Optimal Portfolio Construction

Diss. ETH No. 16514

Optimal Portfolio Construction and Active Portfolio Management Including Alternative Investments

A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Technical Sciences

presented by Simon Theodor Keel Dipl. Masch.-Ing. ETH born 26 January 1976 citizen of Rebstein, SG

accepted on the recommendation of Prof. Dr. H. P. Geering, examiner Prof. Dr. A. J. McNeil, co-examiner Prof. Dr. D. B. Madan, co-examiner

2006

ISBN: 978-3-906483-10-8

IMRT PRESS c/o Measurement and Control Laboratory ETH Zentrum, ML Sonneggstr. 3 CH-8092 Z¨ urich

Acknowledgements

This research was carried out at the Measurement and Control Laboratory (IMRT) at the ETH Zurich, Switzerland from December 2002 to January 2006. The confidence and support of my supervisor, Prof. Dr. H. P. Geering, is gratefully acknowledged. Furthermore, I would like to thank Prof. Dr. A. J. McNeil, ETH Zurich, and Prof. Dr. D. B. Madan, University of Maryland, for accepting to be my co-examiners. I am indebted to the members of the Financial Control Group at the IMRT, Dr. Gabriel Dondi and Dr. Florian Herzog, who supported me throughout my work. Their help and support during the whole thesis is greatly appreciated. I am particularly grateful to Dr. Lorenz Schumann for the challenging discussions, which were of invaluable help and always encouraging. I would also like to thank the entire staff of the Measurement and Control Laboratory, especially Mikael Bianchi. Finally, my thanks go to my parents who helped me in every possible way throughout my time at the ETH. Last but not least, I would like to express my sincerest gratitude to Sonja for all her wonderful support and encouragement.

Zurich, May 2006

Abstract

One aspect of financial engineering is the development of portfolio management strategies. The research field of optimal stochastic control is well suited for the derivation of these strategies in a dynamic environment. It is the aim of this work to explore and extend optimal portfolio construction techniques currently found in the literature. A special emphasis is given to alternative investments. In order to derive an optimal asset allocation strategy, a risk measure has to be introduced and the asset price dynamics have to be modeled. This results in dynamic optimal control problems, which are well studied in control engineering. However, the main emphasis of control engineering is given to deterministic models. Since the prices of financial assets are predominantly driven by randomness, the concepts and techniques of control engineering have to be extended to the stochastic case. The first step of the elaboration of an asset allocation strategy is the definition of the risk measure. However, not all risk measures are well suited for the derivation of optimal asset allocation strategies. Therefore, the terms coherent and convex risk measures are discussed in detail. For the modeling of asset prices, the statistical properties of asset returns have to be taken into account. Several distributions are investigated which are better suited than the typically found normal distribution. Since the literature is mainly concerned with the univariate case, special consideration is given to the multivariate case. It is found that the distribution called generalized hyperbolic and some of its limiting cases yield much more realistic models of asset returns than the normal distribution. In addition to parametric distributions, semi-parametric models including elliptical copulas are analyzed. Particularly, the event of concurrent extreme losses of different financial assets is considered. This work includes an in-depth study of alternative investments. Special consideration is given to their statistical properties. Hedge funds make use of dynamic asset allocation

IV

strategies and may have a large investment universe. Therefore, hedge funds need special attention with respect to risk management. The specific structure and properties of hedge funds are elaborated and discussed. The process of investing in hedge funds is analyzed in detail. A wide range of different statistical properties among the different hedge funds styles is found. Therefore, a universal treatment of hedge fund returns as such is not possible. Following the analysis of the static and dynamic statistical properties of asset returns, optimal asset allocation strategies are derived. At first, a framework of continuous-time stochastic differential equations is considered. The stochastic differential equations are driven by Brownian motion. Again, alternative investments are analyzed in particular. A closed-form solution of an investment strategy with common asset classes is derived. Furthermore, the optimal asset allocation is investigated for the case in which the asset price models contain unknown parameters or processes. It is shown that this problem can be transformed into one in which all parameters and processes are measurable. The properties of the Kalman filter are used for the derivation. The results of these theoretical investigations are tested in a detailed case study including alternative investments. Finally, the topic of active portfolio management is discussed. The importance of the benchmark for active portfolio management is highlighted. A deeper systematic treatment of active portfolio management has not been carried out because there exist neither a generally accepted terminology nor a unified framework for comparing different strategies. A specific active portfolio management problem is presented as well as a procedure for obtaining a solution for a single-period and a multi-period formulation. The single-period solution is backtested with historical data. The very last part of this work considers the use of L´evy processes for the construction of optimal portfolios. The multivariate L´evy measures of the generalized hyperbolic L´evy process and its limiting cases are presented and derived for one limiting case. The work concludes with the presentation of optimal portfolio strategies derived with L´evy processes.

Zusammenfassung

Portfolio Management ist ein wichtiger Aspekt des Fachgebietes Financial Engineering. Die optimale, stochastische Regelung bietet die hierf¨ ur notwendigen mathematischen Grundlagen. Ziel dieser Arbeit ist es, die momentan in der Literatur vorhandenen Techniken f¨ ur die Portfolio Konstruktion zu erweitern. Im speziellen werden alternative Anlagen untersucht. Um optimale Portfolio Management Strategien herzuleiten, muss vorab ein Risikomass bestimmt und die Dynamik der Preise der Anlagem¨oglichkeiten modelliert werden. Hieraus ergeben sich optimale Regelungsprobleme, welche im entsprechenden Fachgebiet bereits gr¨ undlich erforscht wurden. Leider sind aber viele Resultate nur f¨ ur den deterministischen Fall gefunden worden. Da aber bei Finanzproblemen die betrachteten Systeme haupts¨achlich vom Zufall getrieben werden, m¨ ussen die Konzepte auf den stochastischen Fall erweitert werden. Der erste Schritt f¨ ur die Entwicklung einer Portfolio Management Strategie ist die Einf¨ uhrung eines Risikomasses. Es sind jedoch nicht alle Risikomasse gleichermassen geeignet. Koh¨arente und konvexe Risikomasse besitzen f¨ ur die betrachteten Problemstellungen geeignete Eigenschaften. Die Modelle f¨ ur die Renditen von Wertpapieren sollen deren statistische Eigenschaften in realistischer Weise ber¨ ucksichtigen. Hierf¨ ur werden mehrere Distributionen untersucht, welche die h¨aufig angetroffene Normalverteilung ersetzen. Da in Studien oft nur der eindimensionale Fall behandelt wird, wird besonderes Augenmerk auf den mehrdimensionalen Fall gelegt. Die Distribution, welche unter dem Namen Generalized Hyperbolic in der Literatur zu finden ist, kann die betrachteten Renditen sehr viel realistischer beschreiben als die Normalverteilung. Dies gilt auch f¨ ur einige Grenzf¨alle der Generalized Hyperbolic Verteilung. Zus¨atzlich werden elliptische Copulas untersucht.

VI

Diese Arbeit enth¨alt eine ausf¨ uhrliche Untersuchung von alternativen Anlagen. Im speziellen werden deren statistische Eigenschaften untersucht. Hedge Funds verfolgen in der Regel dynamische Anlagestrategien, was im Risikomanagement ber¨ ucksichtigt werden muss. Hierf¨ ur werden die spezifischen Eigenschaften von Hedge Funds untersucht und der Anlageprozess analysiert. Die Eigenschaften von Hegde Funds variieren enorm f¨ ur die verschiedenen Hedge Fund Stile. Deshalb k¨onnen keine universellen Aussagen u ¨ber die statischen und dynamischen Eigenschaften von Hedge Funds gemacht werden. Der erste Teil der Arbeit konzentriert sich auf die statische und die dynamische Modellierung von Anlagem¨oglichkeiten. Im zweiten Teil werden aufgrund der erarbeiteten Modelle optimale Anlagestrategien entwickelt. Als erstes werden Modelle betrachtet, welche auf stochastischen Differentialgleichungen fussen. Als Zufallsprozesse in diesen werden Brownsche Bewegungen eingef¨ uhrt. Auch alternative Anlagen werden als ein solches System modelliert und eine optimale Anlagestrategie in geschlossener Form hergeleitet. Zus¨atzlich werden Modelle betrachtet, welche f¨ ur den Investor unbekannte Parameter und Prozesse enthalten. Um dieses Problem zu l¨osen, wird ein Kalman Filter eingesetzt, die Resultate werden in einem Anwendungsbeispiel getestet. Der letzte Teil dieser Arbeit besch¨aftigt sich mit aktivem Portfolio Management. Die zentrale Bedeutung des Benchmarks f¨ ur das aktive Portfolio Management wird diskutiert. Da das aktive Portfolio Management kein eigentliches Forschungsgebiet darstellt, ist jedoch nur eine oberfl¨achliche Abhandlung m¨oglich. Nichtsdestotrotz wird ein spezifisches aktives Portfolio Management Problem diskutiert und werden zwei m¨ogliche L¨osungsans¨atze pr¨asentiert. Einer dieser L¨osungsans¨atze wird mittels historischer Daten verifiziert. Der letzte Abschnitt dieser Arbeit besch¨aftigt sich mit L´evy Prozessen im Zusammenhang mit Portfolio Konstruktion. Die multivariate L´evy Dichte f¨ ur einen Grenzfall des Generalized Hyperbolic L´evy Prozesses wird hergeleitet. Die Arbeit wird mit der Betrachtung von L´evy Prozessen f¨ ur die Berechnung von optimalen Portfolios abgeschlossen.

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1 Financial Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Financial Assets and Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Financial Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The Asset Allocation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Risk Management and Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1 The Concept of Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Financial Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3

Modeling of Financial Assets and Financial Optimization . . . . . . . . . . . . 21 3.1 Statistical Properties of Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Stylized Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Univariate Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.3 Methodology and Results for the Univariate Case . . . . . . . . . . . . . . . . . 29 3.1.4 Multivariate Properties and Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.5 Results for the Multivariate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Dynamic Models of Financial Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Financial Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4

Alternative Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.1 Hedge Fund Fee Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.2 Hedge Fund Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

VIII

Contents

4.1.3 Hedge Fund Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.4 Funds of Hedge Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.5 Hedge Fund Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Systematic Risks of Hedge Funds and Risk Management . . . . . . . . . . . . . . . . 57 4.2.1 Systematic Risks of Hedge Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.2 Risk Management for Hedge Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.3 Non-linearities in Hedge Fund Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Statistical Properties of Hedge Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.1 Univariate Properties of Hedge Fund Returns . . . . . . . . . . . . . . . . . . . . . 66 4.3.2 Multivariate and Dependence Properties of Hedge Fund Returns . . . . 68 4.4 Hedge Fund Investing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5

Optimal Portfolio Construction with Brownian Motions . . . . . . . . . . . . . 77 5.1 The Full Information Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.1.2 Optimal Asset Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.3 Case Study with Alternative Investments . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2 The Partial Information Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2.2 Estimation of the Unobservable Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.3 Portfolio Dynamics and Problem Transformation . . . . . . . . . . . . . . . . . . 100 5.2.4 Optimal Asset Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.5 Case Study with a Balanced Fund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6

Active Portfolio Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.1 Sector Rotation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 Portfolio Management with L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7

Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.1 Moments of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.2 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.2.1 Normal Mean-Variance Mixture Distributions . . . . . . . . . . . . . . . . . . . . . 133

Contents

IX

A.2.2 Univariate Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.2.3 Multivariate Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2.4 Bessel Functions and Modified Bessel Functions . . . . . . . . . . . . . . . . . . . 139 B GARCH Models for Dynamic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.1 Univariate GARCH Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.2 Multivariate GARCH Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 C Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 C.1 Tail Dependence within a t Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 C.2 Transformation from Partial to Full Information . . . . . . . . . . . . . . . . . . . . . . . 146 C.3 L´evy Density of the Multivariate VG L´evy Process . . . . . . . . . . . . . . . . . . . . . 149 D Additional Data for the Sector Rotation Case Study . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

List of Figures

2.1

Asset allocation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1

Classes of distributions in finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2

Density estimates for the daily Dow Jones returns. . . . . . . . . . . . . . . . . . . . . . 28

3.3

Logarithmic density estimates for the daily Dow Jones returns. . . . . . . . . . . 28

3.4

The model predictive control concept in finance . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1

Assets under management by the hedge fund industry. . . . . . . . . . . . . . . . . . . 49

4.2

Serial correlation of the Tremont convertible arbitrage index. . . . . . . . . . . . . 62

4.3

Dynamic standard deviation of the Tremont long/short equity index. . . . . . 63

4.4

Non-linearities of hedge fund returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5

Kernel regression of lagged S&P 500 returns vs. Tremont fixed income arbitrage returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6

Density estimates for the monthly Tremont convertible arbitrage index returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.7

Correlation of Tremont Hedge Fund Indices with stocks and bonds. . . . . . . . 72

4.8

Hedge fund portfolio construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.9

The hedge fund selection process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1

Asset allocation strategy under full information for γ = −10. . . . . . . . . . . . . 92

5.2

Asset allocation strategy performance under full information for γ = −10. . 93

5.3

Estimations of the short rate and the unobservable factors α and µ. . . . . . . 111

5.4

Asset allocation strategy under partial information for γ = −10. . . . . . . . . . 112

5.5

Asset allocation strategy performance under partial information for γ = −10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

XII

List of Figures

6.1

An MPC approach for the sector rotation problem. . . . . . . . . . . . . . . . . . . . . . 120

6.2

Performance of the sector rotation asset allocation strategy. . . . . . . . . . . . . . 122

List of Tables

2.1

Financial assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Financial risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1

Distributions for daily Dow Jones returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2

Distributions for equity index returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3

Distributions for commodity returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4

Distributions for bond total return index returns. . . . . . . . . . . . . . . . . . . . . . . 31

3.5

Multivariate distributions for equity indices returns. . . . . . . . . . . . . . . . . . . . . 37

3.6

Multivariate distributions for commodity returns. . . . . . . . . . . . . . . . . . . . . . . 38

3.7

Multivariate distributions for a typical portfolio. . . . . . . . . . . . . . . . . . . . . . . . 39

3.8

Copula estimations for asset returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.9

Tail dependence coefficients of weekly world equity indices returns. . . . . . . . 41

3.10 Tail dependence coefficients in a typical portfolio. . . . . . . . . . . . . . . . . . . . . . . 42 4.1

Hedge fund styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2

Common risk factors of hedge funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3

Distributions for monthly Tremont hedge fund indices returns. . . . . . . . . . . . 66

4.4

Tail dependence coefficients for Tremont hedge fund styles. . . . . . . . . . . . . . . 68

4.5

Tail dependence coefficients for Tremont hedge fund styles with common risk factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6

Multivariate distribution models for a portfolio including hedge funds. . . . . 70

5.1

Typical values for the estimated parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2

Key figures for the asset allocation strategy under full information . . . . . . . 94

5.3

Key figures for the asset allocation strategy under partial information. . . . . 113

XIV

List of Tables

D.1 Factors for the sector rotation case study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

List of Symbols and Notation

α

Excess return

β

Factor exposure

γ1

Skewness

γ2

Kurtosis

I

Identity matrix

P

Probability measure

F

Sigma algebra

Ft

Filtration

N

Normal (Gaussian) distribution

µ

Expected return

ν(dx)

L´evy measure

ν(t)

Estimation error

Ω

Sample space

σ

Volatility

Σ(t)

Instantaneous covariance matrix per unit time

1{A}

Identicator function of the set A

C

Copula function

c

Density of a copula

C Ga

Gaussian copula

Ct

t copula

%

Risk measure

L0

Set of all almost surely finite random variables

L1n

Set of n-dimensional integrable functions

P

Asset price

XVI

List of Tables

r

Risk-free interest rate

u

Control vector, asset allocation strategy

V

Investor’s wealth, value of a portfolio

W

Brownian motion

x

Observable factor

y

Unobservable factor

AIC

Akaike Information Criterion

ARCH

Autoregressive conditional heteroskedasticity

BIS

Bank for International Settlements

CCC

Constant conditional correlation

CRRA

Constant relative risk aversion

CVaR

Conditional Value at Risk

DCC

Dynamic conditional correlation

EMH

Efficient market hypothesis

GARCH

Generalized ARCH

GH

Generalized Hyperbolic distribution

GIG

Generalized Inverse Gaussian

HJB

Hamilton-Jacobi-Bellman

i.i.d.

independent, identically distributed

ML

Maximum loss

MM

Method of Moments

NIG

Normal Inverse Gaussian distribution

s-t

Skewed t distribution

SDE

Stochastic differential equation

SP

Shortfall probability

TARCH

Threshold GARCH

VaR

Value at Risk

1 Introduction

Copy from one, it’s plagiarism; copy from two, it’s research. Wilson Mizner

This work explores the possibilities and limits of the use of control engineering methods and techniques in finance. This chapter presents the motivation and goals of this work and the conceptual strategies involved. The application of control engineering methods and techniques to financial problems is called financial engineering. It makes use of engineering tools, i.e., it obtains quantitative results for models and problems developed in research fields such as economics, mathematics, and econometrics. The results in economics and mathematical finance are often of a theoretical or qualitative nature and cannot be used quantitatively as such. The results from the area of econometrics give an indication as to which models are quantitatively applicable. As in engineering problems, the problems considered in this work are solved in two stages: first, modeling of the problem and then computation of its optimal solution. Therefore, the aim of the thesis is to apply improved financial models to optimal portfolio construction problems. In the modeling part of the thesis, the goal is to improve the asset models used most often today. These are discussed in detail in Chapter 3. An important point to be noted is that modeling and optimization are not independent of each other. In general, the more complicated the underlying model is, the more involved the necessary optimization becomes. The models considered in this work are always chosen with the caveat of the existence of a solution for the resulting optimization problem. As in many other research areas, we face the tradeoff between complexity and solvability of the problems posed. In the optimization part, we consider two important topics: reasonable objective functions, i.e., risk measures, and multi-period optimization problems. Various objective functions for investors are explored and their implications for the problems posed are discussed. In

2

1 Introduction

addition, we analyze the advantages and drawbacks of the use of multi-period optimization techniques for investment problems. We obtain a multi-period optimization when it is possible to change the portfolio composition before the end of the problem. However, applying the multi-period optimal solution is not the same as applying optimal singleperiod solutions sequentially, in general.

1.1 Financial Engineering Financial engineering is defined as the use of mathematical finance and modeling to make pricing, hedging, trading, and portfolio management decisions. We mainly consider portfolio management decisions. By definition, a portfolio is a collection of investments held by an institution or an individual. Holding a portfolio with different investments instead of a single one is reducing the investor’s risk and is called diversification. In order to have a model of the portfolio return, we have to model the individual assets as well as their dependencies. Based on these models, we compute the portfolio return and its characteristics. A portfolio optimization is only possible once we have a model of the portfolio return. The investment decisions are derived from the portfolio optimization. We therefore aim to control the financial risk that an investor takes. This raises the question of how to define financial risk, which is still an open issue in theory and in practice. Many different risk measures have been proposed so far, but no risk measure is well suited for all problems arising in the area of financial engineering. This topic is discussed in Chapter 2. Control engineering in technical problems plays a similar role as financial engineering does in finance problems. The use of feedback control strategies, i.e., making use of new information arriving in time is standard in technical problems, but not for financial problems. This topic is a subject of heated debates among scholars and practitioners. The dispute is about the efficient market hypothesis (EMH), proposed in Samuelson (1965) and Fama (1965, 1970a). The efficient market hypothesis states that security prices fully reflect all the information available. There are several forms of the efficient market hypothesis, where the strongest formulation states that all investors have the same information available and behave in the same economic optimal fashion, i.e., investors are rational. From this form, some relaxed forms of the EMH have been derived. According to the EMH, only a buy-and-hold investment strategy can be optimal. However, we doubt that only buy-and-hold investment strategies are optimal under all circum-

1.1 Financial Engineering

3

stances. Among the most important reasons for this statement are: the market behavior is non-stationary, the market has some kind of inertia, not all investors have the same information, and since investors are not always rational, the techniques underlying the investment strategies differ, they provide advantages or disadvantages to investors. We will now discuss these points in more detail. Economies go through phases, such as the well-known bull and bear market phases. In addition, we observe long periods of time during which we cannot distinguish a market direction, i.e., when the market sustains its level. We may speak of different regimes in the market. As a matter of fact, optimal investment strategies in these regimes cannot be the same. Therefore, since a buy-and-hold strategy would just average over the different regimes, it cannot be optimal in either regime. Investors with a buy-and-hold investment strategy have to leave the portfolio unchanged for a considerable amount of time. Only then the optimization of the buy-and-hold investment strategy makes sense. For most investors, this is not a feasible strategy since they are constrained by liabilities and consumption. It is a fact that when markets are in a stress situation, the dependence properties of assets usually change. Assets which reasonably could be considered independent may drop at the same time. This pattern has been observable in every crash that has occurred so far. These facts and many other (empirical) facts show that asset prices are dynamic in their nature and that their properties change over time. The reader is referred to Campbell, Lo and MacKinlay (1997) for more details on this topic. As a matter of fact, financial return series are not independent. This property can easily be verified by examining the serial correlation of squared returns. Therefore, investment decisions taken in the past may no longer be optimal when the market has altered its behavior. The fact that not every investor has the same information available is obvious. The research area of behavioral finance provides strong evidence that for economic and financial theories, the assumption of rational investors is rather bold. We stress the fact the quantitative models used by investors differ tremendously in their degree of sophistication. This leads to further advantages and disadvantages among them. In terms of market paradigms, we agree with the adaptive market hypothesis (AMH) of which the properties are described in Lo (2004). They agree with the statements made so far in this chapter. One implication of the AMH is that a relation between risk and reward exists, but it is

4

1 Introduction

unlikely to be stable over time. A second one is that arbitrage opportunities arise from time to time. See Cvitani´c, Lazrak, Martellini and Zapatero (2004) for a definition of arbitrage. A third implication is that investment strategies which perform well in certain environments may perform poorly in other environments. A fourth implication is that innovation is the key to survival, which is the only objective that matters. The main conclusion of this section is that a portfolio has to be actively managed. The most important reasons, as mentioned, are upcoming liabilities and consumption, changing market behavior, and the advances in research which lead to new tools and methods. Up to this point, we have not discussed the case of arbitrageurs who seem to persistently outperform the market. The money inflows to the hedge fund industry may be considered as evidence, as they have steadily increased over the last years. If there are any legal arbitrage opportunities, they tend to diminish after a reasonable time once they are discovered by others. Therefore, arbitrageurs usually do not provide details about the arbitrage possibilities they have identified and how they are exploiting them. As a consequence, any systematic treatment of this subject is impossible. It is not the purpose of this work to identify arbitrage possibilities but rather to show that quantitative methods can produce added value in a portfolio.

1.2 Structure of the Thesis Chapter 2: Financial Assets and Risk Management In Chapter 2, the financial assets considered are introduced. They are categorized into traditional and alternative investments. The traditional assets are cash, fixed-income investments, equity (stocks), real estate, and foreign exchange. The alternative investments are hedge funds, managed futures, private equity, physical assets (e.g., commodities), and securitized products (e.g., mortgages). A detailed description of the asset allocation process is given. The main levels of the asset allocation process are the strategic asset allocation, the investment analysis, the tactical asset allocation, and the monitoring of the portfolio. We introduce the concepts of risk, risk management, and utility functions. A sound understanding of risk is necessary in order to successfully elaborate a dynamic asset allocation strategy. Therefore, risk measures and their properties are analyzed in detail.

1.2 Structure of the Thesis

5

An overview of financial risks and their classification are presented and the literature on the good properties of risk measures is reviewed. Risk measures with favorable properties in terms of risk management are introduced as coherent and convex risk measures. Finally, the topic of dynamic risk measures is briefly discussed. Chapter 3: Modeling of Financial Assets and Financial Optimization Chapter 3 starts with a brief historical survey of important asset price models proposed so far. The main part of the chapter is devoted to the investigation of the statistical properties of asset returns. The stylized facts of asset return distributions are listed. First, the unconditional properties of univariate asset returns are analyzed. The models proposed in the literature are reviewed. Three main classes of distributions are considered. These are the elliptical distributions, the stable distributions, and the normal mean-variance mixture distributions. In particular, distributions of the generalized hyperbolic (GH) type are investigated. We find that the GH class of distributions fits univariate returns very well. The distributions of the GH class account for the stylized facts which are observed with real-world data. In addition, the GH class contains many important distributions in form of special and limiting cases. Having investigated the univariate case, the next part of the chapter is devoted to the multivariate case. The multivariate version of the GH distribution also offers the best fits in most of the cases considered. Apart from the fully parametric distributions, we investigate the concept of copulas. A copula is a function which ties together univariate distributions to a fully multivariate distribution. Copulas allow for constructing a dependence structure among totally different kinds of marginal distributions. We choose non-parametric models for the margins and only consider elliptical copulas in detail. In particular, the Gaussian and the t copula are investigated. We find that the t copula fits the data considerably better than the Gaussian copula. It is a well-documented fact that correlation is not always sufficient for describing the dependence among asset returns. Therefore, we present some alternative dependence measures commonly found in the literature. We are particularly interested in the measure called tail dependence. Tail dependence describes the limiting proportion of exceeding one margin over a certain threshold, given that the other margin has already exceeded that threshold. Tail dependence is a copula property and independent of the margins. We

6

1 Introduction

find considerable tail dependence among popular asset classes. However, stocks and bonds offer good diversification properties with respect to concurrent extreme losses. After having analyzed the static properties of asset returns in detail, dynamic properties and models of asset returns are briefly reviewed. In particular, factor models and various forms of GARCH models, which are frequently found in the literature, are discussed. The section concludes with an overview of optimization techniques in finance. The most common dynamic optimization technique in finance is stochastic dynamic programming. The model predictive control approach for solving stochastic control problems is briefly described. The main advantage of model predictive control is that constraints on the decision variables can be taken into account. Chapter 4: Alternative Investments In Chapter 4, the topic of alternative investment is discussed. Only the case of hedge funds is considered in detail. First, a brief history and an overview of the current state of the hedge fund industry are given. The investment vehicle hedge fund is formally defined. The special fee structure of hedge funds and its implications for investors are discussed. It is found that high watermarks as well as a considerable amount of the investor’s own money in the fund are favorable for protecting the investor’s interests. The terms alpha and beta are introduced which are often found in the realm of hedge funds. A survey of different hedge fund styles found in the literature is presented. The advantages and disadvantages of funds of hedge funds are discussed. We find the most severe disadvantage of funds of hedge funds to be the double layer of fees. The performance of hedge funds is reviewed and the inherent problems of the performance measurement are highlighted. Because of several biases in the available hedge fund databases, an accurate assessment of the performance of hedge funds is difficult. The most common biases such as survivorship bias, selection bias, and backfill bias, are discussed in detail. The literature on the quantifications of these biases is reviewed. All reviewed publications on this topic find considerable biases in common hedge fund databases. It is also found that the most popular risk-adjusted performance measure for hedge funds is the Sharpe ratio, although the deficiencies of the Sharpe ratio are notorious. Systematic risks are an important input for the risk management of hedge funds. Therefore, the role of the idiosyncratic risk for hedge funds is analyzed. It is observed that

1.2 Structure of the Thesis

7

the variance of a hedge fund portfolio is decreased by combining an increasing number of hedge funds. In contrast to variance, the kurtosis is increased when the number of hedge funds in the portfolio is increased. This is a very unfavorable behavior. However, by combining a sufficiently large number of hedge funds, only the systematic part of risk is expected to remain. The systematic risk is described by a factor model. The most common systematic risk factors for hedge funds are summarized. These risk factors also include non-linear dependencies with respect to traditional asset classes. Sometimes, option-like pay-off structures to traditional assets are found for hedge funds. The risk management of hedge funds demands far more sophisticated methods than traditional assets do. This is due to the fact that the statistical properties of hedge funds are quite different from those of traditional assets. In particular, the topic of tail risk has to be considered carefully. Some returns of hedge fund styles show serial correlation and volatility clustering effects. Market frictions such as illiquidity are the reason for the serial correlations in hedge fund returns. Volatility clustering may be caused by a higher risk-taking of the hedge fund manager because of incurred losses. As for traditional assets, the univariate and multivariate statistical properties of hedge fund returns are analyzed. We find that the results vary considerably among the different hedge fund styles. As in the case of traditional assets, the GH distribution is found to be well suited for describing hedge fund returns. Concerning dependence, the t copula gives far better fits than the Gaussian copula. Finally, the process of hedge fund investing is described. The approaches for constructing a fund of hedge funds portfolio as well as the embedding of hedge funds in the traditional portfolio are discussed. We find that the correlation properties of some hedge fund styles with respect to traditional assets such stocks and bonds are not stable over time. Chapter 5: Optimal Portfolio Construction with Brownian Motions In Chapter 5, dynamic asset allocation strategies are developed for asset prices modeled as continuous-time stochastic differential equations (SDEs) driven by Brownian motion. The main advantage of using the continuous-time framework is that to a high degree the optimal control problem can be solved analytically. In some cases, even closed-form solutions may be derived. This gives more insights into the mechanics of an optimal asset alloca-

8

1 Introduction

tion strategy than a numerical approximation could. However, the modeling properties are rather limited for continuous-time stochastic processes with Brownian motion. The use of factors for explaining expected returns of assets is common in finance. Two different types of problems are considered. We consider the case in which all factors which are explaining the return of assets are known, i.e., measurable. The second case considers the situation where not all of the factors explaining returns are observable. This problem is called optimal asset allocation under partial information. The optimal asset allocation strategies are derived with a stochastic dynamic programming approach. This is done by solving the Hamilton-Jacobi-Bellman (HJB) equation. The HJB equation is a nonlinear partial differential equation, which is very hard to solve if the control variable is constrained. For problems in higher dimensions, it is virtually impossible to find solutions for the constrained case. This fact and the limited possibilities for modeling asset returns are the main disadvantages of modeling assets in a continuous-time stochastic differential framework with Brownian motion. The portfolio dynamics can be derived once the dynamics of the considered assets are defined. The portfolio is modeled to be self-financing, i.e., there are no external in- or outflows of money. Two types of investors are considered who are characterized by their corresponding utility functions. On the one hand, we are considering the popular case of constant relative risk aversion (CRRA). On the other hand, we are considering the case of constant absolute risk aversion (CARA). The problems are solved by using Bellman’s optimality principle. For the partial information case we show that the separation theorem is no longer valid, i.e., we cannot separate the estimation from the optimization. This means that we cannot simply estimate the unobservable quantities and then treat them as if they were known exactly. The general solutions are analyzed in two case studies, one for the full information case and one for the partial information case. The former is simpler to analyze than the latter. The model used for the full information case study is simpler than the one for partial information. However, the full information problem possesses a closed-form solution. This is not the case for the partial information problem. In both case studies, the opportunity set of the investor consists of a bank account, stocks, bonds, and an alternative investment. The resulting dynamic trading strategies are backtested with historical data, for which the parameters are adapted in every step. In both cases, the resulting risk-adjusted returns

1.2 Structure of the Thesis

9

are higher for the actively managed portfolio than for the passive investments. It is found that the partial information approach is superior to the full information approach in the chosen investment framework. Chapter 6: Active Portfolio Management Chapter 6 discusses the role of active portfolio management as well as implementation examples. First, a formal definition of active portfolio management is given and the importance of the definition of the benchmark is highlighted. The key components of active portfolio management are found to be the investment universe and the investment strategy. A crude classification of active portfolio management strategies is given. We differentiate between security selection and market timing. However, there is neither a generally accepted terminology nor a unified framework to compare different strategies. Therefore, a deeper systematic treatment of this topic is not possible. A case study concerning the sector rotation problem is presented and implemented with historical data. The S&P 500 index with its ten sector indices is considered. The active portfolio management strategy is presumed to beat the S&P 500 by over- and under-weighting the single sectors. Two implementation possibilities are presented, i.e., a multi-period and a single-period environment. The actual implementation of the strategy with historical data is done with the single-period strategy. The conditional value at risk (CVaR) is used as risk measure. In both settings, GARCH models for modeling dynamic volatility are used. The ten-dimensional return vector of the sector indices is assumed to have a multivariate normal inverse Gaussian distribution. An adaptive factor model is used to predict the returns of the different sectors. The implementation of the mean-CVaR optimization in an out-of-sample manner is run for the period from 1999 to 2005. The results are promising; we observe an alpha of 5% and an information ratio 0.96. Finally, the use of L´evy processes for optimal portfolio construction is discussed. For the description of asset returns in Chapter 3, we have found that the generalized hyperbolic (GH) distribution and its limiting cases are well suited. Because the GH distribution is infinitely divisible, we may construct a L´evy process whose increments have a GH distribution. Therefore, we can construct dynamic models in continuous time which take the statistical properties of asset returns well into account. The limiting cases of the GH distribution such as the normal inverse Gaussian (NIG) and the variance gamma (VG)

10

1 Introduction

distribution are also well suited for optimal portfolio construction. The necessary L´evy densities are given for describing the corresponding L´evy processes. The study of L´evy processes is one of the most promising research areas in mathematical finance because of the fine properties of L´evy processes.

2 Financial Assets and Risk Management

Being a language, mathematics may be used not only to inform but also, among other things, to seduce. Benoˆıt Mandelbrot

2.1 Financial Assets A financial investment, contrary to a real investment which involves tangible assets such as land or factories, is an allocation of money with contracts whose values are supposed to increase over time. Therefore, a security is a contract to receive prospective benefits under stated conditions like stocks or bonds. The two main attributes that distinguish securities are time and risk. Usually, the interest rate or rate of return is defined as the gain or loss of the investment divided by the initial value of the investment. An investment always contains some sort of risk. Therefore, the higher an investor considers the risk of a security, the higher the rate of return or premium the investor demands, see Sharpe, Alexander and Bailey (1998) for details. We divide financial assets in two main categories, i.e., traditional and alternative investments. Table 2.1 summarizes the considered assets. The main traditional assets are Table 2.1. Financial assets Traditional

Alternative

Cash

Hedge Funds

Fixed-Income

Managed Futures

Equity (stocks)

Private Equity

Real Estate

Physical Assets (Commodities, Art, Wine, ...)

Foreign Exchange

Securitized Products (Mortgages, Loans, ...)

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2 Financial Assets and Risk Management

cash, fixed-income securities, and stocks. We assume that cash is stored in some kind of bank account, the interest rate on this account is often referred to as risk-free interest rate. We briefly describe fixed-income securities. For short-term borrowing, governments and corporations issue securities with a year or less to maturity. This market, where governments and corporations manage their short-term cash needs, is called money market. Two important money market interest rates are the London Interbank Offered Rate (LIBOR) and the interest rate on Treasury Bills. Treasury Bills, in the U.S., are issued by the New York Federal Reserve Bank in weekly auctions. The large banks in London are willing to lend money to each other at the LIBOR rate. The long-term borrowing needs of corporations and governments are met by issuing bonds. A bond contract provides periodic coupon payments and redemption value at maturity to the bondholder. Bonds are either traded over-the-counter or in secondary bond markets. For more details on fixed-income securities, the reader may refer to Fabozzi (2005). Stocks are issued by corporations, which convey rights to the owner. The stock owners elect the board of directors and have claims on the earnings of the company. The stock holders are compensated with cash dividends, whose amount is determined by the board of directors. When we refer to stocks, we mean public stocks. Public trading of stocks (shares) is regulated by the government. The process of arranging the public sale of stocks of a private firm is called initial public offering (IPO). In this context, privately held stocks are referred to as private equity. Real estate investments are also usually found in institutional portfolios, either direct or indirect via investment trusts. Since the end of the Bretton-Woods agreement for fixed exchange rates in 1973, foreign exchange or derivatives on foreign exchange rates are also found in portfolios. This is usually the case for international investors who want to hedge against currency risks. As alternative investments we consider hedge funds, managed futures, private equity, physical assets (e.g. commodities), and securitized products (e.g. mortgages). Alternative investments are discussed in detail in Chapter 4.

2.2 The Asset Allocation Process Obviously, the asset allocation process refers to the process of investing money in different financial assets. There is no generally accepted methodology for this problem. However,

2.2 The Asset Allocation Process

13

there are many keywords describing different stages of the asset allocation process, e.g., strategic and tactical asset allocation. We consider the asset allocation process as an iterative process since a continuous monitoring of the portfolio characteristics is essential. We consider the assets of Table 2.1 as investment opportunities. Note that the iterative nature of the asset allocation process implies active portfolio management.

Strategic Asset Allocation

Investement Philisophy Investment Objectives and Benchmarks ¾ Investment Universe Risk tolerance

Investment Analysis

Traditional Assets: Security-, Sector-, Global-Market Selection ¾ Alternative Investments: Style Analysis, Manager Selection

Tactical Asset Allocation

Modelling the Investment opportunities ¾ Define Mathematical Risk Measure Implement Strategy

?

?

? Monitoring

Strategy Reassessment

Review of Investment Analysis

Update of the Model

Analysis of realized returns Benchmark comparison

Fig. 2.1. Asset allocation process

In Figure 2.1, the asset allocation process is shown graphically. The asset allocation process starts with the strategic asset allocation. The strategic asset allocation is the most important part of a successful investment strategy. It defines the investment objectives, the way risk is measured, gives the set of investment opportunities, and sets the constraints on the single investment positions. The strategic asset allocation should be based on a long-term focus. Therefore, the outermost feedback loop in Figure 2.1, representing the process of the strategy reassessment, has a much lower frequency than the other loops. The next stage is the investment analysis. It may be regarded as a filter for the next step of the asset allocation process. The main task is the further containment of the investment universe. This step includes the fundamental analysis of countries, sectors,

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2 Financial Assets and Risk Management

companies, commodities, hedge fund managers, etc. If the investment opportunities do not comply with the investment philosophy or are unfavorable in some kind of fashion, they are excluded from the investment universe. As the investment strategy, the investment analysis has to be reviewed at a reasonable frequency. This is symbolized by the middle feedback loop in Figure 2.1. After the investment analysis, the definitive investment universe is defined and the actual portfolio construction can be conducted. This part of the asset allocation process is called tactical asset allocation. It has to comply with the constraints and rules of the strategic asset allocation. If the strategic asset allocation and the investment analysis are carried out accordingly, the tactical asset allocation solely consist of the statistical modeling and the mathematical optimization problem. Investment analysis and tactical asset allocation are often combined in the same step. The portfolio construction may be altered at a predefined frequency, usually defined in the strategic asset allocation. This is the innermost feedback loop in Figure 2.1. It has the highest frequency of the three feedback loops. The last step of the asset allocation process is the monitoring of the portfolio and its single positions. The new information about the evolvement of the prices of the different assets is incorporated in the optimization problem, i.e., the model parameters are updated. In addition, the performance in comparison to the benchmark is analyzed. If the risk tolerance is violated, the portfolio composition has to be altered. If the expected, additional gains by changing the portfolio positions are lower than the transaction costs, the portfolio should be left unchanged.

2.3 Risk Management and Risk Measures There are many examples where improper risk management led to huge losses. Some examples are Metallgesellschaft in 1993, Barings Bank in 1995, and Long Term Capital Management (LTCM) in 1998. In each case, catastrophic losses occurred. These cases highlight the importance of proper risk management. Obviously, we first need an understanding of risk before the topics of risk management and risk measures can be addressed. The main problem is that there is no universal definition of risk and neither are there generally accepted definitions for risk in specific environments. There is a close relation between risk and uncertainty. Because of the above

2.3 Risk Management and Risk Measures

15

mentioned points we do not state a rigorous definition of risk. For our purposes we may define risk as follows: Definition 2.1 (Risk). Risk is the exposure to some uncertain future event. The probabilities of the different outcomes of this future event are assumed to be known or estimable. The mathematical tool to describe problems including uncertainty is probability theory. The term exposure in Definition 2.1 states that a certain system only contains the risks of the uncertain events it is exposed to. In a financial context, these uncertain events are often called risk factors. Therefore, only events which have a dependence on the considered system may influence its risk. In a financial model with risk factors, the return of an asset only depends on the considered risk factors. It is common to model stocks with two risk factors. The first factor represents market risk, the second risk factor is the idiosyncratic risk of the company. We are only considering risks involved in the realm of investing. Mathematically speaking, risk is a random variable, mapping the future states of the world into monetary gains or losses. The key for every successful investment strategy is a sound risk management. From this statement the question arises what good risk management is. The two main components of financial risk management are the modeling of the assets and the definition of the risk measure. Once these two elements are defined, risk management becomes a formal, logical process. The first key factor, i.e., the modeling of the assets, is the subject of Chapter 3. The topic of risk measures is discussed in the following. 2.3.1 The Concept of Utility In economics, the concept of utility has been introduced centuries ago. Utility is a measure of the happiness or satisfaction gained from goods or services in an economic context. For financial problems, the argument of utility function usually is money (consumption). The first systematic description of risk for financial problems is the concept of risk aversion. It is introduced in Morgenstern and Neumann (1944) which contains an axiomatic extension of the ordinal concept of utility to uncertain payoffs. We therefore consider the concept of risk aversion as the first form of a risk measure. For a risk averse investor, a utility function U must fulfill certain properties:

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2 Financial Assets and Risk Management

• A utility function must be an increasing continuous function: U 0 > 0.

• A utility function must be concave: U 00 < 0.

The first property makes sure that an investor prefers always more wealth to less wealth. The second property captures the principle of risk aversion. Some commonly used utility functions include 1. the exponential function (a > 0): U (x) = −e−ax . 2. the logarithmic function: U (x) = ln(x). 3. the power functions (b < 1 and b 6= 0): U (x) = 1b (x)b . 4. the quadratic functions (x <

a ): U (x) 2b

= ax − bx2 .

All of these utility functions capture the principle of risk aversion. This is accomplished whenever the utility function is concave. We will not get into the details of utility theory, for more details the reader is referred to Luenberger (1998), Cvitani´c and Zapatero (2004), and Panjer (1998). Since we are usually not interested in the absolute values of utility functions but rather in its shape, Pratt and Arrow have independently developed measures for risk aversion. Let U (x) be a utility function, then the Arrow-Pratt measures for absolute and relative risk aversion are defined as follows: 00

(x) • the Arrow-Pratt measure of absolute risk aversion: a(x) = − UU 0 (x) .

• the Arrow-Pratt measure of relative risk aversion: b(x) = −x

U 00 (x) . U 0 (x)

The main critique on utility theory is that humans are not always rational. We do not discuss this topic since we do not derive economic or financial models based on this assumption. Here, we investigate the performance of rational investment strategies. 2.3.2 Financial Risk The actual return of every security is always uncertain and therefore full information of the underlying risks in a portfolio means knowing the exact distribution of the portfolio return. In addition, one wants to know how the portfolio return distribution is affected by altering the positions in the portfolio. This is very difficult when dealing in a complex stochastic environment. Even for single securities it may be hard to find a suitable distribution, e.g., for illiquid securities. Therefore, different risk-measures as a single quantity have been established. These risk measures are characteristic quantities of a probability density function.

2.3 Risk Management and Risk Measures

17

In the realm of financial markets, risk describes the uncertainty of the future outcome of a current decision or situation. This is put in a more formal manner by introducing the random variable X, defined on a probability space (Ω, F, P), which denotes the profit or loss of a financial position. Therefore, X is a real-valued function on the set Ω of possible scenarios. By L0 we denote the set of all random variables X : Ω → R, which are almost

surely finite. A quantitative measure of risk is given by a mapping % from the set L 0 to the real line. Formally, the definition of a quantitative risk measure is given as:

Definition 2.2 (Risk measure). A risk measure is a function % : L0 → R. The Bank for International Settlements (BIS) is an international organization fostering the cooperation of central banks and international financial institutions. Its classification of financial risks is summarized in Table 2.2. Table 2.2. Financial risks Market Risk

The risk associated with the uncertainty of the value of traded assets

Credit Risk

The risk associated with the uncertainty of the default of debtors.

Operational Risk

The risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems, or from external events.

Liquidity Risk

The risk that positions cannot be liquidated quickly enough at critical times.

Model Risk

The risk of using inaccurate or wrong models for risk budgeting.

Event Risk

The risk of extreme event.

Reputational Risk

The risk of losing ones reputation as investment manager.

In this work, the main emphasis will be on dealing with market risk. Single Period Risk Measures The systematic treatment of risk measures was introduced in the seminal paper of Artzner, Delbaen, Eber and Heath (1998), where the properties of good risk measures are described by some axioms. A risk measure fulfilling these axioms is called a coherent risk measure. Let the two random variables X and Y denote the profit or loss of two assets. The axioms for a coherent risk measure % are (r denotes the risk-free rate of interest): • Subadditivity: ∀X, Y : %(X + Y ) ≤ %(X) + %(Y ) • Positive-homogeneity: ∀X : c ≥ 0 : %(cX) = c%(X)

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2 Financial Assets and Risk Management

• Translation invariance: ∀X : c ∈ R : %(X + cr) = %(X) − c • Monotonicity: ∀X, Y : X ≤ Y : %(X) ≥ %(Y ) The subadditivity property ensures that diversification reduces risk. The positivehomogeneity property, together with subadditivity, implies that the risk measure is convex. A risk measure which is translation invariant and monotone is called monetary. Ziemba and Rockafellar (2000) and F¨ollmer and Schied (2002) introduce the concept of convex monetary risk measures. This concept is a generalization of the more restrictive concept of coherent risk measures. The axioms for convex monetary risk measures are • Convexity: ∀X, Y : %(cX + (1 − c)Y ) ≤ c%(X) + (1 − c)%(Y ),

c ∈ [0, 1]

• Translation invariance: ∀X : c ∈ R : %(X + cr) = %(X) − c • Monotonicity: ∀X, Y : X ≤ Y : %(X) ≥ %(Y ) In most of the cases it is no loss of generality to assume that a given monetary risk measure satisfies %(0) = 0. In Ziemba and Rockafellar (2000), modified risk measures are described as convex monetary risk measures including the properties %(0) = 0 and for X < 0: %(X) > 0. Some examples of single period risk measures are • Maximum Loss (ML) The maximum loss risk measure is intuitive and needs no further explanation. Note that the maximum loss is unbounded and therefore useless when the return distribution is neither truncated nor discrete. Sufficient historical data has to be available for the use of this risk measure. • Shortfall Probability (SP) A shortfall is the event when the return of a portfolio drops below a given threshold. A portfolio manager may not be allowed to drop below a certain performance level; therefore the manager is interested in minimizing the probability to perform below this level. Formally speaking: %SP (a) = P (X ≤ a). • Method of Moments (MM) Since the introduction of mean-variance portfolio theory, moments of return distributions are used as risk measures. The variance is still the most widely used measure to quantify risk. This has obvious disadvantages, e.g., the sign of the return is not taken

2.3 Risk Management and Risk Measures

19

into account. The method of partial moments overcomes the problem of symmetry. As an example, the partial variance for a given threshold a is formally %PV (a) = E[X 2 |X < a]. • Value at Risk (VaR) Value at risk is, besides variance, the most-used quantitative analysis tool in risk management today. The reason for this is because the BIS has introduced VaR as standard risk measure for banks in the new Basel Capital Accord (Basel II). It is intuitively understood and characterized by a confidence level α and the time period considered. This leads to the following formal expression %VaR (α) = inf{x ∈ R|P (X ≥ x) ≥ α}, where X denotes the return of a financial position. The VaR-quantity is the maximum loss over the next time period that will not be exceeded with probability α. The major drawback of VaR is that it does not state what the outcome is once an extreme event has happened, i.e., it contains no information about the tail of the distribution. Furthermore, VaR is not a coherent measure because it is not subadditive, i.e., diversification does not necessarily reduce risk, see Embrechts (2004) for an illustrative example. • Conditional Value at Risk (CVaR) A possible, coherent extension to VaR is the conditional value at risk (CVaR). CVaR is also known as expected shortfall (ES) and is defined as %CVaR (α) = E[X|X ≤ %VaR (α)]. Again, a confidence level α is specified and the returns a characterized for a given time period. CVaR is the expected loss once a return below the VaRα occurs. Informally, CVaR states “how bad is bad?”. CVaR has the appealing property that it is coherent in a single-period setting. Note that, from a regulatory point of view, coherent risk measures should be the preferred choice. From the point of view of an investment manager it is more comfortable to work with convex monetary risk measures in order to more accurately model the investment problem.

20

2 Financial Assets and Risk Management

Dynamic Risk Measures The area of dynamic risk measures is still immature and there is no generally accepted standard. A dynamic risk measure is necessarily a stochastic process. One of the first publications on this subject is Cvitani´c and Karatzas (1999). Formal treatments are found in Balbas, Garrido and Mayoral (2002), Riedel (2004), and Boda and Filar (2005). In these publications, coherent risk measures within a dynamic environment are presented. The axioms for the dynamic case resemble those of the static case. In Cheridito, Delbaen and Kupper (2004), dynamic risk measures are investigated for processes which are rightcontinuous with left limits. The connection between Bellman’s principle and dynamic risk measures is also found in these publications. Riedel (2004) introduces the concept of dynamic consistency, which is an important concept in connection with active portfolio management. A dynamically consistent risk measure rules out contradictory investment decisions over time. Therefore, if two portfolios have the same risk tomorrow in every scenario, then these portfolios should have the same risk today. Note that CVaR needs not to be time consistent in a dynamic environment, as shown in Boda and Filar (2005). For more details on the subject of dynamic risk measures, the reader is referred to the publications mentioned above.

3 Modeling of Financial Assets and Financial Optimization

Young man, in mathematics you don’t understand things, you just get used to them. John von Neumann

The choice of asset models is an important success factor of a quantitative investment strategy. The more realistic the asset prices are modeled, the better the investment strategy performs. In addition, the more accurately the asset returns are reflect by the chosen distribution, the better the actual risk exposure can be calculated. Therefore, we are interested in distributions which can take the stylized facts of asset returns into account. Obviously, we do not want to underestimate the taken risks. However, the overestimation of risk is also unfavorable because this reduces the risk capacity and therefore results in lower returns. We are interested in models for the financial assets discussed in Chapter 2. We provide a short overview of the economic and financial models developed so far. We attribute the first analytic and systematic treatment to Harry Markowitz. In his seminal publication, Markowitz (1952) models asset returns as multivariate random variables. Asset returns are modeled as multivariate Gaussian random variables and the investor’s utility function is quadratic. Therefore, it is often referred to as mean-variance model. The reader is referred to Panjer (1998) for more details on the mean-variance model. Sharpe, Lintner, and Mossin have, based on the assumptions of Markowitz, derived the capital asset pricing model (CAPM), see Sharpe (1964). The CAPM is one of the first factor models. The importance of the CAPM stems also from its terminology, i.e., the use of the Greek letters α and β, which are widely used in portfolio management contexts today. The reader is referred to Sharpe et al. (1998) for more details on the CAPM.

22

3 Modeling of Financial Assets and Financial Optimization

A further milestone in financial modeling is the arbitrage pricing theory (APT) of Ross (1976). The APT framework is essentially a multifactor model which rules out arbitrage possibilities. Fama and French (1993) was one of the first publications, giving empirical evidence that factors explain average returns of stocks and bonds. Treynor and Black (1973) pioneered the area of systematic active portfolio management. Their ideas have been refined in Black and Litterman (1991, 1992) by introducing uncertainty about the model parameters. Besides the single-period models mentioned above, there is the branch of continuoustime finance. The breakthroughs of continuous-time finance are the seminal publications of Black and Scholes (1973) and Merton (1973b). These papers consider the problem of pricing contingent claims. The continuous-time extension of the CAPM is found in Merton (1973a). The models of Black, Scholes, and Merton are based on Brownian motion which implies that returns are normally distributed. The concept of risk-neutral valuation was introduced by Cox and Ross (1976). Short-rate models are frequently used for modeling fixed-income securities, see Vasicek (1977) for an example. Concerning the modeling of the whole term structure, the pioneering work of Ho and Lee (1986) considers the discrete-time case. The continuous-time case is studied in Heath, Jarrow and Morton (1992). Besides the just mentioned economic and financial models, a major advance in volatility modeling is called autoregressive conditional heteroskedasticity (ARCH), introduced by Engle (1982). This topic is discussed in Section 3.2. The drawbacks of the continuoustime models of Black, Scholes, and Merton are that returns are normally distributed. This deficiency is overcome by replacing the Brownian motion with a L´evy process. A L´evy process is a continuous-time process with independent and stationary increments, based on a more general distribution than the normal distribution. However, in order to define such a stochastic process with independent and stationary increments, the distribution has to be infinitely divisible. L´evy processes take the stylized facts of asset returns much better into account than Brownian motion. The reader is referred to Schoutens (2003) for details on L´evy processes in finance. Of course, this short overview is far from complete. It should serve the reader as an overview of the models and methods used in this work.

3.1 Statistical Properties of Asset Returns

23

3.1 Statistical Properties of Asset Returns In Chapter 2, the rate of return is defined as the monetary gain or loss of the investment divided by the initial value of the investment. This concept is called arithmetic return, sometimes also denoted as simple return. The return of an asset may also be defined as the continuous-compounded or log-return. The numerical differences between simple and log-returns are usually small for high frequencies of data. Both concepts have their advantages and disadvantages in terms of portfolio and time aggregation. If not stated otherwise, we usually work with the log-return. The reader is referred to Tsay (2001) for details. In order to describe asset returns, the distribution of the asset returns has to be specified. The distribution can either be parametric, semi-parametric, or non-parametric. Whilst the fully parametric models are most vulnerable to modeling errors, their mathematical use for further calculations is far richer. For instance, portfolio optimizations are by far easier with parametric models than with semi- or non-parametric models. Figure 3.1 gives an overview of important parametric models in finance.

Elliptical

Stable Distributions

Distributions N

Normal MeanVariance Mixture Distributions

Fig. 3.1. Classes of distributions in finance

Elliptical distributions are often reasonably good models for financial return data and have very pleasing properties. For instance, taking linear combinations of elliptical random vectors results in an elliptical random vector of the same type. The marginal and conditional distributions of elliptical distributions are again elliptical. Popular elliptical distributions are the normal and the t distribution. For more details on elliptical distributions, the reader is referred to McNeil, Frey and Embrechts (2005).

24

3 Modeling of Financial Assets and Financial Optimization

Stable distributions were introduced by Paul L´evy in 1925. Note that stable distributions are also called α-stable, stable Paretian, or L´evy stable distributions. The sum of two independent random variables having the same stable distribution is again a random variable with the same stable distribution. Note that, in general, there exists no closed form formula for the density of the the stable distribution. This makes the maximum likelihood estimation computationally tedious because numerical approximations have to be used. Stable distributions have infinite variance, in general, which is also an unpleasant property. A normal mean-variance mixture distribution is a generalization of the normal distribution. The generalization stems from a positive mixing variable, introducing randomness into the mean vector and the covariance matrix of a multivariate normal random variable. Let U be a random variable on [0, ∞), Σ ∈ Rn×n a covariance matrix, and µ, γ ∈ Rn two arbitrary vectors. The random variable X| U = u ∼ N (µ + uγ, uΣ) is said to have a normal mean-variance mixture distribution. This distribution is elliptical for γ = 0 and is called normal variance mixture in this case. The most important normal mean-variance mixture distribution we consider in this context is the generalized hyperbolic distribution (GH). The mixing variable for the GH distribution is the generalized inverse Gaussian (GIG) random variable. The reader is referred to Appendix A for the technical details on normal mean-variance mixture distribution. Having described the important classes of distributions found in Figure 3.1, the role of the normal distribution, denoted by N , becomes apparent. It is the only distribution which is found in every of the three classes. Therefore, the normal distribution is usually considered as benchmark for the modeling of financial assets. In the sequel of this chapter, the univariate and the multivariate properties of asset returns are explored. The data in this chapter is obtained from the Datastream database of Thomson Financial. The datasets range from 1990 to 2005. 3.1.1 Stylized Facts There are many publications on the subject of the stylized facts of asset returns. Since theses stylized facts are observed empirically, they are now more or less accepted. We list the most important stylized facts.

3.1 Statistical Properties of Asset Returns

25

• Equity returns show little or no serial correlation although they are not independent. • Equity returns are fat-tailed and skewed. • Squared or absolute equity returns are serially correlated. • Volatility is time-varying and appears in clusters. For high frequency data and their properties see Cont (2001) and the references therein. A standard reference on high frequency finance is Dacorogna, Gencay, Muller, Olsen and Pictet (2001). For more details on the stylized facts of asset returns the reader is referred to McNeil et al. (2005), Ziemba (2003), Campbell et al. (1997), and Campbell and Viceira (2002). 3.1.2 Univariate Properties We explore the (unconditional) univariate properties of asset returns. Therefore, we consider the set of univariate distributions. The starting point is the normal distribution, which is the most popular in portfolio construction since Markowitz (1952). For finance in general, the normality assumption is found in most models. The first appearance of the normal distribution in finance dates back to Bachelier (1900). Another reason for the popularity of the normal distribution is because of the use of Brownian motion in finance. Although Brownian motion has been mathematically rigorously introduced in 1923 by Norbert Wiener, the Brownian motion first shows up in finance in Osborne (1959). A lot of continuous-time finance results have emerged from Samuelson (1969) and Merton (1969). The central limit theorem makes the normal distribution the most important distribution in probability. Some similar phenomenon may also be observed for equity returns. The lower the frequency of the returns, the more the distribution of the returns resembles a normal distribution. This means that we may reasonably model yearly returns as normal. However, daily returns cannot be assumed normal, statistically. One of the first published doubts about the normality assumption of asset returns are Mandelbrot (1963) and Fama (1965). Since then, many more publications on this subject have appeared. Motivated by the fact that financial returns are skewed and leptokurtic (fat-tailed), we want to investigate suitable extensions of the normal distribution. Figure 3.1 shows promising extensions of the normality assumption proposed so far.

26

3 Modeling of Financial Assets and Financial Optimization

Standard references for continuous distributions are Johnson, Kotz and Balakrishnan (1995a, 1995b). Many distributions are not considered because of their unpleasant properties. The Laplace and the exponential distribution have not been considered because of their shape, the Cauchy distribution has not been considered because its mean is not defined. We consider the log-normal, gamma, generalized inverse Gaussian, chi-square, Weibull, beta, and F distributions as candidates for price distributions, but not for return distributions. There are many publications treating skewness and kurtosis values of asset returns. All of them report that real-world return series are leptokurtic and skewed. Therefore, we are interested in distributions which are skewed, have fat tails, or both. A possible extension of the normal distribution is its skewed version, introduced by Azzalini (1985). The estimation in its original form is inconvenient, we therefore use the methods described in Pewsey (2000). The results for the skewed normal distribution are rather disappointing since the skewed normal distribution does not account for fat tails. Therefore, it is not investigated any further. We may state, as a rule of thumb, that the inclusion of heavy-tails in return distributions is more important than the skewness aspect. A fat-tailed extension of the normal distribution is the t distribution. The t distribution converges to the normal distribution as the parameter ν tends to infinity, see (A.2) for details. Therefore, a large value of ν indicates that the considered random variable may also be considered normal. A further extension would be the skewed t distribution. We use the method of Fernandez and Steel (1998) to extend the t distribution to be skewed, see (A.1) for details. The skewness is measured by the parameter γ ∈ (0, ∞). We have no skewness for γ = 1 which results in the ordinary t distribution. The skewed t distribution obviously has the properties of being leptokurtic and skewed. Note that Hansen (1994) also introduces a skewed version of the t distribution. The generalized hyperbolic (GH) distribution is introduced by Barndorff-Nielsen (1977), although not in a financial context. Eberlein and Keller (1995) use the GH distribution to describe financial return data and also suggest a hyperbolic L´evy motion. The GH distribution is a very flexible distribution and is well suited for describing return data. It contains many important special and limiting cases. Among these are the hyperbolic, normal inverse Gaussian (NIG), a version of the skewed t, variance gamma, t, and the normal distribution. All these distributions are proposed as financial return models

3.1 Statistical Properties of Asset Returns

27

in the literature. For more details on the GH distribution in finance see McNeil et al. (2005), Knight and Satchell (2000), Prause (1999), Raible (2000), Rydberg (1998), and Barndorff-Nielsen and Shepard (2001). The density functions of the GH family are found in Appendix A.2.3. In his publication, Mandelbrot (1963) finds that the stable distribution is well suited for describing asset returns. As their name suggests, these distributions have the pleasing property of being stable. That is, the sum of two independent random variables characterized by the same stable distribution is itself characterized by the same stable distribution. Besides this appealing property, the problem with the stable distribution is that it has infinite second and higher moments. This is in contrast with empirical observations which have finite second moments. Madan and Seneta (1987) introduce the variance gamma distribution. A financial application of the variance gamma distribution is found in Madan and Seneta (1990). Note that Eberlein and von Hammerstein (2004) show that the variance gamma distribution is a limiting case of the generalized hyperbolic distribution. Carr, Gemna, Madan and Yor (2002) give a generalization of the variance gamma distribution, called CGMY. The CGMY distribution is infinitely divisible and therefore also suited for building a corresponding L´evy process. Geman (2002) shows that the GH- and CGMY distribution are well suited for describing asset returns. We distinguish between two main classes to model asset returns more realistically. These classes are the GH class and the class of stable distributions. We investigate further models of the GH class. The reasons for this are manifold. One important reason is that, using multivariate distributions of the GH class, the distribution of the portfolio is easily calculated. Another reason is that for the stable distribution, there exists, in general, no closed-form of its density. Therefore, the GH distribution is much more convenient to work with. The most important reason is that various empirical studies, e.g., Akgiray and Booth (1988), rule out infinite variance of asset returns and therefore also stable distributions. We investigate the following univariate distributions: normal, t, normal inverse Gaussian (NIG), skewed t, and generalized hyperbolic (GH). Apart from these parametric distributions we also consider kernel density estimates. The corresponding kernels are always chosen to be Gaussian, the bandwidth is optimized with the leave-one-out method,

28

3 Modeling of Financial Assets and Financial Optimization Normal GH Kernel density

70 60

density

50 40 30

PSfrag replacements 20 10 0

−0.06

−0.04

−0.02

0 return

0.02

0.04

0.06

Fig. 3.2. Density estimates for the daily Dow Jones returns.

see H¨ardle (1992) for details. Figure 3.2 shows the density estimates for the daily Dow Jones log-returns from 1990 to 2005. The GH distribution gives the best parametric fit in terms of the log-likelihood value. In this case, the deficiency of the normal distribution is that it does not account for the fat tails and the thin middle.

0

log(density)

−5

−10

PSfrag replacements −15

Normal GH Kernel density

−20 −0.06

−0.04

−0.02

0

0.02

0.04

return Fig. 3.3. Logarithmic density estimates for the daily Dow Jones returns.

3.1 Statistical Properties of Asset Returns

29

In order to analyze the tails, the logarithmic density estimates are plotted in Figure 3.3. Obviously, the normal distribution fits very poorly in the tails, therefore considerably underestimating the events of extreme losses. 3.1.3 Methodology and Results for the Univariate Case The distributions are fitted to the return time series by a maximum likelihood approach. For the model selection part, we use the method of information criteria. In this work, we use the concept of Akaike (1974). An alternative approach is suggested in Schwarz (1978), which is more restrictive with respect to higher order models. Accordingly, we chose the distribution with the lowest information criterion as the best model. By y we denote the geometric returns of the price data. The parameters of the distribution are assembled in ˆ The log-likelihood value of the the vector θ, the estimated parameters are denoted by θ. ˆ estimation is denoted by l(θ|y). The Akaike information criterion is defined as ˆ + 2q, AIC = −2l(θ|y)

(3.1)

where q is the number of parameters of the distribution. The distribution which minimizes the Akaike information criterion is considered as the best model. As an example, we give the detailed results for the Dow Jones Industrials index. The results from the maximum likelihood estimation for a daily frequency are shown in Table 3.1, the best results are shown in bold numbers. Note that the normal distribution gives the worst fit for daily returns. Having inspected Figure 3.2 this result is expected. Table 3.1. Distributions for daily Dow Jones returns. Distribution AIC value log-likelihood value GH

-25496.33

12753.17

NIG

-25482.79

12745.39

t

-25468.92

12737.46

Skewed t

-25467.84

12737.92

Normal

-24914.26

12459.13

In Table 3.1, the GH density has the highest log-likelihood value and therefore fits the data best. If the number of parameters is taken into account, i.e., we use the AIC criterion for model selection, the GH distribution still is the best model in this particular case.

30

3 Modeling of Financial Assets and Financial Optimization

Table 3.2 reports the results for different equity indices with data from 1990 to 2005. The GH, NIG, and the skewed t (s-t) fit the data best in terms of the maximum likelihood value. In addition, the skewness γ1 and the kurtosis γ2 are given. The normal distribution is the best model for monthly Nikkei 225 return data. This result is supported by the values of γ1 and γ2 for the monthly Nikkei 225 returns. The considered stock indices, in general, have fat tails and are skewed to the left. This is seen from the values of γ 1 , which are all negative, and from the values of γ2 , which are all larger than three. For daily returns, the GH and the NIG distribution are the best models in terms of the AIC value. In terms of the maximum likelihood value, the GH distribution fits best. Table 3.2. Distributions for equity index returns. monthly

weekly

Equity index

min(AIC)

S&P 500

NIG

-0.36 3.43

NIG

-0.41 5.83

GH

-0.10 6.89

Dow Jones

s-t

-0.27 3.71

s-t

-0.40 6.34

GH

-0.23 7.69

Nasdaq

GH

-0.53 4.22

NIG

-0.45 6.35

GH

-0.02 8.74

FTSE 100

NIG

-0.38 3.75

NIG

-0.33 5.97

NIG

-0.09 6.14

CAC 40

s-t

-0.46 3.50

t

-0.23 4.70

NIG

-0.09 5.83

DAX 30

s-t

-0.76 4.33

NIG

-0.47 5.86

NIG

-0.21 6.87

SMI

s-t

-0.71 5.47

NIG

-0.69 7.34

GH

-0.25 8.21

Nikkei 225

N

-0.13 3.43

t

0.02 4.71

GH

0.20 6.35

S&P Global 1200

s-t

-0.41 3.57

NIG

-0.46 4.81

NIG

-0.19 6.96

γ1

γ2

min(AIC)

daily

γ1

γ2

min(AIC)

γ1

γ2

Table 3.3 reports the results for some commodities. The results for the daily returns are similar to the ones in Table 3.2, the GH and the NIG distribution are the best models. For monthly and weekly returns, the t distribution is often the best choice. Note that the t distribution is fat-tailed whereas the GH distribution has semi-heavy tails, see Prause (1999) for details. Commodity returns, in contrary to equity index returns, may be significantly skewed to the right. The high values of the kurtosis give evidence that commodity returns have fatter tails than the equity indices in Table 3.2 and the bond indices in Table 3.4. Daily returns on oil are significantly more non-normal than the returns on gold, indicated by the corresponding values of the skewness and kurtosis. Table 3.4 reports the results for fixed income indices. We consider US Government Bond indices with different maturities. The indices are total return indices and are calculated by Thomson Financial. These data sets seem to have thinner tails than the asset returns

3.1 Statistical Properties of Asset Returns

31

Table 3.3. Distributions for commodity returns. monthly

weekly

daily

Commodity (index)

min(AIC)

Gold

t

0.60 6.46

t

0.31 8.39

GH

-0.08 14.34

Oil (West Texas Int.)

t

0.07 3.85

s-t

-0.45 8.72

GH

-1.47 29.40

Platinum (London)

t

-0.1 3.75

t

-0.02 6.78

NIG

-0.24 11.19

Moody’s Commodities Index

N

0.30 3.64

t

0.12 4.00

GH

0.04 11.82

GS Commodities Index

t

0.14 3.56

t

-0.59 8.67

GH

-1.02 19.95

GS Energy Index

t

0.26 3.80

t

-0.04 7.43

GH

-0.16 6.42

γ1

γ2

min(AIC)

γ1

γ2

min(AIC)

γ1

γ2

considered so far. For monthly returns with a maturity of less than seven years, the normal distribution is the best model in terms of the AIC value. We observe that the longer the maturity of the bond index, the more the return distribution deviates from the normal distribution. The NIG distribution is particularly well suited for describing daily bond index returns. Besides the normal and the NIG distribution, the skewed t distribution gives the best fits. Table 3.4. Distributions for bond total return index returns. monthly

weekly

FI index

min(AIC)

US Govt. 1-3 years

N

0.008 3.18

t

-0.04 3.88

NIG

0.06 7.05

US Govt. 3-5 years

N

-0.26 3.32

NIG

-0.28 3.66

NIG

-0.22 5.67

US Govt. 5-7 years

N

-0.26 3.32

s-t

-0.39 3.78

NIG

-0.29 5.17

US Govt. 7-10 years

s-t

-0.38 3.71

s-t

-0.45 3.88

NIG

-0.36 5.23

US Govt. >10 years

s-t

-0.49 3.83

s-t

-0.41 3.84

NIG

-0.33 4.67

US Govt. all mat.

s-t

-0.40 3.56

s-t

-0.42 3.73

NIG

-0.32 4.88

γ1

γ2

min(AIC)

daily

γ1

γ2

min(AIC)

γ1

γ2

For the sake of brevity, we only analyzed the univariate properties of some selected equities. However, the results for other assets in these classes are similar. Summarizing, we find that the GH class with its limiting cases offers a fairly good choice for modeling univariate distributions for asset returns. 3.1.4 Multivariate Properties and Dependence A portfolio, by definition, consists of more than one asset. Therefore, asset returns must be modeled as multivariate random variables. The previous section investigated the (unconditional) marginal distributions of different kinds of assets. A portfolio is the linear

32

3 Modeling of Financial Assets and Financial Optimization

combination of the components of a multivariate random variable or a random vector, respectively. As a consequence, the dependencies among the different components significantly influence the properties of the portfolio distribution. The dependence concept used most commonly is correlation, i.e., the linear dependence of multivariate random variables. If we are considering the mean-variance framework, correlation suffices since the correlation matrix of a multivariate normal distribution fully describes its dependence. A standard reference on dependence is Joe (1997). It is documented that correlation is a questionable dependence measure when distributions are not elliptical, see Embrechts, McNeil and Straumann (2002) and Embrechts, Lindskog and McNeil (2003) for more details. Therefore, we investigate for dependence measures beyond correlation. Most of the distributions introduced in the previous section have a multivariate version, see Appendix A.2.3. However, instead of introducing a multivariate distribution function, there exists another approach to construct a multivariate distribution: a copula is a function which ties together univariate distributions to a fully multivariate distribution. Therefore, copulas provide a wide range of possible dependence structures. Copulas allow for constructing a dependence structure among totally different kinds of margins. This is not possible with multivariate distributions whose margins usually are of the same type. Copulas In the area of finance, risk management and diversification play central roles. Obviously, reasonable risk management is not possible without a sound knowledge of the dependencies of the risky assets. Without this knowledge, good diversification cannot be achieved. The copula approach allows us to separate the modeling of the univariate margins and their dependencies. A comprehensive introductory paper for copulas in finance is Bouye, Durrleman, Nikeghbali, Riboulet and Roncalli (2000). We briefly review the most important concepts of copulas used in this context. The reader is referred to the literature for details. A standard reference on copulas is Nelsen (1998), more recent publications are Mari and Kotz (2004) and McNeil et al. (2005).

3.1 Statistical Properties of Asset Returns

33

A copula is a function C that links the univariate margins with the cumulative distribution functions Fi of the random variables X1 , X2 , ..., Xn to a full multivariate distribution F . Therefore, an n-copula is a function C from [0, 1]n to [0, 1] with the properties • C is grounded and n-increasing. • C has margins Cn which satisfy Cn (u) = C(1, ..., 1, u, 1, ..., 1) = u, ∀u ∈ [0, 1]. Sklar proved in 1959 that the copula C is unique for a multivariate distribution function F with continuous margins. Its importance stems from the fact that marginal distributions and their dependence can be separated. Theorem 3.1 (Sklar’s Theorem). Let F be an n-dimensional distribution function with continuous margins F 1 , . . . , Fn . Then, F has a unique copula representation F (x1 , . . . , xn ) = C(F1 (x1 ), . . . , Fn (xn )). Proof. See Nelsen (1998). With the assumptions of Sklar’s theorem we may also state that C(u1 , . . . , un ) = F (F1−1 (u1 ), . . . , Fn−1 (un )). The modeling process within the copula framework has two levels. The first level consists of modeling the marginal distributions. The second level consists of modeling the dependence of the margins, i.e, of choosing the appropriate copula. In both levels we may either use parametric or non-parametric models. The marginal densities of the random variables X i are denoted by fi . The density f of the multivariate distribution function F is given by f (x1 , . . . , xn ) = c(F1 (x1 ), . . . , Fn (xn ))

n Y

fi (xi ),

(3.2)

i=1

where c is the density of the copula given by c(u1 , . . . , un ) =

∂C(u1 , . . . , un ) . ∂u1 · · · ∂un

We only consider elliptical copulas. Of course, there are other classes of copulas such as Archimedean copulas, extreme value copulas, or Marshall-Olkin copulas which are not further investigated. Elliptical copulas are simply the copulas of elliptic distributions. Archimedean copulas are also popular in finance. The Gumbel copula is an Archimedean

34

3 Modeling of Financial Assets and Financial Optimization

copula which is often found in financial studies. Other popular Archimedean copulas are the Frank and the Clayton copula. See Schmidt, Hrycej and St¨ utzle (2003) for the GH copula. As mentioned in the previous section, the multivariate normal distribution is still the most popular distribution when modeling asset returns. This makes the Gaussian copula the most used copula in finance. The multivariate normal or Gaussian copula is given by CRGa (u1 , . . . , un ) = ΦR (Φ−1 (u1 ), . . . , Φ−1 (un )), where Φ denotes the standard normal distribution and ΦR the n-dimensional normal distribution with correlation matrix R. The corresponding density is (ζ i = Φ−1 (ui )) c(u1 , . . . , un ) = p

1 det(R)

1 T (R−1 −I

e− 2 ζ

n )ζ

,

where In denotes the n-dimensional identity matrix. Another copula frequently found in financial applications is the multivariate t copula, t −1 CR,ν (u1 , . . . , un ) = tR,ν (t−1 ν (u1 ), . . . , tν (un )),

where tR,ν denotes the standardized multivariate t distribution with ν degrees of freedom and shape matrix R. Additionally, tν denotes the standard univariate t distribution with ν degrees of freedom. The corresponding density is t cR,ν (u1 , . . . , un )

Qn ζi2 ν+1 2 )[Γ ( ν2 )]n−1 Γ ( ν+n fR,ν (ζ1 , . . . , ζn ) i=1 (1 + ν ) 2 (3.3) =p = Qn ν+n , ν+1 det(R)[Γ ( 2 )]n (1 + ν1 ζ T R−1 ζ) 2 i=1 f1,ν (ζi )

where ζi = t−1 ν (ui ) and fR,ν denotes the density of a tn (ν, 0, R) distributed random variable, see Appendix A.2.3. The properties of the t copula and related copulas are extensively studied in Demarta and McNeil (2005). Malevergne and Sornette (2003) investigate elliptical copulas for financial assets. They consider the Gaussian and the t copula and find that it may be dangerous to blindly assume a Gaussian copula. Breymann, Dias and Embrechts (2003) and Dias and Embrechts (2004) analyze elliptical as well as Archimedean copulas for their use in modeling of financial data. We will investigate the normal and the t copula. The normal copula is considered because of its importance in finance. The t copula is investigated because it is a natural extension of the normal copula, i.e., the normal copula is a limiting case of the t copula. Furthermore, Malevergne and Sornette (2003) and Dias and Embrechts (2004) find very promising results for the t copula also in comparison to Archimedean copulas.

3.1 Statistical Properties of Asset Returns

35

Finally, we examine the statistical inference of copulas. For the non-parametric case we may use the concept of empirical copulas, see Bouye et al. (2000) for details. For non-parametric margins we may either use empirical distributions or kernel regression techniques. For details on the method of kernel regression see H¨ardle (1992). The two main concepts for statistical inference are maximum likelihood and method of moments. We will use the maximum likelihood approach since, in general, it is more accurate than the method of moments. Suppose we have sample of size N , denoted by X . By θ we denote parameters of the model, θc denotes the parameters of the copula, and θi corresponds to the parameters of the i-th margin. From (3.2) we immediately obtain the log-likelihood function l(θ) as l(θ|X ) =

N h X i=1

n ³ ¡ ¡ ¢i ¢´ X (t ) (t ) log fj (xj i |θj ) . log c F1 (x1 i |θ1 ), . . . , Fn (xn(ti ) |θn )|θc +

(3.4)

j=1

If we have a parametric model for the margins and for the copula, (3.4) has to be maximized with respect to all of the elements in θ. However, one could also, in a first step, estimate the parameters of the margins. After having estimated the parameters θˆi (i = 1, . . . , n) of the margins, the parameters of the copula are estimated separately. This procedure could also be applied to a non-parametric estimation of the margins. In this case we call the procedure a pseudo-maximum-likelihood estimation or inference functions for margins method (IFM). In both ways, the second term in (3.4) does not affect the estimation of the copula parameters. Therefore it is often omitted in the literature. By using the IFM method, the inference of the marginal properties and the dependence properties are separated. Dependence Measures We give a brief overview of measures of dependence. For more details on the topic dependence measures, the reader is referred to McNeil et al. (2005), Embrechts et al. (2003), and Embrechts et al. (2002). In the following, we make use the concept of concordance. Informally speaking, the concordance of two random variables is the property that large and (t )

(t )

(t )

(t )

small outcomes of two random processes occur together. Let (x1 i , x2 i ) and (x1 j , x2 j ) be two observations of a random vector (X1 , X2 ) of continuous random variables. We (t )

(t )

(t )

(t )

(t )

(t )

(t )

(t )

say that (x1 i , x2 i ) and (x1 j , x2 j ) are concordant if (x1 i − x1 j )(x2 i − x2 j ) > 0, and (t )

(t )

(t )

(t )

discordant if (x1 i − x1 j )(x2 i − x2 j ) < 0.

36

3 Modeling of Financial Assets and Financial Optimization

• Correlation The most used dependence measure in practice is the correlation. Note that zero correlation is a necessary but not a sufficient condition for independence. • Kendall’s tau Kendall’s tau is defined as the difference between the probability of concordance minus the probability of discordance. Note that Kendall’s tau is a copula property and therefore independent of the margins, see Nelsen (1998) for details. • Spearman’s rho Spearman’s rho is defined to be proportional to the probability of concordance minus the probability of discordance. As Kendall’s tau, Spearman’s rho is a copula property and therefore independent of the margins, see Nelsen (1998) for details. • Tail Dependence The concept of tail dependence describes the dependence for extreme values. Loosely speaking, tail dependence describes the limiting proportion of exceeding one margin over a certain threshold given that the other margin has already exceeded that threshold. We differentiate between lower and upper tail dependence. For elliptical distributions, these two measures are the same. In a financial context we are mainly interested in the lower tail dependence since this means concurrent extreme losses. Definition 3.2 (Lower Tail Dependence). If, for a bivariate copula C, the limit λl = limu→+ 0

C(u,u) u

exists, then C has lower tail

dependence for λl ∈ (0, 1] and no lower tail dependence for λl = 0. As Kendall’s tau and Spearman’s rho, tail dependence is a copula property and therefore independent of the margins. The Gaussian copula has no tail dependence. This is the main deficiency of the Gaussian copula since we know from the empirical properties of asset returns that asset prices may have concurrent extreme losses. The tail t dependence of a bivariate t copula CR,ν is s ³ (ν + 1)(1 − ρ) ´ , λ = 2 tν+1 − 1+ρ

(3.5)

where ρ is the off-diagonal element of R and tν+1 denotes the t distribution with ν + 1 degrees of freedom. For a proof see Embrechts et al. (2002). Note that we can compute the tail dependence coefficients in any dimension of the t copula in an analogous way. For the proof see Appendix C.1.

3.1 Statistical Properties of Asset Returns (t )

(i)

(t )

(j)

(t )

37 (t )

i and x2 i , Let {(x1 i , x2 i )}N i=1 denote a sample of size N and R1 , R2 the rank of x1

respectively. Empirically, the tail dependencies can be calculated as n

X ˆl = 1 λ 1 (i) . (i) k i=1 {R1 ≤k,R2 ≤k} The parameter k has to be defined. It reasonable the set k = αn, where α describes the percentage of the area under the distribution that is considered as tail. 3.1.5 Results for the Multivariate Case We fit the considered multivariate models to various typical combinations of assets by maximum likelihood and analyze the multivariate properties of the asset returns. First, we analyze the results of different multivariate distributions of asset returns. In a second step, we elaborate the tail dependencies among several asset classes. Multivariate Parametric Distributions As in Section 3.1.3, we aim to select the most suitable model for asset returns. We consider the multivariate versions of the normal, t, skewed t, NIG, and the GH distribution. The distribution functions are given in Appendix A.2.3. Again, we make use of the Akaike information criterion as defined in (3.1). We fit the equity indices of Table 3.2 to the multivariate distributions mentioned above. The results are shown in Table 3.5; the best results are shown in bold numbers. The GH distribution gives the best fit in terms of the log-likelihood value. Surprisingly, the multivariate t distribution is the best model in terms of the AIC criterion, although it is not frequently chosen for the univariate case in Table 3.2. We find that the normal distribution fits the data significantly worse than the other distributions. Table 3.5. Multivariate distributions for equity indices returns. monthly

weekly

daily

Distribution

log-l

AIC

log-l

AIC

log-l

AIC

Normal

2835.5

-5583

16274.8

-32462

106129

-212170

2866.7 -5643 16607.6 -33125 108279

-216467

Skewed t

2871.5

-5637

16614.6

-33123

108285

-216464

NIG

2871.4

-5637

16612.3

-33119

108286

-216467

GH

2871.6 -5635 16615.0 -33122 108291 -216474

t

38

3 Modeling of Financial Assets and Financial Optimization

Next, we fit the commodities found in Table 3.5 to the parametric multivariate distributions considered. Table 3.6 shows the results; the best results are shown in bold numbers. As for the equity indices returns, the multivariate t distribution offers a good fit in terms of the AIC value. This time, the choice of the multivariate t distribution is less surprising since the t distribution is also frequently chosen in the univariate case. The best fits for the log-likelihood value are the multivariate skewed t and the GH distribution. Table 3.6. Multivariate distributions for commodity returns. monthly

weekly

daily

Distribution

log-l

AIC

log-l

AIC

log-l

AIC

Normal

2127.0

-4200

11918.4

-23783

76832

-153609

t

2147.9 -4240 12332.0 -24608 80443 -160829

Skewed t

2149.7

-4231 12334.0 -24600 80447 -160825

NIG

2149.7

-4231

12319.1

-24570

80414

-160761

GH

2149.8 -4230

12333.9

-24598

80445

-160821

We do not give the detailed results of the bond indices of Table 3.4. However, the results are similar to those obtained so far. Again, the multivariate t distribution offers the best fit in terms of the AIC value. Although that the univariate distributions for monthly returns are reasonably modeled with the normal distribution, this is no longer the case for multivariate distributions. We interpret this finding as the lack of the multivariate normal distribution to model the dependence structure of bond index returns. In addition to the deficiencies of the univariate normal distribution, the multivariate normal has further disadvantages because of its dependence structure. So far we have only considered multivariate distributions of asset returns belonging to the same asset class. We consider the portfolio of an US investor. The portfolio of this investor consists of the S&P 500, Nasdaq, S&P Global 1200, GS Commodities Index, GS Energy Index, and the DS US Government all maturities index. In this example, there are no foreign exchange or real estate assets in the portfolio. Neither are there any alternative investments. The results are shown in Table 3.7, the best results are shown in bold numbers. Again, the results are similar to the multivariate examples analyzed so far. The GH and the skewed t distribution offer good fits for the considered multivariate asset returns. The results for the portfolio substantiate the fact that assuming a normal

3.1 Statistical Properties of Asset Returns

39

Table 3.7. Multivariate distributions for a typical portfolio. monthly

weekly

daily

Distribution

log-l

AIC

log-l

AIC

log-l

AIC

Normal

2351.4

-4649

13147.2

-26241

84841

-169628

2376.6 -4697 13532.7 -27009 87751

-175447

t Skewed t

2377.7

-4687 13536.4 -27005 87759 -175450

NIG

2377.4

-4687

13518.0

-26968

87724

-175381

GH

2377.7 -4685

13536.3

-27003

87758

-175447

distribution for portfolio construction can underestimate the probability of extreme losses severely. For the sake of brevity, we have only analyzed some selected groups of equities. As expected, the GH family and its limiting cases offer a considerable improvement to the normal distribution. Multivariate Semi-Parametric Distributions In the previous section, we have analyzed the goodness-of-fit of parametric distribution models. In this section, we consider semi-parametric models. The models are semiparametric because we chose non-parametric distributions for the margins. These are tied together with a copula function in order to have a multivariate distribution function. We either choose the empirical distribution or the kernel density for the estimates of the margins. The main advantage of the empirical distribution is that it is trivial to compute. The main deficiency of using empirical distributions is that the tails may not reflect the true tails of the underlying distribution. One possibility to overcome this problem is to use kernel densities. Another possibility would be having parametric tails with models from extreme value theory and using the empirical distribution for the body of the distribution. We investigate the Gaussian copula and make comparisons with the t copula. The copulas are fitted to the same groups of assets as in the previous section. We consider the equities studied in Table 3.5, the commodities of Table 3.6, and the portfolio of the US investor found in Table 3.7. Only the copula parameters are estimated. For the margins, either the empirical distribution or the kernel density is used, i.e., we only consider the left term in (3.4) for the maximum likelihood estimation. We find that the log-likelihood values for kernel den-

40

3 Modeling of Financial Assets and Financial Optimization

sity estimates are significantly higher than for the empirical estimates. Therefore, we use kernel density estimates for the margins. Table 3.8 shows the result for the three considered cases. The rather low estimates for the degree of freedom parameter ν indicate that there is considerable tail dependence for the correlated assets. The higher the frequency of the data, the lower the degree of the freedom parameter of the t copula becomes. Therefore, a high frequency of return data implies more tail dependence. By comparing the log-likelihood values of the semiTable 3.8. Copula estimations for asset returns with kernel estimations for the margins. monthly log-likelihood Gaussian

weekly

daily

t

Gaussian

t

Gaussian

t

Equity indices

2876

2889 (ν=15)

16556

16710 (ν=8)

107891

108706 (ν=6)

Commodities

2156

2163 (ν=12)

12296

12441 (ν=11)

80604

81263 (ν=9)

Portfolio

2382

2395 (ν=11)

13497

13638 (ν=9)

87425

88152 (ν=7)

parametric models with the parametric models, we find that the semi-parametric models with the t-copula give the best fit in every case. The results for the Gaussian copula are mixed. For monthly returns, the semi-parametric Gaussian model is superior to the parametric models. This is not the case for weekly and daily data. The previous section showed that univariate margins, in general, cannot reasonably assumed to be normal. This section indicates that the Gaussian copula is usually not suitable for describing multivariate asset returns as well. The interpretation of the results in Table 3.8 is that the Gaussian copula cannot account for tail dependence, although the low values of ν indicate significant tail dependence. As the multivariate normal distribution, the Gaussian copula underestimates extreme losses in a portfolio because it does not account for concurrent extreme losses of assets in a portfolio. Tail Dependencies In the previous section, several alternative dependence measures have been introduced. Although these dependence measures are found in many scientific texts, they are hardly found in practice. Nevertheless, Spearman’s rho and Kendall’s tau can easily be calculated. In addition, the interpretation of these measures is rather simple and resembles the one of correlation. In contrast to Spearman’s rho and Kendall’s tau, the tail dependence is not easily calculated. However, we consider tail dependence important since it measures

3.1 Statistical Properties of Asset Returns

41

the degree of diversification in extreme situations. An investor is especially interested in good diversification once extreme losses occur. We investigate the tail dependence among different asset classes. This is done by fitting a t copula to the data and then calculating the tail dependence coefficients accordingly, as in (3.5). We do not make any modeling assumptions about the margins. Therefore, the kernel density is used. Note that the results for the tail dependence do not differ significantly when the empirical distribution is used instead of the kernel density. At first, we analyze global equity indices, i.e., S&P 500, Dow Jones, Nasdaq, FTSE 100, CAC 40, DAX 30, SMI, and Nikkei 225. Table 3.9. Tail dependence coefficients of weekly world equity indices returns. S&P 500 (S&P), Dow Jones (DJI), NASDAQ (Nas), FTSE 100 (FSE), CAC 40 (CAC), DAX 30 (DAX), SMI (SMI), Nikkei 225 (Nik) S&P DJI Nas FSE CAC DAX SMI Nik S&P DJI Nas FSE CAC DAX SMI Nik

1

0.61 0.42 0.21 1

0.22

0.22

0.20 0.08

0.21

0.21

0.21 0.07

0.16

0.17

0.18

0.14 0.08

1

0.30

0.26

0.27 0.08

1

0.37

0.30 0.08

1

0.30 0.08

0.27 0.21 1

1

0.08 1

From Table 3.9 we observe strong tail dependence between the S&P 500 and the Dow Jones. Surprisingly, the S&P 500 and the Nasdaq are not as tail dependent as one would expect. The other numbers are moderate except for the Nikkei, which makes the Nikkei very suitable for global diversification. If we chose a daily frequency for the returns, the tail dependencies become much higher. The tail dependence of the S&P 500 and the Dow Jones for daily returns is 0.93. The Nikkei, for daily returns, has a tail dependence of approximately 0.2 with the other indices. The low tail dependence of the Nikkei with the other indices for daily data may be caused by asynchronous returns, see Audrino and B¨ uhlmann (2003) for details. We analyze the tail dependencies of the commodities found in Table 3.3. The Goldman Sachs (GS) indices show considerable tail dependence with oil. The commodities index and the energy index have a high tail dependence coefficient, indicating that energy prices are considerably influenced by commodity prices. The other tail dependencies are rather

42

3 Modeling of Financial Assets and Financial Optimization

small. The bond indices found in Table 3.4 are very tail dependent. We interpret this result as the fact that the duration does not diversify with respect to extreme losses. Again, we consider the portfolio of an US investor. Recall that this portfolio consists of the S&P 500, Nasdaq, S&P Global 1200, GS Commodities Index, GS Energy Index, and the DS US Government all maturities index. The results for this portfolio are shown in Table 3.10. Table 3.10. Tail dependence coefficients in a typical portfolio. S&P 500 (S&P5), Nasdaq (Nasd), S&P Global 1200 (S&PG), GS Commodities Index (GSCI), GS Energy Index (GSEI), and the DS US Government all maturities index (USFI). S&P5 Nasd S&PG GSCI GSEI USFI S&P5 Nasd S&PG GSCI

1

0.39

0.52

0.01

0.01

0.02

1

0.03

0.01

0.01

0.01

1

0.01

0.01

0.02

1

0.70

0.01

1

0.01

GSEI USFI

1

We observe that the S&P 500 and the S&P Global 1200 have considerable tail dependence, indicating the global importance of the US economy. However, the Nasdaq and the S&P Global 1200 have almost no tail dependence. The large tail dependence between the commodities index and the energy index has already been mentioned. The other tail dependencies are very small. Therefore, the basic portfolio containing stocks and bonds makes extreme losses less severe. The US Government all maturities index has low tail dependence to all other assets, making it very suitable for diversification.

3.2 Dynamic Models of Financial Assets Having investigated static asset return models in detail, the dynamic models of financial assets used in this work are briefly discussed. We do not give technical details about the models. They are either found in the literature or in the corresponding applications later on. In Section 3.1, the statistical properties of financial assets are studied. These properties are inherently static and unconditional. However, by inspection of the evolution of asset prices, it is natural to model them as stochastic processes.

3.2 Dynamic Models of Financial Assets

43

The autoregressive moving average (ARMA) model is popular for financial time series in discrete time. From the stylized facts of asset returns, we know that returns, in general, are not autoregressive. Therefore, we do not aim to model asset returns as ARMA processes. For more details on ARMA models, the reader is referred to Hamilton (1994). The stylized facts of asset returns show that squared returns are autocorrelated. In the introduction of this chapter we mentioned the concept of autoregressive conditional heteroskedasticity (ARCH). Often, we encounter processes of a generalized ARCH type, abbreviated by GARCH. There exists a wide range of possible extensions of GARCH models such as threshold GARCH, denoted by TARCH, and many others. The TARCH model is particularly well suited for modeling equity returns. For more details on ARMA and GARCH models in finance the reader is referred to Alexander (2001), Tsay (2001), or McNeil et al. (2005). A good comparison of different volatility models is found in Sadorsky (2004). The univariate case for GARCH models is usually not sufficient for interesting applications. Therefore, we are interested in multivariate models. Popular multivariate models are the vector GARCH model (VEC) and the BEKK model of Baba, Engle, Kroner, and Kraft. In this work we will make use of the constant conditional correlation (CCC) and the dynamic conditional correlation (DCC) GARCH model. The CCC model was proposed by Bollerslev (1990). The DCC model was introduced in Engle and Sheppard (2001) and Engle (2002). In these models, as for the copula model, the univariate models are separated from the dependence structure. Therefore, the univariate time series are modeled by individual GARCH models and then combined by a dynamic correlation matrix. The DCC GARCH model offers a good tradeoff between model complexity and convenience of the estimation of the parameters, see Engle and Sheppard (2001) for details. We only consider GARCH models in a discrete-time context. For a continuous-time version see Kl¨ uppelberg, Lindner and Maller (2004). Cointegration is a popular model for the dependence of asset return prices and introduced by Engle and Granger (1987). However, it is not further investigated in this context. In general, continuous-time models are mathematically more profound than discretetime models. Stochastic differential equations, driven by Brownian motions, are the most popular continuous-time models. In this type of model, asset returns are assumed to be (conditional) normally distributed. We have seen in this chapter that this is not always appropriate. Brownian motion belongs to the family of L´evy processes which offers much

44

3 Modeling of Financial Assets and Financial Optimization

more degrees of freedom to model asset returns. A L´evy process is a continuous-time stochastic process which is continuous in probability and has stationary, independent increments. Therefore, the increments of a L´evy process have an infinitely divisible distribution. In a L´evy process framework, we may model asset return distributions as members of the GH family. This chapter indicates that this type of distribution is well suited for describing asset returns. Since the introduction of the APT by Ross (1976), factor models are popular for describing asset returns. This often results in a linear regression or in an ARMA model. Factor models represent the thought that asset returns are driven by underlying economic factors such as dividend yields, price-earnings ratios etc. In addition, the use of technical factors for forecasting asset returns is popular, e.g., the momentum. Therefore, we consider the use of factors for two reasons. The first usage of factors is to elaborate the systematic risks of asset returns. By knowing the return drivers of an asset, the investor obtains valuable insights with respect to risk management. This is particularly interesting if an investor is inspecting a financial product. The second purpose of factors is to predict asset returns. The biggest problem thereby is an in-sample overfitting of the model which has no out-of sample prediction ability. To overcome this problem, model selection techniques are necessary. The reader is referred to Burnham and Anderson (1998) for more details on this topic.

3.3 Financial Optimization Techniques As mentioned in the introduction, we attribute the first systematic treatment of the portfolio selection problem to Markowitz (1952). It is still the most popular single-period financial optimization although its deficiencies are widely documented. We already know that the normal distribution is not a suitable model for describing univariate or multivariate asset returns. Further drawbacks of the mean-variance model are that the risk criterion is not coherent and that it is only a single period optimization. Some alternative risk measures to variance in a single-period context are semi-variance (see, e.g., Markowitz (1959)), mean-absolute deviation (see Konno and Yamazaki (1991)), expected regret (see Dembo and King (1992)), and conditional value at risk (see Rockafellar and Uryasev (2000)). In these models, the actual optimization problem becomes either a linear or a quadratic program. These can be solved in very large dimensions.

3.3 Financial Optimization Techniques

45

The single period optimization lacks many important properties which are encountered in real-world investment processes. First of all, it does not account for transaction costs. Second, the possibility of altering the portfolio at different times in the future is not taken into account. In a single period optimization problem, the investment decisions are inherently static. The deficiencies of single period optimizations can only be taken care of if a dynamic optimization approach is used. For deterministic systems as well as stochastic systems, Bellman’s optimality principle and Pontryagin’s minimum principle are often used to find optimal solutions, see Fleming and Rishel (1975) for details. Bellman’s optimality principle is also called dynamic programming and is popular in financial optimization. Some early publications on multi-stage portfolio selection problems are Samuelson (1969), Fama (1970b), and Dantzig and Infanger (1993). When the decision or control variables of the optimization problem are constrained, dynamic optimization problems become very hard to solve. A technique for overcoming this problem is called model predictive control (MPC). The use of MPC in deterministic problems is popular, see Bemporad and Morari (1999) for a survey. This is not the case for stochastic MPC for which many important results have only been found recently, the reader is referred to Herzog (2005) for details. Definition of the model, constraints, investment horizon, optimization criterion.

Estimation of the

r - model parameters based on all 6 available data.

-

Computation of the optimal solution.

Implementation of

- of the optimal solu-

tion and monitoring of the portfolio.

Evolution of the asset prices P (t) → P (t + ∆t) Fig. 3.4. The model predictive control concept in finance

Figure 3.4 shows the MPC strategy conceptually. The crucial idea is that in each step, we solve the whole multi-stage optimization problem but then only apply the current decision variable. The future decision variables are calculated but are not actually implemented since the current decisions variable is recalculated in each decision step. We can either have a fixed or a receding horizon. Besides dynamic programming, there exists a different approach to solving dynamic stochastic optimization problems, called stochastic programming. For introductory text-

46

3 Modeling of Financial Assets and Financial Optimization

books on stochastic programming, the reader is referred to Louveaux and Birge (1997) and Kall and Wallace (1994). For applications of stochastic programming see Ziemba and Mulvey (2001) and Wallace and Ziemba (2005). Further case studies and details on the interplay between dynamic programming and stochastic programming are found in Herzog (2005). For a detailed case study of a stochastic programming approach for the asset and liability management of a Swiss pension fund see Dondi (2005). Finally, we give an important result concerning the interplay between the modeling and optimization of a portfolio. We consider the class of elliptical distributions such as the multivariate normal, t, and symmetric NIG distribution. Suppose we use an arbitrary positive-homogeneous, translation-invariant measure of risk to determine the riskminimizing portfolio with a desired return. Then the portfolio weights are the same as if we used the variance as risk measure. The reader is referred to McNeil et al. (2005) for the proof. This means that, in an elliptical world, the mean-variance efficient portfolio is the same as the mean-VaR efficient portfolio. This section serves as a very brief introduction on the topic of financial optimization. As a matter of fact it is far from complete. However, the literature on this topic is very rich. The reader is referred to Deng, Wang and Xia (2000) for nice overview on models and strategies in portfolio selection. The technical details on the optimization methods used in this work are provided in the applications.

4 Alternative Investments

We must believe in luck. For how else can we explain the success of those we don’t like? Jean Cocteau

4.1 Introduction The traditional portfolio consists of fixed-income securities, stocks, real estate, and cash. In Chapter 2, further types of assets have been introduced, called alternative investments. Alternative assets give investors a further degree of freedom to manage the risk-return characteristics of their portfolios. We call hedge funds, managed futures, private equity, physical assets (e.g. commodities), and securitized products (e.g. mortgages) alternative investments. The statistical properties of some commodities have been studied in Chapter 3. The main emphasis of this chapter is on hedge funds. The inherent properties of hedge funds significantly differ from those of private equity and securitized products. Therefore, the results of this chapter cannot be generalized to private equity investments and securitized products. The first hedge fund is usually credited to Alfred Winslow Jones in 1949. However, Ziemba (2003) finds that already John M. Keynes was using hedge fund techniques in an endowment portfolio in the 1920s. Since these early stages of hedge funds, the hedge fund industry has developed a wide range of different strategies to exploit market inefficiencies. The legal environment for hedge funds differs from the legal environment of traditional investments. Hedge funds are usually not allowed to make public advertisements. Also, hedge funds are often domiciliated in off-shore regions, i.e., in regions which are favorable for tax and legal reasons. Popular off-shore locations are Caribbean islands such as the

48

4 Alternative Investments

British Virgin Islands or the Cayman Islands. Having discussed some issues about hedge funds, we finally give a formal definition of a hedge fund. Definition 4.1 (Hedge Fund). A Hedge Fund is a generally pooled private investment vehicle, often in the form of partnerships or limited companies. It is loosely regulated and not required to have a particular performance objective or fee structure. Hedge Funds are not widely available to the public and may have a limited number of investors. They do not prohibit the use of leverage, short selling, or the use of derivatives. From this definition, the differences of hedge funds compared to mutual funds are obvious. Mutual funds are highly regulated, are neither allowed to use leverage, nor to sell short, or to hold derivatives, and have a given free structure. The differences between hedge funds and mutual funds are discussed in Fung and Hsieh (1997, 1999). Often, high minimum investments are required for investing in hedge funds. In addition, lockup periods prevent investors from withdrawing money quickly. As of December 2005, the hedge fund industry consists of some 8000 funds and manages about US$ 1.1 trillion of assets, according to Hedge Fund Research, a provider of hedge fund data. By considering the evolution of the assets under management in Figure 4.1, we observe a steady growth of the industry. The assets under management have considerably increased since 2002. However, according to the Investment Company Institute, a mutual fund data provider, there are currently US$ 16.1 trillion invested in mutual funds worldwide. This makes hedge fund investments rather small in comparison to mutual fund investments. Seen under the caveat that hedge funds are not open to the public, the figure of US$ 1.1 trillion is still impressive. Because of fraud scandals and their loose regulation, hedge funds are often discussed, in the press as well as in politics. This means that investment managers may commit themselves to a considerable amount of reputational risk by including hedge funds in their portfolio. One of the biggest hedge fund disasters in history, as briefly discussed next, has forced the management of financial institutions to leave the company because of the huge losses incurred. The best known hedge fund disaster is the case of Long Term Capital Management (LTCM), which occurred in 1998. This major event is also observable in Figure 4.1. The literature on LTCM is vast, we only give a few references. A very detailed description of

4.1 Introduction

49

1200

Total Assets [$bn]

1000 800 600 400 200

PSfrag replacements 0 1994

1996

1998

2000

year

2002

2004

(Source: Hedge Fund Research) Fig. 4.1. Assets under management by the hedge fund industry.

the LTCM story is found in Lowenstein (2000). Jorion (2000) analyzes risk management aspects of the LTCM disaster. Because of the legal structure of hedge funds, investors take on more risk of fraud with hedge funds than with mutual funds. The legal environment of hedge funds is most probably subject to changes in the near future. Important issues are transparency and the obligation to register. As the emphasis of this work is not on the legal aspects of hedge funds, although it is an important issue, we will not discuss this topic any further. The reader is referred to Lhabitant (2002) or Cottier (1997) for details on legal and regional aspects. The question whether hedge funds as such are “good” or “bad” is a debate which has been ongoing for years. Opponents of hedge funds blame them for several reasons. One argument is that hedge funds are driving asset prices away from their equilibria because of speculation. Another is that hedge funds are causing financial crises or at least making them worse once they occur. The opposite side claims making markets more efficient, effectively stabilizing markets. Fung and Hsieh (2000a) make an attempt to analyze the market impact of hedge funds. We do not cover this interesting topic in this work because it is less important for portfolio construction and risk management. It is not obvious whether hedge funds are a distinct asset class or not, i.e., a set of assets with stable and homogeneous characteristics. From a conceptual point of view, hedge funds are just a mix of different assets which are actively traded. This does not make hedge funds an asset class of its own. For regulatory and reporting reasons, however, institutional

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investors often consider hedge funds as a distinct asset class. The investment process for hedge funds differs considerably from the case of traditional assets. Additionally, hedge funds have a different risk-return profile than most other assets. This makes them a distinct asset class for portfolio construction. The statistical properties of hedge fund returns demand more sophisticated models than traditional assets. This topic is discussed in more detail later in this chapter. Note that all financial data used in the sequel of this chapter is obtained from the Datastream database of Thomson Financial. The hedge fund data is only available on a monthly basis. All data ranges from 1994 to 2005. 4.1.1 Hedge Fund Fee Structure Hedge funds usually charge a management fee and an incentive fee, whereas mutual funds only charge a management fee. In addition, there is a high watermark included in the fee structure of a hedge fund. This means that the manager will only receive incentive fees if the cumulative returns can make up for previous losses. The incentive fee rewards the hedge fund manager for high absolute returns. In addition, hedge fund managers usually have a considerable amount of their own money invested in the fund. This should motivate the manager to produce high risk-adjusted returns. Kouwenberg and Ziemba (2004) find that incentive fees increase the risk appetite of managers considerably. Only if the manager has a substantial amount of his own money in the fund, the risk taking is reduced. Kouwenberg and Ziemba (2004) also give empirical evidence that hedge funds with high incentive fees have significantly lower mean returns (net of fees) and worse risk-adjusted performance. Liang (1999) finds that a high watermark is very effective in aligning the manager’s interest with the fund performance. Goetzmann, Ingersoll and Ross (2003) find a closed-form expression for the value of a hedge fund manager contract. They state that the value of a hedge fund contract is increased with the variance of the portfolio under certain conditions and provide a discussion of the compensation structure of hedge funds. Hedge fund fee structures are also discussed in Ackermann, McEnally and Ravenscraft (1999). Brown, Goetzmann and Park (2001) find that the risk choice of managers is motivated by industry benchmarks and not by high watermarks.

4.1 Introduction

51

Summarizing, the academic literature on performance fees of hedge funds suggests that high watermarks and a considerable amount of the fund managers own money in the fund are favorable for the investor’s interest. 4.1.2 Hedge Fund Terminology When dealing with alternative investments, we always find the Greek letters α and β. The terminology stems from the capital asset pricing theory (CAPM), introduced by Sharpe (1964). In the CAPM, beta refers to the market risk. This concept has been generalized and is now often found in the literature as r(t) = α(t) +

n X

βi (t)Fi (t) + ξ(t),

(4.1)

i=1

where r(t) refers to the return of the hedge fund, α(t) its intercept, F i (t) an arbitrary factor, βi (t) the factor loading, and ξ(t) is a white noise process. Often, α(t) and β(t) are assumed to be constant. The estimation of the alpha and betas then reduces to a simple linear regression. One of the key questions in hedge fund research is actually the search of alpha and the detection of the betas. However, these goals are either difficult or impossible to achieve. In order to find the manager’s real alpha and the actual betas, one has to know the hedge fund’s investment universe and the financial products used therein. As hedge funds are, in general, reluctant to give the investor insights into these topics, the investor has to use a simplified, hypothetical investment universe of the hedge fund. By introducing the hypothetical investment universe we are speaking of systematic returns of hedge funds and not of the actual sources of returns. However, systematic risk is often found as the term sources of returns in the literature. But without an exact knowledge of the investment positions of a hedge fund over time, the determination of the return drivers of hedge funds is virtually impossible. Note that the estimation of alpha and its significance also depends considerably on the underlying model, see Amenc, Curtis and Martellini (2003) for a detailed study. 4.1.3 Hedge Fund Styles There exists a huge variety of different styles by which hedge fund managers actively manage their portfolios. However, there is neither a generally accepted definition for hedge fund styles, nor is there a general classification of these. Nevertheless, many participants

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in the industry made attempts to identify and classify hedge fund styles. In Table 4.1, a broad overview of common hedge fund styles is shown. Note that the directional style is often described as opportunistic. Table 4.1. Hedge fund styles Directional

Relative Value

Event Driven

Others

Global Macro

Convertible Arbitrage

Long/Short Equity

Equity Market Neutral

Distressed Securities

Equity Market Timing

Dedicated Short Bias

Fixed Income

Regulations D

Equity non Hedge

Emerging Markets

Relative Value Arbitrage

Multi-Strategy

Multi-Strategy

Risk (Merger) Arbitrage Funds of Hedge Funds

The long/short equity style was already used by Alfred Winslow Jones in 1949. It hedges market risk and derives its returns from equity selection skills. The return sources of the common directional style are the correct predictions the movements of security prices. Leverage is sometimes used to increase returns. The relative value style seeks out relative pricing discrepancies between related instruments in equity and fixed income markets. An event driven manager invests in corporations involved in special situations such as mergers and acquisitions. Multi-strategy funds do not have a pure style but rather have a mixture of several styles. Funds of hedge funds invest in several hedge funds and are therefore very different to the styles described so far. Funds of hedge funds will be discussed later in this section. A detailed description of the different hedge styles is found in Cottier (1997), Lhabitant (2002, 2004), and Jaeger (2002). An important topic in the realm of hedge funds is style analysis. Investors are interested in which strategy a hedge fund is following in order to allocate their money or to assess risk. The manager’s investment strategy and the investment universe determine the major part of the risk profile of the hedge fund. Note that hedge funds are classified by styles, however, this does not necessarily describe their strategy. The strategy, in contrary to the style, is described by the investor’s degree of freedom to invest in the hedge funds investment universe. Nevertheless, investors are interested whether a hedge fund applies its reported style and whether this style is consistent over time. Bares, Gibson and Gyger (2004) find that style consistency can significantly affect the survival probability of a hedge fund. This corroborates the style analysis for hedge funds. Style analysis for investment funds and traditional portfolios was pioneered by Sharpe (1992). This style analysis is basically a factor model with the constraint that the factors

4.1 Introduction

53

are returns on asset classes. Since hedge funds make use of dynamic trading strategies, the concept of Sharpe may not be applied to hedge funds as such. Not only do we not know what the exact investment universe of the hedge fund is; we also do not know how the investment positions change over time. Obviously, hedge fund mangers are reluctant to give this kind of information since it would reduce their edge. Because of their dynamic behavior, hedge funds returns may have a non-linear relationship to the returns of other asset classes. The classical style analysis is usually extended by more factors than just the returns on asset classes in order to account for the special properties of hedge funds. These additional factors are supposed to explain the trading style and the leverage decisions of hedge fund managers. These factors are discussed in detail later in this chapter. Fung and Hsieh (1997) state that managers having the same style will generate correlated returns. They use principal components and factor analysis to extract style factors. By this, they circumvent the linear structure of classical style analysis by introducing these new factors. Fung and Hsieh essentially find five dominant investment styles. Brown and Goetzmann (2003) use a generalized least squares procedure to identify different hedge fund styles. They find eight different hedge fund styles. However, from Table 4.1, we find much more conceptual hedge fund styles, which are obviously not all distinguishable by numerical analysis. These findings show that there are probably more qualitative styles than quantitative styles. Another approach to analyze the styles of hedge funds is described in Lhabitant (2002). The main idea in this kind of style analysis is to apply classical style analysis with the nine Tremont hedge fund indices as asset class substitutes; see Table 4.3 (page 66) for the list of the Tremont hedge fund indices. This also circumvents the non-linear exposure of hedge fund returns to traditional asset classes. 4.1.4 Funds of Hedge Funds Funds of hedge funds, as their name suggests, allocate money among several hedge funds. In the sequel, we use the terminology fund of funds instead of fund of hedge funds. A fund of funds may either use a top-down or a bottom-up approach to select different hedge funds. In the top-down approach, the style diversification is established first. In a second step, hedge fund managers from the corresponding style classes are chosen. The

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problem of the top-down approach is that there may not be sufficiently many good hedge fund managers for a particular style available. In the bottom-up approach, the hedge fund universe is screened for the best managers and then the fund of funds portfolio is constructed accordingly. This may result in a significant exposure to a specific risk factor or group of risk factors, i.e., a clustering of risks. The importance of style diversification is also documented in Brown and Goetzmann (2003). The main advantages of fund of funds are the diversification of risk, the lower minimum investment requirement, and the access to closed funds. Fund of funds can, at least to some extent, reduce the idiosyncratic risk of hedge funds by diversification, making the hedge fund investment less risky. Due to minimum investment requirements for single hedge funds, it may be difficult for a single investor to hold a diversified hedge fund portfolio by himself. In addition, the managers of the fund of funds usually have a better understanding of the industry and a broader manager network than the investor. Because of the larger size of the fund of funds, the managers get more insight into the single hedge funds, sometimes even privileged access. Having described the advantages of fund of funds, one might wonder why someone would invest in single hedge funds at all. Unfortunately, there is also the flip side of the coin. We list some severe disadvantages of investing in fund of funds: the most important disadvantage is the second layer of fees. Brown, Goetzmann and Liang (2004) and Amin and Kat (2003) find that the fees on fees in fund of some funds are too high and therefore offer the investor no added value. The second disadvantage lies in liquidity. Fund of funds usually offer better liquidity than the funds. This may lead to severe problems when too many investors want to withdraw their money at the same time. As a last disadvantage we mention the lack of control of the investor. If the investor disagrees with the inclusion of new hedge fund in the fund of funds, the only possibility for the investor is to sell all his shares in the fund of funds. 4.1.5 Hedge Fund Performance Besides the risk, the performance of an asset class is a crucial input for the strategic asset allocation. Measuring the performance of traditional asset classes is conceptually not difficult, because traditional investments are sufficiently regulated and are well studied. In addition, there are several de facto benchmarks. For equities, the MSCI World Index

4.1 Introduction

55

is often considered as benchmark for world wide equities. The S&P 500 index is often considered as benchmark for US equities. In contrast to the performance measurement for traditional assets, the performance of hedge funds is very difficult to measure. The main reasons for this is that hedge funds are not obliged to report. Nevertheless, the hedge fund industry has launched several indices to meet investors’ demand for benchmarks. The basis for the calculation of these indices are hedge fund databases which include style and performance information. Since there is no generally accepted definition for hedge fund styles, the categorization of hedge fund performance figures is difficult. Therefore, the comparison of hedge fund indices among various hedge fund data providers leads to confusion because of the lack of standardization. For a list of various hedge fund data providers see Lhabitant (2002). Since hedge fund managers do not have to report, all hedge fund databases have the same problem of not being complete. Whether these databases are still representative for the whole industry is, again, a difficult question and is not addressed in this context. The reporting of hedge fund performance data is usually not audited. This leads to several biases in hedge fund databases; see Liang (2003) for a discussion on the importance of audited hedge fund data. We list the most important systematic biases which are present in hedge fund databases. • Survivorship Bias In some databases, funds which do no longer report their performance figures are excluded from the database. Poor performance is one reason for hedge fund managers to stop reporting. However, if a manager closes the fund for new investors because the optimal size of assets under management has been reached, the manager may also stop reporting performance figures. Since the reasons for a manager to stop reporting are manifold, survivorship bias is hard to quantify. See Ibbotson and Chen (2005), Bares et al. (2004), Liang (2000), Fung and Hsieh (2000b), Ackermann et al. (1999) for estimations of the survivorship bias. However, the estimated numbers differ significantly because of the already mentioned discrepancies in different hedge fund databases. Estimated numbers of survivorship bias range from 1.5% to 3%. Survivorship bias is considerably easier to estimate for mutual funds. • Selection Bias Every database can only consist of a subset of all existing hedge funds. The first type

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of selection bias stems from the fact that hedge fund managers do not have to report. This is the same mechanism that also causes survivorship bias. The reasons for hedge fund managers to report have already been discussed in the topic of survivorship bias. The second type of selection bias arises from data vendors having additional selection criteria to add hedge funds to their databases. • Backfill (Instant History) Bias Hedge funds usually undergo an incubation period with seed money before they are offered to a broader audience. Once they are included in a database, their previous returns are included into the database as well. Obviously, only successful funds survive the incubation phase. Therefore, the database gives a too optimistic picture of the hedge fund universe. Fung and Hsieh (2000b) estimate the backfill bias by 1.4% on major hedge fund databases. Ibbotson and Chen (2005) find a backfill bias of 4.8%. These biases are just an excerpt of the most important biases. Another bias is reporting bias. Reporting bias originates from hedge funds which report the performance manner not in an objective fashion but favorable for their track record. This may be the case when a hedge fund is invested in instruments which are hard to price, e.g., illiquid assets. These biases may also affect the statistical properties of hedge funds. Ackermann et al. (1999) show that survivorship bias can effect the first two moments and the correlations of hedge fund returns. As mentioned, hedge fund indices are calculated from hedge fund databases. Since these databases are biased, hedge fund indices inhere these biases by construction. Amenc and Martellini (2003) discuss the problem of hedge fund indices and construct pure style indices using either a Kalman filter or a principal component approach. A detailed performance measurement for various different models is conducted. They find a wide range of different alpha estimates for different models. This underlines the inherent problems of accurate performance measurement of hedge funds. Nevertheless, performance measurement remains an important topic for hedge funds. An interesting aspect of performance measurement is performance persistence. The investor is only interested in hedge funds which persistently deliver good performance. Gibson, Bares and Gyger (2003) find shortterm performance persistence but no significant long-term persistence. See also Amin and Kat (2003), Agarwal and Naik (2000), and Lochoff (2002) for more details on performance persistence.

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57

The actual performance of hedge funds should be measured by risk-adjusted performance measures. The most popular risk-adjusted performance measure for traditional investments in the Sharpe ratio, introduced by Sharpe (1966). Having its origins in the CAPM, the Sharpe ratio relies on variance as a risk measure, whose deficiencies are well known. Nevertheless, the Sharpe ratio is also the most widely used risk adjusted performance measure for alternative investments, as found in Amenc, Martellini and Vaissie (2002). Because of the non-normality of at least some hedge fund styles, several further risk adjusted performance measures have been developed. These risk measures are usually based on the left-hand side of the distribution, therefore called downside risk measures. Popular risk measures based on downside risk are the Sortino, the Sterling, and the Burke ratio. Also popular are the return-on-VaR measure and the Omega performance measure. The reader is referred to Amenc, Malaise and Vaissie (2005) for details on these risk measures and how to use them in practice. Note that these measures are closely related to the ones used in active portfolio management, where the benchmark plays a central role. Obviously, the returns of the active portfolio are compared to the returns of the benchmark. The comparison between the active portfolio and the benchmark is usually done by considering the differences between the corresponding returns. The mean of this difference is the average outperformance and often called alpha. The standard deviation of the difference is called tracking error. The ratio between alpha and tracking error is a revised Sharpe ratio, often called information ratio. The problem with all these risk-adjusted performance measures is that the inherent risk measures are all not coherent. Therefore, extreme risks are not sufficiently taken into account. The considered performance measures cannot protect the portfolio against large losses in market stress. Therefore, a conditional risk measure such as CVaR should also be considered for risk-adjusted performance measures.

4.2 Systematic Risks of Hedge Funds and Risk Management We aim to discuss two closely related topics in this section. First, we discuss the topic of systematic risks of hedge funds. Second, important topics for the risk management of hedge funds are discussed. The connection between these two topics stem from the fact that the systematic risks determine an important part of the risk profile of a hedge fund or a fund of funds. It is always possible to analyze unconditional hedge fund returns, but

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knowing the underlying risk exposures gives the investor more insight. Consequently, the knowledge of the risk exposures of all assets in a portfolio allows the investor to analyze the behavior of a combination of different assets. As for traditional assets, we differentiate between systematic and idiosyncratic (nonsystematic) risks. Systematic risk is defined as the part of risk which is not diversifiable. The other part of risk is called idiosyncratic risk. It is standard in finance to assume that the investor is only compensated for systematic risk and not for idiosyncratic risk. The most prominent model for this theory is the CAPM of Sharpe (1964). Whilst systematic risks for traditional assets, e.g., stocks, are well understood, this is not the case for hedge funds. From the classification of hedge fund styles, we know the conceptual return drivers of the corresponding funds. As mentioned, hedge funds usually give no details about their investment positions. However, this information would be needed in order to exactly identify the specific risks of a hedge fund. This problem worsens once we combine several hedge funds of different styles in a portfolio of hedge funds. Idiosyncratic risks of hedge funds are risks such as operational risk, model risk, or the risk of fraud. We do not discuss idiosyncratic risk of hedge funds in detail. It is a very difficult question how idiosyncratic risk of hedge funds is to be quantified. Therefore, due diligence is as important as a quantitative analysis when assessing idiosyncratic risk of hedge funds. Having introduced the concept of separating risk into systematic and idiosyncratic risk, we are interested whether this concept is also found empirically. A straight-forward idea to measure the idiosyncratic risk of hedge funds would be to use the same approach as for stocks, i.e., to analyze the combination of different assets. Fung and Hsieh (2002) build randomly selected portfolios of hedge funds and analyze the variance for an ever larger number of hedge funds. They confirm that risks of hedge funds may be separated into an idiosyncratic and systematic part. However, it takes a lot of hedge funds (more than 120) for the idiosyncratic risk to finally diminish. Therefore, Fung and Hsieh suggest considering systematic risk style by style, which does make sense. Similar results for systematic risk are also found in Patel, Krishnan and Meziani (2002). However, there are also disadvantages when building portfolios of hedge funds. Lhabitant and Learned (2003) find that the more funds are assembled into a portfolio, the higher the correlation with the equity market becomes. In addition, the kurtosis rises with the number of funds combined. Because of this finding, extreme losses cannot be reduced through diversification of idiosyncratic risks.

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The contrary is the case, the higher kurtosis even indicates that extreme losses actually become worse through diversification. Lhabitant and Learned suggest not to have a too high number of funds combined (5-10), because the added value in the overall portfolio is successively decreased otherwise. By isolating the systematic risk through a combination of a large number of hedge funds, we can examine the properties of systematic hedge fund risk. We want to know whether an investor is exchanging idiosyncratic hedge fund risk for systematic exposure to traditional risks by diversifying among hedge funds. This is a very crucial question because the investor is certainly not interested in paying hedge fund managers fees for risk exposures already present in the original portfolio. We have already discussed the topic of the sources of returns of hedge funds. Recall that this terminology may lead to confusion. If we knew the sources of returns, the exact risk profile of the hedge fund in connection with the portfolio would be known. Unfortunately, this is not feasible because we virtually never have full information about the positions of a hedge fund. However, for portfolio construction, we are interested in the systematic risks of hedge funds and not in the actual underlying sources of returns. Of course, systematic risks do only reflect an abstract part of the real sources of returns. Nevertheless, knowing the systematic returns of hedge funds allows the investor to more accurately identify the risk profile of the portfolio. This is of special importance when hedge funds are combined with different types of assets. 4.2.1 Systematic Risks of Hedge Funds We have argued previously in this chapter that hedge funds are conceptually not a distinct asset class but rather a dynamic combination of traditional assets and derivatives. However, by inspection of hedge fund returns, we observe that the statistical properties of hedge fund strategies are different from those of traditional assets. In addition, the legal environment of hedge funds is way different from the one of traditional assets. In contrast to mutual funds, the returns of hedge funds do not only depend on the investment universe of the fund. The returns of hedge funds do also depend on the dynamic trading strategy of the fund, which also includes the leverage decisions of the manager. Once the idiosyncratic risk of hedge funds is eliminated by constructing a fund of funds with sufficiently many hedge funds, only the systematic risk remains. We are interested

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in the nature and the profile of systematic risk. One way to explore the systematic risks of hedge funds is to consider hedge fund indices. Conceptually, hedge fund indices are the same as a large fund of hedge funds. Therefore, the idiosyncratic part of risk in hedge fund indices is considered negligible. We are particularly interested if these systematic risks have an exposure to other risks. The literature on propositions of risk factors explaining systematic hedge fund risk is vast. We review some of the literature and give a collection of the most common risk factors. In particular see Chan, Getmansky, Haas and Lo (2005), Schneeweis and Spurgin (1998, 1999), Amenc, Bied and Martellini (2002), Agarwal and Naik (2004), and Jaeger (2003). Table 4.2 gives a summary of the risk factors found in these publications. Table 4.2. Common risk factors of hedge funds Equity markets

Fixed income related

Commodities

Various

US equity index

Term spread

Oil

Volatility

World equity index

Credit spread

Gold

Currency factor

Emerging markets index

US government bond index GS Com. Index

Momentum factors

Value minus growth

US corporate bond index

Moving averages of indices

Small cap minus large cap

Treasury bills

Lagged returns of equity indices

Bank index

Treasury bills minus LIBOR

Absolute returns of indices

Dividend yields

High yield bonds

Option based risk factors

We give the economic interpretation for some of the factors: Treasury bills stand for future economic activity, the credit spread for the default premium, the oil price for short-term business cycles, and the bank index for the supply of liquidity for hedge funds. Option-based risk factors are essentially non-linear dependence properties of hedge fund returns to returns of traditional assets. For more details on the factors of Table 4.2 and their explanatory power, the reader is referred to the literature. The risk factors for explaining hedge fund returns are also found as asset-based style factors in the literature. In the realm of hedge funds, asset-based style factors are rulebased replications of hedge fund styles using traditional assets and derivatives. The reader is referred to Fung and Hsieh (2004) for asset-based style factors for different hedge fund styles. The option-like pay-off structure to traditional asset classes is documented in Agarwal and Naik (2004), Lhabitant (2004) and many other publications. For instance, Fung and Hsieh (2001) model trend-following strategies as lookback straddles.

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4.2.2 Risk Management for Hedge Funds We have argued that mean-variance optimization cannot capture the statistical properties of traditional assets for portfolio construction. If hedge funds are included in the investment universe, the notorious weaknesses of the mean-variance approach become even more apparent. This is also widely documented in the literature, e.g., see Lhabitant (2004), Lo (2001), or Fung and Hsieh (1998). The main reason for this is that correlation is no longer sufficient to measure diversification. Hedge funds have usually low correlations with traditional assets and a better Sharpe ratio than traditional assets, making them very favorable in terms of the mean-variance framework. This, however, leads to over-allocation of money to hedge funds in a mean-variance framework. The problem lies in the fact that hedge funds incur extreme losses when other assets have extreme losses as well. Therefore, when diversification is needed most, it essentially vanishes. Lo (2001) may be considered as a primer for the risk management for hedge funds. The problems of the use of classical risk management tools for hedge funds are highlighted. Additionally, the need for the special consideration of risk management for extreme events, i.e., tail risk, is stressed and illustrated in an example. Lo also discusses the problems of the use of value at risk in the realm of hedge funds. The ease of building an impressive track record with options is highlighted. This can be done by using derivative securities to mimic dynamic trading strategies. This subject is extensively discussed in Haugh and Lo (2001). An example of such a strategy would be the shorting of out-of-the-money put options. This strategy gives small positive returns most of the time. However, if a negative return of this strategy once in a while occurs, this loss is extreme. The additional danger with this type of strategy is that it is hardly revealed from the hedge funds balance sheet, even with full position transparency. Anson and Ho (2003) find that event driven strategies actually have this type of risk exposure to the equity market. Because of this behavior, hedge funds are sometimes classified as short-volatility strategies. Therefore, the investor has to take these extreme events into special consideration when managing risk of hedge funds. A major source of risks for hedge funds are the credit and liquidity risks. Illiquid assets are known to be hard to price, giving the hedge fund manager some degrees of freedom for the reporting of the performance. Calculated or even manipulated prices are an explanation for the sometimes smooth and persistent hedge fund returns, as reported in

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Kat and Brooks (2001) and Agarwal and Naik (2000). Asness, Krail and Liew (2001) find that hedge funds probably price their securities at a lag. One possibility for this are illiquid or otherwise hard to price assets. After eliminating the serial correlation in the considered return series, Asness et al. (2001) find that the broad hedge fund universe and many subcategories no longer offer return and diversification benefits. Kat and Brooks (2001) analyze the statistical properties of asset returns and their implications for investors. Among other findings they find that hedge fund returns are significantly autocorrelated and that returns of hedge funds are smoothed. We know from the stylized facts of asset returns that returns do not have serial correlation. In an efficient market, serial correlation of asset returns is not possible. However, market frictions such as transaction costs, borrowing constraints, restrictions on trading, and costs of information do exist. These frictions all contribute to the possibility of serial correlation in asset returns, which cannot by exploited because of the presence of these frictions. If these frictions would not exist, traders would immediately make use of the serial correlation, which in turn would eliminate this phenomenon. Getmansky, Lo and Makarov (2004) suggest that the serial correlation of hedge fund returns is most probably caused by illiquid assets. 1

PSfrag replacements

Serial correlation

0.8 0.6 0.4 0.2 0 −0.2 −0.4 0

5

10

Lag

15

20

Fig. 4.2. Serial correlation of the Tremont convertible arbitrage index.

Therefore, the Tremont hedge fund indices found in Table 4.3 (page 66) are analyzed for serial correlation. Figure 4.2 shows the sample serial correlation of the Tremont convertible arbitrage index. The convertible arbitrage index shows significant serial correlation for the first two lags. Besides this hedge fund index, the event driven, the emerging market, the equity market neutral, and the fixed-income arbitrage indices also show significant serial

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correlation. This indicates that many hedge fund managers have illiquid assets in their portfolios. From the stylized facts of traditional asset returns it is known that volatility appears in clusters. We are interested if this is also the case for hedge funds. We investigate for GARCH effects in the Tremont hedge fund indices of Table 4.3. The technical details for GARCH processes are found in Appendix B. There are highly significant (99%) GARCH effects for the styles emerging markets, equity market neutral, global macro, and long/short equity. A possible explanation for this would be that the manager is increasing the risk tolerance because of incurred losses. A high watermark will compensate the manager with incentive fees only if the cumulative returns can make up for previous losses. This is comparable to the St. Petersburg concept of betting, see Weisman (2002) for details. Figure 4.3 shows the dynamic standard deviation of the Tremont long/short equity index, resulting from a GARCH model with t distributed innovations. Note that the results from Table 4.3 indicate that the t distribution is suitable for describing Tremont long/short equity index returns. We observe considerable volatility clustering in the year

Standard deviation

2000. In addition, the hedge fund crisis of 1998 is clearly seen in Figure 4.3, where the

PSfrag replacements

0.06 0.05 0.04 0.03 0.02 Jun94

Oct95

Mar97

Jul98

Dec99

Apr01

Sep02

Jan04

May05

Fig. 4.3. Dynamic standard deviation of the Tremont long/short equity index.

volatility more than doubles in one month. Hedge fund styles play an important role for the risk management of hedge funds. The use of style information is manifold. As mentioned, Bares et al. (2004) find that the survival probability of a hedge fund is significantly affected by style consistency. Therefore, hedge funds should be constantly monitored whether they keep following their reported style or not. This can be either done be the approach suggested by Lhabitant (2002) or the approach found in Fung and Hsieh. Concerning styles, Brown and Goetzmann (2003)

4 Alternative Investments

discuss the importance of style diversification for risk management. Their finding makes the top-down portfolio construction superior to the bottom-up approach. Having mentioned the importance of style diversification, the reader has to be aware of the fact these results are only of indicative nature. Obviously, the style of a hedge fund only gives a very limited description of how the fund actually makes profit. However, what determines the risk-return profile of a hedge fund are the investment universe and the strategies implemented in these. The investment universe defines the opportunity sets of the hedge fund manager, the strategy defines the degrees of freedom therein. Therefore, styles are a classification scheme for hedge funds, hedge fund strategies describe the degrees of freedom of a manager. 4.2.3 Non-linearities in Hedge Fund Returns Since hedge funds use dynamic strategies, their exposure to traditional assets is unlikely to be linear. The returns of hedge fund styles in comparison to the equity market are analyzed. Therefore, the scatter plot of various hedge fund styles versus the the S&P 500 are considered. Mitchell and Pulvino (2001) find that risk arbitrage has a similar return profile to the S&P 500 as the one of the short position of a naked put option on the S&P 500. This result is confirmed by performing a kernel regression on the S&P 500 returns versus the Tremont event driven index returns, as seen in Figure 4.4 (b). (a)

0.1 0 −0.1 −0.2 −0.3 −0.2

0.05

−0.1 0 0.1 S&P500 returns

0.2

(b) Fixed Inc. Arbitrage returns

0.2

Event Driven returns

Emerging Markets returns

Sfrag replacements

64

0 −0.05 −0.1 −0.15 −0.2

−0.1 0 0.1 S&P500 returns

0.2

0.04

(c)

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.2

−0.1 0 0.1 S&P500 returns

0.2

Fig. 4.4. Non-linearities of hedge fund returns. Kernel regressions of S&P 500 returns vs. different Tremont hedge fund indices returns.

The value of R2 for this regression is 77%, i.e., 77% of the variation of Tremont event driven index returns are explained by the kernel regression. Note that R 2 denotes the

4.3 Statistical Properties of Hedge Funds

65

percentage of variation explained by the predictor variables, see Hamilton (1994) for more details on linear regression. By performing a kernel regression on the S&P 500 returns versus the Tremont emerging market index returns, as seen in Figure 4.4 (a), the relationship is similar to the eventdriven case. In this case, 86% of the variations may be explained by the regression function. It would be tempting to state that the Tremont fixed income arbitrage index has the shape of a short call option, as shown in Figure 4.4 (c). However, the R 2 of this regression is only 1%, making this statement insignificant. Nevertheless, if the relationship of the fixed income arbitrage index returns is analyzed with respect to the one month lagged S&P 500 returns, the picture changes dramatically. Instead of a short call option profile as in Figure 4.4 (c), we again observe a short put option profile for the one-month lagged S&P 500 returns, see Figure 4.5. This regression function explains 66% of the variations of the Tremont fixed income arbitrage index.

Fixed Income Arbitrage returns

0.04 0.02 0 −0.02 −0.04 −0.06

PSfrag replacements

−0.08 −0.2

−0.1 0 0.1 1 month lagged S&P500 returns

0.2

Fig. 4.5. Kernel regression of lagged S&P 500 returns vs. Tremont fixed income arbitrage returns.

4.3 Statistical Properties of Hedge Funds The statistical properties of hedge fund returns are analyzed and the implications of the results discussed. Obviously, the quality of the data is crucial for a statistical analysis of hedge fund returns. Liang (2003) finds that the data quality of audited funds is of much better quality than of non-audited funds. Therefore, audited funds should be preferred to

66

4 Alternative Investments

non-audited funds. First, the univariate properties of hedge funds are analyzed. Second, the multivariate properties are discussed. 4.3.1 Univariate Properties of Hedge Fund Returns Two important properties of univariate hedge fund returns have already been discussed. These are the serial correlation and the volatility clustering of hedge fund returns, examples are shown in Figures 4.2 and 4.3. In this section, the (unconditional) properties of univariate hedge fund returns are examined. The methodology is as in Chapter 3, the reader is referred to Appendix A.2.3 for the technical details about the fitted distributions. Table 4.3. Distributions for monthly Tremont hedge fund indices returns. Tremont Hedge Fund Indices min(AIC)

γ1

γ2

Hedge Fund Index

GH

-0.03 5.22

Convertible Arbitrage

GH

-1.39 6.08

Dedicated Short Bias

GH

0.61

Emerging Markets

GH

-1.12 9.19

Equity Market Neutral

Normal

Event Driven

GH

-3.83 30.70

Fixed Income Arbitrage

NIG

-3.23 20.12

Global Macro

GH

-0.20 5.79

Long/Short Equity

t

-0.02 6.88

Managed Futures

Normal

-0.10 3.44

Multi-Strategy

NIG

-1.28 6.58

0.31

4.08

3.27

Table 4.3 shows the detailed results for the Tremont hedge fund index and its subindices including the sample skewness γ1 and kurtosis γ2 . We find that the GH model fits the data best in terms of the log-likelihood value. Table 4.3 shows how the statistical properties hugely differ among the different styles. This is seen from diversely selected distributions according to the AIC value and the very diverse values of γ1 and γ2 . By inspection of Table 4.3, it is seen that the normal distribution is well suited for the case of equity market neutral and managed futures returns. In contrary to this, the normal distribution is very bad for describing returns of the styles event driven and fixed income arbitrage. Figure 4.6 shows the detailed results for the Tremont convertible arbitrage index. The return distribution is considerably skewed to the left, also extreme returns are present in the lower tail. The normal distribution is by far not able to model these observed

4.3 Statistical Properties of Hedge Funds

67

properties. The GH distribution handles the asymmetry of the hedge fund index returns fairly well and can also account for the fat tails.

60

Normal GH Kernel density

50

density

40

PSfrag replacements

30

20

10

0 −0.05

−0.04

−0.03

−0.02

−0.01 0 return

0.01

0.02

0.03

Fig. 4.6. Density estimates for the monthly Tremont convertible arbitrage index returns.

The literature is vast on descriptions of the (unconditional) hedge fund returns. They confirm the findings of this section, see Kat and Brooks (2001) for instance. In the previous section it has been found that GARCH effects may occur for hedge fund returns. Threshold ARCH (TARCH) processes are an extension of GARCH processes and are well suited for describing volatility clustering in declining markets. Therefore, the hedge fund indices of Table 4.3 are tested for significant (99%) TARCH effects. There are significant TARCH effects for the convertible arbitrage and the fixed-income arbitrage index. The univariate properties of hedge fund returns may summarized as follows: • The results vary considerably among the different styles. • The returns may have a high excess kurtosis and negative skewness. • The GH distributions usually is the best model for monthly hedge fund returns, but some styles may even be reasonably modeled as normal. • Some styles have significant serial correlation, indicating illiquid assets. • Some styles show volatility clustering, possibly caused by an increased risk tolerance of the manager because of performance fees.

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4 Alternative Investments

We conclude this section with the following remarks. Since the univariate properties of hedge fund returns differ considerably among the various styles, a universal treatment of hedge fund returns as such it not possible. Every fund, in contrast to most traditional assets, has to be analyzed in detail on its own. 4.3.2 Multivariate and Dependence Properties of Hedge Fund Returns The first aspect of hedge funds, i.e., their claimed outperformance or alpha in comparison to traditional assets, has been discussed already. A further aspect to include hedge funds in a portfolio is because of their benefits for diversification. First, the tail dependence of the different hedge fund styles is investigated. In most cases, there is no significant tail dependence. In particular, the styles convertible arbitrage, dedicated short bias, equity market neutral, fixed income arbitrage, managed futures, and multi-strategy show, if at all, only little tail dependence among each other and to the other styles. The styles which show tail dependence among each other are shown in Table 4.4. Tail dependence coefficients for Tremont hedge fund styles, i.e., Tremont Hedge Fund Index (THFI), Emerging Markets (EmMa), Event Driven (EvDr), Global Macro (GlMa), Long/Short Equity (LSEq). THFI EmMa EvDr GlMa LSEq THFI EmMa EvDr

1

0.28

0.32

0.41

0.43

1

0.24

0.14

0.22

1

0.15

0.31

GlMa

0.13

LSEq

1

Table 4.4. The Tremont hedge fund index has large tail dependence coefficients with the emerging markets, the event driven, the global macro, and the long/short equity index. This may indicate that these styles are dominating the Tremont hedge fund index. Since the Tremont index is asset weighted, these styles may have more assets under management than the other styles in the database which Tremont uses, i.e., TASS. However, because this information is not publicly available, this remains a supposition. The long/short equity style has rather high tail dependence with the event driven style. It is plausible to state that these styles have a considerable amount of capital invested in similar equities. Next, we are interested in the tail dependence coefficients of the different hedge fund styles versus some of the risk factors of Table 4.2. There is no significant tail dependence

4.3 Statistical Properties of Hedge Funds

69

of hedge fund indices with commodities and interest rates. The other tail dependence coefficients are shown in Table 4.5, correlations are given in brackets. The S&P 500 is fairly tail dependent with many styles. The dedicated short bias is not tail dependent on any of the considered risk factors except for the value minus growth index. The fixed income arbitrage index has only considerable tail dependence with the Citigroup US Corporate Bond index with maturities 3-7 years. Table 4.5. Tail dependence coefficients for Tremont hedge fund styles with common risk factors. Dow Jones is

CG

US

Cor

p. 3 -7y

ks i nde x FTS E

US

Ban

th DJ Val ue-G row

DJ Sma ll-la rge cap

kets Mar DJ Em erg.

DJ Wo rld

S&P

500

ex U S

abbreviated by DJ, Citigroup is abbreviated by CG. Correlations are given in brackets.

Hedge Fund Index

0.25 (0.44) 0.23 (0.42) 0.17 (0.44) 0.11 (0.26) 0.02 (-0.12) 0.37 (0.25)

0 (0.24)

Convertible Arbitrage

0.10 (0.11) 0.08 (0.07) 0.09 (0.11) 0.02 (0.15) 0.01 (0.12) 0.19 (0.15)

0 (0.19)

Dedicated Short Bias

0 (-0.70)

Emerging Markets

0 (0.45)

Equity Market Neutral Event Driven

0 (-0.65)

0 (-0.62)

0.16 (0.22) 0.02 (-0.41) 0 (-0.05)

0.23 (0.53) 0.27 (0.75) 0.06 (0.25) 0.06 (-0.06) 0.22 (0.33)

0.08 (0.39) 0.01 (0.27) 0.04 (0.30) 0.03 (-0.04)

0 (0.14)

0.35 (0.53) 0.27 (0.56) 0.18 (0.62) 0.08 (0.30) 0.02 (0.05) 0.23 (0.44)

0 (0.09)

0 (0.09)

0 (0.05)

0 (0.28)

Long/Short Equity

0.30 (0.54) 0.00 (0.59) 0.02 (0.54) 0.15 (0.43) 0.10 (-0.20) 0.38 (0.27)

0 (0.16)

Managed Futures

0.07 (-0.17) 0.03 (-0.05) 0.02 (-0.07) 0.00 (-0.03) 0.01 (0.12) 0.01 (-0.15)

0 (0.22)

0.00 (0.07) 0.13 (0.09) 0.10 (-0.08)

0 (0.07)

0 (0.10)

0 (0)

0.02 (0.04) 0.16 (0.2) 0.20 (0.16)

Multi-Strategy

0.13 (0.20) 0.02 (0.10) 0.07 (0.15) 0.04 (0.06)

0 (-0.01)

0 (0.02)

0.1 (0.33)

Fixed Income Arbitrage 0.09 (0.02) 0.02 (0.01) 0.06 (0.03) Global Macro

0 (-0.33)

0 (-0.01)

0.04 (-0.04)

The FTSE US Banks index has considerable tail dependence to many hedge fund styles. In particular, the style with the highest exposure is long/short equity. This underlines the importance of liquidity for hedge funds, provided by the banks. The emerging markets index has no tail dependence with the US market, but a significant one with the Dow Jones emerging markets index. However, there is considerable tail dependence between the Dow Jones World index ex US and the emerging markets index. This indicates that the emerging market index is only tail-independent with the US market but not with the other developed markets. The fact that correlation does not explain the dependence for extreme events is seen from the results for the Tremont hedge fund index versus the FTSE US Banks index and the Citigroup US corporate bond index. The correlation is

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4 Alternative Investments

in both cases almost the same, however, there is a huge difference in the tail dependence coefficient. Finally, the impact of the inclusion of hedge funds in a portfolio is discussed. We consider a portfolio with equities, bonds, commodities, and hedge funds. As representative for these asset classes we choose the S&P 500, the Citigroup US big corporations 3-7 years index, the Lehman government 1-3 years index, the Goldman Sachs commodities index, and the Tremont hedge fund index. Again, the methodology is the same as in Chapter 3. Table 4.6. Multivariate distribution models for a portfolio including hedge funds. Distribution

log-l

Normal

1897

-3753.9 Gaussian 1932.91

t

1916.5

-3791.1 t (ν=8.6) 1943.28

Skewed t

Copula

log-l

1922.4 -3792.9

NIG GH

AIC

1922

-3791.9

1922.4 -3790.9

Table 4.6 gives the results, the best results are shown in bold numbers. Concerning the parametric distributions, the skewed t distribution gives very promising results. However, the copula models offer better log-likelihood values. As for the case with only traditional assets, the t copula offers the best fit. However, optimal portfolio construction with the fully parametric distributions is much more convenient than with a copula model.

4.4 Hedge Fund Investing We distinguish between two main types of hedge fund investing problems. The first is the problem of building a fund of hedge funds portfolio. The second is the problem of embedding hedge funds in a traditional portfolio. These two problems may also be modeled as dependent of each other. Nevertheless, the investor has to make sure that the hedge fund or the hedge fund portfolio provides a significant alpha for the current portfolio. In addition, the systematic risks of the hedge fund part have to be analyzed in order not to interfere with the systematic risks already present in the portfolio.

4.4 Hedge Fund Investing

71

Embedding Hedge Funds in Traditional Portfolios The reasons for including hedge funds in a traditional portfolio are manifold. Once the decision to include hedge funds in a portfolio has been made, the next problem is to decide how much of the portfolio should consist of alternative investments. We have already encountered the problems when trying to answer this question with the mean-variance approach. Kat and Brooks (2001) and Fung and Hsieh (1998) find that mean-variance portfolio construction with hedge funds is not suitable as well. Cvitani´c, Lazrak, Martellini and Zapatero (2003) solve the problem of hedge fund allocation in a dynamic framework and with model uncertainty. They find that the presence of model risk significantly decreases the amount of wealth invested in hedge funds. Therefore, the over-allocation to hedge funds can be eliminated by introducing model uncertainty. For instance, if the alpha of hedge funds is modeled as random variable rather than a deterministic value, the amount of capital allocated to hedge funds is reduced. The topics of risk and performance measurement have already been discussed. Also, dynamic and static possibilities to model asset returns have been sketched. Once the investor has determined a hedge fund which provides alpha for the existing portfolio, the investor then has to make sure that the hedge fund has no undesirable systematic risk exposures. There exists a huge variety of different approaches to construct a portfolio, e.g., the core-satellite strategy. However, all these different methods of portfolio construction lead to the same optimization problem if the models for the assets and the risk measure are the same. The resulting optimization problems then only differ in the constraints on the decision variables. The term portable alpha is often encountered in the context of alternative assets in traditional portfolios. Concerning hedge funds, the portable alpha is closely connected to the systematic risk of hedge funds. The systematic risks of hedge funds have already been discussed in detail. These risks may also contain traditional systematic risks which are unfavorable for the investor because these are already present in the traditional portfolio. However, these exposures should be small since the investor is not interested in paying fees for traditional risks. One way to dispose the systematic risk exposures to traditional risks is by hedging these risks. Recall the hedge fund terminology of (4.1). By hedging the beta-exposures, what remains is the pure alpha. Of course, the isolation of alpha comes at a cost, i.e., the cost of hedging the unpleasant beta-exposures. Because the alpha is

4 Alternative Investments

now isolated from the risks of the investors portfolio, it is called portable. The investor will not alter the risk profile of the portfolio by introducing such a portable alpha. The return of the portfolio, in contrast to the risk of the portfolio which remains the same, is increased by the portable alpha. We have emphasized the importance of tail dependence for risk measurement. However, this does not make the correlation of hedge funds with traditional assets less important. The correlation is of importance because of the return maximization in normal market situations. Since hedge funds make use of dynamic trading strategies, the correlation properties with traditional asset classes may be dynamic as well. The correlation properties of hedge fund indices versus stocks and bonds are calculated. The dynamic conditional correlation (DCC) GARCH model is used to find the dynamic correlations. The technical details can be found in Appendix B.

1

Tremont Hedge Fund Index

1

0.8

0.8

0.6

0.6

Correlation with bonds

Correlation with bonds

Sfrag replacements

72

0.4 0.2 0 −0.2 −0.4

0.4 0.2 0 −0.2 −0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

0.5

Correlation with stocks

1

Tremont Equity Market Neutral Index

−1 −1

−0.5

0

0.5

Correlation with stocks

1

Fig. 4.7. Correlation of Tremont Hedge Fund Indices with stocks and bonds.

Figure 4.7 shows the dynamic correlations of the Tremont hedge fund index and the Tremont equity market neutral index with stocks and bonds. The stock returns are substituted by the S&P 500 index, the bond returns by the Lehman government bond 1-3 years index. The Tremont hedge fund index shows rather stable correlation with the equity market (around 50 %), the correlation with the bond market is not stable and varies between ± 40 %. The Tremont equity market neutral index correlation is unstable with respect to stocks and bonds. The results for all other Tremont hedge fund indices are similar to the results shown in Figure 4.7. This essentially means that the correlation

4.4 Hedge Fund Investing

73

properties of hedge fund indices are not always stable with respect to traditional assets. This underlines the advantage of active portfolio management, also when hedge funds are part of a portfolio. Hedge Fund Selection What remains is the selection of appropriate hedge funds once the investor has decided how much capital of the portfolio is allocated to hedge funds. However, the engagement in a hedge fund differs enormously from an engagement in a traditional asset. We have argued that risk management is systematic and rational. The same should also apply for the construction of the alternative part of a portfolio. In Section 4.1.4, the topic of funds of hedge funds has been discussed and these results also apply for the hedge fund selection problem. The two main approaches are the top-down and the bottom-up approach. These

Hedge Fund Top-down

Styles

Approach

Hedge Fund Universe

Style Composition

Risk- and Portfolio Management

Manager Selection Bottom-up Approach

Hedge Fund Managers

Fig. 4.8. Hedge fund portfolio construction

two concepts are shown graphically in Figure 4.8. The advantages and disadvantages have been discussed in Section 4.1.4. Therefore, the three main elements for hedge fund investing are: • Style Diversification • Hedge Fund Selection

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4 Alternative Investments

• Risk Monitoring In general, the asset allocation process of Figure 2.1 still applies. In the strategic asset allocation, the hedge fund styles used for diversification have to be defined. The selection of the hedge fund or managers is performed in the investment analysis and the tactical asset allocation. The risk monitoring is performed in connection with the risk management for the other assets. Style diversification has been discussed in Section 4.2. The style diversification is conceptually not different from diversification for traditional investments. What differentiates hedge fund engagements from those in traditional investments is the hedge fund selection process. Figure 4.9 gives a schematic overview. In the first step, potential candidates have Hedge Fund Universe (Database)

Initial

-

Filtering

Potential Candidates

Qual. Analysis

- Short-list

Quant. Analysis

Due

-

Dilligence

Hedge Fund Selection

Fig. 4.9. The hedge fund selection process

to be found by screening the hedge fund universe. For this, several commercial hedge fund databases should be filtered for funds with adequate properties. The elaboration of the list with potential candidates usually depends heavily on the investor’s preferences and less on subjective quantitative criteria. The next step is the qualitative and the quantitative analysis of the funds. This results in a short list of suitable funds. The steps performed thus far can be performed in an automated fashion. The next step, i.e., the due diligence of the funds, analyzes the fund in more details. Due diligence is in a sense a qualitative analysis, but much more comprehensive than the one of the previous step. The due diligence usually includes a visit to the fund managers by experienced professionals. Among other things, the strategy of the fund, the people involved, the infrastructure, and the processes are reviewed and analyzed in detail. Therefore, it is often called operational and structural due diligence. After the individual funds have been selected, the hedge fund portfolio has to be constructed accordingly. We have argued that the mean-variance approach is not appropriate for solving this problem. Some alternative approaches to solve this task are found in the literature. Krokhmal, Uryasev and Zrazhevsky (2002) analyze linear rebalancing strategies for hedge fund portfolios using different risk measures such as CVaR and condi-

4.4 Hedge Fund Investing

75

tional drawdown-at-risk. Agarwal and Naik (2004) build mean-CVaR optimal portfolios and compare these with mean-variance portfolios. Agarwal and Naik find that the meanvariance significantly underestimates the tail risk of the portfolio. Amenc and Martellini (2002) present an improved estimator for the out-of-sample covariance matrix of hedge fund index returns. Summarizing, in the realm of hedge funds, qualitative aspects play a much more important role than for traditional investments. However, this does not make quantitative aspects less important. Hedge fund portfolio construction asks for coherent risk measures which take into account the tails of the return distributions.

5 Optimal Portfolio Construction with Brownian Motions

Opportunity is missed by most people because it is dressed in overalls and looks like work. Thomas Edison

In Section 3.3, the optimization techniques in finance have been reviewed. In this chapter, dynamic asset allocation strategies are developed for asset prices, modeled as continuoustime stochastic differential equations (SDEs) driven by Brownian motion. In this type of model, the conditional distribution of asset returns is normal. Therefore, asset prices are log-normally distributed. Note that the unconditional distribution of returns in this framework is not necessarily the normal distribution. The main advantage of using the continuous-time framework is that the optimal control problem can be solved analytically to a high degree. In some cases, even closed-form solutions may be derived. This gives more insight into the mechanics of an optimal asset allocation strategy than a numerical approximation thereof. However, the modeling properties are rather limited for continuous-time stochastic processes with Brownian motion. We will make use of factors for explaining expected returns of assets. Two types of problems are considered in this chapter. We consider the case were all factors which are explaining the returns of assets are known, i.e., measurable. The second case considers the situation where not all of the factors explaining returns are observable. This problem is called optimal asset allocation under partial information. The optimal asset allocation strategies are derived with a stochastic dynamic programming approach. Therefore, the Hamilton-Jacobi-Bellman (HJB) equation has to be solved. The HJB equation is a non-linear partial differential equation, which is very hard to solve if the control variable is constrained. For problems in higher dimensions, it is virtually impossible to find

78

5 Optimal Portfolio Construction with Brownian Motions

analytical solutions for the constrained case. This fact and the limited possibilities to model asset returns are the main disadvantages of modeling assets in continuous-time stochastic differential framework with Brownian motion. In this chapter, when we speak of continuous-time finance, we actually refer to a continuous-time stochastic differential framework with Brownian motion. After having introduced the dynamics of the considered assets, we are able to define the wealth dynamics of our investor. The investor’s portfolio is self-financing, i.e., there are no external in- or outflows of money. We are considering two types of investors in this chapter. They are characterized by their corresponding utility functions. On the one hand, we consider the popular constant relative risk aversion (CRRA) case. On the other hand, we consider the constant absolute risk aversion case (CARA). The problems are solved by using Bellman’s optimality principle. For the partial information case, we show that the separation theorem is not valid anymore, i.e., we cannot separate the estimation from the optimization. This means that we cannot simply estimate the unobservable quantities and treat them afterwards as if they were known exactly. The asset allocation strategies are all backtested with real market data. The data in this chapter is obtained from the Datastream database of Thomson Financial.

5.1 The Full Information Case The cornerstones of continuous-time finance are the publications of Samuelson (1969) and Merton (1969, 1971). Since then, these results have been extended to a wide range of improved models and applications. The most popular application of continuous-time finance is the pricing of contingent claims by Black and Scholes (1973). The modeling of fixed-income securities is also popular in the continuous-time framework. The reader is referred to Shreve (2004) and Bj¨ork (1998) for a detailed treatment of various topics in continuous-time finance. We are less interested in solely modeling in continuous time but rather in the solution of optimal asset allocation problems resulting thereof. The solutions of optimal asset allocation problems can be used either for asset pricing or for optimal portfolio construction. In the developments of Samuelson and Merton, the optimization problem is solved under full information. For the full-information case, several closed-form solutions have been derived. Since the literature on this topic is vast, we can only mention a few. Among them

5.1 The Full Information Case

79

are Kim and Omberg (1996), Browne (1999), Korn and Kraft (2001), Herzog, Dondi, Geering and Schumann (2004), Wachter (2002), Keel, Herzog and Geering (2004), and Munk and Sorensen (2004). A recent publication on this subject is Schroder and Skiadas (2005). Kim and Omberg (1996) find a closed-form solution for a single risky asset with stochastic risk premium. In addition, conditions for the solvability of the problems are discussed, although not in sufficient detail. Browne (1999) gives the analytical solution for the problem of beating a stochastic benchmark. Thereby, the problem of maximizing the expected discounted reward of outperforming the benchmark, as well as the minimization of the discounted penalty paid by being outperformed by the benchmark is discussed. Wachter (2002) solves the optimal portfolio choice problem for an investor with utility over consumption under mean-reverting returns in closed form. The asset allocation problem for investing among stocks, bonds, and cash is solved for a power-utility investor with meanreverting returns and interest rate uncertainty in Munk and Sorensen (2004). However, the excess return is perfectly negatively correlated with the asset prices, which is a limiting assumption. Korn and Kraft (2001) analyze the problem of utility maximization over terminal wealth for stochastic interest rates. The investment opportunities are a savings account, stocks, and bonds. A verification theorem without the usual Lipschitz assumptions is proved. Keel et al. (2004) consider the application of optimal portfolio construction with interest rate risk, market risk, and the risks introduced by an alternative investment. Schroder and Skiadas (2005) introduce a class of recursive utility, which includes additive exponential utility. The solutions, including convex trading constraints, are obtained by solving a constrained forward-backward stochastic differential equation. Since the work of Merton, it is obvious that optimal investment strategies depend on the investment horizon. Therefore, a long-term strategy differs usually in a significant way from the myopic optimization. The long-term optimization problem contains, in addition to the myopic policy, the intertemporal hedging demand. This kind of problem is also called strategic asset allocation and is introduced in Brennan, Schwartz and Lagnado (1997). The strategic asset allocation problem including an arbitrary number of factors explaining mean returns of assets is described in Herzog, Dondi, Geering and Schumann (2004). Similar results are also found in Liu (2005).

80

5 Optimal Portfolio Construction with Brownian Motions

By introducing multiple factors, one has to solve a high-dimensional, non-linear partial differential equation (PDE) to compute the solution. The use of numerical dynamic programming with discrete state approximation suffers from Bellman’s curse of dimensionality and is therefore restricted to very few factors. For an illustration of the problems involved with numerical dynamic programming, the reader may refer to Peyrl, Herzog and Geering (2004). Further advances in numerical methods for solving partial differential equations may make this approach also applicable to higher dimensions. There are many publications on the issue of using explanatory factors for improving the asset allocation. In Fama and French (1993), five common factors for stocks and bonds are identified. For the stock market, an overall market factor, a factor related to the firm size, and book-to-market equity are considered. For the bond market, two factors, related to maturity and default risk, are considered. Some additional publications on this subject are Moskowitz and Grinblatt (1999), Jegadeesh (1990), Bossaerts and Hillion (1999), and Rosenberg, Reid and Lanstein (1985). These papers suggest that the momentum of equities possesses predictive power. In Bossaerts and Hillion (1999), evidence is given that the relation between factors and asset returns is non-stationary. This is a further motivation for having a dynamic prediction model. In Amenc, Bied and Martellini (2003), the predictability of hedge fund returns is discussed. In Campbell and Hamao (1992) and Harvey (1995), possible factors for an international diversified portfolio are discussed. There are various extensions to the standard problems discussed in the literature. The problem of transaction costs is considered in Oksendal and Sulem (2002) and Janecek and Shreve (2004). However, the resulting problems are no longer analytically solvable. Bertsimas and Lo (1998) derive dynamic optimal trading strategies that minimize the expected cost of trading a large block of equity over a fixed time horizon. Portfolio choice problems with stochastic volatility are considered in Fleming and Hernandez-Hernandez (2003) and Chacko and Viceira (2005). Since stochastic dynamic volatility is a stylized fact of asset returns, these results are more realistic than the ones with deterministic volatility. 5.1.1 The Model The investment opportunities are modeled as stochastic differential equations. They are assumed to behave as local geometric Brownian motions. The drift terms of the asset price

5.1 The Full Information Case

81

dynamics are modeled as affine functions of explanatory factors. We model the factors via stochastic differential equations as well. The diffusions of the n risky price processes are driven by an n-dimensional standard Brownian motion process Wp . Similarly, the diffusions of the m factor processes are driven by an m-dimensional standard Brownian motion process Wx . In addition to the risky asset, there is a risk-free asset (or bank account) whose instantaneous return is deterministic. Nevertheless, the return on the risk-free asset may be time-dependent or even stochastic. The Brownian motion Wp of the price processes and the Brownian motion Wx of the factor processes are defined on a fixed, filtered probability space (Ω, F, {Ft }t≥0 , P) with Ft satisfying the usual conditions. The Brownian motions Wp and Wx do not need to be independent. Factor Dynamics In order to improve the asset allocation, factors are included in the asset models. We expect the factors to have some predictive power for the returns of the investment opportunities. The m factors x(t) ∈ Rm are modeled by the following stochastic vector process dx(t) = [Ax (t)x(t) + a(t)]dt + σx (t)dWx (t),

(5.1)

x(0) = x0 , where Ax (t) ∈ Rm×m , a(t) ∈ Rm , σx (t) ∈ Rm×m , and Wx (t) ∈ Rm . The matrix and vector functions Ax (t), a(t), and σx (t) are deterministic functions. In addition, σx (t) is assumed to have full rank for all t. The factor process allows us to model different variables affecting the mean return of the risky assets. Asset Price Dynamics The set of investment opportunities of our investor consists of n ≥ 1 risky assets. The

asset price processes P = (P1 (t), P2 (t), . . . , Pn (t)) ∈ Rn of the risk-bearing investments satisfy the stochastic differential equations, where diag(P (t)) denotes the diagonal matrix of the vector P (t), dP (t) = diag(P (t)) {µ(x(t), t) dt + σp (t)dWp (t)} ,

(5.2)

P (0) > 0 , where µ(x(t), t) ∈ Rn , σp (t) ∈ Rn×n , and Wp (t) ∈ Rn . The matrix function σp is assumed to be deterministic. From the diffusion matrix σp we get the instantaneous covariance

82

5 Optimal Portfolio Construction with Brownian Motions

matrix per unit time as Σ(t) = σp (t)σp (t)T . The matrix function Σ has to be invertible, therefore σp has to have full rank. The correlation matrix ρ ∈ Rn+m×n+m of the Brownian

motion W = [WpT , WxT ]T is defined as



ρ(t) = 

I1 ρ(t) T

ρ (t) I2



,

where I denotes the identity matrix and ρ ∈ Rn×m , I1 ∈ Rn×n , and I2 ∈ Rm×m . In addition, there exists a risk-free investment opportunity B(t) with instantaneous rate of return r(x(t), t) ∈ R: dB(t) = B(t)r(x(t), t)dt ,

(5.3)

B(0) > 0 . The scalar function r is deterministic. The drift terms of the risk-free asset and the riskbearing assets depend on the m factors x. Furthermore, we assume that the drift terms in (5.2) and (5.3) are affine functions of the factor levels, as given by µ(x(t), t) = G(t)x(t) + f (t) ,

(5.4)

r(x(t), t) = F0 (t)x(t) + f0 (t) ,

(5.5)

where G(t) ∈ Rn×m , f (t) ∈ Rn , F0 (t) ∈ R1×m , and f0 (t) ∈ R. The matrix and vector functions G(t), f (t), F0 (t), and f0 (t) are all deterministic. Portfolio Dynamics We assume that the investor’s portfolio is self-financing, i.e., there are no exogenous inor outflows of money, e.g., consumption. The dynamics of the investor’s wealth V may be expressed as n ³ dB(t) X dPi (t) ´ + ui (t) dV (t) = V (t) u0 , B(t) Pi (t) i=1

V (0) > 0,

where u1 (t), . . . , un (t) denotes the fraction of wealth invested in the corresponding risky asset and u0 (t) accordingly in the risk-free asset at time t. The proof is either found in Merton (1992) or Bj¨ork (1998). Because the portfolio is self-financing, we have the P constraint ni=0 ui (t) = 1. Inserting the definitions of the asset price dynamics of (5.2) and (5.3) in (5.6), we arrive at the following wealth dynamics:

5.1 The Full Information Case

dV = {uT [µ(x, t) − 1n r(x, t)] + r(x, t)}dt + uT σp dWp , V

83

(5.6)

where 1n is defined as the vector 1n = (1, 1, . . . , 1)T ∈ Rn and u(t) = [u1 (t), . . . , un (t)]T . 5.1.2 Optimal Asset Allocation Optimal Asset Allocation with a CRRA Utility Function The optimal asset allocation is derived for an investor having constant relative risk aversion (CRRA). Therefore, the expected power utility over terminal wealth is maximized. The definitions of µ and r, found in (5.4) and (5.5), are inserted into the wealth dynamics of 5.6. The investor’s optimization problem under full information becomes (with time arguments omitted for better readability) max1

u(·)∈Ln

s.t.

E

h1 γ

V (T )γ

i

dV = V {uT [F x + f ] + F0 x + f0 }dt + V uT σp dWp

(5.7)

dx = [Ax x + a]dt + σx dWx dWp dWx = ρ dt, with V (0) = V0 and x(0) = x0 . The parameter γ < 1 is the coefficient of risk aversion. The deterministic matrix functions F ∈ Rn×n and f ∈ Rn of the investors wealth dynamics are defined as F = G − 1 n F0 ,

(5.8)

f = f − 1 n f0 .

(5.9)

The deterministic matrix functions G, f , F0 , and f0 are defined in (5.4) and (5.5), the vector 1n is defined as in (5.6). In order to solve this problem with Bellman’s optimality principle, we introduce the value or cost-to-go function as J(t, V, x) = max1 E u(·)∈Ln

h1 γ

i V (T )γ .

The optimal asset allocation strategy is the solution of the Hamilton-Jacobi-Bellman equation. For a proof in the recent literature see Yong and Zhou (1999). The HamiltonJacobi-Bellman partial differential equation for this problem is

84

5 Optimal Portfolio Construction with Brownian Motions

h

Jt + maxn V {uT [F x + f ] + F0 (t)x(t) + f0 (t)}JV + (Ax x + a)T Jx u∈R i 1 1 + V 2 uT ΣuJV V + V uT Ψ JV x + tr{Jxx σx σxT } = 0, 2 2 with Ψ = σp ρ σxT and terminal condition J(T, V (T ), θ(T )) = γ1 V (T )γ . This kind of problem is solved in Herzog, Dondi, Geering and Schumann (2004). The reader may consult this publication for the proof. The optimal control law u∗ of this problem is given by ´ ³ 1 −1 F x + f + Ψ [K1 x + k2 ] , Σ u = 1−γ ∗

(5.10)

where k2 (t) ∈ Rm and K1 (t) ∈ Rm×m are the solutions of two matrix Ricatti equations. The differential equation for the matrix K1 (t) is given by

−

γ (γ − 1)

K˙ 1 + K1 σx σxT K1 + K1 A + AT K1 ³ T ´ T F Σ −1 F + F Σ −1 Ψ K1 + K1 Ψ T Σ −1 F + K1 σx ρT ρσxT K1 = 0,

(5.11)

with terminal condition K1 (T ) = 0. The differential equation for the vector k2 (t) is given by

−

γ (γ − 1)

k˙2 + γF0T + K1 σx σxT k2 + AT k2 + K1 a ³ T ´ T F Σ −1 f + F Σ −1 Ψ k2 + K1 Ψ T Σ −1 f + K1 σx ρT ρσxT k2 = 0,

(5.12)

with terminal condition k2 (T ) = 0. We summarize the results in the following lemma. Lemma 5.1 (Full Information, CRRA Case). The optimal asset allocation strategy under full information for the CRRA case (5.7) is given by (5.10), where K1 (t) and k2 (t) are the solutions of two matrix Ricatti equations (5.11) and (5.12), respectively. Proof. See Herzog, Dondi, Geering and Schumann (2004). Optimal Asset Allocation with a CARA Utility Function The optimal asset allocation is derived for an investor having constant absolute risk aversion (CARA). Therefore, the expected exponential utility over terminal wealth is maximized. For the problem to be solvable, we model the risk-free interest rate r(t) as a function of time only, i.e., independent of x(t). Therefore, we set F0 ≡ 0 in (5.5). We proceed as in the CRRA case and state the investors optimization problem as

5.1 The Full Information Case

max1

u(·)∈Ln

s.t.

85

i h 1 E − e−γV (T ) γ

dV = V {uT [Gx + f ] + r}dt + V uT σp dWp

(5.13)

dx = [Ax x + a]dt + σx dWx dWp dWx = ρ dt, with V (0) = V0 and x(0) = x0 . The parameter γ > 0 is the coefficient of risk aversion, f is as defined in (5.9). The value or cost-to-go function for this problem is i h 1 −γV (T ) . J(t, V, θ) = max1 E − e u(·)∈Ln γ

This results in the following Hamilton-Jacobi-Bellman partial differential equation for this problem h

Jt + maxn V {uT [Gx + f ] + r}JV + (Ax x + a)T Jx u∈R i 1 1 2 T + V u ΣuJV V + V uT Ψ JV x + tr{Jxx σx σxT } = 0. 2 2

This type of HJB partial differential equation is solved in Herzog, Dondi, Geering and Schumann (2004). The optimal solution u∗ is given by ´ ³ 1 u∗ = − Σ −1 Gx + f + Ψ [K1 x + k2 ] . γV k3

(5.14)

It remains to define the functions K1 (t), k2 (t), and k3 (t). The matrix function K1 (t) is the solution of the following linear matrix differential equation K˙ 1 + K1 σx σxT K1 + K1 A + AT K1 − F T Σ −1 F − F T Σ −1 Ψ K1 −K1 Ψ T Σ −1 F − K1 σx ρT ρσxT K1 = 0,

(5.15)

with terminal condition K1 (T ) = 0. The vector function k2 (t) is the the solutions of the following differential equation k˙2 + K1 σx σxT k2 + AT k2 + K1 a − F T Σ −1 f − F T Σ −1 Ψ k2 −K1 Ψ T Σ −1 f − K1 σx ρT ρσxT k1 = 0,

(5.16)

with terminal condition k2 (T ) = 0. Finally, the scalar function k3 (t) is the solution of the following differential equation k˙3 + f0 (t)k3 = 0,

k3 (T ) = −γ .

We summarize the results in the following lemma.

(5.17)

86

5 Optimal Portfolio Construction with Brownian Motions

Lemma 5.2 (Full Information, CARA Case). The optimal asset allocation strategy under full information for the CARA case (5.13) is given by (5.14), where K1 (t), k2 (t), and k3 (t) are the solutions of the differential equations (5.15), (5.16), and (5.17), respectively. Proof. See Herzog, Dondi, Geering and Schumann (2004). 5.1.3 Case Study with Alternative Investments We consider an investor having three risky investment opportunities. These are the stock market, the bond market, and alternative investments. Each of the three investment opportunities offers a different risk-return profile. For the fixed income part, the short rate model of Vasicek (1977) is used. As second investment opportunity, we consider the stock market. We chose a stock market index as a proxy for the market portfolio. It is modeled by a geometric Brownian motion. Its drift and diffusion are constant. For the hedge fund, we use a model originating from Sharpe’s capital asset pricing model including the Greek letters α and β. However, we only use the terminology of the CAPM but do not need the assumptions of the CAPM. Since the model is in continuous time, the intertemporal capital asset pricing model (ICAPM) of Merton (1973a) is used. Cvitani´c et al. (2003) use a similar model for hedge funds. As a consequence, the alternative investment does not have a constant risk premium. This is also the case for the market portfolio since the risk-free interest rate is not constant. The investment opportunities are modeled by appropriate stochastic differential equations. The investor’s utility function is chosen to have constant relative risk aversion. The Model In order to derive the optimal investment strategy, we first need to model the three considered investment opportunities. The Brownian motions of the continuous-time stochastic differential equations involved are defined on a fixed, filtered probability space (Ω, F, {Ft }t≥0 , P) with Ft satisfying the usual conditions. As mentioned, a short rate model is used for the fixed income part. The investor is able to put money into a bank account. The bank account has an interest rate equivalent to the short rate. We have the following SDE for the short rate r:

5.1 The Full Information Case

dr = κ(θ − r)dt + σr dWr ,

87

(5.18)

r(0) = r0 , where κ, θ, σr ∈ R are the constant parameters of the short rate. Given the short rate, we solely need to determine the price of risk λ to determine the dynamics of the bond B with maturity T , ³ λσr ´ σr dB = B r + aT dt − B aT dWr , κ κ B(T ) = 1,

(5.19)

where the scalar function aT (t) is defined as aT (t) = 1 − e−κ(T −t) .

(5.20)

The reader is referred to Vasicek (1977) for details. The second investment opportunity is a passive equity fund, regarded as a proxy of the market portfolio. The passive fund S has the SDE dS = Sµs dt + Sσs dWs ,

(5.21)

S(0) = S0 , where µs , σs ∈ R are the constant parameters of the model. Furthermore Ws is assumed to be independent of Wr . As a last step, the model for the alternative asset remains to be introduced. The price of the alternative asset, denoted by A, is modeled by dA = A(r + β(µs − r) + α)dt + AσA (ρdWs + A(0) = A0 ,

p 1 − ρ2 dWA ),

where β, α, σA , ρ ∈ R are the constant parameters of the model. Furthermore WA is assumed to be independent of Wr and Ws . In this context, r + β(µs − r) describes the risk adjusted return of the asset with respect to the market, whereas α denotes the outperformance of the alternative asset. The β parameter is defined to be β=

cov(dS/S, dA/A) ρσA = , 2 σS σS

(5.22)

where ρ denotes correlation of the return of the market portfolio and the return of the alternative asset. We introduce a three-dimensional control vector u. The three components of u represent the percentage of total wealth invested in the respective investment category. In our case, the wealth equation becomes

88

5 Optimal Portfolio Construction with Brownian Motions

h

i

T

dV = V u µ(r, t) + r dt + V uT σdW, where u ∈ R3 , W = [Wr , Ws , WA ]T ∈ R3 and initial condition V (0) = V0 . The vector µ(r, t) is 

  µ(r, t) = F r + f (t) =  

λσr a (t) κ T

µs − r β(µs − r) + α



  , 

whereas the matrix σ(t) is defined to be   σr − a (t) 0 0   κ T   σ(t) =  . 0 σs 0   p 2 0 σ A ρ σA 1 − ρ

The matrix σ(t)σ T (t) has to be invertible and therefore |ρ| < 1. Solution to Asset Allocation with CRRA Utility The portfolio choice problem is to maximize the expected power utility defined over terminal wealth. Furthermore, we assume that leveraging, short-selling, and borrowing at the risk-free rates are unrestricted. Mathematically, the problem statement is max u(·)

s.t.

n1 o E V γ (T ) γ

h i dV = V uT µ(r, t) + r dt + V uT σdW dr = κ(θ − r)dt + σr dWr ,

with initial conditions V (0) = V0 and r(0) = r0 . The time horizon is denoted by T and γ < 1 denotes the coefficient of risk aversion. The solution of this problem, given Σ(t) = σ(t)σ(t)T and e1 = [1, 0, 0]T , is u∗ (t, r) =

´ ³ 1 σ2 Σ(t)−1 µ(t, r) − r aT (t)[k1 (t)r(t) + k2 (t)]e1 . 1−γ κ

(5.23)

The two functions k1 (t) and k2 (t) are the solutions of two coupled ordinary differential equations (ODEs). The ODE for k1 (t) is σr2 2 k − h1 = 0, k˙1 − 2κk1 + 1−γ 1

k1 (T ) = 0 .

(5.24)

5.1 The Full Information Case

89

The only unknown in the ODE for k1 (t) is the constant h1 , which is defined by h1 =

γ . (γ − 1)σs2

The ODE for k1 is independent of k2 and can be therefore solved independently. Because of the form of (5.24), there exists a closed-form solution. Define the function Υ (t) to be s n ³ κ ´o h1 σr2 Υ (t) = tanh (T − t)δ + atanh , δ= + κ2 . δ 1−γ We finally have for the solution of k1 k1 (t) =

´ 1 − γ³ κ − δΥ (t) . σr2

The remaining unknown of the solution is k2 (t). From the general solution we know that k2 (t) is the solution of an ODE which is dependent of k1 (t). The ODE for k2 (t) is, for our specific case, given by ³ ´ σr2 ˙ k2 − k2 κ − k1 + k1 h3 + h2 = 0, 1−γ

k2 (T ) = 0 .

(5.25)

The constants h2 and h3 are found to be h 2 = γ + h 1 µS

,

h3 = κθ −

γ σr λ. γ−1

Again, we can give an analytical solution, but the form of the solution of (5.25) is more complicated than for k1 (t), i.e., k2 (t) = C

p

1−

Υ (t)2

´ 1−γ 1³1 − γ + κh3 + h2 Υ (t) − h3 2 . 2 δ σr σr

The integration constant C has to be chosen such that the terminal condition k2 (T ) = 0 is met, therefore we get C=

√

κh2 ´ δ 2 − κ2 ³ h3 (1 − γ) . − δ σr2 δ 2 − κ2

The conditions for existence of a solution are σr 6= 0, σs 6= 0, and γ < 0. Since we want to optimize a portfolio with alternative investments, we are especially interested in the third component of u∗ in (5.23). It reflects the fraction of wealth allocated to the alternative investment and is given by u∗3 =

α . (1 − γ)(1 − ρ2 )σA2

90

5 Optimal Portfolio Construction with Brownian Motions

The amount of capital invested in the alternative investment increases linearly with α. The closer the variance of A is to zero, the more is invested in the alternative investment. At first sight, it is counter-intuitive that the larger the absolute value of the correlation ρ, the more is invested in the alternative asset. But if we take a look at u∗2 , the fraction of wealth invested in the market, we observe that, for large absolute values of ρ, the value of u∗2 changes significantly. The asset allocation rule exploits this correlation property by taking much more extreme positions when a large positive or negative correlation is present, u∗2 =

1 σA (µS − r) − u∗3 ρ. 2 (1 − γ)σS σS

The first term of u∗2 is seen to be the well known solution of Merton (1992), whereas the second term depends on the amount of wealth invested in the alternative investment u∗3 . If the correlation ρ is equal to zero, the position in the market is the same as in the standard Merton case. If ρ is positive, the position in the market is reduced in favor of the position in the alternative investment (assuming a positive α). The interesting property lies in the fact that, if the correlation is negative, the optimal weight in the market is larger than in the positively correlated case. The lower the correlation, the more the downturns of the alternative investment are hedged by the position in the stock market. One reason to include hedge funds in portfolios is because of their benefits of diversification, i.e., low correlation. In the perfectly independent case, i.e., ρ = 0, the fraction of wealth invested in the market remains unaffected in terms of the Merton solution. Because of the lack of dependence between the bond market and the stock market and the alternative investment, respectively, the amount of wealth invested in the bond is independent of the characteristics of the market and the alternative investment. Analysis and Backtest of the Asset Allocation Strategy with US Data The optimal control vector u(t) is computed for real market conditions. We use US stock market data and the Tremont hedge fund index. For the equity fund, i.e., the substitute for the market portfolio, the S&P 500 is used. As a proxy for the short rate, we use three month Treasury bills, which have interest rates close to the ones paid on a money market account. For the bond portfolio part, the Datastream USA Total 3-5 years bond index is used. In order to account for the coupon payments, the total return index data is used

5.1 The Full Information Case

91

which is a suitable approximation for the zero coupon bond. The data is obtained on a weekly basis except for the Tremont hedge fund index which is only available on a monthly basis. The three-month Treasury bills are used as a proxy for the short rate for two reasons. The first is because of its long availability (since 1972), which is important for the estimation of the short rate parameters. The second reason is that the federal fund rate is locally deterministic and therefore not suited as proxy for the short rate. The resulting control vector u(t) crucially depends on the parameters chosen. The parameters used for the market portfolio can be estimated with long time series of data and are therefore reliable long term estimates. This is also the case for the fixed income security. The stochastic differential equation for the short rate (5.18) is discretized with the method of Euler, see Kloeden and Platen (1999) for details. We get √ rt+1 = κθ∆t + (1 − κ∆t)rt + σr ∆t ξr , where ∆t is the time increment and ξr is a standard normal white noise process. The parameters of the short rate are estimated by doing an ordinary least squares estimation on the discrete version of the short rate. We assume that the bond has a fixed duration, i.e., having a roll-over bond portfolio part in the entire portfolio. This can be achieved by changing the time-varying function aT (t) in (5.20), to be a function of the duration τ of the bond portfolio part only. We discretize the stochastic differential equation of the logarithmic bond prices (5.19) with the method of Euler and get ´2 ´ ³ σ √ 1 ³ σr σr r aT (τ ) ∆t + aT (τ ) ∆t ξB , ln(Bt+1 ) − ln(Bt ) − rt ∆t = λ aT (τ ) + κ 2 κ κ where ∆t is the time increment and ξB is a standard normal random variable. The duration τ and the price of risk λ of the bond index are estimated by estimating mean and variance of the series above. The drift and the diffusion of the market portfolio are computed in the same way as the bond prices. The price process (5.21) is transformed with the natural logarithm. The resulting stochastic differential equation is then used in its discrete version, using the method of Euler. This gives the relationship √ ¡ 1 ¢ ln(St+1 ) − ln(St ) = µS − σS2 ∆t + σS ∆t ξS , 2

92

5 Optimal Portfolio Construction with Brownian Motions

where ∆t is the time increment and ξS is a standard normal random variable. The drift and the diffusion of the Tremont hedge fund index are computed analogously to the market portfolio. The correlation is estimated by calculating the correlation of the residuals of ln(A) and ln(S). With these estimates at hand, α can be estimated by subtracting r + β(µS − r) from the mean of the Tremont hedge fund index returns. Table 5.1. Typical values for the estimated parameters. Parameter

value

std. error t stat. Parameter

value

std. error t stat.

κ

0.26

0.05

4.8

µS

0.1 p.a.

0.008

11.7

θ

0.07 p.a.

0.005

13

σS

0.16 p.a.

0.004

38.8

σr

0.02 p.a.

0.0007

27.5

ρ

0.49

0.07

6.9

λ

0.56

0.04

14.8

α

0.05 p.a.

0.014

3.27

τ

3.66 years

0.4

8.9

σA

0.09 p.a.

0.006

14.1

Using the time series of the market and the short rate from 1972 to 2005, the bond index from 1980 to 2005, and the alternative asset from 1994 to 2005, we estimate the parameters of the stochastic processes. Table 5.1 shows the results. By considering the t statistics of all parameters, it is seen that they are all significant.

2.5 2

PSfrag replacements

Portfolio weights ui

1.5 1 0.5 0 −0.5 −1

u1 : Bond u2 : Stock market u3 : Alternative inv. Short rate

−1.5 −2 0

1

2

3

4

5

Time [years]

6

7

8

9

10

Fig. 5.1. Asset allocation strategy under full information for γ = −10.

Note that price of risk λ for the bond has a positive sign as in the original Vasicek (1977) model although it is often found to be introduced with a negative sign in recent

5.1 The Full Information Case

93

texts. In Figure 5.1, the asset allocation for an investor with a risk aversion coefficient γ = −10 is shown. We see that for the observed parameter values, the weight in the bond is always the biggest. This may change by choosing different parameter values. The bigger γ, the more aggressive the asset allocating strategy becomes. Because of our model, only the bond part of the portfolio is time-dependent for a given parameter set. However, in the implementation of the strategy, also the other portfolio weights vary because the parameters are constantly updated. Finally, the asset allocation strategy is implemented. In order to have reasonable estimates for all parameters, the investment strategy is implemented starting in January 1997. The portfolio is adjusted every month.

2.8 2.6 2.4

Portfolio DS bond index 3-5 years Stock market Alternative Investment

Value

2.2 2 1.8

PSfrag replacements 1.6 1.4 1.2 1 Mar97

Jul98

Dec99

Apr01

Time

Sep02

Jan04

May05

Fig. 5.2. Asset allocation strategy performance under full information for γ = −10.

The investment horizon of the investor ends in December 2005. As new observations are available, the model parameters are recalculated using all past data available. The application of the data is done as in Brennan et al. (1997) and Bielecki, Pliska and Sherris (2000). The investment strategy is always implemented in an out-of-sample manner. Figure 5.2 shows the results for γ = −10. The investment strategy outperforms the S&P 500 and the Tremont hedge fund index by far. Table 5.2 shows the key figures of the considered time series. The Sharpe ratio of the equity market is rather poor compared to the others. This is due to the bear market from 2000 to 2003. It is noteworthy that the Sharpe ratio of our portfolio does not change

94

5 Optimal Portfolio Construction with Brownian Motions

significantly for different γ’s. The high Sharpe ratios in Table 5.2 give evidence that the risk adjusted returns of the investment strategy are superior to the ones of the single assets. Table 5.2. Key figures for the asset allocation strategy under full information return volatility Sharpe (p.a.)

(p.a.)

ratio

γ = −5

0.17

0.12

0.93

γ = −10

0.12

0.09

0.93

γ = −20

0.08

0.04

0.90

DS 3-5 years

0.06

0.04

0.62

S&P500

0.07

0.16

0.21

Tremont

0.10

0.08

0.84

In the phase of July 1998 to December 1999, the investment strategy does not show a good performance. This because of the enormous drop of the hedge fund index in 1998 which causes the controller to significantly reduce its position in the alternative asset. In the beginning of 2001, the controller is starting to take short positions in the market, which is still the case at the end of the considered time period. For more details about this case study, the reader is referred to Keel et al. (2004).

5.2 The Partial Information Case Asset price models usually contain unknown parameters. This fact is not taken into account in general. In the application of optimal asset allocation strategies, the estimated parameters are often treated as deterministic. This may lead to falsified investment decisions when the uncertainty on the parameters is significant. In order to improve the asset allocation, we do not only consider the problem of unknown parameters but also include external factors in the model. By carefully choosing explanatory variables for the asset returns, we expect the performance of the optimal asset allocation strategy to improve. The dynamics of these factors are modeled as well and have to be included in the optimization. The number and nature of the factors are crucial for the quality of the results of the optimization. We consider two kinds of factors. On the one hand, we consider observable factors, whose current values are known to the investor, i.e., no estimation is necessary. On the other hand, we consider unobservable factors which are, technically speaking, not

5.2 The Partial Information Case

95

measurable with respect to the investor’s filtration. Since the values of these factors are not known, the investor has to estimate their values. In this framework, the methods of the Kalman filter are used. The usage of factors with predictive power is motivated by the literature. Linear filtering is a well-established research area since the work of Kalman (1960) and Kalman and Bucy (1961). The Kalman filter is well-suited for the investor’s dynamic inference problem and is used as maximum likelihood estimator of the current values of the unobservable processes as well as predictor for their further evolvement. In order to work with the Kalman filter, we need a priori estimates of the starting values of the unobservable processes as well as their error covariance matrix. From the dynamics of the Kalman filter, we may deduce the future evolvement of the expected values as well as their error covariance matrices. The first results for economic problems under incomplete information are found in Detemple (1986), Dothan and Feldman (1986), and Gennotte (1986). The results in these papers are rather specific, i.e., closed-form solutions are derived for logarithmic utility functions. Brennan (1998) analyzes the case of one risky asset with a constant, unknown drift parameter. The investor is assumed to have constant relative risk aversion. Numerical results are presented for this case. Brennan (1998) also shows that the myopic investor, having a logarithmic utility function, does not hedge against unfavorable changes of the drift of the risky asset. Therefore, the myopic investor is not able to improve the asset allocation strategy by inspecting asset prices over time. Xia (2001) analyzes the case of dynamic unobservable factors and finds the opportunity costs of ignoring the predictability of asset returns as substantial. A closed-form solution for one single risky asset is also found in Rogers (2001). Rogers observes that the impact of parameter uncertainty is more severe than the problem of infrequent trading, i.e., the investor cannot continuously change the composition of the portfolio. In Brennan and Xia (2001b), the same methodology is used to derive a general equilibrium model for stock prices. The non-observability of the expected dividend growth rate is used in order to derive a representative agent model with rational behavior. The results are tested numerically as well. In Brennan and Xia (2001a), an optimal portfolio strategy is developed for an investor who has detected a price anomaly. Cvitani´c et al. (2003) analyze the optimal asset allocation for portfolios including hedge funds, the proofs

96

5 Optimal Portfolio Construction with Brownian Motions

are found in Cvitani´c et al. (2004). The authors derive an analytical solution in the case of unobservable market returns as well as unobservable abnormal or excess returns of the hedge fund. As in the papers of Brennan and Xia, the results are applied to financial data. The model for the problem in this chapter is similar to the model in Sekine (2001). However, Sekine (2001) uses the convex-duality method to derive the solutions whereas we us a dynamic programming approach. This results again in two matrix Riccati equations, which express the solution of the primal and the dual problem. The dual optimizer defines the equivalent martingale measure. For a comparison of the two methods, the reader may refer to Runggaldier (2003). The infinite-time horizon problem is considered in Bielecki and Pliska (1999) and Nagai and Peng (2002). For a survey on the topic of optimal control under uncertainty, the reader may refer to Runggaldier (1998), who also considers the case of discrete-time problems. 5.2.1 The Model The investment opportunities are modeled via stochastic differential equations. They are assumed to behave as local geometric Brownian motions. The drift terms of the asset price dynamics are modeled as affine functions of explanatory factors. The values of these factors are either known or unknown to the investor. We model the observable and unobservable factors as stochastic differential equations as well. In the case of the unknown factors, the investor has to estimate current values of the factors. Because of our model, we may use model-based filtering, i.e., we make use of the Kalman filter. The diffusions of the n risky price processes are driven by an n-dimensional standard Brownian motion process Wp . Similarly, the diffusions of the m observable and the k unobservable factors processes are driven by the m- and k-dimensional standard Brownian motion processes W x and Wy , respectively. The drift terms of observable and unobservable factors are linearly dependent. In addition to the risky asset, there is a risk-free asset (or bank account) whose instantaneous return is deterministic. However, the return on the risk-free asset may be time-dependent or even stochastic. The Brownian motion Wp of the price processes and the Brownian motions Wx and Wy of the factor processes are defined on a fixed, filtered probability space (Ω, F, {F t }t≥0 , P) with Ft satisfying the usual conditions. The Brownian motions Wp , Wx , and Wy do not need to be independent.

5.2 The Partial Information Case

97

Factor Dynamics In order to improve the asset allocation, factors are included in the investors model. We expect the factors to have some predictive power for the returns of the investment opportunities. We consider two kinds of factors. On the one hand, there are observable factors whose values are known to the investor. On the other hand, there are unobservable factors whose values are unknown to the investor. The m observable factors x(t) ∈ R m are modeled as the following stochastic vector process dx(t) = [Ax (t)x(t) + Ay (t)y(t) + a(t)]dt + σx (t)dWx (t)

(5.26)

x(0) = x0 , where Ax (t) ∈ Rm×m , Ay (t) ∈ Rm×k , a(t) ∈ Rm , σx (t) ∈ Rm×m , and Wx (t) ∈ Rm . The matrix and vector functions Ax (t), Ay (t), a(t), and σx (t) are all deterministic functions. In addition, σx (t) is assumed to have full rank. The k unobservable factors y(t) ∈ Rk are modeled similarly as dy(t) = [Cx (t)x(t) + Cy (t)y(t) + c(t)]dt + σy (t)dWy (t)

(5.27)

y(0) = y0 , where Cx (t) ∈ Rk×m , Cy (t) ∈ Rk×k , c(t) ∈ Rk , σy (t) ∈ Rk×k , and Wy (t) ∈ Rk . The matrix and vector functions Cx (t), Cy (t), c(t), and σy (t) are deterministic functions. Since y(0) is not known, we need an a priori estimate of y(0), denoted by m(0). The error variance of this estimation is denoted by the covariance matrix ν(0). Asset Price Dynamics The set of investment opportunities of our investor consists of n ≥ 1 risky assets. The

asset price processes P = (P1 (t), P2 (t), . . . , Pn (t)) ∈ Rn of the risk-bearing investments satisfy the stochastic differential equations, where diag(P (t)) denotes the diagonal matrix of the vector P (t), dP (t) = diag(P (t)) {µ(x(t), y(t), t) dt + σp (t)dWp (t)}

(5.28)

P (0) > 0 , where µ(x(t), y(t)) ∈ Rn , σp (t) ∈ Rn×n , and Wp (t) ∈ Rn . The matrix function σp is assumed to be deterministic. The drift vector µ contains the relative expected instantaneous changes of the prices P . From the diffusion matrix σp , we get the instantaneous

98

5 Optimal Portfolio Construction with Brownian Motions

covariance matrix per unit time as Σ(t) = σp (t)σp (t)T . The matrix function Σ has to be invertible, therefore σp has to have full rank. In addition, there exists a risk-free investment opportunity B(t) with instantaneous rate of return r(x(t)): dB(t) = B(t)r(x(t), t)dt

(5.29)

B(0) > 0 . The scalar function r is deterministic. The drift terms of the risk-free asset and the riskbearing assets depend on the m observable factors x. In addition, the drift terms of the risk-bearing assets depend on the k unobservable factors y. Furthermore, we assume that the drift terms in (5.28) and (5.29) are affine functions of the factor levels, as given by µ(x(t), y(t), t) = G(t)x(t) + H(t)y(t) + f (t) r(x(t), t) = F0 (t)x(t) + f0 (t) ,

(5.30) (5.31)

where G(t) ∈ Rn×m , H(t) ∈ Rn×k , f (t) ∈ Rn , F0 (t) ∈ R1×m , and f0 (t) ∈ R. The matrix and vector functions G(t), H(t), f (t), F0 (t), and f0 (t) are all deterministic. 5.2.2 Estimation of the Unobservable Factors Since the investor cannot observe the values of y, some kind of estimation (filtering) of the values of y is needed. The estimation with the introduced dynamics is a standard filtering problem and can be solved with the methods of the Kalman filter. The asset prices depend on the unknown values of y(t). Because of this, we cannot derive the optimal asset allocation strategy based on these asset price dynamics. This would not result in an admissible investment process, see Bielecki and Pliska (1999) for details. Therefore, the asset price dynamics need to be transformed as well. The main idea to overcome this problem is to transform the factor and the asset price dynamics such that they are independent of y. For notational convenience, we introduce the vector ξ = [log(P T ), xT ]T ∈ Rn+m , containing the logarithmic prices and the observable factors. We define the Brownian motion Wξ = [WpT , WxT ]T ∈ Rn+m , where Wp is as in (5.28) and Wx as in (5.26). The diffusion

term σξ ∈ Rn+m×n+m and the matrix Dy ∈ Rn+m×k are defined as     σp 0 H , σξ =  Dy =   . 0 σx Ay

5.2 The Partial Information Case

99

The Brownian motions Wp , Wx , and Wy need not be independent. The correlation matrix of the n + m + k dimensional Brownian motion W = [WyT , WpT , WxT ]T ∈ Rn+m+k is defined as 



  I ρyp (t) ρyx (t)  1  I1 ρyξ (t)   , ρ(t) =  ρTyp (t) I2 ρpx (t)  =  T   ρyξ (t) ρξ (t) ρTyx (t) ρTpx (t) I3

where I denotes the identity matrix and ρyp ∈ Rk×n , ρyx ∈ Rk×m , ρpx ∈ Rn×m , I1 ∈ Rk×k , I2 ∈ Rn×n , and I3 ∈ Rm×m . In order to be regular, the correlation matrix ρ needs to be symmetric and positive-definite for all t. We summarize the result in the following lemma. Lemma 5.3 (Transformation of the Price Dynamics). Let the price dynamics evolve as introduced in (5.28) and the factors as in (5.26) and (5.27). The function µ(x, y) is defined in (5.30). Summarized, dP = diag(P ) {µ(x, y) dt + σp dWp } dx = [Ax x + Ay y + a]dt + σx dWx dy = [Cx x + Cy y + c]dt + σy dWy . Then, there exists a Brownian motion Wξ0 ∈ Rn+m which generates the same information as [WpT , WxT ]T . The estimate of y with the Kalman filter is denoted by m, the estimation

error by ν. The transformed dynamics with the new Brownian motion Wξ0 are dP = diag(P ){µ(x, m)dt + σ p dWξ0 } dx = [Ax x + Ay m + a]dt + σ x dWξ0 dm = [Cx x + Cy m + c]dt + [B1 + νDyT ]B2 σ ξ dWξ0 ν˙ = Cy ν + νCyT + B3 − [B1 + νDyT ]B2 [B1 + νDyT ]T ν(0) = E[(y(0) − m(0))(y(0) − m(0))T ]. where B1 = σy ρyξ σξT , B2 = [σξ ρξ σξT ]−1 , and B3 = σy σyT . The volatility matrices are obtained from (5.32). Proof. See Appendix C.2. The volatility matrices σ ξ ∈ Rn+m×n+m , σ p ∈ Rn×n+m , and σ x ∈ Rm×n+m are obtained by the relationship for the following block matrices

100

5 Optimal Portfolio Construction with Brownian Motions

 

σ y σ yξ 0 σξ





=

σy 0 0 σξ



 ρ 12 ,



σξ = 

σp σx

1



.

(5.32) 1

The Cholesky factorization of ρ is a good choice for ρ 2 , i.e., ρ 2 is an upper triangular matrix. The transformation of the price dynamics of the investment opportunities is necessary to transform the partial-information problem into a full-information problem. 5.2.3 Portfolio Dynamics and Problem Transformation We assume that the investor’s portfolio is self-financing, i.e., there are no exogenous inor outflows of money, e.g., consumption. The dynamics of the investor’s wealth V may be expressed as n ³ dB(t) X dPi (t) ´ dV (t) = V (t) u0 ui (t) , + B(t) P (t) i i=1

(5.33)

V (0) > 0,

where u1 (t), . . . , un (t) denotes the fraction of wealth invested in the corresponding risky asset and u0 (t) accordingly in the risk-free asset at time t. The proof is either found in Merton (1992) or Bj¨ork (1998). Because the portfolio is self-financing, we have the P constraint ni=0 ui (t) = 1. By using Lemma 5.3, we can transform the investor’s wealth

dynamics into a full information process.

Theorem 5.4 (Transformation of the Wealth Dynamics). Let the price dynamics evolve as in (5.28), where the factors evolve as in (5.26) and (5.27). Define 1n as the vector 1n = (1, 1, . . . , 1)T ∈ Rn . Then the investors wealth dynamics under partial information evolve like dV = {uT [µ(x, y, t) − 1n r(x, t)] + r(x, t)}dt + uT σp dWp V dx = [Ax x + Ay y + a]dt + σx dWx dy = [Cx x + Cy y + c]dt + σy dWy V (0) > 0. The investors wealth dynamics under full information, written in terms of W ξ0 , are identical to those under partial information. The same notation and definitions apply as in Lemma 5.3. The wealth dynamics under full information are

5.2 The Partial Information Case

101

dV = {uT [µ(x, m, t) − 1n r(x, t)] + r(x, t)}dt + uT σ p dWξ0 V dx = [Ax x + Ay m + a]dt + σ x dWξ0 dm = [Cx x + Cy m + c]dt + [B1 + νDyT ]B2 σ ξ dWξ0 V (0) > 0. Proof. The proof is straightforward. The results of Lemma 5.3 are applied to the selffinancing portfolio dynamics introduced in (5.33). By using Theorem 5.4 we may transform a partial-information problem into a fullinformation problem as shown in the next section. 5.2.4 Optimal Asset Allocation Optimal Asset Allocation with a CRRA Utility Function The optimal asset allocation is derived for an investor having constant relative risk aversion (CRRA). The expected power utility over terminal wealth is maximized. We may state the investor’s optimization problem under full information by using Theorem 5.4. Therefore, the problem is formulated in terms of the Wiener process Wξ0 , the time arguments are omitted for better readability, as max

u(·)∈L1n

s.t.

E

h1 γ

V (T )

γ

i

dV = V {uT [µ(x, m, t) − 1n r(x, t)] + r(x, t)}dt + V uT σ p dWξ0

(5.34)

dx = [Ax x + Ay m + a]dt + σ x dWξ0 dm = [Cx x + Cy m + c]dt + [B1 + νDyT ]B2 σ ξ dWξ0 V (0) > 0. The parameter γ < 1 is the coefficient of risk aversion. The covariance matrix of the estimation error ν is also present in the optimization problem. Since ν is the solution of the ordinary differential equation (see Lemma 5.3), it is a deterministic function. The matrices B1 and B2 are also defined in Lemma 5.3. For convenience, we introduce θ = [xT , mT ]T ∈ Rm+k and rewrite the problem above as

102

5 Optimal Portfolio Construction with Brownian Motions

max1

u(·)∈Ln

i h1 E V (T )γ γ

s.t. dV = {uT [F θ + f ] + F 0 θ + f0 }dt + uT σ p dWξ0 , V dθ = [Aθ + b]dt + σθ dWξ0 ,

(5.35)

V (0) > 0. The deterministic matrix functions F ∈ Rn×m+k , F 0 ∈ R1×m+k , and f ∈ Rn of the investors wealth dynamics are defined as h i F = G − 1 n F0 , H h i F 0 = F0 , 0 f = f − 1 n f0 .

(5.36) (5.37) (5.38)

The deterministic matric functions G, F0 , f , f0 , and H are defined in (5.30) and (5.31), for the vector 1n we have 1n = (1, 1, . . . , 1)T ∈ Rn . Additionally, the deterministic matrix functions A ∈ Rm+k×m+k and b ∈ Rm+k of the dynamics of θ are     Ax Ay a , A= b =  . Cx Cy c

(5.39)

The components of A and b are defined in (5.26) and (5.27). The diffusion matrix of θ, denoted by σθ ∈ Rm+k×n+m , is only used in this form for the proof and is defined as   σx . (5.40) σθ =  (B1 + νDyT )B2 σ ξ

In order to solve this problem with Bellman’s optimality principle, we introduce the value or cost-to-go function as J(t, V, θ) = max1 E u(·)∈Ln

h1 γ

i V (T )γ .

(5.41)

The optimal asset allocation strategy is the solution of the Hamilton-Jacobi-Bellman equation, which for this problem is h Jt + maxn V (uT [F θ + f ] + F 0 θ + f0 )JV + (Aθ + b)T Jθ u∈R i 1 1 + V 2 uT ΣuJV V + V uT σ p σθT JV θ + tr{Jθθ σθ σθT } = 0, 2 2

(5.42)

5.2 The Partial Information Case

103

with terminal condition J(T, V (T ), θ(T )) = γ1 V (T )γ . The optimal control law u∗ of this problem is given by ³ ´ 1 −1 T Σ F θ + f + σ p σθ [K1 θ + k2 ] . u = 1−γ ∗

(5.43)

We want to eliminate the parameters introduced by the filtering. Therefore, we introduce two new matrices Ψ ∈ Rn×m+k and Γ ∈ Rm+k×m+k with the following definitions Ψ = σ p σθT = [σp ρpx σxT , σp ρTyp σyT + Hν T ]   T T T T σx σx σx ρyx σy + Ay ν , Γ = σθ σθT =  T T T T T σy ρyx σx + νAy (B1 + νDy )B2 (B1 + νDy )

(5.44) (5.45)

where B1 are B2 are defined as in Lemma 5.3. We can finally give the solution of the optimal asset allocation strategy (θ = [xT , mT ]T ∈ Rm+k ), u∗ =

³ ´ 1 Σ −1 F θ + f + Ψ [K1 θ + k2 ] , 1−γ

(5.46)

where K1 (t) ∈ Rm+k×m+k and k2 (t) ∈ Rm+k are the solutions of two matrix Ricatti equations. The differential equation for the matrix K1 (t) is given by 1 K1 Γ K 1 + K 1 A + A T K1 1−γ ³ ´ F T Σ −1 F + F T Σ −1 Ψ K1 + K1 Ψ T Σ −1 F = 0, K˙ 1 +

−

γ γ−1

(5.47)

with terminal condition K1 (T ) = 0. The differential equation for the vector k2 (t) is given by

−

γ γ−1

1 T K1 Γ k2 + AT k2 + K1 b k˙2 + γF 0 + 1−γ ³ ´ F T Σ −1 f + F T Σ −1 Ψ k2 + K1 Ψ T Σ −1 f = 0,

(5.48)

with terminal condition k2 (T ) = 0. We summarize the results in the following theorem. Theorem 5.5 (Partial Information, CRRA Case). The optimal asset allocation strategy under partial information for the CRRA case (5.34) is given by (5.46), where K1 (t) and k2 (t) are the solutions of two matrix Ricatti equations (5.47) and (5.48), respectively. Proof. The problem in (5.34) is the full-information representation of the partial-information problem. The transformation is based on Theorem 5.4. The solution of the full-information problem is found in Lemma 5.1.

104

5 Optimal Portfolio Construction with Brownian Motions

Note that (5.47) is a well-known Riccati type equation. In the case where all the parameters are constant, the Riccati equation (5.47) possesses a positive-definite and finite solution for T → ∞, if the following two conditions are met: ´ ³ 1 ´ 12 i h³ γ Ψ T Σ −1 F − A , Γ γ−1 1−γ

(5.49)

´ 12 ³ γ ´i h³ γ T −1 T −1 F Σ F , Ψ Σ F −A γ−1 γ−1

(5.50)

is controllable, and

1

1 is observable in the control engineering sense, see Xu and Lu (1995). The matrix ( 1−γ Γ )2 1

γ 1 is a full rank factorization of ( 1−γ Γ ) and the matrix ( γ−1 F T Σ −1 F ) 2 is a full rank factorγ γ 1 F T Σ −1 F ). Additionally, the matrix ( γ−1 F T Σ −1 F ) and the matrix ( 1−γ Γ) ization of ( γ−1

must be positive-definite. The conditions for solving matrix Riccati equations are wellknown in control engineering. For further details the reader may refer to Anderson and 1 Moore (1990). If the matrix ( 1−γ K1 Γ + A T −

γ F T Σ −1 Ψ ) γ−1

is invertible and (5.47) pos-

sesses a positive-definite finite solution, then (5.48) possesses a finite solution for T → ∞. Note that for γ > 0, the Riccati equation (5.47) possesses no finite solution, Kim and Omberg (1996) call the solution for this case nirvana solution. Optimal Asset Allocation with a CARA Utility Function The optimal asset allocation is derived for an investor having constant absolute risk aversion (CARA). The expected exponential utility over terminal wealth is maximized. For the problem to be solvable, we model the risk-free interest rate r(t) only as a function of time and independent of x(t). Therefore, we set F0 ≡ 0 in (5.31). We proceed as in the CRRA case and therefore state the investors optimization problem as h 1 i −γV (T ) max E − e u(·)∈L1n γ s.t. dV = V {uT [µ(x, m, t) − 1n r] + r}dt + V uT σ p dWξ0

(5.51)

dx = [Ax x + Ay m + a]dt + σ x dWξ0 dm = [Cx x + Cy m + c]dt + [B1 + νDyT ]B2 σ ξ dWξ0 V (0) > 0. The parameter γ > 0 is the coefficient of risk aversion. We introduce θ = [xT , mT ]T ∈ Rm+k for convenience and rewrite the problem above as

5.2 The Partial Information Case

max1

u(·)∈Ln

s.t.

105

i h 1 E − e−γV (T ) γ

dV = V {uT [F θ + f ] + f0 }dt + V uT σ p dWξ0

(5.52)

dθ = [Aθ + b]dt + σθ dWξ0 V (0) > 0. The deterministic matrix function F ∈ Rn×m+k is defined as h i F = G, H ,

(5.53)

f = f − 1 n f0 .

(5.54)

The deterministic matrix functions A ∈ Rm+k×m+k and b ∈ Rm+k are defined in (5.39). The diffusion matrix σθ ∈ Rm+k×n+m is, again, only used for the proof and is defined as

in (5.40). The value or cost-to-go function for this problem is defined as h 1 i J(t, V, θ) = max1 E − e−γV (T ) . u(·)∈Ln γ

(5.55)

The Hamilton-Jacobi-Bellman partial differential equation for this problem is h Jt + maxn V {uT [F θ + f ] + f0 }JV + (Aθ + b)T Jθ u∈R i 1 1 2 T T T T + V u ΣuJV V + V u σ p σθ JV θ + tr{Jθθ σθ σθ } = 0. 2 2

(5.56)

The optimal solution u∗ is given by u∗ = −

´ ³ 1 Σ −1 F θ + f + Ψ [K1 θ + k2 ] , γV k3

(5.57)

where Ψ is defined in (5.44). What remains to be done is to derive the functions K1 (t), k2 (t), and k3 (t). The matrix function K1 (t) is the solution of the following linear matrix differential equation K˙ 1 + K1 A + AT K1 − F T Σ −1 F − F T Σ −1 Ψ K1 − K1 Ψ T Σ −1 F = 0

(5.58)

K1 (T ) = 0 . The vector function k2 (t) is the the solutions of the following differential equation k˙2 + AT k2 + K1 b + F T Σ −1 f − F T Σ −1 Ψ k2 + K1 Ψ T Σ −1 f = 0 k2 (T )= 0 .

(5.59)

106

5 Optimal Portfolio Construction with Brownian Motions

Finally, the scalar function k3 (t) is the solution of the following differential equation k˙3 + f0 (t)k3 = 0,

k3 (T ) = −γ .

(5.60)

We summarize the results in the following theorem. Theorem 5.6 (Partial Information, CARA Case). The optimal asset allocation strategy under partial information for the CARA case (5.51) is given by (5.57), where K1 (t), k2 (t) and k3 (t) are the solutions of the differential equations (5.58), (5.59), and (5.60), respectively. Proof. The proof is analogous to the proof of Theorem 5.5. For the conditions for solvability of the CARA problem, the reader may use the same methodology as for the CRRA case. 5.2.5 Case Study with a Balanced Fund The previously obtained results are applied in a balanced fund example. The balanced fund consists of cash, bonds, stocks, and an absolute return product, which is deemed to be a hedge fund. The investment opportunities are modeled similarly to the ones in the case study for the full information case. However, the model is improved in some aspects. The drift of the equity market is stochastic and not known exactly to the investor. We model the excess return α of the absolute return product as an unknown constant. Asset Models We first introduce the dynamics of the short rate (or money market interest rate), denoted by r. Let r be a mean-reverting process, dr = κ(θ − r)dt + σr dWr ,

(5.61)

r(0) = r0 , where κ, θ, σr ∈ R are the constant parameters of the short rate. This process can be observed in continuous-time. The dynamics of the fixed income security is derived from the short rate, based on the methods of Vasicek (1977). We therefore get for the zerocoupon bond dynamics

5.2 The Partial Information Case

³ σr λσr ´ aT dt − B aT dWr , dB = B r + κ κ B(T ) = 1,

107

(5.62)

where the scalar function aT (t) is defined as aT (t) = 1 − e−κ(T −t) .

(5.63)

The dynamics of the bond price B(t) are derived from the short rate by no-arbitrage reasoning. As already mentioned in the introduction, we do not consider a single bond but rather a bond index. We model the equity market index by its stochastic differential equation as given by dS = Sµdt + Sσs dWs ,

(5.64)

S(0) = S0 , where σs ∈ R is the constant parameter of the model. The drift of the equity market is assumed to be stochastic. It is modeled as mean-reverting process and satisfies the following stochastic differential equation, µ(0) = µ0 , dµ = ζ(η − µ)dt + σµ dWµ ,

(5.65)

dWs dWµ = ρ˜dt, where ζ, η, σµ ∈ R are the constant parameters of the model. The correlation of the two Brownian motions Ws and Wµ is denoted by ρ˜ ∈ R. Note that we have an incomplete market model for |˜ ρ| 6= 1. Since we do not deal with the problem of pricing contingent claims, this is not a limiting assumption. It remains to model the alternative asset in order to analyze the balanced fund. As mentioned above, we use a model resembling to Merton’s intertemporal capital asset pricing model (ICAPM). Therefore, the price of the alternative asset, denoted by A(t), evolves like dA = A(r + β(µ − r) + α)dt + A σA (ρdWs + A(0) = A0 ,

p 1 − ρ2 dWA ),

(5.66)

where β, σA , ρ ∈ R are the constant parameters of the model. The excess return of the alternative investment, denoted by α ∈ R, is assumed to be an unknown constant.

108

5 Optimal Portfolio Construction with Brownian Motions

The excess return is unknown to the investor, i.e., not measurable by the investor. The value of α describes the ability of hedge fund managers to outperform the market. In order to comply with the general model, the Brownian motion Wα ∈ R is introduced. Since Wα does not influence α, it does not affect the solution of the problem. It is therefore assumed to be independent of the other dynamic processes. The main reason to model α as unknown constant is the lack of available data for alternative investments. We do not want to build a model for which we cannot obtain reasonable parameter estimates. Furthermore, the Brownian motions Wr , Ws , Wµ , WA , and Wα are assumed to be mutually independent except that Ws and Wµ are correlated with ρ˜. Therefore, the correlation matrix ρyx is a zero matrix for this case study. Again, r + β(µs − r) describes the risk adjusted return of the asset with respect to the market, whereas α denotes the outperformance of the alternative asset. The β parameter is defined to be β=

ρσA cov(dS/S, dA/A) = , 2 σs σs

(5.67)

where ρ denotes correlation between the diffusions of the equity market index and the diffusion of the alternative asset. We introduce a three-dimensional control vector u. The three components of u represent the percentage of total wealth invested in the corresponding investment category. Optimization Problem The observable factors consist of r, defined in (5.61). Therefore we set x = r. This results for the matrices Ax , Ay , a, and σx in the model of (5.26) in h i Ax = −κ ,

h Ay = 0

i

0 ,

h i a(t) = κθ ,

h i σ x = σr .

(5.68)

The driving Brownian motion Wx of the observable factors is defined as Wx = Wr . The unobservable factors consist of µ, as defined in (5.65), and α, which is an unknown constant. The vector y is thus defined as y = [µ, α]T , and the matrices Cx , Cy , c, and σy in (5.27) are   0 Cx =   , 0



Cy = 

−ζ

0

0

0



,



c=

ζη 0



,

The Brownian motion Wy is defined as Wy = [Wµ , Wα ]T .



σy = 

σµ 0 0 0



.

(5.69)

5.2 The Partial Information Case

109

The vector of asset price processes of (5.28) is P = [B, S, A]T . The drift terms of the risky investment opportunities are defined in (5.30). We therefore need to define the matrices G, H, and f in this equation, which    1 0       G =  0 , H = 1    1−β β

are  0   0 ,  1



  f = 

λσr a κ T

0 0



  . 

(5.70)

The drift term of the risk-free asset is defined in (5.31). For its determination we need to

define the terms F0 and f0 , which are F0 = 1,

f0 = 0.

(5.71)

The diffusion of the risky investment opportunities of (5.28) are determined by the covariance matrix σp . From the asset price dynamics of (5.62), (5.64), and (5.66) we get   σr 0 − a (t) 0   κ T   (5.72) σp (t) =  . 0 σs 0   p 0 σ A ρ σA 1 − ρ 2

The Brownian motion Wp is defined as Wp = [Wr , Ws , WA ]T . The correlation matrix is defined as a block matrix. The three block matrices in this case are       1   0 ρ˜ 0 0   , ρyp =  ρyx =   , ρpx =  0  .   0 0 0 0 0

(5.73)

Recall that m(t) denotes the expected values of the unobservable factors. We may now use Theorem 5.5 to derive the optimal solution for this problem, where θ = [xT , mT ]T , as ³ ´ 1 −1 ∗ Σ F θ + Ψ [K1 θ + k2 ] . u = 1−γ The matrix function F in the equation above is defined in (5.36). The deterministic matrix

functions Ψ and Γ are defined in (5.44) and (5.45), respectively. The matrix and vector function K1 and k2 are the solutions of the matrix Riccati equations, as given in (5.47) and (5.48). Backtest with Historical Data We finish the case study by applying the derived results to historical US data. The data is the same as in the case study for the full information case. Therefore, the three-month

110

5 Optimal Portfolio Construction with Brownian Motions

Treasury bills are chosen as a substitute for the short rate, the S&P 500 for the market portfolio, the Datastream USA Total 3-5 years for the bond index, and the Tremont hedge fund index for the alternative part of the portfolio. Again, we assume that the bond has a fixed duration. Therefore, the time dependent function aT (t) is redefined as a function of the duration of the bond τ only. The parameters of the fixed-income model are estimated by using regression techniques with the discrete versions of the stochastic differential equations. The parameters of the short rate are estimated by doing an ordinary least squares estimation on the discretetime version of the short rate. We discretize the stochastic differential equation for the short rate, √ rt+1 = κθ∆t + (1 − κ∆t)rt + σr ∆t ξr , where ∆t is the time increment and ξr is a Gaussian white noise process. With the same procedure we arrive at the parameters of the bond index. ´2 ´ ³ σ √ 1 ³ σr σr r aT (τ ) ∆t + aT (τ ) ∆t ξr . ln(Bt+1 ) − ln(Bt ) − rt ∆t = λ aT (τ ) + κ 2 κ κ

The duration τ and the price of risk λ of the bond index are estimated by estimating mean and variance of the series above. The problem of parameter estimation for the equity market index is more involved than for the bond index because of the unobservable drift process µ of (5.65). We use the Kalman filter as a tool for parameter identification. The methodology of using the Kalman filter for parameter identification is discussed in Hamilton (1994). For parameter identification in financial problems, the reader may consult Kellerhals (2001). We do not give full details of the parameter estimation but, nevertheless, give the problem statement. Define p to be the logarithmic stock prices, i.e., p = log(S). The discrete dynamics with sampling time ∆t then are           √ pk+1 σs 0 ξ 1 ∆t pk − 21 σs2 ∆t   s, =  + ∆t    +  p ρ˜σµ 1 − ρ˜2 σµ µk+1 ζη∆t ξµ 0 1 − ζ∆t µk

where ξs and ξµ are two independent Gaussian white noise processes. We observe a large negative correlation of the equity index S and its drift µ, denoted by ρ˜. This behavior is also found in Wachter (2002) and the references therein. Having estimated the parameters of the equity market index, the parameters for the alternative investment may be estimated. The parameters are not estimated altogether

5.2 The Partial Information Case

111

because only monthly data is available for the hedge fund index. We estimate the excess or abnormal return α and the correlation coefficient ρ again with the methods of the Kalman filter. Define φ to be the logarithmic prices of the alternative investment, i.e., φ = log(A). The discrete dynamics with sampling time ∆t are        − 12 σA2 ∆t φk 1 β∆t (1 − β)∆t ∆t φk+1                µk+1   0 1 − ζ∆t 0 0   µk   ζη∆t    +  =           rk+1   0 0 1 − κ∆t 0   rk   κθ∆t         αk 0 0 0 1 αk+1 0    p ξ ρσ 1 − ρ 2 σA 0 0  s   A p     √  ρ˜σµ 1 − ρ˜2 σµ 0   ξA  0  , + ∆t      0 0 0 σ r   ξµ     ξr 0 0 0 0

where ξs , ξA , ξµ , and ξr are independent Gaussian white noise processes. In order to have accurate parameters, the parameter estimation is conducted before every adjustment of the portfolio. Figure 5.3 shows the estimated trajectories of α and µ. The time ranges from January 1994 to June 1999 with a monthly sampling frequency. We observe a very strong stock market in this time period. We interpret this as high confidence of the investors that the stock market would rise constantly, as it did in this time period. 0.2 0.18 0.16 0.14 0.12

Market drift µ(t) Short rate r(t) Excess return α(t)

0.1

PSfrag replacements

0.08 0.06 0.04 Jun94

Oct95

Time

Mar97

Jul98

Fig. 5.3. Estimations of the short rate and the unobservable factors α and µ.

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5 Optimal Portfolio Construction with Brownian Motions

The estimated α is a little bit less than 4% for this time period. In Figure 5.4, the optimal asset allocation strategy is plotted for a rather aggressive investor, i.e., γ = −10. Because the involved Riccati equations, the resulting strategy is a non-linear function of time. The investment horizon is a little bit less than twelve years. The strategy of Figure 5.4 would be applied if the investor would not use any further information until the end of the problem at time T . The position in the bond index is highly leveraged and is slowly decreasing in time. The proportion of wealth invested in the stock market is more or less constant, only in the beginning and in the end, minor non-linearities are observed. The position in the hedge fund is increasing with time. The reason for this the high uncertainty of α. The estimated variance of the Kalman filter is decreasing over time and therefore the position in the hedge fund is accordingly increased. Because of the uncertainty of α, the optimal asset allocation strategy allocates less capital to the alternative investment. We generally state that the asset allocation strategy is less aggressive under partial information than under full information. 2 1.5

PSfrag replacements

Portfolio weights ui

1 0.5 0

u1 : Bond u2 : Stock market u3 : Alternative inv. Short rate

−0.5 −1 −1.5 −2 0

2

4

6

Time [years]

8

10

Fig. 5.4. Asset allocation strategy under partial information for γ = −10.

Table 5.3 gives the key figures of the different strategies. The asset allocation is performed for three risk aversion coefficients, i.e., γ = −5, γ = −10, and γ = −20. The return, volatility, and Sharpe ratio are reported for the different time series. The bond index, denoted by DS 3–5 years, has the lowest return but still an attractive Sharpe ratio because of its low volatility. The equity index, in this case the S&P 500, has a poor per-

5.2 The Partial Information Case

113

formance in terms of the Sharpe ratio. The actively managed portfolio has a much better performance than the single investment opportunities. We also observe that the Sharpe ratios differ slightly for the actively managed portfolios. This difference stems from the fact that the Riccati equations depend on the risk aversion coefficient. This causes the relative portfolio weights to differ for different risk aversion coefficients γ. The wealth Table 5.3. Key figures for the asset allocation strategy under partial information. return volatility Sharpe (p.a.)

(p.a.)

ratio

γ = −5

0.25

0.17

1.30

γ = −10

0.17

0.10

1.32

γ = −20

0.12

0.07

1.29

DS 3-5 years

0.06

0.02

0.74

S&P 500

0.06

0.16

0.15

Tremont

0.10

0.08

0.81

evolution for an investor with risk aversion γ = −10 is shown in Figure 5.5. We observe a steady growth with some minor drawdowns. This gives evidence that the dynamic trading strategy is superior to the passive investments. After the year 2000, the stock market has

3.5

Portfolio DS bond index 3-5 years Stock market Alternative Investment

Value

3

2.5

PSfrag replacements 2

1.5

1 Mar97

Jul98

Dec99

Apr01

Time

Sep02

Jan04

Fig. 5.5. Asset allocation strategy performance under partial information for γ = −10.

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5 Optimal Portfolio Construction with Brownian Motions

long time of severe decline in value. The Kalman filter and the investment strategy do correctly identify the directions of market movements and also their degree of uncertainty. If the unobservable factors were modeled as observable, the Sharpe ratio would drop considerably, in the case of γ = −10 to 0.67. The return of the actively managed portfolio is still reasonably good (15% p.a.), however, the variance increases tremendously because the strategy is much more aggressive.

6 Active Portfolio Management

Successful investing is anticipating the anticipations of others. John Maynard Keynes

The reasons for active portfolio management are manifold, as highlighted in Chapter 1. The most important reasons are that the market behavior is non-stationary and investors are constrained by liabilities and consumption. These constraints may also be dynamic and stochastic. The growing demand for absolute return products also gives evidence that many investors are no longer willing to be fully exposed to traditional risks. This chapter introduces the main concepts of active portfolio management and its instruments. We begin with the definition of active portfolio management Definition 6.1 (Active Portfolio Management). Active portfolio management is the implementation of a dynamic investment strategy in order to beat a predefined benchmark at a predefined time in the future. From this definition, the importance of the benchmark for active portfolio management is evident, in terms of risk as well as performance measurement. Not all active managers give a benchmark, making the risk and performance measurement of an investment strategy ambiguous. We denote the residual returns as the portfolio returns minus the corresponding benchmark returns. The mean of the residual returns is usually called alpha, the standard deviation tracking error. The quotient of alpha divided by the tracking error is called information ratio. As for ordinary returns, the variance may not be an appropriate risk measure for analyzing residual returns. Coherent or convex risk measures should be used for analyzing the realized returns. The benchmark can either be stochastic or deterministic. In addition, the benchmark returns may be strictly positive. In this case, the strategy is usually termed an absolute return strategy.

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6 Active Portfolio Management

The returns of an actively managed portfolio stem from two sources, the systematic risk of the involved assets and the investment strategy. Investment strategies are crudely categorized into security selection and market timing. Summarizing, the key components of active portfolio management are: • The Investment universe • The Investment strategy – Security selection – Market timing In order to be representative, the benchmark should consist of all investment opportunities present in the investment universe. In addition, the benchmark should be investable, i.e., the investment positions of the benchmark at time t are non-anticipating. Mathematically speaking, the investment positions at time t are measurable with respect to the investor’s filtration Ft . The exposure to the single securities in the investment universe is usually constrained within the investment strategy. In general, the actual investment strategy is not known by the investor since the active manager is reluctant to give insights into the production of alpha. In some cases, the investor does not even know the investment universe as such, which is the case for some hedge funds. Obviously, the outperformance of the active portfolio stems for the under-weighting of poorly performing assets and over-weighting of well performing assets. The security selection approach, in its simplest form, only identifies the forthcoming winners and losers. Correspondingly, the manager takes long positions in the winners, may be also short position in the losers. The amount of capital allocated to each asset needs not be sophisticated, e.g., equally weighted. However, if the mean of the residual returns is positive, the manager has actually beaten the benchmark. Whether this outperformance is significant or not is reflected by the information ratio. Therefore, the information ratio should rather be seen as an indicator of the manager’s skill than as performance measure. There are other risk measures than variance which are much better choices for measuring risk, as already discussed. We aim to put the above statements into a more mathematical framework. Let the benchmark returns r (B) and the returns of the actively managed portfolio r (P ) be defined by the following factor models:

6 Active Portfolio Management

r

(B)

(t) =

n X

(B) βi (t)ri (t),

r

(P )

(t) =

i=1

n X

117

(P )

βi (t)ri (t).

i=1

Therefore, the investment universe consists of n investment opportunities and ri denotes the return of the i-th investment opportunity. Additionally, we impose the constraints Pn Pn (P ) (B) i=1 βi (t) = 1. From these models, the excess return is obtained i=1 βi (t) = 1 and as

α(t) =

n X i=1

(P )

(B)

(βi (t) − βi (t))ri (t).

The mechanics of active portfolio management are easily seen from this equation. For the case ri (t) ≥ 0, the exposure of the manager should at least be as big as the portfolio (P )

(B)

exposure, i.e., βi (t) ≥ βi (t). Accordingly for the case ri (t) ≤ 0, we would like to have (P )

(B)

βi (t) ≤ βi (t). These trivial equations show that, from a mathematical point of view, security selection and market timing are the same because in both cases, the portfolio (P )

manager controls βi (t) by the mechanics described above. This can be either achieved by a discretionary or are systematic approach. However, the crucial point for successful security selection and market timing remains the correct prediction of ri (t). Note that the correct prediction of security returns is not the only source of alpha. We have argued that the significance of the outperformance of an investment strategy is reflected by the information ratio. By now, we have shown that the information ratio can be increased by better predictions of security prices. Obviously, the information ratio may also be increased by a smaller variance of the residual returns, given that the mean of the residual returns remains the same. Loosely speaking, one way to make money is by not losing it. For an illustrative example of the value of superior risk management, see Lo (2001). Again, the stylized facts of asset returns support this claim. For instance, asset returns are not serially correlated, however, squared asset returns are. Therefore, predicting risk is easier than predicting returns. Treynor and Black (1973) is the first systematic active portfolio management approach, coupling the identification of alpha and risk management. These ideas have been refined in Black and Litterman (1991, 1992) by introducing uncertainty about the model parameters. A more recent publication on this topic is Grinold and Kahn (1999). The academic literature usually does not distinguish between active portfolio management and optimal portfolio construction. As for hedge funds, there is neither a generally accepted terminology nor a unified framework to compare different strategies. We do not attempt to fill

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6 Active Portfolio Management

these gaps since this is virtually impossible. However, what all active portfolio strategies have in common is that the outperformance has to be gained through altering investment positions. This is illustrated next in a case study.

6.1 Sector Rotation Example In this case study, we are considering the S&P 500 index with its ten subindices, also called sectors indices. All data used in the sequel of this chapter is obtained from the Datastream database of Thomson Financial. In our case, the data ranges from 1995 to 2005 on a weekly basis. The ten sectors are consumer discretionary, consumer staples, energy, financials, health care, industrials, information technology, materials, telecommunication services, and utilities. The weight of a sector in the index is determined by the corresponding market capitalization. The investor’s goal is to outperform the S&P 500 index by over- and under-weighting the individual sectors. Two implementation possibilities are presented: a multi-period and a single-period environment. The actual implementation of the strategy with historical data is done with the single-period strategy. Asset Return Prediction In order to improve the performance of the strategy, an adaptive factor model is used to predict returns on a short term basis. A fairly large set of factors is chosen. In general, a large set of factor gives good in-sample predictions. However, the resulting out-of-sample prediction is usually poor. This problem is also called over-fitting. Therefore, the number of factors is reduced to avoid this problem. The returns are modeled as a linear function of the factor values. In order to find the corresponding beta values of the individual factors, an ordinary least squares procedure is used, see Hamilton (1994) for details. To avoid the mentioned over-fitting problem, the following methodology is used:

6.1 Sector Rotation Example

119

1. Start with the full model Mm , i.e., chose all m factors 2. Exclude the factor from the current model which increases the sum of squared residuals the least. 3. Repeat step 2 until only one factor remains. This results in a sequence of m models M1 ⊆ M2 ⊆ · · · ⊆ Mm . 4. Chose the model in the sequence M1 ⊆ M2 ⊆ · · · ⊆ Mm which has the smallest information criterion, in this case we chose Akaike (1974).

The chosen factors to start with consist of world equity indices returns, commodities returns, bond indices returns, real estate indices returns, volatility indices, interest rates, dividend yields, price-earnings ratios, and foreign exchange rates. Table D.1 (page 151) shows a detailed list of all the factors used for the return prediction. The momentum is calculated on all equity indices. Therefore, an exponentially weighted moving average has to be calculated using a smoothing factor of 0.96. The reader is referred to Alexander (2001) for details on calculating exponentially weighted moving averages. A Multi-Period Asset Return Model The volatilities of the ten sector indices are modeled as GARCH processes. A fairly simple model for the dependence is used, in this case, the constant conditional correlation (CCC) model is chosen. The technical details are found in Appendix B. Of course, a dynamic conditional correlations (DCC) model could have been chosen as well. By using a CCC model, however, we can make use of the convolution property of the NIG distribution, as found in Appendix A.2.3. Therefore, we use dynamical variances with NIG distributed innovations. For the asset allocation strategy, a model predictive approach is chosen. At time t, the covariance matrix of the ten sectors is estimated by the CCC-GARCH model. This gives the straightforward prediction of the covariance matrix for t + 1. For the predictions of the covariance matrices for t ≥ t + 4, the unconditional covariance matrix is used. As mentioned, the returns are assumed to have a normal inverse Gaussian distribution. The model is summarized in Figure 6.1. Note that the parameter ψ in the NIG distribution is standardized to one and therefore dropped.

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6 Active Portfolio Management Predictions: Adaptive factor model

Predictions:

CCC-GARCH model r(t) ∼ NIGn (χ1 , µ1 , Σ, γ)

Unconditional mean

Unconditional mean

Unconditional Variance

Unconditional Variance

r(t + 1) ∼ NIGn (χ2 , µ2 , Σ, γ) t+1

t

Predictions:

r(t + 4) ∼ NIGn (χ3 , µ3 , Σ, γ) t+4

T

Fig. 6.1. An MPC approach for the sector rotation problem.

The components of the vector π(t) denote corresponding sector weights in the index. The control vector u(t) denotes the fraction of the investor’s wealth, invested in the corresponding assets. This results in the following portfolio return distribution at time T ³ ´ R(T ) ∼ NIGn χ4 , uTt µ1 + uTt+1 µ2 + uTt+4 µ3 , uTt Σut , uTt γ , √ √ √ χ4 = ( χ 1 + χ 2 + χ 3 ) 2 . We have made the assumptions uTt Σut ≈ uTt+1 Σut+1 ≈ uTt+4 Σut+4 and uTt γ ≈ uTt+1 γ ≈ uTt+4 γ. In this setup, transaction costs may trivially be integrated. The inclusion of risk constraints at t + 1, t + 4, and T may also be included. However, the topic of multi-period risk management has not been addressed yet. From the results in Chapter 2, we know that coherent single-period risk measures are not necessarily coherent in a multi-period setting. A possible multi-period solution is the measuring of risk by penalty functions. The reader is referred to Dondi (2005) for a detailed case study. However, we prefer to work with a monetary risk measure. Therefore, we consider a single-period version of the investment strategy. A Single-Period Asset Return Model As in the multi-period case, the volatilities of the ten sector indices are modeled as GARCH processes. However, a more sophisticated model for the correlation is used, i.e., the dynamic conditional correlation (DCC) model is used. The technical details are found in Appendix B. By using a DCC model, we can make use of the dynamic dependence structure of the multivariate asset returns. The NIG distribution is used to model the multivariate asset returns. The DCC-GARCH parameters are estimated by a quasi maximum-likelihood approach, the reader is referred to McNeil et al. (2005) for details. Because the NIG distribution is a normal mean variance mixture distribution, the portfolio distribution is easily calculated;

6.1 Sector Rotation Example

121

see Appendix A for the technical details. As risk measure, the conditional value at risk (CVaR) is used. The asset allocation strategy consists of a mean-CVaR optimization. Let Zt ∈ R10 be a random vector whose components are NIG distributed with zero mean

and unit variance. The sector returns are denoted by the random vector Rt ∈ R10 whose

covariance matrix is denoted by Σt . The return predictions µ ˆ t are obtained as described above. The unconditional mean of the past returns is denoted by µ. In terms of risk management, only the mean of the past returns is used in order not to avoid biases. Mathematically, the problem at time t is formulated as follows: maxn uTt µ ˆt − γCVaR(uTt Rt )

ut ∈R

s.t. Rt = µ + ε t ,

1/2

ε t = Σ t Zt ,

2 2 σt,k = ak0 + ak1 ε2t−1,k + bk1 σt−1,k ,

k = 1, . . . , 10 .

∆t = diag(σt,1 , . . . , σt,10 ) ∈ R10×10 Σt = ∆t R(Qt )∆t , ³ ´ −1 T Qt = 1 − a1 − b1 Qc + a1 ∆−1 t−1 εt−1 [∆t−1 εt−1 ] + b1 Qt−1 . ³ ´ uTt Rt ∼ NIG χ, ψ, uTt µ, uTt Σt ut , uTt γ

The operator R is given in the appendix, see Definition B.1 (page 142). In Figure 6.2, the out-of-sample performance of the active portfolio versus the benchmark is shown for γ = 1. The data ranges from 1995 to 2005, the implementation of the investment strategy starts in 1999 in order to have sufficient data for the parameter estimation. The model selection is repeated every 50 weeks, i.e., the factor list is updated every 50 weeks. The active portfolio obviously outperforms the benchmark considerably. Recall that the vector π(t) denotes the corresponding sector weights in the benchmark. The control vector u(t) denotes the fraction of the investor’s wealth, invested in the different sectors. We impose the lower constraint for u(t) as − 21 π(t) and the upper constraint as 32 π(t). The resulting strategy has an alpha of 4.8% p.a. and a tracking error of 5.0% p.a. This results in an information ratio of 0.96, which is reasonably good. The benchmark has a disappointing performance in the considered time period, the return is -0.2% p.a. and the volatility is 17%. The return of the active portfolio is 4.7% p.a. with a standard deviation of 18%. We do not observe an outperformance if the return prediction is omitted, i.e., the risk management on its own does not generate outperformance in this setup. However, if only

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6 Active Portfolio Management

1.3

Benchmark Active Portfolio

1.2

Value

1.1 1 0.9

PSfrag replacements

0.8 0.7 Dec99

Apr01

Sep02 Time

Jan04

May05

Fig. 6.2. Performance of the sector rotation asset allocation strategy.

the return predictions are used for the asset allocation, the information ratio is decreased by 20%. This case study gives evidence that active portfolio management may produce added value if implemented correctly. Similar case studies are found in Herzog, Dondi and Geering (2004) and Herzog, Geering and Schumann (2004).

6.2 Portfolio Management with L´ evy Processes Stochastic processes in continuous time, driven by Brownian motion, have been extensively discussed in Chapter 5. Most of the disadvantages which are present in this type of model may be overcome by replacing the Brownian motion with a more general process. A L´evy process is a continuous-time stochastic process with independent and stationary increments, and is continuous in probability. L´evy processes belong to the class of semimartingales. The reader is referred to Protter (1990) for details on semimartingales. Brownian motion itself is a L´evy process and is the only one whose sample paths are not only continuous in probability but also with probability one. L´evy processes can take the stylized facts of asset returns much better into account than Brownian motion. The reader is referred is referred to Schoutens (2003) for details on L´evy processes in finance. For more technical details on L´evy processes and semimartingales see Protter (1990), Sato (1999), and Applebaum (2004). We will use the following conventions, ∆X(t) denotes the

6.2 Portfolio Management with L´evy Processes

123

the jump of the process X(t) at time t, X(t−) = lim X(s), s↑t

∆X(t) = X(t) − X(t−).

In order to define a stochastic process with independent and stationary increments, the distribution of the increments has to be infinitely divisible. We briefly discuss infinitely divisible distributions because of their importance in the realm of L´evy processes. We call the logarithm of the characteristic function of a distribution its characteristic exponent. The characteristic exponent of an infinitely divisible distribution may be decomposed by the L´evy-Khinchine representation, see Protter (1990) for details. Because of this property, every L´evy process L may be decomposed by the L´evy-Itˆo decomposition, i.e., every L´evy process L is fully characterized by the L´evy triplet (γ, σσ T , ν). We use the abbreviation Σ = σσ T . The third component ν is called the L´evy measure and describes the expected number of jumps and their sizes. The Poisson random measure N (t, A) denotes the number of jumps of L of size ∆L(t) ∈ A which occur before or at time t. e (dt, dz) = N (dt, dz)−ν(dz)dt is called the compensated Poisson random measure. Here, N

Then we may write L(t) with the L´evy-Itˆo decomposition as Z tZ Z tZ e (dt, dz) + L(t) = γt + σW (t) + zN 0

= γt + σW (t) +

Z tZ 0

|z|
0

zN (dt, dz)

|z|≥R

zN (dt, dz) = γt + σW (t) +

R

X

∆L(s),

0<s≤t

where W (t) is a standard n-dimensional Wiener process, R > 0 an arbitrary constant, and N (dt, dz) =

 N e (dt, dz),  N (dt, dz),

if |z| < R if |z| ≥ R.

The reader is referred to Oksendal and Sulem (2004) and Cont and Tankov (2004) for the technical details. This means that L´evy processes may be decomposed into a deterministic part, a Brownian motion part, and a discontinuous or jump part. Besides optimal portfolio construction, L´evy processes are also well suited for asset pricing, e.g., the pricing of contingent claims is. However, if asset prices are modeled as L´evy processes, the market model is incomplete, in general. This means there is no unique martingale measure, i.e., there exist an infinite number of equivalent martingale measures. The publications on derivative pricing with L´evy processes are numerous, e.g., see Madan

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6 Active Portfolio Management

and Milne (1991), Madan, Carr and Chang (1998), Prause (1999), Schoutens (2003), and Carr and Wu (2004). However, we do not consider derivative pricing with L´evy processes in this context. Since we are more interested in optimal portfolio construction, the literature on this topic is briefly reviewed. Kallsen (2000) considers the multi-dimensional problem of maximizing the expected logarithmic utility from consumption or terminal wealth in a general semimartingale market model. The solution is given explicitly in terms of the semimartingale characteristics of the securities price process. Benth, Karlsen and Reikvam (2001b) consider a similar problem. However, the problem is solved in one dimension by a viscosity solution approach. The NIG L´evy process is applied in Benth, Karlsen and Reikvam (2001a), where the solution is given in one dimension. A multi-dimensional solution is found in Emmer and Kl¨ uppelberg (2004) with a Capital-at-Risk constraint. Optimal portfolio construction with the variance gamma (VG) L´evy process is considered in Cvitani´c, Polimenisz and Zapatero (2005) and Madan and Yen (2005). The Generalized Hyperbolic L´ evy Process and its Limiting Cases We can define a corresponding L´evy process for every infinitely divisible distribution. In Chapter 3, we have found that the generalized hyperbolic (GH) distribution is well suited for describing returns of various assets. We may define a L´evy process whose increments have a GH distribution because Barndorff-Nielsen and Halgreen (1977) proved that the GH distribution is infinitely divisible. The GH distribution contains a huge variety of important distributions in finance as limiting cases. Among these are the hyperbolic, normal inverse Gaussian (NIG), a version of the skewed t, variance gamma, t, and the normal distribution. All these limiting cases are derived in Eberlein and von Hammerstein (2004) for the univariate case. In order to work with the GH L´evy process, we need to know its L´evy triplet. The notation for the GH distribution of Appendix A.2.3 is used. The corresponding L´evy triplet is (γ, 0, νGH (dx)), i.e., there is no Brownian motion part and therefore, the GH L´evy process is a pure jump process. Note that if the L´evy measure is of the form ν(dx) = ν˜(x)dx, we call ν˜(x) the L´evy density. The n-dimensional L´evy measure of the GH L´evy process is

6.2 Portfolio Management with L´evy Processes

νGH (dx) = p where α =

p

2ex

T Σ −1 γ n

n

125

¢ ¡ √ n max(0, λ)α 2 K n2 α xT Σ −1 x

(2π)n det(Σ)(xT Σ −1 x) 4 ¢ ¡p n Z ∞ 2 o (α + 2y) 4 K n2 (α2 + 2y)(xT Σ −1 x) ¡ ¢ + dy) dx, √ √ 2 2 π 2 y J|λ| ( 2χy) + Y|λ| ( 2χy 0

ψ + γ T Σ −1 γ, Jλ and Yλ are Bessel functions, see Appendix A.2.4 for details.

The proof for this result is found in Masuda (2004). Accordingly, the n-dimensional L´evy measure of the NIG distribution is √ T −1 ¢ ¡p 2 χ ex Σ γ ³ ψ + γ T Σ −1 γ ´ n+1 4 n+1 K xT Σ −1 x(ψ + γ T Σ −1 γ) dx, νN IG (dx) = p T −1 2 x Σ x (2π)n+1 det(Σ)

see Masuda (2004) for the proof. Note that Masuda uses the notation of Blaesild and Jensen (1981). The parameters λ and µ are in both cases the same, the correspondences of the others are δ=

√

χ,

α=

p ψ + γ T Σ −1 γ,

Λ = Σ,

β = Σ −1 γ,

γ=

p ψ.

The derivation of the L´evy densities is based on the fact that the GH L´evy process can be introduced via Brownian subordination. The main idea of constructing a L´evy process by a Brownian subordination is to replace calendar time of a Brownian motion with a random time, given by a stochastic process which is called subordinator. For the construction of the GH L´evy process, the generalized inverse Gaussian (GIG) L´evy process is used a subordinator. The proof of this statement is found in Eberlein (2001). Correspondingly, the limiting cases of the GIG distribution may also be used as subordinators. Note that a subordinator is nondecreasing and always of finite variation. The L´evy measure of the n-dimensional VG distribution is T −1 ³ γ T Σ −1 γ + 2 ´ n4 ³r ¡ 2ex Σ γ 2¢´ ν T Σ −1 x γ T Σ −1 γ + n dx, νV G (dx) = p K x 2 xT Σ −1 x ν ν (2π)n det(Σ)

see Appendix C.3 for the proof. By setting γ = θ and Σ = σ 2 , we arrive at the same

L´evy density as in Madan et al. (1998, Equation (14)) for the univariate case. The L´evy measures of the NIG and the VG L´evy process are much more convenient to work with than the L´evy measure of the GH L´evy process. Therefore, they are more often found in financial applications than GH L´evy processes. Portfolio Dynamics and Optimal Control Problem Formulation We are considering a market which contains n investment opportunities. The price process of the i-th investment opportunity is described as

126

6 Active Portfolio Management

Pi (t) = Pi (0)eLi (t) , Pi (0) > 0, where Li (t) is a L´evy process. The returns of the single assets Li are assembled in the n-dimensional vector L´evy process L = (L1 , . . . , Ln )T ∈ Rn . In addition, there exists a risk-free investment opportunity B(t) with instantaneous rate of return r(t) ∈ R: dB(t) = B(t)r(t)dt , B(0) > 0 . Recall that in the Black-Scholes framework, L(t) is a Wiener process. The stochastic processes considered in this context are defined on a fixed, filtered probability space (Ω, F, {Ft }t≥0 , P) with Ft satisfying the usual conditions. In order to account for the stylized facts of asset returns, let L(t) be an n-dimensional L´evy process with characteristic triplet (γ, σσ T , ν), γ ∈ Rn , σ ∈ Rn×n , and ν(dx) ∈ R. The L´evy measure describes the expected number of jumps of a certain height in a time interval of length 1. A L´evy process is of infinite activity if almost all paths of a L´evy process have an infinite number of jumps on every compact interval. The return vector L(t) may have finite, e.g., the Poisson process, or infinite activity, e.g., the generalized hyperbolic L´evy process. If the process is of infinite activity, we may omit the diffusion component. For finite activity jump processes we must not omit the diffusion because this would lead to absence of local activity which is not a reasonable assumption. On the topic of disentanglement of the diffusion and the jump part in a stochastic process, the reader is referred to Ait-Sahalia (2004). We assume that the investor’s portfolio is self-financing and there are no exogenous inor outflows of money (e.g., consumption). We denote by the vector h(t) ∈ Rn the amount of each corresponding asset held by the investor at time t. The dynamics of the investors wealth V are self-financing if the following relation holds dV (t) = hT (t)dP. The reader is referred to Cont and Tankov (2004) for the proof. Note that h(t) is a simple predictable process, see Protter (1990) for details, and therefore the equality h(t−) = h(t) holds. Denote by u(t) the fraction of wealth invested in the corresponding asset. Using Itˆo’s lemma we get the following dynamics for the price processes,

6.2 Portfolio Management with L´evy Processes

127

o diag(Σ)dt + dL + e∆L − 1 − ∆L , 2 Z n£ ¤ 1 {ez − 1n − z}ν(dz) dt = diag(P (t−)) γ + diag(Σ) + 2 |z|<1 Z o z {e − 1n }N (dt, dz) + σdW (t) +

dP (t) = diag(P (t−))

n1

Rn

where diag(P (t−)) denotes the diagonal matrix of the vector P (t−), diag(Σ) denotes the vector of the diagonal elements of the matrix Σ, ez = (ez1 , . . . , ezn )T ∈ Rn , and 1n = (1, . . . , 1)T ∈ Rn . The same result is also derived in Kallsen (2000). The self-financing dynamics of the investors wealth V may be expressed as n

dPi (t) dV (t) dB(t) X ui (t−) = u0 (t−) + , V (t−) B(t) P (t−) i i=1 where u1 (t), . . . , un (t) denotes the fraction of wealth invested in the corresponding risky asset and u0 (t) accordingly in the riskless asset at time t. Because the portfolio is selfP financing, we have the constraint ni=0 ui (t) = 1. We may then rewrite the wealth dynamics as

ª dV (t) © T 1 = u [ diag(Σ) − 1n r] + r dt + uT (t−)(dL + e∆L − 1n − ∆L) V (t−) 2 Z n¡ ¢o 1 T T z = u [γ + diag(Σ) − 1n r] + r + u {e − 1n − z}ν(dz) dt 2 |z|<1 {z } | + uT σdW (t) +

V (0) > 0.

Z

b(u)

Rn

uT {ez − 1n }N (dt, dz)

(6.1)

The second equality is obtained by using the L´evy-Itˆo decomposition. In order for the wealth to remain positive for all t, we impose the constraint uT 1n ≤ 1 for u ≥ 0, i.e.,

u ∈ U = {u ∈ Rn |uT 1n ≤ 1, u ≥ 0}. Having derived the wealth dynamics of the considered investor, we can finally state the optimization problem. We consider the case in which the investor’s utility function of the constant relative risk aversion (CRRA) type. In order to solve this problem with Bellman’s optimality principle, we introduce the value or cost-togo function as J(t, V ) = max1 E u(·)∈Ln

h1 δ

δ

i

V (T ) ,

where δ denotes the coefficient of risk aversion. The Hamilton-Jacobi-Belman equation for this problem is

128

6 Active Portfolio Management

+

Z

Rn

h 1 Jt (t, V ) + max JV (t, V )b(u)V + JV V (t, V )uT ΣuV 2 u∈U 2 i T z T z J(t, V + V u {e − 1n }) − J(V, t) − JV (t, V )V u {e − 1n }ν(dz) = 0

with b(u) as in (6.1) and with the terminal condition J(T, V (T )) = 1δ V (T )δ . Plugging the separation ansatz J(t, V ) = h(t)V (t)δ into the HJB equation results in h ˙h(t)V (t)δ + h(t)V (t)δ max γb(u)δ + 1 δ(δ − 1)uT Σu u∈U 2 Z i (1 + uT {ez − 1n })δ − 1 − δuT {ez − 1n }ν(dz) = 0. + Rn

In order to compute the optimal asset allocation strategy, the constrained optimization problem above needs to be solved numerically. This completes the derivation of the optimal asset allocation strategy for L´evy driven asset prices. The optimal control vector u is fully determined by the L´evy triplet of the risky investment opportunities.

7 Conclusions and Outlook

If we knew what it was we were doing, it would not be called research, would it? Albert Einstein

The aim of this thesis is to shorten the gap between the development of sophisticated financial models and their application in practice. In order to achieve this goal, methods and models from financial engineering are used. However, these are conceptually the same as those found in feedback control theory. Therefore, the ideas of feedback control are applied to financial problems. The main difference between the control of technical systems and the control of financial systems is that financial systems are heavily dominated by randomness. This rules out the use of deterministic models since these cannot take the main properties of asset returns into account. By modeling asset prices as dynamic stochastic models, optimal asset allocation strategies can be derived through dynamic stochastic optimization. In Chapter 2, the different components of asset allocation are presented. The considered financial assets are briefly described, while the asset allocation process as such is described in greater detail. Risk measurement and management skills are key factors for successful active portfolio management. Therefore, the topic of static and dynamic risk measures is discussed. Chapter 3 reviews the current state of asset price modeling and dynamic optimization techniques in finance. Different models are tested for their suitability to describe univariate and multivariate asset returns. The generalized hyperbolic distribution and its limiting cases offer a very rich modeling family for describing asset returns. These types of distributions fit real-world financial data considerably better than the predominantly encountered normal distribution. The dependence properties of various financial assets are analyzed as well. It is found that the simultaneous occurrence

130

7 Conclusions and Outlook

of extreme losses cannot be explained by correlation alone. This type of phenomenon is called tail dependence and empirical evidence is given that investors should be aware of concurrent extreme losses. In Chapter 4, the topic of hedge funds is discussed. Since hedge fund managers actively manage their portfolios, the hedge fund industry is an inspiring area for active portfolio management strategies. A wide range of different styles categorize the strategies of hedge fund managers. However, the performance measurement considering the whole hedge fund industry is difficult because of the lack of transparency. This also complicates the modeling of hedge funds because their the real sources of returns are hard to discover. The statistical properties of hedge funds are analyzed in detail. As for traditional assets, the generalized hyperbolic distribution and its limiting cases are flexible enough to accurately take the possible skewness and fat-tailedness of hedge funds returns into account. However, returns of some styles show volatility clustering and serial correlation, which are both awkward properties of hedge fund returns. A possible explanation for serial correlation are the presence of illiquid assets in the hedge fund manager’s portfolio. Finally, the peculiarities of hedge fund investing and risk management for hedge funds are discussed. Having extensively discussed the modeling properties of financial assets in the first part, the second part is devoted to the conception and implementation of asset allocation strategies. Chapter 5 considers the well-studied continuous-time models driven by Brownian motion. A closed-form solution is derived for an investor with constant relative risk aversion and three risk-bearing investment opportunities. One of the investment opportunities is an alternative investment. The optimal fraction of wealth invested in the alternative investment is analyzed in detail. In the second part of Chapter 5, the optimal asset allocation strategy is derived for the case in which not all factors influencing the return of the risky assets are exactly known to the investor. We call this type of problem optimal asset allocation under partial information. This problem is solved by using the methods of Kalman filtering to shown that the partial information problem can be transformed to a full information problem whose solution is known. The results are applied in a balanced fund case study including alternative investments. Chapter 6 is devoted to active portfolio management, for which a formal definition is given. Active portfolio management, per definition, needs an associated benchmark, otherwise risk and performance measurement are ambiguous. The returns of an active

7 Conclusions and Outlook

131

portfolio management strategy are driven by the investment universe and the investment strategy. An active portfolio management case study concerning the sector rotation problem is conducted. An adaptive factor model is used for the prediction of the returns and a dynamic volatility model with normal inverse Gaussian distributed innovations is used to perform the risk management. The results of an out-of-sample case study are promising, yielding an alpha of 5% and an information ratio of 0.96. The remaining part of the chapter discusses the use of L´evy processes for active portfolio management. The necessary L´evy densities are given for describing the corresponding multivariate L´evy processes. Outlook Since so many research areas such as economics and mathematics intersect in financial engineering, the room for improvement is vast. However, the statistical testing of asset price models as well as the implementation of optimal asset allocation strategies is time consuming. Concerning the statistical properties of asset returns, it would be interesting to compare the α-stable models against the hyperbolic models analyzed in this work. On the subject of copulas, only two members of the class of elliptical copulas are analyzed in this work. However, there is a huge variety of different copulas which may be better suited for describing the dependence of multivariate asset returns. For instance, the copula of the generalized hyperbolic distribution should offer better results than the t copula. Since the introduction of L´evy processes in finance, dynamic asset models in continuous-time have become much more promising for further research. Since the use of L´evy processes for solving optimal asset allocation problems is still a rather young research area, many questions remain unanswered. This is also the case for dynamic risk measures. Dynamic stochastic optimization in continuous-time is a well-known procedure. However, the numerical computation of constrained problems is still limited to small problems. Nevertheless, viscosity solutions offer a powerful tool for solving nonlinear partial differential equations. Their use for optimal portfolio construction would have exceeded the scope of this work. For the unconstrained case, either Bellman’s optimality principle or Pontryagin’s maximum principle can be used to find optimal solutions. The solution of the partial information problem in a general L´evy framework, however, is not straightforward and has not been found yet. The use of copulas in connection with optimal portfolio construction is particularly well suited in a stochastic programming framework. Clearly,

132

7 Conclusions and Outlook

there are still a lot of interesting problems to be solved in financial engineering, and in optimal portfolio construction in particular.

A Probability and Statistics

A.1 Moments of Random Variables We consider an n-dimensional random vector X. The first central moment is the mean vector, denoted by µ = E[X]. The second central moment is the covariance matrix, defined as cov(X) = E[(X − µ)(X − µ)T ]. We consider the skewness and kurtosis instead of the third and fourth central moments. The univariate skewness γ1 and the kurtosis γ2 are defined as γ1 (Xi ) =

E[(Xi − µ)3 ]

3 , (var(Xi )) 2 E[(Xi − µ)4 ] γ2 (Xi ) = , var(Xi )

respectively. In the literature the term excess kurtosis is often found instead of kurtosis. The excess kurtosis is defined as γ2 (Xi )−3. The excess kurtosis to the normal distribution is zero. There also are measures of multivariate skewness and kurtosis, e.g., see Mardia (1970).

A.2 Probability Distributions A.2.1 Normal Mean-Variance Mixture Distributions Let U be a random variable on [0, ∞), Σ ∈ Rn×n a covariance matrix, and µ, γ ∈ Rn two arbitrary vectors. The random variable X| U = u ∼ N (µ + uγ, uΣ)

134

A Probability and Statistics

is said to have a normal mean-variance mixture distribution. This distribution is elliptical for γ = 0 and is called normal variance mixture in this case. The characteristic function of random variable X which has normal mean-variance mixture distribution is φX (y) = E[eiy

TX

´ ³ T ˆ 1 y T Σy − iy T γ , ] = eiy µ H 2

ˆ is the Laplace-Stieltjes transform of the distribution of the mixing variable. See where H Bingham and Kiesel (2002) for the proof. An important property of normal mean-variance mixture distributions is that they are closed under linear operations, i.e., let X as introduced above, and Y = AX + b with A ∈ Rk×n and b ∈ Rk . The characteristic function of Y becomes φY (y) = E[eiy

T (AX+b)

T

T

T

] = eiy b E[ei(y A)X ] = eiy b φX (AT y) ³ ´ T ˆ 1 y T AΣAT y − iy T Aγ . = eiy (Aµ+b) H 2

Therefore, we have for the random vector Y Y | W = w ∼ N (Aµ + b + wAγ, wAΣAT ) The mean and the variance of X are calculated as • E[X] = µ + E[W ]γ,

• cov[X] = E[W ]Σ + var[W ]γγ T . The reader is referred to McNeil et al. (2005) for more details. A.2.2 Univariate Probability Distributions Generalized Inverse Gaussian (GIG) Distribution A random variable X ∈ R has a generalized inverse Gaussian distribution, i.e., X ∼ GIG(λ, χ, ψ), if its density is p ( ψ/χ)λ λ−1 − 1 (χx−1 +ψx) √ f (x) = , x e 2 2Kλ ( χψ)

x > 0,

with χ ∈ R, ψ ∈ R, and λ ∈ R. Kλ denotes the modified Bessel function of the third kind with index λ, see Appendix A.2.4 for details. The parameters satisfy χ > 0, ψ ≥ 0 if λ < 0; χ > 0, ψ > 0 if λ = 0; and χ ≥ 0, ψ > 0 if λ > 0. The GIG density contains the gamma and inverse gamma densities as limiting cases. The non-central moments of the GIG density are:

A.2 Probability Distributions

³ χ ´ α2 K (√χψ) λ+α √ , E[X α ] = ψ Kλ ( χψ)

135

α ∈ R.

ˆ of the X ∼ GIG(λ, χ, ψ) distribution is given by The Laplace transform H ³ ψ ´ λ2 K (pχ(ψ + 2s)) λ ˆ √ H(s) = , s > 0. ψ + 2s Kλ ( χψ) For more details on the GIG distribution and its properties see Jorgensen (1982). q √ ( χψ) √ • E[X] = µ + ψχ KKλ+1 γ λ ( χψ) • var[X] =

√ √ √ 2 χ Kλ+2 ( χψ) Kλ ( √χψ)−Kλ+1 ( χψ) ψ Kλ2 ( χψ)

Inverse Gaussian (IG) Distribution The best-known limiting distribution of the GIG distribution is the inverse Gaussian distribution X ∼ IG(χ, ψ) = GIG(− 21 , χ, ψ), the density of which is √ √ χ 1 3 −1 f (x) = √ e χψ x− 2 e− 2 (χx +ψx) , x > 0. 2π The Laplace transform of the inverse Gaussian distribution is ˆ H(s) =e

√

χψ−

√

χ(ψ+2s)

,

s > 0.

These equations are easily derived from the properties of the modified Bessel function of the third kind, see Appendix A.2.4. q • E[X] = ψχ q • var[X] = ψ1 ψχ Gamma (Γ ) Distribution Another important limiting case of the GIG distribution is, for λ > 0 and χ = 0, the gamma distribution X ∼ Γ (λ, ψ/2) = GIG(λ, 0, ψ). Its density is f (x) =

(ψ/2)λ λ−1 − 1 ψx x e 2 , Γ (λ)

x > 0.

The Laplace transform of the gamma distribution is ˆ H(s) = • E[X] = 2 ψλ

• var[X] = 4 ψλ2

³

ψ ´λ , ψ + 2s

s > 0.

136

A Probability and Statistics

Inverse Gamma (IΓ ) Distribution For λ < 0 and ψ = 0, we get another limiting case of the GIG distribution. This case is called the inverse gamma distribution X ∼ IΓ (λ, χ/2) = GIG(λ, χ, 0). Its density is f (x) =

(χ/2)−λ λ−1 − 1 χx−1 x e 2 , Γ (−λ)

x > 0.

The Laplace transform of the inverse gamma distribution is ¡√ ¢ −λ 2 2χx 2K (χx/2) λ ˆ , x > 0. H(x) = Γ (−λ) χ • E[X] = − 12 λ+1

• var[X] =

χ2 1 , 4 (λ+1)2 (−λ−2)

λ < −2.

Univariate Skewed t There are several methods for constructing a skewed t distribution. One is found in Fernandez and Steel (1998). It is a method for extending a unimodal and symmetric distribution to a skewed version. Let t(ν, µ, σ 2 ) denote the univariate t density. The following density is a skewed version of the univariate t density, f (x) =

´ 2 ³ 2 2 t(ν, (x − µ)/γ, σ )I (x) + t(ν, (x − µ)γ, σ )I (x) . [µ,∞) (−∞,µ) γ + γ1

(A.1)

The skewness is measured by the parameter γ ∈ (0, ∞). We have no skewness for γ = 1. A.2.3 Multivariate Probability Distributions Multivariate Normal A random variable X ∈ Rn has a normal distribution, i.e., X ∼ Nn (µ, Σ), if its density is f (x) = p

1 (2π)n

1

det(Σ)

e− 2 (x−µ)

T Σ −1 (x−µ)

,

with µ ∈ Rn , and Σ ∈ Rn×n . The mean and covariance are E[x] = µ and cov[X] = Σ, respectively.

A.2 Probability Distributions

137

Multivariate t A random variable X ∈ Rn has a t distribution, i.e., X ∼ tn (ν, µ, Σ), if its density is f (x) =

³ ´− 12 (ν+n) Γ ( ν+n ) 1 T −1 2 p 1 + , (x − µ) Σ (x − µ) ν Γ ( ν2 ) (πν)n det(Σ)

(A.2)

with ν ∈ R, µ ∈ Rn , and Σ ∈ Rn×n , Γ denotes the gamma function. The mean and the covariance are E[X] = µ and cov[X] = kurtosis are γ1 (X) = 0 and γ2 (X) =

ν Σ, ν−2

6 ν−4

respectively. The univariate skewness and

+3 =

3(ν−2) , ν−4

respectively. The k-th moment is

not defined for k > ν. For ν → ∞, the t-distribution converges to the normal distribution. Multivariate Generalized Hyperbolic (GH) A generalized hyperbolic random variable has a normal mean-variance mixture distribution, the mixing variable has a GIG distribution. The random variable X ∈ R n has a multivariate generalized hyperbolic distribution, i.e., X ∼ GHn (λ, χ, ψ, µ, Σ, γ), if its density is

f (x) = c

Kλ− n2

³q¡

¢¡ ¢´ T −1 χ + (x − µ)T Σ −1 (x − µ) ψ + γ T Σ −1 γ e(x−µ) Σ γ , ³ ´ n4 − λ2 χ + (x − µ)T Σ −1 (x − µ)

(A.3)

where λ ∈ R, χ ∈ R, ψ ∈ R, µ ∈ Rn , Σ ∈ Rn×n , and γ ∈ Rn . The normalizing constant c is given as

p ¡ ¢n−λ ( ψ/χ)λ ψ + γ T Σ −1 γ 4 2 c= p , √ (2π)n det(Σ) Kλ ( χψ)

Kλ denotes the modified Bessel function of the third kind with index λ. The domain of variation of the parameters is as for the GIG distribution. Many different kind of parametrizations of the generalized hyperbolic distribution are found in the literature. The advantage of the present parametrization is that the GIG parameters are scale and location invariant. Note that we cannot distinguish between the parameterizations GHn (λ, χ/c, cψ, µ, cΣ, cγ) and GHn (λ, χ, ψ, µ, Σ, γ) for an arbitrary c > 0. We therefore introduce the constraint ψ = 1 when fitting the GH distribution. q √ ( χψ) √ • E[X] = µ + ψχ KKλ+1 γ λ ( χψ) √ q √ ( χψ) χ Kλ+1√ χ Kλ+2 ( χψ) • cov[X] = ψ Kλ ( χψ) Σ + ψ

√ √ 2 Kλ ( χψ)−Kλ+1 ( χψ) √ γγ T Kλ2 ( χψ)

138

A Probability and Statistics

Multivariate Normal Inverse Gaussian (NIG) A random variable X ∈ Rn has a multivariate normal inverse gaussian distribution, i.e.,

X ∼ NIGn (χ, ψ, µ, Σ, γ), if its density is a GHn (λ = − 12 , χ, ψ, µ, Σ, γ). The characteristic function of the NIG distribution is, by using the properties the normal mean-variance mixture distributions, φX (t) = E[eit

TX

] = eit

T µ+

√

χψ−

√

χ(ψ+tT Σt−2itT γ)

From this, we immediately see that the class of NIG distributions is closed under convolutions, i.e., NIGn (χ1 , ψ, µ1 , Σ, γ) ∗ NIGn (χ2 , ψ, µ2 , Σ, γ) = NIGn (χ3 , ψ, µ3 , Σ, γ), √ √ where µ3 = µ1 + µ2 and χ3 = ( χ1 + χ2 )2 . q • E[X] = µ + ψχ γ ´ q ³ χ 1 T • cov[X] = ψ Σ + ψ γγ

The NIG distribution is a normal mean-variance mixture distribution with the IG distribution as mixing variable. Multivariate Skewed t (s-t) A random variable X ∈ Rn has a multivariate skewed t distribution, i.e., X ∼ stn (ν, µ, Σ, γ),

if its density is a GHn (λ = − ν2 , χ = ν, ψ → 0, µ, Σ, γ), ν > 0. If the limit ψ → 0 is evaluated in (A.3), we arrive at ³q¡ ´ ¢ T −1 K ν+n ν + (x − µ)T Σ −1 (x − µ) γ T Σ −1 γ e(x−µ) Σ γ 2 , f (x) = c ³¡ ¢´ ν+n 4 T −1 ν + (x − µ) Σ (x − µ)

(A.4)

where ν ∈ R, µ ∈ Rn , Σ ∈ Rn×n , and γ ∈ Rn . The normalizing constant c is given as ν+n

ν

ν+n

ν+n

2(γ T Σ −1 γ) 4 ( ν2 ) 2 2(γ T Σ −1 γ) 4 ( ν2 ) 2 c = ¡ ν ¢p = ¡ ν ¢p . Γ 2 (2π)n det(Σ) Γ 2 (πν)n det(Σ)

The limiting case γ → 0 results in the multivariate t distribution as described in (A.2). The skewed t distribution is a normal mean-variance mixture distribution with the inverse gamma distribution as mixing variable. • E[X] = µ + • var[X] =

ν γ ν−2

ν Σ ν−2

+

2ν 2 γγ T , (ν−2)2 (ν−4)

ν > 4, for γ 6= 0,

ν > 2, for γ ≡ 0.

A.2 Probability Distributions

139

Multivariate Variance Gamma (VG) A random variable X ∈ Rn has a multivariate variance gamma distribution, i.e., X ∼

VG(ν, µ, Σ, γ), if its density is a GHn (λ = ν1 , χ → 0, ψ = ν2 , µ, Σ, γ), ν > 0. If the limit

χ → 0 is evaluated in (A.3), we arrive at ³q¡ ¢´ ¢¡ T −1 (x − µ)T Σ −1 (x − µ) ν2 + γ T Σ −1 γ e(x−µ) Σ γ K1−n ν 2 , f (x) = c ³ ´ n4 − 2ν1 (x − µ)T Σ −1 (x − µ)

where ν ∈ R, µ ∈ Rn , Σ ∈ Rn×n , and γ ∈ Rn . The normalizing constant c is given as 2

c= Note that we have chosen ψ =

2 ν

1

νν

¡2

ν p

+ γ T Σ −1 γ

¢ n4 − 2ν1

(2π)n det(Σ)Γ ( ν1 )

,

to be most conform with the notation in Madan et al.

(1998). To arrive at the same distribution as in Madan et al. (1998) for the univariate case, set γ = θ and Σ = σ 2 . The VG distribution is a normal mean-variance mixture distribution with the gamma distribution as mixing variable. The characteristic function of the VG distribution is, by using the properties the normal mean-variance mixture distributions, φX (t) = E[eit

TX

] = eit

Tµ

³

1+ν

¡1 T ¢´− ν1 t Σt − itT γ 2

From this, we immediately see that the class of VG distributions is closed under convolutions, i.e., ν VG(ν, µ1 , Σ, γ) ∗ VG(ν, µ2 , Σ, γ) = VG( , µ1 + µ2 , 2Σ, 2γ). 2 • E[X] = µ + γ

• cov[X] = Σ + νγγ T A.2.4 Bessel Functions and Modified Bessel Functions Bessel functions are solutions of the following differential equation, x2

d2 f df + x + (x2 − λ2 )f = 0. 2 dx dx

Jλ is called Bessel function of the first kind and has the following series representation Jλ (x) = (x/2)λ

∞ X k=0

(−x/2)2k , k!Γ (λ + k + 1)

| arg(x)| < π.

The Bessel function of the second kind, Nλ (x), is defined as

140

A Probability and Statistics

Nλ (x) =

Jλ (x) cos(λπ) − J−λ (x) , sin(λπ)

| arg(x)| < π.

Modified Bessel functions are solutions of the following differential equation, x2

d2 f df + x − (x2 + λ2 )f = 0. 2 dx dx

Iλ is called modified Bessel function of the first kind and has the following series representation Iλ (x) = (x/2)

λ

∞ X k=0

(x/2)2k . k!Γ (λ + k + 1)

Kλ denotes the modified Bessel function of the third kind with index λ. The modified Bessel function of the third kind is also called MacDonald function. Kλ may be expressed as function of Iλ , i.e., π Iλ (x) − I−λ (x) 1 ³ x ´λ Kλ (x) = = 2 sin(λπ) 2 2

Z

∞

x2

t−λ−1 e−t− 4t dt.

(A.5)

0

A special case is λ = − 21 , the modified Bessel function of the third kind becomes K− 1 (x) = 2

r

π − 1 −x x 2e . 2

Useful properties of the modified Bessel function of the third kind are K−λ (x) = Kλ (x), 2λ Kλ+1 (x) = Kλ (x) + Kλ−1 (x). x From this, we immediately get K 3 (x) = K 1 (x)(1 + x1 ). For the asymptotic expansions for 2

2

x → 0 we get Kλ (x) ∼ 2λ−1 Γ (λ)x−λ , λ > 0,

Kλ (x) ∼ 2−λ−1 Γ (−λ)xλ , λ < 0.

More details are found in Abramowitz and Stegun (1972) and Gradshteyn and Ryzhik (1994).

B GARCH Models for Dynamic Volatility

We follow the notation of Zivot and Wang (2002), see McNeil et al. (2005) for the technical details. The estimations are all done by a maximum likelihood approach.

B.1 Univariate GARCH Processes Univariate GARCH(p,q) Process Let zt be independent, identically distributed (i.i.d.) random variables with zero mean and unit variance. The process yt is a GARCH(p,q) process if it has the following dynamics yt = µ + ε t , ε t = z t σt , q p X X 2 2 2 bj σt−j , ai εt−i + σt = a 0 + i=1

j=1

where a0 > 0, ai ≥ 0, i = 1, . . . , p, and bj ≥ 0, i = 1, . . . , q. Univariate TARCH(p,o,q) Process Let zt be i.i.d. random variables with zero mean and unit variance. The process yt is a TARCH(p,o,q) process if it has the following dynamics yt = µ + ε t , ε t = z t σt , q p o X X X 2 2 2 2 bj σt−j , 1{εt−k <0} γk εt−k + ai εt−i + σt = a 0 + i=1

k=1

j=1

where 1 denotes the identicator function, a0 > 0, ai ≥ 0, i = 1, . . . , p, and bj ≥ 0, i = 1, . . . , q.

142

B GARCH Models for Dynamic Volatility

B.2 Multivariate GARCH Processes The operator R is defined as follows: Definition B.1 (Operator R).

1/2

Let Σ be a positive-definite covariance matrix. By Σd

we define the diagonal matrix

whose elements are the square roots of the diagonal elements of Σ. The operator R is defined as 1/2

1/2

R(Σ) = (Σd )−1 Σ(Σd )−1 , where R(Σ) is the correlation matrix of Σ. Constant Conditional Correlation (CCC) GARCH Process Let Zt ∈ Rn be a random vector whose components are i.i.d. random variables with zero mean and unit variance. The process Yt is a multivariate CCC-GARCH process if it has the following dynamics 1/2

ε t = Σ t Zt ,

Yt = µ + ε t ,

1/2

where Σt

∈ Rn×n is the Cholesky factor of a positive-definite matrix Σt which is mea-

surable with respect to the filtration Ft−1 . Let ∆t ∈ Rn×n be a diagonal matrix whose elements are the square roots of the univariate GARCH(pk ,qk ) processes is of the form 2 σt,k

=

ak0

+

pk X

aki ε2t−i,k

i=1

+

qk X

2 bkj σt−j,k ,

k = 1, . . . , n .

j=1

where, for all k, ak0 > 0, aki ≥ 0, i = 1, . . . , pk , and bkj ≥ 0, i = 1, . . . , qk . The conditional covariance matrix Σt is defined as Σ t = ∆ t Rc ∆ t , where Rc is a positive-definite correlation matrix. Dynamic Conditional Correlation (DCC) GARCH Process Let Zt ∈ Rn be a random vector whose components are i.i.d. random variables with zero mean and unit variance. The process Yt is a multivariate DCC-GARCH process if it has the following dynamics

B.2 Multivariate GARCH Processes 1/2

Yt = µ + ε t ,

1/2

where Σt

143

ε t = Σ t Zt ,

∈ Rn×n is the Cholesky factor of a positive-definite matrix Σt which is mea-

surable with respect to the filtration Ft−1 . Let ∆t ∈ Rn×n be a diagonal matrix whose elements are the square roots of the univariate GARCH(pk ,qk ) processes is of the form 2 σt,k

=

ak0

+

pk X

aki ε2t−i,k

i=1

+

qk X

2 bkj σt−j,k ,

k = 1, . . . , n .

j=1

where, for all k, ak0 > 0, aki ≥ 0, i = 1, . . . , pk , and bkj ≥ 0, i = 1, . . . , qk . The conditional covariance matrix Σt is defined as Σt = ∆t R(Qt )∆t , where R(Qt ) is the conditional correlation matrix and Qt has the dynamics ³

Qt = 1 −

p X i=1

q p q ´ X X X −1 −1 T ai − bi Q c + ai ∆t−i εt−i [∆t−i εt−i ] + bj Qt−j . j=1

i=1

j=1

Qc is a positive-definite covariance matrix. As noted in McNeil et al. (2005), Q c should be estimated in one step by maximum likelihood. However, we use the unconditional covariance of the standardized residuals resulting from the first stage estimation for convenience.

C Proofs

C.1 Tail Dependence within a t Copula t Suppose we have an n dimensional random vector with t copula CR,ν . At first we are

interested in the distribution of the first n − 1 components, therefore we integrate over

ˆ we denote the correlation matrix the whole range of xn in (3.2) for the t copula (3.3). By R which equals R but has the n-th row and column removed. We make use of the integration Rb R g(b) −1 1 n )) = f (t−1 , rule a f (g(x))g 0 (x)dx = g(a) f (u)du, note that ui = Fi (xi ) and d(tνdu(u n (u )) 1,ν

f (x1 , . . . , xn−1 ) = =

Z

∞ −∞

n−1 Y

i=1 n−1 Y

t cR,ν (F1 (x1 ), . . . , Fn (xn ))

fi (xi )

Z

n Y

ν

n

fi (xi )dxn

i=1

1 0

t cR,ν (u1 , . . . , un )dun

Z 1 −1 fR,ν (t−1 1 ν (u1 ), . . . , tν (un )) = dun fi (xi ) Qn−1 −1 f1,ν (t−1 ν (un )) i=1 f1,ν (tν (ui )) 0 i=1 Z ∞ n−1 −1 Y fR,ν (t−1 ν (u1 ), . . . , tν (un−1 )), xn ) fi (xi ) = dxn (C.1) Qn−1 −1 −∞ i=1 f1,ν (tν (ui )) i=1 n−1 Y

−1 −1 fR,ν ˆ (tν (u1 ), . . . , tν (un−1 ))) = fi (xi ) Qn−1 −1 i=1 f1,ν (tν (ui )) i=1

=

n−1 Y

t fi (xi )cR,ν ˆ (F1 (x1 ), . . . , Fn−1 (xn−1 )).

i=1

In order to solve the integral of C.1, we make use of the properties of the multivariate t density, see Kotz and Nadarajah (2004). From this result we can apply the formula for the tail dependence (3.5) for every pair in an n-dimensional t copula. Every pair in a ndimensional t copula has a bivariate t copula with the corresponding correlation coefficient and the same degree of freedom parameter as the n-dimensional copula.

146

C Proofs

C.2 Transformation from Partial to Full Information Let the price dynamics evolve as introduced (5.28), the factors as in (5.26) and (5.27). The function µ(x, y) is defined in (5.30). Summarized, dP = diag(P ) {µ(x, y) dt + σp dWp }, dx = [Ax x + Ay y + a]dt + σx dWx , dy = [Cx x + Cy y + c]dt + σy dWy . We calculate the dynamics of the logarithmic prices p = ln(P ) ∈ Rn with Itˆo’s lemma, where diag{Σ} denotes the vector of the diagonal elements of the matrix Σ. Note that the time arguments are omitted from now on wherever possible. 1 dp = [µ(x, y, t) − diag{Σ}] dt + σp dWp , 2 p(0) = ln(P (0)). For notational convenience, we introduce the new vector ξ = [pT , xT ]T ∈ Rn+m , containing

all the observable processes. We define the Brownian motion Wξ = [WpT , WxT ]T ∈ Rn+m , where Wp is introduced in Equation (5.28) and Wx in Equation (5.26). The filtering problem then consists of dy = [Cξ ξ + Cy y + c] dt + σy dWy , dξ = [Dξ ξ + Dy y + d] dt + σξ dWξ , where the deterministic matrix functions Cξ ∈ Rk×n+m and Dξ ∈ Rn+m×n+m are defined as h

i

Cξ = 0

Cx ,



Dξ = 

0

G

0

Ax



,

The deterministic matrix function Dy ∈ Rn+m×k and the deterministic vector function d ∈ Rn+m are



Dy = 

H Ay



,



d=

f−



1 diag{Σ} 2 

a

.

The matrix and vector functions Cy , c, and σy are defined in (5.27). The diffusion term of ξ, denoted by σξ ∈ Rn+m×n+m , is easily seen to be

C.2 Transformation from Partial to Full Information



σξ = 

σp 0 0 σx



.

147

(C.2)

The Brownian motions Wp , Wx , and Wy need not be independent. The correlation matrix of the n + m + k dimensional Brownian motion W = [WyT , WpT , WxT ]T ∈ Rn+m+k is defined as





    I1 ρyξ (t)   , ρ(t) =  ρTyp (t) I2 ρpx (t)  =    ρTyξ (t) ρξ (t) ρTyx (t) ρTpx (t) I3 I1

ρyp (t) ρyx (t)

(C.3)

where I denotes the identity matrix and ρyp ∈ Rk×n , ρyx ∈ Rk×m , ρpx ∈ Rn×m , I1 ∈ Rk×k , I2 ∈ Rn×n , and I3 ∈ Rm×m . In order to be regular, the correlation matrix ρ needs to be symmetric and positive-definite for all t. Therefore, the matrix square root of ρ, denoted 1

1

1

1

by ρ 2 , exists. Since ρ 2 is the matrix square root of ρ, we have the relationship ρ = ρ 2 (ρ 2 )T . Rt 1 We may construct the new Brownian motions W (t) by W (t) = 0 ρ 2 (s)dW (s). Thus, 1

we have for the increments dW = ρ 2 dW . The n + m + k Brownian motions W are uncorrelated. The filtering problem with uncorrelated Brownian motions then is dy = [Cξ ξ + Cy y + c]dt + σ y dW y + σ yξ dW ξ ,

(C.4)

dξ = [Dξ ξ + Dy y + d]dt + σ ξ dW ξ ,

(C.5)

where the volatility matrices σ y ∈ Rk×k , σ yξ ∈ Rk×n+m , σ ξ ∈ Rn+m×n+m are obtained by the identity of the following block matrices     σ σ σ 0  y yξ  =  y  ρ 12 . 0 σξ 0 σξ

1

(C.6) 1

The Cholesky factorization of ρ is a good choice for ρ 2 , i.e., ρ 2 is an upper triangular matrix. The equality above still holds when the left and the right hand side are multiplied by their transposed. This simple block matrix calculations gives     T T T T T σ σ + σ yξ σ yξ σ yξ σ ξ σ σ σy ρyξ σξ  y y = y y . T T T T T σ ξ σ yξ σξ σξ σξ ρyξ σy σξ ρξ σξ

(C.7)

From Lipster and Shiryaev (2001a, Theorem 10.3) we get the instantaneous changes of the estimation m, or the conditional mean, of the factors y with the Kalman filter as dm = [Cξ ξ + Cy m + c]dt + [B1 + νDyT ]B2 [dξ − {Dξ ξ + Dy m + d}dt] , m(0) = E[y(0)|ξ(0)].

(C.8)

148

C Proofs

The definition of B1 is given in (C.10). The error of the estimation is denoted by ν. In Lipster and Shiryaev (2001a, Theorem 10.3) it is proofed that the variance of m, denoted by ν ∈ Rk×k , evolves according to the following dynamics ν˙ = Cy ν + νCyT + B3 − [B1 + νDyT ]B2 [B1 + νDyT ]T ,

(C.9)

ν(0) = E[(y(0) − m(0))(y(0) − m(0))T ]. The matrices B1 ∈ Rk×n+m , B2 ∈ Rn+m×n+m , and B3 ∈ Rk×k are defined as B1 = σ yξ σ Tξ = σy ρyξ σξT ,

(C.10)

B2 = [σ ξ σ Tξ ]−1 = [σξ ρξ σξT ]−1 ,

(C.11)

B3 = σ y σ Ty + σ yξ σ Tyξ = σy σyT ,

(C.12)

where the second equalities are obtained from (C.7). In the dynamics of m in (C.8), the term dξ is involved. But dξ is a function of the unobservable factors y, therefore we cannot derive the optimal asset allocation strategy in this setup. Like this, we would not get an admissible investment process (see Bielecki and Pliska (1999) for details). To overcome this problem, we introduce the innovation process Wξ0 (t) given by the following stochastic differential equation dWξ0 = σ −1 ξ [dξ − {Dξ ξ + Dy m + d}dt] = σ −1 ξ Dy [y − m]dt + dW ξ .

(C.13)

It is proved in Lipster and Shiryaev (2001b, Theorem 12.5) that Wξ0 is indeed a Wiener process and that ξ0 ,Wξ0

Ftξ = Ft

.

(C.14)

Or, informally speaking, the information generated by ξ is equivalent to the information generated by (ξ0 , Wξ0 ). This new innovation process, as given in (C.13), is inserted into (C.5) and (C.8). This results in dm = [Cξ ξ + Cy m + c]dt + [B1 + νDyT ]B2 σ ξ dWξ0 ,

(C.15)

dξ = [Dξ ξ + Dy m + d]dt + σ ξ dWξ0 . In order to derive the asset price dynamics under the new innovation process Wξ0 , we decompose σ ξ into

C.3 L´evy Density of the Multivariate VG L´evy Process



σξ = 

σp σx



,

149

(C.16)

where σ p ∈ Rn×n+m , and σ x ∈ Rm×n+m . With this decomposition we may also decompose ξ in p and x. This yields the following new dynamics of p and x as 1 dp = [µ(x, m, t) − diag{Σ}]dt + σ p dWξ0 , 2 dx = [Ax x + Ay m + a]dt + σ x dWξ0 .

(C.17) (C.18)

From (C.7) we can identify the relationship σ ξ σ Tξ = σξ ρξ σξT . By inserting (C.2) and (C.16) in this relation we get  

σ p σ Tp σ p σ Tx σ x σ Tp

σ x σ Tx





=

σp σpT

σp ρpx σxT

σx ρTpx σpT

σx σxT



.

(C.19)

We may now state the price dynamics with respect to the Wiener process Wξ0 . Lemma C.1. Let the price dynamics evolve as in (5.28), where the factors evolve as in (5.26) and (5.27). Then the dynamics of the prices dP (t) = diag(P (t)) {µ(x(t), y(t), t) dt + σp (t)dWp (t)} , are the same as written in terms of Wξ0 , i.e., dP (t) = diag(P (t)){µ(x(t), m(t), t)dt + σ p (t)dWξ0 (t)},

(C.20)

where Wξ0 is defined as in (C.13) and the factors x and m are defined as in (C.18) and (C.15), respectively. Proof. The logarithmic prices are defined in (C.17). By using the equality σ p σ Tp = σp σpT = Σ from (C.19) we get (C.20) by using Itˆ o’s lemma. The transformation of the price dynamics of the investment opportunities is necessary to transform the partial-information problem into a full-information problem.

C.3 L´ evy Density of the Multivariate VG L´ evy Process We proceed as in Masuda (2004). First, we make use of the subordination property of the VG distribution and use Sato (1999, Theorem 30.1) or Cont and Tankov (2004, Theorem 4.2). Therefore,

150

C Proofs

νV G (B) =

Z Z B

∞ 0

T

n

−1

1 e x Σ γ s− 2 νΓ (s) p e− 2 (2π)n det(Σ)

¡

1 T −1 x Σ x+sγ T Σ −1 γ s

where νΓ is the L´evy density of the gamma distribution, i.e., νΓ (s) = have set λ =

1 ν

and ψ =

2 ν

¢

dsdx,

1 − ν1 s e , νs

where we

(see Schoutens (2003) for the definition of the L´evy density of

the gamma distribution). Evaluation the integral yields νV G (B) =

Z

ex

B

By using the equality νV G (B) =

Z

ex

B

ν R

T Σ −1 γ

p (2π)n det(Σ) a −b 1s −sc

s e

T Σ −1 γ

ds =

Z

∞

s

−n −1 2

e

−x

T Σ −1 x 1 −s 2 s

0

R ³ 1 ´a+1 c

¡ γ T Σ −1 γ 1 ¢ n2 p + 2 ν ν (2π)n det(Σ)

¡

γ T Σ −1 γ + ν1 2

¢

dsdx.

1

ta e−bc t −t dt we get Z

∞ 0

t

−n −1 − x 2

e

T Σ −1 x 2

¡

γ T Σ −1 γ + ν1 2

¢

1 −t t

dtdx.

Making use of the integral representation of the modified Bessel function of the third kind as given in (A.5) yields νV G (B) =

Z

2ex

B

ν

T Σ −1 γ

p (2π)n det(Σ)

³ γ T Σ −1 γ + 2 ´ n4 ν

xT Σ −1 x

³r ¡ 2¢´ T −1 T −1 n dx. x Σ x γ Σ γ+ K2 ν

D Additional Data for the Sector Rotation Case Study

Table D.1. Factors for the sector rotation case study. S&P 500 Composite Index

FTSE US Real Estate Index

S&P Global 1200 Index

Gold

Dow Jones World Ex US Index

Crude Oil

S&P Europe 350 Index

Platinum

Dow Jones World Emer. Markets Index GSCI Commodity Index Dow Jones Wilshire Small Cap Index

Lehman Corporate A+ Index

Dow Jones Wilshire Large Cap Index

Lehman Corporate Enhanced BB Index

S&P 500 Barra Value Index

Citigroup German 1-3Y Bond Index

S&P 500 Barra Growth Index

Citigroup German 10+Y Bond Index

NASDAQ Composite Index

Citigroup Japan 1-3Y Bond Index

FTSE 100 Index

Citigroup Japan 10+Y Bond Index

France CAC 40 Index

Citigroup US Big Corp. 1-3Y Bond Index

DAX 30 Performance Index

Citigroup US Big Corp. 10+Y Bond Index

AEX Index

Citigroup UK 1-3Y Bond Index

Swiss Market (SMI) Index

Citigroup UK 10+Y Bond Index

TOPIX Index

Citigroup UK All Mat. Bond Index

NIKKEI 225 Index

VIX Volatility Index

ASX All Ordinaries Index

VDAX Volatility Index

HANG SENG Index

3 months US Tr. Bills rate

S&P CNX 500 Index

1 month LIBOR rate

Korea SE Composite (KOSPI) Index

US Treas. 10 yr Bond red. Yield

Brazil Bovespa Index

US Corporate Bond Moody’s AAA rate

Mexico IPC (BOLSA) Index

US Corporate Bond Moody’s BAA rate

UK £ to US $ Exchange Rate

US Corporate Bond middle rate

Euro to US $ Exchange Rate

Dividend yield on S&P 500

Swiss Franc to US $ Exchange Rate

Price/Earnings ratio on S&P 500

US $ to Japanes YEN Exchange Rate

Dividend yield on Dow Jones

Japanese Yen to Euro Exchange Rate

Price/Earnings ratio on Dow Jones

Swiss Franc to US $ Exchange Rate

Dividend yield on FTSE 100

FTSE W Japan Real Estate Index

Price/Earnings ratio on FTSE 100

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Curriculum Vitae

Personal Data Name:

Simon Theodor Keel

Date of birth: January 26, 1976 Parents:

Alex Keel and Margrith Keel-Tanner

Education 1983–1989

Primary school in St. Gallen, Switzerland

1989–1991

Secondary school in St. Gallen, Switzerland

1991–1996

Gymnasium in St. Gallen, Switzerland

1996

Matura certificate, type C

1996–2001

Studies in mechanical engineering, Swiss Federal Institute of Technology (ETH) Zurich, Switzerland

2001

Diploma as Dipl. Masch.-Ing. ETH

2002-2006

Doctoral student and research assistant at the Measurement and Control Laboratory, Swiss Federal Institute of Technology (ETH) Zurich, Switzerland

Professional Experience 2001-2002

IT Consultant for NetArchitects SA (Altran Group), Zurich