Statistical Methods of Valuation and Risk Assessment: Empirical Analysis of Equity Markets and Hedge Fund Strategies

Swiss Federal Institute of Technology, Zürich
University of Zürich, Swiss Banking Institute
Master of Advanced Studies in Finance

Master Thesis

Statistical Methods of Valuation and Risk Assessment:
Empirical Analysis of Equity Markets
and Hedge Fund Strategies

Adam Czub

*****
January 2004


Acknowledgements
A master thesis is usually thought as entirely individualistic work. This is hardly ever the
case. Constant support, understanding and enlightenment are required from different people
during the process. I will be forever grateful for the emotional support received from my
parents, Elzbieta and Wojciech, and the encouragement from my brother Tomasz.
Many thanks to my supervisor, Prof. Alexander McNeil, for his guidance and help on my
master thesis. I would like to thank also my co-supervisors and colleagues Valerie ChavezDemoulin, Bernhard Brabec and Michael Heintze for their explanations and comments.
Finally, thanks to all of those who in one way or another helped me make this master thesis
become a reality.

-1-


Abstract
The purpose of this paper is first to describe a web-based tool called Riskometer. We

designed and implemented its second version which has a statistical methodology
implemented in S-Plus. This tool, called Riskometer, returns the different Value-at-Risk and
related measures of risk (Expected Shortfall, volatility) for major equity market indices using
standard methods as well as the most recent state-of-the-art methods. This internet tool
continually backtests its own performance against the latest data. We analyse the risk
measures calculated by Riskometer on September 24, 2003 and January 9, 2004.
In the second part of the paper, we analyse hedge fund strategies over a six years sample
period using the database of indices compiled by Morgan Stanley Capital International. For a
better understanding about dependence structures in hedge fund strategies we focus on
analysing their bivariate distributions using Archimedean copulas. To identify style exposures
to relevant risk factors we conduct a return-based style analysis of hedge fund strategies by
relaxing the constraints of the Sharpe’s style analysis, and examine the significance of the
style weights. Finally, we compare these results with those obtained by applying the Kalman
filter and smoother technique.

-2-


Contents

1

INTRODUCTION ....................................................................................................................................... 4

2

QUANTIFICATION OF EQUITY MARKET RISK: RISKOMETER................................................. 4
2.1 INTRODUCTION .......................................................................................................................................... 4
2.2 DATA ......................................................................................................................................................... 4
2.3 METHODS .................................................................................................................................................. 4

2.4 RISK MEASURES ........................................................................................................................................ 5
2.4.1
Volatility.......................................................................................................................................... 6
2.5 BACKTESTING............................................................................................................................................ 8

3

EMPIRICAL CHARACTERISTICS OF HEDGE FUND STRATEGIES.......................................... 10
3.1 INTRODUCTION ........................................................................................................................................ 10
3.2 DATA ....................................................................................................................................................... 10
3.3 RISK-RETURN CHARACTERISTICS............................................................................................................ 15
3.4 DEPENDENCE STRUCTURE ANALYSIS ...................................................................................................... 21
3.4.1
Linear Correlation as Dependence Measure ................................................................................ 22
3.4.2
Alternative Correlation Measures................................................................................................. 22
3.4.3
Archimedean Copulas ................................................................................................................... 24
3.4.4
Statistical Significance of the Copula Parameter.......................................................................... 27
3.4.5
Tail Dependences .......................................................................................................................... 28
3.5 GENERALISED STYLE ANALYSIS ............................................................................................................. 30
3.5.1
Return-Based Style Analysis Model .............................................................................................. 30
3.5.2
Statistical Significance of Style Weights........................................................................................ 32
3.5.3
Analysis of Style Weights............................................................................................................... 32
3.6 TIME-VARYING EXPOSURES ANALYSIS ................................................................................................... 34

3.6.1
Kalman Filter and Smoother Algorithm........................................................................................ 35
3.6.2
Graphical Analysis of Time-Varying Exposures ........................................................................... 37

4

CONCLUDING REMARKS .................................................................................................................... 39

5

REFERENCES .......................................................................................................................................... 41

6

GLOSSARY ............................................................................................................................................... 43

7

FITTING COPULAS ................................................................................................................................ 46

8

DIGITAL FILTERING ............................................................................................................................ 49

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1

Introduction

Much of the financial decision making by financial institutions is focuses on risk
management. Measuring risk and analysing ways of controlling and allocating it require a
wide range of sophisticated mathematical and computational tools. Indeed, mathematical
models of modern finance practice contain some of the most complex applications of
probability, optimisation, and estimation theories.
Mathematical models of valuation and risk assessment are at the core of modern risk
management systems. Every major financial institution in the world depends on these models
and none could function without them. Although indispensable, these models are by necessity
abstractions of the complex real world. Although there is continuing improvement in those
models, their accuracy as useful approximations varies significantly across time and situation.

2

Quantification of Equity Market Risk: Riskometer

2.1 Introduction
The ETHZ Riskometer consists of a web-based tool which was designed and implemented
using a statistical methodology implemented in S-Plus. It returns the different Value-at-Risk
and related measures of risk (expected shortfall, volatility) using standard methods and the
most recent state-of-the-art methods. This educational tool continually backtests its own
performance against the latest data.
In the present version Riskometer focuses principally on three major stock indices: the Dow
Jones Industrial Average (DJIA), the Standard&Poors 500 (S&P500), and the Deutsche
Aktienindex (DAX). The data are collected daily and added to the historical daily time series
dataset, providing risk measures that may be interpreted as prognoses for a one-day time
horizon. The underlying methods included in the Riskometer may be applied to any stock
price, exchange rate, commodity price or portfolio comprising combinations of these
underlying risk factors, whether linear or non-linear. They may also be applied to daily data

or to both higher or lower frequency time series data.
In the market risk area, Value-at-Risk estimation involves portfolios of more than one asset.
The Riskometer can be extended to such multivariate series.

2.2 Data
The Riskometer works with return data since the beginning of 1998 what represents
somewhat less than 1500 days of historical returns. We believe that our time window is long
enough for not losing statistical accuracy in measuring risk and includes relevant data
representing a period characterised by important market up and down moves additionally to
high and low volatility times. Furthermore, we make the assumption that our amount of data
may be considered to be a realisation of a stationary time series model.

2.3 Methods
The underlying methods implemented in the Riskometer provide either unconditional quantile
estimation or conditional quantile estimation. Unconditional methods for calculation of
market risk measures are
-4-


1. Plain Vanilla: Variance – Covariance
2. Historical Simulation
And the conditional methods are
3.
4.
5.
6.

EWMA (Exponentially Weighted Moving Average)
GARCH modelling
GARCH-style time series modelling with extreme value theory

Point process approach

Standard methods (variance-covariance, historical simulation, EWMA, and GARCH
modelling) are described in the technical document of Riskmetrics Group, Inc (2001)
available on-line. Concerning more sophisticated methodologies, GARCH-style time series
modelling with extreme value theory, which still needs to be implemented in the present
version of Riskometer, is detailed in the McNeil and Frey (2000) and Point process approach
is detailed in the Chavez-Demoulin, Davison and McNeil (2003).

2.4 Risk Measures
We describe the risk measures calculated by the different methods of the Riskometer on
September 24, 2003 and January 9, 2004. On September 24, 2003 Riskometer gave the results
shown in Table 1. For each index and each method five numbers have been calculated,
excepting for Historical Simulation, GARCH modelling and Point process approach methods
which do not provide volatility figures. The first four are estimates of Value-at-Risk (VaR)
and Expected Shortfall (ES) at probability levels of 95% and 99%, respectively. Each of these
numbers may be interpreted as potential daily percentage losses for September 24, being
based on closing data up to September 23.
Table 1: Risk Measures on September 24, 2003
Method

DJIA

DAX

S&P500

VaR 95%

ES 95%

VaR 99%

ES 99%

Volatility

1
2
3
4
6
1
2
3
4
6
1
2
3
4
6

2.11
1.97
2.04
1.50
1.88
3.06
2.90

3.04
2.67
3.96
2.19
2.15
2.14
1.63
1.99

2.64
2.76
2.56
2.29
2.71
3.84
3.92
3.81
3.94
5.59
2.75
2.93
2.68
2.34
2.81

2.98
3.16
2.89
2.25
3.22

4.34
4.86
4.30
3.87
6.64
3.10
3.62
3.02
2.45
3.30

3.41
4.13
3.31
3.04
4.03
4.97
5.82
4.93
4.97
7.94
3.55
4.21
3.47
3.14
4.12

20.17
NA
19.63

NA
NA
29.67
NA
29.22
NA
NA
21.06
NA
20.56
NA
NA

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If we concentrate on the DJIA, for the Exponentially Weighted Moving Average method
(method no. 3), a 95% VaR number of 2.04% indicates that the estimated 5th percentile of the
predictive return distribution form day was –2.04%; we estimate that there is one chance in 20
that the return is a loss of magnitude greater than 2.04%. An ES number of 2.56% indicates
that, in the event that such a 20-day loss occurs, this will be its expected size. The 99% VaR
and ES estimates are 2.89% and 3.31%, respectively.
The VaR and ES estimates are all driven by the annualised volatility estimate. It is obtained
by taking the standard deviation of the one-day distribution and multiplying it by the square
root of 260 representing the approximate number of trading days in a year. This number is
best interpreted in relation to other annualised volatility numbers and not necessarily as an
absolute measurement. From Table 1 we see that annualised volatility of the DAX on
September 24 is considerably larger than that of the two American indices, showing that the
German market was more turbulent on this date.
Table 2 shows the daily summary for January 9, 2004, by which time American and European

markets had calmed down.
Table 2: Risk Measures on January 9, 2004
Method

DJIA

DAX

S&P500

VaR 95%

ES 95%

VaR 99%

ES 99%

Volatility

1
2
3
4
6
1
2
3
4
6

1
2
3
4
6

2.08
1.96
2.01
1.29
1.71
3.03
2.84
2.99
1.52
2.82
2.16
2.12
2.10
1.30
1.77

2.60
2.73
2.53
2.06
2.42
3.81
3.89
3.74

2.71
3.82
2.71
2.89
2.63
2.06
2.31

2.93
3.16
2.85
1.94
2.85
4.30
4.74
4.22
2.20
4.46
3.06
3.48
2.97
1.95
2.64

3.36
4.13
3.26
2.65
3.57
4.92

5.82
4.84
3.36
5.28
3.50
4.17
3.40
2.67
3.17

19.81
NA
19.36
NA
NA
29.31
NA
28.70
NA
NA
20.70
NA
20.16
NA
NA

All three annualised volatilities have fallen. For instance, the EWMA method indicates that
the Dow Jones annualised volatility has fallen from 19.63% to 19.36%, the DAX volatility
has fallen from 29.22% to 28.70%, and the S&P500 volatility has fallen from 20.56% to
20.16%. Correspondingly, all risk measures are now lower. The 95% VaR for the DJIA is

now 2.01% and the ES 2.53%. The 99% numbers for VaR and ES are 2.85% and 3.26%.
2.4.1 Volatility
Riskometer allows us to explore the recent historical development of the annualised volatility.
Daily annualised volatilities estimates of our equity market indices since the start of 1998 are
graphically represented online and in Figure 1.
-6-


Figure 1: Annualised Volatility Figures of Equity Market Indices

The graph ends with the volatility estimates of January 8, 2004. The forecasts for January 9
are not shown, but the decrease of volatilities for the three indices after an extremely volatile
period is obvious. The peak volatility for the year 2003 for both American indices occurs on
March 24. The peak volatility for the DAX occurs on April 7, so it is clear that the two
American indices follow each other closely, but the DAX has a somewhat different
behaviour. Furthermore, we clearly observe that the DAX has been the most volatile index
during 2003.
Concentrating on the American indices, we see that the relatively calm present period comes
after a long period of extreme volatility. Indeed, it is the first low volatility period since
autumn of 2002. Throughout the second part of 2002 and the first three quarters of 2003,
volatilities attained spectacular levels. The highest peaks on both the DJIA and S&P500
indices occurs on July 29, 2002 and August 8, 2002 respectively. On those days, volatility
figures reached around 40%. Then, during the fourth quarter of 2002 we peaks of 15 October
for S&P500 and October 17 for DJIA. Once again, after a slight decrease, volatility figures
reached levels of 40%.
Concerning DAX, in the second part of 2002, its volatility peaked on August 8, as S&P500
and on October 17, as DJIA with values over 60%. The last high volatility period of March –
April 2003 of American indices has been followed by the DAX around two weeks later.
Indeed, on April 7, the German index annualised volatility attained once again impressing
figures over 50%. Since autumn 2003 Equity markets indices volatilities decayed to more

modest levels and settle again below 20%.

-7-


2.5 Backtesting
Riskometer backtests itself daily and updates a violation count table that allows a user to see
the historical performance of the method since start of 2001.
To appreciate what happens we look again at January 9, 2004. By the end of that day it was
possible to evaluate what had happened on the markets and to compare risk measures with
reality. In fact, all of the indices had decreased in value, by 1.27%, 0.73% and 0.89%,
respectively, for the DJIA, DAX, and S&P500 indices. Thus, for all indices, we observe that
the actual losses did not exceed the VaR estimates.
As an example for DJIA, the violations of method 3 are shown graphically in Figure 2. We
observe that the last violation of the 95% VaR was on May 19, 2003 and this of the 99% VaR
on March 24, 2003. Note that in this picture negative returns are shown as positive values,
and positive returns as negative values.
Figure 2: EWMA 95% and 99% VaR estimates

If VaR is being estimated successfully violations of the 95% VaR should occur once every 20
days on average, and violations of the 99% VaR once every 100 days. Whether this is
approximately true is more easily judged in Table 3.

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Table 3: Backtesting since start of 2001
Method

DJIA

DAX

S&P500

95%
observed

expected

P-value

99%
observed

expected

P-value

1
2
3
4
6
1
2
3
4
6
1

2
3
4
6

44
46
38
32
38
74
62
69
56
59
44
43
40
36
42

36
36
36
36
36
37
37
37
37

37
36
36
36
36
36

0.18
0.10
0.74
0.50
0.74
0
0
0
0
0
0.18
0.24
0.50
1
0.31

14
7
6
6
8
38
14

26
8
16
11
5
5
6
9

7
7
7
7
7
7
7
7
7
7
7
7
7
7
7

0.02
1
0.71
0.71
0.71

0
0.02
0
0.72
0
0.15
0.46
0.46
0.71
0.46

It shows the results of a backtest from the start of 2001. It covers a period in which we might
have expected 36 violations of the 95% VaR for the American indices and 37 violations of the
95% VaR for the German index and 7 violations of the 99% VAR for all indices. In the
95% observed column we see the actual numbers of violations incurred by Riskometer at the
95% level and in 99% observed column we see the actual numbers of violations incurred by
Riskometer at the 99% level. In the case of DJIA for method 3 it is 38 at the 95% level, and 6
at the 99% level. Although the 95% number is slightly higher than expected, we observe that
VaR is being estimated accurately. A binomial test has been carried out and expressed in the
table as a p-value. A p-value less than or equal to 0.05 would be interpreted as evidence
against the null hypothesis of reliable VaR estimation. For our example this is not the case.
In the case of DJIA for method 1 we observe that, for VaR 99%, the binomial test p-value
indicates evidence against the null hypothesis.
For S&P500, we notice that all methods succeed the binomial test. This confirms a general
good performance of the Riskometer methods at the 95% level and at the 99% level.
Finally, for DAX, we observe that only the GARCH modelling method succeed the binomial
test and only at the 99% level. Concerning the remaining VaR estimates we observe that
Riskometer methods perform very badly at the 95% level and at the 99% level presenting a
systematic underestimation of the VaR and too many violations.

-9-


3

Empirical Characteristics of Hedge Fund Strategies

3.1 Introduction
The hedge fund industry has considerably increased over the last ten years. Indeed, it is now a
more than $700 billion industry with more than 7000 funds worldwide1. Seen its important
development and a widespread acceptance, this alternative investment asset class needs
particular attention concerning its risk exposures.
Actually, understanding the risk exposures of hedge fund strategies has become a rather
important area of research for several reasons. First, a better understanding of hedge fund
risks is needed for individuals and institutions desiring to make investment decisions
involving hedge funds. A detailed analysis of hedge fund risks and returns is also important
from the standpoint of asset pricing theory. Understanding the hedge fund risks exposures is
also a key feature to the design of optimal risk-sharing contracts between hedge fund
managers and investors.
Issues regarding the nature of risks associated with different hedge fund strategies are
challenging because of the complex nature of the strategies in particular, since hedge funds
returns exhibit non-linear option-like exposures to traditional asset classes2. Furthermore,
partly due to less stringent disclosure requirements and partly due to the freedom granted to
the manager about his or her investment strategy it is difficulty to obtain detailed information
concerning a particular hedge fund risk exposures. Thus, their identification remains quite
problematic.

3.2 Data
One purpose of hedge fund indices is to serve as proxy for returns to the hedge fund asset
class. Major hedge fund indices particularly known are the Hedge Fund Research Indices

(HFR) and Credit Suisse First Boston/Tremont Indices (CSFB/Tremont Indices). HFR Index
was launched in 1994 with data going back to1990 and the CSFB/Tremont Index, which is an
asset weighted hedge fund index, was launched in 1999 with data going back to 1994.
In this study we analyse hedge fund strategies using the database of indices compiled by
Morgan Stanley Capital International (MSCI). The MSCI Hedge Fund Indices, launched in
July 2002, consist of over 190 indices based on the MSCI Hedge Fund Classification
Standard. The MSCI hedge fund database currently contains more than 1600 hedge funds
representing more than $175 billion in assets.
Our data sample covers a six years period from October 1997 to September 2003 (N=72). As
shown in Figure 1 this period has been characterised by important market up and down moves
additionally to high and low volatility periods.
It is well known that using a specific sample from an unobservable universe of hedge funds
introduces two types of bias, Selection bias and Survivorship bias, characterising main
sources of difference between the performance of hedge funds in the database and the
performance of hedge funds in the population. It is extremely difficult to completely eliminate
the problem of survivorship bias and thus the problem of over-estimation of the true returns in
1
2

See Henessee group research paper (2002)
See Fung and Hsieh (1997)
- 10 -


hedge fund strategies. This difficulty is due to the fact that the database only contains the
returns of the successful funds, or at least of those that are still in existence. In our study, we
focus on style analysis, dependencies between strategies and with traditional asset classes and
less on performance measurement. Therefore, the effect of exclusion of funds that did not
survive becomes relatively small.
We analyse monthly returns of ten hedge fund strategies representing five different process

groups3.
Table 4: Hedge Fund Strategies
Process Group
Directional Trading

Strategy
Discretionary Trading
Systematic Trading

Relative Value

Type
Directional Strategies

Convertible Arbitrage
Fixed Income Arbitrage

Non-Directional
Strategies

Merger Arbitrage
Security Selection

Long Bias
Short Bias

Directional Strategies

Variable Bias
Specialist Credit

Distressed Securities

Multi-Process

Directional Strategies

Event Driven

The selected strategies can be principally segregated in two categories. Thus, six of them are
considered as directional strategies (Discretional Trading, Systematic Trading, Long Bias,
Short Bias, Variable Bias, and Distressed Securities) and three of them as non-directional
strategies (Convertible Arbitrage, Fixed Income Arbitrage, and Merger Arbitrage). Event
Driven may contain elements of directional and non-directional strategies.
Returns of our strategies are captured through the corresponding MSCI indices, which are
equally weighted performance summary of funds from the MSCI database. It is important to
note that equal weighting of returns of hedge funds gives more weight to the smaller funds.
We keep in mind that it could affect our analysis to the extent that the inferred style exposures
represent the risk exposure of smaller funds to a greater extent as compared to those of the
larger funds.
To represent the broad range of asset classes in which hedge funds invest we use global
market indices. Thus, to incorporate the exposure to the US market, European and Japanese
equity markets (developed markets) and emerging equity markets, we include MSCI US
index, MSCI EU index, MSCI JP index, and the IFC emerging markets index. To assess the
exposure to bonds, we use the JP Morgan Government Bonds Index. To account for returns
arising from exposure to currencies and commodities, we include the Trade-Weighted Dollar
3

For definitions of the process groups and strategies see the Glossary
- 11 -

Index and the Goldman Sachs Commodity Index. We also include CSFB High Yield index to
incorporate returns available from investing in high yield securities. Finally we include the
volatility index VIX to incorporate the exposure to volatility. In Tables 5 and 6 we provide
information on the descriptive statistics and the Spearman’s correlation matrix4 of the global
indices. Furthermore, we add pairs scatter plots in Figure 3 for a better understanding of the
different relationships between the asset classes.
Table 5: Descriptive Statistics of Traditional Asset Classes
Index
MSCI US
MSCI EU
MSCI JP
EM
DWI
GSCI
JPMGBI
CSFBHY
VIX

Min

Max

Median

Mean

Stdev

Skewness

Kurtosis

-15.11
-14.19
-12.28
-33.20
-5.50
-18.17
-3.40
-7.03
-37.84

9.42
12.15
15.43
12.60
4.67
16.27
5.50
5.82
57.97

0.38
0.44
-0.93
0.72
0.09
-0.33

0.25
0.68
-0.46

0.05
-0.12
-0.41
-0.29
-0.06
0.20
0.52
0.38
-0.01

5.26
5.28
6.20
8.16
2.27
6.51
2.01
2.24
17.46

-0.44
-0.42
0.27
-1.05
-0.40
0.05

0.40
-0.57
0.55

2.83
3.30
2.33
5.30
2.78
2.99
2.85
4.97
3.67

Table 6: Traditional Asset Classes Rank Correlations
MSCI US

MSCI EU

MSCI JP

EM

DWI

GSCI

JPMGBI

CSFBHY

MSCI US

0.77

1.00

MSCI JP

0.46

0.44

1.00

EM

0.70

0.65

0.51

1.00

DWI

0.03

-0.15

-0.15

0.02

1.00

GSCI

-0.09

-0.09

0.30

0.07

-0.19

1.00

JPMGBI

-0.14

0.00

0.12

-0.18

-0.85

0.18

1.00

CSFBHY

0.33

0.35

0.28

0.46

-0.01

-0.03

0.03

1.00

VIX

4

1.00

MSCI EU

-0.69

-0.51

-0.34

-0.52

-0.02

0.08

0.19

-0.26

VIX

We compute Spearman’s rank correlations according to the definition on page 22
- 12 -

1.00


Figure 3: Relationships between Traditional Asset Classes

Before analysing in more details the risk-return characteristics of the chosen strategies it is

usually good practice to test for stationarity of the time series. Therefore, we use the KPSS5 to
test the null that our time series are integrated processes of order 0, i.e. I(0) in the case of a
constant and time trend. We also test for normality using the Jarque-Bera statistical test and
for the presence of autocorrelation using the Ljung-Box test. Finally we test for the presence
of ARCH effects in the time series using a Lagrange Multiplier test. The different test
statistics and p-values are reported in Table 7.
Results of KPSS test show that the null hypothesis is not rejected at any significant level for
any strategy. For the Jarque-Bera test only in the case of Systematic Trading strategy we do
not reject the null that data is normally distributed at a level of 5%. Furthermore, results of the
Ljung-Box test show that the null of no autocorrelation cannot be rejected at a level of 5% for
any strategy. Additionally, correlograms and partial correlograms in Figure 4 provide more
information about the sample Autocorrelation Function and Partial Autocorrelation Function
of the strategies returns. These diagnostics plots show for instance that for modelling
Convertible Arbitrage and Fixed Income Arbitrage strategies we could fit a AR(1) model.
Finally, LM test, testing for the null that there are no ARCH effects, results in cases of
Discretionary Trading and Merger Arbitrage in a p-value smaller than 0.05. Thus, it rejects
the null hypothesis.

5

Kwiatkowski, Phillips, Schmidt and Shin test. For details see Zivot and Wang (2003)
- 13 -


Table 7: Statistical Tests
Strategy

KPSS

Jarque-Bera6

Ljung-Box7

LM8

p-value

Test stat

p-value

Test stat

p-value

Test stat

p-value

Discretionary Trading

0.12

123.32

0.00

13.07

0.36

25.48

0.01

Systematic Trading

0.75

2.97

0.22

5.58

0.93

6.29

0.90

Convertible Arbitrage

0.58

6.83

0.03

21.02

0.05

11.89

0.45

Fixed Income
Arbitrage

0.35

2280.86

0.00

15.21

0.22

1.42

0.99

Merger Arbitrage

0.11

243.28

0.00

19.97

0.06

30.16

0.00

Long Bias

0.17

7.78

0.02

8.41

0.75

10.59

0.56

Short Bias

0.36

11.19

0.00

10.47

0.57

8.86

0.71

Variable Bias

0.14

12.16

0.00

12.04

0.44

18.06

0.11

Distressed Securities

0.61

155.82

0.00

20.97

0.05

10.89

0.53

Event Driven

0.38

150.43

0.00

13.59

0.32

19.32

0.08

Figure 4: Correlograms and Partial Correlaograms

6

Under the null hypothesis Jarque-Bera test statistic is asymptotically distributed χ2(2)
Under the null hypothesis Ljung-Box test statistic is asymptotically distributed χ2(12)
8
Under the null hypothesis LM test statistic is asymptotically distributed χ2(12)
7

- 14 -


3.3 Risk-Return Characteristics
To obtain a general idea about performances of our hedge fund strategies in good and bad
times, we report their returns during seven large down moves and seven large up moves of the
MSCI World index over the sample period in Tables 8 and 9.
Table 8: Large Down Moves
Aug-98

Feb-01

Mar-01

Sep-01

Jun-02

Jul-02

Sep-02

Average

MSCI World

-14.45

-8.93

-6.97

-9.35

-6.40

-8.89

-11.79

-9.54

Discretionary
Trading
Systematic
Trading

-8.38

-0.30

0.39

-0.89

0.70

-0.01

0.55

-1.13

11.38

0.61

5.93

3.84

8.25

4.02

3.89

5.42

Convertible

Arbitrage

-1.97

1.43

0.95

0.74

1.31

-0.89

1.64

0.46

Fixed Income
Arbitrage

-2.02

0.54

0.77

0.80

0.76

0.85

0.60

0.33

Merger
Arbitrage

-4.69

0.43

-0.72

-1.85

-0.53

-0.85

0.05

-1.17

Long Bias

-10.81

-2.44

-2.65

-4.69

-2.85

-4.72

-3.18

-4.48

Short Bias

21.20

6.35

4.19

7.52

3.32

5.05

4.22

7.41

Variable Bias

-7.34

-1.08

-1.92

-0.96

-0.17

-0.73

-0.34

-1.79

Distressed
Securities

-8.36

1.29

0.06

-0.99

-2.13

-2.69

-1.65

-2.07

Event Driven

-8.57

-0.07

-1.12

-3.11

-2.66

-3.85

-1.99

-3.05

Table 9: Large Up Moves
Feb-98

Oct-98

Dec-99

Mar-00

Apr-01

Oct-02

Apr-03

Average

MSCI World

6.43

8.53

7.69

6.59

6.99

7.02

8.29

7.36

Discretionary
Trading

2.41

3.46

3.66

4.98

0.93

0.71

0.89

2.43

Systematic
Trading

0.58

-0.56

2.09

-1.14

-4.79

-3.93

1.15

-0.94

Convertible
Arbitrage

1.27

-0.81

1.09

1.81

1.66

1.13

1.66

1.12

Fixed Income

Arbitrage

0.72

-8.35

1.64

0.05

1.57

0.02

0.80

-0.51

Merger
Arbitrage

1.85

1.65

0.53

1.09

0.56

0.21

0.90

0.97

Long Bias

4.77

3.01

10.95

2.18

3.07

1.37

4.09

4.21

Short Bias

-6.57

-5.68

-9.11

1.88

-8.29

-3.23

-4.67

-5.09

Variable Bias

5.26

0.65

9.91

1.14

0.73

-0.03

1.11

2.68

Distressed
Securities

1.87

-0.89

2.19

1.50

0.55

-0.28

3.87

1.26

Event Driven

3.20

1.17

1.46

1.58

1.60

0.50

3.97

1.93

- 15 -


Clearly, the returns of non-directional strategies are much less exposed to the impact of
market turbulence than those of their directional counterparts, in line with our findings on the
aggregate return volatility. Moreover, the returns of non-directional strategies are less aligned
with the direction of underlying market move. This provides a first confirmation of the
claimed neutrality with respect to market factors.
The returns of the directional strategies are without exception much larger reflecting the fact
that they carry systematically a net exposure to major market factors. We have to distinguish
however between those strategies which are either predominantly net long like Long Bias or
Distressed Securities or net short from those strategies that constantly change the direction of
their exposure like Systematic and Discretionary Trading. The returns of these strategies are
not or even negatively correlated to the MSCI World Index. Interesting enough, the
Systematic Trading strategies, by focusing predominantly on systematic trend-following
strategies, managed to make substantial positive returns in all down periods and to keep the
losses in up periods under control. This behaviour motivates the adjunction of Systematic
Trading to any hedge-fund portfolio despite their relatively low risk-adjusted return profile.
Table 10 reports the summary statistics for our MSCI Hedge Fund indices. Directional
strategies have higher monthly returns than the non-directional ones but are obviously also
more volatile as measured in terms of the standard deviation of monthly returns. Indeed,
during the sample period, the average monthly return of the directional strategies was 0.93%

and that of the non-directional was 0.76%. The average standard deviation was 3.28% and
1.14% respectively.
We also notice that non-directional strategies show a higher left side asymmetry in the
distribution functions than those of the directional strategies. Indeed, the average skewness of
non-directional strategies is –2.34 and compares to –0.31 for the directional strategies, where
only Discretionary Trading and Distressed Securities show a left side asymmetry in
distribution functions. Concerning the tails of the distribution function, once again nondirectional strategies’ distributions present heavier tails than those of directional strategies’
functions. The average kurtosis of non-directional strategies is 14.71 and this of directional
strategies 5.96. Event Driven strategy characteristics being clearly in between those described
above, it confirms its particular status of a multi-process investment strategy. Please note that
the averages for the statistical parameters mentioned above are taken on very disparate
populations (i.e. number of funds or assets within a strategy class) and thus do not have any
economic meaning. They are given only for illustrative purposes.
Fixed Income Arbitrage and Merger Arbitrage show particular high negative skewness and
high kurtosis values. The high kurtosis is induced mainly by large negative returns. In the
case of Fixed Income Arbitrage, these high figures are associated with a few large return
figures realised during turbulent market situations like the LTCM crisis or the July-August
2003 fixed-income sell-offs. In fact, fixed-income arbitrageurs provide usually relatively
small but regular returns. However, by capturing the small relative movements between
different assets or rates using high leverage, they expose themselves to losses when markets
move beyond the usual fluctuation bands of the asset or interest-rate spreads they arbitrage.
Merger arbitrage captures the price differential between the stock prices of the acquiring
company and the target company with respect to the agreed conversion ratio. The arbitrageur
has a high probability to gain the price difference known in advance. However, he accepts that
in case the merger does not go through, he looses an amount substantially higher than the
- 16 -


premium to be made. The strategy therefore resembles that of a short selling of a put option.
This explains the asymmetry of the return distribution.

Table 10: Descriptive Statistics of Hedge Fund Strategies
Min

Max

Median

Mean

Stdev

Skewness9

Kurtosis10

Discretionary Trading

-8.38

5.09

0.80

1.05

1.97

-1.31

8.85

Systematic Trading

-6.44

11.38

0.68

0.92

3.51

0.47

3.31

Convertible Arbitrage

-2.08

3.06

0.99

0.88

0.99

-0.59

3.94

Fixed Income Arbitrage

-8.35

2.26

0.84

0.65

1.36

-4.53

29.04

Merger Arbitrage

-4.69

2.76

0.83

0.76

1.06

-1.91

11.16

Long Bias

-10.81

11.33

0.96

1.16

3.55

0.03

4.61

Short Bias

-18.42

21.20

0.71

0.67

5.93

0.18

4.90

Variable Bias

-7.34

9.91

0.52

0.98

2.86

0.46

4.79

Distressed Securities

-8.36

3.87

1.12

0.82

1.88

-1.73

9.32

Event Driven

-8.57

4.69

1.09

0.82

1.95

-1.67

9.25

Strategy

Figures 5 to 8 confirm the larger variability measures of directional strategies’ returns than
those of non-directional strategies. Furthermore, comparing with the MSCI World index, we
notice that directional and non-directional strategies’ annualised volatility11 figures are clearly

smaller. The only exceptions are the Security Selection strategies.
The decreasing volatility after Q3 1998 for many of the strategies and Fixed Income Arbitrage
in particular are due to the wide-scale de-leveraging in the industry after the LTCM crisis as
well as increased emphasise on risk management.

9

A negative skewness points to a higher probability for large negative values relative to positive values
A kurtosis value higher than 3 indicates fat-tailed returns relative to the normal distribution
11
Annualised volatilities are computed with the Exponentially Weighted Moving Average method with a decay
parameter lambda equals to 0.94
10

- 17 -


Figure 5: Directional Trading Strategies

Figure 6: Relative Value Strategies

- 18 -


Figure 7: Security Selection Strategies

Figure 8: Specialist Credit and Multi-Process Strategies

- 19 -

To evaluate the risk-return tradeoff, we compute the Sharpe Ratio according to the following
formula

Sharpe Ratio =

1
N

∑ (R
N

t =1

t

− rt rf

)

σS

where Rt represents the strategy return for the month t and rtrf the risk free rate value for the
month t. N represents the total number of months and σS is the standard deviation of the
strategy monthly returns.
Seen its definition, we note that Sharpe Ratio, using a standard deviation as measurement of
volatility to adjust for risk, may actually punish a strategy for a month of exceptionally high
performance.
To avoid this issue we compute also the Sortino Ratio. Thus, instead of using a standard
deviation in the denominator, we use a downside deviation, which is a measurement of

strategy return deviation below a minimal acceptable rate and is computed as follows

∑ (R
N

σ

dd

=

t =1

t

− rt ma

)

2

N

where Rt represents the strategy return for the month t and rtma the minimal acceptable rate
value for the month t. N represents the total number of months.
By σdd, Sortino ratio allows to determine a measurement of return per unit of risk on the
downside.
As risk free rate we use the LIBOR 3 months and as minimal acceptable rates for the Sortino
Ratio we use the LIBOR 3 months and 0%. We report monthly and annualised figures
respectively for both ratios in Table 11.

In terms of Sharpe ratio, we observe that the non-directional strategies exhibit better riskreturn tradeoffs compared to the directional ones. Indeed, the non-directional strategies
exhibit on average an annualised ratio of 1.35 that is almost two times that for the directional
ones which is equal to 0.76. In particular, we observe the highest annualised figure of 1.92 for
the Convertible Arbitrage strategy and in the opposite the lowest annualised figure of 0.19 for
the Short Bias strategy. We notice the relatively high Sharpe ratio for Discretionary Trading
compared to the other directional strategies. Remembering that Discretionary Trading is the
only directional strategy with relatively high negative skewness and kurtosis, the higher return
per unit of volatility might be considered as a compensation for this additional risk.

- 20 -


Table 11: Sharpe and Sortino Ratios
Strategy

Sharpe Ratio

Sortino Ratio
(LIBOR 3M)

(0%)

Discretionary
Trading
Systematic
Trading
Convertible
Arbitrage
Fixed Income
Arbitrage

Merger
Arbitrage

0.35

1.27

0.30

1.06

0.45

1.67

0.16

0.58

0.19

0.69

0.34

1.23

0.54

1.92

0.50

1.79

0.85

3.10

0.22

0.78

0.10

0.37

0.19

0.67

0.38

1.35

0.28

0.99

0.41

1.47

Long Bias

0.23

0.83

0.23

0.85

0.35

1.30

Short Bias

0.05

0.19

0.05

0.19

0.12

0.43

Variable Bias

0.22

0.79

0.27

0.95

0.40

1.45

Distressed
Securities

0.25

0.88

0.21

0.74

0.34

1.23

Event Driven

0.24

0.85

0.18

0.63

0.33

1.20

In terms of Sortino ratio, the non-directional strategies exhibit still better risk-return tradeoffs
compared to the directional ones independently of the minimal acceptable rate. In the case of
LIBOR 3 months as minimal acceptable rate non-directional strategies exhibit on average an
annualised ratio of 1.05 where the directional strategies exhibit a ratio of 0.75 and in the case
of a minimal acceptable rate of 0% non-directional strategies exhibit on average an annualised
ratio of 1.75 where the directional strategies exhibit a ratio of 1.22. In particular, we observe
large decreases in Sortino ratios of Fixed Income Arbitrage and Merger Arbitrage strategies
comparing to their Sharpe ratios. This is mainly due to the important left asymmetries
observed in their distribution functions. This is consistent with the definition and purpose of
the Sortino ratio.
Seen the above risk-return metrics, the non-directional strategies seem to have delivered better
risk-return tradeoffs compared to the directional strategies. We believe, however, that the
Sortino ratio does not entirely capture the embedded risks, in particular event and liquidity
risk, related to these strategies.

3.4 Dependence Structure Analysis

Understanding relationships among hedge fund strategies is a key issue for investors
especially in the context of hedge fund portfolios. To assess this specific problem,
practitioners often use the Pearson’s linear correlation. Although widely applicable, this
method presents several deficiencies. To avoid potential drawbacks and improve our
understanding about dependence structures in hedge fund strategies we focus on analysing the
bivariate distributions of pairs of strategies using copulas. These statistical tools allow us to
separate the dependence and the marginal behaviour of two hedge fund strategies, what is
particularly useful for analysing the behaviour of extreme values.

- 21 -


3.4.1

Linear Correlation as Dependence Measure

To summarise dependence of two variables the most popular tool is the linear correlation.
However, it is well known that it represents only one particular measure of dependency that
does not capture non-linear dependence relationships. Furthermore, it has been proved that
linear correlation presents serious deficiencies when one is working with models other than
the multivariate normal model. Thus, outside the elliptical world correlations must be
interpreted very carefully.
Seen the characteristics of our strategies we can consider that none of the strategies has a
univariate normal distribution. Furthermore, comparing to a normal distribution, our strategies
distributions are characterised by fat tails. In our situation, disadvantages of Pearson’s
correlation as measure of dependence between two strategies could be numerous. For
instance, the linear correlation could be undefined in the case where variances of strategies
were not finite. This problem appears especially when we work with heavy tailed
distributions. Another problem is that independence of two random variables implies they are
uncorrelated but zero correlation does not in general imply independence. Only in the case of

multivariate normal distributions it is permissible to interpret a zero correlation as implying
independence. Finally, a single observation could have an arbitrarily high influence on the
linear correlation which might decrease the robustness of the dependence measure.
3.4.2

Alternative Correlation Measures

To avoid some potential problems concerning the dependence measure we use rank
correlations. Rank correlations are principally based on the concept of concordance and
discordance which informally says that a pair of random variables are concordant if large
values of one tend to be associated with large values of the other and small values of one with
small values of the other.
To analyse relationships among hedge fund strategies we use one rank correlation measure
known as Spearman’s rank correlation. For two random variables X and Y with distribution
functions F1 and F2 and a joint distribution F the sample estimator of Spearman’s rank
correlation ρS(X,Y) is defined as follows
ˆ
ρ S (X ,Y ) =

N
N + 1
N +1
12

∑  rank ( X i ) − 2   rank (Yi ) − 2 
2
N ( N − 1) i =1 



where the rank of an observation Xi gives the position of Xi in the ordered random sample.
Contrary to the linear correlation, it is considered as a measure of the degree of monotonic
dependence between X and Y. The transformation of X1,…, XN to 1,….,N has the consequence
that the marginal distributions of X and Y will be ignored. Therefore Spearman’s rank
correlation is a distribution independent measure.
Other advantages of Spearman’s rank correlation over linear correlation are that it is invariant
under monotonic transformations and the perfect dependence corresponds to correlations of 1
and –1. Spearman’s rank correlation is quite robust against outliers. Nevertheless, Spearman’s
rank correlation do not lend itself to the same elegant variance-covariance manipulations like
linear correlation, it is not a moment-based correlation.
Additionally to the results regarding correlations among hedge fund strategies, reported in
Table 12, we report in Table 13 Spearman’s rank correlations measuring dependence degrees
between hedge fund strategies and traditional asset classes.
- 22 -


Concerning rank correlations among hedge fund strategies we observe that Discretionary
Trading seems to be mainly correlated with Variable Bias and Merger Arbitrage. We also
notice that Systematic Trading present a very low degree of dependence with the other
strategies considered. Regarding the non-directional strategies Convertible Arbitrage and
Fixed Income Arbitrage present a non-negligible dependence with each other. This can be
explained by the fact that both strategies are penalised by turbulent fixed-income markets
often characterised by flight-to-quality situations often generating low market liquidity and
wide bid-offer spreads in a wide range of securities. For similar reasons and given the fact
that Convertible Arbitrage books hold significant credit risk, Convertible Arbitrage is
significantly correlated with Distressed Securities and Event Driven. Merger Arbitrage
presents relatively important degrees of dependence with Long Bias, Variable Bias and Event
Driven. Concerning Fixed Income Arbitrage, it presents very low degrees of dependence with
the other strategies. Security Selection strategies present particularly high degrees of
dependence with Distressed Securities and Event Driven strategies. Finally, without any

surprise, we observe that Short Bias strategy presents strong negative correlations with Long
Bias, Variable Bias, Distressed Securities and Event Driven strategies.
Table 12: Hedge Fund Strategies Rank Correlations
Discretionary
Trading

Systematic
Trading

Convertible
Arbitrage

Fixed
Income
Arbitrage

Merger
Arbitrage

Long Bias

Short Bias

Variable
Bias

Distressed
Securities

Discretionary

Trading

1.00

Systematic
Trading

0.14

1.00

Convertible
Arbitrage

0.39

0.03

1.00

Fixed Income
Arbitrage

0.26

-0.01

0.43

1.00

Merger
Arbitrage

0.48

-0.01

0.25

0.22

1.00

Long Bias

0.41

-0.17

0.21

0.13

0.40

1.00

Short Bias

-0.26

0.16

-0.17

-0.06

-0.21

-0.88

1.00

Variable Bias

0.62

-0.01

0.39

0.28

0.44

0.82

-0.68

1.00

Distressed
Securities

0.22

-0.19

0.41

0.32

0.20

0.63

-0.52

0.54

1.00

Event Driven

0.39

-0.13

0.41

0.27

0.57

0.78

-0.59

0.70

0.64

Event
Driven

- 23 -

1.00


Table 13: Dependence Measures between Hedge Fund Strategies and Traditional Asset Classes
MSCI US

MSCI EU

MSCI JP

EM

DWI

GSCI

JPMGBI

CSFBHY

VIX

Discretionary
Trading

0.22

0.24

0.21

0.33

-0.17

0.26

0.11

0.03

-0.03

Systematic
Trading

-0.29

-0.20

-0.16

-0.16

-0.16

0.23

0.34

-0.09

0.22

Convertible
Arbitrage

0.01

0.14

0.11

0.02

-0.34

0.13

0.30

0.32

0.03

Fixed Income
Arbitrage

-0.09

0.01

0.09

0.16

-0.02

0.19

-0.01

0.26

0.14

Merger
Arbitrage

0.26

0.27

0.17

0.16

0.01

0.08

-0.02

0.15

-0.25

Long Bias

0.77

0.73

0.59

0.79

-0.01

0.13

-0.14

0.47

-0.54

Short Bias

-0.77

-0.69

-0.59

-0.72

0.02

-0.14

0.11

-0.45

0.55

Variable Bias

0.45

0.56

0.59

0.60

-0.21

0.27

0.09

0.37

-0.24

Distressed
Securities

0.35

0.34

0.35

0.51

0.07

-0.02

-0.14

0.67

-0.31

Event Driven

0.56

0.57

0.37

0.60

-0.08

0.06

-0.07

0.54

-0.45

Rank correlations measuring dependence degrees between hedge fund strategies and
traditional asset classes show us that very few alternative investment strategies are correlated
with equity, currency, commodity or bond markets. Thus, Directional Trading and Relative
Value strategies do not present any significant relationship with traditional asset classes. The
low correlation figures between non-directional strategies and the different asset classes
confirm their market-neutrality. It is surprising, however, that we do not detect any significant
correlation between Convertible Arbitrage and the volatility index because most convertible
books are consistently long volatility. Concerning Security Selection strategies we observe
especially high degrees of dependence with equity indices. Furthermore, Long Bias and Short
Bias strategies present significant dependences with high yield and volatility index. In
opposition to Long Bias, Short Bias strategy is being negatively correlated with equity and
high yield indices but positively correlated with the volatility index. Distressed Securities and
Event Driven strategies present both important degrees of dependence with the emerging
market equity and high yield index. Furthermore, Event Driven seem to be significantly
correlated with US and Europe equity indices while it has a negative correlation with the
volatility index.
3.4.3

Archimedean Copulas

We saw that linear and rank correlations are not sufficient to measure dependence among
hedge fund strategies. This is partly due to the absence of models for our strategies. Thus, to
get a deeper idea about hedge funds dependence structures we need to construct bivariate
distributions which are consistent with given marginal distributions and correlations.

Knowing the joint distribution of two strategies provide us with information regarding their
marginal behaviors and allow us to evaluate the conditional probabilities that one strategy
takes a certain value given that the second strategy takes another value. To reach this aim we
determine an appropriate copula function chosen in a class of copulas known as Archimedean
copulas.
- 24 -