Abstract
In this summary paper we show how investors can
neutralise the unwanted skewness and kurtosis
effects from investing in hedge funds by (1) purchasing
outofthemoney equity puts, (2) investing in
managed futures, and/or by (3) overweighting equity
market neutral and global macro and avoiding distressed
securities and emerging market funds. We show
that all three alternatives are up to the job
but also come with their own specific price tag.
1. Introduction
Due to their relatively weak correlation with
other asset classes, hedge funds can play an important
role in risk reduction and yield enhancement strategies.
Recent research, however, has also shown that
this diversification service does not come for
free. Amin and Kat (2003), for example, show that
although the inclusion of hedge funds in a portfolio
may significantly improve that portfolio's meanvariance
characteristics, it can also be expected to lead
to significantly lower skewness and higher kurtosis.
This means that the case for hedge funds is not
as straightforward as is often suggested and includes
a definite tradeoff between profit and loss potential.
The sting of hedge funds is literally
in the tail as in terms of skewness, hedge funds
and equity do not mix very well. When things go
wrong in the stock market, they also tend to go
wrong for hedge funds as a significant drop in
stock prices is typically accompanied by a drop
in market liquidity, a widening of a multitude
of spreads, etc. Equity market neutral and long/short
funds have a tendency to be long in smaller stocks
and short in larger stocks and need liquidity
to maintain market neutrality. As a result, when
the stock market comes down this type of funds
can be expected to have a hard time. Likewise,
when the stock market comes down mergers and acquisitions
will be postponed which will have a negative impact
on the performance of risk arbitrage funds. Problems
are not limited to funds that invest in equity
though. A drop in stock prices will often also
lead to a widening of credit spreads, which in
turn will seriously damage the performance of
fixed income and convertible arbitrage funds.
As they all share it, diversification among different
funds will not mitigate this.
In this article we discuss a
number of ways to solve the above skewness problem
and the associated costs. We will look at the
use of outofthemoney stock index puts, managed
futures and sophisticated strategy selection.
Before we do so, however, we briefly discuss the
exact nature of hedge fund returns and the associated
skewness problem.
2. The effects of introducing
hedge funds in a portfolio
Generally speaking, risk is one word, but not
one number. The returns on portfolios of stocks
and bonds risk are more or less normally distributed.
Because normal distributions are fully described
by their mean and standard deviation, the risk
of such portfolios can be measured with one number:
the standard deviation. Confronted with nonnormal
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 socalled 'skewness', and the
probability of extreme positive or negative outcomes,
as measured by the distribution's 'kurtosis'.
A symmetrical distribution will have a skewness
equal to zero, while a distribution that implies
a relatively high probability of a large loss
(gain) is said to exhibit negative (positive)
skewness. A normal distribution has a kurtosis
of 3, while a kurtosis higher than 3 indicates
a relatively high probability of a large loss
or gain. Since most investors are in it for the
longer run, they strongly rely on compounding
effects. This means that negative skewness and
high kurtosis are extremely undesirable features
as one big loss may destroy years of careful compounding.
Table 1: Average Skewness
and Kurtosis Individual Hedge Fund Returns
Table 1 shows the average skewness
and kurtosis found in the returns of individual
hedge funds from various strategy groups. The
average hedge fund's returns tend to be nonnormally
distributed and may exhibit significant negative
skewness as well as substantial kurtosis. Put
another way, hedge fund returns may exhibit low
standard deviations but they also tend to provide
skewness and kurtosis attributes that are exactly
opposite to what investors desire. It is this
whole package that constitutes hedge fund risk,
not just the standard deviation. Actually, this
is not the whole story yet, as strictly speaking
we should also include the relationship between
the hedge fund return and the returns on other
assets and asset classes in the definition of
risk. We will look at this shortly.
The skewness and kurtosis properties
of hedge funds do not come as a complete surprise.
If we delve deeper into the return generating
process it becomes obvious that most spread trading
and pseudoarbitrage strategies will generate
these features by their very nature as the profit
potential of trades will typically be a lot smaller
that their loss potential. Consider a merger arbitrage
fund for example. When a takeover bid is announced
the share price of the target will jump towards
the bid. It is at this price that the fund will
buy the stock. When the takeover proceeds as planned
the fund will make a limited profit equal to the
difference between the relatively high price at
which it bought the stock and the bid price. When
the takeover fails, however, the stock price falls
back to its initial level, generating a loss that
may be many times bigger than the highest possible
profit. Spread traders are confronted with a similar
payoff profile. When the spread moves back to
its perceived equilibrium value they make a limited
profit, but when the market moves against them
they could be confronted with a much larger loss.
This is why strategies like this are sometimes
referred to as "picking up nickels in front
of a steamroller". Of course, there is no
reason why a trader could not get lucky and avoid
getting hit by the steamroller for a long period
of time. This does not mean that the risk was
never there, however. It always was. It just never
materialised so it does not show from the trader's
track record.
Since individual hedge funds
carry quite some idiosyncratic risk, combining
hedge funds into a basket, as is standard practice
nowadays, will substantially reduce the standard
deviation of the return on that portfolio. However,
it can also be expected to lower the skewness
and raise the correlation with the stock market.
Table 2: Individual Hedge
Fund and Hedge Fund Portfolio Risks
Table 2 shows the standard deviation,
skewness and correlation with the S&P 500
of the average individual hedge fund in the various
strategy groups as well as an equallyweighted
portfolio of all funds in each group. From the
table we see that forming portfolios indeed leads
to a very substantial reduction in standard deviation.
With the exception of emerging market funds, the
portfolio standard deviations are approximately
half the standard deviations of the average individual
fund. Apparently, there are many different ways
in which the same general strategy can be executed.
Contrary to standard deviation, skewness is not
diversified away and drops as portfolios are formed.
With the exception of equity market neutral funds,
the portfolio skewness figures are quite a bit
lower than for the average individual fund, with
especially merger arbitrage and distressed securities
funds standing out. Despite the lack of overall
correlation, it appears that when markets are
bad for one fund, they tend to be bad for other
funds as well. Finally, comparing the correlation
with the S&P 500 of individual funds and portfolios
we clearly see that the returns on portfolios
of hedge funds tend to be much more correlated
with the stock market than the returns on individual
funds. Although individual hedge funds may be
more or less market neutral, the portfolios of
hedge funds that most investors actually invest
in definitely are not.
Table 3: Effects of Combining
Hedge Funds with Stocks and Bonds
So far we have seen that hedge
fund returns tend to exhibit a number of undesirable
features, which cannot be diversified away. Skewness,
kurtosis and correlation with stocks worsen significantly
when portfolios are formed. But we are not there
yet, as we haven't looked at what happens when
hedge funds are combined with stocks and bonds.
Although the inclusion of hedge funds in a portfolio
may significantly improve that portfolio's meanvariance
characteristics, it can also be expected to lead
to significantly lower skewness as well as higher
kurtosis. Table 3 shows what happens to the standard
deviation, skewness and kurtosis of the portfolio
return distribution if, starting with 50% stocks
and 50% bonds, we introduce hedge funds (modelled
by the average equallyweighted random portfolio
of 20 funds) in a traditional stockbond portfolio.
As expected, when hedge funds are introduced the
standard deviation drops significantly. This represents
the still relatively low correlation of hedge
funds with stocks and bonds. This is the good
news. The bad news, however, is that a similar
drop is observed in the skewness of the portfolio
return. In addition, we also observe a rise in
kurtosis.
Especially the skewness effect
goes far beyond what one might expect given the
hedge fund skewness results in Table 2. When things
go wrong in the stock market, they also tend to
go wrong for hedge funds. Not necessarily because
of what happens to stock prices (after all, many
hedge funds do not invest in equity), but because
a significant drop in stock prices will often
be accompanied by a widening of credit spreads,
a significant drop in market liquidity, higher
volatility, etc. Since hedge funds are highly
sensitive to such factors, when the stock market
drops, hedge funds can be expected to show relatively
bad performance as well. Recent experience provides
a good example. Over the year 2002, the S&P
500 dropped by more than 20% with relatively high
volatility and substantially widening credit spreads.
Distressed debt funds, at the start of 2002 seen
by many investors as one of the most promising
sectors, suffered substantially from the widening
of credit spreads. Credit spreads also had a negative
impact on convertible arbitrage funds. Stock market
volatility worked in their favour, however. Managers
focusing on volatility trading generally fared
best, while managers actively taking credit exposure
did worst. Equity market neutral funds suffered
greatly from a lack of liquidity, while long/short
equity funds with low net exposure outperformed
managers that remained net long throughout the
year. As a result, overall hedge fund performance
in 2002 as measured by the main hedge fund indices
was more or less flat.
So here is the main problem.
Individual hedge fund returns tend to exhibit
some negative skewness. When combined into portfolios,
however, this negative skewness becomes worse.
When subsequently those portfolios are combined
with equity, skewness drops even further. The
increase in negative skewness will tend to offset
the lower standard deviation that results from
the inclusion of hedge funds. In other words,
when adding hedge funds, the investor's downside
risk will largely remain unchanged while at the
same time part of his upside potential is diversified
away. Exactly the opposite of what we want a good
diversifier to do.
In the next sections we discuss
three possible ways to reduce the above skewness
effect, as well as the associated costs, while
maintaining the benefits of the lower standard
deviation.
3. Purchasing OutoftheMoney
Puts
Since the increase in negative skewness that tends
to come with hedge fund investing is highly undesirable,
it is important to look for ways to neutralise
this effect. One solution is to buy hedge funds
in 'guaranteed' form only. In essence, this means
buying a put on one's hedge fund portfolio so
that in down markets the link between the hedge
fund portfolio and the stock market is severed.
Unfortunately, the market for put options on (baskets
of) hedge funds is still in an early stage. As
a result, counterparties for the required contracts
are likely to be hard to find as well as expensive.
With hedge funds so closely related to the ups
and especially downs of the stock market, there
is a very simple alternative though: the purchase
of outofthemoney puts on a stock index. As
discussed in Kat (2003), over the last 10 years
a strategy of buying and rolling over outofthemoney
S&P 500 puts would have generated returns
with very high positive skewness. It therefore
makes sense to use this option strategy to neutralise
the negative skewness in hedge funds.
Suppose we added stock index
put options to a portfolio of stocks, bonds and
hedge funds with the aim to bring the skewness
of the overall portfolio return back to what it
was before the addition of hedge funds. Obviously,
there is a price tag attached to doing so. Since
we are taking away something bad (negative skewness),
we will have to give up something good. If we
used leverage to keep overall portfolio volatility
at the same level as before the addition of the
puts, i.e. if we aimed to preserve the volatility
benefit of the addition of hedge funds, this means
we will have to accept a lower expected return.
Economically, this of course makes perfect sense
as the puts that we add won't come for free and,
since they are outofthemoney, are unlikely
to pay off (which of course is just another way
of saying that the option strategy by itself has
a highly negative expected return).
Table 4: Effects of Combining
Portfolios of Stocks, Bonds and
Hedge Funds with Puts and Leverage
Assuming investors can leverage
their portfolio at a rate of 4% and the expected
returns on stocks, bonds and hedge funds are equal
to their historical 10year means, Table 4 shows
the effect of using puts and leverage in a portfolio
of stocks, bonds and hedge funds (with always
equal allocations to stocks and bonds). Starting
with the situation shown in Table 3, adding puts
to bring the skewness of the overall portfolio
back to what it was before the addition of hedge
funds (0.33), while maintaining the volatility
benefit, requires only a small allocation to options.
As is also clear from the change in portfolio
kurtosis, this small allocation, however, goes
a long way in restoring the (near) normality
of the return distribution. Unfortunately, the
costs in terms of expected return (4th column)
are quite significant. For example, with a 25%
hedge fund allocation, investors can expect to
loose 61 basis points in expected return. This
drop in expected return can be interpreted as
the option market's price of the additional skewness
introduced by hedge funds.
Of course, the above conclusion
depends heavily on the assumption that investors
can leverage (either directly or through the futures
market) their portfolios at 4%, which in the current
interest rate environment does not seem unrealistic.
Obviously, if the interest rate were higher, the
costs of the skewness reduction strategy would
be higher as well because the difference between
the expected return on the unlevered portfolio
and the interest rate, i.e. the pickup in expected
return due to the leverage, would be smaller.
A similar reasoning applies in case of a lower
expected return on stocks, bonds and/or hedge
funds.
Another important element of
the above analysis concerns the assumption that
the allocations to stocks and bonds are always
equal. If we assumed that investors always divided
their money in such a way that 1/3 was invested
in stocks and 2/3 in bonds (as opposed to the
'50/50 portfolio' discussed earlier we will refer
to such a portfolio as a '33/66 portfolio'), our
results would of course change. Under the assumptions
made, a portfolio made up of 1/3 stocks and 2/3
bonds has a skewness of 0.03. With 25% hedge funds,
the portfolio's skewness will come to down to
0.43, while with 50% hedge funds it will drop
to 0.75. Because when hedge funds are introduced
skewness for a 33/66 portfolio drops faster than
for a 50/50 portfolio, we will have to buy more
puts and apply more leverage. Because the mean
of the 33/66 portfolio is substantially lower
than that of the 50/50 portfolio, however, the
increased leverage will not be sufficient to rescue
the expected return. As can be seen in the last
column of Table 4, the costs of the skewness reduction
strategy for a 33/66 portfolio are very substantial.
With 25% hedge funds, the costs of skewness reduction
will amount to 3.20%, as opposed to 'only' 0.61%
for the 50/50 portfolio.
In sum, after introducing hedge
funds, purchasing outofthe money puts can restore
the (near) normality of the portfolio return
distribution fairly easily. However, this may
come at a substantial cost to the portfolio's
expected return, especially for investors that
are overweighted in bonds.
4. Investing in Managed
Futures
In principle, any asset or asset class that has
suitable (co)skewness characteristics can be
used to hedge the additional skewness from incorporating
hedge funds. One obvious candidate is managed
futures. Managed futures programmes are often
trendfollowing in nature. What these programmes
do is somewhat similar to what option traders
do to hedge a short call position. When the market
moves up, they increase exposure and vice versa.
By moving out of the market when it comes down,
managed futures programmes avoid being pulled
in. As a result, the (co)skewness characteristics
of managed futures programmes are more or less
opposite to those of many hedge funds.
The term 'managed futures' refers
to professional money managers known as commodity
trading advisors or CTAs who manage assets using
the global futures and options markets as their
investment universe. Managed futures have been
available for investment since 1948 when the first
public futures fund started trading. The industry
did not take off until the late 1970s though.
Since then the sector has seen a fair amount of
growth with currently an estimated $50 billion
under management.
There are 3 ways in which investors
can get into managed futures. First, investors
can buy shares in a public commodity (or futures)
fund, in much the same way as they would invest
in a stock or bond mutual fund. Second, investors
can place funds privately with a commodity pool
operator (CPO) who pools investors' money and
employs one or more CTAs to manage the pooled
funds. Third, investors can retain one or more
CTAs directly to manage their money on an individual
basis or hire a manager of managers (MOM) to select
CTAs for them. The minimum investment required
by funds, pools and CTAs varies considerably,
with the direct CTA route open only to investors
that want to make a substantial investment. CTAs
charge management and incentive fees comparable
to those charged by hedge funds, i.e. 2% management
fee plus 20% incentive fee. Similar to funds of
hedge funds, funds and pools charge an additional
fee on top of that.
Initially, CTAs were limited
to trading commodity futures (which explains terms
such as public commodity fund, CTA and CPO). With
the introduction of futures on currencies, interest
rates, bonds and stock indices in the 1980s, however,
the trading spectrum widened substantially. Nowadays
many CTAs trade both commodity and financial futures.
Many take a very technical, systematic approach
to trading, but others opt for a more fundamental,
discretionary approach. Some concentrate on particular
futures markets, such as agricultural, currencies,
or metals, but most diversify over different types
of markets.
In this study, the asset class
'managed futures' is represented by the Stark
300 index. This assetweighted index is compiled
using the top 300 trading programmes from the
Daniel B. Stark & Co. database. All 300 of
the CTAs in the index are classified by their
trading approach and market category. Currently,
the index contains 248 systematic and 52 discretionary
traders, which split up in 169 diversified, 111
financial only, 9 financial & metals and 11
nonfinancial trading programmes.
As shown in Kat (2004a), historically
managed futures returns have exhibited a lower
mean and a higher standard deviation than hedge
fund returns. However, managed futures exhibit
positive instead of negative skewness and much
lower kurtosis. In addition, the correlation of
managed futures with stocks and hedge funds is
extremely low, which means that managed futures
make very good diversifiers. Table 5 shows the
effect of incorporating either hedge funds or
managed futures in a traditional 50/50 stockbond
portfolio.
Table 5: Return Statistics
Portfolios of Stocks, Bonds and
Either Hedge Funds or Managed Futures
From the table we, again, see
that if the hedge fund allocation increases, both
the standard deviation and the skewness of the
portfolio return drop substantially, while at
the same time the return distribution's kurtosis
increases. With managed futures the picture is
significantly different, however. If the managed
futures allocation increases, the standard deviation
drops faster than with hedge funds. More remarkably,
skewness rises instead of drops while the reverse
is true for kurtosis. Although (assuming future
performance will resemble the past) hedge funds
offer a somewhat higher expected return, from
an overall risk perspective managed futures appear
much better diversifiers than hedge funds.
Table 6: Allocations and
Annualised Change in Expected Return Portfolios
of Stocks, Bonds, Hedge Funds, and Managed Futures
Now suppose we did the same thing
as before: choose the managed futures allocation
such as to bring the skewness of the portfolio
return back to what it was before the addition
of hedge funds (0.33), while at the same time
preserving the volatility benefit of the addition
of hedge funds by the use of some leverage. The
results are shown in Table 6, which shows that
for smaller hedge fund allocations of up to 15%
the optimal managed futures allocation will be
more or less equal to the hedge fund allocation.
Looking at the change in expected return, we see
that as a result of the introduction of managed
futures the expected portfolio return increases
significantly. With a 25% hedge fund allocation
for example, the investor stands to gain 205 basis
points in annualised expected return. This of
course compares very favourably with the results
on outofthemoney puts. One should, however,
always keep in mind that the outcomes of analyses
like this heavily depend on the inputs used. A
lower expected return for managed futures and/or
a higher borrowing rate (used to leverage the
portfolio volatility back to its initial level)
could easily turn these gains into losses. In
addition, we have to keep in mind that although
the expected return does not seem to suffer from
the use of managed futures to neutralise the unwanted
skewness effect from hedge funds, this does not
mean it comes for free completely. Investors pay
by giving up the positive skewness that they would
have had when they had only invested in managed
futures.
5. Smart Strategy Selection
So far we have modelled the asset class 'hedge
funds' as a representative portfolio of 20 different
individual funds; a proxy for the average fund
of funds portfolio. Although this is what most
investors currently invest in, it is interesting
to investigate in how far it is possible to eliminate
the skewness effect of hedge funds by simply choosing
another hedge fund portfolio, i.e. by allocating
differently to the various hedge fund strategies
available. This is the approach taken in Davies,
Kat and Lu (2004). Using a sophisticated optimisation
technique known as Polynomial Goal Programming
(PGP), they incorporate investor preferences for
return distributions' higher moments into an explicit
optimisation model. This allows them to solve
for multiple competing hedge fund allocation objectives
within a meanvarianceskewnesskurtosis framework.
Apart from underlining the existence of significant
differences in the return behaviour of different
hedge fund strategies, the analysis shows that
PGP optimal portfolios for skewnessaware investors
contain hardly any allocations to long/short equity,
distressed securities and emerging markets funds.
Equity market neutral and global macro funds on
the other hand tend to receive very high allocations,
which is primarily due to their low covariance,
high coskewness and low cokurtosis properties.
Looking back at tables 1 and 2, these conclusions
do not come as a complete surprise. The strategies
that the optimiser tends to drop are exactly the
strategies that exhibit the most negative skewness.
Global macro and equity market neutral strategies
on the other hand come with much more desirable
risk characteristics. Global macro funds primarily
act as portfolio skewness enhancers, while equity
market neutral funds act as volatility and kurtosis
reducers (which are especially important given
the relatively high volatility and kurtosis of
global macro).
An interesting byproduct of
the analysis in Davies, Kat and Lu (2004) is that
introducing preferences for skewness and kurtosis
in the portfolio decisionmaking process yields
portfolios that are far different from the meanvariance
optimal portfolios, with less attractive meanvariance
characteristics. This underlines a point made
earlier in Kat (2004b) that using standard meanvariance
portfolio allocation tools when alternative investments
are involved can be highly misleading. It also
shows that in hedge fund diversification there
is no such thing as a free lunch. When substantially
overweighting global macro and equity market neutral
strategies, investors can expect more attractive
skewness and kurtosis, but at the cost of a less
attractive expected return and volatility.
Finally, it is interesting to
note that many global macro funds tend to follow
strategies that are similar to the strategies
typically employed by CTAs. In fact, some of the
largest global macro funds have their origins
in managed futures. The difference between expanding
into managed futures and overweighting global
macro is therefore probably smaller than one might
suspect.
6. Conclusion
Typical hedge fund portfolios' attractive meanvariance
properties tend to come at the cost of negative
skewness and increased kurtosis. We have shown
that investors can neutralise the unwanted skewness
and kurtosis by purchasing outofthemoney equity
puts, investing in managed futures, and/or by
overweighting equity market neutral and global
macro and avoiding distressed securities and emerging
market funds. Hedge fund returns are not superior
to the returns on other asset classes. They are
just different!
References
Amin,
G. and H. Kat (2003), Stocks, Bonds and Hedge
Funds: No Free Lunch!, Journal of Portfolio Management,
Summer, pp. 113120.
Davies, R, H. Kat
and S. Lu (2004), Fund of Hedge Funds Portfolio
Selection: A MultipleObjective Approach, AIRC
Working Paper No. 18 (downloadable from the AIRC
website: www.cass.city.ac.uk/airc).
Kat, H. (2003), Taking
the Sting Out of Hedge Funds, Journal of Wealth
Management, Winter, pp. 110.
Kat, H. (2004a),
Managed Futures and Hedge Funds: A Match Made
in Heaven, Journal of Investment Management, Vol.2,
No. 1, pp. 19.
Kat,
H. (2004b), Hedge Funds vs. Common Sense: An Illustration
of the Dangers of Mechanical Investment DecisionMaking,
Journal of Investment Management, Vol. 2, No.
4, pp. 95104.
