portfolio selection and hedge funds: linearity, heteroscedasticity, autocorrelation ... ·...

i

Portfolio Selection and Hedge Funds: Linearity, Heteroscedasticity, Autocorrelation and Tail-Risk

Robert John Bianchi B.Comm Griff., M.Bus(Res) QUT., F.Fin

Submitted in partial fulfillment of the requirements of the degree of

Doctor of Philosophy

The School of Economics and Finance Queensland University of Technology

Brisbane, Australia

April 2007

ii

Keywords and Abbreviations

• Autocorrelation

• Conditional Value at Risk (CVaR)

• Heteroscedasticity

• Heteroscedasticity and Autocorrelation Consistent (HAC)

• Linearity

• Mean-Conditional Value at Risk (M-CVaR) • Mean-Value at Risk (M-VaR) • Mean Variance Analysis (MVA)

• Modern Portfolio Theory (MPT)

• Portfolio Selection

• Tail-Risk

• Value at Risk (VaR)

iii

Abstract Portfolio selection has a long tradition in financial economics and plays an integral role

in investment management. Portfolio selection provides the framework to determine

optimal portfolio choice from a universe of available investments. However, the asset

weightings from portfolio selection are optimal only if the empirical characteristics of

asset returns do not violate the portfolio selection model assumptions. This thesis

explores the empirical characteristics of traditional assets and hedge fund returns and

examines their effects on the assumptions of linearity-in-the-mean testing and portfolio

selection.

The encompassing theme of this thesis is the empirical interplay between traditional

assets and hedge fund returns. Despite the paucity of hedge fund research, pension

funds continue to increase their portfolio allocations to global hedge funds in an effort to

pursue higher risk-adjusted returns. This thesis presents three empirical studies which

provide positive insights into the relationships between traditional assets and hedge fund

returns.

The first two empirical studies examine an emerging body of literature which suggests

that the relationship between traditional assets and hedge fund returns is non-linear. For

mean-variance investors, non-linear asset returns are problematic as they do not satisfy

the assumption of linearity required for the covariance matrix in portfolio selection. To

examine the linearity assumption as it relates to a mean-variance investor, a hypothesis

test approach is employed which investigates the linearity-in-the-mean of traditional

assets and hedge funds. The findings from the first two empirical studies reveal that

conventional linearity-in-the-mean tests incorrectly conclude that asset returns are non-

linear. We demonstrate that the empirical characteristics of heteroscedasticity and

autocorrelation in asset returns are the primary sources of test mis-specification in these

linearity-in-the-mean hypothesis tests. To address this problem, an innovative approach

iv

is proposed to control heteroscedasticity and autocorrelation in the underlying tests and

it is shown that traditional assets and hedge funds are indeed linear-in-the-mean.

The third and final study of this thesis explores traditional assets and hedge funds in a

portfolio selection framework. Following the theme of the previous two studies, the

effects of heteroscedasticity and autocorrelation are examined in the portfolio selection

context. The characteristics of serial correlation in bond and hedge fund returns are

shown to cause a downward bias in the second sample moment. This thesis proposes two

methods to control for this effect and it is shown that autocorrelation induces an over-

allocation to bonds and hedge funds. Whilst heteroscedasticity cannot be directly

examined in portfolio selection, empirical evidence suggests that heteroscedastic events

(such as those that occurred in August 1998) translate into the empirical feature known

as tail-risk. The effects of tail-risk are examined by comparing the portfolio decisions of

mean-variance analysis (MVA) versus mean-conditional value at risk (M-CVaR)

investors. The findings reveal that the volatility of returns in a MVA portfolio decreases

when hedge funds are included in the investment opportunity set. However, the

reduction in the volatility of portfolio returns comes at a cost of undesirable third and

fourth moments. Furthermore, it is shown that investors with M-CVaR preferences

exhibit a decreasing demand for hedge funds as their aversion for tail-risk increases.

The results of the thesis highlight the sensitivities of linearity tests and portfolio

selection to the empirical features of heteroscedasticity, autocorrelation and tail-risk.

This thesis contributes to the literature by providing refinements to these frameworks

which allow improved inferences to be made when hedge funds are examined in

linearity and portfolio selection settings.

v

Table of Contents Keywords and Abbreviations............................................................................................ ii

Abstract ............................................................................................................................ iii

Table of Contents ...............................................................................................................v

List of Tables ................................................................................................................. viii

List of Figures ....................................................................................................................x

Statement of Original Authorship ................................................................................... xii

Acknowledgements ........................................................................................................ xiii

1. Introduction ....................................................................................................................1

1.1 Overview and Rationale...........................................................................................1

1.2 Key Research Questions ..........................................................................................3

1.3 Research Methodology ............................................................................................5

1.4 Thesis Structure and Research Contributions ..........................................................7

2. Literature Review.........................................................................................................10

2.1 Introduction ............................................................................................................10

2.2 Risk ........................................................................................................................11

2.3 Portfolio Selection..................................................................................................15

2.3.1 Markowitz .......................................................................................................16

2.3.2 Safety-First and Downside-Risk .....................................................................17

2.3.3 Mean-Value at Risk (M-VaR).........................................................................19

2.3.4 Mean-Conditional Value at Risk (M-CVaR) ..................................................20

2.3.5 Empirical Issues in Portfolio Selection ...........................................................20

2.3.6 Serial Correlation in Asset Returns.................................................................23

2.4 Linearity .................................................................................................................24

2.5 Hedge Funds ..........................................................................................................31

2.6 A Synthesis ............................................................................................................39

vi

3. The Linear Behaviour of Stocks and Bonds.................................................................41

3.1 Introduction ............................................................................................................41

3.2 Related Literature...................................................................................................44

3.3 Method ...................................................................................................................49

3.3.1 Univariate Framework.....................................................................................51

3.3.2 Bivariate Framework.......................................................................................52

3.3.3 Keenan (1985) Test ........................................................................................53

3.3.4 Tsay (1986) Test..............................................................................................56

3.3.5 Teräsvirta, Lin and Granger (1993) V23 Test.................................................58

3.3.6 Equality of Two Regression Coefficients Test ...............................................61

3.4 Data .......................................................................................................................62

3.5 Results ...................................................................................................................66

3.5.1 Univariate Results ...........................................................................................66

3.5.2 Bivariate Results .............................................................................................69

3.6 Conclusion.............................................................................................................75

4. The Linear Behaviour of Hedge Funds and Traditional Asset Classes........................96

4.1 Introduction ...........................................................................................................96

4.2 Related Literature..................................................................................................98

4.3 Method ................................................................................................................101

4.4 Data ......................................................................................................................102

4.5 Results .................................................................................................................107

4.5.1 Univariate Results ........................................................................................107

4.5.2 Bivariate Results ..........................................................................................108

4.5.3 Heteroscedasticity and Rare Events in Linearity Tests ................................118

4.6 Conclusions .........................................................................................................121

vii

5. Portfolio Selection and Hedge Funds: Serial Correlation and Tail-Risk Effects.......144

5.1 Introduction ..........................................................................................................144

5.2 Related Literature.................................................................................................146

5.3 Method .................................................................................................................150

5.3.1 Mean Variance Analysis (MVA) Framework...............................................151

5.3.2 Mean-CVaR (M-CVaR) Framework ............................................................151

5.3.3 Autocorrelation Biased Second Sample Moment Adjustment......................152

5.3.4 Transforming Autocorrelated Returns to IID Returns ..................................152

5.3.5 Bayes-Stein Mean Shrinkage Estimation......................................................153

5.4 Data ......................................................................................................................155

5.5 Results ..................................................................................................................159

5.5.1 MVA and the Effects of Serial Correlation...................................................159

5.5.2 Tail-Risk Effects in Portfolio Selection ........................................................165

5.5.3 Extreme Dependence Effects in M-CVaR Portfolio Selection .....................167

5.6 Conclusion ...........................................................................................................170

6. Conclusion .................................................................................................................172

6.1 Introduction ..........................................................................................................172

6.2 Relevance .............................................................................................................174

6.3 Research Contributions ........................................................................................175

6.4 Policy Implications ..............................................................................................176

6.5 Avenues for Future Research ...............................................................................177

6.6 Conclusion ...........................................................................................................178

Appendix A ....................................................................................................................180

Appendix B ....................................................................................................................185

Appendix C ....................................................................................................................190

References ......................................................................................................................198

viii

List of Tables Table 2.1 Definitions of Hedge Funds .............................................................................30

Table 3.1 Summary Statistics...........................................................................................64

Table 3.2 Univariate Linearity-in-the-Mean Tests...........................................................67

Table 3.3 Tsay (1986) Test – Stocks................................................................................70

Table 3.4 Teräsvirta, Lin and Granger (1993) V23 Test – Stocks ..................................72

Table 3.5 Equality of Two Regression Coefficients Test- Stocks ...................................74

Annexure 3.C Keenan (1985) Bivariate Test – Stocks ....................................................88

Annexure 3.D Keenan (1985) Bivariate Tests – Bonds ...................................................89

Annexure 3.E Tsay (1986) Test – Stocks.........................................................................90

Annexure 3.F Tsay (1986) Tests – Bonds........................................................................91

Annexure 3.G Teräsvirta, Lin and Granger (1993) V23 Test – Stocks ...........................92

Annexure 3.H Teräsvirta, Lin and Granger (1993) V23 Tests – Bonds ..........................93

Annexure 3.I Equality of Two Regression Coefficients Test- Stocks .............................94

Annexure 3.J Equality of Two Regression Coefficients Test - Bonds.............................95

Table 4.1 Summary Statistics.........................................................................................103


Table 4.3 Univariate Linearity-in-the-Mean Tests – Hedge Funds ...............................106

Table 4.4 Keenan (1985) Bivariate Test – Stocks..........................................................109

Table 4.5 Tsay (1986) Bivariate Test – Stocks ..............................................................111

Table 4.6 Tsay (1986) Bivariate Tests – Bonds .............................................................112

Table 4.7 Teräsvirta, Lin and Granger (1993) V23 Bivariate Test – Stocks .................114

Table 4.8 Teräsvirta, Lin and Granger (1993) V23 Bivariate Tests – Bonds ................115

Table 4.9 Equality of Two Regression Coefficients Test – Stocks................................117

Table 4.10 Equality of Two Regression Coefficients Test – Bonds ..............................118

Table 4.11 Teräsvirta, Lin and Granger (1993) V23 Bivariate Test (ex August 1998)120

Annexure 4.C Keenan (1985) Bivariate Test – Stocks ..................................................136

Annexure 4.D Keenan (1985) Bivariate Tests – Bonds .................................................137

ix

Annexure 4.E Tsay (1986) Bivariate Test – Stocks.......................................................138

Annexure 4.F Tsay (1986) Bivariate Tests – Bonds......................................................139

Annexure 4.G Teräsvirta, Lin and Granger (1993) V23 Bivariate Test – Stocks..........140

Annexure 4.H Teräsvirta, Lin and Granger (1993) V23 Bivariate Tests – Bonds.........141

Annexure 4.I Equality of Two Regression Coefficients (ETRC) Test – Stocks............142

Annexure 4.J Equality of Two Regression Coefficients (ETRC) Test – Bonds............143


Table 5.2 Mean-Variance Analysis (Original Sample)..................................................157

Table 5.3 Mean-Variance Analysis (Bayes-Stein Mean Estimates) ..............................158

Table 5.4 Mean-Variance Analysis (Blume, Keim and Patel (1991) Adjustment) ......160

Table 5.5 Mean-Variance Analysis (Geltner (1991, 1993) Adjustment)......................161

Table 5.6 Mean-CVaR Portfolio Optimisation (Original Sample) ................................163

Table 5.7 Mean-CVaR Portfolio Optimisation (Geltner (1991, 1993) Adjustment) .....164

x

List of Figures Figure 3.1 MSCI World Equity Index..............................................................................77

Figure 3.2 Standard and Poors (S&P) 500 All Return Index ...........................................77

Figure 3.3 MSCI USA Equity Index................................................................................78

Figure 3.4 Fama-French HML Risk Factor......................................................................78

Figure 3.5 Fama-French SMB Risk Factor ......................................................................79

Figure 3.6 Fama-French UMD Risk Factor .....................................................................79

Figure 3.7 Morgan Stanley World plus Emerging Sovereign Index................................80

Figure 3.8 J.P. Morgan Global Bond Index .....................................................................80

Figure 3.9 Lehman Global Aggregate Index....................................................................81

Figure 3.10 Morgan Stanley US Government Bond Index..............................................81

Figure 3.11 Lehman USA Aggregate Index.....................................................................82

Figure 3.12 S&P500 All Return Index vs. MSCI World Equity Index ...........................83

Figure 3.13 MSCI USA Equity Index vs. MSCI World Equity Index.............................83

Figure 3.14 HML vs. MSCI World Equity Index ............................................................84

Figure 3.15 SMB vs. MSCI World Equity Index.............................................................84

Figure 3.16 UMD vs. MSCI World Equity Index............................................................85

Figure 3.17 MS World Bond Index vs. MSCI World Equity Index ................................85

Figure 3.18 J.P. Morgan Global Bond Index vs. MSCI World Equity Index ..................86

Figure 3.19 Lehman Global Aggregate Index vs. MSCI World Equity Index ................86

Figure 3.20 Morgan Stanley U.S. Govt. Bond Index vs. MSCI World Equity Index......87

Figure 3.21 Lehman U.S. Aggregate Index vs. MSCI World Equity Index ....................87

Figure 4.1 TASS Index...................................................................................................122

Figure 4.2 TASS Multistrategy Index ............................................................................122

Figure 4.3 TASS Long/Short Equity Index....................................................................123

Figure 4.4 TASS Global Macro Index ...........................................................................123

Figure 4.5 TASS Dedicated Short Bias Index................................................................124

Figure 4.6 TASS Managed Futures Index......................................................................124

xi

Figure 4.7 TASS Equity Market Neutral Index .............................................................125

Figure 4.8 TASS Risk Arbitrage Index..........................................................................125

Figure 4.9 TASS Event Driven Index............................................................................126

Figure 4.10 TASS Distressed Securities Index ..............................................................126

Figure 4.11 TASS Fixed Income Arbitrage Index.........................................................127

Figure 4.12 TASS Event Driven Multistrategy Index ...................................................127

Figure 4.13 TASS Convertible Arbitrage Index ............................................................128

Figure 4.14 TASS Emerging Markets Index..................................................................128

Figure 4.15 TASS Index vs. MSCI World Equity Index ...............................................129

Figure 4.16 Multistrategy Index vs. MSCI World Equity Index ...................................129

Figure 4.17 Long/Short Equity Index vs. MSCI World Equity Index ...........................130

Figure 4.18 Global Macro Index vs. MSCI World Equity Index...................................130

Figure 4.19 Dedicated Short Bias Index vs. MSCI World Equity Index.......................131

Figure 4.20 Managed Futures Index vs. MSCI World Equity Index .............................131

Figure 4.21 Equity Market Neutral Index vs. MSCI World Equity Index.....................132

Figure 4.22 Risk Arbitrage Index vs. MSCI World Equity Index .................................132

Figure 4.23 Event Driven Index vs. MSCI World Equity Index ...................................133

Figure 4.24 Distressed Securities Index vs. MSCI World Equity Index .......................133

Figure 4.25 Fixed Income Arbitrage Index vs. MSCI World Equity Index ..................134

Figure 4.26 Event Driven Multistrategy Index vs. MSCI World Equity Index.............134

Figure 4.27 Convertible Arbitrage Index vs. MSCI World Equity Index......................135

Figure 4.28 Emerging Markets Index vs. MSCI World Equity Index ...........................135

Figure 5.1 HFR Fund of Funds Index vs. MSCI World Equity Index...........................166

Figure 5.2 HFR Fund of Funds Index vs. Lehman Global Aggregate Index.................166

Figure 5.3 Lehman Global Aggregate Index vs. MSCI World Equity index ................167

xii

Statement of Original Authorship The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature:

Date:

xiii

Acknowledgements The completion of this thesis has been a long journey travelled by many individuals over

the past three years. The following persons and entities deserve my thanks and

appreciation for their support.

I would like to thank my principal supervisor Associate Professor Michael Drew and my

associate supervisor Dr. Adam Clements for wise and reliable academic supervision. I

extend my deepest gratitude to Mike for his guidance, support and encouragement which

has allowed me to complete this thesis. Mike’s passion for research has provided me a

gift for which I am truly thankful. My sincere thanks goes to Adam for his

understanding and patience throughout the development of this thesis. Adam has

provided invaluable guidance over many aspects of this research and I thank him for

this.

I gratefully acknowledge the Australian Research Council (ARC) and the industry

partner, H3 Global Advisors Pty Ltd for their financial support in ARC Linkage project

LP0454161. I would like to extend my appreciation to Andrew, Mathew and Shoky

Kaleel at H3 who have provided continuing support, encouragement and discussion.

My thanks also go to the academic and professional staff at the School of Economics

and Finance at QUT. John Polichronis and Evan Reedman deserve a special mention for

their friendship and the countless cups of coffee.

Finally, I wish to express my heartfelt gratitude to my family who have endured more on

this journey than I. My thanks go to my parents, Ken, Dorothy, Giovanni and Vanda for

their constant love and support. To my children, Nicholas and Sophia, who have

tolerated an absent father with lost weekends and odd working hours. To my wife

Katrina, I thank you for your infinite love, patience and support which has allowed me to

complete this thesis.

1

1. Introduction

1.1 Overview and Rationale

The behaviour of asset returns in a portfolio context has a long and rich history since the

seminal work of Markowitz (1952). The development of modern portfolio theory (MPT)

emerged at a time where the investment universe was dominated by stocks, bonds, real

estate and cash. With the globalisation of funds management, the investment opportunity

set has expanded to include new forms of investment vehicles including alternative

assets. A growing financial sector within alternative assets is the global hedge fund

industry. With an expanding investment opportunity set which includes traditional and

alternative assets, it is important to consider how hedge fund returns interact with

traditional asset classes such as stocks and bonds. In this thesis, we examine the linear

and portfolio behaviour between traditional assets and global hedge fund returns.

The study of the hedge fund industry is motivated by the recent growth of these fund

managers in global financial markets. Laurelli (2007) estimates the funds under

management in the global hedge fund industry at US$1.9 trillion. The U.S. Securities

and Exchange Commission (2003) estimate the size of the hedge fund industry at

approximately $600 to $650 billion in funds under management in the United States

alone.1 In the Australian context, the APRA (2003) reports that superannuation trustees

surveyed have, on average, a 4 per cent allocation to hedge funds and some managers

have allocations as high as 10 per cent.2 As domestic and global pension funds continue

to increase their portfolio weightings to alternative assets, the global hedge fund industry

1 The United States Securities Exchange Commission (S.E.C.) (2003) expects the hedge fund industry to grow to over $1 trillion in the next five to ten years. As at 31 December 2002, the S.E.C.(2003) valued the U.S. stock market at $11.8 trillion, making the size of the hedge fund industry in the U.S. at approximately 5% of the value of the U.S. stockmarket. 2 APRA (2007) reports the size of the Australian superannuation industry at over A$1trillion (or A$1,000 billion).

2

will play a prominent role in portfolio investment decisions made today and in the

future.

The central theme of this thesis focuses on the empirical characteristics of hedge fund

returns and their interaction in two distinct frameworks. In the first setting, we explore

the linear behaviour of traditional asset classes and hedge funds as it relates to a mean-

variance investor. To estimate optimal asset weightings in portfolio selection, asset

returns must satisfy the unstated linearity condition. More specifically, the portfolio

selection framework employs the covariance matrix to adequately describe the linear

association between asset returns. If asset returns do not satisfy the linearity assumption

then the covariance matrix is an inadequate tool to describe the relationship between

asset returns. This would result in erroneous asset weightings being estimated in a

portfolio selection framework.

Motivated by the earlier works of Keenan (1985), Lo (2001), Terasävirta, Lin and

Granger (1993) and Tsay (1986), this thesis examines the linearity condition in

traditional assets and hedge funds by employing a hypothesis test approach to test the

linearity-in-the-mean. The findings show that the empirical characteristics of

heteroscedasticity and autocorrelation in asset returns cause the linearity-in-the-mean

hypothesis tests to be mis-specified resulting in the incorrect conclusion of non-linearity.

To address this problem, this thesis develops an augmented framework which controls

heteroscedasticity and autocorrelation in the underlying hypothesis tests. These adjusted

linearity-in-the-mean tests reveal that traditional assets and hedge funds are actually

linear-in-the-mean.

The second framework in this thesis considers traditional assets and hedge funds in a

portfolio selection setting. More specifically, we investigate the sensitivities of portfolio

selection to the empirical features of autocorrelation, heteroscedasticity and tail-risk.

Whilst there is no direct method to evaluate heteroscedasticity in portfolio selection, the

literature reveals a close association between heteroscedasticity and tail-risk. To

examine tail-risk in portfolio choice we compare and contrast the investment decisions

3

of mean-variance analysis (MVA) versus mean-conditional value at risk (M-CVaR)

investors.

The findings reveal that hedge funds are desirable for MVA investors as they tend to

lower the volatility of portfolio returns at the cost of undesirable third and fourth

moments. This represents a conventional economic trade-off. For M-CVaR investors, it

is shown that as risk aversion increases, the demand for hedge funds decreases. When

the effects of serial correlation are examined, it is shown to cause a downward bias in

the second sample moment in asset returns. This thesis employs two methods from the

literature to adjust this bias and it is shown that portfolio selection models tend to over-

allocate to assets which exhibit significant serial correlation, such as bonds and hedge

funds. When tail-risk is considered in optimal portfolio choice, we find that an investor’s

aversion to tail-risk is associated with decreases in demand for hedge funds. In fact,

investors with a high aversion to tail-risk are found to exhibit zero portfolio weightings

to hedge funds.

1.2 Key Research Questions

The allocation of long-term savings by pension funds to the global hedge fund industry

provides the motivation of this thesis. As pension funds increase their allocation to

hedge funds, the behaviour of hedge funds in an investment portfolio emerges as the

motivating research question. To determine optimal portfolio choice between traditional

asset classes and hedge funds, the empirical characteristics of asset returns must satisfy

the theoretical assumptions of Markowitz (1952). However, researchers have found that

empirical hedge fund returns exhibit striking statistical properties. Lo (2001), Geman

and Kharoubi (2003) and Malkiel and Saha (2005) demonstrate that the distribution of

hedge fund returns does not adhere to the normal distribution. Other studies including

Asness, Krail and Liew (2001) and Getmansky, Lo and Makarov (2004) reveal

statistically significant serial correlation in hedge fund returns. Other researchers

including Agarwal and Naik (2004), Favre and Galeano (2002) and Mitchell and Pulvino

4

(2001) observe that hedge fund returns exhibit a non-linear relationship with traditional

asset classes. Finally, Agarwal and Naik (2004) and Brown and Spitzer (2006) report

tail-behaviour in hedge fund returns that occurs more frequently than a normal

distribution. These empirical features in hedge fund returns are problematic for

researchers as they violate the underlying conditions of the Markowitz (1952) portfolio

selection framework.

These empirical features of hedge fund returns therefore motivate the research in this

thesis to consider the following three questions. First, if hedge fund returns are non-

linear then is the non-linearity a function of the returns of the underlying assets? To

answer this research question, we commence the thesis by examining the linear

behaviour of global stocks and world bonds in both univariate and bivariate settings.

The second research question proceeds to consider the linear behaviour between hedge

funds and traditional asset classes. Whilst the works of Agarwal and Naik (2004), Favre

and Galeano (2002) and Mitchell and Pulvino (2001) suggest that hedge fund returns are

non-linear, none of these studies explain the source of the non-linearity nor do they

employ a hypothesis testing approach to the problem. Therefore, the second research

question considered in this thesis examines the linear behaviour of hedge funds in a

univariate setting and their bivariate relationship with traditional asset classes.

The third and final research question investigates optimal portfolio choice when the

investment opportunity set consists of global stocks, world bond and global hedge funds.

More specifically, this research question examines the sensitivities of optimal portfolio

choice in the presence of serial correlation and tail-risk in asset returns. This research

question has important implications for investors who wish to incorporate hedge fund

returns in a conventional portfolio selection framework.

5

1.3 Research Methodology

A number of quantitative approaches are developed to address the research questions

considered in this thesis. The first empirical chapter examines the linear behaviour of

traditional asset classes only. We examine the linear behaviour of stocks and bonds as it

may impact on the behaviour between hedge funds and traditional assets. The second

empirical chapter then proceeds to examine the linearity-in-the-mean in hedge fund

returns in a univariate setting and their relationship with traditional asset classes in a

bivariate framework. The third and final empirical chapter employs the MVA and M-

CVaR frameworks to examine the effects of serial correlation and tail-risk when hedge

funds are included in the investment opportunity set.

To examine the linear behaviour of traditional assets and hedge funds in the first two

empirical chapters, this thesis adopts the methodological apparatus developed by

Granger and Terasävirta (1993) and Lee, White and Granger (1993). The work of

Granger and Terasävirta (1993) and Tsay (2002) inform us that a test for linearity-in-the-

mean compares a standard linear regression with an auxiliary regression that includes

multiplicative regressors. A comparison of these two regressions is usually estimated

via an F-test or a t-test. When the relationship is linear-in-the-mean then the resulting F-

test (or t-test) is statistically insignificant. Conversely, a statistically significant test

statistic signifies that at least one multiplicative regressor has explanatory power.

To thoroughly examine linearity-in-the-mean, Granger and Terasävirta (1993) note that

linearity-in-the-mean hypothesis tests can be estimated in two settings. The first is

known as the univariate setting whereby current asset returns are regressed against non-

linear lagged variables of itself. Informally, linearity-in-the-mean tests in the univariate

setting are autoregressive (AR) regressions which test if quadratic or cubic lagged

variables possess explanatory power. The second form of linearity-in-the-mean

examines the bivariate linear relationship between two exogenous variables.

6

The first two empirical chapters employ the Keenan (1985), Tsay (1986) and

Terasävirta, Lin and Granger (1993) linearity-in-the-mean tests in both univariate and

bivariate settings. As an additional test in the bivariate setting, this thesis also considers

the Equality Test for Two Regression Coefficients which extends the work proposed in

Lo (2001). Although these tests are well established in the econometrics literature,

Granger and Terasävirta (1993) and Lee et. al., (1993) inform us that these types of tests

are prone to mis-specification if the error disturbances from the underlying regressions

are contaminated with heteroscedasticity and serial correlation. An analysis of the asset

returns in this thesis suggests that stock returns possess heteroscedasticity while bond

and hedge fund returns exhibit autocorrelation in returns. To improve the statistical

inference of the linearity-in-the-mean tests, this thesis develops a novel approach to

augment the hypothesis tests to control for the effects of heteroscedasticity and

autocorrelation.

The third and final empirical chapter examines optimal portfolio choice in an investment

universe of global stocks, world bonds and global hedge funds. To examine the effects

of autocorrelation of returns in portfolio choice, this thesis employs the procedures of

Blume, Keim and Patel (1991) and Geltner (1991, 1993) to adjust for the downward bias

in the second sample moment of asset returns.

To examine optimal portfolio choice and the effects of tail-risk, this thesis compares the

portfolio compositions given by MVA versus M-CVaR. The M-CVaR portfolio

selection model developed by Rockafellar and Uryasev (2000, 2002) is a desirable

framework as it reflects investor preferences who wish to minimise the size of the left-

tail of the distribution of portfolio returns. Clearly, an MVA investor with a preference

to minimise portfolio variance will derive a different portfolio composition to M-CVaR

investors.

7

1.4 Thesis Structure and Research Contributions

The final section of this introductory chapter outlines the structure of the thesis. Chapter

2 begins with a review of the literature which provides the contextual framework and

motivation for the research considered in this thesis. The review in Chapter 2 extends

across the disciplines of financial economics, econometrics and statistics as it considers

the scholarly contributions which inform the linearity and portfolio selection research

questions considered in this thesis.

Chapter 3 is the first empirical contribution which examines the linearity of traditional

asset classes in both univariate and bivariate settings. More specifically, we examine the

linearity-in-the-mean of monthly returns of global stocks and bonds as they are of

interest to mean-variance investors who employ expected mean returns, variances and

covariances in portfolio selection. The findings reveal that conventional linearity tests

tend to over-reject the null hypothesis of linearity-in-the-mean due to the empirical

characteristics of heteroscedasticity and/or autocorrelation in asset returns. More

specifically, the effects of heteroscedasticity and autocorrelation in the error

disturbances of the underlying tests cause an over-rejection of the null hypothesis of

linearity-in-the-mean. When the conventional linearity tests are augmented with a

heteroscedasticity and autocorrelation consistent (HAC) procedure, the findings reveal

that traditional asset classes are actually linear-in-the mean. The primary contribution of

this chapter is the empirical evidence which demonstrates that global stocks and world

bonds are linear-in-the-mean when examining monthly returns.

Chapter 4 of this thesis examines the univariate linear behaviour of hedge funds and the

bivariate linear behaviour between traditional asset classes and hedge fund returns.

Again, we discover that conventional linearity-in-the-mean tests over-reject the null

hypothesis of linearity due to heteroscedasticity and autocorrelation in the error

disturbances. When the tests for linearity-in-the-mean are corrected with HAC

procedures, the findings reveal that hedge funds and traditional asset classes are indeed

linear-in-the-mean. These results are in direct opposition to previous hedge fund

8

linearity studies. Previous hedge fund researchers do not address the effects of

heteroscedasticity and autocorrelation which are a major source of test mis-specification.

The contribution of Chapter 3 is the hypothesis testing approach which reveals that

hedge fund returns are linear-in-the-mean when the effects of both heteroscedasticity

and autocorrelation are controlled for in the underlying hypothesis tests. The implication

from Chapters 3 and 4 reveals that researchers may incorrectly conclude that asset

returns such as hedge funds are non-linear when in fact the empirical features of

heteroscedasticity and autocorrelation may be distorting the power of the linearity tests.

Chapter 5 is the third and final empirical chapter which examines the effects of serial

correlation and tail-risk in optimal portfolio choice. In an investment universe of global

stocks, world bonds and global hedge fund returns, we compare the portfolio

compositions of MVA and M-CVaR investors. The findings of this study show that the

autocorrelation in bond and hedge fund returns causes a downward bias in the second

sample moment (ie. the standard deviation of asset returns). The contribution of this

study demonstrates that this downward bias causes asset returns with serial correlation to

be more desirable in a portfolio selection framework. To examine this effect, the Blume,

Keim and Patel (1991) and Geltner (1991, 1993) procedures are employed to correct the

bias in the second sample moments in asset returns. The resulting portfolio optimisations

reveal a reduction in the optimal allocation to hedge funds of between 3 to 20 per cent.

The second part of the study in Chapter 5 examines the effects of tail-risk in portfolio

choice. It is shown that MVA investors exhibit a high demand for hedge funds which

allows them to reduce the volatility of portfolio returns at the cost of undesirable third

and fourth moments. In the M-CVaR setting, we reveal that investors exhibit a

decreasing demand for hedge funds as their aversion for tail-risk increases. The research

contribution of Chapter 5 highlights the excessive portfolio allocation to hedge funds

when the biases of serial correlation and the effects of tail-risk are not considered by the

MVA investor.

9

Chapter 6 provides a synopsis of the thesis along with the limitations associated with the

research. The main conclusions of the thesis are summarised and avenues for future

research are offered.

10

2. Literature Review

2.1 Introduction

The objective of this chapter is to review the literature which relates to the research in

this thesis. It is important to acknowledge that the literature review in this chapter is not

intended to be comprehensive, but rather, it focuses on the bodies of work which inform

and motivate the research questions considered in this thesis.

Section 2.2 commences the literature review by examining the concept of risk. The term

‘risk’ is one of the most important concepts in financial economics and we provide a

review of its development in the literature. Whilst there are many definitions and types

of risk, this thesis focuses on market risk which defines an investor’s risk preferences in

portfolio selection models.

The historic antecedents of portfolio selection are detailed in Section 2.3 of this chapter.

The third empirical study of the thesis considers traditional assets and hedge funds in

portfolio selection, therefore, a review of portfolio choice is warranted. The review of

the portfolio choice literature commences with the traditional MVA and details the

evolution of portfolio selection models to the present day M-CVaR portfolio framework.

Section 2.4 provides a review of the literature which addresses the concept of linearity.

The definition of linearity is developed and a review of the statistical tests to detect

linear or non-linear behaviour is given. The review of the linearity literature is integral

as it is used to address the research questions developed in the first two empirical

chapters of this thesis.

11

Proceeding the discussion on risk, portfolio selection and linearity, Section 2.5 presents

an encompassing review of the hedge fund literature. A detailed synopsis of the

scholarly contributions in the hedge fund literature is provided as it is the centrepiece of

this thesis. The review of this literature suggests that the introduction of hedge funds in

the investment universe provides researchers with numerous empirical challenges.

Finally, Section 2.6 concludes the literature review with a synthesis of the various

strands of literature presented in this chapter. A summary of the key issues is presented

in order to distill the key motivations, research questions and techniques employed in

this thesis.

2.2 Risk

The concept of risk is important in this thesis as it plays an important role in modern

portfolio theory and its subsequent development since Markowitz (1952, 1959). More

specifically, this section reviews the development of new definitions of risk as it

corroborates with the development of new portfolio selection models which are

reviewed in subsequent sections of this chapter.

The concept of ‘risk’ has had a long and rich history in the financial economics

literature. Despite risk being intuitively understood for many years, it was the influential

contribution of Markowitz (1952, 1959) which allowed risk to be formally defined as the

volatility of returns which is the most commonly used term to describe risk. Bernstein

(1996) highlights the fact that there is no generally accepted definition of risk as new

forms of measuring risk have evolved over time. This section of the literature review

does not attempt to consider every type of risk, but rather, a review of market risk is

summarised which details the types of risk measures considered in this thesis.3

3 There are many types of risk including market, operational, key-person, political and credit risk to name just a few. In this thesis the term ‘risk’ refers to market risk.

12

The modern definition of risk originates from Markowitz (1952, 1959) who numerically

defined it as the second moment of the normal distribution of returns. Variance and its

normalised variant, the standard deviation has become the conventional measures of risk

which have permeated in other empirical finance models including the Lintner (1965),

Mossin (1966) and Sharpe (1964) capital asset pricing model (CAPM) and the Black and

Scholes (1973) option pricing framework.

Whilst the second moment of asset returns remains a fundamental risk metric in finance,

it does suffer from two primary shortcomings. First, the second moment treats both

positive and negative returns in the same way thereby penalising upside movements.

Markowitz (1959) himself acknowledges this problem and proposes the semi-variance

as a viable alternative to variance. The second undesirable attribute is that variance is an

efficient measure of risk when asset returns satisfy the normality assumption, however,

Fama (1965a, 1965b), Officer (1972), Jansen and de Vries (1991) and Poon, Rockinger

and Tawn (2004) have demonstrated that traditional asset returns violate the normality

condition.

To address these shortcomings in the variance metric, scholars have explored alternative

measures of risk. A number of these competing risk measures have been developed to be

consistent with the axioms of von Neumann and Morgenstern (1944) expected utility

maximisation.4 One of the earliest risk measures developed was the lower partial

moment (LPM) which is sometimes referred to as the ‘downside risk’ framework. The

theoretical contributions of Bawa (1975), Bawa and Lindenberg (1977), Fishburn (1977)

and the subsequent empirical works of Leibowitz and Henriksson (1989), Leibowitz and

Kogelman (1991) and Harlow (1991) argue that risk can be defined as the undesirable

returns below a pre-specified threshold level.5 This body of work demonstrated that the

4 Campbell and Viceira (2002) acknowledge that the Markowitz (1952, 1959) framework is actually inconsistent with expected utility maximisation. The utility literature of Pratt (1964) demonstrates that absolute risk aversion is expected to decline or at least not increase with wealth. However, Markowitz (1952, 1959) quadratic utility implies that absolute risk aversion increases with wealth. 5 This strand of literature was inspired by Roy’s (1952) earlier work on portfolio selection. The scholarly contribution of Roy (1952) is detailed in a subsequent section of this chapter.

13

LPM was an alternative measure of risk which was consistent with the theoretical

stochastic dominance literature of Hadar and Russell (1969), Hanoch and Levy (1969),

Rothschild and Stiglitz (1970) and Whitemore (1970). Despite scholars such as Balzer

(1994), Estrada (2006) and Sortino and Forsey (1996) advocating the benefits of

downside risk, there has been limited acceptance of the LPM in the academic literature.

The lack of acceptance of the LPM can be partially attributable to the recent emergence

and popularity of the J.P. Morgan (1995) Value at Risk (VaR).

The introduction of the J.P. Morgan (1995) VaR has been motivated by the paradigm

shift to re-define risk as undesirable left tail-risk in the distribution of returns. In short,

VaR measures risk as large infrequent negative returns in the left-tail of the distribution

of returns rather than the conventional variance metric which measures the dispersion of

returns.6 The global emergence of VaR as a popular risk measure is reflected in its

adoption in global banking regulation via the Basle Committee of Banking Supervision

(1996, 2003). Despite the popularity of VaR, the academic literature by Artzner,

Delbaen, Eber and Heath (1997) has found statistical deficiencies which make it

problematic as an effective risk measure. VaR measures a quantile in the distribution of

returns, however, it cannot control extreme tail losses beyond the VaR estimate. The

work of Artzner et. al., (1997) also find that VaR lacks sub-additivity which contradicts

one of the basic tenets of modern portfolio theory.7

The statistical deficiencies of VaR have motivated researchers to develop more efficient

measures of risk. To evaluate new risk measures, the seminal work of Artzner et. al.,

(1997, 1999) propose a framework to evaluate the validity of ‘coherent’ risk measures.

Artzner et. al., (1997, 1999) argue that ‘coherent’ measures of risk require the four

6 Under the normality condition, VaR has been shown to be consistent with expected utility maximisation. Refer to Alexander and Baptista (2002) and Campbell, Huisman and Koedijk (2001) for further details. 7 The failure of VaR to satisfy the sub-additivity condition means that, at times, the risk of two positions combined can lead to a higher VaR measure than the sum of the two positions separately. To be consistent with modern portfolio theory, two positions with a maximum correlation of 1.0 would result in the total VaR being the sum of the two VaR positions.

14

mathematical properties of sub-additivity, homogeneity, monotonicity and transitional

invariance.8 The Artzner et. al., (1997, 1999) axioms of coherence provides a new

methodology by which researchers can introduce new concepts of risk in the literature.

In an effort to develop a coherent measure of tail-risk, Artzner et. al., (1997, 1999),

Rockafellar and Uryasev (2000, 2002), Acerbi (2002), Acerbi and Tasche (2002) and

Tasche (2002) propose the Conditional Value at Risk (CVaR) as an alternative to VaR.9

The CVaR metric measures the size of the left-tail of the distribution of returns. The

statistical shortcoming of VaR is that it measures the probability of the loss distribution

only and does not consider the shape of the losses to the left of the specified percentile

function. As an alternative, CVaR not only measures VaR but it also describes the first

moment of the shape of the losses which exceed VaR.10 Artzner et. al., (1997, 1999)

detail CVaR’s mathematical properties of coherence while Rockafellar and Uryasev

(2000, 2002) demonstrate its convexity properties which make it appealing in portfolio

choice problems. In short, the literature demonstrates that CVaR is a superior and more

efficient measure of tail-risk than VaR.

8 Artzner et. al., (1997, 1999) argue that a coherent measure of risk must possess the following four mathematical properties. Let X and Y be random variables with ρ as the coherent risk measure. The first mathematical property is Sub-Additivity which is given by )()()( YXYX ρρρ +≤+ . This means that the risk of X and Y combined must equal or be less than the risk of the individual sums of X and Y . The second mathematical property is Homogeneity which refers to )()( XX λρλρ = where 0≥λ which stipulates that the risk of λ financial exposures must be equivalent to λ multiplied by the single financial exposure. The third mathematical property is known as Monotonicity which states that for all losses, X and Y , if YX ≤ , then )()( YX ρρ ≤ . The fourth and final mathematical property is Transitional Invariance which states that there is a reduction in risk when a risk-free return α is introduced to an investment portfolio resulting in αραρ +=+ )()( XX . 9 The term Conditional Value at Risk (CVaR) was originally developed by Rockafellar and Uryasev (2000, 2002), however, it was also independently developed under the term Expected Shortfall (ES) by Acerbi (2002), Acerbi and Tasche (2002) and Tasche (2002). Strictly speaking, CVaR is defined as the weighted average of VaR and the losses which exceed VaR. Other definitions such as Mean Excess Loss and Expected Shortfall measure the expected losses when these losses exceed VaR. Finally, Tail VaR measures the expected losses of VaR and the losses which exceed VaR. Under continuous distributions, the CVaR is the same metric as Expected Shortfall. Refer to Rockafellar and Uryasev (2000, 2002) and Krokhmal, Palmquist and Uryasev (2002) for an introduction to mean-CVaR portfolio optimisation. 10 Similar to VaR, the CVaR metric satisfies expected utility maximisation under the assumptions of normality.

15

To sum up, the risk literature highlights the significant contributions made in the

development and refinement of market risk measures. The literature demonstrates that

undesirable risk such as tail-risk in empirical asset returns can be captured with the

recently developed CVaR. The CVaR risk measure is important in this thesis as the third

empirical chapter compares the portfolio composition of investors with MVA versus

CVaR risk preferences. The development of the risk literature cannot be under-

estimated as it lays the foundation for new empirical based portfolio selection models.

We proceed to review the portfolio selection literature and its development in finance as

it relates to the research questions considered in the third empirical study of this thesis.

2.3 Portfolio Selection

The portfolio selection literature is concerned with optimal portfolio decisions made by

investors given an expectation of future returns and associated risk. Since the seminal

beginnings of Markowitz (1952, 1959), many different approaches have been developed

in the literature. Some of these portfolio frameworks such as Merton (1969), Mossin

(1968) and Samuelson (1969) are tightly grounded within the von Neumann and

Morgenstern (1944) theory of expected utility maximisation. Other scholars such as

Roy (1952) and Rockafellar and Uryasev (2000, 2002) have developed portfolio

selection frameworks which are less constrained by the orthodoxy of utility theory.

The review of the portfolio selection literature in this chapter is not a comprehensive

summary. Instead, the synopsis of the literature in this chapter highlights the major

scholarly contributions which inform the research questions considered in the third

empirical study of this thesis.11 Specifically, we review the development of empirical

11 There are voluminous scholarly contributions in the portfolio selection literature which are not considered in this thesis. We do not review these portfolio selection frameworks as they do not inform the research questions in this thesis. Some of these portfolio selection models include the geometric mean portfolio approach developed by Latane (1959), Young and Trent (1969), Hakansson and Liu (1970) and Hakansson (1971a, 1971b). Another portfolio framework not considered in this thesis is the dynamic programming approach which originates from Mossin (1968), Merton (1969) and Samuelson (1969). Finally, the stochastic dominance framework developed by Quirk and Saposnik (1962) and the subsequent contributions by Hadar and Russell (1969), Hanoch and Levy (1969), Rothschild and Stiglitz (1970) and Whitmore (1970) is another portfolio selection approach not considered in this thesis.

16

based portfolio selection models including MVA, safety-first, downside-risk, mean-

value at risk and mean-conditional value at risk. To commence the review of the

portfolio selection literature, we begin with one of the most important contributions in

the history of modern finance, the work of Markowitz (1952).

2.3.1 Markowitz The MVA framework developed by Markowitz (1952, 1959) was the first cogent model

to mathematically express investor behaviour in a portfolio context. By treating asset

returns as random variables, Markowitz (1952, 1959) unified the concepts of return,

variance and covariance into a quantitative framework. Return was defined as the

weighted expected returns of each security in the portfolio while risk was expressed as

the second moment of portfolio returns. The covariance matrix of asset returns was then

used to capture the linear association between assets in an optimisation process. The

notion that behaviour of the overall investment portfolio is more important than the

behaviour of each individual security was the seminal contribution of Markowitz (1952,

1959) and the birth of MPT as we know it today.12 Influential economists of our time

such as Miller (1999) refer to the Markowitz (1952) publication in the Journal of

Finance as the ‘big bang’ in modern finance.

Although the MVA framework earned Harry Markowitz the 1990 Nobel Prize in

Economic Sciences, two major criticisms have been directed at the model’s underlying

assumptions. The first critique of the Markowitz (1952, 1959) framework is its

assumption of the Gaussian distribution. It is well accepted in financial economics that

financial market returns violate the normality assumption. Bawa (1978), Chamberlain

(1983), Frankfurter and Lamoureux (1987) and Jobson and Korkie (1980) fear that the

chasm between the normality assumption of MPT and the empirical characteristics of

asset returns can lead to inefficient portfolio construction.

12 As stated in Lochoff (2002) and Campbell and Viciera (2002), the Markowitz (1952, 1959) model is the dominant portfolio selection framework employed in the global funds management industry.

17

The second major critique of Markowitz (1952, 1959) is its quadratic utility framework.

The classic work on utility theory by Pratt (1964) demonstrates that absolute risk

aversion should decline or at least not increase with wealth. However, absolute risk

aversion increases with wealth under quadratic utility. In short, the assumption of

quadratic utility in Markowitz (1952, 1959) is inconsistent with the axioms of von

Neumann and Morgenstern (1944) expected utility maximisation. This critique of the

Markowitz (1952, 1959) framework has been acknowledged in the finance literature by

numerous scholars including Sarnat (1974) and Campbell and Viceira (2002). Despite

this technical inconsistency, Markowitz (1952, 1959) and MPT still remains one of the

foundations of modern finance.

Markowitz’s (1952, 1959) restrictive assumptions of quadratic utility and multivariate

normality provides scholars with the motivation to explore alternative portfolio selection

frameworks. Of note, scholars have pursued the objective of developing new portfolio

frameworks that are consistent with expected utility maximisation and/or capture the

true risks which are associated with empirical asset returns. It is in this spirit that we

proceed to consider the scholarly contributions in alternative portfolio selection

frameworks over the last fifty-five years.

2.3.2 Safety-First and Downside-Risk One of the first portfolio models to compete with Markowitz (1952) was the Roy (1952)

safety-first portfolio selection framework. The Roy (1952) framework was similar to

Markowitz (1952) with the exception that risk was defined as the probability of a bad

outcome or an undesirable event. Roy (1952) refers to the example of some form of

losses or the probability of not earning a minimum rate of return.13 The work of Roy

(1952) and the subsequent contributions by Telser (1955) and Kataoka (1963) were the

first to consider risk as an undesirable outcome rather than employing the conventional

13 Markowitz (1999) himself describes Roy (1952) as one of the fathers of modern portfolio theory.

18

variance metric. Another striking feature of the Roy (1952) safety-first approach was the

development of a portfolio selection framework outside the utility paradigm.14

The work of Roy (1952) and the subsequent contributions by Telser (1955) and Kataoka

(1963) led Baumol (1963) to develop the first value-at-risk (VaR) based portfolio

selection model. The Baumol (1963) portfolio model was based on minimising the

probability of a predetermined loss within a specific statistical confidence level. The

common characteristic of the safety-first approach to portfolio selection is its

implementation outside the von Neumann and Morgenstern (1944) expected utility

maximisation framework. More recently, the safety-first approach has regained

prominence in the portfolio literature through the emergence of VaR and other tail-risk

measures such as CVaR.

The rudiments of Markowitz (1952) and Roy (1952) not only motivated the safety-first

literature, they also provoked scholars to consider alternative forms of ‘downside-risk’.

The concept of downside-risk originates from Markowitz (1959) and relates to the

notion that risk should be measured by undesirable returns or outcomes rather than on a

symmetrical measure such as variance. The most recognised measure of downside-risk

in the portfolio selection literature is the semi-variance which measures variance below

the mean return. Markowitz (1959) himself advocated semi-variance as a viable

substitute for variance as he argued that a mean/semi-variance framework was more

rational for economic agents than MVA. Unfortunately, a mean/semi-variance

framework was not pursued in the 1950s due to the computational limitations at the

time.

Despite the introduction of the mean/semi-variance approach by Markowitz (1959), the

development of the downside-risk portfolio approach remained dormant until Porter

(1974) revealed that a target return/semi-variance approach yielded more efficient

14 Roy (1952) was one of the first to critique expected utility theory in the portfolio selection framework with the famous quote, ‘A man who seeks advice about his actions will not be grateful for the suggestion that he maximise expected utility [pp.433]’.

19

portfolios than an expected mean/semi-variance approach. The work of Porter (1974)

motivated Arzac and Bawa (1977) and Fishburn (1977) to develop a theoretical

downside-risk portfolio framework which was consistent with expected utility

maximisation. The subsequent studies by Leibowitz and Henriksson (1989), Leibowitz

and Kogelman (1991) and Harlow (1991) demonstrate that less risky assets such as

bonds are more desirable in a downside-risk portfolio selection framework. In recent

years, the downside-risk portfolio approach has been made popular with the proliferation

of VaR within the global finance and banking industry. We proceed to detail the M-VaR

portfolio framework and its development in the literature.

2.3.3 Mean-Value at Risk (M-VaR) The emergence of the J.P. Morgan (1995) Value-at-Risk (VaR) via the Basle Committee

of Banking Supervision (1996, 2003) has positioned it as the risk management

benchmark in global banking regulation. The recent popularity of VaR has motivated

scholars to examine its mathematical properties in a mean-value at risk (M-VaR)

portfolio framework.15 The work of Campbell, Huisman and Koedijk (2001) observe

that the M-VaR portfolio framework is consistent with expected utility maximisation

when the normality condition holds. However, a less sanguine view of VaR has emerged

in the portfolio selection literature. Alexander and Baptista (2002) find that M-VaR

investors may hold portfolios with larger standard deviations than MVA investors

resulting in MVA inefficient portfolios. Consigli (2002) documents the difficulty of

mean-VaR optimisation during periods of financial instability. Basak and Shapiro

(2001) reveal that M-VaR investors suffer worse losses when investment conditions

deteriorate. Finally, Maringer (2005) demonstrates that M-VaR portfolios focus on

minimising the single point in the distribution of portfolio returns resulting in a severe

underestimation of risk beyond the VaR estimate. The conclusions drawn from this

15 Baumol (1963) was the first scholar to develop what is today regarded as a mean-VaR portfolio selection model which estimates optimal portfolio choice by minimising the probability of a predetermined loss within a specific statistical confidence level.

20

literature suggests that the VaR metric in M-VaR portfolio selection is an inefficient

measure of risk.

2.3.4 Mean-Conditional Value at Risk (M-CVaR) Whilst the statistical deficiencies of M-VaR continue to draw criticism from scholars,

Rockafellar and Uryasev (2000, 2002) have developed the mean-CVaR (M-CVaR)

portfolio framework as a viable alternative.16 The scholarly contributions of Rockafellar

and Uryasev (2000, 2002) demonstrate that the convex properties of M-CVaR provide

more efficient portfolios than the M-VaR framework. Subsequent studies by Alexander

and Baptista (2004) and Topaloglou, Vladimirou and Zenios (2002) also examine M-

CVaR and their findings were found to be consistent with Rockafellar and Uryasev

(2000, 2002).17 Other studies by Krokhmal, Uryasev and Zrazhevsky (2002) reveal that

portfolio rebalancing methods in a mean-CVaR portfolio setting provide better out-of-

sample performance in comparison to alternative portfolio frameworks. Although CVaR

is not a standard risk metric in global finance, the literature demonstrates that it is a

viable alternative to VaR. In summing up, the discovery of M-CVaR as a meaningful

substitute for M-VaR motivates Chapter 5 of this thesis which compares the portfolio

decisions of MVA and M-CVaR investors.

2.3.5 Empirical Issues in Portfolio Selection The development of the portfolio selection literature has also motivated researchers to

consider the empirical implementation of these frameworks. Scholars have identified

statistical problems which confront investors when optimal portfolio choice is

implemented in an empirical setting. The review of literature in this section summarises

the statistical problems identified in the literature and the solutions that have been

16 Although Rockafellar and Uryasev (2000, 2002) refer to this risk metric as CVaR, Acerbi (2002) independently developed it under the name of Expected Shortfall (ES). 17 Other studies employ CVaR but digress from a typical portfolio selection framework. For instance, Alexander, Coleman and Li (2006) develop a new function to more efficiently minimise CVaR in a portfolio of derivatives. Rockafellar, Uryasev and Zabarankin (2006) employ CVaR to demonstrate the theoretical limitations of portfolio selection when derivatives such as options are introduced.

21

developed to transform modern portfolio theory into practice. This strand of literature is

important due to the emphasis on empirical portfolio selection in the third empirical

study of this thesis.

The literature concerned with the empirical implementation of portfolio selection

originates from Brown (1976), Jobson, Korkie and Ratti (1979), Jobson and Korkie

(1980, 1981a, 1981b, 1982), Klein and Bawa (1976) and Michaud (1989, 1998). These

bodies of work highlight the fact that MVA portfolio optimisation treats input

parameters as true parameters which are known with certainty. However, in reality, the

MVA input parameters are prone to sampling and estimation error. To address these

problems in portfolio optimisation, scholars have proposed various methods to develop

more meaningful inputs for portfolio selection. This section reviews the literature which

focuses on the development of improved portfolio input parameters and the biases which

stem from autocorrelated returns.

The estimate of the expected mean return is probably the most important input in

portfolio selection. This is highlighted by Merton (1980) who shows that the estimation

risk associated with expected mean returns are high. In another work, Chopra and

Ziemba (1993) estimate that the error effects from sample mean returns are a magnitude

higher than errors from variance and covariance inputs. In short, the estimation risk

associated with expected mean returns contributes more error to portfolio selection than

estimation risk of the covariance matrix.

22

To address estimation risk in expected mean returns, the literature has adopted Bayesian

methods to improve portfolio selection inputs in an empirical setting.18 One of the

earliest tools to estimate expected mean returns originate from Jobson, Korkie and Ratti

(1979), Jobson and Korkie (1980, 1981a, 1981b, 1982) and Jorion (1985, 1986) who

employ the James-Stein (1961) estimator which shrinks the estimated mean returns

towards a global mean of asset returns. In an alternative approach, Frost and Savarino

(1986) propose another Stein estimator which shrinks the estimated mean returns

towards an equal-weighted market portfolio. In a related strand of literature, Black and

Litterman (1992) and Polsen and Tew (2000) employ semi-Bayesian and traditional

Bayesian methods in conditional asset allocation frameworks. To sum up, these

academic contributions and the practitioner based works of Michaud (1998) and Scherer

(2002) provide compelling evidence to demonstrate that Bayesian methods in estimating

an asset’s first moment improve the posterior estimation of inputs in empirical portfolio

selection.

The second portfolio selection parameter which is subject to estimation risk is the

covariance matrix. Although Merton (1980) and Chopra and Ziemba (1993)

demonstrate that the estimation risk of the covariance matrix is less significant than

expected means, Jobson and Korkie (1980) and Michaud (1989) find that the covariance

matrix exhibits a high degree of error in the presence of many similarly related securities

such as an equities portfolio. As stated in Ledoit and Wolf (2003), the problem of the

sample covariance matrix is that it imposes too little structure.

To overcome the statistical deficiencies of the sample covariance matrix, scholars have

developed various methods to minimise the estimation risk in the covariance matrix in

portfolio selection. One of the earliest methods to estimate a more efficient covariance

matrix in portfolio selection originates from Elton and Gruber (1973) who developed the 18 The early work of Stein (1955) demonstrated that the sample mean return is inadmissible under general conditions. To address the issue of estimating efficient sample means, James and Stein (1961) developed a simple Bayesian shrinkage operator which shrinks the estimated mean returns towards a global mean. Studies by Efron and Morris (1977) and Copas (1983) have shown that the James and Stein (1961) shrinkage procedure provides efficient estimates of sample means.

23

constant correlation approach which imposes structure on the covariance matrix. This

technique reduces sampling error at the cost of specification error. Other studies such as

Frost and Savarino (1986) employ a Bayesian estimator which shrinks the variances and

covariances towards an identical set of parameters. More recently, Ledoit and Wolf

(2003, 2004) propose Stein procedures for the estimation of large sets of similarly

related securities such as stocks. Ledoit and Wolf (2003) shrinks the covariance matrix

towards a single-factor Sharpe (1963) based matrix while Ledoit and Wolf (2004) shrink

the covariance matrix towards a constant correlation model. The conclusion to be drawn

from this literature is that Bayesian techniques are the method of choice to more readily

estimate the covariance matrix in portfolio selection.

2.3.6 Serial Correlation in Asset Returns Whilst the discussion in the previous section highlights the literature on estimation risk

of the first moment and the covariance matrix, a different strand of literature identifies

serial correlation of asset returns as a potential problem in empirical finance. Early

research as far back as Fisher (1966), Dimson (1979) and Scholes and Williams (1977)

have demonstrated that stock returns which are priced at different times exhibit spurious

serial correlation.19 More recent studies on hedge fund returns (Asness, Krail and Liew

(2001) and Getmansky, Lo and Makarov (2004)), real-estate prices (Geltner (1991,

1993)) and price indices have shown similar effects. Whilst these contributions examine

the serial correlation effects on regression parameters, other scholars have examined

autocorrelation effects and its effect on the second sample moment. Blundell and Ward

(1987) and Geltner (1991, 1993) examine the second sample moment in real estate

returns while Blume, Keim and Patel (1991) examine the same effects on low-grade

bonds. All of these studies demonstrate that the second sample moment tends to be

under-estimated in the presence of serial correlation in returns. In short, the variance

(and the standard deviation) is shown to be biased downwards in the presence of

autocorrelation in returns.

19 Other studies such as Lo and MacKinlay (1988, 1990) and Kadlec and Patterson (1999) discover serial correlation in weekly returns which cannot be explained by nonsynchronous trading.

24

To estimate a more efficient second sample moment, three empirical approaches have

been developed in the literature to control for autocorrelation effects. The first method

by Blundell and Ward (1987) transforms original real estate returns by adjusting them

with the regression coefficient of an autoregressive first-order AR(1) model. Recent

studies by Herold (2005) and Scherer (2002) have applied the Blundell and Ward (1987)

procedure in portfolio selection studies. The second method from Blume et. al., (1991)

develop an alternative method to adjust for spurious autocorrelation caused by stale

pricing in low-grade bond returns. The third method from Geltner (1991, 1993)

transforms the original real estate returns by the autocorrelation coefficient of the first-

order. The Geltner (1991, 1993) method has been employed in the hedge fund literature

by Bacmann and Gawron (2005) and Loudon, Okunev and White (2006) to mitigate the

serial correlation effects in hedge fund returns. The consistent finding revealed in all of

these studies shows that serial correlation of returns induces a downward bias in the

second sample moment resulting in the under-estimation of risk of assets in portfolio

selection.

In conclusion, the literature reveals that many scholars have considered the theoretical

and empirical implications of optimal portfolio choice. This rich literature provides a

number of avenues by which to address the empirical characteristics of hedge fund

returns in the portfolio selection study in Chapter 5. The literature review now proceeds

to consider the one of the key assumptions of portfolio selection which is the linearity

condition.

2.4 Linearity

The literature review so far has examined a number of assumptions which underpin the

Markowitz (1952, 1959) framework. However, a key assumption in portfolio selection

which has received less attention is the linearity condition between asset returns. When

two or more assets are combined in a MVA, the unstated assumption of linearity in

returns is a necessary condition for efficient portfolio construction. A traditional MVA

portfolio optimisation assumes that asset returns can be sufficiently described as a linear

25

relationship through the covariance matrix. If the relationship between asset returns is

non-linear then the covariance matrix is an inadequate description of return behaviour.

By construction, inaccurate optimal portfolio weightings will result from portfolio

selection models that employ the covariance matrix to describe non-linear returns. This

section reviews the linearity literature as it becomes the encompassing theme of the first

two empirical chapters of the thesis.

The assumption of linearity in the Markowitz (1952, 1959) framework is of empirical

interest to mean-variance investors. The linearity condition is an important element in

Markowitz (1952, 1959) and is an unstated assumption in other empirical finance

models including the Sharpe (1964), Lintner (1965) and Mossin (1966) Capital Asset

Pricing model (CAPM), the Ross (1976) Arbitrage Pricing Model and the Fama and

French (1992, 1993) three-factor model. Despite the importance of the linearity

condition in empirical finance, little or no research attention has considered or examined

this assumption in a portfolio theory or asset pricing context.

To review the linearity literature in this thesis, a formal definition of linearity is

required. Scholars have proposed many different types of linearity, however, this thesis

restricts the linearity definition within the context of theoretical and empirical finance.20

Scholars such as Granger and Teräsvirta (1993), Campbell, Lo and MacKinlay (1997)

and Tsay (2002) have defined the concept of linearity in an ordinary least squares (OLS)

setting. As stated in Granger and Teräsvirta (1993), the definition of linearity can be

decomposed into its two elements in a linear OLS regression,

20 The term, ‘non-linearity’ has a long history in the literature across a wide range of research disciplines. The origins of the non-linearity literature began with Bartlett (1947), Cochran (1947) and Eisenhart (1947) who considered the implications to statistical inference when the assumptions of the method of least squares are not met. As a response to these considerations, Tukey (1949) was one of the earliest researchers who defined the term, ‘non-linearity’, as the condition when the assumptions in the method of least squares do not hold. At this point, the term, ‘non-linearity’ described the identification and detection of failures in the assumptions of the least squares method. In the case of Tukey (1949), the definition of non-linearity was narrowly defined as a test for the least squares property of additivity. Since Tukey (1949), the development of non-linearity tests have proliferated in an effort to examine each component of the least squares method.

26

“However, a more useful form, bringing out an explicit innovation is

tttt IhIgy ε)()( 11 −− +=

where tI is the information set ,jtx − ,jt−ε 0≥j . If ty is a vector,

then the thk component will have the representation

kttktkkt IhIgy ε)()( 11 −− +=

where tI is ,jtx − ,jt−ε 0≥j and tx includes ty . If ≠− )( 1tk Ih a

constant, then the series ky is heteroskedastic, otherwise it is

homoskedastic. If )( 1−tIg is linear in the components of ,1−tI then ty

is said to be ‘linear in mean’. If )( 1−tIh is also a constant, then ty has

a complete linear representation in terms of ,1−tI , otherwise it is non-

linear. Granger and Teräsvirta (1993)[pp.7-8]”

The succinct definition of linearity in Granger and Teräsvirta (1993) shows that when

the regression coefficients in ( )g ⋅ are non-linear then ty is said to be non-linear in

mean. When ( )h ⋅ is time-varying, then it is said that ty is non-linear in variance. To

test for the presence of non-linearity in ( )g ⋅ and ( )h ⋅ , two types of linearity tests have

been developed in the literature.21

21 Many forms of linearity tests exist in the mathematics and statistics literature which are not consistent within the Granger and Teräsvirta (1993) linearity framework. For instance, the Brock, Dechert and Scheinkman (1987) BDS test examines the iid assumption of a time series, however, the BDS test cannot assist in defining the specificity of the non-linear relationship between asset returns. The works of Hinich (1982), Subba Rao and Gabr (1984) and Priestley (1988) have developed the Bispectral Test which examines the Fourier transform of its third-order moments. Whilst the above tests are applicable in a mathematics and statistics discipline, there is little rationale which unifies these testing regimes with finance theory.

27

The first type of test examines whether the conditional mean ( )g ⋅ is linear. One of the

earliest tests which examined the linearity of the conditional mean comes from Ramsey

(1969) who proposed the Regression Specification Error (RESET) test to examine mis-

specification of the functional form. The subsequent work by Keenan (1985) then

developed a test which examines the correlation between the estimated residuals te with

the linear model’s squared forecast 2tf . The Keenan (1985) test proposes that any

additional forecasting property between 2tf and te in the form of a correlation test

results in a quadratic non-linear term which is a departure from linearity-in-the-mean.

The Keenan (1985) test was then extended by Tsay (1986) who included

2/)1( +pp cross-product terms of the components of tX , where p is the number of lags

in the )( pAR model. The inclusion of the cross-product terms in Tsay (1986) results in

an improved test with more power than Ramsey (1969) and Kennan (1985).

More recently, in an effort to improve the test for departures in linearity-in-the-mean,

Teräsvirta, Lin and Granger (1993) developed the V23 test. The V23 test not only tests

for the relationship between squared cross-product terms, but it also tests for

relationships in cubed cross-product terms also. The simulations in Teräsvirta et. al.,

(1993) demonstrate that the V23 test is a more robust test than Ramsey (1969), Kennan

(1985) and Tsay (1986) when attempting to detect unspecified non-linearity. The

Teräsvirta et. al., (1993) V23 test provides a robust test for linearity-in-the-mean which

has good power properties and is regarded as the most powerful linearity-in-the-mean

test in the literature to date.

In the empirical literature, the linearity of the conditional mean has been examined by a

number of scholars. Scheinkman and LeBaron (1989), Hsieh (1991), Opong,

Mulholland, Fox and Farahmand (1999) and Poshakwale (2002) show that developed

and emerging market stock returns are non-linear in a univariate setting. In the interest

rate literature, Ait-Sahalia (1996) and Stanton (1997), Chapman and Pearson (2000) and

Jones (2003) reveal mixed findings. Despite the research attention towards the linearity-

28

in-the-mean in the univariate setting, very few studies have considered the linearity-in-

the-mean in a bivariate framework. Some of the few empirical bivariate studies include

Boudoukh, Richardson and Whitelaw (1997) who detect non-linear behaviour between

the equity risk premium and the term structure while Desai and Bharati (1998) find

similar results between US stocks and bonds.

The second type of non-linearity test examines the time-variant characteristics of the

conditional variance term ( )h ⋅ . The most recognised tests in the literature include the

seminal Engle (1982) autoregressive conditional heteroscedastic (ARCH) and Bollerslev

(1986) generalised autoregressive conditional heteroscedastic (GARCH) frameworks.22

Another framework which has received attention but is less significant comes from

McLeod and Li (1983) who implement Ljung-Box statistics to squared residuals to

examine linear model mis-specification.

Whilst this thesis employs the same definition of linearity as Granger and Teräsvirta

(1993), Campbell, Lo and MacKinlay (1997) and Tsay (2002), other frameworks have

been employed to identify non-linearity. For instance, Boudoukh, Richardson and

Whitelaw (1997) and Mitchell and Pulvino (2001) employ a piecewise regression

framework to identify non-linearity between asset returns. Boudoukh, Richardson and

Whitelaw (1997) detect non-linearity between the equity risk premium and the term

structure while Mitchell and Pulvino (2001) discover non-linear behaviour between

hedge fund merger arbitrage and S&P 500 index returns. In a different approach, Lo

(2001) divides the up and down months of hedge fund and S&P 500 returns into separate

regressors and examine their statistical significance. Lo (2001) reports that the

statistically significant regressors indicate that a non-linear relationship exists between

hedge funds and the S&P 500 returns. To sum up, the literature highlights that there are

numerous frameworks that can be adopted to identify non-linearity in asset returns.

22 Refer to Bollerslev, Chou and Kroner (1992) for a review of the (G)ARCH literature.

29

Despite the various linearity testing frameworks, the problem of test mis-specification

has emerged as a growing concern in the literature. Granger and Teräsvirta (1993)

highlight the first empirical concern which is the effect of heteroscedasticity in linearity

testing. Empirical studies including Hsieh (1991), Opong et. al., (1999), Poshakwale

(2002) and Yadav, Paudyal and Pope (1999) have demonstrated that ARCH effects in

stock returns can partially explain non-linear behaviour. The second empirical concern

comes from Lee, White and Granger (1993) who recognise that serial correlated error

disturbances may in fact cause an over-rejection of the null hypothesis of linearity.23

This literature clearly highlights that ARCH and autocorrelation effects distort the size

of test statistics resulting in the mis-specification of linearity tests.

To summarise the linearity literature, many scholars have developed a number of tests

which define and identify linearity. In this thesis, the definition of linearity is restricted

to the concept proposed by Granger and Teräsvirta (1993). Furthermore, the importance

of the covariance matrix to mean-variance investors motivates this thesis to consider

tests of linearity-in-the-mean rather than linearity-in-the variance. We proceed to review

the hedge fund literature and how it informs this thesis in the context of examining asset

return behaviour in a linearity framework and in the portfolio selection setting.

23 Since Newey and West (1987), the literature has long recognised the statistical inference problems associated with error disturbances which are serially correlated. The implications of autocorrelated error disturbances in a linearity testing regime is highlighted in Lee, White and Granger (1993).

30

Table 2.1 Definitions of Hedge Funds This table presents a variety of hedge fund definitions proposed by various regulatory authorities and scholars. To appreciate the evolutionary understanding of hedge funds over time, these definitions are presented in date chronological order. It is important to acknowledge that the description of hedge funds is not limited to the following seven definitions as other scholars and regulatory authorities may employ alternative terminology to define the global hedge fund industry.

Author and Year Definition

International Monetary Fund (1994) ‘The term hedge fund carries no formal definition in securities law, and the private investment vehicles that make up this industry are extremely diverse. … The fund’s investment portfolio could span government securities, foreign exchange, financial futures and options, commodities, real estate, mergers and acquisitions arbitrage, mortgage-backed securities, or even other hedge funds. … A key question is what is special about hedge funds? … First, hedge funds are less regulated than other large players in financial markets. … Second, hedge fund investors are wealthier and presumably have a higher tolerance for risky investments than the public at large. … Third, hedge funds are generally regarded as the most leveraged players in major financial markets. Yet a fourth distinguishing characteristic is their superior performance.[pp.7-8]’

International Monetary Fund (1998) ‘Hedge funds are collective investment vehicles, often organized as private partnerships and resident offshore for tax and regulatory purposes. Their legal status places few restrictions on their portfolios and transactions, leaving their manager free to use short sales, derivative securities, and leverage to raise returns and cushion risk.[pp.1]’

The President’s Working Group on

Financial Markets (1999)

‘Although it is not statutorily defined, the term (hedge fund) encompasses any pooled investment vehicle that is privately organised, administered by professional investment managers, and not widely available to the public.[pp. 1]’

Cottier (2000) ‘All forms of investment funds, companies and private partnerships that 1. use derivatives for directional investing and/or; 2. are allowed to go short and/or; 3. use significant leverage through borrowing.[pp.17]’

United States Securities Exchange

Commission (2003)

‘Although there is no universally accepted definition of the term “hedge fund”, the term is generally used to refer to an entity that holds a pool of securities and perhaps other assets, whose interests are not sold in a registered public offering and which is not registered as an investment company under the Investment Company Act.[pp.3]’

Australian Prudential Regulation

Authority (APRA) (2003)

‘Although the term is widely used, there is no clear definition of what constitutes a hedge fund. For the purposes of this article, hedge funds are regarded as funds that have some of the following characteristics, (i) funds that rely heavily on a single strategy, with broad delegations for the use of gearing and derivatives, (ii) funds that have a reliance upon a single individual to execute the investment management process, (iii) a relatively short trading history; and/or (iv) target an absolute return rather than a benchmark return.[pp.1]’

Lhabitant (2004) ‘Hedge funds are privately organized, loosely regulated and professionally managed pools of capital not widely available to the public.[pp.4]’

31

2.5 Hedge Funds

This thesis considers whether hedge funds are linearly related to traditional asset classes

and the effects of hedge fund returns in portfolio selection. It therefore seems logical

that a review of the hedge fund literature is warranted. We begin by first considering a

formal definition of the term, hedge fund. The term originates as far back as Loomis

(1966) who coined the term to describe the investment strategy of Tasmanian born

American fund manager Alfred W. Jones who constructed portfolios consisting of long

undervalued stocks and short overvalued stocks. Both academia and industry recognise

Alfred Jones as the first individual to have commenced and managed a hedge fund.24

From these humble beginnings, the size of the global hedge fund industry has grown to

8,000 hedge funds managing over US$1.4 trillion of funds under management according

to Tremont (2006). Since its introduction by Loomis (1966), the term hedge fund is now

employed as a catch-all phrase commonly used to describe a wide and heterogeneous

group of fund managers. Table 2.1 provides a number of hedge fund definitions

proposed by scholars and regulatory bodies. Table 2.1 presents the reader with a broad

set of definitions which allow the reader to better understand hedge funds and how they

are defined.

Table 2.1 demonstrates that there is no formal or universally accepted definition of

hedge funds. For the purposes of this thesis, we can more narrowly define hedge funds

as managed investment vehicles who classify themselves as participants in the hedge

fund industry by voluntarily reporting their performance to the Lipper/Reuters/TASS

(TASS) or Hedge Fund Research (HFR) databases.

24 Ziemba (2003) argues that the famous economist John Maynard Keynes was the first known hedge fund manager based on the research by Chua and Woodward (1983). J.M. Keynes was the First Bursar of the Chest Fund at King’s College, Cambridge from 1927 to 1945. The fund held positions in common stocks, currencies and commodity futures. In today’s terminology, the Chest Fund would have been regarded as a global macro hedge fund. The research on the Chest Fund’s 1927-1945 investment performance by Chua and Woodward (1983) suggests that Keynes was an astute investment manager.

32

Whilst there is no clear definition of what constitutes a hedge fund, a debate exists on

whether hedge funds are an asset class. Industry professionals such as Swensen (2000)

and Man Investments (2005) consider hedge funds as a separate asset class as they

provide uncorrelated returns in comparison to conventional asset classes. Opponents of

this view include Anson (2002) and Oberhofer (2001) who regard hedge funds as

unconventional investment strategies within existing asset classes. A detailed

examination of the literature highlights Oberhofer (2001) who argues that six criteria

must be satisfied to justify the existence of an asset class.25 A synopsis of the asset class

debate suggests that the arguments of uncorrelated returns are not enough for hedge

funds to be considered as a separate asset class. Instead, Anson (2002) and Oberhofer

(2001) provide compelling rationales to support the argument that hedge funds are not a

homogenous asset class, but rather, they are a diverse range of investment strategies

across various asset classes and markets.

The dizzying array of hedge fund investment strategies has motivated scholars to

examine them in an investment style analysis framework.26 As stated in Brown and

Goetzmann (2003), no general accepted principle exists to formally describe investment

style analysis in the global hedge fund industry. The review of the literature provides

two approaches taken to describe investment styles in the global hedge fund industry.

The first method is peer-group self-classification while the second method relies on

returns-based statistical techniques developed in the academic literature. The first

25 Oberhofer (2001) argues that an asset class must satisfy the following six characteristics: (i) The investments in an asset class need to consist of conceptually similar securities. (ii) A high correlation should exist between the investments in the asset class. (iii) An asset class should represent a reasonable fraction of the investment opportunity set. (iv) An asset class should have a reliable set of data. (v) An investor should be able to passively replicate the asset class. (vi) Finally, an asset class should be new and exclusive. Oberhofer (2001) argues that hedge funds fail in almost all of these requirements and therefore should not be regarded as an asset class. 26 The seminal investment style framework is the Sharpe (1992) model which is based on identifying investment styles by mapping fund returns with associated risk factors. The key assumption to the efficient implementation of Sharpe (1992) is that the associated risk factors in the model must be known. In contrast, the innovation of the Brown and Goetzmann (1997, 2003) model is that risk factors are not required and can remain unknown. As there is no general consensus as to the common risk factors which drive global hedge fund returns, the Brown and Goetzmann (1997, 2003) framework becomes the model of choice when considering hedge fund investment style analysis.

33

approach to hedge fund investment style analysis is the self-classification method

advocated by investment professionals and industry participants. Hedge fund database

vendors collect performance information from as many hedge funds as possible and then

sell the database information to hedge fund investors who pursue the analysis of hedge

fund investments. In this environment, hedge fund database vendors develop their own

classification methodologies to group hedge funds into investment styles. Some database

vendors such as TASS segregate the hedge fund industry into 11 styles while others such

as HFR have as many as 30 investment style classifications. Appendices A to C at the

end of the thesis lists the various investment style classifications of three well recognised

hedge fund database vendors. Despite the acceptance of peer-group classification by

industry participants, this method of investment style analysis lacks scientific rigour.

The second approach to hedge fund style analysis provides a more mathematical

approach to the problem. The seminal contribution of the Sharpe (1992) style model

provides investors with the research technology to examine investment style analysis for

any type of fund manager. The Sharpe (1992) framework employs a constrained OLS

regression which relates funds with their factor loadings in order to determine style

attributes. The Sharpe (1992) framework assumes full knowledge of the risk factors and

unconditional linearity in the relationship between fund returns and the common style

factor loadings. The Sharpe (1992) model remains the dominant investment style

framework in the investment style literature. Whilst the underlying assumption of full

knowledge of risk factors is an appropriate condition for the Sharpe (1992) model in the

mutual fund literature, this condition makes it difficult to implement in the hedge fund

context as the underlying risk factors of the global hedge fund industry are not well

known. Recent studies such as Capocci and Hubner (2004) and Bianchi, Drew,

Veeraraghavan and Whelan (2006) identify common risk factors which explain hedge

fund returns, however, this strand of the literature is at its infancy.

34

To overcome the Sharpe (1992) requirement of known risk factors in style analysis,

Brown and Goetzmann (1997) introduce a new style model known as the Generalized

Style Classification (GSC) framework which is a cluster analysis that is consistent with

modern asset pricing theory. The work of Brown and Goetzmann (1997) was extended

to a hedge fund framework in Brown and Goetzmann (2003) which revealed eight

investment styles in the global hedge fund industry.27 In a subsequent study, Bianchi,

Drew, Veeraraghavan and Whelan (2006) utilise the Tibshirani, Walther and Hastie

(2001) Gap Statistic with the Brown and Goetzmann (1997, 2003) model to estimate

only three global hedge fund investment styles. In another framework, Fung and Hsieh

(2001) employ principal component analysis to demonstrate that there are five to eight

broad-based hedge fund investment styles.

The conclusions drawn from the hedge fund style literature indicate that the global

hedge fund industry can be segregated into various investment styles depending on the

classification process employed. The research findings in the hedge fund investment

style debate do not provide a clear direction in terms of how hedge funds should be

classified, however, it is clear that the number of styles estimated by scholars is less than

the number proposed by industry professionals. In the context of this thesis, we are more

interested in examining the systematic returns of the global hedge fund industry rather

than considering the return/risk profiles of the various hedge fund investment styles.

With this objective, we proceed to review the literature which summarises the most

accurate methods of estimating the returns from the global hedge fund industry.

To employ accurate global hedge fund returns in this thesis, we must first review the

literature which informs us of the data biases associated with hedge fund returns.

Brown, Goetzmann and Ibbotson (1999) documents that hedge fund returns are prone to

27 Various style-specific studies include Fung and Hsieh (1997a, 2001, 2002a) who examine trend following CTAs, Mitchell and Pulvino (2001) evaluate a hedge fund investment strategy referred to as merger or risk arbitrage, Fung and Hsieh (2002), Kao (2002) and Loudon, Okunev and White (2006) investigate the fixed income hedge fund style.

35

numerous data biases due to the voluntary reporting regime of the industry. The

literature by Bianchi and Drew (2006), Brown et. al., (1999), Edwards and Caglayan

(2001), Fung and Hsieh (2000), Liang (2000, 2001) and Malkiel and Saha (2005)

suggest that the most common forms of data biases in hedge fund returns are (i)

survivorship bias, (ii) backfilling/instant history bias and (iii) selection/self-reporting

bias.28

As a partial solution to the hedge fund data bias problem, Fung and Hsieh (2000, 2004)

propose the use of fund of hedge fund (FOF) index returns. The work of Fung and Hsieh

(2000, 2004) has industry support from Man Investments (2005) who argue that FOF

index returns are the most representative of the global hedge fund industry. Three

rationales exist which support the use of FOF index returns to estimate the return and

risk of the global hedge fund industry. First, FOFs have minimal survivorship bias

because FOFs may invest in funds which may cease reporting to databases, however,

these returns are still reflected in the underlying returns of the FOF and in the FOF

index. Second, FOFs have minimal backfilling bias because the historical track record

of a new hedge fund is not included in the performance of a FOF and the FOF index

return. Third, selection bias is reduced in FOF index returns because they invest in

hedge funds who may not report to commercial database vendors resulting in FOF and

FOF index returns reflecting hedge fund returns which would otherwise not be reported

to commercial database vendors.

The hedge fund literature clearly indicates that data biases must be properly addressed in

this thesis. The review of the literature suggests that FOF index returns are the most

reliable source to measure the returns and risks of the global hedge fund industry.

Consistent with Fung and Hsieh (2000, 2004), this thesis will employ FOF index returns

in the portfolio selection chapter in order to examine the returns and risks of the global

hedge fund industry. 28 Brown et. al., (1999), Fung and Hsieh (2000) and Liang (2001) estimate hedge fund survivorship bias at 3.00, 3.00 and 2.43 per cent per annum, respectively. Fung and Hsieh (2000) and Bianchi and Drew (2006) estimate instant history bias at 1.67 and 1.40 per cent per annum, respectively. In terms of selection/self-reporting bias, no method exists to effectively measure this bias.

36

Whilst the Fung and Hsieh (2000, 2004) method of FOF index returns aids in the

accurate measurement of global hedge fund returns, the assessment of hedge fund risk

has seen various contributions in the literature. For instance, Geman and Kharoubi

(2003) show that hedge fund risk is not captured in the second moment of returns as

hedge funds tend to exhibit low standard deviation in returns. Instead, Geman and

Kharoubi (2003) reveal that hedge fund risk tends to be located in the third and fourth

moments in the distribution of returns. In another setting, Agarwal and Naik (2004)

reveal that hedge fund returns exhibit tail-risk which is heavier (or thicker) than

expected in a normal distribution. As support for the findings in Agarwal and Naik

(2004), the work of Brown and Spitzer (2006) discover that market neutral hedge funds

exhibit extreme tail behaviour during periods of financial market volatility. In short,

market neutral funds are unable to maintain their beta neutral exposure during periods of

financial market turbulence. Finally, the studies by Asness et. al., (2001) and

Getmansky et. al., (2004) reveal severe serial correlation in hedge fund returns due to

illiquid exposures and smoothed returns. These studies reveal that serial correlation

masks the true beta of hedge fund returns.

Overall, the literature suggests that many complexities exist when examining hedge fund

risk. Furthermore, these studies show that the statistical and time series characteristics

of hedge fund returns make it difficult to employ conventional finance and econometric

models. In the context of this thesis, we concentrate on how scholars have considered

hedge funds in optimal portfolio choice. This is an important part of the literature review

as the third empirical chapter of the thesis examines hedge funds and traditional asset

classes in a portfolio selection framework.

The body of work which examines hedge funds in portfolio selection is an innovative

strand of literature. The central theme in this literature is the emphasis on portfolio

selection frameworks which can incorporate the empirical features of hedge fund

returns. Some of the earliest works which have considered hedge funds in portfolio

selection originate from Lintner (1983) and Elton, Gruber and Rentzler (1987). These

37

early studies examined the role of commodity pools in traditional investment

portfolios.29 Lintner’s (1983) findings advocated the use of publicly listed managed

futures investments as viable economic complements in a portfolio while Elton et. al.,

(1987) revealed the opposite finding from a larger sample of funds. Subsequent studies

by Schneeweis and Spurgin (1998) and Edwards and Liew (1999) included hedge funds

and CTAs in a MVA and found that alternative asset classes dominate the investment

opportunity set.

The favourable view of hedge funds in the portfolio selection literature changed

dramatically in 1998 with the orchestrated bailout of Long-Term Capital Management

(LTCM) by the U.S. Federal Reserve. The events of LTCM highlighted the inherent

risks in hedge funds and their dominance in an MVA portfolio selection became a

controversial finding. It was clear that the Markowitz (1952, 1959) MVA was not fully

capturing the inherent risk in hedge fund investments. In short, MVA was ignoring

hedge fund risk which was located at the extreme left-tail of the distribution of returns.

In an effort to understand this hedge fund dynamic, Amin and Kat (2003) examined

portfolio selection in the presence of stocks, bonds and hedge funds. Amin and Kat

(2003) demonstrated that the inclusion of hedge funds in portfolio selection causes a

lower standard deviation of portfolio returns at the cost of undesirable third and fourth

moments.30 Consistent with the findings of Geman and Kharoubi (2003), the work by

Amin and Kat (2003) show that the two-parameter Markowitz (1952, 1959) framework

does not capture the inherent risk in hedge funds which is located in the third and fourth

moments of portfolio returns.

The works of Amin and Kat (2003) and Geman and Kharoubi (2003) have motivated

other scholars to examine hedge funds in alternative portfolio selection frameworks. For 29 Commodity pools are a sub-set of the commodity trading advisor (CTA) industry which is now more commonly referred to as the managed futures industry. 30 Amin and Kat (2003) discovered that the introduction of hedge funds to the investment opportunity set lowers the skewness and increases the kurtosis of portfolio returns.

38

instance, Cremers, Kritzman and Page (2005) estimate optimal portfolios of hedge funds

by comparing MVA with full-scale optimization for log utility and S-based (ie. prospect

theory) utility investors.31 To capture the tail-risk of hedge fund returns, Krokhmal et.

al., (2002) employ CVaR and drawdown based measures to construct portfolios of

individual hedge funds. Krokhmal et. al., (2002) find that CVaR is an effective risk

measure when constructing fund of hedge funds. To address the non-normality of

returns, Morton, Popova and Popova (2006) construct portfolios of hedge funds with a

specialised stochastic programming technique known as normal-to-anything (NORTA).

Finally, Giamouridis and Vrontos (2007) employ time-varying volatility and correlation

measures in portfolio selection to also construct portfolios of hedge funds. A synopsis of

the above literature shows that CVaR based portfolio frameworks tend to be an efficient

method at capturing a large proportion of the risks in hedge fund returns.

The review of the hedge fund based portfolio selection studies reveals an interesting

observation. With the exception of Amin and Kat (2003), the empirical studies have had

a narrow focus on employing hedge fund returns to construct portfolios of hedge funds

only. In short, little research attention has been paid in examining portfolio selection

between hedge funds and traditional asset classes. This gap in the literature provides the

motivation of this thesis to examine portfolio selection when both traditional assets and

hedge funds are included in the investment opportunity set.

The final section of the hedge fund literature review examines the scholarly

contributions that have evaluated hedge fund non-linearity. A review of hedge fund

linearity is warranted as the second empirical chapter of this thesis focuses on this

research question. The literature reveals six known studies which examine hedge fund

linearity. Fung and Hsieh (2001) sparked the debate on hedge fund non-linearity by

providing graphical evidence to show that trend-following managed futures funds

exhibit U-shaped payoffs with traditional asset returns. More specifically, Fung and

Hsieh (2001) report positive returns for trend-followers when traditional assets report

31 The S-based utility function originates from Kahneman and Tversky (1979) who employed it to describe investors with prospect theory utility preferences.

39

severe positive and negative monthly returns. In another study, Mitchell and Pulvino

(2001) examine the merger arbitrage investment strategy in the United States and show

that it is equivalent to a non-linear sold put option strategy in the US stockmarket. In

another approach, Lo (2001) reports non-linearity in the entire hedge fund industry by

dividing hedge fund and S&P 500 returns into separate up and down regressors and

estimating their statistical significance. In an unconventional approach, Favre and

Galeano (2002) employ a loess regression approach to demonstrate that hedge fund

returns are non-linear. In another study, Agarwal and Naik (2004) report statistically

significant option-based risk factors in a multi-factor regression model as further

evidence that hedge fund returns are non-linear. Finally, Huber and Kaiser (2004)

explain the variation of hedge fund returns with option-based risk factors in a Sharpe

(1992) model.32

The empirical literature on hedge fund linearity suggests that hedge fund returns are

non-linear. However, a closer inspection of the literature reveals that little attention is

paid to address the empirical features of heteroscedasticity and autocorrelation as

highlighted in Granger and Teräsvirta (1993) and Lee et. al., (1993). This gap in the

hedge fund literature provides the motivation in this thesis to consider hedge fund

linearity in a framework which addresses and controls heteroscedasticity and

autocorrelation. By addressing these empirical characteristics of hedge fund returns, this

thesis will provide a new dimension to the hedge fund linearity debate.

2.6 A Synthesis

The literature review in this section is by no means exhaustive. The scholarly

contributions reviewed in this chapter draw upon the interrelated but distinct disciplines

of finance, economics and mathematical statistics. In the context of this thesis and

within the finance discipline, the key issues to emerge from the literature are the

32 A criticism of Huber and Kaiser (2004) is that the Sharpe (1992) model assumes unconditional linearity between the dependent variable (ie. hedge fund returns) and the independent variables (ie. the associated risk factors). Despite employing option-based investment strategies as independent risk factors, Huber and Kaiser (2004) ignore this important assumption in their study.

40

following. First, the linearity condition between asset returns must hold in order to

derive valid estimates in MVA portfolio selection. Second, to obtain precise statistical

inference of linearity, the empirical features of heteroscedasticity, autocorrelation and

tail-risk in asset returns must be examined and controlled in the linearity hypothesis test

framework. Third, the empirical features of heteroscedasticity, autocorrelation and tail-

behaviour in asset returns must affect optimal portfolio choice as conventional MVA is

estimated from the first two moments of portfolio returns.

This thesis aims to examine the sensitivities of linearity testing and portfolio selection to

the empirical characteristics of heteroscedasticity, autocorrelation and tail behaviour.

More specifically, the empirical studies in this thesis aim to examine linearity and

portfolio selection when both traditional assets and hedge funds are in the investment

opportunity set. Fortunately, the review of literature provides a number of avenues to

address these issues. This thesis employs a hypothesis testing approach to examine and

control the effects of heteroscedasticity, autocorrelation and tail behaviour in linearity

testing. Furthermore, we examine how serial correlation and tail-risk affect the portfolio

investment decisions of MVA and M-CVaR investors. This thesis demonstrates that the

empirical features of heteroscedasticity, autocorrelation and tail behaviour can

substantially affect the findings of linearity tests and portfolio selection.

The remaining chapters of the thesis are structured as follows. Chapter 3 examines the

linearity assumption between traditional asset classes only. The logical first step of

examining linearity-in-the-mean is to assess traditional asset classes only before

introducing hedge funds to the problem. Chapter 4 extends the previous empirical study

by examining the linearity between traditional asset classes and hedge funds. Chapter 5

presents the third and final empirical study which explores the interaction between

traditional asset classes and hedge funds in a variety of portfolio selection frameworks.

Chapter 6 provides concluding remarks and offers avenues for future research.

41

3. The Linear Behaviour of Stocks and Bonds

3.1 Introduction

One of the debates in financial economics relates to the issue of linearity of asset returns.

Finance theory provides little or no guidance to the a priori expectation of asset returns

and whether it is expected that they are linear or non-linear. Whilst the assumption of

linearity is not a necessary condition in finance theory, the linearity condition is an

integral element in empirical finance. It is within the empirical framework where

researchers have examined whether asset returns are linear or non-linear. In this study,

we examine whether empirical stock and bond returns are linear-in-the-mean.

The review of linearity in Section 2.4 shows that Granger and Teräsvirta (1993) and

Campbell, Lo and MacKinlay (1997) define asset returns as linear when a model

exhibits a linear conditional mean with constant error disturbances over time. These two

components of a linear framework provide the empirical setting to identify and detect

non-linear dependence in asset returns. Whilst considerable research has examined the

time variation of error disturbances (such as autoregressive conditional

heteroscedasticity (ARCH) effects), the conditional mean in asset returns has received

less attention.33 A linear conditional mean is an important element in empirical finance

as the adequacy of portfolio and asset pricing models such as Markowitz (1952), Sharpe

(1964), Ross (1976), Fama and French (1992, 1993) and Carhart (1997) rely on the

linearity assumption. If the conditional mean in asset returns is a non-linear function,

then empirical finance models may require re-specification in a more complex non-

linear framework. This study examines the conditional mean in the two most important

asset classes in the world, stocks and bonds. 34

33 Refer to Bollerslev, Chou and Kroner (1992) and Campbell et. al., (1997) for a survey of the literature on nonconstant variance. 34 The Bank of International Settlements (ie. BIS (2006)) estimates the value of the world and US debt markets as at 31 December 2005 at US$44,991.7 billion and US$20,554.8 billion, respectively. The

42

The first objective of this study is to examine the conditional mean of stock and bond

returns in a univariate setting.35 Given the size of world and US stock and bond markets,

we know surprisingly little about the linear behaviour of the monthly returns of these

asset classes in a univariate setting. An important element in this study is the

identification and control of the stylised empirical features of financial market returns,

namely, heteroscedasticity and autocorrelation. Granger and Teräsvirta (1993) and Lee,

White and Granger (1993) warn that error disturbances which are not independent and

identically distributed (i.i.d.) may cause erroneous results in linearity tests. To

accurately examine the conditional mean and to accommodate heteroscedasticity and

autocorrelation in the error disturbances, this study proposes a heteroscedasticity and

autocorrelation consistent (HAC) approach to isolate these effects from the linearity tests

so that robust statistical inference can be made. Many studies do not control for

heteroscedasticity and autocorrelation in linearity tests resulting in spurious non-linearity

in the conditional mean. To the best of the author’s knowledge, this is the first known

empirical study to explicitly control both heteroscedasticity and autocorrelation in

linearity-in-the-mean tests in stock and bond returns in both univariate and bivariate

settings.

This study finds that stock and bond returns are actually linear-in-the-mean in a

univariate setting. These results challenge the findings of previous linearity studies from

Hsieh (1991), Opong, Mulholland, Fox and Farahmand (1999) and Yadav, Paudyal and

Pope (1999). We demonstrate that conventional linearity tests have a tendency to detect

spurious non-linearity caused by the heteroscedasticity and autocorrelation in the error

disturbances which contaminates the underlying hypothesis test. By controlling both

heteroscedasticity and autocorrelation in the hypothesis tests, we demonstrate that stock

and bond returns are indeed linear-in-the-mean in a univariate setting.

World Federation of Exchanges (2006) values the world and US equity market capitalization as at 31 December 2005 at US$40,987.1 billion and US$17,000.8 billion, respectively. 35 According to Poshakwale (2002), this type of investigation can be interpreted as a form of random walk hypothesis (RWH) test which examines whether the behaviour of current asset returns can be explained by a non-linear function of past returns of itself.

43

The second goal of this study is to examine the bivariate linear behaviour between two

assets as it relates to a mean-variance investor. Rational agents making investment

decisions in a Markowitz (1952) framework assume that asset returns are

unconditionally linearly associated with each other. If asset returns are found to be non-

linear-in-the-mean in a bivariate setting, then mean-variance investors may need to re-

specify optimal portfolio choice frameworks in a more complex setting.

The findings from this study reveal that stock and bond returns are also linear-in-the-

mean in a bivariate setting. Again, these results are in conflict with the findings of other

researchers such as Boudoukh, Richardson and Whitelaw (1997) and Desai and Bharati

(1998). We show that standard linearity tests detect erroneous non-linearity when the

relationship between stock and bond returns are examined. The difference between our

results and the findings of past studies is that the past literature either does not control

for heteroscedasticity or researchers partially control for it but they do not isolate and

control autocorrelation in the test residuals. Hence, by not directly controlling both

effects, previous studies have incorrectly concluded that non-linearity in the conditional

mean is present in the bivariate relationship between stock and bond returns.

A number of important implications stem from this study. First, this study shows that the

univariate and bivariate behaviour of stock and bond returns is linear-in-the-mean, while

the non-linearity, if any, can be identified and isolated in the error disturbances of a

linear model. The consequence of this finding suggests that researchers should direct

their research attention towards the refinement of linear-based models that accommodate

the dynamic behaviour of error disturbances rather than the development of specific

non-linear models.36 Second, this study highlights the pronounced effects of

heteroscedasticity and autocorrelation on tests of linearity-in-the-mean. By highlighting

these effects in this study, researchers can more readily understand the role that these

36 There is an emerging debate in the literature as to the very existence of non-linearity and the validity of non-linear modelling. For instance, scholars who advocate the use of non-linear time series models include Hamilton (1989), Teräsvirta (1994, 1998), Taylor and Peel (2000) and Taylor, Peel and Sarno (2001). In contrast, scholars such as Buncic (2006), Chapman and Pearson (2000) and Jones (2003) argue that the presence of non-linearity is due to model mis-specification rather than true non-linear behaviour.

44

empirical features have in the future development of portfolio selection and asset pricing

frameworks.

The rest of the study is organised as follows. In Section 3.2 we provide a brief review of

the related literature. Section 3.3 documents the methods employed to examine the

assumption of linearity-in-the-mean. Section 3.4 describes the data employed in this

study. Section 3.5 examines the results while Section 3.6 offers concluding remarks.

3.2 Related Literature

Although finance theory does not impose either linear or non-linear conditions on asset

returns, a number of theoretical rationales have been developed to explain the presence

of non-linearity. The first theoretical underpinning comes from the concept of market

equilibrium with transaction costs and market frictions. Dumas (1992), He and Modest

(1995) and Sercu, Uppal and Van Hulle (1995) postulate that transaction costs and

market frictions give rise to small deviations in asset prices which result in partial

mispricings from market equilibrium. These misalignments persist until the size and

deviation of the mispricing is large enough for arbitrageurs to enter the market and cause

a non-linear adjustment of prices back to equilibrium. Although this theoretical construct

is valid in a microstructure setting, it is less supportive in an asset allocation framework

whereby lower frequency samples such as monthly returns are examined.

The second theoretical rationale seeks to explain non-linearity in asset returns without

the presence of transaction costs. The works of Black and McMillan (2004) and

McMillan (2005) propose that the behavioural finance literature of cognitive biases and

the limits of arbitrage theory may provide an explanation for non-linearity. Black and

McMillan (2004) and McMillan (2005) argue that cognitive biases in investment

behaviour may not be consistent with expected utility maximisation thus causing non-

linear asset price deviations. As a second argument, Black and McMillan (2004) and

McMillan (2005) argue that the limits of arbitrage proposed by Shleifer and Vishny

45

(1997) may also explain non-linearity in asset returns. The work by Shleifer and Vishny

(1997) postulate that arbitrage forces may be ineffective during extreme market

conditions due to capital constraints thereby resulting in non-linear deviations of asset

prices from their true value. The reversal of these market inefficiencies occur when

arbitrageurs believe that price misalignments are at levels where mean reversion

strategies can be rewarded.

Overall, we can see that the theoretical rationales which explain non-linearity exist in the

microstructure literature. However, these rationales provide little theoretical guidance

for rational agents (such as pension funds) examining the linearity of monthly asset

returns in portfolio selection frameworks. It is in this low-frequency setting of monthly

returns which is the focus of this linearity study.

The allure of non-linear modeling is motivated by theorists and empiricists to uncover

the unexplained variations of expected returns in portfolio selection and asset pricing.

However, Granger and Teräsvirta (1993) caution researchers on the haste and mis-use of

non-linear modeling without first considering testing for non-linearity in the data. Thus,

the decision to choose between a linear or non-linear model is of primary importance. It

therefore seems logical that the linear dependence of asset returns be empirically

examined in order to avoid model mis-specification in portfolio selection and asset

pricing.

In the econometrics and statistics literature, many tests have been developed to examine

linearity. A key feature of the literature is the wide and varied frameworks that have

been developed by scholars. One of the earliest tests to detect non-linearity is the

Regression Specification Reset Test (RESET) from Ramsey (1969) which examines

non-linearity in the functional form of a linear model. The Ramsey (1969) framework

was subsequently re-specified in Keenan (1985) in a more simplified framework to

remove multicollinearity. As an extension of Keenan (1985) the contributions of Tsay

(1986) and Teräsvirta, Lin and Granger (1993) develop tests to examine multiplicative

forms of non-linearity in the mean by employing quadratic and cubic terms as well as

46

cross-products. To examine the linear behaviour of the error disturbances of linear

models, the seminal work of Engle (1982) and McLeod and Li (1983) developed

autoregressive conditional heteroscedasticity (ARCH) based frameworks. As a more

general test of linearity, Brock, Dechert and Scheinkman (1987) developed the BDS test

to examine the i.i.d. assumption in a time series. As a specific form of non-linear testing,

Lo (2001) proposes a test which segregates asset returns into up and down regressors

which examine if the beta coefficients are statistically significant. Other studies such as

Boudoukh, Richardson and Whitelaw (1997) and Mitchell and Pulvino (2001) employ a

piecewise regression to evaluate non-linearity. These research contributions represent

some of the many linearity tests developed in the literature, however, many more exist

which have not been cited but are outside the scope of this study. Overall, the literature

informs us that the concept of non-linearity is a loosely defined term and many tests

have been developed to identify and detect various forms of it.

The abovementioned tests of linearity have been applied to various empirical settings.

The empirical studies that have examined non-linearity in asset returns can be divided

into two strands of literature, namely univariate and bivariate studies. In the univariate

setting, researchers have examined the autoregressive (AR) process of asset returns to

see whether current returns can be explained by the non-linear behaviour of past returns.

In the stockmarket literature, the key findings in Scheinkman and LeBaron (1989),

Hsieh (1991), Opong, Mulholland, Fox and Farahmand (1999) and Poshakwale (2002)

show that stock returns in developed and emerging markets are non-linear in the

univariate setting. In the interest rate literature, the evidence of univariate linearity is

mixed and inconsistent. Ait-Sahalia (1996) and Stanton (1997) find non-linearity in

short-rates while Chapman and Pearson (2000) and Jones (2003) challenge these

findings.

Despite the scholarly contributions in the univariate framework, little research attention

has examined non-linearity in the bivariate setting. The contributions of bivariate

linearity is important because they consider the assumption of linearity-in-the-mean

between two exogenous variables. Bivariate linearity studies are of particular interest to

47

mean-variance investors who assume linearity when combining two or more assets in a

portfolio selection setting. The work of Boudoukh, Richardson and Whitelaw (1997)

detect non-linearity between the equity risk premium and the term structure by

employing a piecewise linear regression model. In another study, Desai and Bharati

(1998) detect non-linearity between stock and bond returns by employing a variety of

linearity tests. These bivariate studies provide the motivation to examine the linear

behaviour between stocks and bonds as it relates to a mean-variance investor.

The current state of the literature reveals the following issues that need to be addressed.

First, the literature documents various studies which consider linearity in a univariate

setting, however, little research other than Boudoukh, Richardson and Whitelaw (1997)

and Desai and Bharati (1998) consider the bivariate behaviour between stocks and

bonds. The linear behaviour between stocks and bonds is a necessary condition for

mean-variance investors operating in a portfolio selection framework, therefore, it is

surprising that very few studies investigate this important research question. The

paucity of research which considers linearity between stocks and bonds provides the

motivation to better understand linearity-in-the-mean from the perspective of a mean-

variance investor. Therefore, a large component of this study considers linearity-in-the-

mean in a bivariate setting.

Second, the general approach in Granger and Teräsvirta (1993), Campbell et. al., (1997)

and Tsay (2002) shows that any relationship with a non-constant variance (ie. ARCH

effects) can be technically defined as non-linear. As heteroscedasticity is a stylised

feature of financial market returns, research should be concentrated towards the second

form of non-linearity which is the linearity-in-the-mean. Third, whilst specific critiques

can be made on the various linearity tests in the literature, one specific criticism is the

loose treatment of heteroscedasticity and autocorrelation effects in hypothesis tests of

past studies. Granger (1993), Granger and Teräsvirta (1993) and Lee et. al.,(1993)

caution the use of linearity-in-the-mean tests in the presence of heteroscedasticity and

autocorrelation as it has been found that these features distort the power and robustness

of these tests. To ensure precise statistical inference, tests for linearity-in-the-mean must

48

be formulated to control for these effects.37 This study proposes tests for linearity-in-

the-mean that can be augmented to explicitly control for the empirical features of both

heteroscedasticity and autocorrelation. Fourth, many studies examine non-linearity in a

univariate setting, however, few studies consider the bivariate relationship between stock

and bond returns. The study of linearity-in-the-mean in a bivariate setting is important

for mean-variance investors who assume that asset returns are linearly associated in a

portfolio selection context. To accommodate this research question, we examine the

linearity-in-the-mean in stock and bond returns in both univariate and bivariate settings.

This study differs from previous scholarly contributions in a number of ways. First, this

study examines the linearity-in-the-mean between asset returns from a mean-variance

investor perspective. We are motivated in determining whether the linearity-in-the-

mean condition holds when combining two of the most important asset classes in the

world (ie. stocks and bonds) in an empirical portfolio selection framework. Second, to

the best of the author’s knowledge, this is the first study which examines linearity-in-

the-mean in world and US stock and bond returns in both univariate and bivariate

settings.38 Third, the bivariate tests in this study are formulated in a framework which is

consistent in the way that a mean-variance investor would make portfolio selection

decisions. Fourth, we develop an innovative approach which allows the empirical

features of both heteroscedasticity and autocorrelation to be explicitly controlled while

examining the linearity-in-the-mean assumption of the conditional mean in asset returns.

We proceed to detail the methodology employed in this study.

37 Hsieh (1991), Opong et. al., (1999), Yadav et. al., (1999) and Poshakwale (2002) show that ARCH effects can partially explain the non-linearity, however, heteroscedasticity and autocorrelation are not explicitly controlled in the linearity tests. 38 Desai and Bharati (1998) perform general linearity tests on a variety of US based stock and bond indices only, however, they do not examine world stock and bond returns. A critique of Desai and Bharati (1998) shows that they consider test mis-specification bias caused by heteroscedasticity, however, they do not address test mis-specification caused by autocorrelation.

49

3.3 Method

This study examines the linearity of the conditional mean by employing the general

methodological apparatus from Granger and Teräsvirta (1993), Campbell et. al., (1997)

and Tsay (2002). To test for linearity-in-the-mean in a univariate and bivariate setting,

we can represent a linear relationship in the following generalised forms

t

p

iitit yy εϕϕ ∑

=− ++=

10 (3.1)

0 1 2 ...t t t i t ty x x xϕ ϕ ϕ ϕ ε= + + + + + (3.2)

where ty is the return of the dependent variable, tx is the return of the independent

variables, 0ϕ is the regression intercept, iϕ are the regression slope coefficients, p is

the lag order, tε is the random error disturbances and T is the sample size.

The univariate model in (3.1) and the bivariate framework in (3.2) are considered ‘linear

in mean’ when the inclusion of a non-linear parameter iϕ , where 0>i results in no

statistical improvement in model inference. As with all estimations in the ordinary least

squares (OLS) framework, the statistical inference of the overall model may be

susceptible to mis-specification or bias given the behaviour of the error disturbances tε .

It is possible that non-constant variance effects (such as (G)ARCH) or serial correlation

in tε may result in error disturbances which are not i.i.d. Teräsvirta and Granger (1993)

and Lee et. al., (1993) remind us that error disturbances which are not i.i.d. may result in

the incorrect rejection of the null hypothesis (ie. Type I error) in linearity-in-the-mean

tests. It is therefore imperative that the effects of the error disturbances tε be explicitly

isolated and controlled from ( )iϕ i and the underlying linearity-in-the-mean tests.

50

The general hypothesis test considered in this study can therefore be stated as

0H : We cannot reject the null hypothesis of linearity-in-the-mean in ( )iϕ i

after the adjustment for heteroscedasticity and autocorrelation in tε .

1H : At least one non-linear parameter in ( )iϕ i increases the overall statistical

significance of a model after the adjustment for heteroscedasticity and

autocorrelation in tε .

To examine linearity-in-the-mean in ty whilst controlling heteroscedasticity and

autocorrelation in tε , this study proposes to employ the following linearity-in-the-mean

hypothesis tests:

• Keenen (1985) test

• Tsay (1986) test

• Teräsvirta, Lin and Granger (1993) V23 test

The Keenan (1985), Tsay (1986) and Teräsvirta et. al., (1993) V23 tests belong to a

family of hypothesis tests that examine non-linearity in the form of multiplicative terms

in a linear model. These hypothesis tests employ a restricted least squares approach via

an F-test to compare the sum of squared residuals (SSR) from an original unrestricted

model (where ( )f x is a quadratic and/or cubic function of x ) versus the sum of squared

residuals from a simpler model such as (3.1) or (3.2) in which the null hypothesis is

assumed to be true. The F-test determines if the model with non-linear functional form

has more statistical power than the restricted linear model. A common feature of all of

these tests is that they have some power against general non-linear alternatives.

51

The selection of these linearity-in-the-mean tests in this study are motivated by their

ability to isolate the effects of heteroscedasticity and autocorrelation in the error terms

from the hypothesis test. For comparative purposes only, the Equality Test for Two

Regression Coefficients is also considered in the bivariate framework. We proceed to

detail the mathematical specifications of the various linearity-in-the-mean tests

employed in both univariate and bivariate settings.

3.3.1 Univariate Framework We examine linearity-in-the-mean by first considering the hypothesis testing framework

in a univariate setting. The univariate framework considers whether current asset returns

can be explained by a non-linear function of lagged variables of itself. A number of

rationales exist which motivate this form of analysis. First, asset returns in portfolio

selection are assumed to satisfy the linearity condition. Assets returns which reject the

null hypothesis of linearity-in-the-mean in a univariate setting may cause spurious

results in portfolio optimisations. It is therefore appropriate that the linearity of the

conditional mean is examined in a univariate setting. The second rationale for linearity-

in-the-mean tests in a univariate setting is that the behaviour of these asset returns may

have spillover effects in subsequent bivariate tests considered in later sections of this

study. In short, univariate tests of linearity-in-the-mean serve to provide a reference

prior to the introduction of complexities such as an exogenous variable in a bivariate

linearity-in-the-mean test.39 After the comprehensive examination of linearity-in-the-

mean in a univariate setting, this study will proceed to consider the same test in a

bivariate framework.

39 Linearity tests in a univariate setting have been motivated in various ways in the literature. For instance, Poshakwale (2002) considers it as a test of the random walk hypothesis (RWH) which considers if the behaviour of current asset returns can be explained by a non-linear function of lagged returns of itself.

52

3.3.2 Bivariate Framework Granger and Teräsvirta (1993), Campbell et. al., (1997) and Tsay (2002) show that many

linearity tests are generally specified in a univariate setting, however, they can be easily

re-formulated into a bivariate framework to include exogenous variables. In this study,

MPT serves as the framework to consider linearity-in-the-mean in a bivariate setting. In

an asset allocation setting, an investor has to consider the investment opportunity set

available to determine optimal portfolio choices at time t only. This means that mean-

variance investors are unable to capture the linear dependence of past returns by

assuming that they can earn the lagged returns of exogenous variables.

The divergent assumptions of modern portfolio theory (MPT) and the implementation of

non-linear tests in the econometric framework provides a conundrum. Granger and

Teräsvirta (1993) notes that an efficient test for non-linearity in the econometrics

literature includes all possible lagged endogenous and exogenous variables in the linear

functional form. However, an investor in a portfolio selection framework does not have

access to asset returns of lagged variables. Therefore, a test of linearity-in-the-mean in a

MPT framework is restricted to including asset returns from an investment opportunity

set available at time t only, and is therefore restricted from considering lagged

endogenous and exogenous variables from nttt −−− ...,,2,1 and so forth.40 This

study therefore re-formulates the Keenan (1985), Tsay (1986) and Teräsvirta et.

al.,(1993) tests into a bivariate framework similar to (3.2) whereby the independent

variables are specified at time t only so that it relates to the investment decision process

of a mean-variance investor. For comparative purposes, we also estimate the Equality

Test for Two Regression Coefficients in this bivariate setting also. We proceed to detail

the mathematical specifications of each test.

40 This subtle re-specification of the test for non-linearity may create a circumstance whereby non-linearity is not detected in the linear functional form proposed in this study, however, non-linearity may exist if lagged variables are employed. This study does not explore the non-linear possibilities with lagged variables as it is outside the typical portfolio decision making process of a mean-variance investor.

53

3.3.3 Keenan (1985) Test The Keenan (1985) framework is a linearity test against model mis-specification of

quadratic functional form. In terms of this study, the Keenan (1985) test examines

whether a quadratic fitted regression estimate from the original regression improves the

statistical significance of the underlying model.41

3.3.3.1 Univariate Test The Keenan (1985) univariate test employs the estimate ty from (3.1) in the following

regression

t

p

iitit uyy ++= ∑

=−

10

2ˆ ϕϕ (3.3)

and

ttt vu += ˆˆ αε (3.4)

where 2ˆ ty is the fitted squared value of ty from (3.1), 0ϕ is the regression intercept, iϕ

represents the regression slope coefficients, p is the lag order, tu is the random error

term estimated in (3.3) and tε is the random error term estimated in (3.1). The

regression in (3.3) is estimated to remove the linear dependence of 2ˆ ty on the regressors

in (3.1). The regression in (3.4) then employs the estimated residuals from (3.1) and

(3.3) to form the unrestricted sum of squared errors ∑+=

=T

pttvSSR

1

21 ˆ . The Keenan (1985)

null hypothesis is 0H : 0=α which is given by:

)/(

/)(

1

10

gpTSSRgSSRSSR

F−−

−= with 1++= psg (3.5)

41 Granger and Teräsvirta (1993) remind us that the Keenan (1985) test is identical to the Ramsey (1969) RESET test which has been restricted to quadratic terms only and without the problem of multicollinearity.

54

where 0SSR is the restricted sum of squared errors from (3.1), s equates to the number

of powers required greater than one and F is the F-statistic which is approximately

),( gpTgF −− distributed under the null hypothesis.42

3.3.3.2 Bivariate Test The Keenan (1985) bivariate test compares the restricted regression in (3.2) with the

following unrestricted regression

ttt uxy ++= 102ˆ ϕϕ (3.6)

and

ttt vu += ˆˆ αε (3.7)

Again, 2ˆ ty is the fitted squared value of ty from (3.2), 0ϕ is the regression intercept, 1ϕ

is the regression coefficient of the single independent variable and tu is the random

error term estimated in (3.6) and tε is the random error term estimated in (3.2) and tu is

the random error term. Equation (3.6) is estimated to remove the linear dependence of 2ˆ ty on the single regressor in (3.6). The regression in (3.7) is then calculated and the

estimated residuals are employed to form the unrestricted sum of squared errors

∑+=

=T

pttvSSR

1

21 ˆ . The residuals from (3.2) are employed to form the restricted sum of

squared errors 20

1

ˆT

tt p

SSR ε= +

= ∑ . The Keenan (1985) null hypothesis is 0H : 0=α which

is estimated via an F-statistic which is approximately ),( gpTgF −− distributed under

the null hypothesis.

42 Refer to Tsay (2002) for a review of the Keenan (1985) test.

55

3.3.3.3 Wald Tests To control for heteroscedasticity and autocorrelation in both univariate and bivariate

frameworks, the Keenan (1985) F-ratio is re-specified as a Wald statistic in the

following forms:

)0ˆ(ˆ)0ˆ( 1,, −Ω′−= − R

KWKRKWK TW θθ (3.8)

)0ˆ(ˆ)0ˆ( 1,, −Ω′−= − R

KNWKRKNWK TW θθ (3.9)

where WKW , is the White (1980) heteroscedasticity-consistent Wald Statistic from the

Keenan (1985) test, 1,

ˆ −Ω WK is the White (1980) heteroscedasticity-consistent sample

covariance matrix from the Keenan (1985) auxiliary regression, RKθ is the vector of

auxiliary regression estimators, T is the number of observations in the residuals, NWKW ,

is the Newey and West (1987) heteroscedasticity and autocorrelation consistent (HAC)

Wald statistic of the Keenan (1985) test, and finally, 1,

ˆ −Ω NWK is the Newey and West

(1987) heteroscedasticity and autocorrelation consistent (HAC) sample covariance

matrix from the Keenan (1985) auxiliary regressions. The Keenan (1985) based Wald

Test statistic is derived from Greene (2000) which is asymptotically chi -squared under

the null hypothesis with J degrees of freedom.43 We proceed to specify the second test

of linearity-in-the-mean, namely, the Tsay (1986) test.

43 The univariate Keenan (1985) Wald tests for AR(1), AR(2) and AR(3) are estimated with 3, 4 and 5 degrees of freedom, respectively. J degrees of freedom is estimated as 1 plus the number of higher orders (which equals to 1 as our Keenan test is restricted to a quadratic term only) plus the number of lag orders (ie. 1 to 3). For the bivariate Keenan (1985) Wald tests, the J degrees of freedom equates to 3 which was derived as 1 plus the number of higher orders (ie. 1) plus the number of independent variables (ie. 1).

56

3.3.4 Tsay (1986) Test To examine unspecified non-linearity, Tsay (1986) proposes an alternative specification

to Keenan (1985) whereby the auxiliary regressors include quadratic and cross-product

terms. Not only does the Tsay (1986) test examine non-linearity of quadratic terms, it

also evaluates multiplicative terms also which makes it a more powerful test than

Keenan (1985). The Tsay (1986) test is employed in this study because Lee et.

al.,(1993) find reasonable power and robustness from the Tsay (1986) test and it is also

regarded as a benchmark test in the non-linearity literature.

3.3.4.1 Univariate Test In the generalised model in (3.1), the Tsay (1986) test examines if quadratic and

multiplicative auxiliary regressors of ( )iϕ i are statistically significant. In a univariate

setting, the Tsay (1986) test is given by:

∑ ∑∑= = =

−−− +++=p

i

p

i

p

ijtjtitijitit vyyyy

1 10 δϕϕ (3.10)

where ty is the return of the dependent variable, 0ϕ is the regression intercept, iϕ is the

regression coefficients, ijδ is the regression parameter for each auxiliary regressor

possessing quadratic and cross-product terms, p is the lag order, tv is the random error

term and T is the sample size. The Tsay (1986) test examines the null hypothesis

0:0 =ijH δ against 0:1 ≠ijH δ by comparing the 1SSR in (3.10) with the 0SSR in (3.1).

The Tsay (1986) test is calculated as an F-statistic which is approximately

)1,( −−− mpTmF distributed under the null hypothesis with m auxiliary regressors

and 1−−− mpT degrees of freedom. As an example, a lag order of p = 2 would result

in three auxiliary regressors in (3.10) where the term ∑∑= =

−−

p

i

p

jjtitij yy

1 1

δ would consist of

21−ty , 1 2t ty y− − and 2

2−ty .

57

3.3.4.2 Bivariate Test Lee et. al., (1993) and Granger and Teräsvirta (1993) inform us that the Tsay (1986) test

can be re-specified in a bivariate framework so that the linear behaviour between

exogenous variables can be examined. In this study, the Tsay (1986) test is augmented

into a bivariate setting with the restriction to exogenous variables at time t only in the

following expression:

∑∑= =

−− +++=p

i

p

ijtjtitijtt vxxxy

010 δϕϕ (3.11)


variable, 0ϕ is the regression intercept, 1ϕ is the regression coefficient of tx , ijδ is the

regression coefficient of the auxiliary regressor, p is zero, tv is the random error term

and T is the sample size. The Tsay (1986) test examines the null hypothesis

jiH ij ,,0:0 ∀=δ against 0:1 ≠ijH δ by comparing the 1SSR in (3.11) with the 0SSR

in (3.2). The Tsay (1986) test is calculated as an F-statistic which is approximately


and 1−−− mpT degrees of freedom.

3.3.4.3 Wald Tests To control the effects of heteroscedasticity and autocorrelation in the Tsay (1986) test,

this study proposes an F-statistic which is re-specified as a set of Wald tests that are

heteroscedasticity and autocorrelation consistent (HAC). The Wald statistics for the

Tsay (1986) test can be expressed as:

)0ˆ(ˆ)0ˆ( 1,, −Ω′−= − R

TWTR

TWT TW θθ (3.12)

)0ˆ(ˆ)0ˆ( 1,, −Ω′−= − R

TNWTR

TNWT TW θθ (3.13)

58

where WTW , is the White (1980) heteroscedasticity-consistent Wald Statistic of the Tsay

(1986) test, 1,

ˆ −Ω WT is the White (1980) heteroscedasticity-consistent sample covariance

matrix from the residuals of the respective Tsay (1986) test regression, RTθ is the vector

of regression estimators from the respective Tsay (1986) test regression, T is the number

of observations in the residuals, NWTW , is the Newey and West (1987) heteroscedasticity

and autocorrelation consistent (HAC) Wald statistic of the respective Tsay (1986) test

and 1,

ˆ −Ω NWT is the Newey and West (1987) heteroscedasticity and autocorrelation

consistent (HAC) sample covariance matrix from the residuals of the respective Tsay

(1986) test regression. The Tsay (1986) based Wald Test J statistic is asymptotically

chi -squared under the null hypothesis. We proceed to specify the Teräsvirta et. al.,

(1993) V23 test which is the third and final test of linearity-in-the-mean employed in

this study.

3.3.5 Teräsvirta, Lin and Granger (1993) V23 Test Although the Tsay (1986) test is well regarded in the literature, a more powerful test

known as the V23 test developed by Teräsvirta et. al., (1993) examines quadratic, cubic

and relevant cross-product terms of the independent variables. The V23 test is uniquely

different to Keenan (1985) and Tsay (1986) because it considers non-linearity in the

form of cubic terms in addition to the quadratic and cross-product auxiliary regressors

proposed by Tsay (1986). Simulation studies by Teräsvirta et. al., (1993) demonstrate

that the V23 test is a more powerful test in comparison to others when the type of non-

linearity is unspecified. This feature of the V23 test makes it the method of choice for

testing linearity-in-the-mean despite the little research attention that it has received in

the literature.

59

3.3.5.1 Univariate Test In the generalised model in (3.1), the Teräsvirta et. al., (1993) V23 test examines if the

quadratic and cubic terms and multiplicative auxiliary regressors in ( )iϕ i are statistically

significant. In a univariate setting, the Teräsvirta et. al., (1993) V23 test is given by:

∑∑ ∑∑∑∑= = = = =

−−−−−=

− ++++=p

i

p

ij

p

i

p

ij

p

jktktjtitijkjtitij

p

itit vyyyyyyy

1 1110 δδϕϕ (3.14)

where ty is the return of the dependent variable, 0ϕ is the regression intercept, iϕ

represents the regression coefficients, ijδ is the regression parameter for each auxiliary

regressor possessing quadratic and cross-product terms, ijkδ is the regression parameter

for each auxiliary regressor possessing cubic and cubic-based cross-product terms, p is

the lag order, tv is the random error term and T is the sample size. The Teräsvirta et.

al., (1993) V23 test examines the null hypothesis kjiH ijkij ,,,0:0 ∀== δδ against

0:1 ≠ijH δ or 0≠ijkδ by comparing the 1SSR in (3.14) with the 0SSR in (3.1). The

Teräsvirta et. al., (1993) V23 test is calculated as an F-statistic which is approximately


and 1−−− mpT degrees of freedom. As an example, a lag order of p = 2 would result

in seven auxiliary regressors in (3.14) where the term ∑∑= =

−−

p

i

p

jjtitij yy

1 1

δ would consist of

21−ty , 21 −− tt yy and 2

2−ty while ∑∑∑= = =

−−−

p

i

p

ij

p

jkktjtitijk yyy

1

δ would consist of 31−ty , 2

21 −− tt yy ,

221 −− tt yy and 3

2−ty .

60

3.3.5.2 Bivariate Test Similar to Tsay (1986), we can re-specify the original Teräsvirta et. al., (1993) V23 test

to examine the linearity-in-the-mean with a single exogenous variable. The bivariate

Teräsvirta et. al., (1993) V23 test can therefore be expressed as

∑∑ ∑∑∑= = = = =

−−−−− ++++=p

i

p

ij

p

i

p

ij

p

jktktjtitijkjtitijtt vxxxxxxy

0 010 δδϕϕ (3.15)


variable, 0ϕ is the regression intercept, 1ϕ is the regression coefficient, p is zero, ijδ is

the regression parameter for each auxiliary regressor possessing quadratic and cross-

product terms, ijkδ is the regression parameter for each auxiliary regressor possessing

cubic and cubic-based cross-product terms, tv is the random error term and T is the

sample size. The Teräsvirta et. al., (1993) V23 test examines the null hypothesis

kjiH ijkij ,,,0:0 ∀== δδ against 0:1 ≠ijH δ or 0≠ijkδ by comparing the 1SSR in

(3.15) with the 0SSR in (3.2). The Teräsvirta et. al., (1993) V23 test is calculated as an

F-statistic which is approximately )1,( −−− mpTmF distributed under the null

hypothesis with m auxiliary regressors and 1−−− mpT degrees of freedom.

3.3.5.3 Wald Tests To control heteroscedasticity and autocorrelation in the Teräsvirta et. al., (1993) V23

test, we propose that the F-statistic be re-specified as a set of Wald tests which are

heteroscedasticity and autocorrelation consistent (HAC). The Wald statistics for the

Teräsvirta et. al., (1993) V23 test can be expressed as:

)0ˆ(ˆ)0ˆ( 231

,2323,23 −Ω′−= − RVWV

RVWV TW θθ (3.16)

)0ˆ(ˆ)0ˆ( 231

,2323,23 −Ω′−= − RVNWV

RVNWV TW θθ (3.17)

61

where WVW ,23 is the White (1980) heteroscedasticity-consistent Wald Statistic of the

V23 test, 1,23

ˆ −Ω WV is the White (1980) heteroscedasticity-consistent sample covariance

matrix from the residuals derived from the respective V23 test, 23ˆRVθ is the vector of

regression estimators from the respective V23 test, NWVW ,23 is the Newey and West

(1987) heteroscedasticity and autocorrelation consistent (HAC) Wald statistic of the V23

test, 1,23

ˆ −Ω NWV is the Newey and West (1987) heteroscedasticity and autocorrelation

consistent (HAC) sample covariance matrix from the residuals derived from the

respective V23 test.

3.3.6 Equality of Two Regression Coefficients Test The Tsay (1986) and Teräsvirta et. al., (1993) tests examine non-linearity by evaluating

the statistical significance of auxiliary regressors with multiplicative terms in the linear

model. As a comparison, we estimate the Equality of Two Regression Coefficients

(hereafter ETRC) test which is a linearity test in another framework which also allows

heteroscedasticity and autocorrelation in the error disturbances to be controlled.

The estimation of the ETRC test is motivated from the hedge fund study of Lo (2001).

To examine the non-linearity of hedge fund returns, Lo (2001) examines the general

asymmetry of hedge fund returns against traditional asset classes. Lo (2001) sorts

S&P500 and hedge returns into up and down months to form two regression

coefficients. Lo (2001) then estimates the statistical significance of the slope coefficients

for these up and down regressors, however, it is unclear if the asymmetry in the

coefficients reported in Lo (2001) are statistically different.

This study adopts the same methodology as Lo (2001), however, we identify the

presence of linearity by estimating the ETRC hypothesis test. We follow Lo (2001) by

estimating the regression with up and down regressors as:

62

tttt vxxy +++= −−++210 ϕϕϕ (3.18)

where ty is the return of the dependent variable, 0ϕ is the regression intercept, tx is the

return of the independent variable, tt xx =+ if 0>tx or 0 otherwise and tt xx =− if

0≤tx or 0 otherwise, 1ϕ is the regression coefficient for positive returns and 2ϕ is the

regression coefficient for negative returns. Rather than following Lo (2001) and

reporting the statistical significance of the up and down regressors, we propose to report

the statistical significance of the ETRC test. By following Gujarati (1995), the ETRC

test of two regression coefficients of (3.18) examines the null hypothesis 0 1 2:H ϕ ϕ+ −=

against 1 1 2:H ϕ ϕ+ −≠ with the test statistic as

1 2

1 2 1 2

ˆ ˆˆ ˆ ˆ ˆvar( ) var( ) 2cov( , )

t ϕ ϕϕ ϕ ϕ ϕ

+ −

+ − + −

−=

+ − (3.19)

The t-statistic in (3.19) is estimated from the regression in (3.18) and is approximately t-

distributed with 3−T degrees of freedom.

3.4 Data

The data employed in this study consists of investment opportunities from global and

United States (U.S) index returns. We employ continuous compounded excess returns

of various stock and bond indices consisting of 144 monthly observations for the twelve

year period from January 1994 to December 2005. The sample period includes

significant financial market events including the global bear market in bonds of 1994,

the Asian crisis of 1997, the Long Term Capital Management (LTCM) and Russian

bond default crises in 1998, the dot-com boom of 1998 to 2000 and the 9/11 terrorist

attacks in 2001. Monthly excess returns are employed in this study as we are motivated

to examine the linear behaviour between stock and bond returns in a finance framework

as it relates to a mean-variance investor. The sample commences in 1994 as accurate

63

global bond index returns are not available until the 1990s and we wish to compare these

results with reliable hedge fund returns which commence in January 1994.

As a proxy for world stock returns, we employ the Morgan Stanley Commodity Index

(MSCI) All Country World Equity Index. To replicate US stock returns, we utilise the

Standard and Poors (S&P) 500 All Return Index and the MSCI USA Equity Index. To

better understand the linearity-in-the-mean and the variation of stock returns, we also

employ the Fama and French (1992, 1993) and Carhart (1997) risk factors for

comparative purposes.44 We include the Fama and French (1992, 1993) U.S. based

Small-Minus-Big (SMB) and High-Minus-Low (HML) book-to-market risk factors. In

addition, we also employ the Carhart (1997) Up-Minus-Down (UMD) momentum risk

factor, sourced from the Kenneth French data library for comparative purposes only.

To proxy global bond returns, we employ the Morgan Stanley (MS) World plus

Emerging Sovereign Bond Index, the J.P. Morgan Global Government Bond Index and

the Lehman Global Aggregate Index. To examine US bond returns, we utilise the

Morgan Stanley US Government Bond Index and the Lehman US Aggregate Index. The

risk-free rate employed in this study is the Ibbotson and Associates U.S. 1 month

Treasury Bill rate.

44 Not only are we interested in examining whether stock and bond returns are linear-in-the-mean, this study also considers the Fama and French (1992, 1993) and the Carhart (1997) risk factors also. It is well established in the finance literature that the Fama and French (1992, 1993) and Carhart (1997) risk factors may explain a large proportion of the variation of stock returns. For a comprehensive examination of the linearity-in-the-mean between stock and bond returns, we also include the Fama and French (1992, 1993) and Carhart (1997) risk factors in the analysis.

64

Table 3.1 Summary Statistics This table presents the summary statistics of the monthly excess returns of the stocks and bond indices employed in this study. We also include the Fama-French (1992,1993) and Carhart (1997) risk factors for comparative purposes also. Panel A shows the descriptive statistics of the monthly excess returns of the respective indices. Panel B reports the autocorrelation of returns. Panel C shows the autocorrelation of squared returns. Panel D reports the normalised z-scores of the 1st, 2.5th, 5th, 95th, 97.5th and 99th percentiles. The 1%, 2.5%, 5%, 95%, 97.5% and 99% percentiles for a normal distribution are -2.3263, -1.9600, -1.6449, 1.6449, 1.9600 and 2.3263, respectively. The data is sampled monthly from January 1994 to December 2005 consisting of 144 observations. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Stocks Bonds

Sector World USA USA USA USA USA World World World USA USA Variable

MSCI World Index

S&P500 Index

MSCI USA Equity Index

HML

SMB

UMD

MSCI World plus Em. Sov Index

JP Morgan Global Govt Bond Index

Lehman Global Aggregate Index

MSCI US Govt. Bond Index

Lehman USA Aggregate Index

Panel A: Descriptive Statistics Mean 0.360 0.521 0.531 0.493 -0.199 0.698 0.164 0.235 0.219 0.183 0.317 Standard Deviation 4.028 4.280 4.318 3.615 4.182 5.393 1.897 0.900 0.884 1.341 2.524 Skewness -0.754 -0.746 -0.690 0.291 -1.655 -1.216 0.353 -0.275 -0.348 -0.423 -0.556 Kurtosis 4.002 3.967 3.732 4.890 12.140 9.713 3.328 3.335 3.412 3.718 4.139 Median 0.796 1.085 1.100 0.444 -0.200 0.871 0.350 0.310 0.317 0.245 0.584 Maximum 8.455 8.900 9.100 12.848 12.628 16.890 5.490 2.970 2.986 3.520 6.991 Minimum -14.696 -16.020 -15.370 -10.336 -24.680 -28.835 -4.480 -2.410 -2.184 -4.660 -9.714 Jarque-Bera Statistic 19.680 18.967 14.622 21.911 547.384 294.454 3.635 2.483 3.926 7.377 15.194 Jarque-Bera p-value 0.000** 0.000** 0.000** 0.000** 0.000** 0.000** 0.162 0.289 0.140 0.025* 0.001**

Panel B: Autocorrelation (First Moment) AC1 0.016 -0.016 -0.014 0.134 0.185* -0.072 0.203* 0.190* 0.184* 0.087 0.072 AC2 -0.035 -0.027 -0.019 0.019 0.017 -0.099 0.018 0.039 0.002 -0.156* -0.193** AC3 0.055 0.066 0.087 0.039 -0.203* 0.028 0.064 0.129 0.126 0.079 0.073 AC6 0.111 0.086 0.090 0.019 0.077 0.184* -0.031 -0.035 0.006 -0.047 -0.028 AC12 0.086 0.081 0.091 0.109 0.109 0.195* -0.041 -0.170 -0.174* -0.093 -0.106

Panel C: Autocorrelation (Second Moment) AC1 0.035 0.091 0.107 0.314** 0.427** 0.185* -0.037 0.036 0.035 -0.036 -0.041 AC2 0.201* 0.181** 0.181* 0.403** 0.118 0.114 -0.017 -0.021 -0.038 0.134 0.155 AC3 0.059 0.123 0.160 0.465** 0.174* 0.053 -0.050 -0.034 -0.002 -0.017 -0.050 AC6 0.089 0.115 0.109 0.132 -0.025 0.068 -0.002 -0.041 -0.064 -0.061 -0.083 AC12 0.131 0.088 0.095 0.334** 0.006 0.038 -0.056 -0.043 -0.049 0.006 -0.055

Panel D: Standardised Tail Z-Scores 1st Percentile -3.054 -2.905 -2.973 -2.786 -4.382 -3.545 -2.181 -2.624 -2.716 -2.713 -2.533 2.5th Percentile -2.402 -2.170 -2.138 -2.131 -1.598 -1.969 -1.959 -2.137 -2.399 -2.136 -2.215 5th Percentile -1.806 -1.783 -1.813 -1.602 -1.252 -1.780 -1.647 -1.785 -1.677 -1.927 -1.822 95th Percentile 1.436 1.495 1.481 1.584 1.442 1.470 1.791 1.460 1.465 1.630 1.434 97.5th Percentile 1.641 1.596 1.557 2.248 1.640 2.062 2.415 1.781 1.643 1.740 1.874 99th Percentile 1.953 1.826 1.865 3.176 1.857 2.707 2.788 2.109 1.889 2.089 2.067

65

Figures 3.1 to 3.11 in Annexure 3.A (at the end of this chapter) graphically illustrate the

excess monthly returns for each time series employed in this study. Figures 3.1 to 3.3

clearly illustrate the impact of the LTCM and Russian crises in August 1998 on world

and US stock returns. A visual glance of Figures 3.4 to 3.6 show the increase in the

volatility of returns from the Fama-French factors in the year 2000 and the subsequent

reduction thereafter. The graphs from Figures 3.1 to 3.11 demonstrate that linearity-in-

the-mean hypothesis tests need to be robust against tail events (such as August 1998)

and to heteroscedasticity.

Table 3.1 presents the statistical description of the data. We can see that the summary

statistics reflect the salient features of stock and bond index returns. The empirical

characteristics of negative skewness, excess kurtosis and non-normality in most stock

and bond index returns are the dominant features in the data. Another striking feature is

the statistically significant serial correlation in the second moment (ie. non-constant

variance) in stock returns.45 In contrast, world bond returns exhibit statistically

significant serial correlation while US bonds report significant second order negative

correlation.

Overall, the summary statistics in Table 3.1 highlight the serial correlation in the first

and second moments in returns which may affect the inference of the linearity-in-the-

mean tests. It is clear that the linearity-in-the-mean hypothesis tests will be estimated in

the presence of heteroscedasticity and serial correlation in the data. The results of the

hypothesis tests are now considered.

45 The SMB and UMD risk factors report the worst minimum monthly returns which reflect the idiosyncratic risk associated with these factors in comparison to the systematic returns of the entire stock market.

66

3.5 Results

The results section in this study is presented in two parts, namely, univariate and

bivariate tests. The results of both univariate and bivariate tests are considered separately

with conclusions drawn in the subsequent section of this study. There are two key

findings that can be drawn from these results. First, the conventional linearity-in-the-

mean tests detect spurious non-linearity when examining various stock and bond returns.

Second, when the effects of both heteroscedasticity and autocorrelation in the error

disturbances are controlled, we find that stock and bond returns are actually linear-in-

the-mean in both univariate and bivariate settings. The following section details the

findings from linearity-in-the-mean tests calculated in the univariate setting.

3.5.1 Univariate Results Table 3.2 presents the univariate results of the Keenan (1985), Tsay (1986) and

Teräsvirta et. al., (1993) tests for autoregressive models of first, second and third order.

The key finding from Table 3.2 shows that both conventional and heteroscedasticity

adjusted linearity-in-the-mean tests exhibit p-values which are statistically significant.

However, the heteroscedasticity and autocorrelation consistent (HAC) tests report no

significant p-values. The key result from Table 3.2 demonstrates that stock and bond

returns are linear-in-the-mean and that any non-linearity detected in conventional tests

are the result of heteroscedasticity and/or autocorrelation in the error disturbances in the

underlying tests.46

46 While previous studies have partially attributed non-linearity to ARCH effects, some neglected non-linearity has remained unexplained in the literature. Poshakwale (2002) and Yadav, Paudyal and Pope (1999) are examples of studies that have discovered non-linearity which is not attributable to ARCH effects. The results in Table 3.2 suggest that the unexplained non-linearity captured in previous studies may be the result of autocorrelation in the error disturbances.

67

Table 3.2 Univariate Linearity-in-the-Mean Tests This table reports the p-values of the Keenan (1985), Tsay (1986) and the Teräsvirta et. al., (1993) V23 tests in a univariate setting with lag orders of one, two and three, respectively. Three p-values are estimated for each test. The first p-value is estimated from the conventional test. The second p-value is from the test adjusted as a Wald test employing a White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is from the test adjusted as a Wald test employing a Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Test Types Keenan Keenan Keenan Tsay Tsay Tsay V23 V23 V23 Variable AR(1) AR(2) AR(3) AR(1) AR(2) AR(3) AR(1) AR(2) AR(3) Panel A: Stocks MSCI World Equity Idx.

0.959 0.705 0.793

0.977 0.840 0.893

1.000 0.999 0.999

0.529 0.514 0.508

0.410 0.684 0.747

0.478 0.491 0.549

0.721 0.785 0.753

0.361 0.334 0.931

0.271 0.000** 0.815

S&P 500 All Return Idx.

0.925 0.630 0.816

0.715 0.378 0.726

1.000 1.000 1.000

0.406 0.468 0.552

0.304 0.587 0.787

0.350 0.135 0.558

0.749 0.581 0.721

0.281 0.004** 0.874

0.269 0.000** 0.065

MSCI USA Equity Idx.

0.930 0.678 0.833

0.798 0.498 0.766

1.000 1.000 1.000

0.422 0.503 0.579

0.376 0.695 0.822

0.365 0.108 0.532

0.477 0.650 0.742

0.236 0.002** 0.863

0.323 0.997 0.998

HML

0.153 0.070 0.333

0.263 0.164 0.540

0.869 0.734 0.833

0.001** 0.036* 0.192

0.034* 0.097 0.514

0.025* 0.007** 0.824

0.026* 0.004** 0.379

0.006** 0.001** 0.794

0.022* 0.003** 0.888

SMB

0.004** 0.000** 0.409

0.012* 0.000** 0.645

0.064 0.004** 0.748

0.000** 0.000** 0.323

0.003** 0.000** 0.601

0.015* 0.000** 0.846

0.000** 0.000** 0.606

0.043* 0.000** 0.745

0.033* 0.998 0.999

UMD

0.941 0.725 0.782

0.807 0.714 0.582

0.981 0.956 0.898

0.457 0.491 0.581

0.465 0.792 0.691

0.164 0.088 0.653

0.015* 0.000** 0.330

0.043* 0.000** 0.738

0.082 0.000** 0.999

Panel B: Bonds MSCI Wrld plus E.S. Idx.

1.000 0.962 0.974

1.000 1.000 1.000

1.000 0.971 0.982

0.956 0.833 0.840

0.485 0.684 0.691

0.528 0.628 0.664

0.414 0.343 0.513

0.457 0.664 0.898

0.363 0.786 1.000

JPM Global Govt Bond Idx.

0.237 0.012* 0.139

0.237 0.012* 0.263

0.561 0.064 0.337

0.004** 0.038* 0.091

0.081 0.147 0.361

0.210 0.390 0.692

0.048* 0.110 0.223

0.327 0.379 0.851

0.424 0.124 0.971

Lehman Global Agg. Idx.

0.211 0.012* 0.124

0.254 0.026* 0.263

0.749 0.279 0.226

0.003** 0.024* 0.086

0.089 0.123 0.357

0.284 0.421 0.685

0.050* 0.055 0.177

0.337 0.093 0.799

0.425 0.003** 1.000

MSCI US Govt Bond Idx.

0.909 0.602 0.466

1.000 0.994 0.985

1.000 1.000 1.000

0.363 0.384 0.296

0.721 0.895 0.839

0.732 0.942 0.845

0.084 0.258 0.344

0.323 0.279 0.720

0.510 0.476 1.000

Lehman US Agg. Idx.

0.894 0.672 0.451

0.998 0.984 0.956

1.000 0.997 0.992

0.331 0.424 0.251

0.506 0.679 0.573

0.410 0.742 0.716

0.049* 0.190 0.225

0.207 0.081 0.711

0.39 0.987 1.000

68

Table 3.2 reports three p-values for each test. The first p-value is estimated from the

conventional linearity-in-the-mean test. The second p-value is estimated from the White

(1980) heteroscedasticity adjusted test while the third and final p-value is estimated from

the Newey and West (19987) heteroscedasticity and autocorrelation consistent (HAC)

test. The Keenan (1985) test in Table 3.2 report infrequent statistically significant p-

values. In contrast, the more powerful Tsay (1986) and Teräsvirta et. al., (1993) V23

tests in Table 3.2 report statistically significant p-values. The second p-value of each test

is the heteroscedasticity-consistent Wald tests which also report statistically significant

p-values at times. The third p-value of each test is the Newey and West (1987) HAC

result which shows that for all tests, we cannot reject the null hypothesis of linearity-in-

the-mean.

The results in Table 3.2 provide overwhelming empirical evidence to suggest that non-

linearity detected in conventional tests is due to the effects of both heteroscedasticity and

autocorrelation in the error disturbances of the underlying tests. The results also suggest

that heteroscedasticity-consistent Wald tests are also biased due to autocorrelation in the

error disturbances. This effect can be readily seen in world bond index returns whereby

the heteroscedasticity-consistent p-values report spurious over-rejection of linearity-in-

the-mean caused by the serial correlation in the error disturbances. When the

heteroscedasticity and autocorrelation consistent (HAC) hypothesis tests are estimated,

we discover that all p-values are statistically insignificant. The conclusions to be drawn

from Table 3.2 clearly demonstrate that the asset returns employed in this study are

indeed linear-in-the-mean in a univariate setting. We proceed to consider linearity-in-

the-mean hypothesis tests in the bivariate framework.

69

3.5.2 Bivariate Results This section reports the Keenan (1985), Tsay (1986), Teräsvirta et. al., (1993) V23 and

the Equality of Two Regression Coefficients Test results in a bivariate setting. The

bivariate results in this study are presented in the following format. First, a scatterplot of

each data series against global stock returns is presented in Figures 3.12 to 3.21 in

Annexure 3.B (at the end of this chapter) to better understand the forthcoming statistical

results. Second, in the interest of brevity, the p-values presented in this section are

limited to bivariate relationships which are statistically significant. For completeness,

the full set of linearity-in-the-mean bivariate test results are presented in the Annexure

section at the end of this chapter. Similar to the univariate results section, we report

three p-values for each bivariate test, namely, the conventional p-value, the

heteroscedasticity-consistent p-value and the HAC p-value.

The key findings from the bivariate tests are consistent with the univariate results. The

p-values show that, at times, conventional tests over-reject the null hypothesis of

linearity-in-the-mean due to heteroscedasticity and autocorrelation in the error

disturbances in the underlying tests. When the error disturbances of each test are

estimated in the HAC framework, the findings reveal that all p-values are insignificant.

The bivariate test results provide overwhelming evidence to suggest that stock and bond

returns are linear-in-the-mean in a bivariate setting.

70

Table 3.3 Tsay (1986) Test – Stocks

This table presents the p-values of the Tsay (1986) tests with the stock indices and equity risk factors as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively. Sector Global USA USA USA USA Asset Class

Sector

`

Dependent Variable

MSCI World Equity Index

S&P500 All Return Index


HML

UMD

Stocks

Global

MSCI World Equity Idx.

---- NA ----

0.114 0.163 0.143

0.061 0.048* 0.154

0.650 0.881 0.913

0.672 0.927 0.958

Stocks USA S&P500 All Return Idx.

0.179 0.190 0.429

---- NA ----

0.013* 0.202 0.437

0.365 0.566 0.650

0.491 0.756 0.870

Stocks USA MSCI USA Equity Idx.

0.410 0.507 0.641

0.020* 0.187 0.448

---- NA ----

0.388 0.598 0.671

0.489 0.749 0.863

Stocks USA UMD

0.674 0.916 0.911

0.667 0.889 0.887

0.830 0.973 0.966

0.015* 0.307 0.476

---- NA ----

Bonds Global MSCI World plus E.S. Idx.

0.051 0.012* 0.153

0.069 0.029* 0.241

0.076 0.048* 0.236

0.506 0.708 0.623

0.726 0.876 0.884

Bonds Global JPM Global Bond Idx.

0.225 0.374 0.452

0.017* 0.010** 0.189

0.018* 0.011* 0.179

0.481 0.603 0.650

0.328 0.152 0.460

Bonds Global Lehman Global Agg. Idx.

0.386 0.565 0.596

0.042* 0.023* 0.179

0.045* 0.027* 0.175

0.471 0.614 0.632

0.145 0.023* 0.347

Bonds USA MSCI USA Govt. Bond Idx.

0.330 0.488 0.575

0.039* 0.036* 0.183

0.045* 0.046* 0.190

0.417 0.579 0.578

0.143 0.035* 0.293

71

3.5.2.1 Keenan (1985) Bivariate Results Annexure Tables 3.C and 3.D report the bivariate results of the Keenan (1985) test. The

common feature of both tables is the pronounced insignificant p-values reported across

all asset classes. These results can be attributed to one of two rationales. The first

possibility comes from Granger and Teräsvirta (1993) and Lee et. al., (1993) who

suggest that the Keenan (1985) test may lack power in detecting unspecified non-

linearity in comparison to alternatives such as Tsay (1986) and Teräsvirta et. al., (1993).

The second rationale may be attributable to the fact that asset returns may in fact be

linear-in-the-mean. To determine which of these possibilities are correct, we examine

bivariate linearity with the more powerful Tsay (1986) framework.

3.5.2.2 Tsay (1986) Bivariate Results The significant linearity-in-the-mean results of the Tsay (1986) tests are presented in

Table 3.3. The frequency of statistically significant p-values in Table 3.3 suggests that

the Tsay (1986) test is more powerful than Keenan (1985). Table 3.3 shows that the

rejection of the null hypothesis of linearity-in-the-mean occurs when stock returns are

the independent variable. The Tsay (1986) tests in Annexure 3.F report insignificant p-

values when bonds are the independent variable.

Despite the rejection of the null hypothesis of conventional Tsay (1986) tests, the HAC

p-values in Table 3.3 reveal that we cannot reject the null hypothesis of linearity-in-the-

mean in the bivariate setting. The evidence from the Tsay (1986) tests suggest that stock

and bond returns are linear-in-the-mean and that conventional and heteroscedasticity-

adjusted tests incorrectly detect non-linearity due to the autocorrelation effects in the

error disturbances. To confirm the Tsay (1986) bivariate results, we proceed to estimate

the Teräsvirta et. al., (1993) V23 test.

72

Table 3.4 Teräsvirta, Lin and Granger (1993) V23 Test – Stocks This table presents the p-values of the Teräsvirta et. al., (1993) V23 Tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each V23 test. The first p-value represents the original Teräsvirta et. al., (1993) V23 test. The second p-value is the V23 test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the V23 test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent (HAC) covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Independent Variable Sector Global USA USA USA USA USA Asset Class

Sector

Dependent Variable




HML

SMB

UMD

Stocks Global MSCI World Equity Idx.

---- NA ----

0.231 0.144 0.270

0.165 0.027* 0.292

0.814 0.761 0.723

0.367 0.300 0.403

0.309 0.030* 0.592


0.392 0.241 0.462

---- NA ----

0.019* 0.385 0.510

0.432 0.287 0.437

0.880 0.893 0.671

0.493 0.011* 0.628


0.657 0.561 0.610

0.012* 0.222 0.597

---- NA ----

0.426 0.359 0.473

0.945 0.952 0.839

0.540 0.018* 0.643

Stocks USA HML

0.445 0.665 0.709

0.261 0.651 0.696

0.244 0.701 0.721

---- NA ----

0.045* 0.001** 0.404

0.220 0.000** 0.618

Stocks USA SMB

0.859 0.899 0.901

0.027 0.362 0.446

0.020* 0.416 0.493

0.018* 0.650 0.766

---- NA ----

0.018* 0.000** 0.508

Stocks USA UMD

0.305 0.712 0.757

0.682 0.851 0.850

0.735 0.901 0.921

0.027* 0.383 0.470

0.174 0.655 0.670

---- NA ----

Bonds Global MSCI World plus E.S. Idx

0.132 0.016* 0.419

0.186 0.020* 0.435

0.190 0.046* 0.441

0.517 0.580 0.646

0.936 0.911 0.790

0.927 0.818 0.781


0.034* 0.000** 0.465

0.017* 0.000** 0.357

0.014* 0.000** 0.347

0.514 0.441 0.484

0.409 0.000** 0.447

0.555 0.001** 0.496


0.068 0.000** 0.425

0.042* 0.000** 0.331

0.035* 0.000** 0.329

0.368 0.358 0.445

0.370 0.000** 0.458

0.285 0.000** 0.404


0.076 0.008** 0.454

0.060 0.001** 0.322

0.052 0.001** 0.329

0.334 0.415 0.477

0.407 0.000** 0.474

0.320 0.001** 0.371

Bonds USA Lehman USA Agg. Idx. 0.102 0.014* 0.487

0.109 0.000** 0.401

0.096 0.000** 0.403

0.472 0.533 0.645

0.673 0.006** 0.485

0.587 0.007** 0.405

73

3.5.2.3 Teräsvirta, Lin and Granger (1993) V23 Bivariate Results The Teräsvirta et. al., (1993) V23 p-values in Table 3.4 provide empirical support for

the previous Tsay (1986) test results in Table 3.3. The key finding from the Teräsvirta

et. al., (1993) V23 p-values in Table 3.4 demonstrates that stock and bond returns are

linear-in-the-mean in a HAC bivariate setting.

Again, the V23 test results with significant p-values are the conventional V23 tests with

stock returns as the independent variable. Consistent with the Tsay (1986) results, the

V23 tests with bonds as the independent variable report insignificant p-values in

Annexure 3.H. The p-values in Table 3.4 reveal that the Teräsvirta et. al., (1993) V23

test also has a tendency to over-reject the null hypothesis (ie. Type I error). Table 3.4

reports insignificant p-values in conventional tests which become statistically significant

when the heteroscedasticity-consistent p-value is estimated. The change in statistical

significance can be attributed to the autocorrelation in the test residuals. This can be

easily observed in Table 3.4 where statistically significant p-values are reported with the

serially correlated bond returns as the dependent variable. Consistent with the previous

findings, the Teräsvirta et. al., (1993) V23 p-values estimated in the heteroscedasticity

and autocorrelation (HAC) framework are all insignificant.

74

3.5.2.4 Equality of Two Regression Coefficients Test As a final test of linearity, we proceed to consider the Equality of Two Regression

Coefficients (ETRC) test. The significant p-values are presented in Table 3.5 with the

complete set of p-values tabulated in Annexures 3.I and 3.J. The results of the ETRC test

must be interpreted differently to the previous results as it is not a test of linearity based

on multiplicative terms. Instead, the ETRC test examines whether the up-regressor and

the down-regressor are statistically different.

Table 3.5 reveals that the ETRC test shows little variation in the three forms of p-values

that are estimated. This suggests that the ETRC test is not sensitive to the effects of

heteroscedasticity and autocorrelation in the estimated residuals and standard errors.

The second feature can be seen in Annexures 3.I and 3.J which reveal the high

proportion of insignificant p-values. Given the evidence presented so far we can

confidently state that the up and down regressors are symmetric. The third observation

from the ETRC tests is the statistically significant p-values detected in the bivariate

relationship between world and US stock returns. This result shows that world and US

stocks possess an asymmetric relationship with statistical significance at the 5 per cent

level but not at the 1 per cent level.

Table 3.5 Equality of Two Regression Coefficients Test- Stocks This table presents the Equality of Two Regression Coefficients test with positive and negative returns of the excess monthly returns of the independent variable used as separate regressors. Three p-values are presented in the table. The first reported p-value is estimated from the standard errors from a conventional ordinary least squares regression. The second reported p-value is estimated with the White (1980) heteroscedasticity consistent standard errors. The third p-value is estimated with the Newey and West (1987) heteroscedasticity and autocorrelation consistent errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Sector

Dependent Variable




HML

SMB

UMD

Stocks Global MSCI World Equity Idx. NA

NA NA

0.060 0.047* 0.038*

0.047* 0.037* 0.029*

0.442 0.420 0.465

0.834 0.839 0.840

0.905 0.919 0.931

75

Overall, the ETRC test suggests that the bivariate relationships between stock and bond

returns are symmetrical. The exception is the relationship between world and US stock

returns which report statistical significance at the 5 percent level but not at the 1 percent

level. With the completion of the results section, we provide concluding remarks to this

study.

3.6 Conclusion

The linearity condition is fundamentally important in empirical finance. In the context

of portfolio selection and asset pricing, the linearity-in-the-mean assumption must hold

in order for the covariance matrix to be valid in empirical finance settings. If stocks and

bonds are not linear-in-the-mean then the decisions made with these empirical models

may be subject to model mis-specification. This study contributes to the literature by

examining the univariate and bivariate linear behaviour of the two most important asset

classes in the world, stock and bonds, in an investment based framework.

Of the two forms of linearity (ie. linear conditional mean and constant conditional

variance (ie. ARCH)), we examine the linearity-in-the-mean as it is the least researched

in the literature. We develop a formal hypothesis testing approach to measure the

linearity-in-the-mean of asset returns in both univariate and bivariate settings. The

conventional hypothesis tests report that stock and bond returns are non-linear, in both

univariate and bivariate settings. However, on closer examination, we discover that

many of the stock and bond returns exhibit heteroscedasticity and autocorrelation which

contaminate the error disturbances employed in the hypothesis tests. We propose an

approach which controls these effects by augmenting the linearity-in-the-mean tests into

a HAC framework. This chapter demonstrates that previous empirical studies have

over-rejected the null hypothesis of linearity-in-the-mean (ie. Type I error) due to

heteroscedasticity and autocorrelation in the error disturbances in these tests. When both

of these effects are controlled in a hypothesis testing regime, this study overwhelmingly

demonstrates that stock and bond returns are indeed linear-in-the-mean.

76

The discovery that stocks and bonds are linear-in-the-mean provides additional insights

to the behaviour of these important asset classes. First, the evidence that the conditional

mean in stocks and bonds is linear is good news for mean-variance investors making

portfolio investment decisions. Second, this study highlights how heteroscedasticity and

autocorrelation can contaminate the statistical inference of linearity-in-the-mean tests.

Researchers who examine non-linearity must be able to discriminate and isolate the

effects of both heteroscedasticity and autocorrelation from the underlying testing regime.

This study shows that a failure to isolate these effects will result in biased results.

The findings from this study provide several directions for future research. While stocks

and bond returns are linear-in-the-mean, it is worthwhile to consider the same question

for alternative asset classes such as hedge funds. This research question is considered in

the forthcoming chapter. Second, it may be insightful to consider the methodology of

this study in a finer sampling frequency setting such as weekly or daily returns. Third,

Granger and Teräsvirta (1993) and Lee et. al., (1993) have highlighted that

heteroscedasticity and autocorrelation can bias OLS regressions and tests. This study has

empirically demonstrated this effect. In another setting, Dimson (1979), Roll (1981) and

Blume, Keim and Patel (1991) inform us that autocorrelation also causes a downward

bias in the estimate of variance in returns. The autocorrelation effects on variance (and

therefore standard deviation) must cause biases within the portfolio selection framework.

This thesis investigates the autocorrelation effects on variance and portfolio selection in

Chapter 5.

77

Annexure 3.A

Figure 3.1 MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 3.2 Standard and Poors (S&P) 500 All Return Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

78

Figure 3.3 MSCI USA Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 3.4 Fama-French HML Risk Factor

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

79

Figure 3.5 Fama-French SMB Risk Factor

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 3.6 Fama-French UMD Risk Factor

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

80

Figure 3.7 Morgan Stanley World plus Emerging Sovereign Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 3.8 J.P. Morgan Global Bond Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

81

Figure 3.9 Lehman Global Aggregate Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 3.10 Morgan Stanley US Government Bond Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

82

Figure 3.11 Lehman USA Aggregate Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

83

Annexure 3.B

Figure 3.12 S&P500 All Return Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%

MSCI World Equity Index Excess Monthly Returns

S&P5

00 A

ll R

etur

n In

dex

Exce

ss M

onth

ly R

etur

ns

Figure 3.13 MSCI USA Equity Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


MSC

I USA

Equ

ity In

dex

Exce

ss M

onth

ly R

etur

ns

84

Figure 3.14 HML vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Fam

a-Fr

ench

HM

L R

isk

Fact

or

Figure 3.15 SMB vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Fam

a-Fr

ench

SM

B R

isk

Fact

or

85

Figure 3.16 UMD vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Car

hart

UM

D M

omen

tum

Ris

k Fa

ctor

Figure 3.17 MS World Bond Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


MS

Wor

ld p

lus

Emer

ging

Sov

. Ind

ex E

xces

s M

onth

ly R

etur

ns

86

Figure 3.18 J.P. Morgan Global Bond Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


J.P.

Mor

gan

Glo

bal B

ond

inde

x Ex

cess

Mon

thly

Ret

urns

Figure 3.19 Lehman Global Aggregate Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Lehm

an G

loba

l Agg

rega

te In

dex

Exce

ss M

onth

ly R

etur

ns

87

Figure 3.20 Morgan Stanley U.S. Govt. Bond Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


MS

US

Gov

t. B

ond

Inde

x Ex

cess

Mon

thly

Ret

urns

Figure 3.21 Lehman U.S. Aggregate Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Lehm

an U

S A

ggre

gate

Inde

x Ex

cess

Mon

thly

Ret

urns

88

Annexure 3.C Keenan (1985) Bivariate Test – Stocks This table presents the p-values of the Keenan (1985) tests with the stock indices and equity risk factors as the independent variable. This table reports three p-values for each Keenan (1985) test. The first p-value represents the original Keenan (1985) test. The second p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively. Independent Variable Sector Global USA USA USA USA USA Asset Class

Sector

`

Dependent Variable




HML

SMB

UMD

Stocks

Global


---- NA ----

0.642 0.727 0.691

0.470 0.417 0.712

0.995 1.000 1.000

0.916 0.950 0.992

0.996 1.000 1.000


0.772 0.768 0.946

---- NA ----

0.195 0.784 0.949

0.936 0.980 0.990

0.999 1.000 1.000

0.976 0.997 1.000


0.955 0.968 0.990

0.250 0.763 0.952

---- NA ----

0.946 0.985 0.992

1.000 1.000 1.000

0.976 0.997 1.000

Stocks USA HML

0.996 1.000 1.000

1.000 1.000 1.000

1.000 1.000 1.000

---- NA ----

0.991 1.000 1.000

0.958 1.000 1.000

Stocks USA SMB

0.990 0.303 0.999

0.460 0.837 0.786

0.521 0.896 0.839

0.910 1.000 1.000

---- NA ----

0.998 1.000 1.000

Stocks USA UMD

0.996 1.000 1.000

0.996 1.000 1.000

1.000 1.000 1.000

0.194 0.883 0.960

0.480 0.972 0.995

---- NA ----


0.423 0.182 0.709

0.501 0.311 0.828

0.527 0.415 0.822

0.979 0.995 0.988

1.000 1.000 1.000

0.998 1.000 1.000


0.823 0.923 0.953

0.215 0.160 0.766

0.218 0.175 0.751

0.974 0.985 0.990

0.991 0.998 0.998

0.918 0.708 0.956


0.946 0.980 0.984

0.381 0.276 0.752

0.397 0.304 0.746

0.972 0.987 0.989

0.991 0.999 0.999

0.718 0.274 0.909


0.918 0.964 0.981

0.364 0.355 0.757

0.393 0.407 0.768

0.957 0.982 0.982

1.000 1.000 1.000

0.710 0.348 0.874

Bonds USA Lehman USA Agg. Idx.

0.986 0.998 0.999

0.615 0.691 0.898

0.647 0.747 0.903

0.995 0.999 0.999

1.000 1.000 1.000

0.908 0.683 0.894

89

Annexure 3.D Keenan (1985) Bivariate Tests – Bonds This table presents the p-values of the Keenan (1985) tests with the bond indices as the independent variable. This table reports three p-values for each Keenan (1985) test. The first p-value represents the original Keenan (1985) test. The second p-value is the Keenan (1985) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Independent Variable Sector Global Global Global USA USA Asset Class

Sector

Dependent Variable

MSCI World plus Em Sovrgn Index

J.P. Morgan Global Bond Index


MS USA Govt Bond Index



0.717 0.847 0.800

0.986 0.999 1.000

0.987 0.995 0.998

0.793 0.941 0.959

0.989 0.998 0.999


0.850 0.950 0.942

1.000 1.000 1.000

1.000 1.000 1.000

0.873 0.980 0.980

0.999 1.000 1.000


0.813 0.935 0.928

1.000 1.000 1.000

1.000 1.000 1.000

0.878 0.980 0.981

0.999 1.000 1.000

Stocks USA HML

0.897 0.975 0.976

0.986 0.996 0.994

0.993 0.997 0.997

1.000 1.000 1.000

1.000 1.000 1.000

Stocks USA SMB

1.000 1.000 1.000

0.999 1.000 1.000

0.998 1.000 1.000

0.998 1.000 1.000

1.000 1.000 1.000

Stocks USA UMD

0.994 1.000 1.000

0.782 0.817 0.897

0.880 0.781 0.914

0.956 0.994 0.990

0.946 0.994 0.988

Bonds Global MS World plus E.S. Idx.

---- NA ----

0.776 0.948 0.936

0.426 0.966 0.959

0.722 0.896 0.924

0.965 0.992 0.996


1.000 1.000 1.000

---- NA ----

0.963 0.997 0.991

0.799 0.941 0.984

0.486 0.607 0.963


1.000 1.000 1.000

0.906 0.992 0.985

---- NA ----

0.946 0.998 0.999

0.618 0.912 0.987

Bonds USA MS USA Govt. Bond Idx.

0.980 1.000 1.000

1.000 1.000 1.000

1.000 1.000 1.000

---- NA ----

0.902 0.975 0.994


0.955 0.998 0.997

0.972 1.000 1.000

0.981 1.000 1.000

0.308 0.720 0.917

---- NA ----

90

Annexure 3.E Tsay (1986) Test – Stocks This table presents the p-values of the Tsay (1986) tests with the stock indices and equity risk factors as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively. Independent Variable Sector Global USA USA USA USA USA Asset Class

Sector

`

Dependent Variable




HML

SMB

UMD

Stocks

Global


---- NA ----

0.114 0.163 0.143

0.061 0.048* 0.154

0.650 0.881 0.913

0.326 0.440 0.669

0.672 0.927 0.958


0.179 0.190 0.429

---- NA ----

0.013* 0.202 0.437

0.365 0.566 0.650

0.746 0.939 0.909

0.491 0.756 0.870


0.410 0.507 0.641

0.020* 0.187 0.448

---- NA ----

0.388 0.598 0.671

0.834 0.976 0.959

0.489 0.749 0.863

Stocks USA HML

0.669 0.921 0.938

0.914 0.996 0.996

0.881 0.993 0.992

---- NA ----

0.591 0.864 0.885

0.421 0.901 0.841

Stocks USA SMB

0.580 0.860 0.812

0.059 0.250 0.204

0.074 0.326 0.252

0.317 0.897 0.909

---- NA ----

0.724 0.989 0.986

Stocks USA UMD

0.674 0.916 0.911

0.667 0.889 0.887

0.830 0.973 0.966

0.015* 0.307 0.476

0.064 0.525 0.713

---- NA ----


0.051 0.012* 0.153

0.069 0.029* 0.241

0.076 0.048* 0.236

0.506 0.708 0.623

0.871 0.981 0.979

0.726 0.876 0.884


0.225 0.374 0.452

0.017* 0.010** 0.189

0.018* 0.011* 0.179

0.481 0.603 0.650

0.595 0.780 0.798

0.328 0.152 0.460


0.386 0.565 0.596

0.042* 0.023* 0.179

0.045* 0.027* 0.175

0.471 0.614 0.632

0.596 0.805 0.821

0.145 0.023* 0.347


0.330 0.488 0.575

0.039* 0.036* 0.183

0.045* 0.046* 0.190

0.417 0.579 0.578

0.837 0.973 0.969

0.143 0.035* 0.293


0.548 0.783 0.808

0.104 0.143 0.328

0.116 0.176 0.338

0.645 0.832 0.841

0.839 0.969 0.966

0.312 0.139 0.322

91

Annexure 3.F Tsay (1986) Tests – Bonds This table presents the p-values of the Tsay (1986) tests with the bond indices as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Sector

Dependent Variable







0.148 0.261 0.215

0.549 0.814 0.863

0.556 0.716 0.795

0.194 0.417 0.470

0.577 0.793 0.812


0.242 0.441 0.418

0.829 0.980 0.981

0.912 0.991 0.993

0.267 0.565 0.569

0.755 0.939 0.940


0.148 0.261 0.215

0.817 0.976 0.978

0.896 0.987 0.989

0.273 0.568 0.573

0.754 0.937 0.938

Stocks USA HML

0.297 0.540 0.541

0.548 0.735 0.693

0.618 0.749 0.747

0.975 0.999 0.999

0.981 0.999 0.999

Stocks USA SMB

0.834 0.964 0.962

0.784 0.935 0.931

0.717 0.861 0.868

0.724 0.909 0.923

0.811 0.947 0.960

Stocks USA UMD

0.625 0.862 0.885

0.187 0.230 0.328

0.276 0.200 0.357

0.415 0.696 0.644

0.388 0.703 0.628


---- NA ----

0.182 0.434 0.403

0.192 0.497 0.469

0.151 0.325 0.377

0.442 0.670 0.730


0.821 0.969 0.969

---- NA ----

0.435 0.750 0.657

0.199 0.415 0.595

0.065 0.105 0.485


0.912 0.993 0.994

0.310 0.669 0.603

---- NA ----

0.387 0.768 0.817

0.104 0.352 0.615


0.508 0.859 0.858

0.883 0.993 0.993

0.993 0.999 0.999

---- NA ----

0.304 0.538 0.704


0.411 0.770 0.761

0.470 0.878 0.860

0.514 0.896 0.902

0.030* 0.159 0.363

---- NA ----

92

Annexure 3.G Teräsvirta, Lin and Granger (1993) V23 Test – Stocks This table presents the p-values of the Teräsvirta et. al., (1993) V23 Tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each V23 test. The first p-value represents the original Teräsvirta et. al., (1993) V23 test. The second p-value is the V23 test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the V23 test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Sector

Dependent Variable




HML

SMB

UMD


---- NA ----

0.231 0.144 0.270

0.165 0.027* 0.292

0.814 0.761 0.723

0.367 0.300 0.403

0.309 0.030* 0.592


0.392 0.241 0.462

---- NA ----

0.019* 0.385 0.510

0.432 0.287 0.437

0.880 0.893 0.671

0.493 0.011* 0.628


0.657 0.561 0.610

0.012* 0.222 0.597

---- NA ----

0.426 0.359 0.473

0.945 0.952 0.839

0.540 0.018* 0.643

Stocks USA HML

0.445 0.665 0.709

0.261 0.651 0.696

0.244 0.701 0.721

---- NA ----

0.045* 0.001** 0.404

0.220 0.000** 0.618

Stocks USA SMB

0.859 0.899 0.901

0.027 0.362 0.446

0.020* 0.416 0.493

0.018* 0.650 0.766

---- NA ----

0.018* 0.000** 0.508

Stocks USA UMD

0.305 0.712 0.757

0.682 0.851 0.850

0.735 0.901 0.921

0.027* 0.383 0.470

0.174 0.655 0.670

---- NA ----


0.132 0.016* 0.419

0.186 0.020* 0.435

0.190 0.046* 0.441

0.517 0.580 0.646

0.936 0.911 0.790

0.927 0.818 0.781


0.034* 0.000** 0.465

0.017* 0.000** 0.357

0.014* 0.000** 0.347

0.514 0.441 0.484

0.409 0.000** 0.447

0.555 0.001** 0.496


0.068 0.000** 0.425

0.042* 0.000** 0.331

0.035* 0.000** 0.329

0.368 0.358 0.445

0.370 0.000** 0.458

0.285 0.000** 0.404


0.076 0.008** 0.454

0.060 0.001** 0.322

0.052 0.001** 0.329

0.334 0.415 0.477

0.407 0.000** 0.474

0.320 0.001** 0.371

Bonds USA Lehman USA Agg. Idx. 0.102 0.014* 0.487

0.109 0.000** 0.401

0.096 0.000** 0.403

0.472 0.533 0.645

0.673 0.006** 0.485

0.587 0.007** 0.405

93

Annexure 3.H Teräsvirta, Lin and Granger (1993) V23 Tests – Bonds This table presents the p-values of the Teräsvirta et. al., (1993) V23 Tests with the bond indices as the independent variable. This table reports three p-values for each V23 test. The first p-value represents the original Teräsvirta et. al., (1993) V23 test. The second p-value is the V23 test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the V23 test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Sector

Dependent Variable







0.148 0.070 0.206

0.552 0.426 0.648

0.328 0.236 0.457

0.430 0.387 0.360

0.776 0.625 0.490


0.247 0.190 0.325

0.356 0.338 0.600

0.184 0.108 0.385

0.540 0.480 0.448

0.801 0.713 0.623


0.226 0.181 0.324

0.370 0.344 0.606

0.199 0.118 0.399

0.549 0.476 0.446

0.807 0.711 0.624

Stocks USA HML

0.270 0.518 0.580

0.681 0.746 0.670

0.760 0.706 0.571

0.738 0.642 0.498

0.770 0.520 0.443

Stocks USA SMB

0.473 0.528 0.418

0.787 0.780 0.696

0.846 0.733 0.603

0.531 0.727 0.613

0.854 0.897 0.871

Stocks USA UMD

0.757 0.796 0.821

0.322 0.333 0.438

0.482 0.372 0.484

0.264 0.554 0.641

0.153 0.289 0.515


---- NA ----

0.361 0.405 0.578

0.309 0.401 0.597

0.167 0.401 0.492

0.667 0.711 0.787


0.778 0.907 0.907

---- NA ----

0.676 0.817 0.701

0.392 0.450 0.663

0.155 0.207 0.530


0.776 0.884 0.847

0.582 0.736 0.601

---- NA ----

0.300 0.508 0.790

0.146 0.328 0.707


0.715 0.867 0.838

0.989 0.992 0.993

0.883 0.965 0.980

---- NA ----

0.452 0.366 0.749


0.627 0.769 0.724

0.644 0.891 0.871

0.739 0.909 0.917

0.068 0.155 0.449

---- NA ----

94

Annexure 3.I Equality of Two Regression Coefficients Test- Stocks This table presents the Equality of Two Regression Coefficients test with positive and negative returns of the excess monthly returns of the independent variable used as separate regressors. Three p-values are presented in the table. The first reported p-value is estimated from the standard errors from a conventional ordinary least squares regression. The second reported p-value is estimated with the White (1980) heteroscedasticity consistent standard errors. The third p-value is estimated with the Newey and West (1987) heteroscedasticity and autocorrelation consistent errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Sector

Dependent Variable




HML

SMB

UMD

Stocks Global MSCI World Equity Idx. NA

NA NA

0.060 0.047* 0.038*

0.047* 0.037* 0.029*

0.442 0.420 0.465

0.834 0.839 0.840

0.905 0.919 0.931

Stocks USA S&P500 All Return Idx. 0.225 0.195 0.214

NA NA NA

0.065 0.137 0.099

0.278 0.261 0.302

0.659 0.664 0.647

0.865 0.875 0.891

Stocks USA MSCI USA Equity Idx. 0.432 0.392 0.413

0.081 0.153 0.106

NA NA NA

0.281 0.268 0.309

0.625 0.635 0.612

0.807 0.822 0.843

Stocks USA HML 0.859 0.874 0.883

0.972 0.976 0.976

0.869 0.887 0.892

NA NA NA

0.356 0.384 0.433

0.758 0.850 0.840

Stocks USA SMB 0.963 0.967 0.958

0.109 0.144 0.067

0.107 0.145 0.078

0.246 0.452 0.527

NA NA NA

0.812 0.904 0.891

Stocks USA UMD 0.430 0.453 0.460

0.707 0.698 0.697

0.848 0.845 0.845

0.054 0.206 0.211

0.423 0.644 0.622

NA NA NA

Bonds Global MSCI World plus E.S. Idx. 0.169 0.144 0.131

0.097 0.081 0.085

0.108 0.095 0.096

0.577 0.542 0.506

0.470 0.472 0.456

0.390 0.373 0.400

Bonds Global JPM Global Bond Idx. 0.573 0.568 0.556

0.067 0.064 0.068

0.072 0.068 0.072

0.464 0.405 0.422

0.758 0.748 0.744

0.309 0.232 0.269

Bonds Global Lehman Global Agg. Idx. 0.837 0.830 0.830

0.133 0.116 0.122

0.147 0.127 0.133

0.440 0.378 0.390

0.667 0.666 0.664

0.163 0.102 0.128

Bonds USA MSCI USA Govt. Bond Idx. 0.931 0.929 0.932

0.132 0.123 0.136

0.150 0.140 0.153

0.401 0.352 0.363

0.873 0.880 0.874

0.112 0.083 0.105


0.871 0.867 0.872

0.294 0.287 0.300

0.327 0.320 0.332

0.770 0.746 0.756

0.737 0.746 0.734

0.260 0.198 0.215

95

Annexure 3.J Equality of Two Regression Coefficients Test - Bonds This table presents the Equality of Two Regression Coefficients test with positive and negative returns of the excess monthly returns of the independent variable used as separate regressors. Three p-values are presented in the table. The first reported p-value is estimated from the standard errors from a conventional ordinary least squares regression. The second reported p-value is estimated with the White (1980) heteroscedasticity consistent standard errors. The third p-value is estimated with the Newey and West (1987) heteroscedasticity and autocorrelation consistent errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Sector

Dependent Variable







0.102 0.078 0.067

0.505 0.506 0.565

0.465 0.420 0.477

0.170 0.182 0.235

0.476 0.474 0.519


0.202 0.179 0.168

0.652 0.669 0.699

0.635 0.616 0.651

0.182 0.211 0.248

0.557 0.569 0.595


0.175 0.157 0.148

0.641 0.656 0.688

0.612 0.591 0.629

0.188 0.214 0.253

0.551 0.561 0.588

Stocks USA HML

0.132 0.115 0.112

0.772 0.747 0.740

0.760 0.721 0.725

0.789 0.758 0.761

0.774 0.740 0.744

Stocks USA SMB

0.941 0.936 0.932

0.951 0.945 0.944

0.816 0.786 0.784

0.898 0.891 0.890

0.970 0.968 0.969

Stocks USA UMD

0.808 0.801 0.807

0.165 0.115 0.141

0.262 0.187 0.239

0.411 0.392 0.387

0.422 0.430 0.433


NA NA NA

0.103 0.095 0.093

0.079 0.079 0.088

0.061 0.061 0.057

0.348 0.337 0.330


0.579 0.573 0.580

NA NA NA

0.541 0.563 0.537

0.473 0.491 0.496

0.264 0.279 0.281


0.779 0.780 0.777

0.227 0.253 0.233

NA NA NA

0.835 0.856 0.866

0.493 0.552 0.564


0.912 0.923 0.921

0.845 0.870 0.871

0.376 0.475 0.520

NA NA NA

0.371 0.379 0.394


0.810 0.830 0.826

0.629 0.697 0.690

0.946 0.957 0.961

0.051 0.053 0.054

NA NA NA

96

4. The Linear Behaviour of Hedge Funds and Traditional Asset Classes

4.1 Introduction

Chapter 3 demonstrated that stock and bond returns exhibit a linear conditional mean in

both univariate and bivariate settings. This finding suggests that the most studied assets

in the portfolio selection literature, stocks and bonds, are linear-in-the-mean in the

monthly time frequency. In Chapter 4, we extend this theme by considering the linear

behaviour of hedge fund returns. The behaviour of hedge funds may differ to stocks and

bonds because of the dynamic nature of investment strategies in the global hedge fund

industry. Fung and Hsieh (1997) and Lo (2001) reveal that hedge funds employ short-

selling, derivatives and leverage techniques to shift the exposures of their investment

portfolios through time. In light of these dynamic portfolio management techniques, a

study which examines the linear behaviour between hedge funds and traditional asset

classes is therefore warranted.

The recent growth in the global hedge fund industry motivates researchers to examine

these investment vehicles in the literature. Lo (2002), Geman and Kharoubi (2003) and

Getmansky, Lo and Makarov (2004) have examined the statistical and time-series

characteristics of hedge fund returns and how they interact with traditional investment

portfolios. These studies have revealed that hedge fund returns, in general, depart from

normality and possess serial correlation in the first and second moments. In another

strand of literature, Agarwal and Naik (2004), Favre and Galeano (2002), Huber and

Kaiser (2004), Lo (2001) and Mitchell and Pulvino (2001) suggest that hedge fund

returns are non-linear. If the linearity assumption does not hold in the case of hedge

funds, then the development of complex non-linear portfolio and asset pricing

frameworks is required. Chapter 4 of this thesis seeks to examine the statistical

significance of hedge fund non-linearity in a formal hypothesis testing framework.

97

The first objective of this study is to examine the univariate linear behaviour of hedge

fund returns. To the best of the author’s knowledge, this study is the first to examine the

linear behaviour of hedge fund returns in a univariate setting. Similar to Chapter 3, we

investigate the linearity-in-the-mean of hedge fund returns by employing a hypothesis

testing approach.

The findings from this study reveal that hedge fund returns are linear-in-the-mean in a

univariate framework. This is an original research contribution as our findings cannot be

directly compared with any other works. The hypothesis testing approach employed in

this study reveals that the stylised features of heteroscedasticity and autocorrelation in

the error disturbances cause the underlying linearity tests to over-reject the null

hypothesis of linearity-in-the-mean. When these linearity tests are augmented to correct

for heteroscedasticity and autocorrelation, we reveal that hedge fund returns are indeed

linear-in-the-mean in a univariate setting.

The second goal of this study is to examine the linearity-in-the-mean between hedge

funds and traditional asset classes in a bivariate setting. Previous studies have examined

the linear relationship between hedge fund returns with traditional asset classes by

considering various estimation techniques. By contrast, this study employs a hypothesis

testing approach to examine if hedge fund returns are linear-in-the-mean with traditional

asset classes. To the best of the author’s knowledge, this research is an original

contribution as it is the first to examine linearity-in-the-mean between hedge fund and

traditional asset class returns in a bivariate setting.

The findings from the study reveal that hedge funds and traditional asset class returns

are linear-in-the-mean in a bivariate setting. These results are an alternative view to the

current body of knowledge. We demonstrate that conventional bivariate hypothesis tests

detect spurious non-linearity caused by the effects of heteroscedasticity and

autocorrelation in the error disturbances of the underlying hypothesis tests. When these

effects are controlled in augmented linearity-in-the-mean tests, we show that hedge

funds and traditional asset classes are linear-in-the-mean in a bivariate setting.

98

This study provides a number of contributions to the literature. First, the findings

highlight the effects of heteroscedasticity and autocorrelation on linearity-in-the-mean

tests when examining hedge fund returns. This study provides a new dimension to the

hedge fund literature by demonstrating that if non-linearity is present, if at all, then it is

located and isolated in the error disturbances of a linear model and not in the conditional

mean. The findings from this study suggest that hedge fund returns are linear-in-the-

mean and that the necessity of non-linear modelling of hedge fund returns may be

overstated. This study demonstrates that the claim of hedge fund non-linearity may need

to be re-examined in light of the empirical effects of heteroscedasticity and

autocorrelation.

The rest of the study is organised as follows. In Section 4.2 we provide a brief review of

the related literature. Section 4.3 documents the methods employed to examine the

assumption of linearity-in-the-mean. Section 4.4 describes the data in this study. Section

4.5 examines the results while Section 6 offers concluding remarks.


The seminal work of Markowitz (1952) and the development of modern portfolio theory

(MPT) is one of the oldest and important areas in finance. The widespread use of MPT

can be seen in the voluminous academic literature and its use in the investment

profession. A large portion of the MPT literature has examined and developed its

theoretical constructs and the empirical problems associated with implementing MPT in

practice. Despite these various strands of literature, a paucity of research attention has

considered the assumption of linearity between asset returns from a mean-variance

investor perspective.

A handful of empirical studies have examined the linear behaviour between hedge fund

returns and traditional asset classes. The literature shows that a variety of techniques

have been employed to consider this research question. The first studies from Fung and

Hsieh (1997a, 1997b, 2001) have shown that specific hedge funds known as commodity

99

trading advisors (CTAs) have a U-shaped or straddle-like payoff with global stocks.

Whilst the graphical illustrations in these studies may be considered as simplistic, they

provide a compelling argument to suggest that managed futures fund managers in the

global hedge fund industry exhibit non-linear return behaviour.

In a more rigourous framework, Lo (2001) regresses hedge fund index returns against

the S&P500 with up and down months as separate regressors. Lo (2001) finds that the

regression coefficients are statistically significant thereby demonstrating that hedge fund

returns are non-linear.

The third hedge fund linearity study comes from Mitchell and Pulvino (2001) who

analyse the non-linear payoff of the merger arbitrage hedge fund investment strategy.

Mitchell and Pulvino (2001) employ a piecewise linear regression to analyse the linear

behaviour between the C.R.S.P. United States value-weighted stock index and hedge

funds. Mitchell and Pulvino (2001) report that both slope coefficients in the piecewise

regression are statistically significant thereby concluding that hedge fund merger

arbitrage returns are non-linear.

In another framework, Favre and Galeano (2002) analyse hedge fund index returns by

employing loess regressions against Swiss pension benchmark investment portfolios.

Favre and Galeano (2002) find that the curvature of the loess regression support the

claim that hedge fund returns are non-linearly related to traditional asset classes.

In a different setting, Agarwal and Naik (2004) employ the returns of a stockmarket out-

of-the-money (OTM) put option writing strategy as a risk factor in a multi-factor asset

pricing model. Agarwak and Naik (2004) find that the OTM put option writing risk

factor is statistically significant when modeling hedge fund index returns. Agarwal and

Naik (2004) conclude that hedge fund returns exhibit option-like returns and are

therefore non-linear.

100

In the same spirit as Agarwal and Naik (2004), the work of Huber and Kaiser (2004)

model the statistical significance of various option trading strategies in a Sharpe (1992)

investment style analysis framework. Again, Huber and Kaiser (2004) report the returns

of option based trading strategies as statistically significant therefore concluding that

hedge fund returns are non-linear.

The review of the hedge fund linearity literature raises a number of issues that need to be

addressed. It is clear from the academic literature that the concept of hedge fund non-

linearity is loosely defined. The previous studies show that there is no generally

accepted definition of non-linearity in the literature. This study responds to this problem

by employing a strict econometric approach borrowed from Granger and Teräsvirta

(1993), Campbell, Lo and MacKinlay (1997) and Tsay (2002). The framework selected

in this study allows the source of non-linearity (if any) to be specifically identified and

isolated.

Second, previous hedge fund studies have failed to control the joint effects of both

heteroscedasticity and autocorrelation when estimating non-linearity between hedge

funds and other asset returns. The failure to control for these effects in their respective

frameworks may result in the tendency of statistically significant regression slope

parameters to be reported due to heteroscedasticity and autocorrelation in the error

disturbances. This study aims to address this issue by controlling the effects of

heteroscedasticity and autocorrelation in a linearity-in-the-mean hypothesis testing

framework.

Third, previous studies have examined hedge fund non-linearity by examining its

behaviour against traditional asset classes in a bivariate setting. To the best of the

author’s knowledge, the linear behaviour of hedge fund returns in a univariate setting

has never been considered in the literature. This study provides a scholarly contribution

by examining hedge fund linearity in a univariate setting for the first time. The study of

the non-linear functional form of past hedge fund returns provides important information

101

content to the current body of knowledge. We proceed to outline the methodology

employed in this study.

4.3 Method

To examine the linearity-in-the-mean of hedge fund returns, we employ the same

methodology as outlined in Chapter 3 of this thesis. To identify all forms of linearity-in-

the-mean, we perform tests in both univariate and bivariate settings. In the univariate

framework, we employ the Keenan (1985), Tsay (1986) and Teräsvirta et. al., (1993)

V23 tests. In the bivariate setting, we employ the same tests and we also consider the

Equality for Two Regression Coefficients (ETRC) test for comparative purposes only.

As in the previous chapter, modern portfolio theory (MPT) serves as the framework to

consider linearity-in-the-mean in the bivariate setting. In an asset allocation framework,

an investor has to consider the investment opportunity set available to determine optimal

portfolio choices at time t only. We therefore specify linearity-in-the-mean tests in the

bivariate setting with no lagged independent variables.

Heteroscedasticity and autocorrelation are time series characteristics which are stylised

features of financial market returns. These effects have been shown to affect the

statistical inference of the linearity-in-the-mean tests considered in this study. To

control for heteroscedasticity and autocorrelation, the above linearity-in-the-mean tests

are augmented with the White (1980) and Newey and West (1987) procedures. Similar

to Chapter 3, this study reports three p-values for each test which represents the

conventional test, the White (1980) heteroscedasticity adjusted test and the Newey and

West (1987) HAC test. To review the mathematical specifications of the hypothesis

tests in this chapter, refer to Section 3.3 of this thesis.

102

4.4 Data

In this study we analyse a dataset comprising of hedge fund indices and traditional asset

classes. The hedge fund index returns are sourced from TASS-Tremont, a hedge fund

database provider owned by Lipper-Reuters. The selection of these hedge fund indices

is based on the fact they commence in January 1994 and provide some of the longest

index returns available in the global hedge fund industry. The fourteen hedge fund

indices employed in this study are: (i) TASS Index, (ii) Multistrategy, (iii) Long/Short

Equity, (iv) Global Macro, (v) Equity Market Neutral, (vi) Dedicated Short Bias, (vii)

Managed Futures, (viii) Risk Arbitrage, (ix) Event Driven, (x) Distressed Securities, (xi)

Fixed Income Arbitrage, (xii) Event Driven Multi-Strategy, (xiii) Convertible Arbitrage

and (xiv) Emerging Markets.

Of the hedge fund indices listed above, the TASS Index represents the most important

data series in this study. The TASS Index is a value-weighted index of all hedge funds

in the TASS-Tremont database with a minimum of US$50 million of funds under

management, one year track record and current audited financial statements.47

Informally, the TASS Index represents the systematic returns of the entire global hedge

fund industry on a value-weighted basis.

To examine linearity-in-the-mean in the bivariate setting, we consider the linear

behaviour between hedge fund returns and traditional asset classes. In these bivariate

tests, we employ the same traditional asset class returns which are presented and

analysed in Section 3.4 of this thesis.

47 To be included in the TASS indices, the minimum funds under management has recently increased from US$10 million to US$50 million under management.

103

Table 4.1 Summary Statistics This table presents the summary statistics of the monthly excess returns of the hedge fund indices employed in this study. Panel A provides the descriptive statistics of the monthly excess returns of the respective indices. Panel B reports the autocorrelation of returns. Panel C presents the autocorrelation of squared returns. Panel D reports the normalised z-scores of the 1st, 2.5th, 5th, 95th, 97.5th and 99th percentiles. The 1%, 2.5%, 5%, 95%, 97.5% and 99% percentiles for a normal distribution are -2.3263, -1.9600, -1.6449, 1.6449, 1.9600 and 2.3263, respectively. * and ** denote statistical significance at the 5% and 1% levels, respectively. Variable

TASS Index

Multi- Strategy

Long/Short Equity

Global Macro

Dedicated Short Bias

Managed Futures

Panel A: Descriptive Statistics Mean 0.534 0.432 0.624 0.745 -0.484 0.201 Std. Dev. 2.250 1.254 2.936 3.180 4.917 3.475 Skewness -0.129 -1.391 -0.099 -0.286 0.589 -0.097 Kurtosis 5.283 6.811 6.948 5.955 4.100 3.400 Median 0.562 0.602 0.641 0.805 -0.671 -0.045 Maximum 7.824 3.222 11.868 9.622 20.067 9.087 Minimum -8.249 -5.334 -12.543 -12.735 -9.553 -10.291 J-B Stat. 29.686 125.547 89.270 51.434 14.688 0.886 J-B p-value 0.000** 0.000** 0.000** 0.000** 0.000** 0.642 Sharpe Ratio 0.237 0.345 0.213 0.234 0.000 0.058

Panel B: Autocorrelation (First Moment) AC1 0.114 0.005 0.156 0.058 0.103 0.051 AC2 0.023 0.049 0.033 0.030 -0.044 -0.104 AC3 -0.022 0.144 -0.058 0.077 -0.029 -0.021 AC6 -0.026 0.084 0.142 -0.101 0.010 -0.111 AC12 0.016 0.008 -0.046 0.031 -0.189* -0.062 Panel C: Autocorrelation (Second Moment) AC1 0.065 0.201* 0.128 0.090 -0.025 0.039 AC2 0.268** 0.121 0.341** 0.069 0.124 -0.037 AC3 0.007 0.137 0.100 0.075 0.259** 0.051 AC6 0.169** 0.105 0.163 0.189* 0.070 -0.002 AC12 0.066 0.102 0.058 0.087 -0.053 -0.024

Panel D: Standardised Tail Z-Scores 1st Percentile -2.632 -3.487 -3.101 -2.769 -1.826 -2.735 2.5th Percentile -2.283 -2.832 -1.719 -2.139 -1.553 -1.996 5th Percentile -1.610 -1.931 -1.464 -1.748 -1.482 -1.733 95th Percentile 1.814 1.305 1.537 1.649 1.620 1.661 97.5th Percentile 2.353 1.476 1.961 2.659 1.924 1.935 99th Percentile 2.641 1.800 3.281 2.744 2.700 2.437

104

Table 4.2 Summary Statistics This table presents the summary statistics of the monthly excess returns of the hedge fund indices employed in this study. Panel A provides the descriptive statistics of the monthly excess returns of the respective indices. Panel B reports the autocorrelation of returns. Panel C presents the autocorrelation of squared returns. Panel D reports the normalised z-scores of the 1st, 2.5th, 5th, 95th, 97.5th and 99th percentiles. The 1%, 2.5%, 5%, 95%, 97.5% and 99% percentiles for a normal distribution are -2.3263, -1.9600, -1.6449, 1.6449, 1.9600 and 2.3263, respectively. * and ** denote statistical significance at the 5% and 1% levels, respectively. Variable

Market Neutral

Risk Arbitrage

Event Driven

Distressed Securities

Fixed Inc. Arbitrage

Event Driven Multi-Strat

Convertible Arbitrage

Emerging Markets

Panel A: Descriptive Statistics Mean 0.475 0.307 0.590 0.738 0.195 0.508 0.376 0.357 Std. Dev. 0.810 1.196 1.680 1.900 1.100 1.793 1.349 4.835 Skewness 0.160 -1.523 -3.820 -3.269 -3.328 -2.921 -1.503 -1.152 Kurtosis 3.524 10.838 30.671 24.834 21.389 22.251 6.735 9.264 Median 0.427 0.371 0.776 0.943 0.385 0.586 0.646 1.125 Maximum 2.839 3.310 3.384 3.693 1.553 4.192 3.036 14.920 Minimum -1.543 -6.752 -12.927 -13.702 -7.679 -12.644 -5.248 -26.570 J-B Stat. 1.959 409.151 4797.284 3022.270 2224.404 2353.563 132.684 257.071 J-B p-value 0.375 0.000** 0.000** 0.000** 0.000** 0.000** 0.000** 0.000** Sharpe Ratio 0.586 0.257 0.351 0.388 0.177 0.283 0.279 0.074

Panel B: Autocorrelation (First Moment) AC1 0.250** 0.240** 0.332** 0.287** 0.389** 0.329** 0.552** 0.301** AC2 0.147 -0.057 0.134 0.127 0.060 0.142 0.368** 0.030 AC3 0.028 -0.139 0.023 0.015 0.004 0.050 0.112 0.010 AC6 -0.027 0.074 -0.026 -0.047 -0.078 -0.025 -0.004 -0.117 AC12 0.049 -0.069 0.017 -0.024 0.086 0.013 0.031 -0.011 Panel C: Autocorrelation (Second Moment) AC1 0.129 0.020 0.041 -0.013 0.287** 0.129 0.382** 0.064 AC2 0.296** 0.040 -0.021 -0.037 0.056 0.004 0.301** -0.020 AC3 -0.009 -0.009 0.016 0.019 0.020 0.008 -0.040 0.105 AC6 0.226** 0.194* -0.003 -0.005 -0.002 -0.004 -0.050 -0.016 AC12 -0.003 -0.050 -0.032 -0.033 0.030 -0.040 -0.064 -0.001

Panel D: Standardised Tail Z-Scores 1st Percentile -2.271 -2.953 -2.737 -3.045 -4.219 -3.476 -4.101 -2.533 2.5th Percentile -2.200 -2.006 -2.144 -1.827 -2.222 -1.830 -2.789 -1.927 5th Percentile -1.669 -1.492 -1.274 -1.342 -1.631 -1.294 -1.730 -1.780 95th Percentile 1.615 1.361 1.121 1.253 0.955 1.400 1.108 1.345 97.5th Percentile 2.413 1.750 1.384 1.394 1.096 1.653 1.791 1.698 99th Percentile 2.710 2.309 1.532 1.527 1.227 1.892 1.929 2.816

105

For this study, we employ the continuous compounded excess returns of these hedge

fund indices consisting of 144 monthly observations for the twelve year period from

January 1994 to December 2005. As in the previous chapter, the Ibbotson and

Associates US Treasury 1 month Treasury Bill rate is employed as the risk-free rate.

The sample period in this study includes significant financial market events including

the Long Term Capital Management (LTCM) hedge fund bail-out in 1998. We employ

excess returns in this study as we are interested in the linear behaviour of returns in a

finance framework as it relates to a mean-variance investor. Figures 4.1 to 4.14 located

in Annexure 4.A at the end of this chapter illustrate the monthly excess returns of each

index return series. A common feature of these graphs is the substantial negative returns

during the extraordinary financial market events between August to October 1998.48

The TASS Managed Futures Index is the only return series that recorded positive excess

returns during all three months. Another feature of these graphs is the reduction in the

volatility of returns in recent years.

Tables 4.1 and 4.2 provide the statistical description of the excess returns of the hedge

fund indices employed in this study. The striking features of the data includes the

rejection of the null hypothesis of normality and the negative skewness for nearly all

indices. These summary statistics reflect the salient characteristics of hedge fund returns.

All hedge fund indices report some form of statistically significant serial correlation of

the first or second moment with the exception of the TASS Managed Futures Index. The

statistical features of heteroscedasticity and autocorrelation reflect the stylised

characteristics of hedge fund index returns. The dominant feature of Table 4.2 is that all

index returns have a positive serial correlation of order one with statistical significance

at the 1 percent level.

48 This turbulent period in global financial markets was dominated by the Russian bond default on 17th August 1998, the Federal Reserve bailout of Long Term Capital Management (LTCM) on 23rd September 1998 and the inter-FOMC interest rate cut on 15th October 1998.

106

Table 4.3 Univariate Linearity-in-the-Mean Tests – Hedge Funds This table reports the p-values of the Keenan (1985), Tsay (1986) and the Teräsvirta et. al., (1993) V23 tests in a univariate setting with lag orders of one, two and three, respectively. Three p-values are reported in the table for each test. The first p-value is estimated from the conventional test. The second p-value is the test adjusted as a Wald test employing a White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the test adjusted as a Wald test employing a Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively. Keenan Keenan Keenan Tsay Tsay Tsay V23 V23 V23 Variable AR(1) AR(2) AR(3) AR(1) AR(2) AR(3) AR(1) AR(2) AR(3) TASS Index

0.172 0.134 0.608

0.200 0.233 0.790

0.458 0.107 0.466

0.002** 0.012* 0.056

0.073 0.056 0.207

0.073 0.045* 0.609

0.019* 0.036* 0.157

0.028* 0.035* 0.594

0.059 0.071 0.837

Multistrategy

0.940 0.998 0.999

0.825 0.986 0.947

0.938 0.977 0.993

0.454 0.563 0.564

0.297 0.391 0.342

0.131 0.181 0.517

0.101 0.327 0.533

0.072 0.000** 0.534

0.123 0.592 0.657

Long/Short Equity Hedge

1.000 1.000 1.000

1.000 1.000 1.000

0.989 0.987 0.988

0.995 0.967 0.968

0.248 0.752 0.671

0.105 0.046* 0.753

0.085 0.092 0.487

0.076 0.102 0.896

0.012* 0.511 0.913

Global Macro

0.087 0.423 0.522

0.115 0.786 0.767

0.999 0.998 0.996

0.000** 0.020* 0.069

0.018* 0.009** 0.106

0.146 0.070 0.389

0.002** 0.022* 0.106

0.060 0.004** 0.530

0.062 0.000** 0.879

Market Neutral

0.488 0.948 0.940

0.136 0.643 0.901

0.201 0.275 0.470

0.033* 0.397 0.377

0.005** 0.010** 0.458

0.014* 0.025* 0.749

0.101 0.677 0.662

0.069 0.000** 0.853

0.097 0.533 0.964


0.636 0.517 0.987

0.574 0.758 0.994

0.515 0.192 0.874

0.079 0.054 0.355

0.103 0.102 0.672

0.086 0.001** 0.485

0.257 0.001** 0.634

0.340 0.000** 0.854

0.061 0.003** 0.448

Managed Futures

0.829 0.933 0.872

0.993 1.000 0.999

0.995 0.970 0.949

0.225 0.271 0.251

0.271 0.167 0.179

0.446 0.498 0.489

0.238 0.515 0.490

0.281 0.056 0.505

0.242 0.548 0.067

Risk Arbitrage

0.993 0.996 0.999

0.968 0.994 0.998

0.975 0.890 0.924

0.773 0.342 0.421

0.141 0.015 0.292

0.251 0.004** 0.502

0.776 0.333 0.663

0.358 0.000** 0.628

0.490 0.000** 0.970

Event Driven

1.000 1.000 1.000

0.999 1.000 1.000

0.999 0.991 0.997

0.964 0.519 0.667

0.423 0.001** 0.705

0.327 0.000** 0.852

0.519 0.000** 0.374

0.522 0.000** 0.940

0.381 0.000** 0.860


0.952 0.959 0.997

0.960 0.974 0.999

0.955 0.703 0.939

0.497 0.032* 0.424

0.490 0.002** 0.591

0.503 0.001** 0.899

0.361 0.000** 0.549

0.436 0.000** 0.698

0.305 0.000** 0.936

Fixed Income Arbitrage

0.123 0.864 0.994

0.378 0.992 1.000

0.277 0.730 0.940

0.001** 0.513 0.470

0.003** 0.061 0.428

0.013* 0.004** 0.746

0.000** 0.000** 0.572

0.000** 0.000** 0.739

0.001** 0.000** 0.847

Event Driven Multi Strategy

0.844 0.745 0.752

0.961 0.968 0.950

0.959 0.750 0.610

0.246 0.000** 0.220

0.261 0.000** 0.328

0.376 0.000** 0.633

0.444 0.000** 0.225

0.315 0.000** 0.704

0.212 0.005** 0.892


0.460 0.981 0.938

0.593 0.999 0.990

0.999 0.999 0.997

0.027* 0.485 0.310

0.269 0.806 0.806

0.065 0.000** 0.759

0.110 0.691 0.535

0.085 0.162 0.664

0.101 0.288 0.951

Emerging Markets

0.982 0.992 0.992

1.000 1.000 1.000

1.000 0.995 0.987

0.668 0.342 0.159

0.747 0.792 0.812

0.742 0.897 0.948

0.654 0.393 0.359

0.222 0.000** 0.530

0.320 0.000** 0.952

107

Overall, we can see that the summary statistics of hedge fund indices (and traditional

asset classes in the previous chapter) typically reflect the salient features of financial

market returns. It is important to highlight the statistically significant serial

autocorrelation of the first and second moments in the data which can impact on the

efficiency of the linearity-in-the-mean tests. We proceed to detail the findings of the

linearity-in-the-mean hypothesis tests.

4.5 Results

The results section of this study is presented in two parts. The first section reports and

analyses the linearity-in-the-mean of hedge fund excess returns in the univariate setting.

The second part of the results section examines linearity-in-the-mean in the bivariate

framework. The key finding from this study demonstrates that hedge fund returns are

linear-in-the-mean in both univariate and bivariate settings.

4.5.1 Univariate Results Table 4.3 presents the p-values of the univariate test results for Keenan (1985), Tsay

(1986) and Teräsvirta et. al., (1993) for autoregressive models of first, second and third

order. The key finding from Table 4.3 shows that we are unable to reject the null

hypothesis of linearity-in-the-mean after the hypothesis tests have been corrected for

heteroscedasticity and autocorrelation. Informally, the findings reveal that hedge fund

returns are linear-in-the-mean in a univariate setting.

A closer examination of Table 4.3 shows that all of the Keenan (1985) p-values are

statistically insignificant. In contrast, the conventional and heteroscedasticity consistent

Tsay (1986) and Teräsvirta et. al., (1993) V23 tests report numerous p-values which are

statistically significant. However, when these tests are re-estimated in a HAC

framework, Table 4.3 reports statistically insignificant p-values.

108

The conclusions to be drawn from these univariate tests show that non-linear functions

of lagged hedge fund returns provide no statistical inference to current hedge fund

returns. In short, the non-linear functions of lagged returns do not assist in explaining

current hedge fund returns. These results imply that there are no spillover effects of

univariate non-linearity into the forthcoming tests of bivariate linearity-in-the-mean.

4.5.2 Bivariate Results The bivariate test results are presented in the following order. A comprehensive set of

scatterplots are presented in Annexure 4.B to illustrate the excess returns of each hedge

fund index against the MSCI World Equity Index. The graphs provide an aid in

understanding the bivariate relationship between different hedge fund returns. Second,

the Keenan (1985), Tsay (1986), Teräsvirta et. al., (1993) V23 and the ETRC hypothesis

test results are reported in this section. In the interest of brevity, the p-values presented

in this section are those which report statistical significance at the 5 per cent level or

lower. The complete set of p-values are available for all tests in Annexures 4.C to 4.J at

the end of this chapter.

The bivariate results in this study can be directly compared to other bivariate studies of

non-linearity. The key finding from this study is that we are unable to reject the null

hypothesis of linearity-in-the-mean between hedge fund returns and traditional asset

classes. The findings reveal that the effects of heteroscedasticity and autocorrelation

distort statistical inference and cause these tests to incorrectly reject the null hypothesis

of linearity-in-the-mean. Our findings suggest that other studies which claim hedge fund

non-linearity may need to be re-considered in light of the effects of heteroscedasticity

and autocorrelation in these tests of linearity-in-the-mean.

109

Table 4.4 Keenan (1985) Bivariate Test – Stocks This table presents the p-values from the Keenan (1985) tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each Keenan (1985) test. The first p-value represents the original Keenan (1985) test. The second p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively. Dependent Variable

MSCI World Equity

S&P500 All Return

MSCI USA Equity

SMB

UMD

TASS Index

0.392 0.523 0.700

0.315 0.364 0.677

0.417 0.499 0.717

0.048* 0.082 0.556

0.149 0.020* 0.273

Multistrategy

0.994 0.978 0.965

0.999 0.993 0.985

0.999 0.987 0.975

0.131 0.000** 0.382

0.640 0.105 0.286


0.789 0.835 0.865

0.366 0.501 0.658

0.460 0.627 0.703

0.000** 0.027* 0.509

0.129 0.119 0.369

Global Macro

0.721 0.676 0.787

0.931 0.840 0.918

0.967 0.902 0.947

0.958 0.814 0.861

0.244 0.004** 0.277


0.999 0.999 0.999

0.921 0.838 0.832

0.865 0.785 0.777

0.000** 0.000** 0.349

0.599 0.554 0.600

Managed Futures

0.083 0.012* 0.195

0.147 0.042* 0.330

0.203 0.088 0.354

0.781 0.195 0.420

0.998 0.963 0.965

Risk Arbitrage

0.047* 0.361 0.537

0.009** 0.181 0.519

0.016* 0.246 0.529

0.049* 0.058 0.453

0.648 0.501 0.582

Event Driven

0.000** 0.325 0.563

0.000** 0.111 0.509

0.000** 0.178 0.517

0.164 0.060 0.540

0.672 0.132 0.347


0.000** 0.251 0.519

0.000** 0.073 0.459

0.000** 0.116 0.452

0.718 0.412 0.683

0.904 0.340 0.453


0.047* 0.247 0.483

0.046* 0.071 0.481

0.068 0.066 0.475

0.965 0.632 0.729

0.877 0.376 0.596


0.005** 0.403 0.605

0.000** 0.153 0.553

0.000** 0.245 0.574

0.018* 0.005** 0.472

0.398 0.134 0.358


0.510 0.598 0.849

0.506 0.731 0.869

0.713 0.839 0.912

0.005** 0.000** 0.262

0.038* 0.005** 0.097

Emerging Markets

0.185 0.519 0.668

0.027* 0.182 0.559

0.049* 0.286 0.585

0.023* 0.000** 0.437

0.511 0.113 0.395

110

4.5.2.1 Keenan (1985) Bivariate Results Table 4.4 reports the statistically significant Keenan (1985) bivariate tests with hedge

funds as the dependent variable and traditional asset classes as the independent variable.

The complete set of Keenan (1985) bivariate test results are presented in Annexures 4.C

and 4.D at the end of this chapter. The key finding from the Keenan (1985) bivariate

results indicates that hedge fund and traditional asset class returns are linear-in-the-

mean.

A number of observations can be drawn from the Keenan (1985) bivariate tests.

Consistent with previous results in this thesis, Annexure 4.D reports insignificant p-

values for the Keenan (1985) bivariate tests when bond returns are the independent

variable. Table 4.4 and Annexure 4.C appear to reveal conventional p-values which are

statistically significant when stock returns are employed as the independent variable.

When the Keenan (1985) test is repeated in a HAC framework, we find that all p-values

become statistically insignificant.

The overall assessment of the Keenan (1985) bivariate tests appear to indicate that hedge

fund and traditional asset class returns are linear-in-the-mean in a bivariate setting. It is

shown that conventional Keenan (1985) results tend to report spurious linearity-in-the-

mean when heteroscedasticity and autocorrelation in the error disturbances are not

controlled in the underlying tests. To confirm these results, we repeat the same

experiment with the Tsay (1986) linearity-in-the-mean bivariate test.

111

Table 4.5 Tsay (1986) Bivariate Test – Stocks This table presents the p-values from the Tsay (1986) tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Independent Variable

Dependent Variable MSCI World Equity

S&P500 All Return

MSCI USA Equity

HML

SMB

UMD

TASS Index

0.082 0.523 0.700

0.059 0.364 0.677

0.091 0.499 0.717

0.255 0.717 0.833

0.005** 0.082 0.556

0.020* 0.019* 0.273

Multistrategy

0.775 0.978 0.965

0.905 0.993 0.985

0.886 0.987 0.975

0.525 0.750 0.849

0.017* 0.000** 0.382

0.192 0.105 0.286


0.304 0.835 0.865

0.074 0.501 0.658

0.107 0.627 0.703

0.025* 0.500 0.686

0.000** 0.027* 0.509

0.017* 0.119 0.369

Global Macro

0.247 0.676 0.787

0.504 0.840 0.918

0.608 0.902 0.947

0.482 0.764 0.855

0.577 0.814 0.861

0.040* 0.004** 0.277


0.948 0.999 0.999

0.482 0.838 0.832

0.390 0.785 0.777

0.185 0.663 0.768

0.000** 0.000** 0.349

0.169 0.554 0.600

Managed Futures

0.010** 0.012* 0.195

0.020* 0.042* 0.330

0.031* 0.088 0.354

0.820 0.966 0.967

0.296 0.195 0.420

0.840 0.963 0.965

Risk Arbitrage

0.005** 0.361 0.537

0.001** 0.181 0.519

0.001** 0.246 0.529

0.292 0.504 0.580

0.005** 0.058 0.453

0.197 0.501 0.582

Event Driven

0.000** 0.325 0.563

0.000** 0.111 0.509

0.000** 0.178 0.517

0.818 0.954 0.962

0.023* 0.060 0.540

0.212 0.132 0.347


0.000** 0.251 0.519

0.000** 0.073 0.459

0.000** 0.116 0.452

0.538 0.725 0.668

0.244 0.412 0.683

0.450 0.340 0.453


0.005** 0.247 0.483

0.005** 0.071 0.481

0.007** 0.066 0.475

0.863 0.961 0.958

0.601 0.632 0.729

0.406 0.376 0.596


0.000** 0.403 0.605

0.000** 0.153 0.553

0.000** 0.245 0.574

0.316 0.578 0.770

0.002** 0.005** 0.472

0.084 0.134 0.358


0.127 0.742 0.849

0.125 0.731 0.869

0.240 0.839 0.912

0.046* 0.086 0.339

0.000** 0.000** 0.262

0.004** 0.005** 0.097

Emerging Markets

0.028* 0.519 0.668

0.002** 0.182 0.559

0.005** 0.286 0.585

0.272 0.531 0.768

0.002** 0.000** 0.437

0.127 0.113 0.395

112

4.5.2.2 Tsay (1986) Bivariate Results Tables 4.5 and 4.6 present a summary of the statistically significant Tsay (1986)

bivariate test results. The complete set of p-values are available in Annexures 4.E and

4.F at the end of this chapter. Again, the key finding from the Tsay (1986) bivariate tests

appears to show that hedge funds and traditional asset classes are linear-in-the-mean in a

bivariate setting.

Table 4.5 shows that the conventional Tsay (1986) tests report numerous over-rejections

of the null hypothesis of linearity-in-the-mean. Similar to the Keenan (1985) results, the

Tsay (1986) tests report spurious non-linearity due to the effects of heteroscedasticity in

the error disturbances in the underlying tests. When the tests are repeated with the

heteroscedasticity-consistent procedure, we can see many of the p-values become

Table 4.6 Tsay (1986) Bivariate Tests – Bonds This table presents the p-values of the Tsay (1986) tests with bond indices as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Dependent Variable MSCI World plus Em Sovrgn

J.P. Morgan Global Bond

Lehman Global Aggregate

MS USA Govt Bond

Lehman USA Aggregate

TASS Index

0.880 0.992 0.992

0.007** 0.053 0.234

0.024* 0.055 0.252

0.077 0.243 0.193

0.290 0.601 0.524

Global Macro

0.342 0.752 0.781

0.018* 0.054 0.274

0.026* 0.056 0.263

0.221 0.493 0.433

0.495 0.814 0.790

Risk Arbitrage

0.822 0.963 0.959

0.047* 0.288 0.306

0.299 0.489 0.413

0.060 0.236 0.240

0.215 0.444 0.402

Event Driven

0.466 0.728 0.649

0.016* 0.299 0.403

0.184 0.326 0.445

0.091 0.373 0.393

0.332 0.639 0.610


0.668 0.896 0.850

0.046* 0.399 0.463

0.338 0.518 0.606

0.136 0.443 0.420

0.405 0.705 0.662


0.324 0.628 0.580

0.012* 0.234 0.363

0.123 0.229 0.337

0.094 0.326 0.378

0.329 0.593 0.573

113

insignificant. This feature can be readily seen in Table 4.5 with world and US stocks as

the independent variable. The Tsay (1986) tests with the SMB and UMD as the

independent variable report statistically significant p-values for both conventional and

heteroscedasticity-consistent tests. This result is expected due to the statistically

significant serial correlation in the returns of the SMB and UMD risk factors. The p-

value becomes insignificant when the Tsay (1986) test is re-estimated in a HAC

framework.

The Tsay (1986) bivariate tests in Table 4.6 reveal statistically significant p-values when

global bond returns are the independent variable. The complete set of tests in Annexure

4.F show that insignificant p-values are reported for all tests when US domestic bond

index returns are employed as the independent variable. Again, when the tests are

repeated as a heteroscedasticity consistent procedure, we see that all p-values become

insignificant.

Overall, the HAC corrected Tsay (1986) test results are consistent with the previous

findings from the Keenan (1985) tests. The increased power of the Tsay (1986) HAC

test confirms the Keenan (1985) results which suggest that hedge funds and traditional

asset classes are linear-in-the-mean in a bivariate setting. To provide further support for

these findings, we proceed to consider the most powerful test of linearity-in-the-mean,

the Teräsvirta et. al., (1993) V23 bivariate test.

114

Table 4.7 Teräsvirta, Lin and Granger (1993) V23 Bivariate Test – Stocks This table presents the p-values from the Teräsvirta et.al., (1993) tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each Teräsvirta et. al., (1993) test. The first p-value represents the original Teräsvirta et. al., (1993) test. The second p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Dependent Variable

MSCI World Equity

S&P500 All Return

MSCI USA Equity HML SMB UMD

TASS Index

0.159 0.639 0.731

0.138 0.557 0.720

0.212 0.650 0.765

0.457 0.701 0.842

0.002** 0.064 0.649

0.062 0.000** 0.224

Multistrategy

0.279 0.241 0.665

0.670 0.771 0.891

0.844 0.732 0.860

0.061 0.009** 0.321

0.056 0.000** 0.588

0.398 0.004** 0.280


0.584 0.893 0.905

0.204 0.659 0.692

0.269 0.741 0.736

0.072 0.613 0.740

0.000** 0.022* 0.640

0.019* 0.000** 0.190

Global Macro

0.227 0.739 0.812

0.547 0.877 0.914

0.709 0.926 0.948

0.512 0.718 0.830

0.685 0.627 0.699

0.092 0.000** 0.348


0.675 0.851 0.878

0.533 0.796 0.899

0.539 0.829 0.869

0.051 0.594 0.581

0.000** 0.000** 0.613

0.013* 0.045* 0.476

Managed Futures

0.007** 0.002** 0.408

0.011* 0.000** 0.449

0.011* 0.001** 0.457

0.809 0.885 0.886

0.496 0.000** 0.507

0.433 0.008** 0.448

Risk Arbitrage

0.000** 0.321 0.717

0.000** 0.261 0.636

0.000** 0.379 0.620

0.143 0.429 0.552

0.000** 0.000** 0.343

0.057 0.236 0.506

Event Driven

0.000** 0.457 0.669

0.000** 0.195 0.454

0.000** 0.251 0.434

0.704 0.767 0.667

0.001** 0.000** 0.272

0.328 0.007** 0.205


0.000** 0.381 0.527

0.000** 0.156 0.348

0.000** 0.172 0.316

0.316 0.125 0.252

0.010** 0.000** 0.219

0.726 0.319 0.370


0.013* 0.064 0.500

0.012* 0.038* 0.565

0.024* 0.048* 0.564

0.504 0.011* 0.326

0.864 0.247 0.622

0.418 0.000** 0.618


0.000** 0.491 0.764

0.000** 0.235 0.647

0.000** 0.347 0.671

0.607 0.715 0.847

0.000** 0.000** 0.455

0.089 0.000** 0.205


0.031* 0.675 0.724

0.002** 0.335 0.452

0.002** 0.425 0.418

0.116 0.009** 0.364

0.001** 0.000** 0.542

0.003** 0.000** 0.171

Emerging Markets

0.009** 0.618 0.810

0.001** 0.273 0.665

0.002** 0.392 0.712

0.499 0.624 0.815

0.003** 0.000** 0.620

0.249 0.000** 0.264

115

4.5.2.3 Teräsvirta, Lin and Granger (1993) V23 Bivariate Tests Tables 4.7 and 4.8 report the statistically significant results from the Teräsvirta et. al.,

(1993) V23 bivariate test. The complete set of p-values for all tests are presented in

Annexures 4.G and 4.H at the end of this chapter. The key finding from the Teräsvirta

et. al., (1993) V23 tests confirm that hedge funds and traditional asset classes are indeed

linear-in-the-mean in a bivariate setting.

Tables 4.7 and 4.8 show that the conventional Teräsvirta et. al., (1993) V23 test reports

numerous p-values which are statistically significant. The Teräsvirta et. al., (1993) V23

test results appear to report a higher proportion of statistically significant p-values which

reflects the increased power and robustness of the test. As in previous tests, we can see

that the conventional tests detect spurious non-linearity in the conditional mean as many

of these p-values become statistically insignificant when the heteroscedasticity

Table 4.8 Teräsvirta, Lin and Granger (1993) V23 Bivariate Tests – Bonds This table presents the p-values of the Teräsvirta et. al., (1993) tests with bond indices as the independent variable. This table reports three p-values for each Teräsvirta et. al., (1993) test. The first p-value represents the original Teräsvirta et. al., (1993) test. The second p-value is the Teräsvirta et. al., (1993) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.




Lehman Global Agg.

MS USA Govt Bond


TASS Index

0.123 0.517 0.629

0.023* 0.099 0.395

0.035* 0.095 0.394

0.165 0.313 0.358

0.558 0.662 0.621

Managed Futures

0.203 0.512 0.460

0.037* 0.245 0.549

0.028* 0.292 0.605

0.772 0.949 0.962

0.499 0.820 0.855


0.557 0.528 0.586

0.039* 0.079 0.353

0.258 0.083 0.378

0.248 0.030* 0.107

0.591 0.188 0.095


0.701 0.857 0.918

0.447 0.718 0.853

0.804 0.716 0.838

0.691 0.128 0.597

0.582 0.043* 0.535

Emerging Markets

0.162 0.244 0.442

0.207 0.041* 0.388

0.217 0.047* 0.394

0.502 0.172 0.310

0.578 0.226 0.291

116

consistent p-value is calculated. Again, when the p-values are re-estimated with the

HAC procedure, we find that all p-values become statistically insignificant.

The conclusions to be drawn from the Teräsvirta et. al., (1993) V23 bivariate tests

support the previous findings from the Keenan (1985) and Tsay (1986) frameworks.

When these linearity-in-the-mean tests are estimated as a heteroscedasticity and

autocorrelation (HAC) procedure, it is shown that hedge funds and traditional asset class

returns are linear-in-the-mean in a bivariate setting. To provide further support for these

results, we compare these bivariate results with the ETRC test.

4.5.2.4 Equality of Two Regression Coefficients Test The ETRC test is a fundamentally different test to those reported so far in this study.

Whilst the Keenan (1985), Tsay (1986) and Teräsvirta et. al., (1993) V23 tests examine

the statistical significance of multiplicative forms of the linear functional form, the

ETRC test measures whether the up and down regressors are statistically different. The

statistically significant results from the ETRC tests are reported in Tables 4.9 and 4.10

with the complete set of p-values for all tests presented in Annexures 4.I and 4.J at the

end of this chapter.

The striking feature of the HAC consistent ETRC tests in Table 4.9 is the p-values

which reject the null hypothesis of linearity which are statistically significant at the 10

per cent level, but not at the 5 per cent level. Consistent with the previous linearity-in-

the-mean tests in this chapter, there are only a small number of statistically significant p-

values when bond returns are the independent variable. Overall, Tables 4.9 and 4.10 are

consistent with the previous findings which appear to indicate that all bivariate

relationships between hedge funds and traditional asset classes are linear-in-the-mean.

117

Table 4.9 Equality of Two Regression Coefficients Test – Stocks This table presents the p-values of the ETRC tests with stock indices as the independent variable. This table reports three p-values for each ETRC test. The first p-value represents the original ETRC test. The second p-value is the ETRC test with White (1980) heteroscedasticity-consistent standard errors. The third p-value is the ETRC test with Newey-West (1987) heteroscedasticity and autocorrelation consistent standard errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Dependent Variable

MSCI World Equity

S&P500 All Return

MSCI USA Equity SMB UMD

TASS Index

0.160 0.273 0.267

0.155 0.245 0.292

0.197 0.295 0.335

0.153 0.283 0.248

0.036* 0.038* 0.075


0.441 0.561 0.553

0.360 0.476 0.459

0.435 0.555 0.530

0.026* 0.196 0.163

0.019* 0.049* 0.115

Global Macro

0.340 0.430 0.439

0.483 0.540 0.597

0.561 0.605 0.656

0.760 0.746 0.737

0.076 0.047* 0.054

Market Neutral

0.363 0.372 0.366

0.260 0.257 0.248

0.292 0.287 0.274

0.353 0.340 0.331

0.045* 0.012* 0.063


0.817 0.833 0.826

0.936 0.940 0.937

0.821 0.834 0.825

0.005** 0.014* 0.063

0.071 0.095 0.152

Managed Futures

0.030* 0.040* 0.056

0.072 0.092 0.098

0.105 0.131 0.135

0.359 0.285 0.277

0.870 0.858 0.856

Event Driven

0.014* 0.202 0.171

0.001** 0.073 0.086

0.001** 0.083 0.093

0.554 0.601 0.627

0.262 0.148 0.184


0.009** 0.148 0.128

0.000** 0.047* 0.056

0.000** 0.046* 0.055

0.668 0.688 0.699

0.619 0.503 0.524


0.039* 0.161 0.134

0.030* 0.086 0.137

0.037* 0.085 0.134

0.637 0.533 0.526

0.494 0.431 0.445


0.063 0.282 0.237

0.005** 0.110 0.133

0.009** 0.137 0.159

0.105 0.178 0.201

0.087 0.056 0.089


0.799 0.860 0.860

0.752 0.824 0.846

0.929 0.949 0.955

0.003** 0.001** 0.063

0.001** 0.000** 0.061

Emerging Markets

0.133 0.259 0.245

0.038* 0.109 0.130

0.047* 0.131 0.151

0.028* 0.036* 0.059

0.155 0.121 0.180

118

4.5.3 Heteroscedasticity and Rare Events in Linearity Tests It is clear from Granger and Teräsvirta (1993) and in the empirical results of this thesis

that heteroscedasticity plays an important role in the statistical inference of linearity

tests. Studies such as Poon, Rockinger and Tawn (2004) argue that unexpected rare

events are the main source of heteroscedasticity causing spikes in the volatility of

returns. This section of the study examines these unexpected rare events to evaluate their

impact on heteroscedasticity and on the statistical inference of linearity-in-the-mean

tests.

The scatterplots in Figures 4.15 to 4.25 in Annexure 4.B reveal large unexpected

negative returns that may be a primary contributor to the heteroscedasticity observed in

this study. A closer inspection of the returns shows that seven of the fourteen hedge

fund indices recorded their worst monthly return in August 1998. This was an

Table 4.10 Equality of Two Regression Coefficients Test – Bonds

This table presents the p-values of the ETRC tests with bond indices as the independent variable. This table reports three p-values for each ETRC test. The first p-value represents the original ETRC test. The second p-value is the ETRC test with White (1980) heteroscedasticity-consistent standard errors. The third p-value is the ETRC test with Newey-West (1987) heteroscedasticity and autocorrelation consistent standard errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Dependent Variable J.P. Morgan Global Bond

Lehman Global Agg.

MS USA Govt Bond

TASS Index

0.012* 0.018* 0.057

0.019* 0.018* 0.056

0.029* 0.033* 0.051

Global Macro

0.048* 0.053 0.078

0.027* 0.028* 0.057

0.106 0.111 0.138

Risk Arbitrage

0.023* 0.040* 0.065

0.080 0.074 0.077

0.018* 0.032* 0.054

Event Driven

0.013* 0.045* 0.076

0.093 0.090 0.137

0.041* 0.081 0.114


0.031* 0.071 0.104

0.170 0.154 0.206

0.048 0.084 0.105


0.012* 0.036* 0.067

0.072 0.068 0.107

0.062 0.098 0.141

119

extraordinary period in global financial markets as the unexpected Russian government

bond default occurred on 17th August 1998. This event in global financial markets

caused an increase in the volatility of many asset markets resulting in a turbulent period

for investors. This major event in financial markets is clearly illustrated in the univariate

graphs presented in Figures 4.1 to 4.14 in Annexure 4.A.

To shine light on the impact of rare events such as August 1998 and the effect of

heteroscedasticity, we construct a simple experiment by repeating the Teräsvirta et. al.,

(1993) V23 bivariate tests but with the exclusion of the returns of August 1998. The

results are presented in Table 4.11. Given that the results of Table 4.7 report high levels

of heteroscedasticity with stocks, we construct the results of Table 4.11 with stock

returns as the independent variable.

Table 4.11 reports the original Teräsvirta et. al., (1993) V23 bivariate test p-values as in

Table 4.7 and the same test repeated with the truncated dataset which excludes the

returns from August 1998. The p-values of interest are the first and second p-values of

each test which report the conventional and heteroscedasticity-adjusted p-values. The

results in Table 4.11 clearly show that there is a pronounced reduction in statistically

significant p-values in the truncated dataset. By excluding August 1998, there are fewer

heteroscedasticity-adjusted p-values which are statistically significant. Consistent with

Poon et. al., (2004), Table 4.11 reveals that rare events (such as the Russian bond default

in August 1998) are a major source of heteroscedasticity which ultimately affects

linearity-in-the-mean testing.

120

Table 4.11 Teräsvirta, Lin and Granger (1993) V23 Bivariate Test (ex August 1998) This table presents the p-values from the Teräsvirta et.al., (1993) tests with world and U.S. stock indices as the independent variable. Tests are estimated on the full sample and on the full sample with the month of August 1998 excluded. This table reports three p-values for each Teräsvirta et. al., (1993) test. The first p-value represents the original Teräsvirta et. al., (1993) test. The second p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.



MSCI World Equity

S&P500 All Return

S&P500 All Return

MSCI USA Equity

MSCI USA Equity

Full Sample ex. Aug. 1998 Full Sample ex. Aug. 1998 Full Sample ex. Aug. 1998 TASS Index

0.159 0.639 0.731

0.004** 0.057 0.491

0.138 0.557 0.720

0.002** 0.048* 0.457

0.212 0.650 0.765

0.008** 0.048* 0.497

Multistrategy

0.279 0.241 0.665

0.215 0.234 0.563

0.670 0.771 0.891

0.618 0.709 0.845

0.844 0.732 0.860

0.534 0.443 0.588


0.584 0.893 0.905

0.012* 0.037* 0.476

0.204 0.659 0.692

0.002** 0.088 0.446

0.269 0.741 0.736

0.003** 0.061 0.456

Global Macro

0.227 0.739 0.812

0.021* 0.308 0.572

0.547 0.877 0.914

0.021* 0.067 0.430

0.709 0.926 0.948

0.079 0.094 0.490

Market Neutral

0.602 0.792 0.803

0.321 0.465 0.546

0.109 0.415 0.506

0.128 0.259 0.340

0.142 0.483 0.559

0.172 0.300 0.363


0.675 0.851 0.878

0.099 0.158 0.432

0.533 0.796 0.899

0.322 0.478 0.660

0.539 0.829 0.869

0.316 0.463 0.678

Managed Futures

0.007** 0.002** 0.408

0.089 0.198 0.534

0.011* 0.000** 0.449

0.045* 0.190 0.469

0.011* 0.001** 0.457

0.070 0.239 0.526

Risk Arbitrage

0.000** 0.321 0.717

0.705 0.879 0.901

0.000** 0.261 0.636

0.762 0.918 0.922

0.000** 0.379 0.620

0.588 0.847 0.863

Event Driven

0.000** 0.457 0.669

0.358 0.776 0.802

0.000** 0.195 0.454

0.023* 0.428 0.571

0.000** 0.251 0.434

0.020* 0.412 0.549


0.000** 0.381 0.527

0.173 0.601 0.644

0.000** 0.156 0.348

0.006** 0.289 0.441

0.000** 0.172 0.316

0.004** 0.249 0.414


0.013* 0.064 0.500

0.029* 0.442 0.499

0.012* 0.038* 0.565

0.026* 0.452 0.585

0.024* 0.048* 0.564

0.067 0.464 0.617


0.000** 0.491 0.764

0.733 0.855 0.819

0.000** 0.235 0.647

0.304 0.707 0.778

0.000** 0.347 0.671

0.332 0.718 0.758


0.031* 0.675 0.724

0.479 0.539 0.540

0.002** 0.335 0.452

0.449 0.639 0.662

0.002** 0.425 0.418

0.310 0.489 0.558

Emerging Markets

0.009** 0.618 0.810

0.470 0.359 0.737

0.001** 0.273 0.665

0.446 0.448 0.690

0.002** 0.392 0.712

0.566 0.581 0.769

121

4.6 Conclusions

This study examines the linear behaviour of hedge fund returns and evaluates whether

non-linearity exists in the linear functional form in both univariate and bivariate settings.

Given the striking statistical and time series characteristics of hedge fund returns, this

study considers augmented linearity-in-the-mean tests which account for both

heteroscedasticity and autocorrelation. The findings demonstrate that hedge funds

returns are indeed linear-in-the-mean in all of the testing frameworks considered. These

findings are in direct opposition to previous hedge fund linearity studies. This study

provides new empirical evidence to suggest that the non-linearity detected in previous

studies may be the result of spurious non-linearity caused by the effects of

heteroscedasticity and/or autocorrelation rather than a genuine non-linear relationship

between hedge fund and traditional asset class returns.

This study then focuses on heteroscedasticity and the important role it plays in the

statistical inference of linearity-in-the-mean tests. This study examines the source of

heteroscedasticity in stocks returns and we find that the Russian bond default of August

1998 is a large contributor to non-constant variance in stock returns. We re-estimate the

Teräsvirta et. al., (1993) V23 bivariate test on data which excludes the Russian bond

default and we find that August 1998 to be a major source of heteroscedastic error in

linearity-in-the-mean tests.

The findings from this study provide several avenues for future research. Whilst

traditional asset classes and hedge funds are found to be linear-in-the-mean, it may be

worthwhile to consider the same research question in a finer sampling frequency such as

weekly or daily returns. Second, the effects of heteroscedasticity and autocorrelation in

an asset allocation framework may be a worthwhile research theme. Finally, a better

understanding of the effects of rare events (such as August 1998) in asset allocation may

also be a promising area for future research. This thesis considers some of these research

questions in the forthcoming chapter.

122

Annexure 4.A Figure 4.1 TASS Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.2 TASS Multistrategy Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94A

pr-9

4Ju

l-94

Oct

-94

Jan-

95A

pr-9

5Ju

l-95

Oct

-95

Jan-

96A

pr-9

6Ju

l-96

Oct

-96

Jan-

97A

pr-9

7Ju

l-97

Oct

-97

Jan-

98A

pr-9

8Ju

l-98

Oct

-98

Jan-

99A

pr-9

9Ju

l-99

Oct

-99

Jan-

00A

pr-0

0Ju

l-00

Oct

-00

Jan-

01A

pr-0

1Ju

l-01

Oct

-01

Jan-

02A

pr-0

2Ju

l-02

Oct

-02

Jan-

03A

pr-0

3Ju

l-03

Oct

-03

Jan-

04A

pr-0

4Ju

l-04

Oct

-04

Jan-

05A

pr-0

5Ju

l-05

Oct

-05

Exce

ss M

onth

ly R

etur

ns

123

Figure 4.3 TASS Long/Short Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.4 TASS Global Macro Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

124

Figure 4.5 TASS Dedicated Short Bias Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.6 TASS Managed Futures Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

125

Figure 4.7 TASS Equity Market Neutral Index

-30%

-20%

-10%

0%

10%

20%

30%Ja

n-94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.8 TASS Risk Arbitrage Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

126

Figure 4.9 TASS Event Driven Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.10 TASS Distressed Securities Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

127

Figure 4.11 TASS Fixed Income Arbitrage Index

-30%

-20%

-10%

0%

10%

20%

30%Ja

n-94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.12 TASS Event Driven Multistrategy Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

128

Figure 4.13 TASS Convertible Arbitrage Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

Figure 4.14 TASS Emerging Markets Index

-30%

-20%

-10%

0%

10%

20%

30%

Jan-

94

May

-94

Sep

-94

Jan-

95

May

-95

Sep

-95

Jan-

96

May

-96

Sep

-96

Jan-

97

May

-97

Sep

-97

Jan-

98

May

-98

Sep

-98

Jan-

99

May

-99

Sep

-99

Jan-

00

May

-00

Sep

-00

Jan-

01

May

-01

Sep

-01

Jan-

02

May

-02

Sep

-02

Jan-

03

May

-03

Sep

-03

Jan-

04

May

-04

Sep

-04

Jan-

05

May

-05

Sep

-05

Exce

ss M

onth

ly R

etur

ns

129

Annexure 4.B Figure 4.15 TASS Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


TASS

Inde

x Ex

cess

Mon

thly

Ret

urns

Figure 4.16 Multistrategy Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Mul

tistr

ateg

y In

dex

Exce

ss M

onth

ly R

etur

ns

130

Figure 4.17 Long/Short Equity Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Long

/Sho

rt E

quity

Inde

x Ex

cess

Mon

thly

Ret

urns

Figure 4.18 Global Macro Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Glo

bal M

acro

Inde

x Ex

cess

Mon

thly

Ret

urns

131

Figure 4.19 Dedicated Short Bias Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Ded

icat

ed S

hort

Bia

s In

dex

Exce

ss M

onth

ly R

etur

ns

Figure 4.20 Managed Futures Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Man

aged

Fut

ures

Inde

x Ex

cess

Mon

thly

Ret

urns

132

Figure 4.21 Equity Market Neutral Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Equi

ty M

arke

t Neu

tral

Inde

x Ex

cess

Mon

thly

Ret

urns

Figure 4.22 Risk Arbitrage Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Ris

k A

rbitr

age

Inde

x Ex

cess

Mon

thly

Ret

urns

133

Figure 4.23 Event Driven Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Even

t Driv

en In

dex

Exce

ss M

onth

ly R

etur

ns

Figure 4.24 Distressed Securities Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Dis

tres

sed

Secu

ritie

s In

dex

Exce

ss M

onth

ly R

etur

ns

134

Figure 4.25 Fixed Income Arbitrage Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Fixe

d In

com

e A

rbitr

age

Inde

x Ex

cess

Mon

thly

Ret

urns

Figure 4.26 Event Driven Multistrategy Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Even

t Driv

en M

ultis

trat

egy

Inde

x Ex

cess

Mon

thly

Ret

urns

135

Figure 4.27 Convertible Arbitrage Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Con

vert

ible

Arb

itrag

e In

dex

Exce

ss M

onth

ly R

etur

ns

Figure 4.28 Emerging Markets Index vs. MSCI World Equity Index

-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%


Emer

ging

Mar

ket I

ndex

Exc

ess

Mon

thly

Ret

urns

136

Annexure 4.C Keenan (1985) Bivariate Test – Stocks This table presents the p-values from the Keenan (1985) tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each Keenan (1985) test. The first p-value represents the original Keenan (1985) test. The second p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Dependent Variable

MSCI World Equity

S&P500 All Return

MSCI USA Equity

HML

SMB

UMD

TASS Index

0.392 0.523 0.700

0.315 0.364 0.677

0.417 0.499 0.717

0.732 0.717 0.833

0.048* 0.082 0.556

0.149 0.020* 0.273

Multistrategy

0.994 0.978 0.965

0.999 0.993 0.985

0.999 0.987 0.975

0.940 0.750 0.849

0.131 0.000** 0.382

0.640 0.105 0.286


0.789 0.835 0.865

0.366 0.501 0.658

0.460 0.627 0.703

0.172 0.500 0.686

0.000** 0.027* 0.509

0.129 0.119 0.369

Global Macro

0.721 0.676 0.787

0.931 0.840 0.918

0.967 0.902 0.947

0.921 0.764 0.855

0.958 0.814 0.861

0.244 0.004** 0.277

Market Neutral

0.941 0.832 0.732

0.811 0.669 0.485

0.856 0.707 0.549

0.706 0.153 0.252

0.752 0.477 0.521

0.647 0.064 0.328


0.999 0.999 0.999

0.921 0.838 0.832

0.865 0.785 0.777

0.626 0.663 0.768

0.000** 0.000** 0.349

0.599 0.554 0.600

Managed Futures

0.083 0.012* 0.195

0.147 0.042* 0.330

0.203 0.088 0.354

0.997 0.966 0.967

0.781 0.195 0.420

0.998 0.963 0.965

Risk Arbitrage

0.047* 0.361 0.537

0.009** 0.181 0.519

0.016* 0.246 0.529

0.776 0.504 0.580

0.049* 0.058 0.453

0.648 0.501 0.582

Event Driven

0.000** 0.325 0.563

0.000** 0.111 0.509

0.000** 0.178 0.517

0.997 0.954 0.962

0.164 0.060 0.540

0.672 0.132 0.347


0.000** 0.251 0.519

0.000** 0.073 0.459

0.000** 0.116 0.452

0.945 0.725 0.668

0.718 0.412 0.683

0.904 0.340 0.453


0.047* 0.247 0.483

0.046* 0.071 0.481

0.068 0.066 0.475

0.999 0.961 0.958

0.965 0.632 0.729

0.877 0.376 0.596


0.005** 0.403 0.605

0.000** 0.153 0.553

0.000** 0.245 0.574

0.801 0.578 0.770

0.018* 0.005** 0.472

0.398 0.134 0.358


0.510 0.598 0.849

0.506 0.731 0.869

0.713 0.839 0.912

0.267 0.086 0.339

0.005** 0.000** 0.262

0.038* 0.005** 0.097

Emerging Markets

0.185 0.519 0.668

0.027* 0.182 0.559

0.049* 0.286 0.585

0.754 0.531 0.768

0.023* 0.000** 0.437

0.511 0.113 0.395

137

Annexure 4.D Keenan (1985) Bivariate Tests – Bonds This table presents the p-values of the Keenan (1985) tests with bond indices as the independent variable. This table reports three p-values for each Keenan (1985) test. The first p-value represents the original Keenan (1985) test. The second p-value is the Keenan (1985) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Keenan (1985) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.





MS USA Govt Bond


TASS Index

0.999 0.992 0.992

0.066 0.053 0.234

0.168 0.055 0.252

0.377 0.243 0.193

0.775 0.601 0.524

Multistrategy

0.998 0.940 0.935

0.558 0.240 0.435

0.510 0.163 0.443

0.992 0.920 0.915

0.974 0.837 0.841


0.828 0.559 0.613

0.505 0.377 0.392

0.806 0.479 0.570

0.643 0.365 0.347

0.916 0.705 0.691

Global Macro

0.827 0.571 0.781

0.134 0.054 0.274

0.175 0.056 0.263

0.685 0.493 0.433

0.927 0.814 0.790

Market Neutral

0.334 0.168 0.180

0.997 0.971 0.971

0.982 0.896 0.909

0.981 0.888 0.894

0.996 0.967 0.967


0.951 0.731 0.760

0.984 0.932 0.926

0.991 0.929 0.929

0.956 0.828 0.819

0.999 0.981 0.981

Managed Futures

0.405 0.362 0.322

0.992 0.984 0.984

0.998 0.992 0.993

0.992 0.964 0.965

0.908 0.817 0.815

Risk Arbitrage

0.997 0.963 0.959

0.271 0.288 0.306

0.784 0.489 0.413

0.318 0.236 0.240

0.677 0.444 0.402

Event Driven

0.913 0.728 0.649

0.122 0.299 0.403

0.625 0.326 0.445

0.418 0.373 0.393

0.817 0.639 0.610


0.980 0.896 0.850

0.267 0.399 0.463

0.823 0.518 0.606

0.529 0.443 0.420

0.876 0.705 0.662


0.998 0.983 0.982

0.607 0.638 0.691

0.876 0.800 0.825

0.981 0.919 0.888

0.989 0.914 0.878


0.810 0.628 0.580

0.097 0.233 0.363

0.501 0.229 0.337

0.425 0.326 0.378

0.814 0.593 0.573


0.889 0.750 0.725

0.729 0.604 0.691

0.951 0.774 0.797

0.982 0.895 0.882

0.999 0.974 0.963

Emerging Markets

0.762 0.509 0.521

0.434 0.376 0.453

0.676 0.353 0.386

0.741 0.477 0.457

0.965 0.813 0.778

138

Annexure 4.E Tsay (1986) Bivariate Test – Stocks

This table presents the p-values from the Tsay (1986) tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.



S&P500 All Return

MSCI USA Equity

HML

SMB

UMD

TASS Index

0.082 0.523 0.700

0.059 0.364 0.677

0.091 0.499 0.717

0.255 0.717 0.833

0.005** 0.082 0.556

0.020* 0.019* 0.273

Multistrategy

0.775 0.978 0.965

0.905 0.993 0.985

0.886 0.987 0.975

0.525 0.750 0.849

0.017* 0.000** 0.382

0.192 0.105 0.286


0.304 0.835 0.865

0.074 0.501 0.658

0.107 0.627 0.703

0.025* 0.500 0.686

0.000** 0.027* 0.509

0.017* 0.119 0.369

Global Macro

0.247 0.676 0.787

0.504 0.840 0.918

0.608 0.902 0.947

0.482 0.764 0.855

0.577 0.814 0.861

0.040* 0.004** 0.277

Market Neutral

0.527 0.832 0.732

0.325 0.669 0.485

0.377 0.707 0.549

0.235 0.153 0.252

0.271 0.477 0.521

0.197 0.064 0.328


0.948 0.999 0.999

0.482 0.838 0.832

0.390 0.785 0.777

0.185 0.663 0.768

0.000** 0.000** 0.349

0.169 0.554 0.600

Managed Futures

0.010** 0.012* 0.195

0.020* 0.042* 0.330

0.031* 0.088 0.354

0.820 0.966 0.967

0.296 0.195 0.420

0.840 0.963 0.965

Risk Arbitrage

0.005** 0.361 0.537

0.001** 0.181 0.519

0.001** 0.246 0.529

0.292 0.504 0.580

0.005** 0.058 0.453

0.197 0.501 0.582

Event Driven

0.000** 0.325 0.563

0.000** 0.111 0.509

0.000** 0.178 0.517

0.818 0.954 0.962

0.023* 0.060 0.540

0.212 0.132 0.347


0.000** 0.251 0.519

0.000** 0.073 0.459

0.000** 0.116 0.452

0.538 0.725 0.668

0.244 0.412 0.683

0.450 0.340 0.453


0.005** 0.247 0.483

0.005** 0.071 0.481

0.007** 0.066 0.475

0.863 0.961 0.958

0.601 0.632 0.729

0.406 0.376 0.596


0.000** 0.403 0.605

0.000** 0.153 0.553

0.000** 0.245 0.574

0.316 0.578 0.770

0.002** 0.005** 0.472

0.084 0.134 0.358


0.127 0.742 0.849

0.125 0.731 0.869

0.240 0.839 0.912

0.046* 0.086 0.339

0.000** 0.000** 0.262

0.004** 0.005** 0.097

Emerging Markets

0.028* 0.519 0.668

0.002** 0.182 0.559

0.005** 0.286 0.585

0.272 0.531 0.768

0.002** 0.000** 0.437

0.127 0.113 0.395

139

Annexure 4.F Tsay (1986) Bivariate Tests – Bonds

This table presents the p-values of the Tsay (1986) tests with bond indices as the independent variable. This table reports three p-values for each Tsay (1986) test. The first p-value represents the original Tsay (1986) test. The second p-value is the Tsay (1986) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Tsay (1986) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.





MS USA Govt Bond


TASS Index

0.880 0.992 0.992

0.007** 0.053 0.234

0.024* 0.055 0.252

0.077 0.243 0.193

0.290 0.601 0.524

Multistrategy

0.834 0.941 0.935

0.149 0.240 0.435

0.127 0.163 0.443

0.755 0.920 0.915

0.634 0.837 0.841


0.344 0.559 0.613

0.125 0.377 0.392

0.321 0.479 0.570

0.194 0.365 0.347

0.473 0.705 0.691

Global Macro

0.342 0.752 0.781

0.018* 0.054 0.274

0.026* 0.056 0.263

0.221 0.493 0.433

0.495 0.814 0.790

Market Neutral

0.064 0.168 0.180

0.825 0.971 0.971

0.676 0.896 0.909

0.673 0.888 0.894

0.810 0.967 0.967


0.554 0.731 0.760

0.690 0.932 0.926

0.739 0.929 0.929

0.569 0.828 0.819

0.873 0.981 0.981

Managed Futures

0.087 0.362 0.322

0.754 0.984 0.984

0.839 0.992 0.993

0.753 0.964 0.965

0.458 0.817 0.815

Risk Arbitrage

0.822 0.963 0.959

0.047* 0.288 0.306

0.299 0.489 0.413

0.060 0.236 0.240

0.215 0.444 0.402

Event Driven

0.466 0.728 0.649

0.016* 0.299 0.403

0.184 0.326 0.445

0.091 0.373 0.393

0.332 0.639 0.610


0.668 0.896 0.850

0.046* 0.399 0.463

0.338 0.518 0.606

0.136 0.443 0.420

0.405 0.705 0.662


0.831 0.983 0.982

0.174 0.638 0.691

0.405 0.800 0.825

0.669 0.919 0.888

0.727 0.914 0.878


0.324 0.628 0.580

0.012* 0.234 0.363

0.123 0.229 0.337

0.094 0.326 0.378

0.329 0.593 0.573


0.426 0.750 0.725

0.252 0.604 0.691

0.554 0.774 0.797

0.679 0.895 0.882

0.864 0.974 0.963

Emerging Markets

0.279 0.509 0.521

0.097 0.376 0.453

0.215 0.353 0.386

0.262 0.477 0.457

0.601 0.813 0.778

140

Annexure 4.G Teräsvirta, Lin and Granger (1993) V23 Bivariate Test – Stocks

This table presents the p-values from the Teräsvirta et.al., (1993) tests with stock indices and equity risk factors as the independent variable. This table reports three p-values for each Teräsvirta et. al., (1993) test. The first p-value represents the original Teräsvirta et. al., (1993) test. The second p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Dependent Variable

MSCI World Equity

S&P500 All Return


TASS Index

0.159 0.639 0.731

0.138 0.557 0.720

0.212 0.650 0.765

0.457 0.701 0.842

0.002** 0.064 0.649

0.062 0.000** 0.224

Multistrategy

0.279 0.241 0.665

0.670 0.771 0.891

0.844 0.732 0.860

0.061 0.009** 0.321

0.056 0.000** 0.588

0.398 0.004** 0.280


0.584 0.893 0.905

0.204 0.659 0.692

0.269 0.741 0.736

0.072 0.613 0.740

0.000** 0.022* 0.640

0.019* 0.000** 0.190

Global Macro

0.227 0.739 0.812

0.547 0.877 0.914

0.709 0.926 0.948

0.512 0.718 0.830

0.685 0.627 0.699

0.092 0.000** 0.348

Market Neutral

0.602 0.792 0.803

0.109 0.415 0.506

0.142 0.483 0.559

0.498 0.261 0.491

0.462 0.412 0.677

0.343 0.065 0.345


0.675 0.851 0.878

0.533 0.796 0.899

0.539 0.829 0.869

0.051 0.594 0.581

0.000** 0.000** 0.613

0.013* 0.045* 0.476

Managed Futures

0.007** 0.002** 0.408

0.011* 0.000** 0.449

0.011* 0.001** 0.457

0.809 0.885 0.886

0.496 0.000** 0.507

0.433 0.008** 0.448

Risk Arbitrage

0.000** 0.321 0.717

0.000** 0.261 0.636

0.000** 0.379 0.620

0.143 0.429 0.552

0.000** 0.000** 0.343

0.057 0.236 0.506

Event Driven

0.000** 0.457 0.669

0.000** 0.195 0.454

0.000** 0.251 0.434

0.704 0.767 0.667

0.001** 0.000** 0.272

0.328 0.007** 0.205


0.000** 0.381 0.527

0.000** 0.156 0.348

0.000** 0.172 0.316

0.316 0.125 0.252

0.010** 0.000** 0.219

0.726 0.319 0.370


0.013* 0.064 0.500

0.012* 0.038* 0.565

0.024* 0.048* 0.564

0.504 0.011* 0.326

0.864 0.247 0.622

0.418 0.000** 0.618


0.000** 0.491 0.764

0.000** 0.235 0.647

0.000** 0.347 0.671

0.607 0.715 0.847

0.000** 0.000** 0.455

0.089 0.000** 0.205


0.031* 0.675 0.724

0.002** 0.335 0.452

0.002** 0.425 0.418

0.116 0.009** 0.364

0.001** 0.000** 0.542

0.003** 0.000** 0.171

Emerging Markets

0.009** 0.618 0.810

0.001** 0.273 0.665

0.002** 0.392 0.712

0.499 0.624 0.815

0.003** 0.000** 0.620

0.249 0.000** 0.264

141

Annexure 4.H Teräsvirta, Lin and Granger (1993) V23 Bivariate Tests – Bonds

This table presents the p-values of the Teräsvirta et. al., (1993) tests with bond indices as the independent variable. This table reports three p-values for each Teräsvirta et. al., (1993) test. The first p-value represents the original Teräsvirta et. al., (1993) test. The second p-value is the Teräsvirta et. al., (1993) test adjusted as a Wald test employing an adjusted White (1980) heteroscedasticity-consistent covariance matrix. The third p-value is the Teräsvirta et. al., (1993) test re-specified as a Wald test employing an adjusted Newey-West (1987) heteroscedasticity and autocorrelation consistent covariance matrix. * and ** denote statistical significance at the 5% and 1% levels, respectively.




Lehman Global Agg.

MS USA Govt Bond


TASS Index

0.123 0.517 0.629

0.023* 0.099 0.395

0.035* 0.095 0.394

0.165 0.313 0.358

0.558 0.662 0.621

Multistrategy

0.931 0.873 0.896

0.231 0.354 0.575

0.157 0.178 0.580

0.933 0.862 0.864

0.736 0.827 0.846


0.105 0.377 0.456

0.170 0.092 0.411

0.172 0.131 0.466

0.429 0.332 0.279

0.721 0.571 0.364

Global Macro

0.128 0.530 0.696

0.060 0.198 0.461

0.080 0.185 0.415

0.205 0.393 0.552

0.587 0.734 0.777

Market Neutral

0.118 0.254 0.387

0.634 0.714 0.811

0.908 0.939 0.946

0.902 0.914 0.913

0.961 0.971 0.970


0.415 0.222 0.392

0.459 0.514 0.682

0.256 0.346 0.495

0.652 0.647 0.642

0.668 0.638 0.572

Managed Futures

0.203 0.512 0.460

0.037* 0.245 0.549

0.028* 0.292 0.605

0.772 0.949 0.962

0.499 0.820 0.855

Risk Arbitrage

0.567 0.628 0.663

0.113 0.147 0.251

0.358 0.491 0.250

0.158 0.145 0.143

0.419 0.246 0.188

Event Driven

0.509 0.346 0.424

0.056 0.066 0.393

0.267 0.129 0.466

0.241 0.103 0.095

0.589 0.335 0.083


0.482 0.379 0.481

0.131 0.078 0.429

0.332 0.193 0.527

0.327 0.224 0.113

0.666 0.493 0.126


0.862 0.959 0.963

0.267 0.501 0.753

0.290 0.559 0.784

0.699 0.743 0.854

0.517 0.292 0.681


0.557 0.528 0.586

0.039* 0.079 0.353

0.258 0.083 0.378

0.248 0.030* 0.107

0.591 0.188 0.095


0.701 0.857 0.918

0.447 0.718 0.853

0.804 0.716 0.838

0.691 0.128 0.597

0.582 0.043* 0.535

Emerging Markets

0.162 0.244 0.442

0.207 0.041* 0.388

0.217 0.047* 0.394

0.502 0.172 0.310

0.578 0.226 0.291

142

Annexure 4.I Equality of Two Regression Coefficients (ETRC) Test –

Stocks This table presents the p-values from the ETRC test with stock indices and equity risk factors as the independent variable. This table reports three p-values for each ETRC test. The first p-value represents the original ETRC test. The second p-value is the ETRC test with White (1980) heteroscedasticity-consistent standard errors. The third p-value is the ETRC test with Newey-West (1987) heteroscedasticity and autocorrelation consistent standard errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.


Dependent Variable

MSCI World Equity

S&P500 All Return


TASS Index

0.160 0.273 0.267

0.155 0.245 0.292

0.197 0.295 0.335

0.928 0.942 0.946

0.153 0.283 0.248

0.036* 0.038* 0.075

Multistrategy

0.926 0.924 0.933

0.625 0.633 0.595

0.613 0.637 0.638

0.548 0.555 0.567

0.491 0.486 0.484

0.381 0.396 0.395


0.441 0.561 0.553

0.360 0.476 0.459

0.435 0.555 0.530

0.224 0.426 0.442

0.026* 0.196 0.163

0.019* 0.049* 0.115

Global Macro

0.340 0.430 0.439

0.483 0.540 0.597

0.561 0.605 0.656

0.935 0.938 0.945

0.760 0.746 0.737

0.076 0.047* 0.054

Market Neutral

0.363 0.372 0.366

0.260 0.257 0.248

0.292 0.287 0.274

0.169 0.108 0.099

0.353 0.340 0.331

0.045* 0.012* 0.063


0.817 0.833 0.826

0.936 0.940 0.937

0.821 0.834 0.825

0.572 0.631 0.691

0.005** 0.014* 0.063

0.071 0.095 0.152

Managed Futures

0.030* 0.040* 0.056

0.072 0.092 0.098

0.105 0.131 0.135

0.947 0.943 0.944

0.359 0.285 0.277

0.870 0.858 0.856

Risk Arbitrage

0.182 0.355 0.322

0.074 0.229 0.222

0.096 0.261 0.252

0.725 0.718 0.701

0.312 0.433 0.423

0.143 0.172 0.179

Event Driven

0.014* 0.202 0.171

0.001** 0.073 0.086

0.001** 0.083 0.093

0.585 0.568 0.563

0.554 0.601 0.627

0.262 0.148 0.184


0.009** 0.148 0.128

0.000** 0.047* 0.056

0.000** 0.046* 0.055

0.229 0.192 0.192

0.668 0.688 0.699

0.619 0.503 0.524


0.039* 0.161 0.134

0.030* 0.086 0.137

0.037* 0.085 0.134

0.785 0.763 0.764

0.637 0.533 0.526

0.494 0.431 0.445


0.063 0.282 0.237

0.005** 0.110 0.133

0.009** 0.137 0.159

0.935 0.939 0.941

0.105 0.178 0.201

0.087 0.056 0.089


0.799 0.860 0.860

0.752 0.824 0.846

0.929 0.949 0.955

0.095 0.080 0.110

0.003** 0.001** 0.063

0.001** 0.000** 0.061

Emerging Markets

0.133 0.259 0.245

0.038* 0.109 0.130

0.047* 0.131 0.151

0.540 0.559 0.606

0.028* 0.036* 0.059

0.155 0.121 0.180

143

Annexure 4.J Equality of Two Regression Coefficients (ETRC) Test – Bonds

This table presents the p-values of the ETRC tests with bond indices as the independent variable. This table reports three p-values for each ETRC test. The first p-value represents the original ETRC test. The second p-value is the ETRC test with White (1980) heteroscedasticity-consistent standard errors. The third p-value is the ETRC test with Newey-West (1987) heteroscedasticity and autocorrelation consistent standard errors. * and ** denote statistical significance at the 5% and 1% levels, respectively.




Lehman Global Agg.

MS USA Govt Bond


TASS Index

0.924 0.930 0.931

0.012* 0.018* 0.057

0.019* 0.018* 0.056

0.029* 0.033* 0.051

0.119 0.138 0.164

Multistrategy

0.392 0.323 0.428

0.466 0.427 0.481

0.452 0.412 0.443

0.559 0.547 0.526

0.695 0.652 0.617


0.326 0.303 0.280

0.071 0.083 0.088

0.193 0.165 0.190

0.132 0.121 0.141

0.339 0.329 0.352

Global Macro

0.571 0.617 0.642

0.048* 0.053 0.078

0.027* 0.028* 0.057

0.106 0.111 0.138

0.243 0.262 0.289

Market Neutral

0.063 0.057 0.059

0.856 0.853 0.854

0.698 0.693 0.700

0.402 0.391 0.402

0.398 0.413 0.417


0.597 0.541 0.537

0.441 0.471 0.469

0.469 0.460 0.461

0.573 0.581 0.590

0.754 0.754 0.761

Managed Futures

0.079 0.090 0.085

0.366 0.442 0.468

0.609 0.657 0.682

0.941 0.946 0.948

0.543 0.576 0.587

Risk Arbitrage

0.722 0.697 0.702

0.023* 0.040* 0.065

0.080 0.074 0.077

0.018* 0.032* 0.054

0.083 0.108 0.132

Event Driven

0.453 0.418 0.432

0.013* 0.045* 0.076

0.093 0.090 0.137

0.041* 0.081 0.114

0.151 0.203 0.243


0.502 0.465 0.463

0.031* 0.071 0.104

0.170 0.154 0.206

0.048 0.084 0.105

0.165 0.210 0.238


0.763 0.773 0.802

0.132 0.182 0.208

0.332 0.374 0.398

0.456 0.475 0.463

0.373 0.361 0.352


0.479 0.471 0.489

0.012* 0.036* 0.067

0.072 0.068 0.107

0.062 0.098 0.141

0.185 0.221 0.264


0.482 0.483 0.534

0.197 0.214 0.264

0.492 0.468 0.499

0.549 0.540 0.548

0.599 0.576 0.577

Emerging Markets

0.289 0.269 0.274

0.077 0.090 0.122

0.132 0.103 0.132

0.192 0.197 0.211

0.480 0.470 0.482

144

5. Portfolio Selection and Hedge Funds: Serial Correlation and Tail-Risk Effects

5.1 Introduction

Previous chapters have highlighted the effects of autocorrelation, heteroscedasticity and

tail-behaviour (ie. rare events) in tests of linearity-in-the-mean. Chapters 3 and 4 have

demonstrated the sensitivity of linearity-in-the-mean tests when the empirical features of

autocorrelation, heteroscedasticity and tail-behaviour are not controlled. The

conclusions drawn from these chapters suggest that the empirical characteristics in asset

returns can lead researchers to incorrectly conclude that asset returns are not linear-in-

the-mean. In this chapter, we continue this theme by examining how the empirical

characteristics of asset returns affect portfolio selection. More specifically, this chapter

examines the sensitivity of portfolio selection to the effects of serial correlation and tail-

risk when the investment opportunity set consists of traditional assets and hedge funds.

Whilst the Markowitz (1952, 1959) MVA has become a cornerstone of financial

economics, it relies on the assumptions of quadratic utility or multivariate normal

returns. However, voluminous research suggests that the normality assumption is not

easily observed in empirical finance. The first empirical characteristic which violates the

normality assumption is serial correlation in asset returns. Studies from Fama and

French (1989), Ilmanen (1995) and Kihn (1996) provide empirical evidence of serial

correlation in bond returns. In alternative assets, Asness, Krail and Liew (2001),

Getmansky, Lo and Makarov (2004) and Lo (2002) show that serial correlation in hedge

fund returns exist due to illiquidity and smoothed return factors.

The second empirical characteristic which violates the normality assumption is the

emergence of tail-risk in asset returns. That is, the density in the tails of the distribution

of asset returns is thicker than assumed in a normal distribution. Studies by Fama

(1965b), Officer (1972), Jansen and de Vries (1991) and Poon, Rockinger and Tawn

(2004) have demonstrated that traditional asset returns exhibit heavier tails relative to

145

the normal distribution. In the hedge fund literature, Agarwal and Naik (2004) and

Brown and Spitzer (2006) have reported tail-behaviour in hedge fund returns which does

not adhere to the normality assumption.49 In short, the empirical literature suggests that

tail-behaviour is an empirical characteristic in some asset returns.

This portfolio selection study argues that the empirical characteristics of serial

correlation and tail-risk in asset returns have serious implications for investors operating

within the normality assumption of Markowitz (1952, 1959). Studies by Dimson (1979),

Roll (1981) and Blume, Keim and Patel (1991) demonstrate that serial correlation in

returns causes downward biases in the second sample moment of returns. This bias must

surely affect mean-variance investors who employ the second moment as the risk

measure in portfolio optimisation. In terms of tail-risk, Alexander and Baptista (2002)

and Campbell, Huisman and Koedijk (2001) reveal that portfolio compositions of tail-

based portfolio frameworks differ to those constructed under a conventional Markowitz

(1952, 1959) MVA. This study seeks to examine these sensitivities in portfolio selection

when the investment opportunity set comprises of traditional assets and hedge funds.

This chapter provides two important contributions to the literature. First, this study

examines the serial correlation biases in portfolio selection where the investment

opportunity set consists of traditional assets and hedge funds. In this study we reveal that

investors have a tendency to over-weight their portfolio allocations towards assets with

serial correlation such as bonds and hedge funds. Second, this study demonstrates that

the risk in hedge fund investments in a portfolio selection framework is located in their

tail-behaviour. The findings show that M-CVaR investors have a lower demand for

hedge fund investments than MVA investors. In short, this study demonstrates that an

increased aversion to tail-risk results in a decreasing allocation to hedge funds.

49 Other studies in the hedge fund literature such as Lo (2001), Geman and Kharoubi (2003) and Malkiel and Saha (2005) also find that hedge fund returns are not normally distributed.

146

The rest of the chapter is structured as follows. Section 5.2 provides a brief review of the

related literature. Section 5.3 documents the methods employed to examine hedge funds

in the MVA and M-CVaR portfolio selection settings. Section 5.4 describes the data in

this study. Section 5.5 examines the results while Section 5.6 offers concluding remarks.


The empirical characteristic of serial correlation has been explored in numerous strands

of the financial economics literature. The early works by Fisher (1966), Dimson (1979)

and Scholes and Williams (1979) with the subsequent contributions by Atchison, Butler

and Simonds (1987), Shanken (1987) and Kadlec and Patterson (1999) have

demonstrated that nonsynchronous trading and stale pricing contribute to spurious

correlation in equity market microstructure. Such effects have been shown to lead to a

re-assessment of return and risk in empirical finance settings. Ahn, Boudoukh,

Richardson and Whitelaw (2002), Chalmers, Edelen and Kadlec (2001) and Greene and

Ciccotello (2006) document the emerging phenomenon of spurious serial correlation

causing mutual fund valuation mispricings which lead to inappropriate wealth transfer

between informed and uninformed investors.

The spurious effects from serial correlation are not limited to the mutual fund literature.

There is a growing hedge fund literature which reveals similar findings. Asness et. al.,

(2001) show that serial correlation effects result in an under-estimation of the beta

coefficient between hedge funds and stock returns. In another setting, Getmansky et. al.,

(2004) and Lo (2002) demonstrate the upward bias in Sharpe ratio estimates on hedge

fund returns due to serial correlation. The hedge fund literature clearly recognises the

serial correlation effects on beta-coefficients and variance estimates in Sharpe ratios,

therefore, it is likely that these effects must also impact on portfolio selection models

which rely on variance estimations.

147

Whilst Dimson (1979) and Roll (1981) document serial correlation bias, other scholars

have developed methods to address the downward bias on estimating the variance

metric. The first technique by Blume et. al., (1991) propose an adjustment which

transforms the second sample moment into an unbiased standard deviation estimate.

The second method from the real estate literature by Blundell and Ward (1987) proposes

a transformation of the original returns by adjusting them with the regression coefficient

of an autoregressive first-order AR(1) model. Recent studies by Herold (2005) and

Scherer (2002) have applied the Blundell and Ward (1987) procedure in portfolio

selection studies. The third and final method by Geltner (1991, 1993) in the real estate

literature proposes an adjustment method which removes the autocorrelation effect from

each data observation. A number of hedge fund studies including Kat and Lu (2002),

Brooks and Kat (2002), Bacmann and Gowran (2005) and Loudon, Okunev and White

(2006) employ Geltner (1991, 1993) to remove severe autocorrelation in returns. It

appears that the literature provides a number of methodological avenues to address the

serial correlation biases in the portfolio selection context.

Whilst the serial correlation in returns violates the MVA normality condition, the second

empirical feature of tail-behaviour also provides challenges for mean-variance investors.

In the spirit of the Roy (1952) safety-first portfolio approach, scholars such as Baumol

(1963) and Arzac and Bawa (1977) were some of the first researchers to develop

portfolio selection frameworks with tail-behaviour as the risk preference. With the

development of the J.P.Morgan (1995) Value at Risk (VaR), scholars such as Basak and

Shapiro (2001), Campbell, Huisman and Koedijk (2001) and Alexander and Baptista

(2002) have developed the mean-VaR portfolio framework where portfolio investment

decisions are based on minimising VaR. These studies show that M-VaR is consistent

with expected utility maximisation when the normality condition is satisfied, however,

M-VaR portfolios are found to be mean-variance inefficient under less restrictive

assumptions.

148

To address the deficiency of M-VaR, the literature has seen the development of the

Rockafellar and Uryasev (2000, 2002) mean-Conditional Value at Risk (M-CVaR)

portfolio framework. The M-CVaR framework differs to M-VaR and MVA in that it

minimises the area of the left tail of the distribution of portfolio returns.50 M-CVaR

portfolio studies by Rockafellar and Uryasev (2000, 2002), Topaloglou, Vladimirou and

Zenios (2002), Krokhmal, Palmquist and Uryasev (2002) and Krokhmal, Uryasev and

Zrazhevsky (2002) have found that CVaR is a better risk management tool in

comparison to other measures including VaR, drawdown and mean absolute deviation.

The literature appears to suggest that the Rockafellar and Uryasev (2000, 2002) M-

CVaR model is the best framework to examine tail-risk which also adheres to the von

Neumaan and Morgenstern (1944) axioms of expected utility maximisation and to the

Artzner et. al., (1997, 1999) principles of coherence.

Despite the development of M-CVaR, other scholars have proposed alternative portfolio

frameworks to capture hedge fund tail-risk in portfolio selection. For instance, Cremers,

Kritzman and Page (2005) propose full-scale optimisation to construct optimal portfolios

of hedge funds for log and S-based utility investors. Krokhmal, Uryasev and

Zrazhevsky (2002) employ both CVaR and drawdown measures to construct portfolios

of individual hedge funds. Krokhmal et. al., (2002) find that CVaR is an effective risk

measure when constructing fund of hedge funds. In an unorthodox framework, Morton,

Popova and Popova (2006) construct portfolios of hedge funds with a stochastic

programming method known as normal-to-anything (NORTA). Finally, Giamouridis

and Vrontos (2007) employ GARCH based methods to model time-varying volatility

and correlation measures in hedge fund portfolio construction.

50 It is important to note that the M-VaR framework does not minimise the left-tail of the distribution of portfolio returns. For example, the 95 per cent M-VaR framework minimises the 5th percentile of the distribution of portfolio returns, however, it is agnostic to losses beyond the 95 percent confidence interval.

149

Optimal portfolio choice not only requires an appropriate model, it also needs to

incorporate the important concept of estimation risk. The early works of Brown (1976,

1979) and Jobson and Korkie (1979) demonstrate that ex-post mean return estimates are

not admissible in portfolio selection frameworks. The sensitivities of portfolio selection

to changes in mean returns were documented in Best and Grauer (1991) and Chopra and

Ziemba (1993). To address the deficiencies of historical mean returns, Eun and Resnick

(1988), Jorion (1985, 1991) and Topaloglou et. al., (2002) posit the virtues of Bayes-

Stein estimation of future expected returns which has been shown to improve the input

parameters in optimal portfolio choice. The literature on Bayes-Stein estimation clearly

shows that a comprehensive portfolio selection study must address the issue of

estimation risk.

The review of the hedge fund literature highlights a number of important issues which

have not been addressed. First, very few studies exist which examine portfolio selection

between traditional assets and hedge funds. Given the growth of global pension funds

and the increasing demand for hedge fund investments, it seems appropriate that this

chapter considers portfolio selection with traditional assets and hedge funds in the

investment opportunity set. Second, little research attention has considered the

sensitivities of serial correlation bias on MVA and M-CVaR frameworks. Third, very

few studies have examined the portfolio selection effects of minimising tail-risk when

the investment opportunity set is traditional assets and hedge funds. This chapter aims to

address these research questions in order to expand the current body of knowledge in the

hedge fund literature. We proceed to detail the methodological specifications of this

study.

150

5.3 Method

The motivation of this study is to examine the shifts in portfolio composition between

global stocks, bonds and hedge funds. To understand these shifts in portfolio selection,

we examine optimal portfolio choice under the conditions of (i) no riskless lending and

borrowing and (ii) short sales are not allowed as investors cannot readily short hedge

fund investments at this time.51 In short, this study concentrates on the optimal allocation

to risky assets only. Portfolio selection studies which have employed similar

assumptions include Amin and Kat (2003), Black (1972) and Elton and Gruber (1995).

The practical rationale for this approach is to examine the effects on asset allocation

when hedge funds are included in an investment universe consisting of the two most

important asset classes in the world, global stocks and bonds.

This study employs a variety of methodologies to examine the effects of serial

correlation and tail-risk in portfolio selection. To account for serial correlation bias, this

study employs the Blume et. al., (1991) and the Geltner (1991, 1993) methods to adjust

the second sample moment from the biases of serial correlation. To compare and

contrast the effects of tail-risk, we employ the original returns and the serial correlation

adjusted returns in a traditional Markowitz (1952) MVA and the Rockafellar and

Uryasev (2000, 2002) M-CVaR portfolio framework. To accommodate estimation risk

in the analysis, we employ the Bayes-Stein mean shrinkage estimation in a MVA

framework. We now detail the mathematical specifications of the empirical frameworks

employed in this chapter.

51 The work of Jagannathan and Ma (2003) discover that imposing non-negativity portfolio constraints provides substantial net benefits to portfolio construction.

151

5.3.1 Mean Variance Analysis (MVA) Framework The Markowitz (1952) MVA portfolio selection framework can be mathematically

expressed as:

Χmin Var )( pR (5.1)

s.t. ,1,,...,1,01

==≥ ∑=

n

iii xnix and ≥)( pRE Target Return

where pR and Var ΧΧ′= VRp )( are the n -assets portfolio return and variance

respectively, ),...,,( 21 ′=Χ nxxx is the vector containing the asset weights in the

portfolio, V is the n x n covariance matrix, and )( pRE is the expected return of the

portfolio.

5.3.2 Mean-CVaR (M-CVaR) Framework The mean-CVaR portfolio optimisation model employed in this study follows the

convex programming formulation of Rockafellar and Uryasev (2000, 2002). The

portfolio framework can be expressed as:

Χmin CVaR( ),α

pRF (5.2)

s.t. ,1,,...,1,01

==≥ ∑=

n

iii xnix and ≥)( pRE Target Return

where

CVaR( −≤−= ppR RREFp

(),α VaR)(

)()

VaRF

dzzzfR

pR

VaR

p

−−= ∫

−

∞− ,

VaR( ),~ αpRF = 1(1 ),

p pR RF fα−− − and pRF denote the probability density and the

cumulative density of pR , respectively, and α is the probability level.

152

5.3.3 Autocorrelation Biased Second Sample Moment Adjustment To account for biased second moments, this study employs the adjustment procedure

proposed in Blume et. al., (1991). This can be mathematically expressed in the form:

1 21(12 22 )a o

a mσ σ ρ= + (5.3)

and

12

aa am

σσ = (5.4)

where amσ is the first-order autocorrelation adjusted monthly standard deviation, a

aσ is

the first-order autocorrelation adjusted annualised standard deviation, 1ρ is the first-

order autocorrelation coefficient and omσ is the original monthly standard deviation.

Informally, the Blume et. al., (1991) two-step procedure is calculated by converting the

original monthly standard deviation into an annualised measure which takes into account

the first-order autocorrelation. The second step of the procedure is to divide the adjusted

annualised standard deviation by 12 to re-express it into monthly units. The final

estimate from the Blume et. al., (1991) procedure is an adjusted second sample moment

(ie. variance and standard deviation) which can be employed in portfolio selection.

5.3.4 Transforming Autocorrelated Returns to IID Returns As a second method to account for serial correlation, the Geltner (1991, 1993) procedure

is employed to construct i.i.d. returns. The work of Geltner (1991, 1993) from the real

estate literature proposes a transformation of the original data to an unsmoothed time

series based on the level of autocorrelation. The Geltner (1991, 1993) procedure results

in an augmented time series which exhibits an identical first sample moment as the

original returns but with a modified second sample moment and zero autocorrelation.

This method has been previously employed in hedge fund studies including Brooks and

Kat (2002), Kat and Lu (2002), Bacmann and Gawron (2005) and Loudon, Okunev and

White (2006). We follow the work of Geltner (1991, 1993) and mathematically state the

adjustment procedure as:

153

1 1,

11t t

u tr rr ρ

ρ−−

=−

(5.5)

where ,u tr is the adjusted return, tr is the original return and 1ρ is the first-order

autocorrelation coefficient.

5.3.5 Bayes-Stein Mean Shrinkage Estimation To derive Bayesian estimates of expected mean returns, we follow Eun and Resnick

(1988), Jorion (1985, 1991) and Topaloglou et. al., (2002) and calculate the Bayes-Stein

mean estimate as:

0)1(~ ωιρρω +−=r (5.6)

where ρ denotes the sample excess mean-return vector estimated from the historical

observations, 0ρ is the mean return of the minimum variance portfolio based on the

same historical observations, ι is a vector of ones, and ω is the estimated shrinkage

factor for shrinking the sample mean return vector ρ towards 0ρ . The shrinkage

parameter ω is expressed as:

))(2()()1)(2(

)1)(2(

01'

0 ιρριρρω

−−−Σ−+−+−+

= − NTTTNTN (5.7)

where T is the length of the sample observations, N is the number of asset classes and

Σ is the sample covariance matrix estimation from the historical observations. To

account for estimation risk, various covariance matrix Bayesian estimates were

calculated but were found to have no impact on the results.52

52 Bayesian shrinkage estimation of the covariance matrix was also employed in this study. Because global stocks, bonds and hedge funds are not similarly related, standard covariance shrinkage estimation of the covariance matrix proposed by Frost and Savarino (1986) and Ledoit and Wolf (2003, 2004) are not applicable. Due to the unrelatedness of the assets in this study, we employ a block-diagonal covariance

154

Table 5.1 Summary Statistics This table shows the summary statistics of the monthly excess returns of the three risky asset classes employed in this study. The global stock proxy is the MSCI All Country World Equity index. The global bond proxy is the Lehman Brothers Global Aggregate Index. The global hedge fund proxy is the HFR Fund of Funds Index. Panel A shows the descriptive statistics of the monthly excess returns of the three risky asset classes. Panel B reports the autocorrelation of returns. Panel C shows the autocorrelation of squared returns. Panel D reports the normalised z-scores of the minimum, 1st, 2.5th, 5th, 95th, 97.5th , 99th and maximum percentiles. The 1%, 2.5%, 5%, 95%, 97.5% and 99% percentiles for a normal distribution are -2.3263, -1.9600, -1.6449, 1.6449, 1.9600 and 2.3263. The data is sampled monthly from January 1994 to December 2005 consisting of 144 observations. ^ denotes calculations estimated from the Blume, Keim and Patel (1991) two-step procedure. * and ** denote statistical significance at the 5% and 1% levels, respectively.

Original Returns (Geltner) Adjusted Returns Variables

Global Stocks

Global Bonds

Global Hedge Funds

Global Stocks

Global Bonds

Global Hedge Funds

Panel A: Descriptive Statistics Mean 0.360 0.219 0.287 0.360 0.219 0.287 Standard Deviation 4.028 0.884 1.671 4.028 1.068 2.372 Adj. Standard Deviation^ 4.028 1.022 2.122 ---- ---- ---- Skewness -0.754 -0.348 -0.524 -0.754 -0.397 -0.683 Kurtosis 4.002 3.412 7.729 4.002 3.305 7.528 Median 0.796 0.317 0.433 0.796 0.320 0.204 Maximum 8.455 2.986 6.187 8.455 3.432 7.065 Minimum -14.696 -2.184 -8.193 -14.696 -2.781 -11.994 Jarque-Bera Statistic 18.751 3.646 134.655 18.751 4.076 127.462 Jarque-Bera p-value 0.000** 0.162 0.000** 0.000** 0.130 0.000** Sharpe Ratio (monthly) 0.089 0.247 0.171 0.089 0.203 0.120 Adj. Sharpe Ratio (monthly)^ 0.089 0.214 0.135 ---- ---- ---- Panel B: Autocorrelation (First Moment) AC1 0.016 0.184* 0.334** 0.016 0.012 0.013 AC2 -0.035 0.002 0.096 -0.035 -0.053 0.010 AC3 0.055 0.126 -0.026 0.055 0.135 -0.036 AC6 0.111 0.006 -0.018 0.111 0.014 0.005 AC12 0.086 -0.174* -0.056 0.086 -0.138 -0.008 Panel C: Autocorrelation (Second Moment) AC1 0.035 0.035 0.123 0.035 0.053 0.024 AC2 0.201** -0.038 0.143 0.201** -0.061 0.097 AC3 0.059 -0.002 0.006 0.059 -0.003 0.067 AC6 0.089 -0.064 0.048 0.089 -0.047 0.056 AC12 0.131 -0.049 -0.040 0.131 -0.083 0.018 Panel D: Standardised Tail Z-Scores Minimum -3.738 -2.719 -5.073 -3.738 -2.806 -5.176 1st Percentile -3.054 -2.716 -2.652 -3.054 -2.635 -2.715 2.5th Percentile -2.402 -2.400 -1.725 -2.402 -2.226 -1.775 5th Percentile -1.806 -1.677 -1.460 -1.806 -1.737 -1.460 95th Percentile 1.436 1.465 1.421 1.436 1.477 1.585 97.5th Percentile 1.641 1.643 2.252 1.641 1.608 2.020 99th Percentile 1.953 1.889 2.666 1.953 1.835 2.606 Maximum 2.010 3.131 3.530 2.010 3.010 2.859

matrix method based on personal communication with Olivier Ledoit. The portfolio selection results employing the block-diagonal method show that it has little or no impact on the overall findings.

155

5.4 Data

The data employed in this study represents proxies for global stocks, bonds and hedge

fund returns. To minimise the impact of idiosyncratic risk, we employ global index

returns for each investment rather than utilising the returns of individual stocks, bonds or

hedge funds. We employ the Morgan Stanley Capital International (MSCI) All Country

World Equity Index as the proxy for global stocks, the Lehman Brothers Global

Aggregate Index as the proxy for global bonds and the Hedge Fund Research (HFR)

Fund of Funds Index as the proxy for global hedge fund returns. The returns of each data

series are computed as monthly US dollar returns in excess of the risk-free rate of return

which is the one-month US Treasury bill.53 As this study involves the estimation of

multi-asset portfolios, we follow Campbell, Lo and MacKinlay (1997) and employ

arithmetic excess returns rather than the conventional continuous compounded excess

returns when estimating MVA and M-CVaR portfolio choice.

To model global hedge fund returns, we follow Fung and Hsieh (2000, 2004) and

employ the Hedge Fund Research Fund of Fund Index (HFRFOFI). Fung and Hsieh

(2000, 2004) have shown that hedge fund of fund indices have minimal survivorship,

backfilling and selection biases in comparison to other sources of hedge fund returns.

Although the HFRFOFI commences in January 1990, Fung and Hsieh (2004) report less

than 100 funds in the index prior to January 1994. This small number of funds in the

index may not be representative of the returns of global hedge funds therefore we follow

Fung and Hsieh (2004) by employing the index data from January 1994 onwards.

53 The method of employing excess returns allows the portfolio selection procedure to be estimated in an OLS regression framework similar to that proposed in Britten-Jones (1999). The method of examining risky assets only follows similar methodologies employed in Amin and Kat (2003) Black (1972) and Elton and Gruber (1995).

156

The summary statistics in Table 5.1 reflect the salient features of financial market

returns with negative third moments, excessive fourth moments, extreme left tail

behaviour and serial correlation in the first and second moments. The Blume et. al.,

(1991) unbiased standard deviation estimate of the original returns shows that hedge

fund volatility has increased proportionately more than bonds.54 Panel D reports the tail-

behaviour of each asset class which shows that standardised z-scores at and below the

2.5th percentile are more severe than expected from a normal distribution. Panel D also

reports that hedge funds possess the lowest standardised z-score of all asset classes. To

summarise the tail-behaviour in Table 5.1 we can observe that the extreme left tail of

hedge fund returns is more severe than stocks and bonds.

In contrast, the Geltner (1991, 1993) adjusted returns in Table 5.1 report higher

estimates of asset return volatility. In fact, hedge fund returns report a higher variance

under the Geltner (1991, 1993) procedure in comparison to the Blume et. al., (1991)

adjustment.55 These two procedures reveal that the Geltner (1991, 1993) adjustment

penalises hedge fund return volatility more than the Blume et. al., (1991) procedure.

54 Hedge fund standard deviation rose from 1.671 to 2.122 per cent which is a 27 per cent rise in volatility. In contrast, the standard deviation of global bonds rose from 0.884 to 1.022 per cent which is a 15.6 per cent rise in volatility. 55 The Geltner (1991, 1993) procedure reports a 42 per cent rise (from 1.671 to 2.372 per cent) in the estimate of hedge fund volatility while bonds rose only 20.8 per cent (from 0.884 to 1.068 per cent).

157

Table 5.2 Mean-Variance Analysis (Original Sample)

This table presents the mean-variance analysis where the investment universe consists of risky assets only. This asset allocation estimates the portfolio weights with a non-negativity constraint. The mean-variance analysis is performed by minimising portfolio variance for a given level of expected return. The three asset class proxies of global stocks, bonds and hedge funds are the MSCI World Equity Index, the Lehman Brothers Treasury Index and Global Aggregate Index and the HFR Fund of Funds Index. Excess monthly returns were employed for the period January 1994 to December 2005. The heading M1 denotes mean monthly excess return, M2 denotes portfolio standard deviation, M3 denotes skewness and M4 denotes kurtosis of the respective portfolio returns. The range of the required rates of return in the mean-variance analysis are divided into ten decile portfolios to form the efficient set which allows a direct comparison with other investment universes with the same required rate of return. The headings Eq. CVaR denote the equivalent Conditional Value at Risk calculation of each mean-variance portfolio.

2 Asset Universe – Stocks and Bonds 3 Asset Universe – Stocks, Bonds and Hedge Funds

M1

M2

M3

M4 Stocks

(%) Bonds

(%) Eq. CVaR

95% Eq. CVaR

97.5% Eq. CVaR

99%

M1

M2

M3

M4 Stocks

(%) Bonds

(%) HF (%)

Eq. CVaR 95%

Eq. CVaR 97.5%

Eq. CVaR 99%

Panel A: Minimum Variance Portfolio Panel B: Minimum Variance Portfolio 0.227 0.856 -0.354 3.600 5.3 94.7 -1.83 -2.04 -2.23 0.227 0.850 -0.448 3.669 0.0 79.9 20.1 -1.72 -1.97 -2.24

Panel C: Tangent Portfolio Panel D: Tangent Portfolio 0.231 0.864 -0.334 3.531 8.0 92.0 -1.84 -2.01 -2.30 0.231 0.859 -0.421 3.590 1.0 75.5 24.5 -1.70 -1.94 -2.25

Panel E: Efficient set Panel F: Efficient set 0.227 0.856 -0.354 3.600 5.3 94.7 -1.83 -2.04 -2.23 0.227 0.850 -0.448 3.669 0.0 79.9 20.1 -1.72 -1.97 -2.24 0.242 0.965 -0.310 2.928 16.0 84.0 -1.88 -2.08 -2.49 0.242 0.915 -0.392 3.529 2.5 68.5 29.0 -1.77 -2.00 -2.30 0.258 1.229 -0.417 2.647 26.5 73.5 -2.43 -2.62 -2.92 0.258 1.020 -0.458 4.283 7.0 51.2 41.8 -2.12 -2.67 -3.77 0.273 1.571 -0.538 2.924 37.0 63.0 -3.37 -3.67 -4.59 0.273 1.290 -0.617 5.553 11.5 33.8 54.7 -2.67 -3.36 -5.71 0.289 1.951 -0.618 3.247 47.5 52.5 -4.39 -4.92 -6.26 0.289 1.606 -0.739 6.421 16.0 16.4 67.6 -3.36 -4.16 -7.64 0.304 2.350 -0.668 3.497 58.0 42.0 -5.46 -6.19 -7.93 0.304 1.945 -0.822 6.808 21.3 0.0 78.7 -4.17 -5.04 -9.54 0.320 2.761 -0.699 3.676 68.5 31.5 -6.54 -7.46 -9.60 0.320 2.366 -0.894 5.485 41.0 0.0 59.0 -5.46 -6.37 -10.80 0.335 3.178 -0.719 3.805 79.0 21.0 -7.62 -8.74 -11.27 0.335 2.875 -0.861 4.643 60.6 0.0 39.4 -6.80 -8.00 -12.07 0.351 3.599 -0.733 3.900 89.5 10.5 -8.70 -10.01 -12.94 0.351 3.435 -0.801 4.198 80.3 0.0 19.7 -8.27 -9.64 -13.34 0.366 4.024 -0.742 3.971 100.0 0.0 -9.78 -11.28 -14.61 0.366 4.024 -0.742 3.971 100.0 0.0 0.0 -9.78 -11.28 -14.61

158

Table 5.3 Mean-Variance Analysis (Bayes-Stein Mean Estimates)

This table presents the mean-variance analysis where the investment universe consists of risky assets only. This asset allocation estimates the portfolio weights with a non-negativity constraint. The mean-variance analysis is performed by minimising portfolio variance for a given level of expected return. The three asset class proxies of global stocks, bonds and hedge funds are the MSCI World Equity Index, the Lehman Brothers Treasury Index and Global Aggregate Index and the HFR Fund of Funds Index. Excess monthly returns were employed for the period January 1994 to December 2005. The heading M1 denotes mean monthly excess return, M2 denotes portfolio standard deviation, M3 denotes skewness and M4 denotes kurtosis of the respective portfolio returns. The range of the required rates of return in the mean-variance analysis are divided into ten decile portfolios to form the efficient set which allows a direct comparison with other investment universes with the same required rate of return. The headings Eq. CVaR denote the equivalent Conditional Value at Risk calculation of each mean-variance portfolio.


M1

M2

M3

M4 Stocks

(%) Bonds

(%) Eq. CVaR

95% Eq. CVaR

97.5% Eq. CVaR

99%

M1

M2

M3

M4 Stocks

(%) Bonds

(%) HF (%)

Eq. CVaR 95%

Eq. CVaR 97.5%

Eq. CVaR 99%




159

5.5 Results

The results section of this study is presented in three parts. The first section compares

the MVA portfolio selection weightings of the original returns with both Blume et. al.,

(1991) and Geltner (1991, 1993) adjustments. The second section reports M-CVaR

portfolio selection and compares optimal portfolio choice derived from both original

returns and the Geltner (1991, 1993) transformed time series returns. The third section

of the results provides a rationale to explain the findings reported in this study.

5.5.1 MVA and the Effects of Serial Correlation Table 5.2 reports the portfolio compositions of an MVA estimated from the original

returns. The maximum hedge fund allocation reported in Table 5.2 is 79 per cent. Panel

A shows the importance of bonds in the minimum variance portfolio in a 2-asset

universe of stocks and bonds. An unexpected finding in Panel C reveals a small

weighting to stocks in the tangent portfolio. This unusual result reflects the small global

equity premium of 4.3 per cent per year reported over the 1994-2005 sample period.56

The introduction of hedge funds into the opportunity set in Panels B, D and F in Table

5.2 shows their importance in the minimum variance portfolio, the tangent portfolio and

in almost all portfolios across the MVA efficient set. Overall, the introduction of hedge

funds in Table 5.2 reveals a reduction in the volatility of portfolio returns but it comes at

the cost of undesirable third and fourth moments.57 This result is consistent with the

empirical results from Amin and Kat (2003). This finding can be interpreted in an

economic sense as the price for holding hedge funds in the portfolio.

56 The literature reports difficulties in estimating the long-term mean excess returns for global stocks due to the inaccuracy of the available data. Campbell and Viceria (2006) report a mean excess return of 6.31 per cent per year for US stocks from 1952-2002. In another study, Jorion and Goetzmann (1999) estimate the long-term real return of 4.3 per cent for US stocks in comparison to 0.8 per cent for the real returns of global stocks. It is also important to acknowledge that long term studies such as these are also susceptible to structural changes to the return generation process due to the effects of World Wars I and II, the Great Depression and Bretton Woods. 57 Refer to Scott and Horvarth (1980) who develop the theoretical portfolio rationale which demonstrates that rational agents maximising expected utility will have a preference for positive skewness and decreasing kurtosis in portfolio returns.

160

Table 5.4 Mean-Variance Analysis (Blume, Keim and Patel (1991) Adjustment) This table presents the mean-variance analysis where the investment universe consists of risky assets only. This asset allocation estimates the portfolio weights with a non-negativity constraint. The mean-variance analysis is performed by minimising portfolio variance for a given level of expected return. The three asset class proxies of global stocks, bonds and hedge funds are the MSCI World Equity Index, the Lehman Brothers Treasury Index and Global Aggregate Index and the HFR Fund of Funds Index. Excess monthly returns were employed for the period January 1994 to December 2005. The heading M1 denotes mean monthly excess return, M2 denotes portfolio standard deviation, M3 denotes skewness and M4 denotes kurtosis of the respective portfolio returns. The range of the required rates of return in the mean-variance analysis are divided into ten decile portfolios to form the efficient set which allows a direct comparison with other investment universes with the same required rate of return. The headings Eq. CVaR denote the equivalent Conditional Value at Risk calculation of each mean-variance portfolio.


M1

M2

M3

M4 Stocks

(%) Bonds

(%) Eq. CVaR

95% Eq. CVaR

97.5% Eq. CVaR

99%

M1

M2

M3

M4 Stocks

(%) Bonds

(%) HF (%)

Eq. CVaR 95%

Eq. CVaR 97.5%

Eq. CVaR 99%




161

Table 5.5 Mean-Variance Analysis (Geltner (1991, 1993) Adjustment) This table presents the mean-variance analysis where the investment universe consists of risky assets only. This asset allocation estimates the portfolio weights with a non-negativity constraint. The mean-variance analysis is performed by minimising portfolio variance for a given level of expected return. The three asset class proxies of global stocks, bonds and hedge funds are the MSCI World Equity Index, the Lehman Brothers Treasury Index and Global Aggregate Index and the HFR Fund of Funds Index. Excess monthly returns were employed for the period January 1994 to December 2005. The heading M1 denotes mean monthly excess return, M2 denotes portfolio standard deviation, M3 denotes skewness and M4 denotes kurtosis of the respective portfolio returns. The range of the required rates of return in the mean-variance analysis are divided into ten decile portfolios to form the efficient set which allows a direct comparison with other investment universes with the same required rate of return. The headings Eq. CVaR denote the equivalent Conditional Value at Risk calculation of each mean-variance portfolio.


M1

M2

M3

M4 Stocks

(%) Bonds

(%) Eq. CVaR

95% Eq. CVaR

97.5% Eq. CVaR

99%

M1

M2

M3

M4 Stocks

(%) Bonds

(%) HF (%)

Eq. CVaR 95%

Eq. CVaR 97.5%

Eq. CVaR 99%




162

Table 5.3 reports the MVA results with Bayes-Stein mean estimators which allows the

effects of estimation risk to be incorporated in portfolio selection. The Bayes-Stein

portfolios in Table 5.3 reports a maximum allocation to hedge funds of 76 per cent. The

comparison of ex-post mean returns in Table 5.2 versus the Bayes-Stein estimates in

Table 5.3 reveal very little difference in portfolio composition. In light of these results,

we continue to employ ex-post returns in the subsequent sections of this study.58

We examine the serial correlation biases in portfolio selection by reporting the MVA

results with the Blume et. al., (1991) second sample moment and the Geltner (1991,

1993) transformed data series in Tables 5.4 and 5.5, respectively. The Blume et. al.,

(1991) adjusted MVA portfolios in Table 5.4 report a maximum hedge fund allocation

of 59.0 per cent while the Geltner (1991, 1993) MVA portfolio results in Table 5.5 show

a maximum hedge fund allocation of 66.7 per cent.

The overall assessment of Tables 5.3 to 5.5 suggests that serial correlation bias can

cause MVA to over-estimate its optimal portfolio allocation to hedge funds of

approximately 2 to 20 per cent. This discovery reveals that serial correlation biases can

cause rational investors to significantly over-allocate their portfolio allocations to

significant serially correlated asset returns such as hedge funds. While there are no

known studies that explicitly examine the effects of serial correlation of hedge fund

returns on portfolio selection, these results are consistent with Asness et. al., (2001),

Blume et. al., (1991), Dimson (1979), Getmansky et. al., (2004), Lo (2002) and Roll

(1981) who report serial correlation effects in the analysis of variance, beta coefficient

and Sharpe ratios.

58 It is important to acknowledge that one limitation of this study is the short twelve year history of global hedge fund returns from January 1994 to December 2005. Whilst this period provides a long-term history of global hedge fund returns, it is a relatively short period when compared to the available history of stock returns.

163

Table 5.6 Mean-CVaR Portfolio Optimisation (Original Sample) This table presents the mean-CVaR portfolio optimisations in a two and three asset universe. The portfolio weights were constrained to be positive with no short sales allowed. The headings m1 to m4 denote the first four moments of the distribution of the associated portfolio returns. The range of the required rates of return in the mean-variance analysis are divided into ten decile portfolios to form the efficient set which allows a direct comparison with other investment universes with the same required rate of return. Rows with a * denote that these portfolios cannot be directly compared as their required rates of return differ due to the varying assets in the opportunity set.


M1 M2 M3 M4 Stocks (%) Bonds (%) CVaR (%) M1 M2 M3 M4 Stocks (%) Bonds (%) HF (%) CVaR (%)

Panel A: CVaR 95% Panel B: CVaR 95% *0.229 0.859 -0.343 3.575 6.9 93.1 -1.83 *0.234 0.814 -0.417 3.590 0.0 72.9 27.1 -1.69

0.235 0.891 -0.313 3.319 11.2 88.8 -1.85 0.235 0.816 -0.421 3.612 0.0 72.5 27.5 -1.73 0.252 1.110 -0.364 2.652 22.3 77.7 -2.13 0.252 0.945 -0.432 3.424 9.3 62.9 27.7 -1.95 0.268 1.448 -0.500 2.808 33.4 66.6 -3.05 0.268 1.190 -0.572 5.027 10.7 40.7 48.6 -2.46 0.285 1.839 -0.599 3.161 44.5 55.5 -4.09 0.285 1.512 -0.708 6.237 14.5 21.3 64.2 -3.15 0.301 2.257 -0.659 3.446 55.6 44.4 -5.21 0.301 1.868 -0.777 7.122 16.8 0.0 83.2 -3.94 0.317 2.689 -0.695 3.649 66.7 33.3 -6.35 0.317 2.285 -0.892 5.684 37.6 0.0 62.4 -5.23 0.334 3.130 -0.717 3.792 77.8 22.2 -7.49 0.334 2.814 -0.867 4.715 58.4 0.0 41.6 -6.63 0.350 3.575 -0.732 3.895 88.9 11.1 -8.64 0.350 3.402 -0.804 4.217 79.2 0.0 20.8 -8.19 0.366 4.024 -0.742 3.971 100.0 0.0 -9.78 0.366 4.024 -0.742 3.972 100.0 0.0 0.0 -9.78

Panel C: CVaR 97.5% Panel D: CVaR 97.5% *0.232 0.882 -0.317 3.383 10.4 89.6 -1.99 *0.234 0.823 -0.403 3.515 3.0 73.3 23.7 -1.92

0.235 0.891 -0.313 3.319 11.2 88.8 -1.99 0.235 0.828 -0.402 3.504 4.1 71.7 24.2 -1.92 0.252 1.110 -0.364 2.652 22.3 77.7 -2.32 0.252 1.047 -0.398 2.769 18.7 73.6 7.8 -2.29 0.268 1.448 -0.500 2.808 33.4 66.6 -3.27 0.268 1.252 -0.606 3.614 21.0 52.5 26.5 -2.94 0.285 1.839 -0.599 3.161 44.5 55.5 -4.56 0.285 1.556 -0.754 4.723 25.2 33.4 41.4 -3.87 0.301 2.257 -0.659 3.446 55.6 44.4 -5.90 0.301 1.870 -0.825 6.386 22.9 7.1 70.0 -4.82 0.317 2.689 -0.695 3.649 66.7 33.3 -7.25 0.317 2.285 -0.892 5.684 37.6 0.0 62.4 -6.09 0.334 3.130 -0.717 3.792 77.8 22.2 -8.59 0.334 2.814 -0.867 4.715 58.4 0.0 41.6 -7.81 0.350 3.575 -0.732 3.895 88.9 11.1 -9.93 0.350 3.402 -0.804 4.217 79.2 0.0 20.8 -9.54 0.366 4.024 -0.742 3.971 100.0 0.0 -11.28 0.366 4.024 -0.742 3.972 100.0 0.0 0.0 -11.28

Panel E: CVaR 99% Panel F: CVaR 99% 0.221 0.872 -0.356 3.487 1.3 98.7 -2.17 0.221 0.872 -0.356 3.487 1.3 98.7 0.0 -2.17 0.235 0.891 -0.313 3.319 11.2 88.8 -2.37 0.235 0.808 -0.429 3.620 1.4 77.7 20.9 -2.24 0.252 1.110 -0.364 2.652 22.3 77.7 -2.65 0.252 1.096 -0.372 2.670 21.5 76.9 1.6 -2.64 0.268 1.448 -0.500 2.808 33.4 66.6 -4.01 0.268 1.448 -0.500 2.808 33.4 66.6 0.0 -4.01 0.285 1.839 -0.599 3.161 44.5 55.5 -5.78 0.285 1.839 -0.599 3.161 44.5 55.5 0.0 -5.78 0.301 2.257 -0.659 3.446 55.6 44.4 -7.54 0.301 2.257 -0.659 3.446 55.6 44.4 0.0 -7.54 0.317 2.689 -0.695 3.649 66.7 33.3 -9.31 0.317 2.689 -0.695 3.649 66.7 33.3 0.0 -9.31 0.334 3.130 -0.717 3.792 77.8 22.2 -11.07 0.334 3.130 -0.717 3.792 77.8 22.2 0.0 -11.07 0.350 3.575 -0.732 3.895 88.9 11.1 -12.84 0.350 3.575 -0.732 3.895 88.9 11.1 0.0 -12.84 0.366 4.024 -0.742 3.971 100.0 0.0 -14.61 0.366 4.024 -0.742 3.971 100.0 0.0 0.0 -14.61

164

Table 5.7 Mean-CVaR Portfolio Optimisation (Geltner (1991, 1993) Adjustment) This table presents the mean-CVaR portfolio where the investment universe consists of risky assets only. This asset allocation constrains portfolio weights to be positive with no short sales allowed. The asset class proxies for global stocks, bonds and hedge funds were the MSCI World Equity Index, Lehman Brothers Global Aggregate Index and the HFR Fund of Funds Index for the period January 1994 to December 2005. The headings M1 to M4 denote the first four moments of the distribution of the respective portfolio returns. The range of the required rates of return in the mean-variance analysis are divided into ten decile portfolios to form the efficient set which allows a direct comparison with other investment universes with the same required rate of return. Rows with a * denote that these portfolios cannot be directly compared as their required rates of return differ due to the varying assets in the opportunity set.


M1 M2 M3 M4 Stocks (%) Bonds (%) CVaR (%) M1 M2 M3 M4 Stocks (%) Bonds (%) HF (%) CVaR (%)

Panel A: CVaR 95% Panel B: CVaR 95% 0.236 1.103 -0.325 2.957 17.3 82.7 -2.26 0.236 1.054 -0.440 3.212 6.3 76.3 17.4 -2.25 0.246 1.298 -0.405 2.678 26.5 73.5 -2.56 0.246 1.235 -0.460 2.810 20.5 70.1 9.4 -2.50 0.256 1.564 -0.516 2.818 35.7 64.3 -3.30 0.256 1.403 -0.615 3.815 15.7 52.8 31.4 -3.00 0.266 1.872 -0.602 3.094 44.9 55.4 -4.17 0.266 1.654 -0.715 4.862 16.7 38.9 44.4 -3.58 0.276 2.204 -0.659 3.354 54.0 46.0 -5.09 0.276 1.942 -0.767 6.005 15.0 23.5 61.5 -4.21 0.286 2.551 -0.695 3.560 63.2 36.8 -6.02 0.286 2.238 -0.828 6.540 18.0 10.7 71.4 -4.87 0.296 2.908 -0.718 3.717 72.4 27.6 -6.96 0.296 2.536 -0.889 6.646 24.6 0.0 75.4 -5.58 0.306 3.270 -0.732 3.837 81.6 18.4 -7.90 0.306 2.906 -0.928 5.299 49.7 0.0 50.3 -6.75 0.316 3.638 -0.741 3.929 90.8 9.2 -8.84 0.316 3.414 -0.851 4.411 74.9 0.0 25.1 -8.19 0.326 4.008 -0.747 4.000 100.0 0.0 -9.78 0.326 4.008 -0.747 4.000 100.0 0.0 0.0 -9.78

Panel C: CVaR 97.5% Panel D: CVaR 97.5% 0.236 1.103 -0.325 2.957 17.3 82.7 -2.47 0.236 1.062 -0.402 3.110 11.1 79.2 9.7 -2.42 0.246 1.298 -0.405 2.678 26.5 73.5 -2.74 0.246 1.298 -0.405 2.678 26.5 73.5 0.0 -2.74 0.256 1.564 -0.516 2.818 35.7 64.3 -3.60 0.256 1.455 -0.596 3.108 26.2 58.9 14.9 -3.36 0.266 1.872 -0.602 3.094 44.9 55.4 -4.62 0.266 1.686 -0.728 3.898 27.8 45.3 26.9 -4.11 0.276 2.204 -0.659 3.354 54.0 46.0 -5.70 0.276 1.957 -0.816 4.606 30.8 32.6 36.6 -4.95 0.286 2.551 -0.695 3.560 63.2 36.8 -6.82 0.286 2.251 -0.875 5.129 34.6 20.3 45.1 -5.79 0.296 2.908 -0.718 3.717 72.4 27.6 -7.93 0.296 2.547 -0.915 5.713 36.2 6.7 57.1 -6.68 0.306 3.270 -0.732 3.837 81.6 18.4 -9.05 0.306 2.906 -0.928 5.299 49.7 0.0 50.3 -7.88 0.316 3.638 -0.741 3.929 90.8 9.2 -10.16 0.316 3.414 -0.851 4.411 74.9 0.0 25.1 -9.58 0.326 4.008 -0.747 4.000 100.0 0.0 -11.28 0.326 4.008 -0.747 4.000 100.0 0.0 0.0 -11.28

Panel E: CVaR 99% Panel F: CVaR 99% 0.236 1.103 -0.325 2.957 17.3 82.7 -2.56 0.236 1.103 -0.325 2.957 17.3 82.7 0.0 -2.56 0.246 1.298 -0.405 2.678 26.5 73.5 -3.04 0.246 1.254 -0.442 2.750 22.6 71.3 6.1 -2.92 0.256 1.564 -0.516 2.818 35.7 64.3 -4.21 0.256 1.564 -0.516 2.818 35.7 64.3 0.0 -4.21 0.266 1.872 -0.602 3.094 44.9 55.4 -5.69 0.266 1.872 -0.602 3.094 44.9 55.4 0.0 -5.69 0.276 2.204 -0.659 3.354 54.0 46.0 -7.18 0.276 2.204 -0.659 3.354 54.0 46.0 0.0 -7.18 0.286 2.551 -0.695 3.560 63.2 36.8 -8.66 0.286 2.551 -0.695 3.560 63.2 36.8 0.0 -8.67 0.296 2.908 -0.718 3.717 72.4 27.6 -10.15 0.296 2.908 -0.718 3.717 72.4 27.6 0.0 -10.15 0.306 3.270 -0.732 3.837 81.6 18.4 -11.64 0.306 3.270 -0.732 3.837 81.6 18.4 0.0 -11.64 0.316 3.638 -0.741 3.929 90.8 9.2 -13.12 0.316 3.638 -0.741 3.929 90.8 9.2 0.0 -13.12 0.326 4.008 -0.747 4.000 100.0 0.0 -14.61 0.326 4.008 -0.747 4.000 100.0 0.0 0.0 -14.61

165

5.5.2 Tail-Risk Effects in Portfolio Selection We examine the effects of tail-risk by comparing MVA and M-CVaR portfolio selection

results. The M-CVaR portfolio optimisations in Table 5.6 reveal a systematic reduction

in 95 per cent and 97.5 per cent CVaR when hedge funds are included in the investment

opportunity set. However, a striking finding is reported in Panel F of Table 5.6 reveals

that an M-CVaR investor constraining 99 per cent CVaR will exhibit little or no demand

for hedge fund investments. Again, this is consistent with the tail-behaviour of hedge

funds revealed in Table 5.1. The second striking feature of Panel F shows that investors

who seek conservative rates of excess returns will allocate a proportion of their portfolio

to hedge funds.

Overall, the conclusions drawn from Table 5.6 suggest that M-CVaR investors have a

lower demand for hedge funds as their tail-risk aversion increases. Furthermore, as M-

CVaR investors require more conservative rates of excess return, they are exposed to

tail-risk in global bonds. To reduce their tail-risk from a concentration of bonds, the M-

CVaR investor will allocate a proportion of their portfolio to hedge funds.

To examine the biases from serial correlation, Table 5.7 reports the M-CVaR portfolio

results with the Geltner (1991, 1993) adjusted return series.59 A comparison of Panels A,

C and E of Tables 5.6 and 5.7 reveals a systematic decrease in bonds in the two-asset

universe. This is the result of a higher second sample moment in bonds after the

adjustment for serial correlation. Panels B, D and F of Tables 5.6 and 5.7 also reveal an

overall reduction in hedge fund allocations. However, a comparison of Panel F of Tables

5.6 and 5.7 shows a pronounced decrease in the demand for hedge funds when the

effects of serial correlation bias is included. The overall assessment of the M-CVaR

portfolio optimisations suggests that as M-CVaR investors become more tail-risk averse,

the demand for hedge funds disappears. 59 Blume et. al., (1991) only adjusts the second sample moment and therefore cannot be employed in a M-CVaR framework. A researcher could estimate a M-CVaR under the assumption of normality and estimate 1.65 and 1.96 times the standard deviation to estimate 95% and 99% CVaR, however, this imposes the normality condition and defeats the empirical nature of this study.

166

Figure 5.1 HFR Fund of Funds Index vs. MSCI World Equity Index

-15%

-10%

-5%

0%

5%

10%

15%

-15% -10% -5% 0% 5% 10% 15%


HFR

Fun

d of

Fun

ds In

dex

Exce

ss M

onth

ly R

etur

ns

Figure 5.2 HFR Fund of Funds Index vs. Lehman Global Aggregate Index

-15%

-10%

-5%

0%

5%

10%

15%

-15% -10% -5% 0% 5% 10% 15%

Lehman Global Aggregate Index Excess Monthly Returns

HFR

Fun

d of

Fun

ds In

dex

Exce

ss M

onth

ly R

etur

ns

August 1998

August 1998

167

Figure 5.3 Lehman Global Aggregate Index vs. MSCI World Equity index

-15%

-10%

-5%

0%

5%

10%

15%

-15% -10% -5% 0% 5% 10% 15%


Lehm

an G

loba

l Agg

rega

te In

dex

Exce

ss M

onth

ly R

etur

ns

5.5.3 Extreme Dependence Effects in M-CVaR Portfolio Selection The MVA and M-CVaR results in this chapter reveal two puzzling findings. First, the

results show that global hedge funds provide some benefits when portfolio CVaR is

constrained at the 95 and 97.5 per cent confidence levels, however, hedge funds are

undesirable for investors with heightened tail-risk aversion at the 99 per cent CVaR

region. Why is this so? The second puzzling finding is the desirability of hedge funds

for investors who wish to generate conservative portfolio returns. This section provides

a simple and logical rationale to explain these results.

The 99 per cent M-CVaR portfolio selection effectively determines optimal portfolio

choice by selecting assets which meet the required rate of returns and by constraining

portfolio losses at the 99 per cent CVaR level. This dataset consists of 144 monthly

observations therefore the data employed to constrain portfolio risk at the 99 per cent

CVaR level is effectively determined by the single worst observation of each asset

August 1998

168

class.60 We find that global stocks and hedge fund returns possess some form of

extremal dependence structure in the extreme left-tail of the joint distribution. This

dependence between stocks and hedge funds is captured in the mean-CVaR (99 per cent)

portfolio framework which results in a pronounced reduction in the allocation to hedge

funds. Figures 5.1 to 5.3 are presented which illustrate the bivariate relationships

between the three asset classes employed in this study.

Figures 5.1 to 5.3 present the scatterplots of the 144 monthly excess returns between

global stocks, bonds and hedge funds. As the mean-CVaR portfolio selection process

selects asset classes based on required rates of portfolio returns and the minimisation of

the extreme left-tail of portfolio returns (as defined by CVaR), one must carefully

observe the location of negative outliers in these graphs.

The scatterplot in Figure 5.1 shows that the dependence for global stocks and hedge fund

returns is persistent for both positive and negative extreme returns. The bottom left

quadrant of Figure 5.1 reveals that when stocks suffered their worst monthly excess

return of -14.70 per cent, hedge funds also generated their worst monthly excess return

of -8.19 per cent. This striking datapoint occurs in the month of August 1998. Based on

the work of Poon et. al., (2004), this extreme observation suggests that stocks and hedge

fund returns possess some form of asymptotic dependence during rare events.

60 An entire field of study known as Extreme Value Theory (EVT) exists which is interested in the modelling and estimation of the tails of both univariate and bivariate distributions. Refer to Embrechts, McNeil and Straumann (1999), Jansen, Koedijk and de Vries (2000) and Poon et. al.,(2004) for seminal works in this area of research in the financial risk management literature. EVT is not employed in this study for a number of reasons. First, this study consists of a finite sample of only 144 monthly observations. Studies such as Diebold, Schuermann and Stroughair (1998) and Poon et. al., (2004) acknowledge that EVT requires large datasets for accurate estimation. Second, researchers including Diebold, Schuermann and Stroughair (1998) argue that EVT is valid only if returns are IID. The descriptive statistics in this study clearly illustrate that the asset classes in this study do not meet the IID condition. Third, ambiguity exists in EVT in terms of the parameters to be estimated to determine accurate tail index measures. Given the reasons mentioned above, we do not proceed with the field of EVT, however, it does seem to be a promising area for future research when more hedge fund data becomes available.

169

In contrast, Figure 5.3 reports the excess returns between stocks and bonds. A striking

feature of Figure 5.3 is the lack of dependence in the returns between these two asset

classes. Of more economic importance, Figure 5.3 reveals that when stocks reported

their worst monthly return, bonds generated a positive excess return for that month.

The point to be emphasised from Figures 5.1 to 5.3 is that the tail dependence at the 99

per cent confidence level is based on a single observation between these three asset

classes. The rare observations in Figures 5.1 and 5.3 explain the M-CVaR results in this

study. These rare outliers suggest that investors who constrain portfolio M-CVaR at the

99 per cent confidence level will prefer a portfolio combination of stocks and bonds

rather than a portfolio of stocks and hedge funds.61

To explain the desirability of hedge funds for investors who require conservative rates of

return, we turn our attention to Figure 5.2. The scatterplot between bonds and hedge

funds in Figure 5.2 reveals a close and compact relationship between these two asset

classes. Figure 5.2 also shows that when hedge funds recorded their worst monthly

returns, bonds reported consistent positive returns. Furthermore, when bonds reported

their worst monthly excess return of -2.18 per cent in July 2003, hedge funds returned

+0.16 per cent. Given this inverse relationship during extreme negative months in

bonds, M-CVaR investors prefer bond-hedge fund combinations rather than bond-stock

portfolio compositions when pursuing conservative rates of return.

61 Bacmann and Gawron (2005) also observe the extreme tail-behaviour in August 1998, however, they do not find the association between stock and hedge fund returns.

170

5.6 Conclusion

This chapter has continued the theme of this thesis by investigating the empirical

features of asset returns in financial economics. Whereas previous chapters have

examined the impact of heteroscedasticity and serial correlation on linearity-in-the-mean

tests, this chapter has focused on the effects in portfolio selection when hedge funds are

included in the investment opportunity set.

It has been shown that serial correlation bias affects portfolio selection. The empirical

data series employed in this study reveals that serial correlation bias in bond and hedge

fund returns cause an under-estimation in the second sample moment which happens to

be an integral element in portfolio selection modelling. We discover that serial

correlation bias in MVA and M-CVaR causes investors to over-allocate their portfolios

to hedge funds.

The second objective of this chapter considered the effects of tail-risk on optimal

portfolio choice. We find that MVA investors who minimise portfolio volatility do so at

the cost of decreasing third moments and increasing fourth moments. In contrast, M-

CVaR investors have a demand for hedge funds when constraining the left tail of the

distribution of portfolio returns. However, M-CVaR investors exhibit a decreasing

demand for hedge funds as their tail-risk aversion rises. The findings reveal that this

effect is due to extreme tail dependence between stock and hedge fund returns.

In summary, this study contributes to the literature by demonstrating that naïve portfolio

selection can mask some of the inherent risks in hedge fund returns. This chapter

provides investors with a framework to uncover the risks of hedge funds in portfolio

selection by accounting for serial correlation bias and by capturing the left-tail behaviour

of hedge fund returns in a M-CVaR portfolio framework.

171

The outcomes from this study provide a number of avenues for future research. Whilst

this study reveals a decrease in the demand for hedge funds in unconditional MVA and

M-CVaR portfolio choice when serial correlation and tail-risk are taken into account, it

is worthwhile to consider the same question in a conditional portfolio choice framework.

Second, the methodologies to measure serial correlation bias can be easily amended to

examine these biases in other finance frameworks including asset pricing and options.

We leave these worthwhile and thought provoking questions for future research.

172

6. Conclusion

6.1 Introduction

This thesis has examined the behaviour of hedge fund returns and their interaction with

traditional asset classes. The first empirical study considered whether traditional asset

returns exhibit a conditional linear mean in both univariate and bivariate settings. The

second empirical chapter then examined the linear behaviour between traditional assets

and hedge funds. The third and final study investigated the interaction between

traditional assets and hedge fund returns in a portfolio selection framework. Throughout

these three empirical studies, we focused on the empirical effects of hedge fund returns

and the way they influence the results in a linearity-in-the-mean setting and portfolio

selection framework. More specifically, the encompassing theme throughout this body

of work considered the empirical features of heteroscedasticity, autocorrelation and tail-

risk in hedge fund returns. A set of tools are developed in this thesis which allows

investors to control these empirical effects in linearity-in-the-mean testing and in

portfolio selection.

The findings in the first two empirical chapters demonstrated that conventional linearity

tests tend to over-reject the null hypothesis of linearity-in-the-mean. The cause of this

effect is attributable to the presence of heteroscedasticity and autocorrelation in the error

disturbances of the underlying hypothesis tests. As a contribution to the hedge fund

literature, a set of augmented tests were proposed to control for these effects and it was

shown that traditional asset classes and hedge funds are indeed linear-in-the-mean in the

monthly return frequency. The discovery that traditional assets and hedge fund returns

are linear-in-the-mean provides empirical support for mean-variance investors who

employ the covariance matrix to describe the linear relationship between asset returns.

173

Having addressed the concept of hedge fund linearity, the third empirical chapter

considered the interaction between traditional assets and hedge funds in a portfolio

selection setting. The study provided corroborating evidence which showed that mean-

variance investors tend to allocate significant portfolio weightings to hedge funds

resulting in the reduction in the overall volatility of portfolio returns. However, the

benefit of hedge funds to mean-variance investors comes at a cost of undesirable third

and fourth moments.

To more accurately estimate portfolio selection, the study examined the impact of the

empirical feature of serial correlation in hedge fund returns. To contribute to the

literature, it was shown that serial correlation tends to induce a downward bias in the

sample second moment thereby under-estimating the risk of these assets in a portfolio

selection framework. This serial correlation effect was found in global bonds but was

especially pronounced in global hedge fund returns. We employed two techniques to

account for this serial correlation bias and estimated the over-allocation to global hedge

funds in a MVA at approximately 2 to 20 per cent.

Finally, the third empirical chapter considered the impact of heteroscedasticity of hedge

fund returns in portfolio selection. Whilst the impact of heteroscedasticity cannot be

directed measured in unconditional portfolio selection, we revealed that

heteroscedasticity is a major source of tail-risk. To examine the empirical effects of tail-

risk, we evaluated the portfolio decisions of a mean-Conditional Value at Risk (M-

CVaR) investor. The contribution of this study showed that investors who wish to

constrain tail-risk will have a reduced demand for hedge fund investments in comparison

to MVA investors. Furthermore, as tail-risk aversion increases, the demand for hedge

fund investments was found to decrease to zero. The findings also revealed an

association of extreme left tail-behaviour between global stocks and hedge fund returns.

It is the association of this rare event which causes the decrease in the demand for hedge

funds in the M-CVaR framework.

174

6.2 Relevance

The study of the behaviour of traditional assets and hedge funds in a linearity setting and

in a portfolio selection framework is of particular relevance to long-term investors such

as domestic superannuation funds and global pension funds. The findings in this thesis

provide new insights which are relevant to both theoretical and empirical researchers

also.

The finding that traditional assets and hedge funds are linear-in-the-mean has positive

empirical implications for mean-variance investors. This result provides positive support

for the continued use of expected mean returns and covariance matrices as valid input

parameters to portfolio selection. Furthermore, it questions the emerging literature which

claims that hedge fund returns are non-linear. In short, the findings from this thesis

argue that the general claim of hedge fund non-linearity may be contentious and pre-

mature.

An important implication from the portfolio selection study is that an investor’s risk

preference can dramatically alter the allocation to hedge funds. Investors who wish to

minimise the volatility of portfolio returns are motivated to allocate a substantial

proportion of their investment portfolio to hedge funds. Investors with tail-risk

preferences will do the same, with the exception of agents who exhibit a high aversion to

tail-risk. Investors with an acute aversion to tail-risk will reduce their hedge fund

allocations to zero. These findings have negative implications for investors who are

exposed to hedge fund allocations during rare and unexpected events in global financial

markets.

175

6.3 Research Contributions

This thesis makes a number of contributions to the existing hedge fund literature which

are enumerated in Section 1.4. The findings to emerge from this thesis provide the

following research contributions to the literature:

(i) Conventional linearity-in-the-mean tests between global stocks and world

bonds detect spurious non-linearity caused by heteroscedasticity and

autocorrelation effects in the error disturbances of the underlying hypothesis

tests. The thesis proposes a HAC approach to augment these tests so that the

effects of heteroscedasticity and autocorrelation are controlled. The results of

the augmented tests show that the behaviour between global stocks and world

bonds is linear-in-the-mean in the monthly return frequency in both

univariate and bivariate settings. This empirical finding is good news for

mean-variance investors employing expected returns and covariance matrices

as input parameters in portfolio selection frameworks.

(ii) Furthermore, the behaviour between traditional assets and hedge funds

results in the over-rejection of the null hypothesis of linearity-in-the-mean.

Again, this is due to the presence of heteroscedasticity and autocorrelation in

the error disturbances of these tests. The findings from the augmented tests

reveal that hedge funds are indeed linear-in-the-mean with traditional asset

classes in both univariate and bivariate settings. This finding differs from

previous hedge fund studies as researchers do not explicitly control both

heteroscedasticity and autocorrelation in their research methodologies. The

contribution of this research suggests that hedge fund non-linearity is

statistically insignificant and therefore may not be an issue when employing

hedge funds in a portfolio selection framework.

176

(iii) In the portfolio study (Chapter 5), we examined how autocorrelation of asset

returns affects portfolio selection. The study demonstrated that asset returns

with serial correlation exhibit a downward bias in the second sample

moment, thus, making them less risky than they seem. When the adjusted

second sample moments are employed in a MVA, the findings revealed that

investors over-allocate to assets that possess serial correlation in returns.

Furthermore, we find that the serial correlation bias tends to cause a severe

over-allocation to hedge funds.

(iv) The portfolio study also examined the sensitivity of the portfolio

compositions between MVA and M-CVaR investors. It was shown that

investors who wish to minimise portfolio return variance will tend to have a

high demand for hedge funds, but it comes at the cost of less desirable third

and fourth moments. When M-CVaR is evaluated, we find that hedge funds

are also found to be desirable for investors who wish to minimise the size of

the left-tail of the distribution of portfolio returns. However, the study

provides evidence to suggest that as tail-risk aversion increases, the

allocation to hedge funds decrease.

6.4 Policy Implications

The key policy implications from this study relate to the inclusion of hedge funds in a

portfolio selection context. More specifically, the findings from the thesis have

important implications for long-term investors including pension funds and

superannuation trustees. The study provides evidence to suggest that hedge funds are a

desirable asset class in portfolio construction, however, they expose the investor to

unnecessary tail-risk during unexpected rare events in global financial markets. The

evidence presented in this study informs both finance regulators and investors that hedge

funds are an attractive asset class, however, these benefits of minimising portfolio

variance or tail-risk comes at the cost of less desirable third and fourth moments and

adverse tail behaviour during extreme rare events.

177

6.5 Avenues for Future Research

The findings from these studies and the limitations of the thesis provide a number of

avenues for future research. The thesis has focused exclusively on linearity-in-the-mean

tests and portfolio selection decisions employing monthly returns. It is a worthwhile

research question to consider linearity-in-the-mean tests which control both

heteroscedasticity and autocorrelation on daily returns. Whilst this type of research

question is outside the framework of a mean-variance investor, it would more closely

resemble the previous contributions in the econometrics and microstructure literature.

A second research area for future consideration would be to explore the effects of

heteroscedasticity in a portfolio selection framework. In an unconditional portfolio

selection framework, this thesis was unable to directly measure the effects of

heteroscedasticity on optimal portfolio choice. However, it may be possible to examine

the effects of heteroscedasticity in a conditional (ie. time-varying) portfolio choice

framework.

The third and final area for future research would be the empirical implications of the

serial correlation effects on the second sample moment. Whilst this thesis demonstrates

the impact of the second sample moment in portfolio selection, it may be fruitful to

examine its implications in other areas of finance including asset pricing and option

pricing frameworks. The estimation of stock variance and covariances may be under-

estimated for securities with serial correlation in returns. Furthermore, option prices on

assets with serial correlation may be systematically mis-priced if researchers have not

addressed the ex-post serial correlation biases in their a priori estimation of volatility.

178

6.6 Conclusion

In summary, the findings of this thesis are important as they contribute to our empirical

understanding of the behaviour of traditional assets and hedge fund returns. This thesis

has presented new empirical evidence to suggest that global stocks, bonds and hedge

funds are linear-in-the-mean. Whilst this finding contradicts previous studies, we

demonstrate that the empirical characteristics of heteroscedasticity and autocorrelation

contaminate conventional hypothesis tests. By controlling these effects in a set of

augmented tests, this thesis provides new empirical evidence to demonstrate that

traditional assets and hedge funds are indeed linear-in-the-mean in monthly returns.

In the portfolio context, it is shown that the empirical characteristic of autocorrelation

contaminates portfolio selection estimates. This thesis demonstrates that investors under-

estimate the risk of an asset in the presence of statistically significant autocorrelation in

returns. By employing two methods to adjust the serial correlation bias in the volatility

of asset returns, it is shown that investors tend to over-allocate their portfolio weightings

to hedge funds.

The second part of the portfolio study examined the effect of tail-risk. A comparison of

MVA and M-CVaR investors shows that MVA investors have a strong preference for

hedge funds as they lower the volatility of portfolio returns at the cost of undesirable

third and fourth moments. Conversely, M-CVaR investors who wish to minimise the

size of the left tail of the distribution of returns have a decreasing demand for hedge

funds. Furthermore, M-CVaR investors with a heightened aversion to tail-risk will avoid

hedge funds altogether.

In closing, this thesis contributes a number of new findings to the current body of

scholarly work. The implementation of linearity-in-the-mean testing and portfolio

selection must be carefully considered in order to account for the empirical features of

asset returns. As outlined in this thesis, it is important for researchers to develop new

statistical and research methods when the empirical characteristics of asset returns do

179

not adhere to the assumptions of normality. If not, serious implications confront

researchers and investors who ignore the empirical features of heteroscedasticity,

autocorrelation and tail-risk in global stock, bond and hedge fund returns.

180

Appendix A

Credit Suisse/Tremont List of Hedge Fund Investment Styles

1. Credit Suisse/Tremont Hedge Fund Index

Credit Suisse/Tremont Hedge Fund Index is compiled by Credit Suisse Tremont Index

LLC. It is an asset-weighted hedge fund index and includes only funds, as opposed to

separate accounts. The Index uses the Credit Suisse/Tremont database, which track over

4500 funds, and consists only of funds with a minimum of US$50 million under

management, a 12-month track record, and audited financial statements. It is calculated

and rebalanced on a monthly basis, and shown net of all performance fees and expenses.

2. Credit Suisse/Tremont Convertible Arbitrage Index

Convertible Arbitrage managers seek to profit from investments in convertible securities

employing both single security and portfolio hedging strategies. Managers typically

build long positions of convertible and other equity hybrid securities and then hedge the

equity component of the long securities positions by shorting the underlying stock or

options of that company. Interest rate, volatility and credit hedges may also be

employed. Hedge ratios need to be adjusted as markets move and positions are typically

designed with the objective of creating profit opportunities irrespective of market

moves.

3. Credit Suisse/Tremont Dedicated Short Bias Index

Dedicated Short Bias managers seek to profit from maintaining overall net short

portfolios of long and short equities. Detailed individual company research typically

forms the core alpha generation driver of short bias managers, and a focus on companies

with weak cash flow generation is common. Risk management consists of offsetting

long positions and stop-loss strategies. The fact that money losing short positions grow

in size for a short bias manager makes risk management challenging.

181

4. Credit Suisse/Tremont Emerging Markets Index

Emerging Markets managers seek to profit from investments in currencies, debt

instruments, equities and other instruments of “emerging” markets countries (typically

measured by GDP per capita). Emerging Markets include countries in Latin America,

Eastern Europe, Africa, and Asia. There are a number of sub-sectors, including

arbitrage, credit and event driven, fixed income bias, and equity bias.

5. Credit Suisse/Tremont Equity Market Neutral Index

Equity Market Neutral managers seek to profit from exploiting pricing relationships

between different equities or related securities while typically hedging exposure to

overall equity market moves. There are a number of sub-sectors including statistical

arbitrage, quantitative long/short, fundamental long/short and index arbitrage. Managers

often apply leverage to enhance returns.

6. Credit Suisse/Tremont Event Driven Index

Event Driven managers seek to profit from the potential mispricing of corporate

securities. There is a wide range of sub-sectors within the Event Driven sector with a

common theme of corporate activity or creditworthiness. Sub-sectors include mergers

and acquisitions; special situations equity trading, distressed investing and credit

oriented trading. Many managers use a combination of strategies; adjusting exposures

based upon the opportunity sets in each sub-sector.

Risk (Merger) Arbitrage Specialists invest simultaneously long and short in the

companies involved in a merger or acquisition. Risk arbitrageurs are typically long the

stock of the company being acquired and short the stock of the acquirer. By shorting the

stock of the acquirer, the manager hedges out market risk, and isolates his exposure to

the outcome of the announced deal. In cash deals, the manager needs only long the

acquired company. The principal risk is deal risk, should the deal fail to close. Risk

arbitrageurs also often invest in equity restructurings such as spin-offs or ‘stub trades’.

182

Distressed/High Yield Securities Fund managers in this non-traditional strategy invest in

the debt, equity or trade claims of companies in financial distress or already in default.

The securities of companies in distressed or defaulted situations typically trade at

substantial discounts to par value due to difficulties in analyzing a proper value for such

securities, lack of street coverage, or simply an inability on behalf of traditional

investors to accurately value such claims or direct their legal interests during

restructuring proceedings. Various strategies have been developed by which investors

may take hedged or outright short positions in such claims, although this asset class is in

general a long-only strategy.

Regulation D, or Reg. D This sub-set refers to investments in micro and small

capitalization public companies that are raising money in private capital markets.

Investments usually take the form of a convertible security with an exercise price that

floats or is subject to a look-back provision that insulates the investor from a decline in

the price of the underlying stock.

(6a) Credit Suisse/Tremont Distressed Index

Distressed/High Yield Securities Fund managers in this non-traditional strategy

invest in the debt, equity or trade claims of companies in financial distress or

already in default. The securities of companies in distressed or defaulted

situations typically trade at substantial discounts to par value due to difficulties

in analyzing a proper value for such securities, lack of street coverage, or simply

an inability on behalf of traditional investors to accurately value such claims or

direct their legal interests during restructuring proceedings. Various strategies

have been developed by which investors may take hedged or outright short

positions in such claims, although this asset class is in general a long-only

strategy.

183

(6b) Credit Suisse/Tremont Event Driven Multi-Strategy Index

This subset refers to hedge funds that draw upon multiple themes, including risk

arbitrage, distressed securities, and occasionally others such as investments in

micro and small capitalization public companies that are raising money in private

capital markets. Fund managers often shift assets between strategies in response

to market opportunities

(6c) Credit Suisse/Tremont Risk Arbitrage Index

Specialists invest simultaneously in long and short positions in both companies

involved in a merger or acquisition. Risk arbitrageurs are typically long the stock

of the company being acquired and short the stock of the acquiring company.

The principal risk is deal risk, should the deal fail to close.

7. Credit Suisse/Tremont Fixed Income Arbitrage Index

Fixed Income Arbitrage managers seek to profit from relationships between different

fixed income securities; leveraging long and short positions in securities that are related

either mathematically or economically. Many managers trade globally with a goal of

generating steady returns with low volatility. The sector includes yield curve relative

value trading involving interest rate swaps, government securities and futures; volatility

trading involving options; and mortgage-backed securities arbitrage (the mortgage-

backed market is primarily US-based, over-the-counter, and particularly complex).

8. Credit Suisse/Tremont Global Macro Index

Global Macro managers seek to profit from long and short positions in any of the

world’s major capital markets (fixed income, currency, equity, commodity). Managers

typically consider both economic adjustment themes as well as shorter-term technical

conditions when choosing trading positions that anticipate market movements. Managers

often employ a “top-down” global approach and may invest in multiple markets in

anticipation of expected market movements. These movements may result from

forecasted shifts in world economies, political changes or global supply and demand

184

imbalances. Many Global Macro managers primarily trade in more liquid instruments in

order to keep their trading activities flexible.

9. Credit Suisse/Tremont Long/Short Equity Index

Long/Short Equity managers seek to profit from investing on both the long and short

sides of equity markets. Managers have the ability to shift from value to growth, from

small to medium to large capitalization stocks, and from net long to net short. Managers

can change their exposures from net long to net short or market neutral at times. In

addition to equities, long/short managers can trade equity futures and options as well as

equity related securities and debt. Manager focus may be global, regional, or sector

specific, such as technology, healthcare or financials. Managers tend to build portfolios

that are more concentrated than traditional long-only equity funds.

10. Credit Suisse/Tremont Managed Futures Index

Managed Futures managers seek to profit from investments in listed bond, currency,

equity and commodity futures markets globally. Also referred to as Commodity Trading

Advisors (CTA), these managers tend to follow model based systematic trading

programs that largely rely upon historical price data. The most common trading

programs are long-term trend following ones that tend to invest with directional trends

while using stop-loss points to control risk. Other common programs include short-term

counter trend and hybrid systematic/discretionary programs.

11. Credit Suisse/Tremont Multi-Strategy Index

Multi-Strategy managers seek to profit from allocating to a number of different

strategies and adjusting their allocations based upon perceived opportunities. Many

Multi-Strategy managers began as convertible arbitrage managers that diversified into

other strategies. Because each strategy is not in a separate fund, these managers often

have the ability to run higher leverage levels than single strategy managers.

Source: Credit Suisse Tremont documentation at www.hedgeindex.com

185

Appendix B

Hennessee List of Hedge Fund Investment Styles

1. Asia-Pacific

Funds typically have long and short equity positions in companies located in the Pacific

Basin region (i.e. Japan, China, Hong Kong, Taiwan, Korea, Singapore, Thailand,

Malaysia, India, Australia, New Zealand, and other countries in Asia.)

2. Convertible Arbitrage

This type of arbitrage involves the simultaneous purchase of a convertible bond and the

short sale of the underlying stock. Interest rate risk may or may not be hedged.

3. Distressed

Primary investment focus involves securities of companies that have declared

bankruptcy and/or may be undergoing reorganization. Investment holdings range from

senior secured debt (uppermost tier of a company's capital structure) to the common

stock of the company (lower tier of the capital structure).

4. Emerging Markets

This strategy focuses on investing in lesser-developed, non-G7 countries whose financial

markets provide exploitable pricing inefficiencies. Popular geographic regions include

Latin America, Eastern Europe, the Pacific Rim and Africa. Asset classes range from

equities and bonds to local currencies.

5. Europe

Funds typically have long and short equity positions in European companies located in

the United Kingdom, Western Europe, and Eastern Europe.

186

6. Event Driven

This strategy can include merger arbitrage, distressed, liquidations, and spin-offs in

addition to value driven special situation equity investing. Investments are usually

dependent on an "event" as the catalyst to release the position's intrinsic value.

7. Financial Equities

Funds typically have long and short equity positions within the financial sector (banks,

thrifts, brokerage, insurance, etc.)

8. Fixed Income

Funds typically employ a variety of fixed income related strategies ranging from relative

value based trades (basis, TEDs, yield curve, etc.) to directional bets on interest rate

shifts. Style also includes credit related arbitrage, which typically involves the

purchasing (or selling) of corporate issues and the simultaneous selling (or purchasing)

of government issues.

9. Growth

Funds typically have long and short equity positions in companies that exhibit an

acceleration (or deceleration) of earnings growth, revenues, and market share.

10. Healthcare/Biotech

Funds typically have long and short equity positions in medical related stocks, which

include biotechnology, pharmaceuticals, HMO's, medical devices, etc.

11. High Yield

Funds typically have long and short equity positions in non-investment grade corporate

bonds, which offer attractive coupon yields. Interest rate risk may or may not be hedged.

187

12. International

Funds typically have long and short equity positions in the stocks of international

companies. Positions can be either growth or value and, in addition to global

investments, funds typically have exposure to U.S. companies.

13. Latin America

Funds typically have long and short equity and/or debt positions in companies located in

Latin American countries such as Chile, Mexico, Venezuela, Argentina, Brazil, and

Ecuador.

14. Macro

Dominant investment theme is to capitalize on changes in the global macroeconomic

environment through participation in the various capital markets. A top-down

methodology allows managers of this strategy to utilize all asset classes (equities, bonds,

currencies, derivatives) available in the global capital markets.

15. Market Neutral

Funds typically have long and short equity positions with approximately zero net dollar

exposure. In addition, some funds will attempt to be beta, sector, and market cap neutral

to further reduce equity market risk. Funds within this style utilize a range of methods

from quantitative modeling to fundamental pairs trading.

16. Merger Arbitrage

Style typically involves the simultaneous purchase of stock in a company being acquired

and the short sale of stock in the respective acquirer. Many merger arbitrage managers

attempt to mitigate deal risk by engaging only in strategic takeovers after they have been

announced.

188

17. Multiple Arbitrage

Style includes funds that employ more than one arbitrage strategy. The portfolio

manager opportunistically allocates capital among the various strategies in an attempt to

create the best risk/reward profile for the overall fund. Common strategies include

merger arbitrage, convertible arbitrage, fixed income arbitrage, long/short equity pairs

trading, quantitative equity trading, volatility arbitrage, and distressed investments.

18. Opportunistic

Funds typically have long and short equity positions while maintaining a flexible net

exposure to reflect the changing dynamics of the market on a minute-to-minute or day-

to-day basis. Investments can be initiated from technical and/or fundamental analysis

and portfolio turnover is typically high as managers have a short term investment time

horizon.

19. Pipes/Private Financing

PIPEs (private investments in public entities) are transactions by which publicly traded

companies access new capital through the sale of stock directly to private investors.

PIPEs can be transacted with a number of financial instruments, including the issuance

of common stock, convertible securities, or warrants. Private financing includes asset

based lending/acquisitions and direct loan investing such as mezzanine financing, bridge

loans, and debtor in possession financing.

20. Short Bias

Funds typically have long and short equity positions with an overall net short exposure

to the market. Investments can be fundamental, technical, or event driven. This style can

be used as a hedge against long-only portfolios and by investors who feel the market is

approaching or in a bearish cycle.

21. Technology

Funds typically have long and short equity positions in technology-related sectors such

as semiconductors, hardware, software, networking devices, etc.

189

22. Telecom/Media

Funds typically have long and short equity positions in the telecommunication and

media sectors such as telecommunication services, fiber optics, cable services,

publishing, entertainment, programming, broadcasting, etc.

23. Value

Funds typically have long and short equity positions in undervalued companies which

trade below their intrinsic value. Undervalued securities may be defined as, but not

limited to, equities with low price-to-earnings ratios or low price-to-book value ratios.

Managers also focus on companies that generate substantial free cash flow and utilize

cash for debt retirement, share repurchase programs, and other methods utilized to

realize shareholder value.

Source: www.hennesseegroup.com

190

Appendix C

Hedge Fund Research, Inc. List of Hedge Fund Investment Styles

1. Convertible Arbitrage involves purchasing a portfolio of convertible securities,

generally convertible bonds, and hedging a portion of the equity risk by selling short the

underlying common stock. Certain managers may also seek to hedge interest rate

exposure under some circumstances. Most managers employ some degree of leverage,

ranging from zero to 6:1. The equity hedge ratio may range from 30 to 100 percent. The

average grade of bond in a typical portfolio is BB-, with individual ratings ranging from

AA to CCC. However, as the default risk of the company is hedged by shorting the

underlying common stock, the risk is considerably better than the rating of the unhedged

bond indicates.

2. Distressed Securities strategies invest in, and may sell short, the securities of

companies where the security’s price has been, or is expected to be, affected by a

distressed situation. This may involve reorganizations, bankruptcies, distressed sales and

other corporate restructurings. Depending on the manager’s style, investments may be

made in bank debt, corporate debt, trade claims, common stock, preferred stock and

warrants. Strategies may be subcategorized as “high-yield” or “orphan equities.”

Leverage may be used by some managers. Fund managers may run

a market hedge using S&P put options or put options spreads.

3. Emerging Markets funds invest in securities of companies or the sovereign debt of

developing or “emerging” countries. Investments are primarily long. “Emerging

Markets” include countries in Latin America, Eastern Europe, the former Soviet Union,

Africa and parts of Asia. Emerging Markets - Global funds will shift their weightings

among these regions according to market conditions and manager perspectives. In

addition, some managers invest solely in individual regions. Emerging Markets - Asia

involves investing in the emerging markets of Asia. Emerging Markets - Eastern

191

Europe/CIS funds concentrate their investment activities in the nations of Eastern

Europe and the CIS (the former Soviet Union). Emerging Markets - Latin America is a

strategy that entails investing throughout Central and South America.

4. Equity Hedge investing consists of a core holding of long equities hedged at all times

with short sales of stocks and/or stock index options. Some managers maintain a

substantial portion of assets within a hedged structure and commonly employ leverage.

Where short sales are used, hedged assets may be comprised of an equal dollar value of

long and short stock positions. Other variations use short sales unrelated to long

holdings and/or puts on the S&P 500 index and put spreads. Conservative funds mitigate

market risk by maintaining market exposure from zero to

100 percent. Aggressive funds may magnify market risk by exceeding 100 percent

exposure and, in some instances, maintain a short exposure. In addition to equities, some

funds may have limited assets invested in other types of securities.

5. Equity Market Neutral investing seeks to profit by exploiting pricing inefficiencies

between related equity securities, neutralizing exposure to market risk by combining

long and short positions. One example of this strategy is to build portfolios made up of

long positions in the strongest companies in several industries and taking corresponding

short positions in those showing signs of weakness.

6. Equity Market Neutral: Statistical Arbitrage utilizes quantitative analysis of

technical factors to exploit pricing inefficiencies between related equity securities,

neutralizing exposure to market risk by combining long and short positions. The strategy

is based on quantitative models for selecting specific stocks with equal dollar amounts

comprising the long and short sides of the portfolio. Portfolios are typically structured to

be market, industry, sector, and dollar neutral.

7. Equity Non-Hedge funds are predominately long equities although they have the

ability to hedge with short sales of stocks and/or stock index options. These funds are

commonly known as “stock-pickers.” Some funds employ leverage to enhance returns.

192

When market conditions warrant, managers may implement a hedge in the portfolio.

Funds may also opportunistically short individual stocks. The important distinction

between equity non-hedge funds and equity hedge funds is equity non-hedge funds do

not always have a hedge in place. In addition to equities, some funds may have limited

assets invested in other types of securities.

8. Event-Driven is also known as “corporate life cycle” investing. This involves

investing in opportunities created by significant transactional events, such as spin-offs,

mergers and acquisitions, bankruptcy reorganizations, recapitalizations and share

buybacks. The portfolio of some Event-Driven managers may shift in majority

weighting between Risk Arbitrage and Distressed Securities, while others may take a

broader scope. Instruments include long and short common and preferred stocks, as well

as debt securities and options. Leverage may be used by some managers. Fund managers

may hedge against market risk by purchasing S&P put options or put option spreads.

9. Fixed Income: Arbitrage is a market neutral hedging strategy that seeks to profit by

exploiting pricing inefficiencies between related fixed income securities while

neutralizing exposure to interest rate risk. Fixed Income Arbitrage is a generic

description of a variety of strategies involving investment in fixed income instruments,

and weighted in an attempt to eliminate or reduce exposure to changes in the yield

curve. Managers attempt to exploit relative mispricing between related sets of fixed

income securities. The generic types of fixed income hedging trades include: yield-curve

arbitrage, corporate versus Treasury yield spreads, municipal bond versus Treasury yield

spreads and cash versus futures.

10. Fixed Income: Convertible Bonds funds are primarily long only convertible bonds.

Convertible bonds have both fixed income and equity characteristics. If the underlying

common stock appreciates, the convertible bond’s value should rise to reflect this

increased value. Downside protection is offered because if the underlying common stock

declines, the convertible bond’s value can decline only to the point where it behaves like

a straight bond.

193

11. Fixed Income: Diversified funds may invest in a variety of fixed income strategies.

While many invest in multiple strategies, others may focus on a single strategy less

followed by most fixed income hedge funds. Areas of focus include municipal bonds,

corporate bonds, and global fixed income securities.

12. Fixed Income: High-Yield managers invest in non-investment grade debt.

Objectives may range from high current income to acquisition of undervalued

instruments. Emphasis is placed on assessing credit risk of the issuer. Some of the

available high-yield instruments include extendible/reset securities, increasing-rate

notes, pay-in-kind securities, step-up coupon securities, split-coupon securities and

usable bonds.

13. Fixed Income: Mortgage-Backed funds invest in mortgage-backed securities.

Many funds focus solely on AAA rated bonds. Instruments include: government agency,

government-sponsored enterprise, private-label fixed- or adjustable-rate mortgage pass-

through securities, fixed- or adjustable-rate collateralized mortgage obligations (CMOs),

real estate mortgage investment conduits (REMICs) and stripped mortgage-backed

securities (SMBSs). Funds may look to capitalize on security-specific mispricings.

Hedging of prepayment risk and interest rate risk is common. Leverage may be used, as

well as futures, short sales and options.

14. Macro involves investing by making leveraged bets on anticipated price movements

of stock markets, interest rates, foreign exchange and physical commodities. Macro

managers employ a “top-down” global approach, and may invest in any markets using

any instruments to participate in expected market movements. These movements may

result from forecasted shifts in world economies, political fortunes or global supply and

demand for resources, both physical and financial. Exchange-traded and over-the-

counter derivatives are often used to magnify these price movements.

194

15. Market Timing involves allocating assets among investments by switching into

investments that appear to be beginning an uptrend, and switching out of investments

that appear to be starting a downtrend. This primarily consists of switching between

mutual funds and money markets. Typically, technical trend-following indicators are

used to determine the direction of a fund and identify buy and sell signals. In an up

move “buy signal,” money is transferred from a money market fund into a mutual fund

in an attempt to capture a capital gain. In a down move “sell signal,” the assets in the

mutual fund are sold and moved back into the money market for safe keeping until the

next up move. The goal is to avoid being invested in mutual funds during a market

decline.

16. Merger Arbitrage, sometimes called Risk Arbitrage, involves investment in event-

driven situations such as leveraged buy-outs, mergers and hostile takeovers. Normally,

the stock of an acquisition target appreciates while the acquiring company’s stock

decreases in value. These strategies generate returns by purchasing stock of the company

being acquired, and in some instances, selling short the stock of the acquiring company.

Managers may employ the use of equity options as a low-risk alternative to the outright

purchase or sale of common stock. Most Merger Arbitrage funds hedge against market

risk by purchasing S&P put options or put option spreads.

17. Regulation D Managers invest in Regulation D securities, sometimes referred to as

structured discount convertibles. The securities are privately offered to the investment

manager by companies in need of timely financing and the terms are negotiated. The

terms of any particular deal are reflective of the negotiating strength of the issuing

company. Once a deal is closed, there is a waiting period for the private share offering to

be registered with the SEC. The manager can only convert into private shares and cannot

trade them publicly during this period; therefore their investment is illiquid until it

becomes registered. Managers will hedge with common stock until the registration

becomes effective and then liquidate the position gradually.

195

18. Relative Value Arbitrage attempts to take advantage of relative pricing

discrepancies between instruments including equities, debt, options and futures.

Managers may use mathematical, fundamental, or technical analysis to determine

misvaluations. Securities may be mispriced relative to the underlying security, related

securities, groups of securities, or the overall market. Many funds use leverage and seek

opportunities globally. Arbitrage strategies include dividend arbitrage, pairs trading,

options arbitrage and yield curve trading.

19. Sector: Energy is a strategy that focuses on investment within the energy sector.

Investments can be long and short in various instruments with funds either diversified

across the entire sector or specializing within a sub-sector, i.e., oil field service.

20. Sector: Financial is a strategy that invests in securities of bank holding companies,

banks, thrifts, insurance companies, mortgage banks and various other financial services

companies.

21. Sector: Healthcare/Biotechnology funds invest in companies involved in the

healthcare, pharmaceutical, biotechnology, and medical device areas.

22. Sector: Miscellaneous funds invest in securities of companies primarily focused on

miscellaneous sectors of investments, such as precious metals (gold, silver), beverage

companies, retail stores, home improvement outlets, shipping industry, weather/climate

opportunities, or the entertainment/sports industry.

23. Sector: Real Estate involves investing in securities of real estate investment trusts

(REITs) and other real estate companies. Some funds may also invest directly in real

estate property.

24. Sector: Technology funds emphasize investment in securities of the technology

arena. Some of the sub-sectors include multimedia, networking, PC producers, retailers,

semiconductors, software, and telecommunications.

196

25. Short Selling involves the sale of a security not owned by the seller; a technique

used to take advantage of an anticipated price decline. To effect a short sale, the seller

borrows securities from a third party in order to make delivery to the purchaser. The

seller returns the borrowed securities to the lender by purchasing the securities in the

open market. If the seller can buy that stock back at a lower price, a profit results. If the

price rises, however, a loss results. A short seller must generally pledge other securities

or cash with the lender in an amount equal to the market price of the borrowed

securities. This deposit may be increased or decreased in response to changes in the

market price of the borrowed securities.

26. Fund of Funds invest with multiple managers through funds or managed accounts.

The strategy designs a diversified portfolio of managers with the objective of

significantly lowering the risk (volatility) of investing with an individual manager. The

Fund of Funds manager has discretion in choosing which strategies to invest in for the

portfolio. A manager may allocate funds to numerous managers within a single strategy,

or with numerous managers in multiple strategies. The minimum investment in a Fund

of Funds may be lower than an investment in an individual hedge fund or managed

account. The investor has the advantage of diversification among managers

and styles with significantly less capital than investing with separate managers.

27. FOF: Conservative: FOFs classified as "Conservative" exhibit one or more of the

following characteristics: seeks consistent returns by primarily investing in funds that

generally engage in more "conservative" strategies such as Equity Market Neutral, Fixed

Income Arbitrage, and Convertible Arbitrage; exhibits a lower historical annual standard

deviation than the HFRI Fund of Funds Composite Index. A fund in the HFRI FOF

Conservative Index shows generally consistent performance regardless of market

conditions.

197

28. FOF: Diversified: FOFs classified as "Diversified" exhibit one or more of the

following characteristics: invests in a variety of strategies among multiple managers;

historical annual return and/or a standard deviation generally similar to the HFRI Fund

of Fund Composite index; demonstrates generally close performance and returns

distribution correlation to the HFRI Fund of Fund Composite Index. A fund in the HFRI

FOF Diversified Index tends to show minimal loss in down markets while achieving

superior returns in up markets.

29. FOF: Market Defensive: FOFs classified as "Market Defensive" exhibit one or

more of the following characteristics: invests in funds that generally engage in short-

biased strategies such as short selling and managed futures; shows a negative correlation

to the general market benchmarks (S&P). A fund in the FOF Market defensive Index

exhibits higher returns during down markets than during up markets.

30. FOF: Strategic: FOFs classified as "Strategic" exhibit one or more of the following

characteristics: seeks superior returns by primarily investing in funds that generally

engage in more opportunistic strategies such as Emerging Markets, Sector specific, and

Equity Hedge; exhibits a greater dispersion of returns and higher volatility compared to

the HFRI Fund of Funds Composite Index. A fund in the HFRI FOF Strategic Index

tends to outperform the HFRI Fund of Fund Composite Index in up markets and

underperform the index in down markets.

Source: www.hedgefundresearch.com

198

References

Acerbi, C., 2002, Spectral measures of risk: A coherent representation of subjective risk

aversion, Journal of Banking and Finance 26, 1505-1518.

Acerbi, C. and Tasche, D., 2002, On the coherence of expected shortfall, Journal of

Banking and Finance 26, 1487-1503.

Agarwal, V. and Naik, N., 2004, Risks and portfolio decisions involving hedge funds,

Review of Financial Studies 17, 63-98.

Ahn, D., Boudoukh, J., Richardson, M. and Whitelaw, R., 2002, Partial adjustment or

stale prices? Implications from stock index and futures return autocorrelations,

Review of Financial Studies 15, 655-689.

Ait-Sahalia, Y., 1996, Testing continuous time models of the spot interest rate, Review

of Financial Studies 9, 385-426.

Alexander, G. and Baptista, A., 2002, Economic implications of using a mean-VaR

model for portfolio selection: A comparison with mean-variance analysis, Journal

of Economic Dynamics and Control 26, 1159-1193.

Alexander, S., Coleman, T. and Li, Y., 2006, Minimizing CVaR and VaR for a portfolio

of derivatives, Journal of Banking and Finance 30, 583-605.

Amin, G. and Kat, H., 2003, Stocks, bonds and hedge funds, Journal of Portfolio

Management 29, 113-120.

Anson, M., 2002, Handbook of Alternative Assets, John Wiley & Sons, New York,

U.S.A.

199

Artzner, P., Delbaen, F., Eber, J. and Heath, D., 1997, Thinking coherently, Risk 10,

November, 68-70.

Artzner, P., Delbaen, F., Eber, J. and Heath, D., 1999, Coherent measures of risk,

Mathematical Finance 9, 203-228.

Arzac, E. and Bawa, V., 1977, Portfolio choice and equilibrium in capital markets with

safety-first investors, Journal of Financial Economics 4, 277-288.

Asness, C. Krail, R. and Liew, J., 2001, Do hedge funds hedge?, Journal of Portfolio


Atchison, M., Butler, K. and Simonds, R., 1987, Nonsynchronous security trading and

market index autocorrelation, Journal of Finance 42, 111-118.

Australian Prudential Regulation Authority (APRA), 2003, APRA Alerts Super Industry

to the Drawbacks of Hedge Funds, Media Release No. 03.25, Wednesday 5th

March.

Australian Prudential Regulation Authority (APRA), 2007, Statistics: Quarterly

Superannuation Performance, December 2006, Release date: 29th March.

Bacmann, J. and Gawron, G., 2005, Fat-tail risk in portfolios of hedge funds and

Traditional Investments, in Hedge Funds: Insights in Performance Measurement,

Risk Analysis and Portfolio Allocation, by Gregoriou, G., Hubner, G.,

Papageorgious, N. and Rouah. F., John Wiley and Sons. Inc., New Jersey.

Balzer, L., 1994, Measuring investment risk: A review, Journal of Investing 3, 47-58.

200

Bank for International Settlements (BIS), 2006, BIS Quarterly Review September 2006,

Basel.

Bartlett, M., 1947, The use of transformations, Biometrics 3, 39-52.

Basak, S. and Shapiro, A., 2001, Value-at-risk based risk management: Optimal policies

and asset prices, Review of Financial Studies 14, 371-405.

Basle Committee of Banking Supervision, 1996, Amendment to the capital accord to

incorporate market risks, Basle Committee on Banking Supervision, www.bis.org.

Basle Committee of Banking Supervision, 2003, The new Basel capital accord, Basle

Committee on Banking Supervision, www.bis.org.

Baumol, W., 1963, An expected gain-confidence limit criterion for portfolio selection,

Management Science 10, 174-182.

Bawa, V., 1975, Optimal rules for ordering uncertain prospects, Journal of Financial

Economics 2, 95-121.

Bawa, V., 1978, Safety-first, stochastic dominance, and optimal portfolio choice,

Journal of Financial and Quantitative Analysis 14, 255-271.

Bawa. V. and Lindenberg, E., 1977, Capital market equilibrium in a mean, lower partial

moment framework, Journal of Financial Economics 5, 189-200.

Bernstein, P., 1996, Against The Gods, John Wiley & Sons, Inc, New York, U.S.A.

Best, M. and Grauer, R., 1991, On the sensitivity of mean-variance-efficient portfolios

to changes in asset means: Some analytical and computational results, Review of

Financial Studies 4, 315-342.

201

Bianchi, R. and Drew, M., 2006, Hedge fund biases and the survivor premium, Working

Paper, School of Economics and Finance, Queensland University of Technology,

Brisbane, Australia.

Bianchi, R., Drew, M., Veeraraghavan, M. and Whelan, P., 2006, Hedge fund style

analysis with the gap statistic, Working Paper, School of Economics and Finance,

Queensland University of Technology, Brisbane, Australia.

Black, A. and McMillan, D., 2004, Non-linear predictability of value and growth stocks

and economic activity, Journal of Business Finance and Accounting 31, 439-474.

Black, F., 1972, Capital market equilibrium with restricted borrowing, Journal of

Business 45, 444-454.

Black, F. and Litterman, R., 1992, Global portfolio optimization, Financial Analysts

Journal 48, 28-43.

Black, F. and Scholes, M., 1973, The pricing of options and corporate liabilities, Journal

of Political Economy 81, 637-654.

Blume, M., Keim, D. and Patel, S., 1991, Returns and volatility of low-grade bonds

1977-1989, Journal of Finance 46, 49-74.

Blundell, G. and Ward, C., 1987, Property portfolio allocation: a multi-factor model,

Land Development Studies 4, 145-156.

Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal

of Econometrics 31, 307-327.

202

Bollerslev, T., Chou, R. and Kroner, K., 1992, ARCH modeling in finance: A review of

the theory and empirical evidence, Journal of Econometrics 52, 5-59.

Boudoukh, J., Richardson, M. and Whitelaw, R., 1997, Nonlinearities in the relation

between the equity risk premium and the term structure, Management Science 43,

371-385.

Britten-Jones, M., 1999, The sampling error in estimates of mean-variance efficient

portfolio weights, Journal of Finance 54, 655-671.

Brock, W., Dechert, W. and Scheinkman, J., 1987, A test for independence based on the

correlation dimension, unpublished paper, University of Wisconsin at Madison,

University of Houston and University of Chicago.

Brooks, C. and Kat, H., 2002, The statistical properties of hedge fund index returns and

their implications for investors, Journal of Alternative Investments 5, 26-44.

Brown, S., 1976, Optimal portfolio choice under uncertainty: A Bayesian approach,

Ph.D Dissertation, University of Chicago.

Brown, S., 1979, The effect of estimation risk on capital market equilibrium, Journal of

Financial and Quantitative Analysis 14, 215-220.

Brown, S. and Goetzmann, W., 1997, Mutual fund styles, Journal of Financial


Brown, S. and Goetzmann, W., 2003, Hedge funds with style, Journal of Portfolio


Brown, S., Goetzmann, W. and Ibbotson, R, 1999, Offshore hedge funds: Survival and

performance, 1989-95, Journal of Business 72, 91-117.

203

Brown, S. and Spitzer, J., 2006, Caught by the tail: Tail risk neutrality and hedge fund

returns, Working Paper, 19th May, New York University.

Buncic, D., 2006, Dissecting some recent empirical non-linear real exchange rate

models, Working Paper, 21st August, School of Economics, University of New

South Wales, Australia.

Campbell, R., Huisman, R. and Koedijk, K., 2001, Optimal portfolio selection in a

value-at-risk framework, Journal of Banking and Finance 25, 1789-1804.

Campbell, J., Lo, A. and MacKinlay, C., 1997, The Econometrics of Financial Markets,

Princeton University Press, Princeton, New Jersey, U.S.A.

Campbell, J. and Viceira, L., 2002, Strategic Asset Allocation, Oxford University Press.

Campbell, J. and Viceira, L., 2006, The term structure of the risk-return trade-off,

Financial Analysts Journal 61, 34-44.

Cappoci, D. and Hubner, G., 2004, Analysis of hedge fund performance, Journal of

Empirical Finance 11, 55-89.

Carhart, M., 1997, On persistence of mutual fund performance, Journal of Finance 52,

57-82.

Chalmers, J., Edelen, J. and Kadlec, G., 2001, On the perils of financial intermediaries

setting security prices: The mutual fund wild card option, Journal of Finance 56,

2209-2236.

Chamberlain, G., 1983, A characterization of the distributions that imply mean-variance

utility functions, Journal of Economic Theory 29, 185-201.

204

Chapman, D. and Pearson, N., 2000, Is the short rate drift actually nonlinear?, Journal of

Finance 55, 355-388.

Chopra, V. and Ziemba, W., 1993, The effect of errors in means, variances and

covariances on optimal portfolio choice, Journal of Portfolio Management 19, 6-

11.

Chua, J. and Woodward, R., 1983, J.M. Keynes’s investment performance: A note,

Journal of Finance 38, 232-235.

Cochran, W., 1947, Some consequences when the assumptions for the analysis of

variance are not satisfied, Biometrics 3, 22-38.

Consigli, G., 2002, Tail estimation and mean-VaR portfolio selection in markets subject

to financial instability, Journal of Banking and Finance 26, 1355-1382.

Copas, J., 1983, Regression, prediction and shrinkage, Journal of the Royal Statistical

Society (B) 45, 311-354.

Cottier, P., 2000, Hedge Funds and Managed Futures, Verlag Paul Haupt, Bern,

Switzerland.

Cremers, J., Kritzman, M. and Page, S., 2005, Optimal hedge fund allocations, Journal

of Portfolio Management 31, 70-81.

Desai,V. and Bharati, R., 1998, The efficacy of neural networks in predicting returns on

stock and bond indices, Decision Sciences 29, 405-425.

205

Diebold, F., Schuermann, T. and Stroughair, J., 1998, Pitfalls and opportunities in the

use of extreme value theory in risk management, Financial institutions Center, The

Wharton School, University of Pennsylvania.

Dimson, E., 1979, Risk measurement when shares are subject to infrequent trading,

Journal of Financial Economics 7, 197-226.

Dumas, B., 1992, Dynamic equilibrium and the real exchange rate in a spatially

separated world, Review of Financial Studies 5, 153-180.

Edwards, F. and Caglayan, M., 2001, Hedge fund performance and manager skill,

Journal of Futures Markets 21, 1003-1028.

Edwards, F. and Liew, J., 1999, Hedge funds versus managed futures as asset classes,

Journal of Derivatives 6, 45-64.

Efron, B. and Morris, C., 1977, Stein’s paradox in statistics, Scientific American 236,

119-127.

Eisenhart, C., 1947, The assumptions underlying the analysis of variance, Biometrics 3,

1-21.

Elton, E. and Gruber, M., 1973, Estimating the dependence structure of share prices,


Elton, E. and Gruber, M., 1995, Modern Portfolio Theory and Investment Analysis, 5th

ed., John Wiley & Sons, Inc, New York, USA.

Elton, E., Gruber, M. and Rentzler, J., 1987, Professionally managed, publicly traded

commodity funds, Journal of Business 60, 175-199.

206

Embrechts, P., McNeil, A. and Straumann, D., 1999, Correlation and dependence

properties in risk management: Properties and pitfalls, In Risk Management: Value

at Risk and Beyond, ed. M. Dempster, 176-223. Cambridge: Cambridge University

Press.

Engle, R., 1982, Autoregressive conditional heteroskedasticity with estimates of the

variance of UK inflation, Econometrica 50, 987-1008.

Estrada, J., 2006, Downside risk in practice, Journal of Applied Corporate Finance 18,

117-125.

Eun, C. and Resnick, B., 1988, Exchange rate uncertainty, forward contracts, and

international portfolio selection, Journal of Finance 43, 197-215.

Fama, E., 1965a, Portfolio analysis in a stable paretian market, Management Science 11,

404-419.

Fama, E., 1965b, The behaviour of stock market prices, Journal of Business 38, 34-105.

Fama, E. and French, K., 1992, The cross-section of expected stock returns, Journal of

Finance 47, 427-465.

Fama, E. and French, K., 1993, Common risk factors in the returns on bonds and stocks,


Favre, L. and Galeano, J., 2002, An analysis of hedge fund performance using loess fit

regression, Journal of Alternative Investments 14, 8-24.

Fishburn, P., 1977, Mean-risk analysis with risk associated with below-target returns,

American Economic Review 67, 116-126.

207

Fisher, L., 1966, Some new stock market indices, Journal of Business 39, 191-225.

Frankfurter, G. and Lamoureux, C., 1987, The relevance of the distributional form of

common stock returns to the construction of optimal portfolios, Journal of

Financial and Quantitative Analysis 22, 501-511.

Frost, P. and Savarino, J., 1986, An empirical Bayes approach to portfolio selection,


Fung, W. and Hsieh, D., 1997a, Empirical characteristics of dynamic trading strategies:

The case of hedge funds, Review of Financial Studies 10, 275-302.

Fung, W. and Hsieh, D., 1997b, Survivorship bias and investment style in the returns of

CTAs, Journal of Portfolio Management 24, 30-41.

Fung, W. and Hsieh, D., 2000, Performance characteristics of hedge funds and CTA

funds: Natural versus spurious biases, Journal of Financial and Quantitative

Analysis 35, 291-307.

Fung, W. and Hsieh, D., 2001, The risk in hedge fund strategies: Theory and evidence

from trend followers, Review of Financial Studies 14, 313-341.

Fung, W. and Hsieh, D., 2002, Risk in fixed-income hedge fund styles, Journal of Fixed

Income 12, 6-27.

Fung, W. and Hsieh, D., 2004, Hedge fund benchmarks: A risk based approach,


Geltner, D., 1991, Smoothing in appraisal based returns, Journal of Real Estate Finance

and Economics 4, 327-345.

208

Geltner, D., 1993, Estimating market values from appraised values without assuming an

efficient market, Journal of Real Estate Research 8, 325-345.

Geman, H. and Kharoubi, C., 2003, Hedge funds revisited: Distributional characteristics,

dependence structure and diversification, Journal of Risk 5, 55-74.

Getmansky, M., Lo, A. and Makarov, I., 2004, An econometric model of serial

correlation and illiquidity in hedge fund returns, Journal of Financial Economics

74, 529-609.

Giamouridis, D. and Vrontos, I., 2007, Hedge fund portfolio construction: A comparison

of static and dynamic approaches, Journal of Banking and Finance 31, 199-217.

Granger, C., 1993, Strategies for modeling nonlinear time-series relationships, The

Economic Record 69, 233-238.

Granger, C. and Teräsvirta, T., 1993, Modelling Nonlinear Economic Relationships,

Oxford University Press, New York.

Greene, J. and Ciccotello, C., 2006, Mutual fund dilution from market timing trades,

Journal of Investment Management 4, 42-66.

Greene, W., 2000, Econometric Analysis, 4th edition, Prentice Hall, New Jersey.

Gujarati, D., 1995, Basic Econometrics, 3rd edition, McGraw-Hill, Inc, New York.

Hadar, J. and Russell. W., 1969, Rules for ordering uncertain prospects, The American

Economic Review 59, 25-34.

Hakansson, N., 1971a, Capital growth and the mean-variance approach to portfolio

selection, Journal of Financial and Quantitative Analysis 6, 517-557.

209

Hakansson, N., 1971b, Multi-period mean-variance analysis: Towards a general theory

of portfolio choice, Journal of Finance 26, 857-884.

Hakansson, N. and Liu, T., 1970, Optimal growth portfolios when yields are serially

correlated, The Review of Economics and Statistics 52, 385-394.

Hamilton, J., 1989, A new approach to the economic analysis of nonstationary time

series and the business cycle, Econometrica 57, 357-384.

Hanoch, G. and Levy, H., 1969, The efficiency analysis of choices involving risk, The

Review of Economic Studies 36, 335-346.

Harlow, W., 1991, Asset allocation in a downside risk framework, Financial Analysts

Journal 47, 28-40.

He, H. and Modest, D., 1995, Market frictions and consumption based asset pricing,

Journal of Political Economy 103, 94-117.

Herold, U., 2005, Computing implied returns in a meaningful way, Journal of Asset


Hinich, M., 1982, Testing for gaussianity and linearity of a stationary time series,

Journal of Time Series Analysis 3, 169-176.

Hsieh, D., 1991, Chaos and non-linear dynamics: Application to financial markets,


Huber, C. and Kaiser, H., 2004, Hedge fund risk factors with option like structures:

Examples and explanations, Journal of Wealth Management 7, 49-60.

210

Ilmanen, A., 1995, Time-varying expected returns in international bond markets,


International Monetary Fund (IMF), 1994, Role of hedge funds: International capital

markets. Developments, prospects and policy issues, September, Washington D.C.,

U.S.A.

International Monetary Fund (IMF), 1998, Hedge funds and financial market dynamics,

Occasional Paper 166, May, Washington D.C., U.S.A.

J.P. Morgan, 1995, Riskmetrics Technical Manual, J.P. Morgan, New York.

Jagannathan, R. and Ma, T., 2003, Risk reduction in large portfolios: Why imposing the

wrong constraints helps, Journal of Finance 58, 1651-1683.

Jansen, D. and de Vries, C., 1991, On the frequency of large stock returns: Putting

booms and busts into perspective, Review of Economics and Statistics 73, 18-24.

Jansen, D., Koedijk, K. and de Vries, C., 2000, Portfolio selection with limited downside

risk, Journal of Empirical Finance 7, 247-269.

James, W. and Stein, C., 1961, Estimation with quadratic loss, Proceedings of the

Fourth Berkeley Symposium on Probability and Statistics, Berkeley, University of

California Press, 361-379.

Jarque, C. and Bera, A., 1987, A test for normality of observations and regression

residuals, International Statistical Review 55, 163-172.

Jobson, D. and Korkie, B., 1980, Estimation for Markowitz efficient portfolios, Journal

of the American Statistical Association 75, 544-554.

211

Jobson, D. and Korkie, B., 1981a, Putting Markowitz theory to work, Journal of

Portfolio Management 7, 70-74.

Jobson, D. and Korkie, B., 1981b, Estimation for Markowitz efficient portfolios, Journal

of the American Statistical Association 75, 544-554.

Jobson, D. and Korkie, B., 1982, Potential performance and tests of portfolio efficiency,


Jobson, D., Korkie, B. and Ratti, V., 1979, Improved estimation for Markowitz efficient

portfolios using James-Stein type estimators, In Proceedings of the Business and

Economics Statistics Section in Washington D.C., August 13-16, by the American

Statistical Association, 279-284.

Jones, C., 2003, Nonlinear mean reversion in the short term interest rate, Review of

Financial Studies 16, 793-843.

Jorion, P., 1985, International portfolio diversification with estimation risk, Journal of

Business 58, 259-278.

Jorion, P., 1986, Bayes-stein estimation for portfolio analysis, Journal of Financial and

Quantitative Analysis 21, 279-292.

Jorion, P., 1991, Bayesian and CAPM estimators of the means: Implications for

portfolio selection, Journal of Banking and Finance 15, 717-727.

Jorion, P. and Goetzmann, W., 1999, Global stock markets in the twentieth century,


Kadlec, G. and Patterson, D., 1999, A transactions data analysis of nonsynchronous

trading, Review of Financial Studies 12, 609-630.

212

Kahneman, D. and Tversky, A., 1979, Prospect Theory: An analysis of decision under

risk, Econometrica 47, 263-292.

Kao, D., 2002, Battle for alphas: Hedge funds versus long-only portfolios, Financial

Analysts Journal 58, 16-34.

Kat, H. and Lu, S., 2002, An excursion into the statistical properties of hedge fund

returns, Working paper, ISMA Centre, University of Reading, Reading, U.K.

Kataoka, S., 1963, A stochastic programming model, Econometrica 31, 181-196.

Keenan, D., 1985, A Tukey non-additivity type test for time series nonlinearity,

Biometrika 72, 39-44.

Kihn, J., 1996, The financial performance of low-grade municipal bond funds, Financial


Klein, R. and Bawa, V., 1976, The effect of estimation risk on optimal portfolio choice,


Krokhmal, P., Palmquist, J. and Uryasev, S., 2002, Portfolio optimization with

conditional value at risk objective and constraints, Journal of Risk 4, 11-27.

Krokhmal, P., Uryasev, S. and Zrazhevsky, G., 2002, Risk management for hedge fund

portfolios, Journal of Alternative Investments 5, 10-30.

Latane, H., 1959, Criteria for choice among risky ventures, Journal of Political

Economy 67, 144-155.

213

Laurelli, P., 2007, Hedge Fund Industry Asset Flow and Trends Report 2006-2007,

March, in association with Institutional Investor News and Hedgefund.net.

Ledoit, O. and Wolf, M., 2003, Improved estimation of the covariance matrix of stock

returns with an application to portfolio selection, Journal of Empirical Finance 10,

603-621.

Ledoit, O. and Wolf, M., 2004, Honey, I shrunk the covariance matrix, Journal of


Lee, T., White, H. and Granger, C., 1993, Testing for neglected non-linearity in time

series models: A comparison of neural networks and alternative tests, Journal of

Econometrics 56, 269-290.

Leibowitz, M. and Henrickson, R., 1989, Portfolio optimization with shortfall

constraints: A confidence limit approach to managing downside risk, Financial


Leibowitz, M. and Kogelman, S., 1991, Asset allocation under shortfall constraints,


Lhabitant, F., 2004, Hedge Funds: Quantitative Insights, John Wiley & Sons, Ltd, West

Sussex, England.

Liang, B., 2000, Hedge funds: The living and the dead, Journal of Financial and

Quantitative Analysis 35, 309-326.

Liang, B., 2001, Hedge fund performance: 1990-1999, Financial Analysts Journal 57,

11-18.

214

Lintner, J., 1965, Security prices, risk and maximal gains from diversification, Journal

of Finance 20, 587-615.

Lintner, J., 1983, The potential role of managed commodity financial futures accounts

(and/or funds) in portfolios of stocks and bonds. Paper presented at the annual

conference of the Financial Analysts Federation, Toronto.

Lo, A., 2001, Risk management for hedge funds: introduction and overview, Financial


Lo, A., 2002, The statistics of sharpe ratios, Financial Analysts Journal 58, 36-52.

Lo, A. and MacKinlay, A.C., 1988, Stock market prices do not follow random walks:

Evidence from a simple specification test, Review of Financial Studies 1, 41-66.

Lo, A. and MacKinlay, A.C., 1990, An econometric analysis of nonsynchronous-trading,

Journal of Econometrics 45, 181-212.

Lochoff, R., 2002, Hedge funds and hope, Journal of Portfolio Management 28, 92-99.

Loomis, C., 1966, The Jones nobody keeps up with, Fortune Magazine, April, 237-247.

Loudon, G., Okunev, J. and White, D., 2006, Hedge fund risk factors and the value at

risk of fixed income strategies, Journal of Fixed Income 16, 46-61.

Malkiel, B. and Saha, A., 2005, Hedge funds: Risk and returns, Financial Analysts

Journal 61, 80-88.

Man Investments, 2005, Hedge fund data biases Q & A, March, Switzerland.

215

Maringer, D., 2005, Portfolio Management with Heuristic Optimization, Springer,

Dordrecht, Netherlands.

Markowitz, H., 1952, Portfolio selection, Journal of Finance 7, 77-91.

Markowitz, H., 1959, Portfolio Selection: Efficient Diversification of Investment, John

Wiley, New York.

McLeod, A. and Li, W., 1983, Diagnostic checking ARMA time series models using

squared-residual autocorrelations, Journal of Time Series Analysis 4, 269-273.

McMillan, D., 2005, Non-linear dynamics in international stock market returns, Review

of Financial Economics 14, 81-91.

Merton, R., 1969, Lifetime portfolio selection under uncertainty: The continuous-time

case, The Review of Economics and Statistics 51, 247-257.

Merton, R., 1980, On estimating the expected return on the market, Journal of Financial


Michaud, R., 1989, The Markowitz optimization enigma: Is optimized optimal?,


Michaud, R., 1998, Efficient Asset Management, Harvard Business School Press,

Boston, Massachusetts, U.S.A.

Miller, M., 1999, The history of finance, Journal of Portfolio Management 25, 95-101.

Mitchell, M. and Pulvino, T., 2001, Characteristics of risk and return in risk arbitrage,


216

Morton, D., Popova, E. and Popova, I., 2006, Efficient fund of hedge funds under

downside risk measures, Journal of Banking and Finance 30, 503-518.

Mossin, J., 1966, Equilibrium in a capital asset market, Econometrica 34, 768-783.

Mossin, J., 1968, Optimal multiperiod portfolio choices, Journal of Business 4, 215-229.

Newey, W. and West, K., 1987, A simple positive semi-definite, heteroskedasticity and

autocorrelation consistent covariance matrix, Econometrica 55, 703-708.

Oberhofer, G., 2001, Hedge funds – A new asset class or just a change in perspective,

Alternative Investment Management Association (AIMA), Newsletter, December.

Officer, R., 1972, The distribution of stock returns, Journal of the American Statistical

Association 76, 807-812.

Opong, K., Mulholland, G., Fox, A. and Farahmand, K., 1999, The behaviour of some

UK equity indices: An application of Hurst and BDS test, Journal of Empirical

Finance 6, 267-282.

Polson, N. and Tew, B., 2000, Bayesian portfolio selection: An empirical analysis of the

S&P 500 Index 1970-1996, Journal of Business and Economic Statistics 18, 164-

173.

Poon, S., Rockinger, M. and Tawn, J., 2004, Extreme value dependence in financial

markets: Diagnostics, models, and financial implications, Review of Financial

Studies 17, 581-610.

Porter, R., 1974, Semi-variance and stochastic dominance: A comparison, American


217

Poshakwale, S., 2002, The random walk hypothesis in the emerging Indian stock market,

Journal of Business Finance and Accounting 29, 1275-1299.

Pratt, J., 1964, Risk aversion in the small and in the large, Econometrica 32, 122-136.

Priestley, M., 1988, Non-Linear and Non-Stationary Time Series Analysis, Academic

Press, London and San Diego.

President’s Working Group on Financial Market, 1999, Hedge funds, leverage, and the

lessons of Long Term Capital Management, Washington, D.C., April.

Quirk, J. and Saposnik, R., 1962, Admissibility and measurable utility functions, Review

of Economic Studies 29, 140-146.

Ramsey, J., 1969, Tests for specification errors in classical linear least-squares

regression, Journal of the Royal Statistical Society Series B (Methodological) 31,

350-371.

Rockafellar, R. and Uryasev, S., 2000, Optimization of conditional value at risk, Journal

of Risk 2, 21-41.

Rockafellar, R. and Uryasev, S., 2002, Conditional value at risk for general loss

distributions, Journal of Banking and Finance 26, 1443-1471.

Rockafellar, R., Uryasev, S. and Zabarankin, M., 2006, Master funds in portfolio

analysis with general deviation measures, Journal of Banking and Finance 30,

743-778.

Roll, R., 1981, A possible explanation of the small firm effect, Journal of Finance 36,

879-888.

218

Ross, S., 1976, The arbitrage theory of capital asset pricing, Journal of Economic

Theory 13, 341-360.

Rothschild, M. and Stiglitz, J., 1970, Increasing risk I: A definition, Journal of

Economic Theory 2, 225-243.

Roy, A., 1952, Safety first and the holding of assets, Econometrica 20, 431-449.

Samuelson, P., 1969, Lifetime portfolio selection by dynamic stochastic programming,

The Review of Economics and Statistics 51, 239-246.

Sarnat, M., 1974, A note on the implications of quadratic utility for portfolio theory,


Scheinkman, J. and LeBaron, B., 1989, Non-linear dynamics and stock returns, Journal

of Business 62, 311-337.

Scherer, B., 2002, Portfolio Construction and Risk Budgeting, Risk Books, London,

U.K.

Schneeweis, T. and Spurgin, R., 1998, Multifactor analysis of hedge fund, managed

futures and mutual fund return and risk characteristics, Journal of Alternative

Investments, Fall, 1-24,

Scholes, M. and Williams, J., 1977, Estimating betas from asynchronous data, Journal of

Financial Economics 5, 309-327.

Scott, R. and Horvath, P., 1980, On the direction of preference for moments of higher

order than the variance, Journal of Finance 35, 915-919.

219

Sercu, P., Uppal, R. and Van Hulle, C., 1995, The exchange rate in the presence of

transaction costs: Implications for test of purchasing power parity, Journal of

Finance 50, 1309-1319.

Shanken, J., 1987, Nonsynchronous data and the covariance factor structure of returns,


Sharpe, W., 1963, A simplified model for portfolio selection, Management Science 9,

277-293.

Sharpe, W., 1964, Capital asset prices: A theory of market equilibrium under conditions

of risk, Journal of Finance 19, 425-442.

Sharpe, W., 1992, Asset allocation: Management style and performance measurement.”

Journal of Portfolio Management 18, 7-19.

Shleifer, A. and Vishny, R., 1997, The limits of arbitrage, Journal of Finance 52, 35-55.

Sortino, F. and Forsey, H., 1996, On the use and misuse of downside risk, Journal of


Stanton, R., 1997, A nonparametric model of term structure dynamics and the market

price of interest rate risk, Journal of Finance 52, 1973-2002.

Stein, C., 1955, Inadmissibility of the usual estimator of the mean of a multivariate

normal distribution, Proceedings of the Third Berkeley Symposium on Probability

and Statistics, Berkeley, University of California Press.

Subba Rao, T. and Gabr, M., 1984, An Introduction to Bispectral Analysis and Bilinear

Time Series Models, Springer-Verlag, Berlin.

220

Swensen, D., 2000, Pioneering Portfolio Management, The Free Press, New York,

U.S.A.

Tasche, D., 2002, Expected shortfall and beyond, Journal of Banking and Finance 26,

1519-1533.

Taylor, M. and Peel, D., 2000, Nonlinear adjustment, long-run equilibrium and

exchange rate fundamentals, Journal of International Money and Finance 19, 33-

53.

Taylor, M., Peel, D. and Sarno, L., 2001, Nonlinear adjustment in real exchange rates:

Towards a solution to the purchasing power parity puzzles, International


Telser, L., 1955, Safety first and hedging, Review of Economic Studies 23, 1-16.

Teräsvirta, T., 1994, Specification, estimation and evaluation of smooth transition

autoregressive models, Journal of the American Statistical Association 89, 208-

218.

Teräsvirta, T., 1998, Modelling economic relationships with smooth transition

regressions, in Handbook of Applied Economic Statistics, New York, Marcel

Dekker, 507-552.

Teräsvirta, T., Lin, C. and Granger, C., 1993, Power of the neural network linearity test,

Journal of Time Series Analysis 14, 209-220.

Tibshirani, R., Walther, G. and Hastie, T., 2001, Estimating the number of clusters in a

data set via the gap statistic, Journal of the Royal Statistical Society B 63, 411-423.

221

Topaloglou, N., Vladimirou, H. and Zenios, S., 2002, CVaR models for selective

hedging for international asset allocation, Journal of Banking and Finance 26,

1535-1561.

Tremont, 2006, Tremont Asset Flows Report-Third Quarter 2006, Tremont Capital

Management, Corporate Center at Rye, 555 Theodore Fremd Avenue, Rye New

York 10580.

Tsay, R., 1986, Non-linearity tests for time series, Biometrika 73, 461-466.

Tsay, R., 2002, Analysis of Financial Time Series, John Wiley and Sons, Inc., New

York, U.S.A.

Tukey, J., 1949, One degree of freedom for non-additivity, Biometrics 5, 232-242.

U.S. Securities and Exchange Commission., 2003, Implications of the Growth of Hedge

Funds, SEC Special Studies, 29 September.

von Neumann, J. and Morgenstern, O., 1944, Theory of Games and Economic

Behaviour, John Wiley, New York, 1964, 3rd edition.

White, H., 1980, A heteroskedasticity-consistent covariance matrix estimator and a

direct test for heteroskedasticity, Econometrica 48, 817-838.

Whitmore, G., 1970, Third degree stochastic dominance, American Economic Review

60, 457-459.

World Federation of Exchanges, 2006, World Federation of Exchanges Annual Report

and Statistics 2005, March 2006.

222

Yadav, P., Paudyal, K. and Pope, P., 1999, Non-linear dependence in stock returns:

Does trading frequency matter?, Journal of Business Finance and Accounting 26,

651-679.

Young, W. and Trent, R., 1969, Geometric mean approximations of individual security

and portfolio performance, Journal of Financial and Quantitative Analysis 4, 179-

199.

Ziemba, W., 2003, The Stochastic Programming Approach to Asset, Liability and

Wealth Management, The Research Foundation of the Association for Investment

Management and Research (AIMR), Charlottesville, Virginia, U.S.A.

portfolio selection and hedge funds: linearity, heteroscedasticity, autocorrelation ... ·...

Documents