Replication
and Evaluation of Fund of Hedge Funds Returns

Harry M. Kat, Professor
of Risk Management and Director Alternative
Investment Research Centre, Cass Business
School, City University, London
Helder P. Palaro, PhD Student, Cass Business
School, City University, London

February 2006

1. Introduction

With the first hedge fund
said to be dating back to 1949, hedge funds
have been around for quite some time. Academic
research into hedge funds, however, only
took off towards the end of the 1990s when
sufficient data became available. Since
then, and inspired by the strong growth
of the hedge fund industry worldwide, a
respectable number of research papers and
articles have provided insight in many different
aspects of hedge funds. One question largely
remains unanswered though. Do hedge funds
provide their investors with superior returns?
In other words, do hedge funds provide their
investors with returns, which they could
not have obtained otherwise?

According to the hedge fund industry itself,
the answer to the above question is of course
affirmative, although with the recent disappointing
performance of hedge funds, this point is
put forward less often and less forcefully
than it used to. Nowadays, most emphasis
is on the diversification properties of
hedge funds. Various academic studies have
attempted to shed light on the issue of
hedge fund return superiority as well. Most
of these apply traditional performance measures,
such as the Sharpe ratio or factor model
based alphas, to hedge fund returns obtained
from one or more of the main hedge fund
databases. The conclusion is typically that
hedge fund returns are indeed superior.
From other studies, however, it is now well
understood that raw hedge fund return data
may suffer from various biases, which, when
not corrected for, will produce artificially
high Sharpe ratios and alphas. In addition,
hedge fund returns are typically not normally
distributed and may derive from exposure
to very unusual risk factors. This makes
traditional performance measures unsuitable
for hedge funds, as deviations from normality
as well as every risk factor that is incorrectly
specified or left out altogether, will tend
to show up as alpha, thereby suggesting
superior performance where there actually
may be none.

In theory, once the relevant risk factors
have been identified, factor model based
performance evaluation of hedge fund returns
should work well. In practice, however,
we don't know enough about hedge fund return
generation to be certain that all the relevant
risk factors are included and correctly
specified. As a result, factor models typically
explain only 25-30% of the variation in
individual hedge fund returns, which compares
very unfavourably with the 90-95% that is
typical for mutual funds.

Although the procedure
works better for portfolios of hedge funds,
funds of funds and hedge fund indices, where
most of the idiosyncratic risk is diversified
away, the low determination coefficients
of these models make it impossible to arrive
at a firm conclusion with respect to the
superiority of hedge fund returns.

It is quite surprising that so many people,
on the buy-side as well as in academia,
are so eager to believe that the, sometimes
huge, alphas reported for hedge funds are
truly there. Anyone who is well calibrated
to the world we live in and the global capital
markets in particular, knows how difficult
it is to consistently beat the market, ie,
systematically obtain a better return than
what would be fair given the risks taken.
Over time, hundreds, if not thousands, of
studies have confirmed this. Is it therefore
likely that suddenly we are facing a whole
new breed of super-managers; not one or
two, but literally thousands of them? Of
course not! And if anything, the rise of
the hedge fund industry has made markets
more efficient, not less.

Although by far the most popular, factor
models are not the only way to evaluate
hedge fund performance. Based on previous
work by Amin and Kat (2003), Kat and Palaro
(2005), or KP for short, recently developed
a technique that allows the derivation of
dynamic trading strategies, trading cash,
stocks, bonds, etc, which generate returns
with predefined statistical properties.
The technique is not only capable of replicating
(the statistical properties of) fund of
funds returns, but works equally well for
individual hedge fund returns. Since the
KP replicating strategies are explicitly
constructed to replicate the complete risk
and dependence profile of a fund, the average
return on these strategies can be used as
a performance measure. When the average
fund return is significantly higher than
the average return on the replication strategy,
the fund is the most efficient alternative
and vice versa.

The KP replication technique is similar
to that used in Amin and Kat (2003). The
important difference, however, is that the
latter only replicate the marginal distribution
of the fund return, while KP also replicate
its dependence structure with an investor's
existing portfolio. This is a very significant
step forward as most investors nowadays
are attracted to hedge funds because of
their relatively weak relationship with
traditional asset classes, ie, their diversification
potential. Only replicating the marginal
distribution without giving any consideration
to the dependence structure between the
fund and the investor's existing portfolio
would therefore be insufficient.

From a performance evaluation perspective,
replication of a fund's dependence pattern
with other asset classes is a necessity.
According to theory as well as casual empirical
observation, expected return and systematic
co-variance, co-skewness and co-kurtosis
are directly related. In other words, it
is not so much the marginal distribution,
but its dependence structure with other
assets that determines an asset's expected
return. An asset, which is highly correlated
with stocks and bonds, offers investors
very little in terms of diversification
potential. As a consequence, there will
be little demand for this asset. Its price
will be low and its expected return therefore
relatively high. On the other hand, an asset
that offers substantial diversification
potential will be in high demand. Its price
will be high and its expected return relatively
low. Although hedge funds are not priced
by market forces in the same way as primitive
assets are, they do operate in the latter
markets. It therefore seems plausible that
a similar phenomenon is present in hedge
fund returns as well .

2. The KP Efficiency
Measure

Applying the KP replication
technique to hedge funds, the goal is to
create a dynamic trading strategy, which
generates returns with the same statistical
properties as a given hedge fund or fund
of funds, ie, returns that are drawings
from the same distribution as the distribution
from which the actual fund returns are drawn.
The basic idea behind the procedure is straightforward.
From the theory of dynamic trading it is
well known that in the standard theoretical
model with complete markets any payoff function
can be hedged perfectly. This observation
forms the foundation of arbitrage-based
option pricing theory. If it is possible
to find a payoff function which, given the
distribution of the underlying assets, implies
the same distribution as the one from which
the fund returns are drawn, then the accompanying
dynamic trading strategy will generate (returns
that are drawings from) that distribution.

Given the KP replication
technique and following the same reasoning
as in Amin and Kat (2003), we derived the
following evaluation procedure, which consists
of five distinct steps.

Monthly return data
are collected on the fund to be evaluated,
the representative investor's portfolio,
and a so-called reserve asset. The latter
is the main source of uncertainty in the
replication strategy. As we want to know
whether the returns that investors obtain
from hedge funds are superior, fund returns
should be net of all fees.

From the available
return data, the bivariate distribution
of the fund return and the representative
investor's portfolio return is inferred
(KP refer to this as the 'desired distribution').
The same is done for the bivariate distribution
of the investor's portfolio return and
the return on the reserve asset (the 'building
block distribution'). In line with KP,
we allow for 54 different joint distributions,
choosing between them using the Akaike
Information Criterion (AIC) .

Assuming an initial
investment in the fund of 100, we determine
the cheapest payoff function, which is
able to turn the building block distribution
into the desired distribution. This payoff
function is known as the 'desired payoff
function' and lies at the basis of the
KP replication strategies.

The desired payoff
function is priced using the multivariate
option pricing model of Boyle and Lin
(1997), which explicitly allows for transaction
costs. For the pricing of the payoff function,
we estimate the required volatility and
correlation inputs over the period covered
by the track record of the fund being
evaluated. We use the average 1-month
interest rate over the same period for
the interest rate input. We will refer
to the price thus obtained as 'the KP
efficiency measure'.

Finally, we compare
the KP efficiency measure with the 100
initially invested in the fund. If the
efficiency measure is 100 as well, then
the replication strategy and the fund
are equivalent. If the efficiency measure
is less (more) than 100, the strategy
is cheaper (more expensive) than the fund
and the fund therefore inefficient (efficient).

All performance evaluation
studies in finance follow the same general
procedure. First, using a fund's track record
and possibly some additional data over the
same period as well, the fund return is characterised
in some way. With the Sharpe ratio this is
done by calculating the volatility of the
fund return. With alphas this is done by estimating
a fund's exposure to the relevant risk factors.
Second, based on this characterisation, a
benchmark return is determined and compared
with the actual average fund return over its
track record. With the Sharpe ratio the benchmark
return is derived from the average index return
and the volatility of the index, while with
alphas it derives from the average returns
of the risk factors.

Our procedure is not different. We just use
a different characterisation. Where others
use volatility or factor loadings, we use
the desired payoff function. Where others
use the average return on the index or the
chosen risk factors, we use the average interest
rate, building block volatilities and correlation
over a fund's track record to set a benchmark.
What is different, however, is that we do
not need to make any unrealistically strong
assumptions concerning the exact nature of
a fund's risk exposure or the behaviour of
markets in general. As shown by KP, a fairly
limited set of returns will often be enough
to obtain a sufficiently good estimate of
the desired distribution and the efficiency
measure. As such, our procedure is quite robust.

Another point worth noting about the above
evaluation procedure is the fact that it explicitly
takes transaction costs into account by, instead
of a Black-Scholes type option pricing model,
using the Boyle and Lin (1997) model. In factor
model based evaluations, transaction costs
are typically ignored, despite the fact that
maintaining the replicating portfolio's factor
loadings at their desired levels is likely
to require periodic rebalancing. In addition,
when dealing with hedge funds the risk factors
used may be quite unusual and may therefore
be accompanied by significant levels of transaction
costs.

In the evaluations, we
do not use hedge funds' raw returns. The
reason is that, as shown in Brooks and Kat
(2002) and Lo et al. (2004) for example,
monthly hedge fund returns may exhibit high
levels of autocorrelation. This primarily
results from the fact that many hedge funds
invest in illiquid securities, which are
hard to mark to market. When confronted
with this problem, hedge fund administrators
will either use the last reported transaction
price or a conservative estimate of the
current market price. This creates artificial
lags in the evolution of hedge funds' net
asset values, ie, artificial smoothing of
the reported returns. As a result, estimates
of volatility, for example, will be biased
downwards.

One possible method to correct for this
bias is found in the real estate finance
literature. 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. The approach employed
in this literature has been to "unsmooth"
the observed returns to create a new set
of returns which are more volatile and whose
characteristics are believed to more accurately
capture the characteristics of the underlying
property values. Nowadays, there are several
unsmoothing methodologies available. In
this study we use the method originally
proposed by Geltner (1991).

3. An Example

To clarify the above, let's
look at a worked-out example. XYZ is a well-known
fund of hedge funds, which started in 1985.
Given XYZ's monthly, net-of-fee returns
since 1985, the first step is to model the
joint distribution of XYZ and the investor's
portfolio, as well as the joint distribution
of the investor's portfolio and the reserve
asset. Before we can do so we need to decide
what exactly the investor's portfolio and
the reserve asset are, as well as unsmooth
the raw fund return data.

Let's assume that the representative investor's
portfolio consists of 50% S&P 500 and
50% long-dated US Treasury bonds. Let's
also assume that all exposure management
is done in the futures markets. So instead
of investing in the cash market, we will
hold fully collateralised (nearby) futures
contracts. We use nearby Eurodollar futures
as the reserve asset. Futures have several
advantages over cash, in particular high
liquidity and low transaction costs, which
is extremely important given the dynamic
nature of the KP replication strategies.

Table 1: Risk Statistics
XYZ

Standard
Deviation

Skewness

Excess
Kurtosis

1M
Auto
Correlation

XYZ
smooth

0.0370

-1.726

11.505

0.138

XYZ
unsmooth

0.0424

-1.746

11.581

0.008

Table 1 shows the marginal
risk characteristics of the raw and unsmoothed
XYZ returns. From the table, we see that
XYZ's raw returns exhibit negative skewness
and positive autocorrelation. Application
of the unsmoothing procedure eliminates
the autocorrelation and produces returns
with the same degree of skewness, but with
a substantially higher volatility (annualised
14.7% vs 12.8% for the raw returns).

We are now ready to infer the desired and
the building block distribution. Using the
same methodology as KP, we find that the
best fit (according to the AIC) is provided
by the following set of marginals and copulas
:

XYZ: Student-t (µ = 0.0101,
s = 0.0406, df = 4.0544) Portfolio: Normal (µ = 0.0101,
s = 0.0282) Reserve: Johnson (? = 0.0031, ? =
0.0046, ? = -0.60, d = 1.599) Copula (XYZ, portfolio): Normal (?
= 0.754) Copula (portfolio, reserve): Gumbel
(a = 1.3349)

Given the above distributions, we can derive
the desired payoff function following the
methodology developed in KP. The result
is depicted in Figure 1 and shows that the
desired payoff is an increasing function
of both the investor's portfolio and the
reserve asset, implying that the replication
strategy will take long positions in both
assets. Subsequently, we price this payoff
function using the Boyle and Lin (1997)
model, assuming transaction costs in the
futures markets are 1bp one-way. This produces
a value for the KP efficiency measure of
99.53, meaning that, seen over the whole
life of the fund, XYZ's returns are not
as miraculous as many investors may have
thought. Trading S&P 500, T-bond and
Eurodollar futures, investors could have
generated the same risk profile as XYZ and
obtained a higher average return at the
same time.

Figure
1: Desired Payoff Function for Replication
XYZ Returns

Figure 2: Scatter Plot Investor's Portfolio
Returns vs XYZ Returns (left) and
Replicated Returns (right)

To see how well the derived
payoff function succeeds in replicating
the desired distribution, Figure 2 shows
a scatter plot of the investor's portfolio
return versus the XYZ return (left) as well
as a plot of the portfolio return versus
the replicated return (right). The two plots
are very similar, suggesting that the replication
has indeed been successful. We see that
the replication strategy is unable to replicate
the three large losses that XYZ reported
during the sample period. This is not surprising
as these are clearly outliers, which simply
cannot be captured by a parametric model
like ours.

A further indication of the accuracy of
the replication strategy comes from comparing
the mean, standard deviation, skewness and
kurtosis of XYZ's returns with those of
the replicated returns. The latter statistics
can be found in Table 2, together with the
correlation and Kendall's Tau with the investor's
portfolio. Since the XYZ returns exhibit
a few negative outliers, apart from the
standard skewness and kurtosis measures,
we also report a more robust skewness and
kurtosis measure . To test whether the marginal
distribution of the replicated returns and
the joint distribution of the replicated
returns and the investor's portfolio are
significantly different from the original
distributions, we use the univariate and
bivariate Kolmogorov-Smirnov (K-S) tests
.

Comparing the entries in
Table 2, it is clear that the statistical
properties of XYZ's returns have been quite
successfully replicated. The replication
strategy has not only replicated the marginal
distribution of XYZ's returns but also its
relationship with the investor's portfolio.
The same conclusion follows from both the
K-S tests.

4. Distributional Analysis

A crucial stage in the
evaluation procedure is the proper modelling
of the distributional characteristics of
the fund, the investor's portfolio and the
reserve asset. This means that, although
not explicitly designed to do so, the evaluations
provide a wealth of information on the distributional
properties of fund of funds returns. Table
3 summarises how often (out of a total of
485 funds) a given marginal or copula was
used in the evaluations for modelling the
fund return marginal and the joint distribution
of the fund and the investor's portfolio
return.

Table
3: Distributional Characteristics Fund of
Funds Returns

Marginals

No.

Copulas

No.

Normal

145

Normal

88

Student-t

257

Student

49

Johnson

83

Gumbel

36

SJC

40

Cook-Johnson

128

Frank

144

Table 3 confirms that,
despite an often substantial degree of diversification
in the larger funds, the majority of fund
of funds returns are far from normally distributed.
Out of 485 funds, 340 funds' marginal return
is better modelled by a Student-t or Johnson
distribution than a normal distribution.
In addition, for only 88 of the 485 funds
is the relationship with the investor's
portfolio (consisting of 50% S&P 500
and 50% T-bonds) best modelled by the normal
copula. This emphasises once more how important
it is to evaluate hedge fund and fund of
funds performance using a method, which
does not rely on the assumption of normally
distributed returns.

5. Evaluation Results

Having introduced the evaluation
procedure, we now present the evaluation
results. Our total sample consists of 485
funds of funds with a minimum of four years
of history available. All data were obtained
from TASS as of November 2004. Note that
this implies that the again disappointing
results of 2005 were not taken into account
in the evaluation . Funds denominated in
another currency than USD were converted
to USD, ie, the perspective taken is that
of a USD-based investor. Table 4 provides
some information on the starting and end
dates of the track records of the funds
in our sample.

Table
4: Fund of Funds Starting Date and End Date
Details

Jan
1985

Jan
1988

Jan
1991

Jan
1994

Jan
1997

Jan
2000

Jan
2003

Oct
2004

Start
after

479

463

419

329

201

43

0

0

End
before

0

0

0

0

12

53

133

208

Table 4 shows that, reflecting
the increasing popularity of hedge funds
in the second half of the 1990s, the majority
of funds started after 1994. Most hedge
fund databases, including TASS, first started
collecting data around 1994. As a result,
our sample contains no funds that stopped
reporting before that date. We also see
that out of 485, 208 funds stopped reporting
before October 2004. This confirms that,
although lower than for individual hedge
funds, the attrition rate in funds of funds
is still quite high.

Table
5: Length Fund of Funds Track Records

4-5Y

5-6Y

6-7Y

7-8Y

8-9Y

9-10Y

10-12Y

12-14Y

14+

No.
funds

89

81

62

52

43

40

53

31

34

Table 5 provides details
on the length of the available fund of funds
track records. Out of the 485 funds in the
sample, only 118 have more than ten years
of history. This again reflects the fact
that most funds of funds are still relatively
young and attrition can be significant.

As in the example in section 3, in the evaluations
we assume that the representative investor's
portfolio consists of 50% S&P 500 and
50% long-dated US Treasury bonds, with all
exposure management done through fully collateralised
(nearby) futures contracts . Since it is
one of the most traded futures contracts
in the world, we use nearby Eurodollar futures
(trading on the CME) as the reserve asset.
Transaction costs on all futures contracts
are assumed to be 1bp one-way. For the pricing
of the payoff functions, we use 1-month
USD Libor as the relevant interest rate,
while estimating the required volatilities
and correlations over the period covered
by the track record of the fund that is
being evaluated. The interest rate data
was obtained from Datastream, while the
futures data was obtained from Commodity
Systems Inc (CSI).

Figure
3: Scatter Plot Fund vs Replicated Standard
Deviation

Figure
4: Scatter Plot Fund vs Replicated Skewness

Figure
5: Scatter Plot Fund vs Replicated Correlation
with Investor's Portfolio

To get an idea of the typical
accuracy of the replication procedure, Figure
3-5 show scatter plots of the fund standard
deviation (Fig. 3), standard skewness (Fig.
4) and correlation with the investor's portfolio
(Fig. 5) versus the replicated values for
all 485 funds. As is clear from these graphs,
on average the replication of these parameters
is unbiased and quite accurate. Not surprisingly,
the replication of skewness can be difficult
at times as fund returns may contain one
or more outliers, which will have a major
impact on the standard skewness statistic,
but which cannot be replicated. We encountered
this problem before in the example in section
3.

Figures 3-5 also provide additional information
on the risk-return profile of the funds
in our sample. From Figure 4 for example,
we see that for most funds of funds estimated
skewness lies somewhere between -1 and +1.
Likewise, from Figure 5 we see that the
majority of funds are positively correlated
a portfolio of 50% stocks and 50% bonds.
Most correlation coefficients lie between
0 and 0.6, indicating that many funds of
funds' returns are a lot less 'market neutral'
than the term 'absolute returns'suggests
.

Figure
6: Histogram KP Efficiency Measure 485 Funds
of Funds

Figure 6 shows a histogram
of the values of the KP efficiency measure
obtained for the 485 funds of funds in our
sample. From the graph we see that the majority
of funds produce a value for the efficiency
measure that is below 100. The average value
for the KP efficiency measure over all 485
funds is 99.36. We tested the statistical
significance of the above efficiency measure
results by calculating bootstrapped confidence
intervals. We distinguished between three
cases and obtained the following results:

The confidence interval
is entirely lower than 100 - 389 funds

The confidence interval
contains 100 - 41 funds

The confidence interval
is entirely higher than 100 - 55 funds

This confirms the results
in the histogram in Figure 6. The majority
of funds of hedge funds have not provided
their investors with returns, which they
could not have generated themselves in the
futures market.

We sorted the funds in our sample in various
ways depending on starting date and/or end
date to see if there was any indication
of old funds doing better than new funds,
live funds doing better than funds that
closed down, etc. However, none of these
subdivisions produced significant results.

Figure
7: Percentage of Funds that Stopped Reporting
Before October 2004
as Function KP Efficiency Measure

The histogram in Figure
6 is negatively skewed, implying that some
of the funds in our sample have shown extremely
bad performance relative to what could have
been achieved trading S&P 500, T-bond
and Eurodollar futures. Since lack of performance
is one of the main reasons for funds to
close down, Figure 7 shows the percentage
of funds that stopped reporting to the database
before October 2004, as a function of their
KP measure. From the graph we see that there
is a strong relationship. Out of the 93
funds with a KP measure below 99, no less
than 71 (76%) stopped reporting. Out of
the 70 funds with a KP measure higher than
100, only 16 (23%) did so. A similar relationship
is observed in the average KP measures of
live and dead funds. The average KP measure
over the 208 dead funds is 98.97, while
over the 277 funds still alive the average
is 99.65.

6. Conclusion

In this paper we have used
the hedge fund return replication technique
recently introduced in Kat and Palaro (2005)
to evaluate the net-of-fee performance of
485 funds of hedge funds. The results indicate
that the majority of funds of funds have
not provided their investors with returns,
which they could not have generated themselves
by trading S&P 500, T-bond and Eurodollar
futures. Purely in terms of returns therefore,
most funds of funds have failed to add value.

Compared with the various hedge fund performance
evaluation studies that have been carried
out over the last couple of years, our results
are quite unusual. Often, the conclusion
from hedge fund performance studies is that
funds of funds generate superior returns,
not inferior. This emphasises how tricky
factor model based performance evaluation
really is. As long as one can't be sure
that all relevant risk factors are accounted
for, it is impossible to know whether unexplained
returns are indeed true alpha or just unexplained
because one or more risk factors were left
out or specified incorrectly. Our methodology
is a lot more robust, as it relies on only
one simple principle: "if it can be
replicated, it can't be superior".
Of course, we need to make assumptions as
well, but these are a lot less crucial for
the final outcome of the evaluation than
the kind of assumptions required to make
factor model based alphas work.

Should investors rush out to buy into the
funds with the highest KP measures? Although
tempting, the answer is no. By definition,
extreme events only occur infrequently.
With more than 75% of the funds in our sample
having a track record of less than 120 months,
it is therefore hard to identify the presence
of any catastrophic (but compensated) risks
from the available data. A fund may be taking
the most terrible risks, but if so far it
has been lucky, the premium collected for
taking on those risks will show from the
fund's track record, but the risk itself
won't. A high KP measure should therefore
first and foremost be interpreted as an
indication that further due diligence is
required. One can only speak of truly superior
performance if such follow-up research shows
that the manager in question has generated
the observed excess return without taking
any extreme risks.

Finally, it has to be noted
that although in terms of the returns delivered
to investors, funds of funds do not seem
to add value, this does not mean there is
no economic reason for funds of funds to
exist. Most private and smaller institutional
investors do not have the skills and/or
resources required to perform the necessary
due diligence that comes with hedge fund
investment. In addition, given typical minimum
investment requirements, small private investors
will often lack sufficient funds to build
up a well-diversified hedge fund portfolio.
They therefore have no choice. If they want
hedge funds, they will have to go through
a fund of funds.

Large institutions do have a choice. Most
of them, however, still prefer to go the
fund of funds route. This is quite surprising
given the amount of fees that could be saved
by skipping the middlemen. Apart from believing
that fund of funds managers add enough value
to justify their fees (which research has
shown to be unlikely), part of the reason
that many large institutions go for funds
of funds lies in the fact that the interests
of institutional asset managers are typically
not correctly lined up with the interests
of those whose money they manage. As a result,
job protection becomes an important consideration.
By investing in a fund of funds, instead
of picking hedge funds themselves, institutions
avoid having to take responsibility for
the bottom-line fund selection. In the end,
all they can be held responsible for is
the decision to invest in hedge funds and
the selection of the fund of funds that
they invested in; risks, which can easily
be hedged by not making a move until others
do, hiring a big name consultant and a big
name fund manager, as most institutions
do.

The authors like to
thank CSI and Rudi Cabral for allowing them
to access the CSI futures database.

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from www.cass.city.ac.uk/airc).

Kat, H. and H. Palaro (2005). Who Needs
Hedge Funds? A Copula-Based Approach to
Hedge Fund Return Replication, Working paper,
Alternative Investment Research Centre,
Cass Business School, City University London,
(downloadable from www.cass.city.ac.uk/airc).

Lo, A., M. Getmansky and I. Makarov (2004).
An Econometric Analysis of Serial Correlation
and Illiquidity in Hedge Fund Returns. Journal
of Financial Economics, Vol. 74, pp. 529-609.

Footnotes

^{1}
This is confirmed by the results in Kat
and Miffre (2005).

^{2} See Akaike (1973) for details.

^{3} Distributions and copulas as
defined in Kat and Palaro (2005).

^{4 }See Hinkley (1975) and Crow
and Siddiqui (1967) for details.

^{5} See Fasano and Franceschini
(1987) for details.

^{6 }The CISDM Fund of Funds Index
recorded a return of 4.72% over the first
11 months of 2005. Over the same period,
the Barclay/Global HedgeSource Fund of Funds
Index registered 4.81%.

^{7} More in particular, we traded
S&P 500 futures on the CME and T-bond
futures on the CBOT. Both contracts are
in the top 10 of most traded futures contracts
in the US.

^{8} In this context it is important
to note that at least for some of the more
complex distributions encountered (see Table
3), the correlation coefficient will not
be a particularly good measure of dependence
and may underestimate the true level of
dependence.

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