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Time-Varying Coefficient Model for Hedge Funds∗

Gilles CRITON†and Olivier SCAILLET‡

Second version, September 2009

Abstract

We propose a time-varying coefficient model in order to analyze the dynamic in estimated

alpha and betas. We showed that the proportion of “skilled”funds are higher with our model

than with a static linear factor model. Indeed, our time-varying coefficient model captures the

dynamic part of alpha that reflects the dynamic strategy that we can find within Hedge Funds.

Furthermore this model is not only capable of looking into anticipation for Hedge Fund managers

but is equally well suited for the analysis of beta exposure. We show that whatever the strategy,

the increase in risk behavior is mainly concentrated on the credit spread risk factor and bond

risk factor.

∗We thank Jerome Teiletche as well as seminar participants at Pictet, Unigestion, SoFiE (2009).†

‡

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1 Introduction

Nowadays, it is well known that investing in Mutual Funds, on average, underperform passive

investment strategies. One class commonly called Hedge Funds are defined as pooled invest-

ment vehicles that are privately organized, and not widely available to the general investing

public. Hedge Funds are private partnerships that use advanced investment strategies and can

use derivatives, leverage and short selling. Over the last few years, Hedge Funds have become

the favorites of many private as well as institutional investors. Hedge Funds follow investment

strategies that are substantially different from the non-leveraged, long-only strategies conven-

tionally followed by investors. During alternative investment seminars and conferences, Hedge

Fund managers boast about their ability to produce something they refer to as “alpha”or “ab-

solute return”in the sense that performance is not due to primary asset classes performance.

They do not aim to track and try to beat a certain benchmark, but instead are focused on

pure return generation. Hence Hedge Funds generate alpha, as opposed to depending on beta

and performance should normally result from active management decisions combined with the

skills of their advisors. However statistical analysis shows that many of them retain significant

exposure to different types of market risk factors.

Consequently, it is essential for qualified investors to determine whether or not these strategies

are sensitive to the market and whether they can find alpha through the manager’s skill. For

these reasons, an increasing focus has been directed to the performance of Hedge Funds and

their factor exposures.

Mostly, owing to the theory of CAPM or APT, fund performances are assessed by a parametric

model with the hypothesis of normality and linearity of coefficients, as well as, the non-dynamic

behavior of beta. Some researchers expanded the parametric model to the world of Hedge

Funds. Fung and Hsieh (2001, 2004a) developed factors used to replicate trend-following

strategies. Agarwal and Naik (2000) suggested an approach using option-based returns in

order to capture the non-linearity in beta. Recently, Bollen and Whaley (2008) tested for

structural change and used two econometric models in order to capture the dynamic in beta.

In this paper, we attempt to go a step further than the previous research.

First, by developing the result from Bollen and Whaley (2008) by testing up to five multiple

structural changes. We showed that the relative number of breaks by fund has increased over

the last few years, resulting in the increase of the use of leverage.

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Secondly, by developing a new model1 for Hedge Funds which can take into consideration their

distinctly non-normal characteristics, the dynamic in manager skill, as well as, the dynamic

trading represented respectively by alpha and beta. This model allows us to relax traditional

parametric models and to exploit possible hidden structure. For example, what manager skillis the most important? Is it the skill to pick and choose the right stocks, bonds, any other

financial products, or is it the skill to anticipate market events? If we consider that alpha

estimated from a linear model contains the two skills, how can we separate and study them?

Can we find a different proportion of positive-alpha funds? This model specifically allows us to

look into how managers behave in relation to strong events and whether or not they succeed

to generate alpha. Moreover, Hedge Fund managers are likely to change trading strategies

in order to obtain “absolute return”2. Therefore, the exposure to a risk factor can change

during special events and can have some deep implications for a Fund of Funds or a portfolio.

Often it has been merely stated that during an event, risk to a specific factor can increase or

decrease. Our study also portrays how the beta exposure or risk behaves in relation to market

reaction, as well as, alpha during two agitated periods; the equity bubble crisis (the Equity

Bubble hereafter)3 and LTCM crisis (LTCM hereafter) 4.

Thirdly, with the methodology used for mutual Funds by Barras, Scaillet and Wermers (2008)

called False Discovery Rate, we look into the proportion of the fund’s population which has

an increase in different market exposures, as well as, the proportion of skilled, unskilled and

zero-alpha funds.

We showed that the proportion of “skilled”funds are higher with our model than with a static

linear factor model. We explained such a difference by our model’s ability to capture the

dynamic part of alpha that reflects Hedge Fund managers’ market timing ability. We found

that the quantity of new alpha substantially increased the proportion of positive and negative

alpha funds. Nevertheless, we showed that the majority of Hedge Funds are zero-alpha funds

as Barras, Scaillet and Wermers (2008) portrayed for Mutual Funds. Furthermore, we look

into the possibility for a strategy to have a common increasing trend to a market exposure

during a crisis. For all strategies, the main result was a common trend to an increase in the

credit spread risk factor.

1This time-varying coefficient model has been fully explored by the seminal work of Cleveland, Grosse and Shyu(1991).

2positive returns in all market conditions, up or down.3February, March 20004July, Auguste, and September 1998

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To the best of our knowledge, this is the first paper which tries to look into how the risk of

Hedge Funds and CTAs exposure changes. This paper also looks into the skill of manager

anticipation, part of alpha.

The rest of our paper is organized as follows; Section 2 reviews related literature about Hedge

Funds modeling, and the dynamic in beta5. The data is described in section 3. Section 4

summarizes the risk factors defined in Fung and Hsieh’s paper (2001, 2004). Section 5 tests

for structural change by applying the method of Bai and Perron (1998). Section 6 defines

our time-varying coefficient model. Our methodology in which we apply our model is defined

in section 7. Section 8 provides results of our time-varying coefficient model, as well as, the

application of the False discovery Rate (FDR hereafter) to alpha and beta bringing, in this

way, a new tool for Hedge Fund analysis. Section 9 presents our conclusion.

2 Literature review

It is well-known that there are some drawbacks to model Hedge Funds. Various fundamental

and statistical multifactor models for Hedge Funds have been analyzed by Agarwal and Naik

(2000), Edwards and Caglayan (2001), Fung and Hsieh (1997, 2001, 2004a), Lhabitant (2001),

Liang (2001), Schneeweis and Spurgin (1998), among others.

Research has specifically been focused on identifying dynamic trading strategies in order to

mimic the actual trades of Hedge Funds (for example: Fung and Hsieh (1997), Agarwal and

Naik (2000)). Contrary to mutual funds, performance analyzes of Hedge Funds are completely

different. One reason for this, concerns the returns of Hedge Funds which usually have low

correlations with market indices, and therefore a traditional CAPM analysis using the Jensen

measure is typically non-appropriate. Another reason concerns the linear regression which

works well for mutual funds because its strategies tend to follow a buy-and-hold pattern.

Brown, Goetzmann and Ibbotson (1999) have shown that Hedge Funds pursue many different

styles which are completely different to the buy and hold strategy followed by Mutual Funds

and therefore can put the hypothesis of linearity to question. Liang (2001) also documents

that Hedge Fund investment strategies are dramatically different from those of Mutual Funds.

To the best of our knowledge, the first paper which suggested another approach in order to bet-

ter analyze the returns of Hedge Funds was written by Fung and Hsieh (1997). They attempted

5the readers can find a detailed literature review into the book from Agarwal and Naik (2005).

4



to analyze the returns to Hedge Funds by applying the factor or style analysis conducted by

William Sharpe with respect to Mutual funds. As we will be using their methodology in this

paper we will introduce and develop it further later on in our work. In their paper, they found

that the amount of variation in Hedge Fund returns that is explained by asset class return islow 6. Schneeweis and Spurgin (1998) confirm this result by conducting a regression analysis

of the returns of stocks, bonds, commodities, and currency returns. They found that R-square

measures that range from near zero for relative value Hedge Funds to 0.67 for Hedge Funds

that pursue primarily a long equity investment strategy, which concludes that Hedge Fund

return patterns do not map as well to the financial asset as do mutual fund returns. In order

to ameliorate the explanation of Hedge Fund returns by asset class, Fung and Hsieh (1997)

applied a factor analysis based on a Hedge Fund’s trading style. They found that five dif-

ferent trading style (system/opportunistic, global/macro, value, systems/trend following and

distressed) explain about 45 percent of the cross-sectional variation in Hedge Fund returns.

Following Fung and Hsieh (1997), a lot of articles have been developed which seek to under-

stand the trading strategies and characteristics of Hedge Funds by regressing their returns

on explanatory factors (Agarwal and Naik (2000), Brown and al. (2001), Mitchell and Pul-

vino (2001)). Agarwal and Naik (2000) extended this analysis of Hedge Fund performance by

recognizing that funds may follow dynamic non-linear trading strategies. They used stepwise

regression to identify the independent variables, and found that a put or a call option on

an underlying variable is the most significant factor in the case of 54 percent of their funds.

Also, Fung and Hsieh (2002) incorporated option strategies into a Sharpe style model, but

they focused on exploring the risk of fixed income Hedge Fund styles and did not consider

performance explicitly. In contrary to Agarwal and Naik’s (2000) discoverers, Fung and Hsieh

found that in most cases their option strategies only played a marginal role. One reason

given by the authors for the varied results mentioned above is due to the fact that Fung and

Hsieh (2002) used active and advanced straddle strategies. The paper written by Mitchell and

Pulvino (2001) on merger-arbitrage strategies are also begun to produce useful explicit links

between Hedge-Fund strategies and observable asset returns. Mitchell and Pulvino (2001) sim-

ulated the returns of a merger arbitrage strategy applied to announced takeover transactions

from 1968 until 1998. In summary, Fung and Hsieh (1999,2000,2001), Mitchell and Pulvino

(2001) and Agarwal and Naik (2004) show that Hedge Fund returns relate to conventional6R-Square measures were less than twenty five percent for almost half of the Hedge Fund studied

5



asset class returns and option-based strategy returns. They relate another strong difference

between Mutual Funds and Hedge Funds which concerns the problem of managers changing

their investment style over time. This problem is less acute for Mutual Funds than for Hedge-

Funds. Brealey and Kaplanis (2001) presented evidence that within each category funds tendto make similar changes to their factor exposures. Their interest is in the broader issue of how

well data on Fund returns can be used to measure “changes”in factor exposure. Indeed they

found that their CUSUMs’ tests of the stability of factor loadings rejected the null hypothesis

of stable coefficients at the 5 percent level in the case of three quarters of the funds. In a

similar way, Fung et al.(2006) paper was concerned with estimating factor exposures at the

time of particular crises. They study vendor-provided fund-of-fund indices, and performed

a modified-CUSUM test to find structural break points in fund factor loadings. They found

that the break points coincide with extreme market events7. Recently, the paper from Bollen

and Whaley8 studied 2 econometric techniques that accommodate changes in risk exposure.

The primer methodology was developed by Andrews, Lee, and Ploberger (1996). They stud-

ied a class of optimal tests9 for multiple changes. The latter, uses maximum likelihood and

a Kalman filter under an AR(1) model. They found significant changes in the risk factor

parameters in about 40 percent of their sample of Hedge Funds.

Nevertheless the 2 precedent methodologies use a hypothesis of normality which is not adapted

to the world of Hedge Funds. With strong evidence of non normality (Agarwal and Naik, 2001;

Amin and Kat, 2003; Fung and Hsieh, 1999; Lo, 2001), the mean and standard deviations are

not sufficient to describe the return distribution, and higher moments need to be considered.

Kat and Lu (2002), Brooks and Kat (2002) show that although Hedge Funds offer high mean

returns and low standard deviations, the returns also exhibit third and fourth moment at-

tributes10 as well as positive first-order serial correlation11.

Furthermore these results about the distribution of Hedge Funds could be different depending

on their strategies (Anson, 2006).7the collapse of Long-Term Capital Management in September 1998, and the peak of the technology bubble in

March 20008At the moment where we write this article, we do not know when this paper will be publish in Journal of Finance.9In the sense that they maximize a weighted average power, namely the Avg-W and Exp-W.

10Skewness and kurtosis.11They showed (as Lo & al. (2004)) that monthly Hedge Fund returns may exhibit high levels of autocorrelation.

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3 Database

For this study, we use the Center for International Securities and Derivatives Markets (CISDM)

and HedgeFund.Net database. The primer covers the period from January 1994 through to

July 2007 with the advantage of the inclusion of dead funds. One of the main studies in this

article is to analyze whether or not Hedge Funds and CTA were able to generate alpha during

market events or on the contrary, whether they generated negative alpha. In order not to

create a bias we have included all funds in our analysis even though dead funds mean that

they did not necessarily generate alpha or worst, generate negative alpha. The sample with

all funds contains approximately 9800 funds (Hedge Funds, CTA and Fund of Funds), and

approximately 3000 live funds. The latter is the largest commercial database of active Hedge

Fund, Fund of Fund and CTA products with over 8500. It covers the period from May 1975

through to October 2008 but we are only concerned with the period from January 1991 until

October 2008. It has approximately 3000 Funds of Funds, 4900 Hedge Funds, and 600 CTAs.

For every fund, we have collected the returns, the strategy and fund type12. Returns are net

of management and performance based fees. We select funds for a minimum of 30 months

covering the period May 1998 until June 2000. We have created 2 groups in order to analyze

them separately; one for Hedge Funds and the other for CTAs. Nevertheless, it is more

pertinent to study Hedge Funds and CTAs depending on their strategies. There is a vast

selection of academic literature on how to classify Hedge Funds. Fung and Hsieh (1997)

and Brown and Goetzmann (2003) have identified between five and eight investment styles,

whereas, Bianchi Drew, Veeraraghavan, and Whelan (2005) have found the presence of only

three different Hedge Fund styles. In contrast, Hedge Fund database providers classify Hedge

Funds into between 11 and 31 investment styles. Therefore choosing a number of strategies is

not an easy task, especially because we are not persuaded by the precedent results.13. Thus,

it seems better for this study to follow the 23 strategies defined by the provider plus the CTA

as another strategy as well as Fund of Funds.

{please insert Table I here}

HedgeFund.net uses 31 strategies that we have grouped in order to obtain the same 23 strate-

gies from the CISDM.

12

This database combines four main group, Hedge Funds, Funds of Funds, CTA, and CPO.13it will be the topic we are going to turn to next.

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{please insert Table II here}

The main analyzes consider the merge database14 due to the specificity of these products that

they tend to avoid direct regulation (by the SEC or any regulatory authorities).

4 Factors

Hedge Funds can be divided into four main groups, market directional, corporate restructuring

fund, convergence trading fund and opportunistic funds. We can add to each of them a major

risk factor. The exposure to the stock market is the major risk affecting market directional

funds 15. The major risk affecting corporate restructuring funds 16 is exposure to the event

risk17 which is the same for the convergence trading fund 18. Into every group, each strategy

can also have a specific exposure to another risk factors and therefore risk exposure can change

drastically in regards to the strategy.

Consequently, defining factors is not an easy task. Precedent articles have shown that the

relation between Hedge Fund returns and market returns are nonlinear. Moreover, Hedge

Funds have a systematic risk but it is not possible to capture this risk with standard asset

benchmarks. Therefore many researchers have suggested factors in order to explain Hedge

Fund returns (Fung and Hsieh (1999, 2000, 2001, 2004), Mitchell and Pulvino (2001) and

Agarwal and Naik (2004)). Nevertheless, it is not the only problem that we have with factors.

Indeed, knowing the right number of factors is major in order to capture risk correctly. If

some risk factors are missing, the model is misspecified and we can question whether alpha

corresponds to the manager skill. If there are too many factor, we can have a problem of

multicollinearity.

To the best of our knowledge, the most accomplished article about Hedge Fund factors was

written by Fung and Hsieh (2004). They showed that their seven factor model strongly explains

variation in Hedge Fund returns and at the same time avoid multicollinearity19. Moreover,

they managed to obtain similar results using the Agarwal and Naik (2004) option-based factor

14We merged the HedgeFund.Net database to the CISDM and found that they are 925 funds in common.15equity Long/Short, Short selling and equity market timing.16distressed securities, merger arbitrage and event driven.17failure of the proposed transaction.18fixed income arbitrage, convertible bond arbitrage, equity market neutral, statistical arbitrage, and relative value

arbitrage.19We use the diagnostic technique presented in chap 3 of Regression Diagnostic by Belsley, Kuh, and Welsh (1980).

The diagnostic is capable of determining the number of near linear dependencies in a given data matrix X, and thediagnostic identifies which variable are involved in each linear dependency. We do not detect any multicollinearitywith these eight factors.

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model.

Their paper included seven factors and they added an eighth factor on their website 20 that we

have also added.

Therefore we will follow the eight Hedge Fund risk factors defined in Fung and Hsieh’s paper(2004)21.

These factors are :

Three Trend-Following Factors: Bond, Currency and Commodity22 which capture a non linear

exposure.

•• 2 Equity-oriented Risk Factors: S&P500 minus risk free rate23 and Size Spread Factors defined

by the Russell 2000 index monthly total return less S&P500 monthly total return.

• 2 Bond-oriented Risk Factors: a Bond Market Factor represented by the monthly change in the

10-year treasury constant maturity yield, and a Credit Spread Factor formed by the monthly

change in the Moody’s Baa yield less 10-year treasury constant maturity yield.

• One Emerging Market Risk Factor, the MSCI Emerging market minus the risk free rate.

5 Structural Change

Few researchers have tested if there has been a structural change. The tests used considered

only one structural change and most of them needed to know when the break point should

happened. Indeed, the classical test for structural change is typically attributed to Chow

(1960) with the weakness that the breakdate must be known a priori. Quandt (1960) treated

the breakdate as unknown but, in this case, the chi-square critical values are inappropriate.

The problem of critical values was solved simultaneously in the early 1990s by several sets of

authors. Nevertheless, the best known solution was provided by Andrews (1993), and Andrews

and Plobergers (1994). Andrews, Lee and Ploberger (1996) developed the precedent results

into a class of optimal tests for linear models with known variance. This class of optimal

tests allows an arbitrary number of changepoints. However, Bolley and Whaley implemented

this test using just one changepoint. The problem with these tests in the case of multiple

structural change is practical implementation. According to Perron, the Avg-W and Exp-W

20

http://faculty.fuqua.duke.edu/ dah7/.21For more details about the construction of these factors see Fung and Hsieh, 1997, 2001, 2004a.22These factors are downloadable on http://faculty.fuqua.duke.edu/ dah7/DataLibrary/TF-FAC.xls .233-month USD LIBOR

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tests require the computation of the W-test over all permissible partitions of the sample. Bai

and Perron (1998, 2004) solved this problem with a very efficient algorithm which is available

on their website24. Although Bai and Perron (1998, 2004) proposed three different tests25, we

have applied just one of them which is, in our opinion, the most significant for this analysis.This is the second test provided by Bai and Perron, called Double Maximum Test. It is a

test of no structural breaks against an unknown number of breaks given some upper bound

(M = 5 in our application). In this category, there are two tests, the UDmax and the WDmax

which differ by their weight methodology26.

The results are conclusive.

{please insert Table III here}

The majority of Hedge Funds present structural breaks. By strategy, the minimum percentage

of Hedge Funds with breaks is 31 percent (Other Relative Value) and the maximum is 70

percent for Event Driven Multi Strategy. If we consider the most representative strategy

(Equity Long/Short, 2142 Hedge Funds), 62 percent present some structural changes. All

these precedent results took into consideration track records with more than 30 months. Bai

and Perron’s tests also give the break’s date. Therefore another interesting view is to look at

the frequency of break’s dates in the Hedge Fund’s universe by considering the number of the

breaks at time t; relative to one fund. Indeed, the increase in the amount of Hedge Funds is

considerable and we must take this huge growth into consideration within our analysis. For

this reason, we have suggested to create a ratio called Rbreaks, which is defined as the number

of breaks at time t divided by the number of funds at time t.

{please insert Graph I,II,III,IV here}

Once again, the results are conclusive. Whatever the strategy, we have noted an increase in

the ratio over the period 2002-2007 as well as an increase in the frequency of breaks.

There are two interesting things to say about the two crisis. In August 1998, the Russian gov-

ernment defaulted on the payment of its outstanding bonds. This default caused a worldwide

liquidity crisis with credit spreads expanding rapidly all around the globe. The Russian debt

24http://people.bu.edu/perron/. This code is a companion to the paper: Estimating and testing structural changesin multivariate regressions (Econometrica, 2007) (developed by Zhongjun Qu).

25The primer test considers no break versus fixed number of breaks (up to 5 breaks)This test considers the sup Ftype test of no structural break (m = 0) versus the alternative hypothesis that there are m = 1,..., 5 breaks. It is ageneralization of the sup F test considered by Andrews (1993). The latter is a test of the null hypothesis of l breaksagainst the alternative that an additional break exists. This sequential procedure estimates each break one at a time.

It stops when the sup F (l + 1|l) test is not significant.26See Bai and Perron (1998) for a full explanation on the different weights.

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crisis (LTCM) was a crisis that materially affected Hedge Fund returns that was confirmed by

the results from Rbreaks. Furthermore, the precedent results for a majority of strategies were

confirmed.

Surprisingly, we noticed the opposite for the equity bubble crisis which corroborated the con-

clusions of Brunnermeier and Nagel (2002).

The conditions for financial markets in 1999 were very good, especially for riskier assets.

During this period a bubble developed. According to Brunnermeier and Nagel (2002) most

Hedge Funds, despite irrational levels of valuation, decided to ride the bubble rather than

clear their positions. They explained that Hedge Funds heavily tilted their portfolios towards

technology stocks without offsetting this long exposure by short or derivatives. They concluded

that Hedge Funds deliberately held technology stocks and were able to exploit this opportunity.

These arguments are confirmed by the fact that Rbreaks is very low, for the majority of the

strategies, showing fewer structural changes.

This section has shown that it is not enough to purely take into consideration different dis-

tributions but also that alpha and beta could be dynamic and consequently depend on time.

Furthermore, we showed that the risk in Hedge Funds increased over the last few years es-

sentially due to the use of leverage in order to reach the investors desired return target. The

next section presents an answer to the problem of dynamic for alpha and beta. We suggest a

time-varying coefficient model with the Fung and Hsieh factors which are, in this way, better

suited for Hedge Funds.

6 Our Model

Stone (1977) introduced local linear least squares kernel estimators as a regression estimator

which has been generalized by Cleveland (1979)27.

Stone (1980, 1982) used local linear least squares kernel and its generalization to higher-order

polynomials to show the achievement of his bounds on rates of convergence of estimators of a

function m and its derivatives. Fan (1992, 1993) showed in the univariate case that another

important advantage of local linear least squares kernel estimators is that the asymptotic bias

and variance expressions are particularly interesting and appear to be superior to those of

the Nadaraya-Watson or Gasser-Muller kernel estimators. Furthermore, Kernel estimators27The robust local regression estimators.

11



have the advantage of being simple to understand and globally used by researchers. The

mathematical analyze and the implement into a computer is easy, and they are consistent for

any smooth m, provided the density f of X is satisfies certain assumptions.

The varying coefficient model assumes the following conditional linear structure:

Y t =

p j=1

β j(t)X jt + εt = α(t) + Xβ (t) + εt

for a given covariates (t, X 1,...,X p) and variable Y . See appendix for more details on the

Time-Varying Coefficient Model.

Thus β i(t) depend on t, this hypothesis can significantly reduce the modeling bias and espe-

cially avoid the “curse”of dimensionality28.

To practice statistical inferences such as the construction of confidence interval for β i(t) dif-

ferent methods have been suggested. Coling and Chiang (2000)29 suggest a naive bootstrap

procedure that we have applied in this article.

The main advantage of this naive bootstrap procedure is that it does not rely on the asymp-

totic distributions of β i(t). Coling and Chiang (2000) recommend another alternative boot-

strap procedure suggested by Hoover et al. (1998), which relies on normal approximations

of the critical values30. According to the authors, both bootstrap procedures may lead to

good approximations of the actual (1 − α) confidence intervals when the biases of β i(t) are

negligible31.

It is well-known in kernel regression that the selection of bandwidths is more important than

the selection of kernel function. In practice, bandwidth may be selected by examining the plots

of the fitted curves. Moreover, for this study we need an automatic bandwidth selection. Colin,

Wu and Chiang (2000) suggest that we apply the choice of bandwidth “leave-one-subject-

out”cross-validation32. We illustrated the performance of our estimator by a simulation study.

We used the bandwidth defined above and a number of data equal to 50 in order to respect

28Kernel depends, in this context, only on t.29Another paper of Galindo, Kauermann, and Carroll (2000) suggest another bootstrap method based on the wild-

bootstrap of Hardle and Marron (1991)30Construct pointwise intervals of the form

β i(t) ± z(1−α/2) se∗B(t),

where se∗B(t) is the estimated standard error of β i(t) from the B bootstrap estimators and z(1−α/2) is the (1− α/2th)

percentile of the standard Gaussian distribution.31They point out that theoretical properties of these bootstrap procedures have not yet been developed.32Appendix I gives a summary to the algorithm.

12



the average size of Hedge Fund tracks (see appendix).

We regress the net-of-fee monthly excess return (in excess of the risk-free rate) of a Hedge

Fund on the excess returns earned by traditional buy and hold and primitive trend following

strategies defined above33. We have grouped the estimates together into the 23 strategies

defined by the CISDM. In each group obtained, we have formed 2 groups. The first is composed

by all tracks record which is superior to 30 months. The second has the first requirement (more

than 30 months) but only has funds which were available during the 2 crisis.

{please insert Table I here}

Indeed, we are concerned with the 2 different aspects. The first focusses on the security

selection ability ( α(t)). The second aspect treats the ability to anticipate market events or

to cope with them (i.e. robust ability). In order to capture anticipation in alpha, we are

going to look at 2 strong market events: the LTCM crisis and the Equity Bubble crisis. To

properly cover these periods, we have selected 3 consecutive months. For the LTCM period we

will focus on July, August, and September 1998, and for the Equity Bubble period, February,

March and April 200035. Using these 3 months’ data, we have built 2 indicators for α36 and

2 indicators for β i, (i = 1, ..., 8). The 2 primers are the difference between the second month

and the first month and the difference between the last month and the second month.37 The 2

indicators for beta38 have the same differences but are augmented by 10 percent39. By doing

this, we were able to determine whether or not managers were able to react quickly and also

if we can capture change exposure superior to 10%. In each case, we have used the t-statistic, ti,α = αi/ σ αi and ti,β = β i/ σ β ifor every α j(t) and β ij(t), i = 1, ..., 8. For every indicator

33Kat and Lu (2002), Brooks and Kat (2002) show that the net-of-fees monthly returns of the average individualHedge Funds exhibit positive first-order serial correlation which is due, according to the authors, to marking-to-marketproblems. We have removed serial correlation by applying the same methodology as used in Brooks and Kat’s paper(2001), called the simple Blundell-ward filter34.

The observed (or smoothed) value V ∗t of a Hedge Fund at time t could be expressed as a weighted average of the true

value at time t, V t and the smoothed value at time t − 1, V ∗t−1:

V ∗t = αV t + (1 − α)V ∗t−1,

rt =r∗t − αr∗t−1

1− α.

35End of month of July, Auguste, and September and end of month of February, March, and April.36αt − αt−137These two indicators were created in order to show a new methodology that this model bring to the analysis of

Hedge Funds. Nevertheless, some others indicators could be more appropriate for a specific analysis of these marketevents.

38β t − (1.1) × β t−139

This methodology is simply a linear relation between 2 independent variables which, under the condition of normality for β jt j = 1,...N j ; N j being the number of funds, assure that the linear relation follows also a normaldistribution.

13



covering the 2 periods i.e. 4 indicators by fund for α j(t) and 32 indicators by fund for β ij(t).

Once it was done, we calculated the p-value for every indicator and the precedent means.40

If we examine just 1 fund, we only have to test a null hypothesis H 0 versus an alternative H 1

based on a statistic (let’s say X ). We have 2 possibilities for a given rejection region Γ. Either

we reject H 0 when X ∈ Γ or we accept H 0 when X ∈ Γ. Nevertheless, there is the possibility

that there is an error in the test. On the one hand, the test can reject H 0 whereas H 0 is true,

called type I error. On the other hand, the test can accept H 0 whereas it is H 1 which is true.

This error is called a type II error. Now, if we want to test for numerous Hedge Funds this

problem becomes much more complicated. Testing numerous Hedge Funds is to do, in fact, a

multiple-hypothesis testing.

A first approach was suggested by Kosowski, Naik and Teo (2005), using a bootstrap proce-

dure; They analyzed whether or not Hedge Fund performance can be explained by luck and if

Hedge Fund performance persists at annual horizons. Their methodology tested the skills of

a single fund that was chosen from the universe of alpha-ranked funds. Barras, Scaillet, and

Wermers (2008) suggested another approach which is notably informative with regards to the

prevalence of outstanding managers in the whole fund population. Their approach simultane-

ously estimated the prevalence and location of multiple outperforming fund within a group;

examining fund performance, in this manner, from a more general perspective. Therefore we

have used their methodology called False Discovery Rate (FDR hereafter) in our multiple-

hypothesis test.

Step 1, for each strategy, we have applied the FDR on α (security selection ability) in order to

determine the proportion of zero-alpha funds and skilled and unskilled funds; but also for the

indicator defined above αt − αt−1 (timing ability) for the 2 crisis. In this way, we were able

to determine, for each strategy, the proportion of the security selection ability as well as theproportion of the timing ability.

Step 2, we applied the FDR on the “beta” indicators defined above during the 2 crisis in order

to determine if we could observe a common increasing trend for change in market exposure i.e.

a common increase in beta value of more than 10%. Moreover, we calculated the median of the

percentage change for each beta of each strategy in order to give an “I.D.” of the possibility

change in market exposure.

40

The t-statistic distributions for individual Hedge Funds are generally non-normal. In order to overcome the non-normality, we use the same approach as Barras, Scaillet, and Wermers (2008), consisting of the use of a bootstrap tomore accurately estimate the distribution of t-statistics for each Hedge Funds (and their associated p-values).

14



With these results, we brought 2 different views about exposure change. Firstly, a view about

common trend change and secondly about what are the highest impacted beta.

7 Results

Generally speaking, Mutual Fund managers use a buy and hold strategy consisting of buying

a range of financial products according to their investment strategy and then they hold them

according to the time horizon (or investment horizon)41. Therefore Mutual Funds are often

associated with Funds that have a relative performance. Unsurprisingly, Barras, Scaillet and

Wermers (2008) have shown that only 0.6 percent of Mutual Funds generate positive alpha

and a majority of them can also be considered as zero-alpha funds. So the legitimate question

is: Are the results relatively the same for Hedge Funds or are they completely different?

On the one hand, if we apply a static linear factor model such as the Newey-West (1987)

heteroscedasticity and autocorrelation consistent estimator, we determine a “static”alpha that

does not capture the particularities of Hedge Fund Strategies. And therefore we have showed

that, whatever the strategies, the majority of Hedge Funds are zero-alpha funds.

On the other hand, when we applied our time-varying coefficient model, we were able to

capture other skills from Hedge Fund managers, resulting in obtaining us a non negligible

increase of positive alpha funds.

{please insert Table V here}

Concerning the risk behavior, we showed that the majority of Hedge Funds have a tendency

to get an increase in credit spread as well as bond market risk factors. We are going to look

into the results for seven out of the twenty four strategies in the following part of this paper 42.

Results for the CTA

We observed a strong difference for estimated alpha between the static factors model (SFM

hereafter) and our time-varying coefficient model (TVCM hereafter). The SFM gave a very

small percentage of positive and negative alpha funds with two percent and one percent respec-

tively. Whereas the TVCM roughly gave nineteen and twenty eight percent. The estimated

alpha from the two different crisis are interesting to point out. Although The CTA coped

41refer to the time between making an investment and needing the funds.42The results and the graphs for the seventeen remaining strategies are available upon request.

15



during the two crisis it obtained a strong percentage of positive alpha funds during the Equity

Bubble Crisis where approximately twenty seven percent were positive alpha funds.

During the Summer of 1998, CTAs had one of its best performances43 while all other Hedge

Fund strategies were struggling.Unsurprisingly, during the two events, the CTA had a tendency to show a slight increase in

the credit spread risk factor and the emerging market risk factor. Nevertheless this sensitivity

concerned only a small percentage of our population. The majority of CTAs had a relatively

stable exposure.

{please insert Graph XI here}

Emerging Markets

The emerging market strategy showed a good proportion of stock-picker skilled funds where

approximately twelve percent generated positive-alpha which pointed out that the majority of

managers were fundamental bottom-up stock-pickers. We obtained the same results using the

SFM. It indicated that the estimated alpha was more “static”than with other strategies. The

proportion of dynamic skilled funds was very good during the two crisis with a huge number of

roughly forty percent, of positive alpha funds. These results confirmed that emerging market

equity hedge fund managers perceived the high volatility of emerging markets as an asset.

During the Equity Bubble and LTCM we noticed a greater dynamic strategy than previously

seen where approximately twenty-five and fifty percent of our population showed an increase

in two risk factors: the credit spread risk factor and the bond market risk factor. The credit

spread risk factor was the most sensitive factor during LTCM, whereas, four out of the eight

factors showed a sensitivity during the Equity Bubble crisis.

{please insert Graph V here}

Equity Long/Short

Equity Long/Short obtained approximately the same results as the CTA apart from the Equity

Bubble crisis. It obtained a better percentage with twenty four percent using the TVCM. The

SFM gave a small four percent and three percent of negative alpha funds. Certain Equity

Long/Short specialize in a specific sector like technology, and, unsurprisingly, the timer ability

had been more impact during the Equity Bubble than during LTCM. Generally speaking, the

43Approximately 10 percent in August and 7.5 percent in September according to CSFB/Tremont Managed Futures

16



proportion stayed relatively consistent pointing-out their ability to switch from the short to

the long position and visa versa.

A small percentage of the population showed an increase or a decrease in exposure during the

two crisis. We still noticed a sensitivity to the credit spread risk factor and to the commoditiesfactors during LTCM. Whereas emerging risk factor was the most sensitive during the Equity

Bubble crisis.

{please insert Graph VI here}

Equity Market Neutral

This strategy produced a very interesting result where the proportion of stock-picker skilled

funds was three percent higher than the estimated proportion using the SFM. Therefore the

two results were relatively closer than with the other strategies. Furthermore this result was

confirmed by the obtained percentage of estimated alpha during LTCM with zero percent. It

didn’t cope as well during the equity bubble. We noticed a tiny four percent during the first

period which reached a strong sixteen percent in the second period.

{please insert Graph VII here}

It was the most robust strategy in terms of change in exposure. Nevertheless, for a minorityof the population we noticed that the credit spread and the commodity showed the biggest

sensitivity during LTCM and the emerging market factor during the Equity Bubble.

Event Driven Multi Strategy

Event Driven Multi Strategy obtained the worst percentage of skilled-funds with zero percent

for the SFM and a small two point five percent for the TVCM. The result was confirmed

during the two crisis with zero percent during LTCM. This was surprising because the two

crisis created several opportunities44, and only the second crisis was more profitable. On the

other hand it was the strategy which obtained the smallest proportion of unskilled funds.

Therefore we can define it as a zero-alpha fund strategy.

{please insert Graph VIII here}

Another strength, for this strategy, concerned the percentage of the population to show a

variation in factor exposure. A negligible part showed an instability during the crisis. The44Invests in mergers, spin-offs, reorganizations, and other announced events.

17



factors concerned were the credit spread the emerging market factor and the commodity factor

for LTCM and the emerging risk factor for the equity bubble.

Global Macro

Global Macro showed a proportion of stock-picker skilled funds equal to five percent using

TVCM and one percent using the SFM. It obtained one of the bigger groups of unskilled

funds with roughly twenty eight percent. Unsurprisingly, the percentage of positive alpha

during LTCM increased up to twenty six percent. These results confirmed that global macro

managers have the most extensive investment universe and that they were able to find some

arbitrages. The Equity Bubble crisis also gave a good percentage of positive alpha funds with

eighteen percent.

Less than ten percent of our population showed an increase or decrease in our general expo-

sure. Credit Spread risk factor stayed the most sensitive during LTCM. Whereas, the size

spread factor, the emerging market risk factor and the credit spread risk factor (slightly) were

concerned about the change in exposure.

{please insert Graph IX here}

Short Bias

Step one, it is important to say that the number of Hedge Funds following this strategy was

relatively small, so the result, in our opinion, could be debatable.

The SFM gave five percent of skilled funds whereas the TVCM gave thirty five percent.

Furthermore, this strong percentage was less than the seventy one percent of positive alpha

funds found during LTCM. The Equity Bubble obtained a tiny twenty seven percent: This

impressive result confirmed the strong dynamic within the strategy.

Moreover, the short bias produced the best percentage of variation exposure in. More than

sixty percent of our population showed an increase in different market risk factors. Credit

spread was still present during LTCM while Size-spread risk factor had an non negligible

sensitivity to bond, commodity and emerging market risk factors during the Equity Bubble.

{please insert Graph X here}

18



Robustness

The goal of this section is to demonstrate that our model is robust to structural change and

that we succeeded to capture a jump in estimated alpha or beta. For this, we created a track

of return where the two betas had a structural change. We allowed for three different sizes:

fifty, one hundred and one hundred and fifty months which akin the size of Hedge Funds that

we can find in different databases.

The main problem with structural change concerns some technical conditions. Mainly that the

second derivative is required to be continuous in a neighborhood of t. As showed in appendix

I, we managed to obtain some very good results; even in the short sample.

Furthermore, in order to show that our results were independent to our model we created

another time-varying coefficient model which was also based on the work of Fan and Zhang

(1999). Nevertheless in second step to our estimator we used a B-spline smoothing instead of

a local regression. We retested the ability to capture a structural break (see appendix I). The

results were also very good.

We used one of the same example as in Zhang, Lee and Song (2002). Our result are less

accurate than Zhang, Lee and Song due to three reasons.

Firstly, our sample contains between 50 and 150 values in opposite to the range in 250-1000.Secondly, we did not apply a monte carlo study in order to find the optimal bandwidth (or

knot for the second model). Because of the size of our database, the use of a second bootstrap

algorithm for each fund (the first algorithm calculates the p-value) would have made the

estimation process too costly.

Thirdly, we used the S&P500 and the monthly change in the 10-year treasury constant maturity

yield as factors which are not a perfect factors set. And therefore these differences slightly

alter the performance of our estimator. For these reasons the different graphics truly represent

the obtained result for each fund.

In a second step, we applied the false discovery rate to the estimated alpha and betas obtained

by B-spline. The results gave approximately the same percentage of unskilled, zero and skilled

funds for our population.

19



8 Conclusion

Hedge Funds cover many different strategies which radically vary in terms of market exposure

and risk. However there are some common characteristics. Hedge Fund managers try to focus

on positive return (whatever the market conditions), the use of leverage and their structural

fees. If we want to model Hedge Funds, these characteristics imply a model which takes

some new assumptions into consideration. No assumption about statistical distribution, the

dynamic in beta as well as non linearity exposure to the market.

In order to overcome these drawbacks, we used a time-varying coefficient model as well as

the whole factors; defined in Fung and Hsieh’s paper (1997, 2001, 2004). The model allowed

us to postulate that alpha and beta are a function which depended on time and avoided

parametric distribution. The use of the factors from Fung and Hsieh allowed us to capture

the non-linearity in beta and therefore gave the best overall risk factors.

This model allowed us to separate manager skill into two components pointed-out by the (stock

or bond or funds)-picking and the ability to anticipate market events. It also allowed us to see

what were the changes in beta exposure or the managers’ reactions according to the market’s

conditions.

Whereas Barras, Scaillet and Wermers (2008) showed us that only 0.6 percent of Mutual

Funds generated positive alpha and therefore a majority of them can be considered as zero-

alpha funds. We found a different result for Hedge Funds. Hedge Fund managers seek absolute

returns and try to outperform the market whatever the market conditions are. We showed

that this dynamic was not well-captured by a static factor model where the results considered

Hedge Funds as zero-alpha funds. Whereas our model found a better proportion of positive

alpha funds but also, unfortunately, a proportion of negative alpha funds.

Another advantage of this model was its capacity to analyze change in factorial exposure. We

saw three main results. Firstly, that a majority of Hedge Funds with a track record superior

to 30 months had a minimum of one structural break. Furthermore we noticed that the

frequency of breaks has increased over the last few years. Secondly, by strategy, we examined

the percentage change between our eight factors and saw if there was a persistent for beta

parameters. Our results suggest that the common increasing percentage change of different

Hedge Funds in the down-state of the market is strongly represented by one exposure. it

was the credit spread risk factor. Last but not least, we applied the FDR to determine the

20



proportion of the fund’s population which had an increase, a decrease, or stayed constant

in the market’s exposure during the two crisis (LTCM and the Equity Bubble). We show

that within a strategy, Hedge funds have a tendency to have an increase in market exposure.

The changes in factorial exposure and the proportion of funds give us a strong tool for riskmanagers and specifically for stress-testing.

21



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27



Table I

See Appendix I for definitions of fund types. The funds used at minimum cover the LTCM and bubble period which

represent a track record of a minimum of 30 months. Certain of the funds used are considered as dead funds i.e. they

stopped their activities. The database marked by a asterisk give the number of funds covering the 2 specific periods

i.e, LTCM period (July, Auguste, and September 1998) and the Equity Bubble years (February, March, and April 2000)

Strategies Number of Hedge Funds

CISDM merge CISDM* merge*

Equity Long/Short 2141 3519 1561 2525

Multi Strategy 251 734 142 462

Emerging Markets 445 635 331 461

Sector 387 630 296 468

Equity Market Neutral 322 698 227 519

Event Driven Multi Strategy 224 384 168 293Global Macro 303 534 205 377

Equity Long Only 148 276 99 174

Single Strategy 114 112 50 50

Fixed Income 153 326 109 260

Distressed Securities 162 268 131 232

Fixed Income Arbitrage 190 314 143 237

Convertible Arbitrage 207 271 181 227

Relative Value Multi Strategy 89 179 74 137

Fixed Income - MBS 74 98 59 69Option Arbitrage 29 126 17 76

Merger Arbitrage 126 139 111 122

Other relative Value 16 32 7 17

Short bias 51 74 32 56

Regulation D 16 56 13 44

Capital Structure Arbitrage 21 30 13 21

Market Timing 2 3 1 1

Unclassified 58 521 37 369

CTAs (systematic) 1003 759

CTAs (discretionary) 283 1915 202 1394

FoHFs (multi strategy) 1837 1390

28



Table II

See Appendix I for definitions of fund types. This table shows how we have grouped the 31 strategies from Hedge-

Fund.Net and the 23 strategies from CISDM .

CISDM database HedgeFund.Net database23 Strategies 31 Strategies

Multi StrategyMulti Strategy

Statistical Arbitrage

Equity Long/Short Equity Long/Short

Short bias Short bias

Event Driven Multi Strategy Event Driven

Emerging Markets Emerging Markets

Merger Arbitrage Merger (risk) Arbitrage

Fixed Income Fixed Income (non arbitrage)

Equity Market Neutral Market Neutral Equity

Global Macro Macro

Relative Value Multi Strategy Value

Sector

Small/Micro Cap

Finance Sector

Technology Sector

Energy Sector

Healthare Sector

Equity Long Only Long Only

Distressed Securities Distressed

Single Strategy FoF Market Neutral

Unclassified

Asset Based Lending

country specific

Special situations

short-term trading

Fixed Income - MBS Mortgage

Convertible Arbitrage Convertible Arbitrage

Fixed Income Arbitrage Fixed Income Arbitrage

Other relative Value Other Arbitrage

Market Timing Market Timer

Option Arbitrage Option Strategies

Regulation D Regulation D

Capital Structure Arbitrage Capital Structure Arbitrage

29



Table III: Test of Multiple Structural Changes

See Appendix I for definitions of fund types. The funds used at minimum cover the LTCM and bubble period which

represent a track record a minimum of 30 months. Listed is a test provided by Bai and Perron (1998) to analyze

whether Hedge Funds have some structural. We note the percentage of Hedge Funds significant to the test after

applying the False Discovery Rate methodology . The test considers tests of no structural break against an unknownnumber of breaks given some upper bound (m = 5).

Strategy Double Maximum test

UDmax WDmax

Equity Long/Short 38% 38%

Multi Strategy 46% 46%

Emerging Markets 34% 34%

Sector 41% 41%

Equity Market Neutral 49% 49%

Event Driven Multi Strategy 35% 35%

Global Macro 42% 42%

Equity Long Only 53% 53%

Single Strategy 71% 71%

Fixed Income 40% 40%

Distressed Securities 20% 20%

Fixed Income Arbitrage 41% 41%

Convertible Arbitrage 41% 41%

Relative Value Multi Strategy 30% 30%

Fixed Income - MBS 58% 58%

Option Arbitrage 57% 57%

Merger Arbitrage 27% 27%

Other relative Value 75% 75%

Short bias 51% 51%

Regulation D 29% 29%

Capital Structure Arbitrage 53% 53%

Market Timing 100% 100%

Unclassified 38% 38%

Fund of Hedge Funds 65% 66%

CTA Systematic 70% 70%

CTA Discretionary 66% 66%

30



T a b l e V : e s t i m a t e d p r o p

o r t i o n s o f z e r o - a l p h a , u n s k i l l e d , a n d s k i l l e

d f u n d s f r o m

m e r g e d a t a b a s e

S e e A p p e

n d i x I f o r d e fi n i t i o n s o f f u n d t y p e s .

T h i s t a b l e d i s p l a y s t h e e s t i m a t e d p r o p o r t i o n s o f z e r o - a l p h a , u n s k i l l e d , a n

d s k i l l e d f u n d f o r e a c h s t r a t e g y a f t e r

a p p l y i n g t h e f a l s e

D i s c o v e r y

r a t e m e t h o d o l o g y d e v e l o p e d b y B a r r a s , S c a i l l e t a n d W e r m e r s ( 2 0 0 8 ) . T h e s e r e s u l t s c o m e f r o m t h e m e r g e b e t w

e e n t h e C I S D M a n d t h e H e g d e F u n d . n e t d a t a b a s e s . T h e

f u n d s u s e

d a t m i n i m u m c o v e r t h e L T C M a n d b u b b l e p e r i o d w h i c h r e p r e s e n t a t r a c k

r e c o r d a m i n i m u m o f 3 0 m o n t h s . W e e s t i m a t e a l p h a w i t h t h e t i m e - v a r y i n

g c o e ffi c i e n t m o d e l

d e fi n e d i n

s e c t i o n 6 . W i t h t h e e s t i m a t e s , w e c r e a t e t h e m e a n ( α ) w h i c h r e p r e s e n t s t

h e s t o c k - p i c k e r a b i l i t y a n d t h e i n d i c a

t o r s d e fi n e d i n s e c t i o n 7 d u r i n g t h e 2

c r i s i s : α L 1

a n d α L 2

f o r L T C M

a n d α B 1

a n d α B 2

f o r t h e E q u i t y B

u b b l e w h i c h r e p r e s e n t s t h e d i ff e r e n t m a r k e t t i m e r a b i l i t i e s . W e a l s o g a v e

t h e r e s u l t u s i n g a l i n e a r f a c t o r m o d e

l : t h e N e w e y - W e s t

( 1 9 8 7 ) h e

t e r o s c e d a s t i c i t y a n d a u t o c o r r e l a t i o n c o n s i s t e n t e s t i m a t o r . W e d o n o t g i v e

s o m e r e s u l t s f o r S i n g l e S t r a t e g y b e c a u s e o f t o o f e w d a t a . F u r t h e r m o r e t h

e r e s u l t s a b o u t t h e

o t h e r s t r a t e g i e s a r e a v a i l a b l e u p o n r e q u e s t .

S t r a t e

g y

M o

d e

l

α

α L 1

α L 2

α B 1

α B 2

u s e

d

π + A

π 0

π − A

π + A

π 0

π − A

π + A

π 0

π − A

π + A

π 0

π − A

π + A

π 0

π − A

E q .

L o

n g

/ S h o r t

T V C M

1 8 . 6 %

5 8 . 8

%

2 2 . 6

%

1 6 . 0

%

6 1 . 9

%

2 2 . 1

%

2 2 . 8

%

6 2 . 2

%

1 5 . 0

%

1 7

. 0 %

5 8 . 8

%

2 4 . 2 %

1 7

. 0 %

5 8 . 8

%

2 4

. 2 %

N w e s t

3 . 8

% %

9 3 . 5

%

2 . 7

% %

E m e r g

i n g

M a r k e t s

T V C M

1 2 . 6 %

4 1 . 5

%

4 5 . 9

%

4 4

. 2 %

4 2 . 6

%

1 3 . 2

%

3 8 . 9

%

4 3 . 8

%

1 7

. 3 %

4 4

. 4 %

3 9 . 2

%

1 6 . 4 %

4 3 . 6

%

3 9 . 2

%

1 7

. 3 %

N w e s t

1 2 . 3 %

8 7

. 5 %

0 . 2

%

E q .

M

a r k e t N e u t r a l

T V C M

1 1 . 6 %

7 4

. 1 %

1 4

. 2 %

0 . 0

%

9 0 . 4

%

9 . 6

%

0 . 0

%

9 2 . 2

%

7 . 8

%

4 . 3

%

8 3 . 2

%

1 2 . 5 %

1 6 . 4

%

8 3 . 2

%

0 . 5

%

N w e s t

8 . 6 %

8 9 %

2 . 4

%

E v e n t

D r i v e n

M .

S .

T V C M

2 . 5 %

9 7

. 5 %

0 . 0

%

0 . 0

%

9 7

. 5 %

2 . 5

%

0 . 0

%

9 9 . 8

%

0 . 2

%

0 . 9

%

9 0 . 7

%

8 . 4 %

0 . 9

%

9 0 . 7

%

8 . 4

%

N w e s t

0 %

1 0 0 %

0 %

G l o b a l M a c r o

T V C M

5 . 2 %

6 7

. 0 %

2 7

. 8 %

2 6 . 1

%

6 9 . 1

%

4 . 8

%

2 6 . 2

%

6 6 . 8

%

7 . 0

%

1 8 . 5

%

5 9 . 9

%

2 1 . 6 %

1 8 . 5

%

5 9 . 9

%

2 1 . 6

%

N w e s t

1 %

9 6 %

3 %

S h o r t

b i a s

T V C M

3 5 . 2 %

9 . 5

%

5 5 . 2

%

7 1 . 4

%

0

. 0 %

2 8 . 6

%

7 1 . 4

%

0 . 0

%

2 8 . 6

%

4 9 . 0

%

1 9 . 0

%

3 1 . 9 %

4 9 . 0

%

1 9 . 0

%

3 1 . 9

%

N w e s t

5 . 3 %

6 3 . 6

%

3 1 . 2

%

C T A

T V C M

1 8 . 8 %

5 3 . 1

%

2 8 . 1

%

1 7

. 0 %

6 0 . 2

%

2 2 . 8

%

1 7

. 0 %

6 0 . 2

%

2 2 . 8

%

2 7

. 2 %

5 5 . 3

%

1 7 . 5 %

2 7

. 2 %

5 5 . 3

%

1 7

. 5 %

N w e s t

2 %

9 7 %

1 %

31



Figure I:

3 1− Ma r− 19 97 3 0− No v− 19 98 3 1− Ju l− 20 00 3 1− Ma r− 20 02 3 0− No v− 20 03 3 1− Ju l− 20 05 3 1− Ma r− 20 070

0.2

0.4

0.6

0.8

1LTCM Equity Bubble

Rbreaks

Capital Structure Arbitrage

31−Dec−1993 31−Aug−1995 30−Apr−1997 31−Dec−1998 31−Aug−2000 30−Apr−2002 31−Dec−2003 31−Aug−2005 30−Apr−20070

0.05

0.1

0.15

0.2LTCM Equity Bubble

Rbreaks

Convertible Arbitrage


0.05

0.1

0.15


Rbreaks

CTA


0.05

0.1

0.15


Rbreaks

Distressed Securities


0.05

0.1

0.15

0.2


Rbreaks

Emerging Markets


0.1

0.2

0.3


Rbreaks

Equity Long Only

See Appendix I for definitions of fund types. The number of hedge Funds has increased drastically since 2000 thereforethe number of breaks in 1995, for instance, has not the same significance as the number of breaks in 2002. In orderto overcome this problem and to study correctly the dynamic in beta we define a ratio: Rbreaks which is equal to

the number of funds at time t divided by the number of break at time t. In this way, we are able to study how thedynamic in hedge Funds behaves during a long period.

The bar figures illustrate the number of breakdates relative to one fund.

32



Figure II:


0.05

0.1

0.15


Rbreaks

Equity Long/Short


0.05

0.1

0.15


Rbreaks

Equity Market Neutral


0.05

0.1

0.15

0.2


Rbreaks

Event Driven Multi Strategy


0.1

0.2

0.3

0.4


Rbreaks

Fixed Income − MBS


0.1

0.2

0.3


Rbreaks

Fixed Income Arbitrage


0.05

0.1

0.15

0.2


Rbreaks

Fixed Income




33



Figure III:


0.05

0.1

0.15


Rbreaks

Global Macro


0.1

0.2

0.3


Rbreaks

Merger Arbitrage


0.1

0.2

0.3

0.4


Rbreaks

Multi Strategy

− −1993 30−Nov−1995 30−Sep−1997 31−Aug−1999 30−Jun−2001 31−May−2003 31−Mar−2005 28−Feb−2007 31−Dec−1993

LTCM Equity Bubble

Option Arbitrage


LTCM Equity Bubble

Other Relative Value


LTCM Equity Bubble

Regulation D




34



Figure IV:


0.05

0.1

0.15

0.2


Rbreaks

Relative Value Multi Strategy


0.05

0.1

0.15

0.2


Rbreaks

Sector


0.1

0.2

0.3

0.4


Rbreaks

Short Bias

28−Feb−1994 31−Jan−1996 31−Dec−1997 30−Nov−1999 30−Sep−2001 31−Aug−2003 31−Jul−2005 30−Jun−2007 28−Feb−0

0.2

0.4

0.6

0.8

1LTCM Equity Bubble

Rbreaks

Single Strategy


0.05

0.1

0.15


Rbreaks

unclassified




35



F i g u r

e V :

L T C M

f i r s t P e r i o

d

0 . 0

0 9 6

0 . 0

1 9

0 . 0

2 9

0 . 0

3 8

0 . 0

4 8

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

S e c o n d P e r i o d

0 . 0

0 9 9

0 . 0

2

0 . 0

3

0 . 0

4

0 . 0

4 9

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

E q u i t y B u b b l e F i r s t P e r i o d

0 . 0

0 5

0 . 0

1

0 . 0

1 5

0 . 0

2

0 . 0

2 5

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

E q u i t y B u b b l e S e c o n d P e r i o d

0 . 0

0 4 1

0 . 0

0 8 2

0 . 0

1 2

0 . 0

1 6

0 . 0

2

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

s e c o n d p e r i o d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M

E q u i t y B u b b l e f i r s t p e r i o d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E

q u i t y B u b b l e s e c o n d p e r i o d

f a c t o r

π A −

π 0

π A +

E m e r g i n g M a r k e t s

L e g e n d :

1 ) T F B

: B o n d T r e n d − F o l l o w i n g f a c t o r

2 ) B T C o

: C o m m o d i t y T r e n d − F o l l o w i n g f a c t o r

3 ) T F C u

: C u r r e n c y T r e n d − F o l l o w i n g f a c t o r

4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d

: S i z e S p r e a d f a c t o r

6 )

B M

: B o n d m a r k e t f a c t o r

7 )

C S

: C r e d i t S p r e a d f a c t o r

8 )

E M

: E m e r g i n g M a r k e t f a c t o r

S e e A p p e

n d i x I f o r d e fi n i t i o n s o f f u n d t y p e s . T h e n u m b e r o f f u n d s c o v e r i n g t h e p e

r i o d i s e q u a l t o 4 6 1 . T h e b a r fi g u r e s

i l l u s t r a t e t h e p r o p o r t i o n o f f u n d s h a

v i n g m o r e t h a n 1 0

p e r c e n t d

e c r e a s e ( b l u e o r π − A ) , c o n s t a n t ( w h i t e

o r π 0 ) , a n d i n c r e a s e ( r e d o r π

+ A ) m a r k

e t e x p o s u r e d u r i n g t h e 2 c r i s i s . E a c h c r i s i s i s d i v i d e d i n 2 p e r i o d ( s e e s e c t i o n 7 : m e t h o d o l o g y ) .

. T h e s e c

o n d fi g u r e s ( F a c t o r R a d a r C h a r t ) i n d

i c a t e s t h e s t r a t e g y ’ s s e n s i t i v i t i e s ( p e r c e n t a g e c h a n g e ) t o v a r i o u s f a c t o r s i n

r e g a r d s t o t h e 2 c r i s i s .

36



F i g u r

e V I :

L T C M

f i r s t P e r i o

d

0 . 0

0 4 3

0 . 0

0 8 7

0 . 0

1 3

0 . 0

1 7

0 . 0

2 2

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


0 . 0

0 4 1

0 . 0

0 8 2

0 . 0

1 2

0 . 0

1 6

0 . 0

2 1

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 5 7

0 . 0

1 1

0 . 0

1 7

0 . 0

2 3

0 . 0

2 8

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 5 8

0 . 0

1 2

0 . 0

1 7

0 . 0

2 3

0 . 0

2 9

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E


f a c t o r

π A −

π 0

π A +

E q u i t y L o n g / S h o r t

L e g e n d :

1 ) T F B


2 ) B T C o


3 ) T F C u


4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d


6 )

B M


7 )

C S


8 )

E M


S e e A p p e


T h e n u m b e r o f f u n d s c o v e r i n g t h e p

e r i o d i s e q u a l t o 2 5 2 5 . T h e b a r fi g u r e s i l l u s t r a t e t h e p r o p o r t i o n o f f u n d s

h a v i n g m o r e t h a n

1 0 p e r c e n t d e c r e a s e ( b l u e o r π − A ) , c o n s t a n t ( w h i t e o r π 0 ) , a n d i n c r e a s e ( r e d o r π

+ A ) m a r k e t e x p o s u r e d u r i n g t h e 2 c r i s i s . E a c h c r i s i s i s d i v i d e d i n 2 p e r

i o d ( s e e s e c t i o n 7 :

m e t h o d o l o g y ) . . T h e s e c o n d fi g u r e s ( F a c t o r R

a d a r C h a r t ) i n d i c a t e s t h e s t r a t e g y ’ s s e n s i t i v i t i e s ( p e r c e n t a g e c h a n g e ) t o v a r i o u s f a c t o r s i n r e g a r d s t o t h e 2 c r i s i s .

37



F i g u r

e V I I :

L T C M

f i r s t P e r i o

d

0 . 0

0 9 1

0 . 0

1 8

0 . 0

2 7

0 . 0

3 6

0 . 0

4 5

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


0 . 0

0 7 9

0 . 0

1 6

0 . 0

2 4

0 . 0

3 2

0 . 0

3 9

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 4 1

0 . 0

0 8 1

0 . 0

1 2

0 . 0

1 6

0 . 0

2

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 4 3

0 . 0

0 8 5

0 . 0

1 3

0 . 0

1 7

0 . 0

2 1

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E


f a c t o r

π A −

π 0

π A +

E q u i t

y M a r k e t N e u t r a l

L e g e n d :

1 ) T F B


2 ) B T C o


3 ) T F C u


4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d


6 )

B M


7 )

C S


8 )

E M


S e e A p p e





p e r c e n t d



+ A ) m a r k


. T h e s e c




38



F i g u r

e V I I I :

L T C M

f i r s t P e r i o

d

0 . 0

0 5 7

0 . 0

1 1

0 . 0

1 7

0 . 0

2 3

0 . 0

2 9

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


0 . 0

0 4 2

0 . 0

0 8 5

0 . 0

1 3

0 . 0

1 7

0 . 0

2 1

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 8 4

0 . 0

1 7

0 . 0

2 5

0 . 0

3 4

0 . 0

4 2

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 8 4

0 . 0

1 7

0 . 0

2 5

0 . 0

3 4

0 . 0

4 2

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E


f a c t o r

π A −

π 0

π A +

E v e n t D r i v e

n M u l t i S t r a t e g y

L e g e n d :

1 ) T F B


2 ) B T C o


3 ) T F C u


4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d


6 )

B M


7 )

C S


8 )

E M


S e e A p p e





p e r c e n t d



+ A ) m a r k


. T h e s e c




39



F i g u r

e I X :

L T C M

f i r s t P e r i o

d

0 . 0

0 6 7

0 . 0

1 3

0 . 0

2

0 . 0

2 7

0 . 0

3 4

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


0 . 0

0 6

0 . 0

1 2

0 . 0

1 8

0 . 0

2 4

0 . 0

3

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 4 3

0 . 0

0 8 5

0 . 0

1 3

0 . 0

1 7

0 . 0

2 1

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 3 4

0 . 0

0 6 9

0 . 0

1

0 . 0

1 4

0 . 0

1 7

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E


f a c t o r

π A −

π 0

π A +

G l o b a l M a c r o

L e g e n d :

1 ) T F B


2 ) B T C o


3 ) T F C u


4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d


6 )

B M


7 )

C S


8 )

E M


S e e A p p e





p e r c e n t d



+ A ) m a r k


. T h e s e c




40



F i g u r

e X :

L T C M

f i r s t P e r i o

d

0 . 0

1

0 . 0

2

0 . 0

3

0 . 0

4

0 . 0

5

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


0 . 0

0 6 7

0 . 0

1 3

0 . 0

2

0 . 0

2 7

0 . 0

3 3

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 5 1

0 . 0

1

0 . 0

1 5

0 . 0

2 1

0 . 0

2 6

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 5 4

0 . 0

1 1

0 . 0

1 6

0 . 0

2 2

0 . 0

2 7

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E


f a c t o r

π A −

π 0

π A +

S h o r t B i a s

L e g e n d :

1 ) T F B


2 ) B T C o


3 ) T F C u


4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d


6 )

B M


7 )

C S


8 )

E M


S e e A p p e


T h e n u m b e r o f f u n d s c o v e r i n g t h e p e r i o d i s e q u a l t o 5 6 . T h e b a r fi g u r e s



p e r c e n t d



+ A ) m a r k


. T h e s e c




41



F i g u r

e X I :

L T C M

f i r s t P e r i o

d

0 . 0

0 3 8

0 . 0

0 7 6

0 . 0

1 1

0 . 0

1 5

0 . 0

1 9

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


0 . 0

0 3 9

0 . 0

0 7 7

0 . 0

1 2

0 . 0

1 5

0 . 0

1 9

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 6 4

0 . 0

1 3

0 . 0

1 9

0 . 0

2 6

0 . 0

3 2

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M


0 . 0

0 6 2

0 . 0

1 2

0 . 0

1 8

0 . 0

2 5

0 . 0

3 1

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M

f i r s t p e r i o

d

f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M

L T C M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S

P 5 0 0

S S d

B d M t

C S

E M


f a c t o r

0 %

5 0 %

1 0 0 %

T F B

T F C u

T F C o

S P 5 0 0

S S d

B d M t

C S

E M E


f a c t o r

π A −

π 0

π A +

C T A s M e r g e

L e g e n d :

1 ) T F B


2 ) B T C o


3 ) T F C u


4 ) S P 5 0 0 : S & P 5 0 0

5 )

S S d


6 )

B M


7 )

C S


8 )

E M


S e e A p p e


T h e n u m b e r o f f u n d s c o v e r i n g t h e p

e r i o d i s e q u a l t o 1 3 9 4 . T h e b a r fi g u r e s i l l u s t r a t e t h e p r o p o r t i o n o f f u n d s

h a v i n g m o r e t h a n

1 0 p e r c e n t d e c r e a s e ( b l u e o r π − A ) , c o n s t a n t ( w h i t e o r π 0 ) , a n d i n c r e a s e ( r e d o r π

+ A ) m a r k e t e x p o s u r e d u r i n g t h e 2 c r i s i s . E a c h c r i s i s i s d i v i d e d i n 2 p e r

i o d ( s e e s e c t i o n 7 :

m e t h o d o l o g y ) . . T h e s e c o n d fi g u r e s ( F a c t o r R

a d a r C h a r t ) i n d i c a t e s t h e s t r a t e g y ’ s s e n s i t i v i t i e s ( p e r c e n t a g e c h a n g e ) t o v a r i o u s f a c t o r s i n r e g a r d s t o t h e 2 c r i s i s .

42



Appendix I

Definition of Strategies

• The emerging markets strategy attempts to capture gains from inefficiencies in emergingmarkets.

• The Equity Long/Short strategy refers to taking both long and short positions in equities.

• The Market Timer focus on securities associated with companies that will soon experiencea significant event.

The Distressed Securities Strategy focuses on asset of distressed companies. Buys equity,debt, or trade claims at deep discounts of companies in or facing bankruptcy or reorganization.

• The Merger Arbitrage Strategy also called risk arbitrage strategy exploit pricing inef-ficiencies associated with a merger or acquisition.

• The event driven multi-strategy can use both the distressed securities style and/or themerger arbitrage style.

• The Relative Value arbitrage style take positions in 2 securities that are mispriced relativeto each other, with the expectation that their prices will converge to appropriate values in thefuture(Arbitrage, Market neutral.

• The arbitrage involves simultaneously purchasing and selling related securities that are mis-priced relative to each other.

• Convertible Arbitrage Strategy can be described by taking a long position in a convertiblebond and sells short the associated stock. Convertible arbitrage - exploit pricing inefficiencies

between convertible securities and the corresponding stocks.

• Fixed Income Arbitrage Strategies encompass a wide range of strategies in both domesticand global fixed-income markets. Fixed income arbitrage - exploit pricing inefficiencies betweenrelated fixed income securities.

• Equity Market Neutral style creates a position that attempts to hedge out most marketrisk by taking offsetting positions. This strategy exploits the mispricing between a stockwhich is overvalued and one that is undervalued such that beta of the combined position iszero. Statistical arbitrage - equity market neutral strategy using statistical models.

• Index arbitrage style generally attempts to exploit mispricing between an index and deriva-

tives on that index.

• Mortgage-backed securities arbitrage style exploit the mispricing of mortgage-backedassets relative to Treasury securities.

• multi-strategy style uses different styles and may change exposures to different styles basedupon changing market conditions.Multi strategy in Macro strategy - combination of discre-tionary and systematic macro. Multi strategy in FoHF - a hedge fund exploiting a combinationof different hedge fund strategies to reduce market risk.

• dedicated short selling style only takes short equity positions.

• Global Macro Discretionary macro - trading is done by investment managers instead of

generated by software.Systematic macro (Systematic diversified) - trading is done mathematically, generated bysoftware without human intervention.

43



• Sector funds - expertise in niche areas such as technology, health care, biotechnology, phar-maceuticals, energy, basic materials.

• Fundamental value - invest in undervalued companies.

• Fundamental growth - invest in companies with more earnings growth than the broad equity

market.

• Quantitative Directional, statistical arbitrage - equity trading using quantitative tech-niques.

• Multi manager - a hedge fund where the investment is spread along separate sub managersinvesting in their own strategy.

• Trend following - long-term or short-term. Non-trend following (Counter trend) - profitfrom trend reversals.

• Regulation D - specialized in private equities.

• Credit arbitrage or fixed income arbitrage strategy - specialized in corporate fixedincome securities.

• Fixed Income asset backed - fixed income arbitrage strategy using asset-backed securities.

• Volatility arbitrage - exploit the change in implied volatility instead of the change in price.

• Yield alternatives - non fixed income arbitrage strategies based on the yield instead of theprice.

• Capital Structure Arbitrage - involves taking long and short positions in different financialinstruments of a company’s capital structure, particularly between a company’s debt and

equity product.

44



Two-step Time-Varying Coefficient Model

The varying coefficient model assumes the following conditional linear structure:

Y t =

p

j=1

β j(t)X jt + εt = α(t) + Xβ (t) + εt

for a given covariates (t, X 1,...,X p) and variable Y with

E [ε|t, X 1,...,X p] = 0,

V ar[ε|t, X 1,...,X p] = σ2(t),

In this study, we took X 1 = 1 as the intercept term and t = time.

if we consider that β i depends on t: (β i(t)), we can approximate the function locally asβ i(t) ≈ ai + bi(t − t0). This leads to the following local least-squares problem:

minimizen

i=1

Y i −

p j=1

{a j + b j(T i − t0)}X ij

2 K h(T i − t0)

for a given kernel function K and bandwidth h, where K h(.) = K (./h)/h.In matrix notation:

Let

X0 = X 11 X 11(T 1 − t0) . . . X 1 p X 1 p(T 1 − t0)

... ... . . . ... ...X n1 X n1(T n − t0) . . . X np X np(T p − t0)

Y = (Y 1,...,Y n) and W 0 = diag(K h0(T 1 − t0),...,K h0(T n − t0))

Then the solution to the least-squares problem can be expressed as:

a j,0 = e2 j−1,2 p(X0W 0X0)−1X

0W 0Y

With these estimates,

a j,0, a local least-square regression is fitted again via substituting the

initial estimate into the local least-squares problem:

ni=1

Y i −

p−1 j=1

a j,0(T i)X ij −

a p + b p(T i − t0) + c p(T i − t0)2 + d p(T i − t0)3

X ip

2

×K h2(T i−t0)

where h2 is the bandwidth in the second step. In this way, a two-step estimator is obtained.Fan and Zhang showed that the bias of the two-step estimator is of O(h4

2) and the varianceO

(nh2)−1

45



Two-step Time-Varying Coefficient Model using B-splines

As mentioned by Fan and Zhang (1999), other techniques such as smoothing splines can alsobe used in the second stage of fitting. Therefore we built the second two-step estimator basedon the same article but we used during the second step a smoothing splines instead of local

regression.From the first step, we obtained the estimates:

a j,0 = e2 j−1,2 p(X0W 0X0)−1X

0W 0Y

In a second step, knowing that we can approximate each β p(t) by a basis function expansion

β p(t) K k=0

γ ∗ pkB pk(t),

We can now minimized in order to estimate γ ∗kp:

ni=1

wi

Y i −

p−1 j=1

a j,0(T i)X ij −

K k=0

γ kpBkp

X ip

2

then it is natural to estimate β p(t) by

β p(t) =K k=1

γ kpBkp(t).

46



Robustness: local and B-spline Time-Varying Coefficient Model

The following example will be used to illustrate the performance of our estimator. We created two “betas”(called β icreated) which represent a possible structural change for a Hedge Fund. We put up a simulated HedgeFund track (RHF ) in this manner:

RHF = β 1createdX1 + β 2createdX2 + ε

where X1, X2 are the S&P500 and the monthly change in the 10-year treasury constant maturity yield respec-tively. The random variable ε follows a normal distribution with mean zero and variance 1.We called the local time-varying coefficient model: L-TVCM and the B-spline time varying coefficient model:B-TVCM.

Short sample: 50 months

0 20 40 60−2

−1

0

1

2

β 1 and true β 1 using L-TVCM

months

0 20 40 60−5

0

5

10


months

0 20 40 60−2

0

2

4

β 2 and true β 2 using B-TVCM

months

0 20 40 60−2

0

2

4

6


months

FIG.: Comparison of the performance between the one-step estimator (long-dashed curve) and the true coeffi-cient function (the solid curve).

47



Medium sample: 100 months

0 50 100−2

−1

0

1


months

0 50 100−2

0

2

4


months

0 50 100−1

−0.5

0

0.5

1


months

0 50 100−2

0

2

4

6


months


Long sample: 150 months

0 50 100 150−2

−1

0

1


months

0 50 100 150−2

0

2

4


months

0 50 100 150−1

−0.5

0

0.5

1


months

0 50 100 150−2

−1

0

1

2


months

true β

β


48



Bootstrap Confidence intervals

We summarize the methodology from Colin and Chiang (2000) in order to create confidenceregions.

According to the author this following naive bootstrap procedure can be used to constructapproximate pointwise percentile confidence intervals for β i(t):

• 1) Randomly sample n subjects with replacement from the original data set, and let{(t∗ij, X∗

i ), Y ∗ij ; 1 = i = n, 1 = j = ni} be the longitudinal bootstrap sample.

• 2) Compute the kernel estimator β booti (t)

• 3) Repeat the above 2 steps B times, so that B bootstrap estimators β booti (t) of β i(t) areobtained.

• 4) Let Lα/2(t) and U (α/2)(t) be the (α/2)th and (1 − α)th i.e. lower and upper (α/2th) per-

centiles, respectively, calculated on the B bootstrap estimators. An approximate (1 − α)bootstrap confidence interval for β i(t) is given by (L(α/2)(t), U (α/2)(t)).

bandwidth ”leave-one-subject-out” cross-validation methodology

The leave-out method is based on regression smoothers in which one, say the jth, observationis left out. So, for N values,

• 1) Compute the leave-out estimate mh,j(X j) = n−1

i= j

W hi(X j)Y i.

• 2) Construct the cross validation function CV (h) = n−1n

j=1(Y j − mh,j(X j))2w(X j). where w

denotes a weight function.

• 3) With this N CV(h), we can, now, define the automatic bandwidth as h = arg minh∈H n

[CV (h)]

criton scallet time varying coefficients hf e4b

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