The performance of hedge funds in an institutional portfolio context

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The performance of hedge funds in an institutional portfolio

context

Frederik Tijhuis

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Supervisor: dr. R.M. Salomons

Abstract

This thesis investigates whether hedge funds add value in institutional portfolios. To investigate this, optimal portfolios are constructed. I find that hedge funds do add value, because optimal portfolios always include hedge funds. Optimal allocations to hedge funds are relatively large, usually more than 50%. This has important implications for financial institutions. They could optimize their portfolios by allocation (more) to hedge funds. This finding is robust across different risk measures, such as standard deviation, downside deviation and VaR. Moreover, this finding is robust across time.

JEL classification: G23

Keywords: hedge funds, institutional investors

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University of Groningen, July 2014 Student number: S2026457

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1. Introduction

The first known hedge fund was created by Alfred W. Jones in 1949. Jones used a combination of long and short positions to decrease market exposure. Moreover, he used leverage to increase returns. His approach was so successful that soon many others followed, and the hedge fund industry grew rapidly (Stulz, 2007, p. 176; Ineichen, 2000, p.153).

Initially the industry growth was fuelled by high net worth individuals, who liked the promise of superior returns. Hedge fund managers’ superior skills and extensive investment experience were seen as a virtual guarantee for superior performance. In the late 1990s, however, the superior returns argument began to change into a diversification argument (Kat and Palaro, 2005). Then hedge funds promised that they had low correlations with traditional asset classes, and therefore were a valuable addition to investment portfolios (Fung and Hsieh, 1997). From then on, institutional flows into hedge funds increased rapidly. This study investigates whether hedge funds are still a valuable addition to institutional investment portfolios nowadays.

There are two reasons why they could be a valuable addition. As mentioned before the first reason is superior, risk-adjusted returns. And the second reason is the diversification argument; hedge funds have low correlations with other asset classes. This leads to the questions: do hedge funds really offer superior risk-adjusted returns, and do they have low correlations? In the data analysis section these questions will be addressed by investigating the Sharpe ratio, as a proxy for absolute, risk-adjusted returns and by looking at the various correlations.

A third argument for hedge funds could be that they are relatively safe. The first hedge fund originated as a portfolio which consisted of both long and short positions, and therefore it was relatively safe (Stulz, 2007, p. 176; Ineichen, 2000, p.153). That is also how these funds got their name. They used to be ‘hedged’. In order to investigate whether hedge funds are relatively safe, various risk measures will be investigated and compared across different asset classes and hedge fund categories. Some strategies, such as equity market neutral, actively try to be independent of the equity market. This should lead to low correlations. But besides correlations this paper investigates other risk measures, such as downside deviation and Value at Risk (VaR). Downside deviation uses only the negative side of the distribution. So, it will take into account negative outliers/extremes, and will not penalize for positive outliers. Besides that, VaR is included to control for extreme outcomes. It is possible that hedge funds generate stable superior returns, but at the cost of higher risk of extreme negative outcomes, e.g. because of the use of (extreme) leverage. This is comparable to a company which sells insurance. They generate a very steady stream of returns, but sometimes incur an extreme loss.

After looking at the data, the actual effect of adding hedge funds to traditional portfolios will be

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be optimized with and without hedge funds. This will be done based on mean-variance analysis. The reason for this is that portfolio optimization takes into account both the risk-return characteristics and the correlations of an asset. Thus, the optimizations will show the combined effect of the changes in the different hedge fund characteristics on portfolio compositions. However, because the data is not normally distributed, and because there might be negative outliers (negatively skewed) for hedge funds, other risk measures will be used. The portfolios will also be constructed based on mean-downside deviation and mean-VaR analysis. Besides that, the dataset will be split into two periods, to check for consistency through time. The portfolio optimizations will be repeated for the two subsets. If all these optimizations significantly allocate to hedge funds, it might be safe to assume that hedge funds do add value in an institutional portfolio.

First, there will be a short literature review. This paper contributes to the research, which mostly dates from the late 1990s and early 2000s, by extending the dataset till 2014. Moreover, this paper also adds to the current body of research, because it looks at the changes in hedge fund performance over time and across different risk measures. After the literature review, there will be a data section, where potential biases will be discussed, and an extensive data analysis will be presented. The next section will present the findings of several portfolio optimizations based on different risk measures. The last section concludes.

2. Literature

There is a vast body of hedge fund research. This thesis focusses on the field of hedge fund

performance in a portfolio context. Within this field there are several important papers, which will be discussed in this section.

Early research (late 1990s, early 2000s) finds that hedge funds have low volatility, low downside deviation and high Sharpe ratios compared to standard equity indices (Goldman Sachs & Co., 1998). Morover, researchers find that hedge funds have relatively low correlations with traditional asset classes, and therefore provide excellent opportunities for diversification (Agarwal and Naik, 2000; Ineichen, 2000; Fung and Hsieh, 1997). McFall Lamm (1999) finds that hedge funds enter the efficient frontier across all risk levels, even when low returns are assumed. Hedge funds enter the frontier mostly at the expense of bonds. These results were later confirmed by Amin and Kat (2003).

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variance analysis will not capture all the effects of hedge funds. And other risk measures, which take skewness and kurtosis into account, should be used.

Kouwenberg (2003) finds that in general, hedge funds have positive alpha’s, and therefore are a useful contribution to institutional investment portfolios.

In general, the early research into hedge funds is rather positive about hedge funds, and concludes that hedge funds improve the mean-variance characteristics when added to a portfolio. However, McFall Lamm (1999) finds a negative trend in hedge fund returns since the 1980s. If this trend has continued, it could mean that hedge funds are less attractive nowadays.

Al the literature discussed so far is based on hedge fund data from the 1990s. This was a limited time period, and in general during that period there was a bull market. I want to add to this body of research by investigating whether these results still hold for a larger sample period, 1990-2014, which also includes a bear market (2007-2009).

More recent research focusses less on the performance of the aggregate hedge fund industry and more on specific issues. Gupta and Liang (2005) find evidence of non-normality in the returns of hedge funds. And therefore suggest that value-at-risk type measures are used to account for this non-normality. Further, Liang and Park (2010) find that downside risk measures are better at predicting hedge fund failure than traditional risk measures, such as the standard deviation. Therefore, they suggest that downside deviation is a more informative measure than the standard deviation, for

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3. Data and methodology

The data set for this thesis consists of monthly returns for the hedge fund indices provided by Hedge Fund Research (HFR). Besides that, the data set includes the ‘traditional’ asset classes such a stocks, bonds and commodities. Because this thesis is from the perspective of an American institutional investor, the proxy’s for asset classes will also be American. The S&P 500 index is used to represent stocks. To represent government bonds, the US 10 year government bonds will be used. And to represent corporate bonds, the Barclays corporate bond indexes for Investment Grade and High Yield will be used. And finally, as a proxy for commodities, the S&P GSCI Commodity index will be used. All this data is downloaded from Datastream.

HFR provides monthly indices which are designed to represent the performance of the hedge fund industry. These so-called HFRI indices are constructed by equally weighting the constituent funds. The HFRI indices vary in depth from the industry level, HFRI fund weighted composite index, to category level, such as event driven, macro, relative value, etc., and even to specific strategy levels, such as convertible arbitrage, short bias, equity market neutral, etc.2

For this thesis I will use the HFRI fund weighted composite and the HFRI fund of funds composite for the industry level. Besides that, I will use the emerging markets, equity hedge, event-driven, macro and relative value indices for the category level. And finally, I will include a few specific strategies, based on prior research and personal expectations. Ineichen (2000) states that relative value and event driven strategies are most likely to have sustainable performance. I will investigate this by including specific strategies for these categories. For relative value, fixed income/convertible arbitrage will be included and for event driven, merger arbitrage will be included. Besides that, I expect these strategies to be least dependent on the stock market and therefore, they should do well during the crisis. To investigate the value added of these specific strategies, they are included in the data set. Connolly and Hutchinson (2012) show that dedicated short bias still adds value nowadays, and is a valuable diversifier.

Moreover, short bias is also expected to do well during the crisis. In order to test this, short bias is also included. And finally, equity market neutral is included for the same reasons.

3.1. Biases

Hedge fund data are likely to suffer from biases because of the way the data is collected. Hedge fund reporting is voluntarily and data is collected by private data collectors, such as Hedge Fund Research. Since hedge funds are not allowed to use marketing, they can use the reporting of good returns as a form of indirect marketing. But this means that only funds that perform well will report to a database. This leads to a self-selection bias. Besides that, there is also a selection bias in the funds that stop reporting to a database. Funds could stop reporting because they no longer need to attract capital, or

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because they want to protect their investment strategy. But, on the other hand, they could also stop reporting because of bad returns, or because they go out of business. This leads to a self-selection bias, and might mean that the database is not representative of the entire hedge fund industry (Fung and Hsieh, 2000, p. 293). In that case, conclusions are only valid for the hedge funds within that specific database. Besides that, hedge funds are allowed to backfill their returns. This leads to a backfill bias, because only firms with a good performance will backfill their returns. Authors agree that there is a backfill bias in hedge fund databases, but they do not agree on the extent of the bias. For example, Fung and Hsieh (2000) find that the backfill bias is 1.4% per annum, whereas Posthuma and van der Sluis (2003) find that it is 4% per annum. Survivorship bias results from only analyzing surviving funds. Authors agree that there is a survivorship bias, However, the extent of the survivorship bias remains unclear. Brown et al. (1999) find a bias of 3% for offshore funds. Fung and Hsieh (2000) find a bias of 3% per annum in the TASS database. Liang (2000) investigates the survivorship bias in different databases. For the TASS database he finds that the bias exceeds 2% per annum, whereas for the HFR database, this bias is 0.6% per annum. Ackermann, McEnally and Ravenscraft (1999) suggest that the survivorship and the self-selection bias offset each other. On the one hand, the database may overestimate hedge fund returns, because only survivors are analyzed. On the other hand, the database may underestimate the returns, when the excellent funds decide not to report.3

Besides these biases, which result from the way in which the data is collected, the actual collected data might also be biased. Hedge Funds often have exotic assets and derivatives, for which no market value is available. This means that hedge fund managers have flexibility with regard to the valuation of these assets.

Getmansky, Lo, and

Makarov (2004) find that managers use this flexibility to smooth their returns in order to present a picture of low risk and consistent performance.4 Besides that, it is also very remarkable that Santa Claus is very kind to hedge funds. The average monthly returns in December are more than twice the returns for the rest of the year (Agarwal, Daniel, and Naik, 2005). So, there is evidence that managers manage their returns for window dressing purposes.

To be clear, the data will not be adjusted for potential biases, because it remains uncertain how, and to what extent the data should be adjusted. It must be noted that therefore the findings may potentially be biased.

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Fung and Hsieh (1997) find anecdotal evidence for this. For example George Soros’s Quantum Fund had an excellent performance record, but did not report to a database.

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3.2. Methodology

Hedge funds claim to be a valuable addition to portfolios because of high risk-adjusted returns and low correlations. Besides that, hedge funds claim to be relatively safe compared to other asset classes, in terms of various risk measures, such as downside deviation, maximum drawdown, VaR, number of negative months, etc. Al these claims will be investigated in the data analysis section. First, the Sharpe ratio will be used to check whether hedge funds offer superior risk-adjusted returns. The rolling Sharpe ratio will be used to see how the risk-adjusted returns evolve over time. Second, the correlations between hedge funds and other asset classes will be investigated to see if hedge funds really have low correlations. Correlations will also be calculated on a rolling basis, to see how they evolve over time. And finally, the different risk measures will be investigated, to see if hedge funds are really safer than other asset classes. Some of these risk measures will be calculated on a rolling basis to check for consistency over time. These measures are the downside deviation, the Sortino ratio, which is based on the downside deviation, and VaR, because these measures will be used in the next section where portfolios will be optimized.

The effects of the changes in risks, returns and correlations will be tested by using portfolio

optimization. Optimized portfolios will show the real added value of hedge funds, because they include both the risk-return and the correlation characteristics.

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3.3. Data analysis

This section will start with descriptive statistics. Then, (rolling) Sharpe ratios and correlations will be analyzed. And finally, various (rolling) risk measures will be analyzed.

3.3.1. Descriptive statistics

Table 1

Descriptive statistics5

This table shows the descriptive statistics of the monthly returns of several asset classes and hedge fund categories for the period 2/7/1990 till 3/7/2014, which includes 290 months. All asset classes and hedge fund categories had a Jarque-Bera value which was significant at the 1% level, and therefore, normality is rejected for all asset classes.

Mean Median

Standard

Deviation Kurtosis Skewness Minimum Maximum S&P 500 0.88% 1.26% 4.87% 3.26 -0.31 -21.06% 19.61%

US govt. bonds 0.54% 0.59% 2.20% 1.19 0.08 -6.69% 10.85%

US inv. grade 0.67% 0.70% 1.80% 2.81 -0.59 -9.05% 6.48%

US high yield 1.02% 1.12% 3.84% 9.98 0.38 -15.45% 23.91%

Commodity 0.55% 0.76% 6.49% 2.16 -0.31 -26.00% 26.63%

Hedge Fund total 0.89% 1.08% 1.99% 2.55 -0.69 -8.70% 7.65%

Fund of funds 0.61% 0.77% 1.65% 4.05 -0.68 -7.47% 6.85% emg. mkt. 1.03% 1.48% 4.04% 3.74 -0.83 -21.02% 14.80% equity hedge 1.03% 1.21% 2.62% 1.89 -0.26 -9.46% 10.88% event driven 0.94% 1.24% 1.94% 4.17 -1.31 -8.90% 5.13% macro 0.94% 0.72% 2.15% 1.07 0.57 -6.40% 7.88% relative value 0.81% 0.88% 1.25% 13.90 -2.14 -8.03% 5.72% market neutral 0.55% 0.53% 0.92% 1.67 -0.26 -2.87% 3.59%

fixed inc./conv. arb. 0.71% 0.94% 1.88% 29.53 -3.09 -16.01% 9.74%

merger arb. 0.68% 0.81% 1.15% 9.10 -2.08 -6.46% 3.12%

short bias 0.02% -0.34% 5.30% 2.37 0.26 -21.21% 22.84%

The descriptive statistics clearly show that hedge funds do in fact offer superior risk-adjusted returns. For example when comparing the S&P 500 with the aggregate of the hedge fund industry, HF Total, one can see that the returns are similar, while the standard deviation of HF Total is less than half the standard deviation of the S&P 500. Therefore, the Sharpe ratio of hedge funds must be more than twice the Sharpe ratio of the S&P 500. This confirms the findings of Goldman Sachs & Co. (1998).

Besides superior risk-adjusted returns, hedge funds also have relatively high kurtosis and negative skewness. This means that hedge fund returns are skewed to the left, and have relatively fat tails. This confirms the findings of Amin and Kat (2003).

And finally, hedge funds (except for short bias) have lower standard deviations than most asset classes. This can also be seen in the smaller minimum and maximum returns. These findings are also in line with Goldman Sachs & Co. (1998).

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3.3.2. Sharpe ratio

This section will analyze the risk-adjusted returns of hedge funds, and whether they change over time. This will be done by analyzing the rolling Sharpe ratio.

Figure 1: This figure shows the rolling monthly Sharpe ratio for all asset classes in the data set. The rolling Sharpe ratio is calculated by dividing the average returns over the past 60 month by the standard deviation over the past 60 months. As can be seen in Figure 1, the Sharpe ratio for HF Total clearly shows a decreasing trend. This means that in risk-adjusted terms, the aggregate hedge fund industry has become less attractive. The five year rolling Sharpe ratio has decreased from about 1.0 in the late 1990s to less than 0.4 since 2008.

Nowadays, the Sharpe ratio of hedge funds is comparable to that of other asset classes. In 2008 the rolling Sharpe ratio suddenly drops. This is very likely to be the result of the financial crisis. The decreasing trend in the Sharpe ratio of hedge funds is due to decreasing returns. The returns have decreased from about 1.5% per month on average in 1995 to about 0.5% in 2013, whereas the standard deviation has not increased significantly.

Based on this, one can conclude that the hedge fund industry is becoming less attractive in risk-adjusted return terms, since the Sharpe ratio is less than half of what it used to be.

-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1995 1995 1996 1997 1998 1999 2000 2000 2001 2002 2003 2004 2005 2005 2006 2007 2008 2009 2010 2010 2011 2012 2013 2014

5Y rolling Sharpe ratio for asset classes

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3.3.3. Correlations

In this section the correlation between hedge funds and other asset classes will be investigated. Hedge funds claim that they provide relatively uncorrelated returns and, therefore, add value in a portfolio. A quick glance at the correlation matrix (Appendix A) shows that this is in fact the case. Most

correlations between hedge funds and other asset classes are below 0.5.

In order to test the consistency of these correlations over time, the rolling correlations are calculated.

Figure 2: this figure shows the five year rolling correlation between HF Total and all other asset classes in the data set. Figure 2 shows that before the financial crisis, hedge funds had quite constant correlations with different asset classes. However, since the crisis all correlations (except for government bonds) suddenly increase and converge. This is the exact opposite of what hedge funds claim. They claim to be a safe, uncorrelated asset class. But if the low correlations disappear when they are really needed, i.e. during a crisis, then hedge funds add little value. After 2013, the correlations appear to be diverging again. The three year rolling correlations diverge in the direction of their pre-crisis levels (see Appendix B). It remains unclear whether they will eventually reach those lower levels again. Thus, future research will have to show whether the effect of the crisis, i.e. converging correlations, is permanent or temporary.

The increase in correlations between HF Total en the S&P 500 is driven by all hedge fund categories, except macro. These findings are shown in Figure 3.

-0,60 -0,40 -0,20 0,00 0,20 0,40 0,60 0,80 1995 1996 1997 1998 2000 2001 2002 2003 2005 2006 2007 2008 2010 2011 2012 2013

5Y rolling correlation between asset classes and HF Total

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Figure 3: this figure shows the five year rolling correlation between the different hedge fund categories and the S&P 500. Figure 3 shows an interesting pattern. All categories, except macro, show more or less the same pattern since 2008. After October 2008, the 5 year rolling correlation changes from about 0.25 to 0.50 in one month, for emerging markets, equity hedge, and event driven. For relative value, it changes from 0.17 to 0.54. In this month, the S&P returned -19.68%. Only the macro funds seem to be unaffected by this sudden market drop. What is also interesting is that the other four categories show very similar patterns after the crisis. For some reason they behave more or less the same since the crisis.

The analysis above is also conducted for specific strategies. The results are quite similar. All strategies investigated, except short bias, show increased correlations since 2008. Short bias also shows an increased correlation, but then a negative correlation. An increased negative correlation is positive, and indicates that these funds do an even better job in bear markets.

All in all, there is a decreasing trend in Sharpe ratios and an increasing trend in correlations, for the hedge fund industry in general. So far it seems that hedge funds are becoming less attractive over time. However, to further investigate this, different risk measures are analyzed in the following section.

3.3.4. Various risk measures

Beside high risk-adjusted, uncorrelated returns hedge funds also claim to be relatively safe. For example, they claim to have lower drawdowns, fewer negative months, a lower VaR, etc. To

investigate this, several risk measures will be calculated for all hedge fund categories, and compared to traditional asset classes. The results are shown in table 2. After that, three of these measures will be calculated on a rolling basis, to check for consistency over time. These three measures are: downside deviation, Sortino ratio, and VaR. These measures are chosen because they will be used in the next section where portfolios will be optimized.

-0,20 -0,10 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 1995 1995 1996 1997 1998 1999 2000 2000 2001 2002 2003 2004 2005 2005 2006 2007 2008 2009 2010 2010 2011 2012 2013 2014

5Y rolling correlation between hedge fund categories and S&P500

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Various risk measures

This table shows various risk measures for all asset classes and hedge fund categories based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. Sortino ratio is the return divided by the downside deviation. Downside deviation is the standard deviation of returns below zero. Downside capture ratio first measures the total return per asset class during the negative months of the S&P 500, and then calculates those returns as a percentage of the S&P 500 returns during those months. Downside correlation measures the correlation between the asset classes and the S&P 500, for those months in which the S&P was down. VaR measures loss which will (at least) be incurred in 5% of the time. Maximum drawdown measures the maximum loss from a certain peak, until a new peak is reached. % of negative months measures the number of negative months as a percentage of total months.

Skewness Kurtosis Mean

standard deviation Sharpe ratio Sortino ratio Downside deviation Downside capture ratio (compared to S&P 500) Downside correlation (with S&P 500) Value at Risk (5%) max drawdown % of negative months Minimum monthly returns S&P 500 -0.31 3.26 0.88% 4.86% 0.18 0.25 3.58% 100% 1.00 -6.66% 55% 36% -21.06% US Govt bonds 0.08 1.19 0.54% 2.20% 0.25 0.42 1.28% -19.69% -0.25 -2.97% 12% 39% -6.69% US inv. grade -0.59 2.81 0.67% 1.80% 0.37 0.51 1.32% -4.74% 0.21 -2.41% 14% 33% -9.05% US high yield 0.38 9.98 1.02% 3.84% 0.27 0.28 3.61% 25.25% 0.43 -4.05% 38% 25% -15.45% Commodity -0.31 2.16 0.55% 6.48% 0.08 0.12 4.64% 6.12% 0.21 -10.11% 68% 44% -26.00% HF total -0.69 2.55 0.89% 1.98% 0.45 0.60 1.48% 3.53% 0.43 -2.61% 21% 29% -8.70% FOF -0.68 4.05 0.61% 1.65% 0.37 0.45 1.34% -0.32% 0.36 -2.31% 22% 30% -7.47% emg. Mkt -0.83 3.74 1.03% 4.04% 0.26 0.32 3.20% 22.08% 0.33 -5.50% 43% 33% -21.02% equity hedge -0.26 1.89 1.03% 2.61% 0.40 0.59 1.76% 5.81% 0.43 -3.73% 31% 30% -9.46% event driven -1.31 4.17 0.94% 1.94% 0.48 0.52 1.79% 1.77% 0.39 -2.33% 25% 24% -8.90% macro 0.57 1.07 0.94% 2.15% 0.44 0.90 1.04% -14.26% 0.18 -2.11% 11% 36% -6.40% relative value -2.14 13.90 0.81% 1.25% 0.65 0.51 1.59% -10.21% 0.36 -1.05% 18% 16% -8.03% market neutral -0.26 1.67 0.55% 0.92% 0.60 0.82 0.67% -11.38% 0.24 -1.15% 9% 23% -2.87%

fixed inc./conv. Arb. -3.09 29.53 0.71% 1.88% 0.38 0.28 2.52% -3.25% 0.24 -2.11% 35% 20% -16.01%

merger arb. -2.08 9.10 0.68% 1.15% 0.59 0.52 1.31% -5.79% 0.35 -1.25% 8% 19% -6.46%

short bias 0.26 2.37 0.02% 5.29% 0.00 0.01 3.31% -76.58% -0.26 -7.53% 62% 55% -21.21%

The results in table 2 show that in general, hedge funds are more attractive than other asset classes based on these risk measures. In general, hedge funds have lower standard and downside deviations than traditional asset classes. Moreover, they have higher Sharpe and Sortino ratios. Besides that, in general, they have lower VaRs, lower drawdowns, fewer negative months and lower maximum losses. As can be seen in Table 2, all hedge fund categories, except relative value, have a higher mean return than the S&P500, with a lower standard deviation. So it is true that hedge funds offer equity-like returns with bond-like standard deviations.6 This means that for all hedge fund categories and

strategies, except short bias, the Sharpe ratios are higher than for the S&P500. The same counts for the Sortino ratio, which is based on the downside deviation. All hedge funds categories have a downside deviation which is lower than the S&P500. In general the results based on standard deviation are quite similar to the results based on downside deviation. The downside deviations of asset classes are usually in line with their standard deviation. Except for macro hedge funds. Macro has a relatively high

standard deviation compared to other hedge fund categories, but a relatively low downside deviation.

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This means that most of the deviation of macro funds is positive. This can also be seen from the skewness, macro is the only positively skewed category.

Concluding, one could say that based on all these different risk measures, in general, hedge funds are more attractive than traditional asset classes.

In order to test the consistency of these findings over time, rolling statistics are calculated for the various risk measures.

Sortino Ratio

Figure 4: this figure shows the five year rolling Sortino ratio for the different asset classes. The rolling Sortino ratio is calculated by dividing the five year mean return by the five year downside deviation.

Figure 4 shows that the Sortino ratio for HF Total has decreased over time. This pattern is due to decreased returns and not due to increased downside deviations. There is no clear increasing trend in downside deviation (see Figure 5). These findings are similar to those for the Sharpe ratio (Figure 1). The shape of the HF Total line is mostly due to the emerging markets and event driven strategies. For a more extensive analysis, please refer to Appendix D.

In general, Sortino ratios show a decreasing trend for HF Total, due to decreasing returns.

-0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 1995 1995 1996 1997 1997 1998 1999 1999 2000 2001 2001 2002 2003 2003 2004 2005 2005 2006 2007 2007 2008 2009 2009 2010 2011 2011 2012 2013 2013

5Y rolling Sortino ratio of asset classes

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13 Downside deviation

Figure 5: this figure shows the downside deviation of the different asset classes in the data set. Downside deviation is calculated as the standard deviation of the returns below zero.

As can be seen in Figure 5, the downside deviation of hedge funds has remained rather constant over time. Whereas the downside deviation of commodities, the S&P 500, and high yield bonds has increased over time.

In general, there is no clear trend in the downside deviation of hedge funds. Also when looking at the specific categories, no significant pattern emerges (see appendix C).

Value at Risk

The VaR does not show a clear pattern over time (see Figure 6). The aggregate index, HF total, shows an increase in the VaR from -1.5% to -3%. This means that the extreme losses which occur 5% of the time have increased. They used to be at least 1.5%, now they are at least 3%. However, compared to other asset classes this increase in relatively small. In general, the VaR of hedge funds is very low, and it does not increase significantly over time.

0% 1% 2% 3% 4% 5% 6% 7% 8% 1995 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2006 2007 2008 2009 2010 2011 2012 2013

5Y rolling downside deviation of asset classes

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Figure 6: this figure shows the five year rolling Value at Risk (VaR) at 5% for all asset classes in the data set. VaR is calculated as the loss which is (at least) incurred 5% of the time. For example a VaR of -3% means that 5% of the time (i.e. one out of 20 months) hedge funds will incur a loss of at least 3%.

3.3.5. Summary/conclusion

Concluding, all these rolling risk measures show a certain pattern. Hedge fund returns show a decreasing trend, whereas hedge fund variance does not show a clear trend. This in turn also leads to decreased Sharpe and Sortino ratios. However, on the other hand, the risk measures do not show a clear trend. The correlation between hedge funds and stocks is increasing. But the downside correlation between hedge funds and stocks is decreasing (Appendix E). This is a good thing, which means that hedge funds profit when the market goes up, but do not suffer when the market goes down. In addition to that, the downside deviation remains more or less constant, which means that hedge funds have not become riskier over time.

So basically, hedge funds have remained equally risky (volatile) over the past 290 months, however, their returns are decreasing. So maybe the retention of low risks comes at a cost of lower returns. It could be that because of the increased competition, it is harder to generate alpha. And hedge funds managers want to remain ‘safe’, so they trade higher returns for more stability/safety.

-18% -16% -14% -12% -10% -8% -6% -4% -2% 0% 1995 1995 1996 1997 1997 1998 1999 1999 2000 2001 2001 2002 2003 2003 2004 2005 2005 2006 2007 2007 2008 2009 2009 2010 2011 2011 2012 2013 2013

5Y rolling VaR (5%) of asset classes

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15 4. Results

This section will provide the results of all portfolio optimizations. Portfolios will be optimized by different constraints, e.g. all hedge funds strategies included, as well as per strategy. Also, the

allocation to hedge funds will be constrained at 15% in order to create a more realistic result. After that, portfolios will be optimized based on downside deviation and VaR, in order to investigate if hedge funds allocations differ depending on the risk measure used. And finally, the dataset will be split into two subsets and the analysis will be repeated for the subsets. This is done to test the robustness of the findings across time.

4.1. Mean-variance optimization

In this section, portfolios are optimized based on their mean return and standard deviation. Table 3 shows the unconstrained allocations. The first section includes all different hedge fund categories and strategies, the second column only includes the aggregate hedge fund index, and the third column does not include hedge funds.7

Figure 7 shows the efficient frontiers for these portfolios. From table 3 and Figure 7 it becomes clear that hedge funds do certainly add value in a portfolio. Also, it is clear that funds of funds do not add much value. The Sharpe ratio increases somewhat, but the return does not increase. Funds of funds perform less than the aggregate of equally weighted hedge funds. This means that it is better to equally weight all hedge funds than to pay someone to pick the best hedge funds for you. Because, net of fees, he will not add value. However, it must be noted that funds of funds are not necessarily bad; they just perform a slightly different function. Funds of funds allow for more diversification, and grant smaller investors access to the hedge fund industry. This creates an extra layer of fees, which reduces returns. Another remarkable finding is that unconstrained allocations, including all hedge funds, will allocate almost 90% to hedge funds. On the one hand, this is a very interesting finding, which shows that hedge funds are a valuable addition to an institutional portfolio. On the other hand, however, this is a really unrealistic picture, because most institutions constrain their allocation to alternatives. Therefore, the analysis above will be repeated, but this time with hedge fund allocation constrained at 15%.

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16 Table 3

mean-variance optimizations

This table shows unconstrained mean-variance portfolio optimizations of different asset and hedge fund classes, based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. The portfolios are constructed by optimizing the Sharpe ratio and by minimizing the variance. The first two columns include all hedge funds categories and strategies. The second two columns only include the aggregate hedge fund index, HF Total. The last columns are the benchmark and do not include any hedge funds. Below the allocations, the descriptive statistics for the optimized portfolios are shown.

All Hedge Funds Included Only with HF total No Hedge Funds at all

max Sharpe ratio minimum variance max Sharpe ratio minimum variance max Sharpe ratio minimum variance

S&P 500 1.6% 1.81% 0.0% 0.0% 6.45% 7.91% US Govt bonds 9.1% 9.59% 30.3% 41.8% 30.84% 46.49% US inv. grade 0.0% 0.00% 10.1% 6.6% 41.38% 31.40% US high yield 0.0% 0.00% 0.0% 0.0% 17.28% 6.45% Commodity 0.0% 0.00% 0.2% 3.0% 4.05% 7.75% HF total 0.0% 0.00% 59.4% 48.5% 0.00% 0.00% FOF 0.0% 0.00% 0.0% 0.0% 0.00% 0.00% emg. mkt 0.0% 0.00% 0.0% 0.0% 0.00% 0.00% equity hedge 0.0% 0.00% 0.0% 0.0% 0.00% 0.00% event driven 0.0% 0.00% 0.0% 0.0% 0.00% 0.00% macro 6.0% 0.44% 0.0% 0.0% 0.00% 0.00% relative value 30.9% 17.65% 0.0% 0.0% 0.00% 0.00% market neutral 23.4% 38.21% 0.0% 0.0% 0.00% 0.00%

fixed inc./conv. Arb. 0.0% 0.00% 0.0% 0.0% 0.00% 0.00%

merger arb. 22.5% 25.44% 0.0% 0.0% 0.00% 0.00% short bias 6.5% 6.87% 0.0% 0.0% 0.00% 0.00% mean 0.65% 0.60% 0.76% 0.72% 0.70% 0.64% Std. Dev. 0.72% 0.69% 1.40% 1.36% 1.73% 1.65% Sharpe Ratio 0.906 0.868 0.544 0.528 0.405 0.387 skewness -1.009 -0.508 -0.577 -0.534 -0.469 -0.501 kurtosis 4.007 2.316 1.389 1.091 2.543 0.874 minimum -3.06% -2.73% -5.11% -5.12% -7.83% -5.88% maximum 2.41% 2.94% 4.25% 4.05% 7.26% 4.98%

Figure 7: this figure shows the efficient frontiers for optimized portfolios with and without hedge funds.

-0,01 -0,005 0 0,005 0,01 0,015 0,02 0 0,02 0,04 0,06 0,08 0,1 0,12 M o n th ly R e tu r n Standard Deviation Efficient frontiers

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The optimizer always allocates the maximum amount possible (15%) to hedge funds (see Appendix F). This finding has significant implications for institutional investors. If hedge funds are currently not (a large) part of their portfolio, they should consider adding (more) hedge funds to their portfolio. Another important finding is that the constraint significantly reduces the Sharpe ratios. Especially for the first column, including all different hedge funds, the Sharpe ratio is halved. It is understandable that there are reasons why institutions will not allocate 90% to hedge funds, but this analysis shows that it might be very profitable to ease the current constraints.

Besides the aggregate hedge fund industry, the individual strategies are also analyzed, to investigate whether they add value on a stand-alone basis. Table 4 shows the allocations to individual hedge fund classes. As can be seen, relative value is the most attractive category on a stand-alone basis, directly followed by market neutral and merger arbitrage. Both in terms of allocation percentage, and in terms of Sharpe ratio. However, this comes at the cost of lower skewness and higher kurtosis.8

Table 4

Different strategies (maximized Sharpe ratio's)

This table shows optimal portfolios based on mean-variance analysis, based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. The portfolios are constructed by optimizing the Sharpe ratio. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total a nd Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

Base case (No HF's) HF total FOF Emg. Mkt Equity Hedge Event driven Macro Relative Value Market Neutral Fixed inc./conv. Arb. Merger arb. short bias S&P 500 6.45% 0.00% 2.08% 3.26% 0.00% 0.00% 4.22% 0.00% 0.46% 4.14% 0.00% 9.96% US Govt bonds 30.84% 28.55% 31.67% 33.07% 34.77% 34.33% 8.57% 21.76% 14.32% 38.58% 14.02% 27.84% US inv. grade 41.38% 12.14% 5.22% 33.61% 14.42% 1.62% 34.14% 0.00% 0.00% 0.00% 8.53% 27.76% US high yield 17.28% 0.00% 10.24% 10.20% 5.84% 0.00% 3.54% 0.00% 7.66% 5.12% 0.00% 19.75% Commodity 4.05% 0.06% 0.00% 2.70% 0.00% 0.67% 0.60% 0.00% 0.00% 0.29% 0.62% 3.29% HF total 0.00% 59.24% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% FOF 0.00% 0.00% 50.79% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% emg. Mkt 0.00% 0.00% 0.00% 17.17% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% equity hedge 0.00% 0.00% 0.00% 0.00% 44.97% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% event driven 0.00% 0.00% 0.00% 0.00% 0.00% 63.38% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% macro 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 48.94% 0.00% 0.00% 0.00% 0.00% 0.00% relative value 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 78.24% 0.00% 0.00% 0.00% 0.00% market neutral 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 77.57% 0.00% 0.00% 0.00%

fixed inc./conv. Arb. 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 51.87% 0.00% 0.00%

merger arb. 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 76.84% 0.00% short bias 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 11.40% mean 0.70% 0.76% 0.64% 0.73% 0.81% 0.79% 0.81% 0.75% 0.59% 0.67% 0.66% 0.65% Std. Dev. 1.73% 1.40% 1.29% 1.69% 1.60% 1.36% 1.56% 1.02% 0.86% 1.36% 1.01% 1.50% 8

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4.2. Mean-downside deviation optimization

In this section, portfolios will be optimized based on their mean returns and downside deviation. The downside deviation will only take into account the negative side of the distribution. Therefore, it will only take into account negative extremes and will not penalize for positive outliers. Since most hedge funds are negatively skewed, the downside deviation might be more informative than the standard deviation (Liang and Park, 2010). Portfolios will be optimized based on their Sortino ratio, which takes return and downside deviation into consideration. Table 5 shows optimal portfolios for these

optimizations.

Table 5

mean-downside deviation optimizations (maximizing Sortino ratio)

This table shows optimal portfolios based on mean-downside deviation analysis, based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. These portfolios are constructed by optimizing the Sortino ratio and by minimizing the downside deviation. Every column presents an optimal portfolio. The first two columns include all hedge fund strategies and categories. The second two columns only include the aggregate for the hedge fund industry, HF Total. The third two columns do not include hedge funds and represent traditional asset classes, i.e. ‘the market’. Below the allocations, descriptive statistics for the optimal portfolios are shown.

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As can be seen in Table 5, there are still significant allocations to hedge funds. So, in mean-downside deviation space hedge funds also add value. Table 6 investigates the different strategies on a stand-alone basis in means-downside deviation space. Market neutral had the highest allocation, and also the highest Sortino ratio. Moreover, market neutral also has the highest Sharpe ratio. So, market neutral is a very attractive strategy, both based on variance and based on downside deviation.

Table 6

Different strategies (maximized Sortino ratio's)

This table shows optimal portfolios for mean-downside deviation analysis, based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. These portfolios are constructed by optimizing the Sortino ratio. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

Base case (No HF's) HF total FOF Emg. Mkt Equity Hedge Event driven Macro Relative Value Market Neutral Fixed inc./conv.

Arb. Merger arb. short bias S&P 500 2.6% 0.0% 0.9% 1.7% 0.0% 0.0% 1.3% 0.9% 0.0% 9.4% 0.1% 8.9% US Govt bonds 67.0% 45.5% 56.3% 63.8% 47.1% 57.0% 11.5% 48.5% 22.9% 68.8% 39.9% 60.1% US inv. grade 6.9% 0.0% 0.0% 5.6% 0.0% 0.0% 8.3% 0.0% 0.0% 0.0% 9.0% 0.0% US high yield 23.5% 0.0% 20.3% 17.8% 0.0% 0.0% 5.0% 0.0% 1.9% 16.4% 0.0% 20.7% Commodity 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 2.3% 0.0% 0.0% 0.0% 0.0% 0.0% HF total 0.0% 54.5% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% FOF 0.0% 0.0% 22.4% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% emg. Mkt 0.0% 0.0% 0.0% 11.2% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% equity hedge 0.0% 0.0% 0.0% 0.0% 52.9% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% event driven 0.0% 0.0% 0.0% 0.0% 0.0% 43.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% macro 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 71.7% 0.0% 0.0% 0.0% 0.0% 0.0% relative value 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 50.5% 0.0% 0.0% 0.0% 0.0% market neutral 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 75.3% 0.0% 0.0% 0.0%

fixed inc./conv. Arb. 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 5.3% 0.0% 0.0%

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4.3. Mean-VaR optimization

The previous analysis will be repeated, but this time based on mean return and value at risk (VaR). Table 7 shows the optimal portfolios for the mean-VaR optimizations and VaR only optimizations. The mean-VaR optimizations are done by minimizing mean divided by VaR. This approach is similar to optimizing the Sharpe or the Sortino ratios. Table 7 shows that optimizing on VaR only, i.e.

minimizing VaR, provides similar results than when optimizing both VaR and mean return at the same time.

Table 7

mean-VaR optimizations

This table shows optimal portfolios based on mean-VaR analysis. The data is based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. These portfolios are constructed by optimizing the mean/VaR ratio and by minimizing

VaR. It turns out that both approaches produce the same result. Every column presents an optimal portfolio. The first two columns include all hedge fund strategies and categories. The second two columns only include the aggregate for the hedge fund

industry, HF Total. The third two columns do not include hedge funds and represent traditional asset classes, i.e. ‘the market’. Below the allocations, descriptive statistics for the optimal portfolios are shown.

min mean/VaR

min.

VaR min mean/VaR min. VaR min mean/VaR min. VaR

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As can be seen in table 7, hedge funds also add value in mean-VaR space. The solver allocates more than 50% to hedge funds in all scenarios.

Table 8

Also on an individual level, all hedge fund classes add value to an institutional portfolio by reducing VaR, as can be seen in Table 8.

Concluding, the analysis above shows that hedge funds do add value in an institutional portfolio. Optimal portfolios always have more than 50% allocated to hedge funds, regardless of the risk measure

Different strategies (minimized VaR)

This table shows optimal portfolios based on mean-VaR analysis. The data is based on monthly returns for the period 2/7/1990 till 3/7/2014, which includes 290 months. These portfolios are constructed by minimizing VaR. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

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used. Also the individual categories and strategies do add value. The solver always allocates a positive amount to hedge funds, for all strategies and across all risk measures. Moreover, this amount is often quite large, most of the time it is greater than 50%.

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4.4. Variations across time; analysis of the subsets

In this section the findings above will be checked for consistency through time. In order to do this, the data set will be split in two equal subsets of 145 months. The first period will be from 7/2/1990 till 7/2/2002 and the second period will be from 7/3/2002 till 7/3/2014. For both subsets optimal portfolios will be created based on the three risk measures: standard deviation, downside deviation, and VaR. For every risk measure, the findings of subset one and two will be compared, to see if allocations vary in across time.

4.4.1. Mean-variance analysis

Table 9

Mean-variance optimal portfolios for subset 1 (7/2/1990 - 7/2/2002)

This table shows optimal portfolios based on mean-variance analysis. The analysis is based on monthly returns for the period 7/2/1990 - 7/2/2002, which includes 145 months. The portfolios are constructed by optimizing the Sharpe ratio. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

Base case (No HF's) All HF's HF total FOF Emg. Mkt Equity Hedge Event driven Macro Relative Value Market Neutral Fixed inc./conv. Arb. Merger arb. short bias S&P 500 9.29% 3.03% 0.00% 2.19% 5.54% 0.00% 0.00% 6.75% 0.00% 3.04% 0.12% 0.00% 15.04% US Govt bonds 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 2.71% 0.00% 0.00% 0.00% 0.00% US inv. grade 69.84% 2.33% 45.40% 42.27% 69.10% 52.23% 40.03% 46.62% 17.52% 10.27% 15.83% 27.39% 50.10% US high yield 12.63% 0.00% 0.00% 5.50% 6.61% 0.00% 0.00% 0.30% 0.00% 4.34% 0.00% 0.00% 15.97% Commodity 8.24% 0.00% 2.50% 1.51% 7.14% 1.89% 2.55% 4.69% 0.56% 0.75% 1.01% 2.31% 7.40% HF total 0.00% 0.00% 52.10% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% FOF 0.00% 0.00% 0.00% 48.53% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% emg. Mkt 0.00% 0.00% 0.00% 0.00% 11.61% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% equity hedge 0.00% 9.58% 0.00% 0.00% 0.00% 45.89% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% event driven 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 57.42% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% macro 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 41.64% 0.00% 0.00% 0.00% 0.00% 0.00% relative value 0.00% 18.48% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 79.20% 0.00% 0.00% 0.00% 0.00% market neutral 0.00% 27.61% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 81.60% 0.00% 0.00% 0.00%

fixed inc./conv. Arb. 0.00% 21.64% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 83.03% 0.00% 0.00%

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25 Table 10

Mean-variance optimal portfolios for subset 2 (7/3/2002 - 7/3/2014)

This table shows optimal portfolios based on mean-variance analysis. The analysis is based on monthly returns for the period 7/3/2002 - 7/3/2014, which includes 145 months. The portfolios are constructed by optimizing the Sharpe ratio. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive st atistics for the optimal portfolios are shown.

Base case (No HF's) All HF's HF total FOF Emg. Mkt Equity Hedge Event driven Macro Relative Value Market Neutral Fixed inc./conv. Arb. Merger arb. short bias S&P 500 5.15% 2.60% 0.35% 2.22% 1.44% 1.96% 0.49% 2.57% 1.97% 0.00% 4.93% 0.00% 5.94% US Govt bonds 61.33% 12.24% 40.85% 42.97% 54.34% 53.11% 40.46% 29.90% 27.88% 23.56% 55.27% 19.74% 55.93% US inv. grade 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% US high yield 30.82% 0.60% 10.55% 17.30% 20.00% 20.83% 6.76% 16.30% 0.00% 11.36% 21.38% 2.53% 30.13% Commodity 2.70% 0.00% 0.00% 0.00% 0.00% 0.39% 0.00% 0.00% 0.00% 0.00% 1.32% 0.00% 2.45% HF total 0.00% 0.00% 48.26% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% FOF 0.00% 0.00% 0.00% 37.51% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% emg. Mkt 0.00% 0.00% 0.00% 0.00% 24.22% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% equity hedge 0.00% 0.00% 0.00% 0.00% 0.00% 23.72% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% event driven 0.00% 0.39% 0.00% 0.00% 0.00% 0.00% 52.29% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% macro 0.00% 7.70% 0.00% 0.00% 0.00% 0.00% 0.00% 51.23% 0.00% 0.00% 0.00% 0.00% 0.00% relative value 0.00% 25.95% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 70.14% 0.00% 0.00% 0.00% 0.00% market neutral 0.00% 0.86% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 65.09% 0.00% 0.00% 0.00%

merger arb. 0.00% 38.32% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 77.73% 0.00% short bias 0.00% 11.34% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 5.55% mean 0.62% 0.37% 0.54% 0.50% 0.65% 0.57% 0.58% 0.55% 0.53% 0.36% 0.58% 0.42% 0.58% std dev 1.74% 0.61% 1.30% 1.29% 1.68% 1.53% 1.29% 1.23% 1.02% 0.78% 1.58% 0.81% 1.60% skewness -0.337 -1.442 -1.065 -1.259 -1.013 -0.787 -1.168 -0.173 -2.763 -1.407 -1.027 -0.657 -0.294 kurtosis 2.913 5.518 2.959 3.923 3.119 2.731 4.134 1.124 14.789 3.772 4.582 1.351 3.232 minimum -5.68% -2.61% -5.28% -5.25% -6.89% -5.83% -5.82% -3.24% -6.26% -3.24% -6.85% -2.57% -5.02% maximum 7.67% 1.73% 3.21% 3.74% 4.89% 5.25% 3.59% 4.47% 2.47% 2.20% 5.72% 2.22% 7.35% Sharpe ratio 0.359 0.613 0.417 0.386 0.386 0.373 0.449 0.450 0.514 0.455 0.367 0.511 0.360

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Sharpe ratio of fixed income/convertible arbitrage. Because of decreased returns, and an increased standard deviation (see Appendix I).

Apart from hedge fund allocations, there is also a remarkable pattern in the allocation of the traditional assets. In the first period there are high allocations to investment grade bonds, whereas in the second period there are no allocations to investment grade bonds. This means that the market itself, i.e. the traditional asset classes, is very different in the two periods. So it could be that the change in allocations to hedge funds is actually due to the change in the characteristics of the traditional asset classes, instead of a change in hedge fund returns.

There are no real changes in the Sharpe ratios of government and investment grade bonds. However, the rolling correlations change very differently (see Appendix K). The correlation with stocks (S&P500) and hedge funds (HF Total) increases significantly for investment grade bonds for period two. While the correlations for government bonds decrease in period two. Therefore, government bonds become relatively more attractive. Another remarkable finding with regard to government and investment grade bonds is that their correlation with each other used to be very close to 1 (near perfect correlation) until 2008. This means that until 2008 the two types of bonds could be viewed as

substitutes. When optimizing portfolios, the optimizer will choose the asset with the highest Sharpe ratio, because the correlations are similar. That is why the optimizer allocates significant amounts to investment grade bonds in the first period. After 2008, the correlation between government and investment grade bonds reduced to less than 0.4. The correlation between government bonds and other asset classes decreased, whereas the correlation between investment grade bonds and other asset classes increased. Therefore, after 2008, they can no longer be viewed as substitutes. In the second period the optimizer allocates more to government bonds, because of the decreased correlations with other asset classes. And the optimizer allocates nothing to investment grade bonds because of the increased correlations with other asset classes.

Apparently the crisis has shown that investment grade bonds are riskier than expected, and that relatively safe companies can get into trouble all over sudden. And government bonds have proven to be the only real safe, risk-free asset. But that is just a guess, and beyond the scope of this research. For now it is important to note that not only hedge funds, but also the market conditions have changed significantly over time.

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4.4.2. Mean-downside deviation analysis

Table 11

Mean-downside deviation optimal portfolios for subset 1 (7/2/1990 - 7/2/2002)

This table shows optimal portfolios based on mean-downside deviation analysis, based on monthly returns for the period 7/2/1990 - 7/2/2002, which includes 145 months. The portfolios are constructed by optimizing the Sortino ratio. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

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Mean-downside deviation optimal portfolios for subset 2 (7/3/2002 - 7/3/2014)

This table shows optimal portfolios based on mean-downside deviation analysis, based on monthly returns for the period 7/3/2002 - 7/3/2014, which includes 145 months. The portfolios are constructed by optimizing the Sortino ratio. Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

fixed inc./conv. Arb.

0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% merger arb. 0.00% 18.69% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 73.35% 0.00% short bias 0.00% 17.23% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 7.37% mean 0.59% 0.35% 0.49% 0.59% 0.59% 0.59% 0.53% 0.54% 0.50% 0.32% 0.59% 0.41% 0.53% std dev 1.76% 0.75% 1.33% 1.76% 1.76% 1.76% 1.37% 1.28% 1.33% 0.84% 1.76% 0.83% 1.59% skewness -0.232 0.257 -0.302 -0.232 -0.232 -0.232 -0.226 0.124 -0.260 -0.276 -0.232 -0.332 -0.138 kurtosis 0.711 0.073 1.250 0.711 0.711 0.711 1.264 0.418 1.463 1.073 0.711 0.744 1.107 minimum -4.38% -1.46% -4.00% -4.38% -4.38% -4.38% -4.36% -3.00% -4.37% -2.24% -4.38% -2.29% -3.71% maximum 6.54% 2.37% 4.91% 6.54% 6.54% 6.54% 5.04% 4.77% 5.22% 3.43% 6.54% 2.71% 6.61% Downside deviation 1.10% 0.32% 0.90% 1.10% 1.10% 1.10% 0.89% 0.65% 0.87% 0.53% 1.10% 0.55% 0.98% Sortino ratio 0.534 1.102 0.546 0.534 0.534 0.534 0.591 0.831 0.570 0.594 0.534 0.748 0.542

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significantly increased in the second period. For example event driven goes from 1.01% to 40.05% and merger arbitrage goes from 15.00% to 73.35%. Overall, there is no clear pattern, some allocations increase, whereas others decrease. But the average allocations remain similar.

Another finding is that all asset classes, and especially investment grade bonds, had significant

allocations in the first period. Whereas, in the second period there are (almost) no allocations to stocks, investment grade bonds, and commodities. The optimizer allocates most to government bonds. And when there are no hedge funds included, the optimizer also allocates to high yield bonds. Thus, hedge funds and high yield bonds seem to be substitutes in the second period.

This can be explained due to the changing correlations. As mentioned above, the correlations between government bonds and other asset classes have decreased over time (see Appendix K). Moreover, government bonds have the highest Sortino ratio of all asset classes in the second period (see Appendix I). This causes government bonds to be very attractive in terms of mean-downside deviation

optimization in the second period. This explains the high allocations to government bonds. High yield bonds have a relatively low correlation with government bonds and a relatively high Sortino ratio, compared to other asset classes, in the second period. Therefore, high yield bonds are added to the government bonds. When hedge funds are added to the portfolio, they enter the optimal portfolio at the expense of high yield bonds. Because hedge funds, i.e. HF Total, have similar correlations as high yield bonds, but have a higher Sortino ratio (see Appendix I & K).

The changes in the allocations to individual strategies can be explained by the changes in Sortino ratios of those strategies. Which in turn are due to the changes in downside deviation. For all those strategies the mean return goes down over time. But for equity hedge and fixed income/convertible arbitrage the downside deviation goes up over time, whereas for event driven and merger arbitrage it goes down (see Appendix I).

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4.4.3. Mean-VaR analysis

Table 13

Mean-VaR optimal portfolios for subset 1 (7/2/1990 - 7/2/2002)

This table shows optimal portfolios based on mean-VaR analysis, based on monthly returns for the period 7/2/1990 - 7/2/2002, which includes 145 months. The portfolios are constructed by minimizing VaR.9 Every column presents an optimal portfolio. The first column represents the traditional assets classes, i.e. ‘the market’. The second column includes all hedge funds categories and strategies. The third and fourth columns include the aggregate hedge funds industry, represented by HF Total and Funds of Funds (FOF) respectively. The other columns include individual hedge fund categories or strategies. Below the allocations, descriptive statistics for the optimal portfolios are shown.

fixed inc./conv. Arb.

0.00% 33.39% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 73.91% 0.00% 0.00% merger arb. 0.00% 24.26% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 88.79% 0.00% short bias 0.00% 11.62% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 13.67% mean 0.77% 0.99% 1.07% 0.87% 0.90% 1.22% 1.11% 1.20% 1.02% 0.89% 0.92% 0.95% 0.76% std dev 1.60% 0.74% 1.53% 1.37% 1.74% 1.73% 1.54% 1.96% 0.98% 0.88% 0.93% 1.18% 1.55% skewness -0.246 -1.169 -0.635 -0.720 -0.164 -0.041 -1.282 -0.026 -1.193 0.016 -1.206 -2.952 -0.024 kurtosis -0.131 3.784 1.362 2.757 -0.108 0.539 4.672 0.320 7.662 0.629 3.242 13.047 0.137 minimum -4.33% -2.58% -5.39% -5.27% -4.17% -4.24% -6.66% -4.86% -4.72% -2.12% -2.94% -5.94% -3.22% maximum 4.08% 2.93% 4.13% 4.16% 4.92% 6.16% 4.49% 6.30% 4.40% 3.16% 3.35% 2.71% 5.30% Value at Risk -1.85% -0.08% -1.26% -1.01% -1.64% -1.22% -0.98% -1.60% -0.32% -0.36% -0.48% -0.61% -1.38% mean/VaR -0.41 -13.04 -0.85 -0.86 -0.55 -1.00 -1.13 -0.75 -3.16 -2.52 -1.92 -1.56 -0.55 9