Hedge Fund Diversification:
How Much Is Enough?
François-Serge Lhabitant, Michelle Learned*
July 2002
Abstract: There are many benefits to investing in hedge
funds, particularly when using a diversified multi-strategy approach.
Over the recent years, multi-strategy funds of hedge funds have flourished
and are now the favorite investment vehicles of institutional investors
to discover the world of alternative investments. More recently, funds
of hedge funds that specialize within an investment style have also
emerged. Both types of funds put forward their ability to diversify
risks by spreading them over several managers. However, diversifying
a hedge fund portfolio also raises a number of issues, such as the optimal
number of hedge funds to really benefit from diversification, and theinfluence
of diversification on the various statistics of the return distribution
(e.g. expected return, skewness, kurtosis, correlation with traditional
asset classes, value at risk and other tail statistics). In this paper,
using a large database of hedge funds over the 1990-2001 period, we
study the impact of diversification on naively constructed (randomly
chosen and equally weighted) hedge fund portfolios. We also provide
some insight into style diversification benefits, as well as the inter-temporal
evolution of diversification effects on hedge funds.
* Both authors are at Thunderbird, the American Graduate School of International Management (AZ, USA). FranŽois-Serge Lhabitant is also a FAME Research Fellow, Head of Risk Management and Quantitative Analysis at the Alternative Asset Management Group of Union Bancaire Priv˙e (Geneva) and Visiting Professor of Finance at HEC University of Lausanne. Any comment is welcome. Contact: francois@lhabitant.net or michellelearned@global.t-bird.edu
Executive Summary
After two years of bear markets, traditional assets
such as equities, fixed income and even real estate are no longer considered
to generate attractive returns. Family offices, discretionary portfolio
managers, institutional investors, high-net work individuals and the private
banks that manage their own funds are therefore looking for more sophisticated
ways to reach their investment objectives. Most of these investors have
now allocated up to 5% or more of their portfolios to alternative strategies,
and even the most conservative ones are dipping a toe in the water. Among
the set of potential candidates, hedge funds have progressively gained
acceptance as a core asset class, thanks to their consistent absolute
returns and low correlation with traditional assets. This makes them a
valuable tool for the diversification of conventional equity market risk.
Consequently, the hedge fund industry is getting larger and larger every
day, to the point that the capacity of hedge fund managers to digest the
flows of new money regularly pouring into their funds is regularly questioned.
Despite this apparent success, there is still a remarkable lack of understanding
and information about the hedge funds industry amongst individual investors,
advisers, and institutions. Keynes' observation that diversification is
protection against ignorance is best illustrated by the fact that most
institutions prefer gaining their exposure through funds of hedge funds,
which give them instant diversification Ū and free them from the responsibility
of monitoring individual managers. However, the proliferation of funds
of hedge funds - which vary greatly in the number of underlying managers
(5 to 100), the strategies on which they focus (diversified vs. sector
or geographically focused) as well as their asset allocation strategy,
if any - should not hide the fact that diversification in hedge funds
is not as easy as it seems. In particular, diversification in hedge funds
can be made at two levels. Diversification by investment style involves
investment in a number of strategies (long-short, global macro, convertible
arbitrage, etc.) to reduce exposure to individual style exposures, while
diversification by judgment recommends that investors diversify across
a number of managers within a particular investment style, in order to
avoid performance solely being determined by one manager's skills. While the benefits of diversification have been widely
studied and documented for traditional assets, the research on hedge fund
diversification has been rather scarce. For a long time, it was limited
to measuring the effects of including a hedge fund index in a traditional
strategic asset allocation. It is only recently that a few papers started
investing in the issue of how many hedge funds were required in a hedge
fund portfolio to efficiently reduce volatility. The answers vary greatly
depending on the sample considered and the time-period investigated. In
this paper, we therefore reconsider the problem of hedge fund diversification.
Our approach relies on the naive diversification strategies that were
suggested in the early 1950s: we simply build equally weighted portfolios
of randomly selected hedge funds. By repeating the process several times
and studying the characteristics of the resulting portfolios (50,000 in
total), we are able to study the impact of naively increasing the number
of hedge funds in a portfolio. Our first empirical findings tend to demonstrate that
diversification works well in a meanvariance space. That is, increasing
the number of hedge funds in a hedge fund portfolio decreases the portfolio's
volatility, while maintaining its average return level. Downside risk
statistics (such as maximum monthly loss, maximum drawdown or value at
risk) are also reduced in larger-size hedge fund portfolios. This seems
to validate the existence of funds of funds as useful investment vehicles.
However, when one goes beyond the mean-variance framework and considers
additional factors such as skewness and kurtosis, diversification is far
from being a free lunch. For several strategies, diversification reduces
positive skewness, may even generate negative skewness, and increases
kurtosis, i.e. there is a trade-off between profit potential and reduced
probability of loss. In addition, the correlation with the S&P 500 of
large-sized hedge fund portfolios increases, which clearly evidences the
dangers of diversification overkill, that is, the attempt of advisors
to incorporate an unwieldy number of hedge funds in their portfolio construction
process. Since most of the diversification benefits are reached for small-sized
portfolios (typically 5 to 10 hedge funds), it therefore seems that hedge
fund portfolios should rather be cautious on their allocations past this
number of funds. Our empirical findings also illustrate the difference
between diversification by investment style and diversification by judgement.
Clearly, the benefit of increasing managers within a strategy is a function
of the homogeneity or heterogeneity of the sample from which the managers
are drawn. Style diversification (diversification by investment style)
obviously provides better opportunities for diversification than diversification
by judgment. In that respect, it also seems that information about investment
style reported by fund managers should be used in the portfolio construction
process, albeit naive, to increase the diversification benefits. "If you have a harem of 40 women, you never get to
know any of them very well." The world's most famous value investor 1. Introduction They were exotic products created by unregulated,
renegade stock-pickers and exclusively held by a private club of high-net-worth
individuals for financial snobbery motives. They have progressively become
the darlings of the investment industry, as evidenced by financial publications,
analysts' reports, boardrooms and even happy hour cocktails. Their success
was fuelled by the wealth created by the long bull equity market of the
1990s, and is now supported by the difficult and highly volatile environment
that has prevailed since the early 2000s. Indeed, by focusing on absolute
performance and abstracting from benchmarks, hedge funds are able to generate
superior returns in virtually all types of market environments. Consequently,
they offer the much-needed diversification to portfolios invested in traditional
asset classes such as equities and bonds. This provides a strong argument
for using them in wealth management and contributes to making this new
asset class - or new way to manage traditional asset classes - an increasingly
popular investment choice. However, we all know that there is "no such thing
as a free lunch" in finance. Thus, private and institutional investors
willing to include hedge funds in their portfolios must realize that to
deliver their favorable return/risk characteristics, hedge funds must
carry additional risks, which are not common to traditional stock and
bond investments. These risks are inherent to the strategies pursued,
the instruments and markets used, the amount of leverage employed, and
last, but not least, the specific skills of the selected hedge fund managers.
Since choosing a bad manager may easily wipe out all
the benefits of a hedge fund allocation, investing in only one hedge fund
is likely to be sub-optimal. The reasons are threefold. Firstly, dramatic
performance differentials between competing funds raise the issue of whether
a single investment instrument can deliver consistent returns close to
those of the broad hedge-fund indices that are used at the strategic asset-allocation
level. Secondly, a number of individual hedge funds have collapsed under
the weight of spectacular frauds or investment debacles (Manhattan Capital
Management, Maricopa Investment Corporation, Lipper Convertible Arbitrage,
etc.). This has raised concerns among investors, who often lack sufficient
information to evaluate comparative hedge fund performance and to perform
the necessary exhaustive on-site due diligence checks. Finally, investing
only with managers who have a good reputation and an established track
record does not provide a complete hedge, as illustrated by the debacle
of the brain trust that was Long Term Capital Management LP. Consequently, risk-conscious investors are coming
back to the central tenet of modern portfolio theory, namely, diversification.
By combining several hedge funds with differing return distributions and
risk profiles in a portfolio, investors are able to diversify specific
risk away and ensure a more disciplined exposure to the overall hedge
fund asset class. This is likely to result in better long-term risk-adjusted
returns. Those willing to avoid the logistical problems and record-keeping
headaches of tracking several hedge funds may even delegate the portfolio
construction and monitoring activities to a fund of hedge funds. This
is the preferred investment structure for most institutional investors,
since it gives them instant diversification and frees them from the responsibility
of monitoring managers1. Funds of hedge funds were initially diversified across
investment styles, sectors and/or regions. However, more recently, funds
of hedge funds focusing on a single investment style, a particular asset
class, a single sector or region have also emerged. They are usually more
concentrated in terms of risks as well as in terms of number of underlying
hedge funds than the larger diversified funds of funds. Their adage could
be: "Put all your eggs in one basket, but then don't take your eyes off
that basket". Nevertheless, focused-hedge fund managers still invest in
more than one underlying fund and therefore rely on the risk-reduction
power of diversification. 1 There are in fact other benefits of funds of hedge funds, such as lower minimum investments, access to proven and successful hedge fund managers - including those that have closed their fund to new investors - as well as reduced lock-up periods and better liquidity. On the downside, funds of hedge funds add a layer of fees, which reduces the overall return. For an exhaustive discussion of these advantages and disadvantages, see Lhabitant (2002b).
Intuitively, the existence of hedge fund diversification benefits will
depend upon the number of hedge funds in a portfolio. Beyond the agreement
that holding only a few funds may imply under-diversification, exposure
concentration, and, therefore, too much risk, while holding too many funds
may result in over-diversification, the dilution of each fund's contribution
and the neutralization of most diversification benefits, there seems to
be no consensus on the optimal number of funds. On the academic side,
the literature suggests that approximately eight to ten managers should
be sufficient to reduce significantly the overall risk of the portfolio
- see Billingsley and Chance (1996) for managed futures, Henker and Martin
(1998) for CTAs and Henker (1998) for hedge funds. However, Amin and Kat
(2000) show that one has to hold at least twenty funds to fully realize
the diversification potential in hedge funds. From the practitioner's
perspective, the consensus seems to be that at least twenty to thirty
managers are necessary to diversify effectively, as shown by the information
released by funds of hedge funds. The short note by Ruddick (2002) evidences
that the maximum benefits of diversification are reached with around 20
funds, and that it is still possible to have them at around 40 funds if
the quality of new additions can be maintained. Most of the attention is dedicated to the diversification
benefits accrued by adding hedge funds to traditional asset portfolios.
There has thus far been very little large-scale research done on the topic
of pure alternative assets diversification. For example: do the diversification
benefits, if any, go beyond volatility and also affect other important
statistics (e.g. average returns, skewness2, kurtosis, correlation
with other asset classes, value at risk, maximum drawdown, etc.)? Do they
differ within styles and across styles? These questions are still largely
unanswered. In this paper we aim, therefore, to fill the gap. We believe
our contribution is unique for three primary reasons. Firstly, our database
is more comprehensive than any of those used in previous studies. Secondly,
we include trials of the various investment styles followed by hedge fund
managers. Thirdly, we analyze the inter-temporal variations in the diversification
benefits. The structure of this paper is as follows. Section 2 reviews
the various arguments for diversification, both in the world of traditional
investments and that of the hedge funds. Section 3 describes our methodology
and discusses our major findings. Finally, Section 4 draws conclusions
and opens the way for further research. 2Skewness is a measure of the symmetry of a return distribution around its mean. If the return distribution is skewed to the left (returns lower than the mean have higher probability), skewness is negative, while when the return distribution is skewed to the right (returns higher than the mean have higher probability), skewness is positive. As a reference, the standard normal distribution is perfectly symmetrical and has a skewness of zero. Kurtosis refers to the weight of the tails of a distribution, or "peakedness". Distributions where a large proportion of the observed values lie towards the extremes are said to be "platykurtic" and display positive kurtosis. If, on the other hand, the observed values are bunched near the mean, the distribution is said to be "leptokurtic" and kurtosis is negative. A normal distribution is said to be "mesokurtic" and has a kurtosis equal to 0.
2. Diversification: from stocks to hedge funds Portfolio diversification - the practice of spreading one's
money among many different investments - is a common sense concept that
has many parallels in popular language and culture3. Its theoretical
foundations were introduced in the normative work of Harry Markowitz (1952,
1959), and later confirmed by the work of William Sharpe (1964). 2.1 The pioneersMarkowitz's initial assumption was that risk-averse, mean-variance
utility agents were concerned with only two elements of their portfolios
- the expected return, as measured by the mean rate of return, and the
risk, as quantified by the standard deviation or variance of the mean
rate of return. When risky assets are aggregated, their correlation often
determines the majority of the total risk rather than individual volatilities.
Consequently, the total risk of a carefully constructed portfolio should
be less than the sum of the risks in the portfolio's component pieces.
Markowitz thus suggested a quadratic programming algorithm to calculate
the optimal combination of assets. Given the lack of computing power that existed in
the 1950s, Markowitz's approach was considered as a poignant but useless
exercise. Nevertheless, his formulation contributed directly to the subsequent
Capital Asset Pricing Model (CAPM) of Sharpe (1964), Lintner (1965), and
Mossin (1966). Their respective works clearly delineated the partitioning
of risk into two parts. The systematic part is the portion that is unavoidable
once the investor invests in a particular asset class, while the unsystematic,
or specific, part can be reduced, or even eliminated at the portfolio
level, by creating a mixed-asset portfolio. Since diversification is inexpensive,
specific risk is not rewarded and should be eliminated while systematic
risk remains and will be rewarded by markets. 3For example, "Don't put all your eggs in one basket."
Until the late 1980s, these results remained somewhat
abstruse bits of theory that were admired by academics but had only limited
influence on practitioners. Then, the situation changed dramatically,
thanks to the availability of computing power and the development of econometric
methods. Diversification is now shaping the way sophisticated investors
view their portfolios. 2.2 The magic number: between 8 and 40The issue of how many assets should be included in a diversified
portfolio has been extensively debated in the finance literature for over
30 years; see for instance Elton and Gruber (1977), Evans and Archer (1968),
Latane and Young (1969), Fischer and Lorie (1970), Mokkelbost (1971),
Wagner and Lau (1971), Johnson and Shannon (1974), Lorie (1975), Upson,
Jessup and Matsumoto (1975), Lloyd, Hand and Modani, (1981), Tole (1982),
Statman (1987), Newbould and Poon (1993) or O'Neal (1997), among others.
Nevertheless, the definitive answer to that question has not yet been
found. Even worse, several studies contradict each other. As an illustration,
Evans and Archer (1968) observed that most of the effects of diversification
take place with the aggregation of eight to ten securities and raised
doubts about the usefulness of increasing portfolio sizes beyond that
point, while Statman (1987) concluded that at least thirty to forty stocks
were needed to achieve sufficient portfolio diversification. In our opinion, the truth is likely to be somewhere
in between the two. It depends on the country under investigation, the
types and individual characteristics of the assets considered, the market
conditions and the level of transaction costs. To get a precise answer,
investors should perform a marginal analysis of the costs and benefits
associated with increased diversification. This is precisely what Markowitz
did when developing the modern portfolio theory. 2.3 Diversification in practice: Naive Diversification Although diversification has become the unquestioned
bedrock of investment allocation decisions, in practice, it is frequently
the case that asset allocation does not just blindly apply the diversification
principles set forth by Markowitz. For example, very few investors effectively
take correlations (that is, the non-linearity of risk) into account when
making complex portfolio decisions. Rather, they prefer to allocate assets
using simpler rules, such as dividing allocations evenly among the assets
available. This approach, also known as the "1/N heuristics" or "naive
diversification", has a long history in asset allocation. In fact, it
was even recommended in the Talmud. Writing in about the 4th century,
a certain Rabbi Issac bar Aha gave the following asset allocation advice:
"A man should always place his money, a third into land, a third into
merchandise, and keep a third on hand". Harry Markowitz reported that
he used this rule himself and justified his choice on psychological grounds:
"My intention was to minimize my future regret. So I split my contributions
fifty-fifty between bonds and equities". Simply stated, naive diversification is a protection
against ignorance. It aims to spread assets evenly in the portfolio in
order to reduce overall risk, while at the same time ignoring the mathematical
complexities underlying modern portfolio theory. According to the latter,
naive diversification does not give proper consideration to the correlations
among the assets and should therefore result in sub-optimal portfolios.
However, in practice, naive diversification usually results in reasonably
diversified portfolios that are surprisingly close to some point on the
efficient frontier4. In contrast, the consequences of fuelling
an optimizer with an incorrect number can be potentially quite significant
in terms of lost benefits, as evidenced by Brennan and Torous (1999).
The implementation of a naive diversification strategy, however, is likely
to revolve around transaction and portfolio management costs. There are
diminishing marginal returns, and eventually, absolute returns when increasing
portfolio size, since transaction costs remain relatively constant while
incremental reductions in portfolio risk get smaller. Therefore, the question
of the number of optimal securities necessary in a naively diversified
portfolio is still open. 4 See for instance Fisher and Statman (1997).
2.4 From mutual funds to hedge fundsAt about the same time Markowitz was polishing his portfolio
diversification theory, Alfred Winslow Jones was working on exactly the
opposite objective - namely, the isolation of specific risk and the elimination
of market risk. Jones was convinced that he had superior stock-selection
ability, but no market-timing skills. Therefore, his strategy consisted
of combining long positions in undervalued stocks with short positions
in overvalued ones. This allowed him to make a (small) net profit in all
markets, capitalizing on his stock-picking abilities while simultaneously
reducing overall risk through lesser net-market exposure. To magnify his
portfolio's returns, Jones added leverage, that is, he used the proceeds
from his short sales to finance the purchase of additional long positions.
This provided the basic principles for what was the first hedge fund.
More than half a century later, hedge funds have significantly
evolved from the original model. Indeed, most of them do not actually
hedge anything. Nowadays, the term "hedge fund" is applied somewhat indiscriminately
and beyond the scope of its original meaning. It refers to any pooled
investment instrument that is not a conventional investment fund - that
is, any fund using a strategy or set of strategies other than investing
long in bonds, equities, money markets, or a mix of these assets. Consequently,
hedge funds are better identified by their common structural characteristics
than by their "hedged" nature. These characteristics include, but are
not limited to, active management, long-term commitment of investors,
use of incentive fees, leverage, broad discretion over the investment
styles, asset classes and investment vehicles, as well as the aim of insulating
the skills of their advisors against the vagaries of the market. Given the diversity of their strategies, hedge fund
returns generally display moderate to low correlations with traditional
equity and bond indices. In addition, hedge fund strategies have moderate
to low correlations with each other. The idea of diversifying among loosely
correlated funds is therefore very natural. But which approach should
be selected: Markowitz's, or naive? Intuitively, it appears that very
few hedge fund portfolios are optimized along the lines of Markowitz's
recommendations. The reasons are threefold: o Non-normality. Hedge fund return distributions
tend to exhibit skewness and fat tails. Both characteristics are important
for investors' portfolio decisions5, but are superbly ignored
by mean-variance optimizers who consequently produce myopic and biased
results. o Econometric difficulties. Mean-variance optimizers
require precise forecasts of risks, returns, and correlations to put together
optimum, forward-looking portfolios. One commonly employed simplification
is to focus on expected returns and accept historic risk and correlation
as reasonable approximations of the future6. However, while
the quantity and quality of hedge fund information has improved considerably
in recent years, adequate time-series are still largely unavailable to
obtain robust estimates of historical hedge fund data. In addition, hedge
fund returns and strategies are not necessarily stable over time, which
complicates the forecasting of future returns and questions the use of
historical estimates as a forecast. o Operational difficulties. The process of hedge
fund selections has been essentially qualitative so far, due to the specificities
of the individual vehicles (e.g. minimum investments, lock-up periods,
exit notifications, restrictions to access, etc.). Most optimizers are
unable to incorporate these constraints, and therefore, their use often
result in non-feasible portfolios. A recent survey by Arthur Andersen (2002) of Swiss
hedge fund investors and fund of hedge funds managers confirms our intuition.
It appears that most participants do not use a quantitative approach for
their asset-allocation strategy. Many respondents even admitted to having
no asset-allocation strategy at all! naive diversification is therefore
well suited to provide, at least, a starting point that is much better
than simply leaving things to intuition. It also simplifies the asset-allocation
process while still enjoying the benefits of risk reduction. 5 See for instance Scott and Horvath
(1980), Kane (1982), Kraus and Litzenberger (1976) or Lhabitant (2000,
2002). 6 As evidenced by Chopra (1993), the return forecast is the
most critical aspect of constructing efficient portfolios 3. Empirical tests3.1 Data and methodology The data we use in this paper are taken from a series of historical
quarterly snapshots by a number of hedge fund information providers (Managed
Account Reports, Hedge Fund Research, TASS+ and Evaluation Associates
Capital Management), as well as from data received directly from several
hedge fund administrators. In total, our database contains 6,985 distinct
hedge funds, with no restriction whatsoever on their assets under management
or their lifespan. In particular, it includes a large number of now defunct
funds for the duration of their existence. Since we want to test naive diversification of hedge
funds, we proceed by using Monte Carlo simulation based on the effective
hedge fund returns. We create equally weighted portfolios of increasing
size (N=1, 2, Ū 50) by randomly selecting hedge funds from our data set.
For each portfolio, we build a time series of returns and use it to generate
various statistics (average return, volatility, etc.). For each portfolio
size, this process is repeated 1,000 times to obtain 1,000 observations
of each statistic. This is necessary to estimate the "typical" behavior
of a portfolio of size N, but also to build confidence intervals for our
results. It is important to note that each portfolio is independently
assembled, that is, a portfolio of size N+1 is not built from the portfolio
of size N obtained at a previous stage of the simulation. Furthermore,
each portfolio sampling is conducted without replacement to effectively
ensure equal weighting of the hedge funds in the portfolio. In order to assess the diversification benefits for
different strategies, we split our sample arbitrarily into 10 representative
investment styles, which can be described as follows: o Convertible arbitrage hedge funds focus on
the mis-pricing of convertible bonds. Their rationale is that (i) since
convertibles are hybrid in nature, they do not attract pure bond and pure
stock investors, so that price discrepancies are frequent; and (ii) convertible
securities often contain several call, put or exercise-date options that
are often neglected by the market. A typical position involves a long
position in the convertible bond and a short position in the underlying
asset. Credit risk and interest rate risk of the convertible position
may also be hedged using adequate instruments. o Fixed income arbitrage hedge funds tend to
profit from price anomalies between related securities and/or bet on the
evolution of interest rates spread. Typical trading strategies are butterfly-like
structures (e.g. sell expensive 3-year and 5-year bonds while buying a
cheap 4-year bond), cash/futures basis trading strategies (the cheapest
to deliver often trades either rich or cheap to the fitted yield curve),
or relative swap spread trades (e.g. buy a government bond and short a
strip of euro-dollar contracts to buy the Treasury/Euro-Dollar spread).
o Event-driven hedge funds focus on price movements
generated by an anticipated corporate event, such as a merger, an acquisition,
a bankruptcy, etc. The funds in thiscategory cover a wide range of strategies.
The most popular ones are distressed securities (bets on the likelihood
of a bankruptcy and the estimation of the recovery rate) and mergerarbitrage
(bets on the completion of announced mergers). In the latter category,
some of the funds get involved in the restructuring process and are at
the frontier of private equity,while others only invest in the most liquid
and senior part of the debt. Note that severalfunds in the latter category
had to reorient themselves to the recent drastic diminution ofthe merger
activity. o Long-short equity hedge funds represent the
original hedge fund model. They invest in equities, both on the long and
the short sides, and generally have a small net long exposure. They are
genuinely opportunistic strategies capitalizing on the stock picking abilities
of their managers and could be classified as "double alpha, low beta"
funds. o Market neutral hedge funds that seek to neutralize
certain market risks by taking offsetting long and short positions in
instruments with actual or theoretical relationships. Most of them are
in fact essentially long-short equity hedge funds that maintain long and
short portfolios of the same size and/or risk, so that they have no exposure
to the stock market. o Dedicated short bias hedge funds are essentially
long-short equity hedge funds that maintain a consistent net short exposure.
Very few of them are only short, as pure short sellers were progressively
eliminated during the bull market of the 1990s. o Emerging market hedge funds invest in equities
and fixed-income securities of emerging markets around the world. Because
shorting is not permitted in many emerging markets, and there are no viable
futures or other derivative products with which to hedge, emerging market
hedge fund managers must often go to cash or other markets when valuations
make being long unattractive. o Global macro funds take leveraged views on
overall market directions as influenced by major economic trends and/or
events. Their existence was revealed to the public in 1992 when G. Soros
and his Quantum Fund cashed in a gain of several billion by forcing the
British pound to exit the European Monetary System. o Managed futures funds implement discretionary
or systematic trading in listed financial, commodity and currency futures
around the world. The managers of these funds are also known as commodity
trading advisors (CTAs). o Multi-strategy funds recently emerged as a
new concept. They implement the diversification concept within a single
investment vehicle in the sense that they regroup managers acting in several
of the above-mentioned strategies. The next step was to group our hedge funds according
the above-mentioned classification. We decided to rely on the self-proclaimed
strategy stated by the fund managers. Although there is increasing evidence
that hedge fund managers are sometimes subject to style drifts and do
not fully disclose the true nature of their strategies, we believe that
relying on what the manager says is the only classification scheme that
is effectively accessible to most investors. Indeed, it would be rather
surprising to see an investor capable of performing sophisticated techniques,
such as style analysis or cluster partitioning, while at the same time
relying on naive diversification for his portfolio construction! Once our funds were classified, we then ran the tests
described above for each investment style, as well as for the overall
data set without any style consideration (the true "naive diversification"
across styles). We also tested a more elaborate strategy that we call
"smart diversification". The latter consists in selecting hedge funds
naively, but while imposing diversification across styles. That is, a
portfolio of size N will contain randomly selected and equally weighted
funds, but the number of funds in the most represented investment style
should not exceed by more than one the number of funds in the least represented
investment style. This corresponds to the strategy of an investor that
would attempt to use the available information on fund styles. In order to assess the inter-temporal variations of
diversification benefits, we also split our sample in three periods, namely,
1990-1993, 1994-1997 and 1998-2001, and ran the above tests on each sub-sample.
The issue of survivorship bias was resolved by creating two different
samples for each period. One contains all the funds that existed at some
point during that period. Assets are then balanced anew each time a new
fund is included in, or excluded from, the portfolio. The second one contains
only hedge funds that survived the entire period. Hereafter, we will present
and discuss the results for the first sample. The results for the survivor-funds
sample will be commented later on.
3.2 Results Despite its numerous deficiencies, Markowitz's venerable mean-variance
framework is still widely used to assess the performance of hedge funds.
We, therefore, began by focusing on the average return and the volatility
of the hedge fund portfolio. Return and Volatility Figure 1 and Table 1 show the evolution of the average return
of a hedge fund portfolio as a function of its number of underlying funds.
Similarly to Amin and Kat (2002), we observe that the mean portfolio return
does not seem significantly affected by the number of hedge funds it contains.
This result is not surprising, due to the linearity of the average operator7.
Of course, the mean return diverges widely across strategies and over
time. During the first period (1990-1993), the Global Macro style provides
an impressive 22% to 23% return. However this strategy loses favor during
the latter half of the decade. Convertible Arbitrage too does well during
the first period with returns around 20%, but drops to around 10% by the
end of the decade. More consistent styles are Long/short Equity with around
16 to 17% return on average, while Emerging Markets hovers at the bottom
of the grouping with returns around 8% to 9%. Figure 2 and Table 2 show the evolution of the average
volatility of a hedge fund portfolio as a function of the number of its
underlying funds. Whatever the period and the investment style, as the
number of funds increases, volatility decreases and then stabilizes. Once
again, this result is not surprising, since it just depends on the correlation
between the underlying funds (which cannot exceed one). Our result is
therefore similar to the ones traditionally observed for stocks. The number of hedge funds necessary to effectively
diversify is surprisingly low. Whatever the time period and the style,
it seems that between 5 and 10 hedge funds are sufficient to eliminate
75% of the specific risk in the portfolio. This raises the question of
the number of hedge funds in fund of funds portfolios, since the latter
is usually much larger. The drop in the overall volatility is sharpest
for Market Neutral and Short Selling, whereas the volatility of Convertible
Arbitrage decreases the least, and the others reach a diversification
benefit in the range of about 23%. Whereas in the first two periods, the
volatility among the different styles is more evenly distributed in the
total range of about 4%-16%, the third time period show two groupings
of styles: those in the range of 16%-22% (Emerging, Multi-strategies,
Long/short), and those in the 5%-11% range (all remaining strategies).
We also observe that smart diversification outperforms the naive diversification
in terms of risk reduction. Skewness and Kurtosis Our previous results suggest that increasing the number of
hedge funds in a portfolio reduces volatility while maintaining the expected
return. However, hedge funds have properties that go beyond these two
parameters. In particular, their returns do not conform to the normal
(bell-curve) distribution, but tend to display asymmetries and fat tails.
Therefore, higher distribution moments, such as skewness and kurtosis,
should also be included in the analysis. Skewness is expected to disappear
with diversification as funds with negative skewness are mixed with positively
skewed ones so that, at the aggregate level, these individual effects
are cancelled. For the same reason, we expect excess kurtosis to be somehow
reduced by diversification. Figure 3 and Table 3 show the evolution of the average
skewness of a hedge fund portfolio as a function of the number of its
underlying funds. The graph has several interesting features. In particular,
it shows clearly that skewness variations are not uniform across styles.
For instance, when the number of funds increases, the skewness drops systematically
and is negative for Fixed Income Arbitrage, Convertible Arbitrage and
Event Driven strategies, while it increases slightly and is positive for
Short Sellers, Managed Futures and Long Short Equity strategies. It also
demonstrates that the observed range of skewness widens, particularly
on the downside. While the skewness of all strategies remains between
-1 and 0.75 in the 1990-1993 and 1994-1997 periods, it jumps to -4.5 to
1.25 during the 1998-2001 period. 7The average operator is linear. In a sense, the figure we obtain for a one-fund portfolio is the average of 1,000 hedge funds returns, while the figure for a two-fund portfolio will simply be the average of 2,000 hedge funds. The number will therefore rapidly converge to the sample average.
Figure 4 and Table 4 show the evolution of the average
kurtosis of a hedge fund portfolio as a function of the number of its
underlying funds. We find that most kurtosis results tend to be concentrated
in the -0.5 to +0.5 range. Changes in kurtosis tend to be less predictable,
and differ widely over time and across investment styles. For instance,
the 1994-1997 period seems to validate the positive influence of diversification
on kurtosis, except for the Fixed Income Arbitrage style. However, the
situation changes in 1998-2001, when several style portfolios exhibit
an increase in their kurtosis as the number of funds increases. Note however
that the dramatic Fixed Income Arbitrage pattern may be due to the LTCM
crisis pattern (the more funds one invests in, the more likely the crisis
will affect the portfolio), while the Event Driven pattern may result
from the failure of several mega-mergers (e.g. Alcatel-Honeywell is a
good example of a deal whose spreads diverged instead of narrowed). Given these results, assessing hedge funds based on
return and volatility criteria may be misleading because of the potential
underestimation of return volatilities8. In particular, diversification
within some hedge fund strategies may appear highly attractive in meanvariance
terms, but this is much less so when skewness and kurtosis are taken into
account. Downside risk statistics Since downside risk is of great concern to many investors,
we have also examined the impact of hedge fund diversification on three
commonly accepted downside risk measures, namely, the largest monthly
loss, the maximum draw-down, and the value at risk (VaR). The largest
monthly loss is the greatest decline in net asset value for a particular
hedge fund for any one month period over the period considered. The maximum
draw-down is the biggest percentage-losing period ("peak to valley") experienced
by a particular fund, regardless of whether or not the draw-down consisted
of consecutive months of negative performance. It corresponds to the loss
that an investor would experience buying shares at the highest net asset
value and selling them at the lowest subsequent net asset value over a
period. Finally, the value at risk (VaR) is an estimate of the maximum
amount a particular fund could lose over a one-month period in normal
market conditions. In our case, we defined "normal market conditions"
as being 95% of the time and we calculated value at risk by simply taking
the 5% percentile of the empirical return distribution over the considered
period.8See Lhabitant (2000)
Figure 5 and Table 5 show the evolution of the average
worst-month return of a hedge fund portfolio as a function of its number
of underlying funds. The worst returns and the lowest benefits from diversification
are observed for the Emerging Markets investment style, which is not surprising
given the contagion effect that affected these markets during all past
crises (e.g. Asia in 1997, Russia in 1998, etc.) thus mitigating the gains
from portfolio diversification. Otherwise, increasing the number of funds
in a portfolio is obviously beneficial for all strategies. Looking at
the temporal variations is also quite instructive. It is very clear that
the most recent period we considered (1998-2001) was a period with an
increased downside risk for hedge funds as an asset class, and particularly
for directional strategies such as Dedicated Short Bias and Emerging Markets.
Risk reduction opportunities also seem to be shrinking in some arbitrage
strategies, such as Event Driven and Fixed Income Arbitrage. This results
from an increasing number of hedge funds arbitraging a limited pool of
arbitrage situations, so that these funds tend to be invested in the same
deals. If one of these deals collapses, most of the funds acting in the
sector will be negatively affected. This was the case with recent mega-mergers
that failed (e.g. the $42 billion deal between General Electric and Honeywell),
as well as for distressed situations that went bankrupt (e.g. Kmart, Global
Crossing, WorldCom, and Qwest Communications) or the planned elimination
of the 30-year T-Bond contract. It seriously raises the issue of the existence
of any benefits linked to diversifying within a style for arbitrage funds.
Fortunately, it seems that diversifying across styles still provides important
benefits, particularly when selecting the smart diversification approach.
Figure 6 and Table 6 show the evolution of the maximum
draw-down of a hedge fund portfolio as a function of its number of underlying
funds, while Figure 7 and Table 7 list the same information for the empirical
value at risk. For both measures, diversification seems to work well in
terms of downside risk reduction. However, once again, we observe that
most of the diversification benefits are obtained with a number of funds
that varies between 5 and 10, depending on the strategy. Adding more funds
still provides benefits, but the gains seems marginal compared to the
drawbacks of managing the corresponding portfolio (minimum requirements,
multiple lock-up periods, etc.). Once again, we also observe that smart
diversification outperforms the naive diversification in terms of downside
risk reduction, whatever the performance measure considered. Comparing the worst monthly return (Table 5) with
the value at risk (Table 6) is noteworthy. Here, we can see that investment
styles that display a negative skewness tend to have a worst month far
below the value at risk. Consider for instance the Fixed Income Arbitrage
strategy over the 1998-2001 time-period. This strategy has an average
value at risk of -0.54% (the lowest of the group) while at the same time
displaying the highest average worst-case return (-10.85%) and the lowest
average skewness (-4.40). A similar point can be made about the Event-Driven
strategy over the same period, with an average value at risk of -1.54%
(among the lowest ones of the group) while at the same time displaying
a large average worst-case return (-10.55%) and the second lowest average
skewness (-2.54). This is a clear illustration that arbitrage strategies
in general tend to have low risk statistics, but are significantly affected
by a single tail event when the latter occurs ("the LTCM syndrome"). Correlation statistics
Finally, the last statistic we examined was correlation.
On the one hand, sophisticated investors are increasingly looking to alternative
investments such as hedge funds to provide diversification in their traditional
portfolios, particularly the equity-oriented ones. On the other hand,
these investors prefer to take a diversified approach and allocate their
capital in more than one hedge fund. It is, therefore, essential to examine
the impact of the number of funds in a portfolio on its correlation attributes
with equities (represented hereafter by the S&P 500). Figure 8 and Table 8 summarize our findings. It appears
that diversification within a style leads to a small increase in the absolute
value of the correlation with the S&P 500. That is, positive correlations
with the S&P 500 tend to increase, while negative correlations with the
S&P 500 tend to decrease as the number of funds increases. However, note
that the correlation coefficient is not stable over the three periods
we considered. As an illustration, the Managed Futures and the Multi-strategy
categories exhibited a continuous decrease of their correlation with the
S&P 500 over the three periods, while the Market Neutral category exhibits
exactly the opposite trend. The final correlation figure of 0.79 for a
50-fund portfolio invested in the latter category raises serious doubts
about the validity of the self-proclaimed market-neutral classification
for this group, which is more likely to be made of long-short equity managers.
It also questions any conclusion drawn from an asset allocation that would
assume constant correlations or rely on the stability of historical estimates
of correlation coefficients, as this is obviously not the case. It is interesting to observe the correlation effects
when diversifying across all styles. Although the average correlation
of a single-fund portfolio with the S&P 500 is still low (0.33 to 0.38,
depending on the period), it tends to increase rapidly with the number
of funds in the portfolio. As one could expect, portfolios built according
to the smart diversification strategy tend to be less correlated with
the S&P 500 than equally-sized naively built portfolios, but the correlation
increase is still perceivable. This is a strong case against over-diversifying
a hedge fund portfolio, as the result would be a dramatic reduction of
the potential diversification benefits when mixing this hedge fund portfolio
with traditional assets. It is therefore essential to specify the desired
final correlation properties of a hedge fund portfolio prior to building
it rather than just relying on the fact that the portfolio constituents
taken individually are loosely correlated with traditional assets and
should transfer this property to the portfolio. Since many investors use hedge-fund indices at the
strategic asset-allocation level and switch to hedge funds in the effective
implementation of their decisions, another essential point is therefore
the correlation that exists between the hedge-fund indices and the portfolio
of hedge funds. For example, how many funds in a portfolio are necessary
for the correlation of the portfolio with a hedge fund index to be sufficiently
high? That is, what is the necessary number of funds for a portfolio to
be sufficiently correlated with a hedge fund index? Figure 9 provides
the answer, in the case of the HFR indices (which use equal weights to
build their indices). The results are sound: the larger the number of
funds, the higher is the correlation with the corresponding strategy index.
However, some strategies (e.g. Market Neutral, Global Macro) are hard
to track, since even a 50-fund portfolio provides a loose correlation
with the Market Neutral or Global Macro index. Other strategies (e.g.
Long Short Equity or Event Driven) are easier to track, even with a 10
or 20-fund portfolio. Over time, we also observe an increase in the correlation
properties for most strategies. That is, it appears that correlation coefficients
with the HFR indices seem to increase. Survivorship bias
As mentioned previously, we also investigated the
impact of hedge fund survivorship bias on our previous findings by running
the same analysis over a set of hedge funds that survived each considered
period. Given the comprehensive nature of our database, it is clear that
most hedge funds that disappear from our universe do it for poor performance
reasons. Indeed, if a manager with extraordinary good performance stops
reporting to hedge fund information providers, but still runs its fund
with its existing investors (e.g. Monroe Trout's Trout Trading Fund in
1993), we would still obtain the information about the fund from its offshore
administrator. Therefore, it is not surprising that our results on the
surviving sample were in line with our expectations. Overall, we obtained the same behavior as with the
complete sample: increasing the number of hedge funds in a portfolio increased
the portfolio's average return, decreased its volatility, skewness and
kurtosis, significantly reduced its downside risk, and increased correlations
with both traditional and alternative equity indices. The magnitude of
these changes was somehow comparable across the two samples, although
the initial value of these statistics was different. Indeed, surviving
funds tend to display higher average return, lower volatility, limited
skewness and kurtosis, lower downside risk and stronger correlation with
both traditional and alternative equity indices. However, these desirable
features are impossible to arbitrage ex-ante, since survivorship is by
definition an ex-post feature. 4. ConclusionsThere is a wide array of barriers to hedge fund investing for
consultants and investors alike, such as the scarcity of historical data,
the complexity of strategies, the lack of meaningful benchmarks, poor
transparency of the industry and the clear limitation of mean-variance
optimization tools. Nevertheless, diversification is still a good tool
against risk. In this paper, we offer a fresh look at hedge fund diversification
and go a step further than other studies. We find that naively adding
more funds to a portfolio tends to leave returns stable, decrease the
standard deviation, and reduce downside risk. Thus, diversification should
be increased as long as the marginal benefits of adding a new asset to
a portfolio exceeds the marginal cost. Of course, we do not advocate that diversification
should be implemented in such a naive approach. Hedge funds are complex
instruments, which are very different from what most investors are used
to. Investment decisions should therefore be more elaborate than a simple
coin-toss approach. As an illustration, naive diversification has important
- and sometimes dramatic - side effects on skewness, kurtosis, as well
as on correlation properties with respect to both traditional and alternative
indices. Hedge fund portfolio construction should therefore include more
research - not just into the performance of individual funds, but into
the effects of a given hedge fund allocation upon an entire portfolio
structure. Otherwise, if investors diversify naively, they may wrongly
believe that they hold diversified portfolios. In addition, naive diversification (the 1/N heuristics)
as implemented so far suffers from a well-known bias: allocation choices
are dependent on the set of hedge funds provided to the investors. For
example, given the popularity of macro strategies in the early 1990s and
the explosion of long/short strategies in the late 1990s, the naively
built portfolios would tend to hold more funds of these two styles during
their popular years. It is therefore imperative to carefully choose the
hedge funds that are included in portfolios to achieve adequate diversification.
In other words, the investor must seek out varied styles despite the limited
choice of the "flavor of the day". With that respect, the number of funds
in a portfolio is not the sole determinant of the degree of diversification
and randomly picking hedge funds is not the most intelligent way to diversify.
An investor could significantly reduce risk in his portfolio with fewer
hedge funds if he chooses them wisely across different stated investment
styles, so they do not move up and down in lockstep. In practice, this
approach can also be implemented naively by randomly selecting each additional
fund in an investment style not yet represented in the portfolio9.
We call this approach "skilled naive diversification". Since the stated
style is publicly available information, all but the most naive diversification
strategies should attempt to make some use of this information. More sophisticated
investors should also investigate the homogeneity or heterogeneity of
the sample from which the hedge fund managers are drawn. If a sample of
managers is relatively style pure, then a fewer number of managers will
minimize the unsystematic risk of that style. On the contrary, if the
sample is really heterogeneous, increasing the number of managers may
still provide important diversification benefits. Nevertheless, beyond the diversification strategy
selected, our results raise an interesting question: why are funds of
hedge funds so diversified? With a few exceptions, they seem to hold between
15 and 40 underlying funds, while our results indicate that 5 to 10 would
provide most of the diversification benefits, even, in certain cases,
avoiding some of the negative side effects. We will leave that question
open for further research. 9 If all investment styles are equally represented, the fund is selected randomly without any style consideration.
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