes for free completely. Investors pay by giving up the positive skewness that they would have had when they had only invested in managed futures.

5. Smart Strategy Selection

So far we have modelled the asset class 'hedge funds' as a representative portfolio of 20 different individual funds; a proxy for the average fund of funds portfolio. Although this is what most investors currently invest in, it is interesting to investigate in how far it is possible to eliminate the skewness effect of hedge funds by simply choosing another hedge fund portfolio, i.e. by allocating differently to the various hedge fund strategies available. This is the approach taken in Davies, Kat and Lu (2004). Using a sophisticated optimisation technique known as Polynomial Goal Programming (PGP), they incorporate investor preferences for return distributions' higher moments into an explicit optimisation model. This allows them to solve for multiple competing hedge fund allocation objectives within a mean-variance-skewness-kurtosis framework. Apart from underlining the existence of significant differences in the return behaviour of different hedge fund strategies, the analysis shows that PGP optimal portfolios for skewness-aware investors contain hardly any allocations to long/short equity, distressed securities and emerging markets funds. Equity market neutral and global macro funds on the other hand tend to receive very high allocations, which is primarily due to their low co-variance, high co-skewness and low co-kurtosis properties. Looking back at tables 1 and 2, these conclusions do not come as a complete surprise. The strategies that the optimiser tends to drop are exactly the strategies that exhibit the most negative skewness. Global macro and equity market neutral strategies on the other hand come with much more desirable risk characteristics. Global macro funds primarily act as portfolio skewness enhancers, while equity market neutral funds act as volatility and kurtosis reducers (which are especially important given the relatively high volatility and kurtosis of global macro).

An interesting by-product of the analysis in Davies, Kat and Lu (2004) is that introducing preferences for skewness and kurtosis in the portfolio decision-making process yields portfolios that are far different from the mean-variance optimal portfolios, with less attractive mean-variance characteristics. This underlines a point made earlier in Kat (2004b) that using standard mean-variance portfolio allocation tools when alternative investments are involved can be highly misleading. It also shows that in hedge fund diversification there is no such thing as a free lunch. When substantially overweighting global macro and equity market neutral strategies, investors can expect more attractive skewness and kurtosis, but at the cost of a less attractive expected return and volatility.

Finally, it is interesting to note that many global macro funds tend to follow strategies that are similar to the strategies typically employed by CTAs. In fact, some of the largest global macro funds have their origins in managed futures. The difference between expanding into managed futures and overweighting global macro is therefore probably smaller than one might suspect.

6. Conclusion

Typical hedge fund portfolios' attractive mean-variance properties tend to come at the cost of negative skewness and increased kurtosis. We have shown that investors can neutralise the unwanted skewness and kurtosis by purchasing out-of-the-money equity puts, investing in managed futures, and/or by overweighting equity market neutral and global macro and avoiding distressed securities and emerging market funds. Hedge fund returns are not superior to the returns on other asset classes. They are just different!

References

Amin, G. and H. Kat (2003), Stocks, Bonds and Hedge Funds: No Free Lunch!, Journal of Portfolio Management, Summer, pp. 113-120.

Davies, R, H. Kat and S. Lu (2004), Fund of Hedge Funds Portfolio Selection: A Multiple-Objective Approach, AIRC Working Paper No. 18 (downloadable from the AIRC website: www.cass.city.ac.uk/airc).

Kat, H. (2003), Taking the Sting Out of Hedge Funds, Journal of Wealth Management, Winter, pp. 1-10.

Kat, H. (2004a), Managed Futures and Hedge Funds: A Match Made in Heaven, Journal of Investment Management, Vol.2, No. 1, pp. 1-9.

Kat, H. (2004b), Hedge Funds vs. Common Sense: An Illustration of the Dangers of Mechanical Investment Decision-Making, Journal of Investment Management, Vol. 2, No. 4, pp. 95-104.