**ABSTRACT **

Over the last 20 years, investors have come to approach investment decision-making in an increasingly mechanical manner. Optimisers are filled up with historical return data and the 'optimal' portfolio follows almost automatically. In this paper we argue that such an approach can be extremely dangerous, especially when alternative investments such as hedge funds are involved. Proper hedge fund investing requires a much more elaborate approach to investment decision-making than currently in use by most investors. The available data on hedge funds should not be taken at face value, but should first be corrected for various types of biases and autocorrelation. Tools like mean-variance analysis and the Sharpe ratio that many investors have become accustomed to over the years are no longer appropriate when hedge funds are involved as they concentrate on the good part while completely skipping over the bad part of the hedge fund story. Investors also have to find a way to figure in the long lock-up and advance notice periods, which make hedge fund investments highly illiquid. In addition, investors will have to give weight to the fact that without more insight in the way in which hedge funds generate their returns it is very hard to say something sensible about hedge funds' future longer-run performance. The tools to accomplish this formally are not all there yet, meaning that more than ever investors will have to rely on common sense and doing their homework.

**1. INTRODUCTION **

Hedge funds are on their way to become the next big thing in investment management. New funds start up every day, hedge funds are marketed aggressively to institutions and, under pressure to make up for recent losses, many institutional investors are showing serious interest. The amount of assets under management by hedge funds has grown from around $40 billion in 1990 to an estimated $600 billion in 2003. In line with this, the number of hedge funds worldwide has grown to around 6000. In the early days not much was known about hedge funds. Since 1994, however, a number of data vendors, hedge fund advisors and fund of hedge funds operators have been collecting performance and other data on hedge funds. This has allowed researchers to take a more serious look at hedge funds. Although research in this area is still in its infancy, it has become clear that hedge funds are a lot more complicated than common stocks and investment-grade bonds and may not be as phenomenally attractive as many hedge fund managers and marketers want investors to believe. Hedge fund investing requires a much more elaborate approach to investment decision-making than what most investors are used to. Mechanically applying the same decision-making processes as typically used for stock and bond investment may lead to some very nasty surprises.

**2. MODERN PORTFOLIO THEORY IN ACTION
**

Before the arrival of finance theory as we known it today, finance was a very practical discipline. Finance students would spend their time studying accounting, taxation, law and the writings of Benjamin Graham and others. With the arrival of Harry Markowitz's mean-variance analysis, Bill Sharpe's CAPM and Gene Fama's efficient market hypothesis things changed, however. From a practical discipline, finance very quickly reinvented itself as a branch of neo-classical price theory, concentrating on the analysis of abstract little 'toy worlds' where most of what makes life complicated is simply assumed away in an attempt to come to the essence of things. At the same time, computers and databases started to evolve up to a point where nowadays every investor has access to extensive computing power and market data. All this has had a profound impact on the way investment decisions are being taken. Mean-variance optimisers play an important part in modern investment management and essentially take over a substantial portion of the responsibility for the asset allocation decision. Likewise, performance evaluation relies heavily on theoretical concepts such as the Sharpe ratio and Jensen's alpha. All these tools have become so common that many investors tend to apply them in a purely mechanical fashion, giving little or no thought to their underlying assumptions. The same is true for the required inputs. Often, means, and especially variances and correlations coefficients are simply calculated from downloaded historical return data with little or no consideration for the sometimes very specific factors that generated the data in question.

The way in which the basic concepts of modern finance are used in practice leaves a lot to be desired. Despite this, their application in the stock and bond markets does not appear to be without some merit. There are a number of reasons for this. First, return data on stocks and bonds often covers a long time period and tends to be of good quality. Second, stock and bond returns tend to exhibit statistical characteristics that are very much in line with what is assumed in theory. Third, stock and bond markets typically tend to offer relatively good liquidity. When we move away from stocks and bonds and into the realm of alternative investments then the situation changes dramatically, however. Serious data problems, complex return generating processes, non-normal return distributions, low transparancy and substantial illiquidity all have to be taken into account properly. If not, investors risk self-deception. They will see miracles where there are none and vice versa. In the sections that follow we will discuss these matters in greater detail focusing on hedge funds and the typical way in which 'sophisticated' investors look at them.

**3. HEDGE FUND DATA **

With the hedge fund industry still in its
infancy and hedge funds under no formal
obligation to disclose their results, gaining
insight in the performance characteristics
of hedge funds is not straightforward. Fortunately,
many funds nowadays release performance
as well as other administrative data to
attract new and to accommodate existing
investors. These data are collected by a
number of data vendors and fund advisors,
some of which make their data available
to qualifying investors and researchers.
The available data on hedge funds are not
without problems though. Here are some of
them:

__An unknown universe.__

Most hedge funds only report into one or two databases. As a result, every database covers a different subset of the hedge fund universe and different researchers may arrive at quite different conclusions simply because different databases were used.

__No independent auditing. __

Most databases are of relatively low quality as most data vendors simply pass on the data supplied by the fund managers and their administrators without any independent verification. This means that before any serious research can take place, one must check the data for a number of possible errors and either correct these or delete the funds in question altogether.

__Backfill bias.__

Hedge fund databases tend to be backfilled,
i.e. although typically funds only start
reporting to a database some time after
their actual start-up, when they do, their
full track record is included in the database.
Since only funds with good track records
will eventually decide to report, this means
that the available data sets are overly
optimistic about hedge fund performance.
As shown in Posthuma and Van der Sluis (2003),
on average actual hedge fund returns may
be 4% per annum lower than reported.

*Survivorship bias.*

Most data vendors only supply data on funds that are still in operation. However, disappointing performance is a major reason for hedge funds to close down. As shown in Amin and Kat (2003), this means that the data available to investors will overestimate the returns that investors can realistically expect from investing in hedge funds by 2-4% per annum. In addition, concentrating on survivors only will lead investors to underestimate the risk of hedge funds by 10-20%.

__Marking-to-market problems.__

Since many hedge funds invest in illiquid assets, their administrators have great difficulty generating up-to-date valuations of their positions. When confronted with this problem, administrators will either use the last reported transaction price or a conservative estimate of the current market price, which creates artificial lags in the evolution of these funds' net asset values. As we will discuss in more detail in section 4, this will lead to very substantial underestimation of hedge fund risk, sometimes by as much as 30-40%.

__Limited data.__

Since most data vendors only started collecting data on hedge funds around 1994, the available data set on hedge funds is very limited. The available data on hedge funds also span a very special period: the bull market of the 1990s and the various crises that followed combined with the spectacular growth of the hedge fund industry itself. This sharply contrasts with the situation for stocks and bonds. Not only do we have return data over differencing intervals much shorter than one month, we also have those data available over a period that extends over many business cycles. This has allowed us to gain insight into the main factors behind stock and bond returns and also allows us to distinguish between normal and abnormal market behaviour. The return generating process behind hedge funds on the other hand is still very much a mystery and so far we have little idea what constitutes normal behaviour and what not.

With institutional interest in hedge funds on the increase another question that arises is when the hedge fund industry will reach capacity. While the industry has experienced strong growth over the last five years in terms of assets under management, hedge funds themselves are showing lower returns every year. This could be an indication that there are no longer enough opportunities in the global capital markets to allow hedge funds to continue to deliver the sort of returns that we have seen so far.

**4. HEDGE FUND RISK **

Marking-to-market problems tend to create lags in the evolution of hedge funds' net asset values, which statistically shows up as autocorrelation in hedge funds' returns. As discussed in Brooks and Kat (2002) for example, this autocorrelation causes estimates of the standard deviation of hedge fund returns to exhibit a systematic downward bias. The second column in table 1 shows the average 1-month autocorrelation found in the returns of individual hedge funds in some of the usual strategy groups over the period 1994-2001. The table shows that the problem is especially acute for convertible arbitrage and distressed securities funds, which makes sense as these funds' assets will typically be the most difficult to value. One way to correct for the observed autocorrelation is to 'unsmooth' the observed returns by creating a new set of returns which are more volatile but whose other characteristics are unchanged. One method to do so stems from the real estate finance literature, where due to smoothing in appraisals and infrequent valuations of properties, the returns of direct property investment indices suffer from similar problems as hedge fund returns (see Geltner (1991, 1993)). The third and fourth column of table 1 show the average standard deviations of the original as well as the unsmoothed returns on individual hedge funds belonging to the different strategy groups. From the table we see that the difference between the observed and the true standard deviation can be very substantial. For distressed securities funds the true standard deviation is almost 30% higher than observed. For convertible arbitrage funds the difference is even higher.

**Table 1: Average 1-Month Autocorrelation and Standard Deviations Original and Unsmoothed Individual Hedge Fund Returns**

AC (1) | Original SD | Unsmoothed SD | |
---|---|---|---|

Merger Arbitrage | 0.13 | 1.75 | 2.02 |

Distressed Securities | 0.25 | 2.37 | 3.05 |

Equity Market Neutral | 0.08 | 2.70 | 3.04 |

Convertible Arbitrage | 0.30 | 3.01 | 4.00 |

Global Macro | 0.03 | 5.23 | 5.37 |

Long/short Equity | 0.09 | 5.83 | 6.37 |

Emerging Markets | 0.15 | 8.33 | 9.75 |

A second reason why many investors think hedge funds are less risky than they really are results from the use of the standard deviation as the sole measure of risk. 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 indeed be measured with one number. 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 2: Average Skewness and Kurtosis of Individual Hedge Fund Returns**

Skewness | Kurtosis | |
---|---|---|

Merger Arbitrage | -0.50 | 7.60 |

Distressed Securities | -0.77 | 8.92 |

Equity Market Neutral | -0.40 | 5.58 |

Convertible Arbitrage | -1.12 | 8.51 |

Global Macro | 1.04 | 10.12 |

Long/short Equity | 0.00 | 6.08 |

Emerging Markets | -0.36 | 7.83 |

Table 2 shows the average skewness and kurtosis found in the returns of individual hedge funds from various strategy groups. From the table it is clear that the average hedge fund's returns tend to be far from normally distributed and may exhibit significant negative skewness as well as substantial kurtosis. Put another way, hedge fund returns may exhibit relatively 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.

The skewness and kurtosis properties of hedge funds should 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 will generate these features by their very nature as the profit potential of trades will typically be a lot smaller than 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 materialized so it does not show from the trader's track record.

**5. HEDGE FUND SHARPE RATIOS **

To evaluate hedge fund performance many investors use the Sharpe ratio, which is calculated as the ratio of the average excess return and the return standard deviation of the fund being evaluated. When applied to raw hedge fund return data, the relatively high means and low standard deviations offered by hedge funds lead to Sharpe ratios that are considerably higher than those of most benchmarks. Whilst this type of analysis is widely used, it is not without problems. First, survivorship bias, backfill bias and autocorrelation will cause investors to overestimate the mean and underestimate the standard deviation. Second, the Sharpe ratio does not take account of the negative skewness and excess kurtosis observed in hedge fund returns. This means that the Sharpe ratio will tend to systematically overstate true hedge fund performance. There tends to be a clear relationship between a fund's Sharpe ratio and the skewness and kurtosis of that fund's return distribution. High Sharpe ratios tend to go together with negative skewness and high kurtosis. This means that the relatively high mean and low standard deviation offered by hedge funds is not a free lunch. Investors simply pay for a more attractive Sharpe ratio in the form of more negative skewness and higher kurtosis.

**Figure 1: Trade-off between Median, Standard Deviation and Skewness.**

**Figure 2: Sharpe Ratios
for Different Skewness Levels.**

Another way to look at this is to use ordinary put and call options to create distributions that have exceedingly non-normal characteristics. Starting with the (approximately normal) return distribution of the index we can increase the skewness of that distribution by buying puts on that index for example. Likewise, we can reduce skewness by selling calls on that index. When we calculate the median (for skewed distributions this is a better measure of location than the mean), standard deviation and skewness of such distributions and plot those graphically we obtain a graph as in figure 1, which shows the median return as a function of the standard deviation and skewness. From the graph we see that for a given level of standard deviation lower (higher) skewness produces a higher (lower) median. Alternatively, we could of course say that for a given median lower skewness produces a lower standard deviation and vice versa. From the graph in figure 1 we can derive (median-based) Sharpe ratios for different skewness levels. The result is shown in figure 2, where the slope of each line equals the Sharpe ratio for the given level of skewness. Obviously, the lower the skewness level, the higher the Sharpe ratio will be. This shows how wrong it can be to evaluate fund managers that produce returns with different degrees of skewness with the same benchmark Sharpe ratio. Different skewness levels require different benchmark Sharpe ratios, higher when skewness is negative and lower when skewness is positive. If not, the good guys, that produce positively skewed returns, will end up being punished while the bad guys are rewarded.

**6. HEDGE FUND ALPHAS**

Another performance measure often used is Jensen's alpha. The idea behind alpha is to first construct a portfolio that replicates the sensitivities of a fund to the relevant return generating factors and then compare the fund return with the return on that portfolio. If the fund produces a higher average return, this can be interpreted as superior performance since both share the same return generating factors. The main problem with this approach lies in the choice of return generating factors. As mentioned before, we have little idea what factors really generate hedge fund returns. As a result, investors that calculate hedge funds' alphas are likely to leave out one or more relevant risk factors. This will produce excess return where in reality there is none. Good examples of often forgotten but extremely important risks are credit and liquidity risk. So far, no study of hedge fund performance has correctly figured in credit or liquidity risk as a source of return, despite the fact that some hedge funds virtually live off it.

**Table 3: Regression Individual Hedge Fund Alphas on Autocorrelation Coefficients**

Average Alpha | Average AC (1) | Regression Coefficient | |
---|---|---|---|

Merger Arbitrage | 1.20 | 0.13 | 1.1356 |

Distressed Securities | 0.89 | 0.25 | 0.8720 |

Equity Market Neutral | 0.40 | 0.08 | 0.3112 |

Convertible Arbitrage | 0.97 | 0.30 | 1.2975 |

Global Macro | 0.26 | 0.30 | 0.2864 |

Long/short Equity | 0.94 | 0.09 | 0.8954 |

Emerging Markets | 0.33 | 0.15 | 0.3680 |

Providing liquidity can be expected to be compensated by a higher average return. However, when this is not taken into account, we will find alpha where there is in fact none. Table 3 provides a simple example. For individual funds in the various strategy groups, table 3 shows (a) the average alpha assuming the stock and bond market are the only relevant risk factors, and (b) the average 1-month autocorrelation coefficient found in these funds monthly returns. Since the autocorrelation found in hedge fund returns primarily stems from marking-to-market problems in illiquid markets, we can use the autocorrelation coefficient as a measure of the liquidity risk taken on by a fund. From the table we see that there tends to be a positive relationship between alpha and autocorrelation. This is also confirmed by the last column, which shows the results of regressing individual funds' alphas on their autocorrelation coefficients. All regression coefficients are positive and significant (t-values not reported), meaning that in every category the funds that take most liquidity risk also tend to be the funds with the highest alphas.

The above makes it very clear that when it comes to hedge funds, traditional performance evaluation methods like the Sharpe ratio and alpha can be extremely misleading. A high Sharpe ratio or alpha should not be interpreted as an indication of superior manager skill, but first and foremost as an indication that further research is required. One can only speak of superior performance if such research shows that the manager in question generates the observed excess return without taking any unusual and/or catastrophic risks. Unfortunately, simply studying a manager's past returns will not be enough. Apart from the fact that most hedge fund managers do not have much of a track record to study, extreme events only occur infrequently so that it is hard if not impossible to identify the presence of catastrophic risk from a relatively small sample of returns. Consider the following example. A substantial portion of the outstanding supply of catastrophe-linked bonds is held by hedge funds. These bonds pay an exceptionally high coupon in return for the bondholder putting (part of) his principal at risk. Since the world has not seen a major catastrophe for some time now, these bonds have performed very well and the available return series show little skewness. However, this does not give an accurate indication of the actual degree of skewness as when a catastrophe does eventually occur, these bonds will produce very large losses.

There are no shortcuts to hedge fund selection. After properly controlling for the risks involved, small funds do not perform better than larger funds, young funds do not perform better than older funds, closed funds do not perform better than open funds, etc. Proper hedge fund selection is first and foremost a matter of asking the right questions and doing one's homework. There are various due diligence question lists available on the internet and more and more institutions develop their own. The use of such lists harbours its own risks, however. First, it may lead to a more and more mechanical application where getting all the questions answered becomes more important than correctly interpreting the answers. Second, in an attempt to offload as much responsibility and job risk as possible, especially institutional investors will add more and more questions to the list, thereby wasting more and more of hedge fund managers' time. When setting up a due diligence procedure, investors must remember that one of the most important goals is to obtain proper insight in the true risk-return profile (including the relationship with other asset classes) of the strategy followed. This means asking a lot of detailed questions about the strategy and risk management procedures followed, going home and studying the latter under many different scenarios. Common sense and doing one's homework are crucial in alternative investments.

**7. HEDGE FUND DIVERSIFICATION**

For risk-averse investors, diversification is often said to be the only true free lunch in finance. Unfortunately, this does not include hedge funds. Although combining hedge funds into a basket will substantially reduce the standard deviation of the return on that portfolio, it can also be expected to lower the skewness and raise the correlation with the stock market.

**Table 4: Individual Hedge Fund and Hedge Fund Portfolio Risks**

Individual Hedge Funds | Portfolio of Hedge Funds | |
---|---|---|

Standard Deviation | Skewness | |

Merger Arbitrage | 1.75 | -0.50 |

Distressed Securities | 2.37 | -0.77 |

Equity Market Neutral | 2.70 | -0.40 |

Convertible Arbitrage | 3.01 | -1.12 |

Global Macro | 5.23 | 1.04 |

Long/short Equity | 5.83 | 0.00 |

Emerging Markets | 8.33 | -0.36 |

Table 4 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 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. This signals that the degree of correlation between funds in the same strategy group must be quite low. 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 actually drops further as portfolios are formed. With the exception of equity market neutral funds, the portfolio skewness figures are 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 hedge 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.

**8. HEDGE FUNDS AND EQUITY**

It is often argued that given their relatively weak correlation with other asset classes, hedge funds can play an important role in risk reduction and yield enhancement strategies. Again, this diversification service does not come for free, however. 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 5 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 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.

**Table 5: Effects of Combining Hedge Funds with Stocks and Bonds**

% HF | SD | Skewness | Kurtosis |
---|---|---|---|

0 | 2.49 | -0.33 | 2.97 |

5 | 2.43 | -0.40 | 3.02 |

10 | 2.38 | -0.46 | 3.08 |

15 | 2.33 | -0.53 | 3.17 |

20 | 2.29 | -0.60 | 3.28 |

25 | 2.25 | -0.66 | 3.42 |

30 | 2.22 | -0.72 | 3.58 |

35 | 2.20 | -0.78 | 3.77 |

40 | 2.18 | -0.82 | 3.97 |

45 | 2.17 | -0.85 | 4.19 |

50 | 2.16 | -0.87 | 4.41 |

Especially the skewness effect goes far beyond what one might expect given the hedge fund skewness results in table 4. 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.

**9. HEDGE FUNDS AND MEAN-VARIANCE ANALYSIS
**

When studied in a mean-variance framework, the inclusion of hedge funds in a portfolio appears to pay off impressive dividends: equity-like returns with bond-like risk. Since mean-variance analysis only looks at the mean and standard deviation, however, it skips over the fact that with hedge funds more attractive, mean-variance attributes tend to go hand in hand with less attractive skewness and kurtosis properties. In addition, it skips over the significant co-skewness between hedge funds and equity.

**Table 6: Mean-Variance Optimal Portfolios**

Stocks and Bonds Only | |
---|---|

Std. Dev. | Mean |

2 | 0.77 |

2.5 | 0.95 |

3 | 1.10 |

3.5 | 1.23 |

4 | 1.36 |

Stocks, Bonds and Hedge Funds | |

2 | 0.92 |

2.5 | 1.06 |

3 | 1.20 |

3.5 | 1.30 |

4 | 1.39 |

We performed two standard mean-variance optimisations; one with only stocks and bonds and one with stocks, bonds and hedge funds as the available asset classes. The results of both optimisations can be found in table 6. Starting with the case without hedge funds (top panel), we see that moving upwards over the efficient frontier results in a straightforward exchange of bonds for stocks. Since stocks have a higher mean than bonds, the mean goes up. While this happens, the skewness of the return distribution drops in a more or less linear fashion as stock returns are more negatively skewed than bond returns. The kurtosis of the return distribution remains more or less unchanged. Next, we added hedge funds and recalculated the efficient frontier (bottom panel). Moving over the efficient frontier, we see that at first bonds are exchanged for stocks while the hedge fund allocation remains more or less constant. When the bond allocation is depleted, the equity allocation continues to grow but now at the expense of the hedge fund allocation. Similar to the case without hedge funds, if we increase the standard deviation, the mean goes up, while the skewness of the return distribution goes down. Unlike what we saw before, however, skewness drops as long as bonds are being replaced by equity but rises again as hedge funds start to be replaced by equity. The lowest level of skewness is reached when the bond allocation reaches 0%, which is in line with our earlier observation that in terms of skewness hedge funds and equity are not a good mix.

Comparing the case with and without hedge funds we see a significant improvement in the mean, especially for lower standard deviations. However, we also see a major deterioration in skewness and kurtosis, with the largest change taking pace exactly there where the mean improves most. From this it is painfully clear that standard mean-variance portfolio decision-making is not appropriate when hedge funds are involved as it completely ignores these effects. When hedge funds are involved investors need a decision-making framework that also incorporates the skewness and kurtosis of the portfolio return distribution. Since portfolios with low skewness will tend to exhibit the most attractive mean-variance properties, mean-variance optimisers are essentially nothing more than skewness minimizers.

When bringing together different assets or asset classes in a mean-variance framework, we implicitly assume that these are comparable in terms of liquidity and the quality of the inputs used. With stocks and bonds this assumption is often justified. When alternative investments are introduced, however, this is no longer true. Many hedge funds employ long lock-up and advance notice periods. Such restrictions are not only meant to reduce management costs and cash holdings but also allow managers to aim for longer-term horizons and invest in relatively illiquid securities. As a result, hedge fund investments are substantially less liquid than stocks or bonds. In addition, since the available hedge fund data cover such a short and exceptional period we have little idea what the return generating process behind hedge funds looks like and what constitutes normal behaviour and what not. This illiquidity and additional uncertainty should be properly incorporated in the portfolio optimization process. If not, hedge funds are artificially made to look good and consequently too much money will be allocated to them.

**10. CONCLUSION **

Proper hedge fund investing requires a much more elaborate approach to investment decision-making than currently in use by most investors. The available data on hedge funds should not be taken at face value, but should first be corrected for various types of biases and autocorrelation. Tools like mean-variance analysis and the Sharpe ratio that many investors have become accustomed to over the years are no longer appropriate when hedge funds are involved as they concentrate on the good part while completely skipping over the bad part of the hedge fund story. Investors also have to find a way to figure in the long lock-up and advance notice periods, which make hedge fund investments highly illiquid. In addition, investors will have to give weight to the fact that without more insight in the way in which hedge funds generate their returns it is very hard to say something sensible about hedge funds' future longer-run performance. The tools to accomplish this formally are not there, meaning that more than ever investors will have to rely on good old-fashioned common sense and doing their homework. Given the lemming-like behaviour of especially institutional investors, however, for most this may well turn out to be too much to ask for.

It has also become clear that hedge funds are not the miracle cure that many investors think or have been told they are. Again, this boils down to a matter of common sense. Anyone who is well calibrated to the world we live in will have extreme difficulty believing that there is a significant (and growing) number of people that are able to systematically beat the market to such an extent that even after deducting "2 plus 20" or even more the investor is left with a superior return. Hedge funds offer investors a way to obtain a lower standard deviation and/or higher expected return but only at the cost of lower skewness and higher kurtosis. Whether the resulting portfolio makes for a more attractive investment than the original is purely a matter of taste, not a general rule.