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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 ISBN: 978-3-906483-10-8 accepted on the recommendation of Prof. Dr. H. P. Geering, IMRT PRESS c/o Measurement and Control Laboratory ETH Zentrum, ML examiner Prof. Dr. A. J. McNeil, coSonneggstr. 3 CH-8092 Zurich examiner Prof. Dr. D. B. Madan, co-examiner 2006

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 condence 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 coexaminers. 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 sta 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 nancial engineering is the development of portfolio management strategies. The research eld 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 nancial assets are predominantly driven by randomness, the concepts and techniques of control engineering have to be extended to the stochastic case. The rst step of the elaboration of an asset allocation strategy is the denition 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 dierent nancial assets is considered.

This work includes an in-depth study of alternative investments. Specialconsideration is given to their statistical properties. Hedge funds make use of dynamic asset allocation

strategies and may have a large investment universe. Therefore, hedge funds needspecial attention with respect to risk management. The specic 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 dierent statistical properties among the dierent 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 rst, a framework of continuous-time stochastic dierential equations is considered. The stochastic dierential 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 lter 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 unied framework for comparing dierent strategies. A specic 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 Levy processes for the construction of optimal portfolios. The multivariate Levy measures of the generalized hyperbolic Levy 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 Levy processes.

Zusammenfassung

Portfolio

Management

ist

ein

wichtiger

Aspekt

des

Fachgebietes

Financial

Engineering. Die optimale, stochastische Regelung bietet die hierfur notwendigen mathematischen Grundlagen. Ziel dieser Arbeit ist es, die momentan in der Literatur vorhandenen Techniken fur 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 Anlagemoglichkeiten modelliert werden. Hieraus ergeben sich optimale Regelungsprobleme, welche im entsprechenden Fachgebiet bereits grundlich erforscht wurden. Leider sind aber viele Resultate nur fur den deterministischen Fall gefunden worden. Da aber bei Finanzproblemen die betrachteten Systeme

hauptsachlich vom Zufall getrieben werden, mussen die Konzepte auf den stochastischen Fall erweitert werden. Der erste Schritt fur die Entwicklung einer Portfolio Management Strategie ist die Einfuhrung eines Risikomasses. Es sind jedoch nicht alle Risikomasse

gleichermassen geeignet. Koharente und konvexe Risikomasse besitzen fur die betrachteten Problemstellungen geeignete Eigenschaften. Die Modelle fur die Renditen von Wertpapieren sollen deren statistische Eigenschaften in realistischer Weise berucksichtigen. Hierfur werden mehrere Distributionen untersucht, welche die haug angetroene Normalverteilung ersetzen. Da in Studien oft nur der eindimensionale Fall behandelt gelegt. wird, Die wird besonderes welche Augenmerk unter dem auf den

mehrdimensionalen

Fall

Distribution,

Namen

Generalized Hyperbolic in der Literatur zu nden ist, kann die betrachteten Renditen sehr viel realistischer beschreiben als die Normalverteilung. Dies gilt auch fur einige Grenzfalle der Generalized Hyperbolic Verteilung. Zusatzlich werden elliptische Copulas untersucht.

Diese Arbeit enthalt eine ausfuhrliche Untersuchung von alternativen Anlagen. Im speziellen werden deren statistische Eigenschaften untersucht. Hedge Funds verfolgen in der Regel dynamische Anlagestrategien, was im Risikomanagement berucksichtigt werden muss. Hierfur werden die spezischen Eigenschaften von Hedge Funds untersucht und der Anlageprozess analysiert. Die Eigenschaften von Hegde Funds variieren enorm fur die verschiedenen Hedge Fund Stile. Deshalb konnen keine universellen Aussagen uber 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 Anlagemoglichkeiten. Im zweiten Teil werden aufgrund der erarbeiteten Modelle optimale Anlagestrategien entwickelt. Als erstes werden Modelle betrachtet, welche auf stochastischen Dierentialgleichungen fussen. Als Zufallsprozesse in diesen werden Brownsche Bewegungen eingefuhrt. Auch alternative Anlagen werden als ein solches System modelliert und eine optimale Anlagestrategie in geschlossener Form hergeleitet. Zusatzlich werden Modelle betrachtet, welche fur den Investor unbekannte Parameter und Prozesse enthalten. Um dieses Problem zu losen, wird ein Kalman Filter eingesetzt, die Resultate werden in einem Anwendungsbeispiel getestet. Der letzte Teil dieser Arbeit beschaftigt sich mit aktivem Portfolio Management. Die zentrale Bedeutung des Benchmarks fur das aktive Portfolio Management wird diskutiert. Da das aktive Portfolio Management kein eigentliches Forschungsgebiet darstellt, ist jedoch nur eine oberachliche Abhandlung moglich. Nichtsdestotrotz wird ein spezisches aktives Portfolio Management Problem diskutiert und werden zwei mogliche Losungsansatze prasentiert. Einer dieser Losungsansatze wird mittels historischer Daten veriziert. Der letzte Abschnitt dieser Arbeit beschaftigt sich mit Levy Prozessen im Zusammenhang mit Portfolio Konstruktion. Die multivariate Levy Dichte fur einen Grenzfall des Generalized Hyperbolic Levy Prozesses wird hergeleitet. Die Arbeit wird mit der Betrachtung von Levy Prozessen fur die Berechnung von optimalen Portfolios abgeschlossen.

Contents

1 Introduction ......................................................... 1 1.1 Financial Engineering .............................................. 2 1.2 Structure of the Thesis ............................................. 4 2 Financial Assets and Risk Management .............................. 11 2.1 Financial Assets ................................................... 11 2.2 TheAssetAllocationProcess........................................ 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 FundsofHedgeFunds......................................... 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 EstimationoftheUnobservableFactors.......................... 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 PortfolioManagementwithLevyProcesses............................ 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.2.2UnivariateProbabilityDistributions............................. 134 A.2.3MultivariateProbabilityDistributions ........................... 136 A.2.4 Bessel Functions and Modied Bessel Functions . . . . . . . . . . . . . . . . . . . 139 B GARCH Models for Dynamic Volatility ............................. 141 B.1 Univariate GARCH Processes ....................................... 141 B.2MultivariateGARCHProcesses...................................... 142 C Proofs ............................................................... 145 C.1 Tail Dependence within a t Copula ................................... 145 C.2 Transformation from Partial to Full Information ....................... 146 C.3 Levy Density of the Multivariate VG Levy Process ..................... 149 D Additional Data for the Sector Rotation Case Study ................. 151 References .............................................................. 153 Curriculum Vitae ....................................................... 165

List of Figures

2.1 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 4.5

Asset allocation process ............................................ 13 Classes of distributions in nance .................................... 23 Density estimates for the daily Dow Jones returns. ..................... 28 Logarithmic density estimates for the daily Dow Jones returns. . . . . . . . . . . 28 The model predictive control concept in nance........................ 45 Assets under management by the hedge fund industry................... 49 Serial correlation of the Tremont convertible arbitrage index. . . . . . . . . . . . . 62 Dynamic standard deviation of the Tremont long/short equity index. . . . . . 63 Non-linearities of hedge fund returns.................................. 64 Kernel regression of lagged S&P 500 returns vs. Tremont xed income

arbitrage returns................................................... 65 4.6 Density estimates for the monthly Tremont convertible arbitrage index

returns. .......................................................... 67 4.7 4.8 4.9 5.1 5.2 93 5.3 5.4 5.5 Correlation of Tremont Hedge Fund Indices with stocks and bonds.. . . . . . . 72 Hedge fund portfolio construction.................................... 73 The hedge fund selection process .................................... 74 Asset allocation strategy under full information for = 10. ............ 92 Asset allocation strategy performance under full information for = 10.. Estimations of the short rate and the unobservable factors and . ...... 111 Asset allocation strategy under partial information for = 10. ......... 112 Asset allocation strategy performance under partial information for =

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 Distributionsforequityindexreturns................................. 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 coecients of weekly world equity indices returns. . . . . . . . 41 3.10 Tail dependence coecients in a typical portfolio. ...................... 42 4.1 Hedge fund styles.................................................. 52 4.2 Common risk factors of hedge funds.................................. 60 4.3 4.4 4.5 Distributions for monthly Tremont hedge fund indices returns. . . . . . . . . . . . 66 Tail dependence coecients for Tremont hedge fund styles. . . . . . . . . . . . . . . 68 Tail dependence coecients for Tremont hedge fund styles with common

riskfactors........................................................ 69 4.6 5.1 Multivariate distribution models for a portfolio including hedge funds. . . . . 70 Typical values for the estimated parameters. .......................... 92

5.2 Key gures for the asset allocation strategy under full information . . . . . . . 94 5.3 Key gures for the asset allocation strategy under partial information. . . . . 113

XIV List of Tables

D.1 Factorsforthesectorrotationcasestudy.............................. 151

List of Symbols and Notation

Excess return Factor exposure 1 Skewness 2 Kurtosis I Identity matrix P Probability measure F Sigma algebra Ft Filtration N Normal (Gaussian) distribution Expected return (dx) Levy measure (t) Estimation error Sample space Volatility (t) Instantaneous covariance matrix per unit time 1{A} Identicator function of the set A C Copula function c Density of a copula

C C

Ga t

Gaussian copula t copula

Risk measure L0 Set of all almost surely nite random variables L1 n

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 Investors 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 E?cient 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 di?erential equation SP Shortfall probability TARCH Threshold GARCH VaR Value at Risk

Introduction

Copy from one, its plagiarism; copy from two, its research. Wilson Mizner

This work explores the possibilities and limits of the use of control engineering methods and techniques in nance. This chapter presents the motivation and goals of this work and the conceptual strategies involved. The application of control engineering methods and techniques to nancial problems is called nancial engineering. It makes use of engineering tools, i.e., it obtains quantitative results for models and problems developed in research elds such as economics, mathematics, and econometrics. The results in economics and mathematical nance 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: rst, modeling of the problem and then computation of its optimal solution. Therefore, the aim of the thesis is to apply improved nancial 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 tradeo 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 single-period solutions sequentially, in general.

1.1 Financial EngineeringFinancial engineering is dened as the use of mathematical nance and modeling to make pricing, hedging, trading, and portfolio management decisions. We mainly consider portfolio management decisions. By denition, a portfolio is a collection of investments held by an institution or an individual. Holding a portfolio with dierent investments instead of a single one is reducing the investors risk and is called diversication. 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 nancial risk that an investor takes. This raises the question of how to dene nancial risk, which is still an open issue in theory and in practice. Many dierent risk measures have been proposed so far, but no risk measure is well suited for all problems arising in the area of nancial engineering. This topic is discussed in Chapter 2. Control engineering in technical problems plays a similar role as nancial engineering does in nance 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 nancial problems. This topic is a subject of heated debates among scholars and practitioners. The dispute is about the ecient market hypothesis (EMH), proposed in Samuelson (1965) and Fama (1965, 1970a). The ecient market hypothesis states that security prices fully reect all the information available. There are several forms of the ecient 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

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 dier, 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 dierent 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 dierent 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 buyand-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, nancial return series are not independent. This property can easily be veried 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 nance provides strong evidence that for economic and nancial theories, the assumption of rational investors is rather bold. We stress the fact the quantitative models used by investors dier 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 Cvitanic, Lazrak, Martellini and Zapatero (2004) for a denition 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 inows 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 identied 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 ThesisChapter 2: Financial Assets and Risk Management In Chapter 2, the nancial assets considered are introduced. They are categorized into traditional and alternative investments. The traditional assets are cash, xedincome 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

An overview of nancial risks and their classication 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 briey 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 nd that the GH class of distributions ts 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 oers the best ts 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 dierent 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 nd that the t copula ts the data considerably better than the Gaussian copula. It is a well-documented fact that correlation is not always sucient 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

nd considerable tail dependence among popular asset classes. However, stocks and bonds oer good diversication 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 briey 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 nance. The most common dynamic optimization technique in nance is stochastic dynamic programming. The model predictive control approach for solving stochastic control problems is briey 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 dened. 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 investors own money in the fund are favorable for protecting the investors interests. The terms alpha and beta are introduced which are often found in the realm of hedge funds. A survey of dierent hedge fund styles found in the literature is presented. The advantages and disadvantages of funds of hedge funds are discussed. We nd 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 dicult. The most common biases such as survivorship bias, selection bias, and backll bias, are discussed in detail. The literature on the quantications of these biases is reviewed. All reviewed publications on this topic nd considerable biases in common hedge fund databases. It is also found that the most popular riskadjusted performance measure for hedge funds is the Sharpe ratio, although the deciencies 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

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 suciently 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-o 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 dierent 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 eects. 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 nd that the results vary considerably among the dierent 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 ts 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 nd 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 dierential 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 nance. Two dierent 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 dierential equation, which is very hard to solve if the control variable is constrained. For problems in higher dimensions, it is virtually impossible to nd 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 dierential framework with Brownian motion. The portfolio dynamics can be derived once the dynamics of the considered assets are dened. The portfolio is modeled to be self-nancing, i.e., there are no external in-or outows 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 Bellmans 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 closedform 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

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 denition of active portfolio management is given and the importance of the denition 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 classication of active portfolio management strategies is given. We dierentiate between security selection and market timing. However, there is neither a generally accepted terminology nor a unied framework to compare dierent 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 tendimensional 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 dierent 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 Levy 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 innitely divisible, we may construct a Levy 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 Levy processes. The study of Levy processes is one of the most promising research areas in mathematical nance because of the ne properties of Levy processes.

Financial Assets and Risk Management

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

2.1 Financial AssetsA nancial 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 benets 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 dened 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 nancial assets in two main categories, i.e., traditional and alternative investments. Table 2.1 summarizes the considered assets. The main traditional assets areTraditional Table Alternative 2.1. Financial assets Cash Hedge Funds Fixed-Income Managed Futures Equity (stocks) Private Equity Real Estate Physical Assets (Commodities, Art, Wine, ...) Foreign Exchange Securitized Products (Mortgages, Loans, ...)

12 2 Financial Assets and Risk Management

cash, xed-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 briey describe xed-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 Oered 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 xed-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 rm is called initial public oering (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 xed 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 ProcessObviously, the asset allocation process refers to the process of investing money in dierent nancial assets. There is no generally accepted methodology for this problem. However,

2.2 The Asset Allocation Process

there are many keywords describing dierent 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 ffd8ffe000104a46494600010201012c012c0000ffe20c584943435f50524f46494c450001 0100000c484c696e6f021000006d6e74725247422058595a2007ce000200090006003100 00616373704d5346540000000049454320735247420000000000000000000000000000f 6d6000100000000d32d4850202000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000116370727400000150000 0003364657363000001840000006c77747074000001f000000014626b70740000020400 0000147258595a00000218000000146758595a0000022c000000146258595a000002400 0000014646d6e640000025400000070646d6464000002c400000088767565640000034c 0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040 c0000002474656368000004300000000c725452430000043c0000080c67545243000004 3c0000080c625452430000043c0000080c7465787400000000436f707972696768742028 63292031393938204865776c6574742d5061636b61726420436f6d70616e79000064657 3630000000000000012735247422049454336313936362d322e31000000000000000000 000012735247422049454336313936362d322e310000000000000000000000000000000 00000000000000000000000

Investmen t Analysis

Tactical Asset Allocation

Monitoring

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 denes 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 lter 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,

14 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 denitive investment universe is dened 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 predened frequency, usually dened 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 dierent 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 MeasuresThere 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 rst 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 denition of risk and neither are there generally accepted denitions for risk in specic environments. There is a close relation between risk and uncertainty. Because of the above

2.3 Risk Management and Risk Measures

mentioned points we do not state a rigorous denition of risk. For our purposes we may dene risk as follows: Denition 2.1 (Risk). Risk is the exposure to some uncertain future event. The probabilities of the dierent 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 Denition 2.1 states that a certain system only contains the risks of the uncertain events it is exposed to. In a nancial context, these uncertain events are often called risk factors. Therefore, only events which have a dependence on the considered system may inuence its risk. In a nancial 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 rst 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 nancial risk management are the modeling of the assets and the denition of the risk measure. Once these two elements are dened, risk management becomes a formal, logical process. The rst 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 nancial problems, the argument of utility function usually is money (consumption). The rst systematic description of risk for nancial 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 payos. We therefore consider the concept of risk aversion as the rst form of a risk measure. For a risk averse investor, a utility function U must fulll certain properties:

1 16 2 Financial Assets and Risk ManagementA utility function must be an increasing continuous function: U > 0. 0 < 0. A utility function must be concave: U The rst 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

1the exponential function (a> 0): U(x)= eax . 2the logarithmic function: U(x) = ln(x). 3the power functions (b< 1 and b b (x)b

2.3 Risk Management and Risk Measures

In the realm of nancial 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, dened on a probability space (, F, P), which denotes the prot or loss of a nancial position. Therefore, X is a real-valued function on the set of possible scenarios. By L0 we denote the set of all random variables X : R, which are almost surely nite. A quantitative measure of risk is given by a mapping from the set L0 to the real line. Formally, the denition of a quantitative risk measure is given as: = 0): U(x)= . Denition 2.2 (Risk measure). 0 A risk measure is a function : a): R. ax bx2 . L 4. the quadratic functions (x< 2b U(x)= All The Bank for International Settlements (BIS) is an of risk aversion. This is of these utility functions capture the principle international organization accomplishedcooperation thecentral banks and international nancial institutions.the fostering the whenever of utility function is concave. We will not get into Its details of utility nancialfor more details the reader is2.2. classication of theory, risks is summarized in Table referred to Luenberger (1998), Cvitanic and Market Risk (2004), and Panjer (1998). Since we are usually not Zapatero interested in the absolute values of utility functions but rather in its shape, Pratt and Credit Risk Arrow have independently developed measures for risk aversion. Let U(x) be a utility Operational Risk function, then the Arrow-Pratt measures for absolute and relative risk aversion arefailed dened as follows: internal processes, people and systems, or from external events. Liquidity Risk " "( (x) The risk that positions cannot be liquidated quickly enough atUcritical times. U " x) Model Risk the Arrow-Pratt measure of absolute risk aversion: a(x)= U"" (x) . The the Arrow-Pratt risk of using of relative risk aversion: b(x)= x. U" (x) measure inaccurate or wrong models for risk budgeting. The risk of direct or indirect loss resulting from inadequate or The risk associated with the uncertainty of the default of debtors. Table 2.2. Financial risks The risk associated with the uncertainty of the value of traded assets1

The main

Event Risk The risk of extreme event. Reputational Risk critique on utility theory is that humans are not always The risk of losing ones reputation as investment manager.

rational. We do not

discuss this topic since we do not derive economic or nancial models based on this assumption. Here, we investigate the performance of rational investment strategies. In this work, the main emphasis will be on dealing with market risk. 2.3.2 Financial Risk Single Period Risk Measures The actual return of every security is always uncertain and therefore full information The systematic treatment of risk measures was introduced in the seminal paper of of the underlying risks in a portfolio means knowing the exact distribution of the Artzner, Delbaen, Eber and Heath (1998), where the properties of good risk measures portfolio return. In addition, one wants to know how the portfolio return distribution is are described by some axioms. A risk measure fullling these axioms is called a aected by altering the positions in the portfolio. This is very dicult when dealing in coherent risk measure. Let the two random variables X and Y denote the prot or loss a complex stochastic environment. Even for single securities it may be hard to nd a of two assets. The axioms for a coherent risk measure are (r denotes the risk-free suitable distribution, e.g., for illiquid securities. Therefore, dierent risk-measures as rate of interest): r a single quantity have been )established. ) Subadditivity: VX, Y : (X + Y : (X)+ (Y These risk measures are characteristic Positive-homogeneity: VX : c 0: (cX)= quantities of a probability density function. c(X)

1 18 2 Financial Assets and Risk ManagementTranslation invariance: VX : c R : (X + cr)= (X) c Monotonicity: VX, Y : X : Y : (X) (Y )

The subadditivity property ensures that diversication reduces risk. The positivehomogeneity property, together with subadditivity, implies that the risk measure is convex. measure which is translation invariant and monotone is called monetary. A risk Ziemba and Rockafellar (2000) and Follmer 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

r risk measures are

Convexity: VX, Y : (cX + (1 c)Y ) : c(X) + (1 c)(Y ),c [0, 1] 1 Translation invariance: VX : c R : (X + cr)= (X) c Monotonicity: VX, Y : X : Y : (X) (Y )

2.3 Risk Management and Risk Measures

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 |X 0. Some examples of single period risk about the tail of the distribution. Furthermore, VaR is not a coherent measure because it is not Maximum Loss (ML) subadditive, i.e., diversication does not necessarily reduce risk, see Embrechts (2004) for an illustrative example. is intuitive and needs no further explanation. The maximum loss risk measure Note that the maximum loss is unbounded and therefore useless when the return Conditional Value at Risk (CVaR) distribution is neither truncated nor discrete. Sucient historical data has to be A possible, coherent extension to VaR is the conditional value at risk (CVaR). CVaR available for the use of this risk measure. is also known as expected shortfall (ES) and is dened as Shortfall Probability (SP) CVaR()= E[X|X : VaR()]. 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 Again, a condence level is specied and the returns a characterized for a given performance level; therefore the manager is interested in minimizing the time period. CVaR is below this level. Formally speaking: below the VaR occurs. probability to perform the expected loss once a return Informally, CVaR states how bad SP(a)= P (X : a). has the appealing property that is bad?. CVaR it is coherent in a single-period setting. Method of Moments (MM) Note that, from a regulatory point of view, coherent risk measures should be the Since the introduction of mean-variance portfolio theory, moments of return preferred choice. From therisk measures. The variance is still the most widely more distributions are used as point of view of an investment manager it is used comfortable to work withrisk. This has obvious measures in order to moresign of the measure to quantify convex monetary risk disadvantages, e.g., the accurately model the investment problem. return is not taken

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 rst publications on this subject is Cvitanic 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 right-continuous with left limits. The connection between Bellmans 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.

Modeling of Financial Assets and Financial Optimization

Young man, in mathematics you dont 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 reect 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 nancial assets discussed in Chapter 2. We provide a short overview of the economic and nancial models developed so far. We attribute the rst 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 investors 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 rst factor models. The importance of the CAPM stems also from its terminology, i.e., the use of the Greek letters and , 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 nancial 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 rst 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 rened 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 continuous-time nance. The breakthroughs of continuous-time nance 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 xed-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 nancial 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 continuous-time models of Black, Scholes, and Merton are that returns are normally distributed. This deciency is overcome by replacing the Brownian motion with a Levy process. A Levy 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 dene such a stochastic process with independent and stationary increments, the distribution has to be innitely divisible. Levy 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 Levy processes in nance. 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

3.1 Statistical Properties of Asset ReturnsIn Chapter 2, the rate of return is dened 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 dened as the continuous-compounded or log-return. The numerical dierences 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 specied. The distribution can either be parametric, semi-parametric, or nonparametric. 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 nance. ffd8ffe000104a46494600010201012c01 2c0000ffe20c584943435f50524f46494c45 00010100000c484c696e6f021000006d6e 74725247422058595a2007ce0002000900 0600310000616373704d5346540000000 049454320735247420000000000000000 000000000000f6d6000100000000d32d48 502020000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 011637072740000015000000033646573 Normal Mean63000001840000006c77747074000001f0 00000014626b707400000204000000147 258595a00000218000000146758595a00 00022c000000146258595a00000240000 00014646d6e640000025400000070646d Fig. 3.1. Classes of distributions in nance 6464000002c4000000887675656400000 34c0000008676696577000003d4000000 246c756d69000003f8000000146d656173 Elliptical distributions 0000040c0000002474656368000004300 are often reasonably good models for nancial return data 000000c725452430000043c0000080c675 and have very pleasing properties. For instance, taking linear combinations of 452430000043c0000080c6254524300000 elliptical random vectors 43c0000080c7465787400000000436f707 same type. The results in an elliptical random vector of the 972696768742028632920313939382048 marginal and conditional distributions of elliptical distributions are again elliptical. 65776c6574742d5061636b61726420436f 6d70616e7900006465736300000000000 Popular elliptical distributions are the normal and the t distribution. For more details 00012735247422049454336313936362d 322e31000000000000000000000012735 on elliptical distributions, the reader is referred to McNeil, Frey and Embrechts 247422049454336313936362d322e3100 (2005). 000000000000000000000000000000000 0000000000000000000

24 3 Modeling of Financial Assets and Financial Optimization

Stable distributions were introduced by Paul Levy in 1925. Note that stable distributions are also called -stable, stable Paretian, or Levy 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 innite 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, ), R and , R two arbitrary vectors. The random variable X| U = u N ( + u, u) is said to have a normal mean-variance mixture distribution. This distribution is elliptical for = 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 nancial 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.n nn

a covariance matrix,

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 3.1 Statistical Properties of Asset ReturnsEquity returns show little or no serial correlation although they are not independent. Equity returns are fat-tailed and skewed. E Squared or absolute equity returns are serially correlated. S Volatility is time-varying and appears in clusters.

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 dened. We consider the log-normal, gamma, generalized For high frequency data and their properties see Cont (2001) and the references inverse Gaussian, chi-square, Weibull, beta, and F distributions as candidates for therein. A standard reference on high frequency nance is Dacorogna, Gencay, price distributions, but not for return distributions. Muller, Olsen and Pictet (2001). For more details on the stylized facts of asset returns There are many to McNeil et al. (2005), Ziemba and kurtosis values of asset the reader is referredpublications treating skewness (2003), Campbell et al. (1997), returns. All of them report that real-world return series are leptokurtic and skewed. and Campbell and Viceira (2002). 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 3.1.2 Univariate Properties Azzalini (1985). The estimation in its original form is inconvenient, we therefore use We explore the (unconditional) univariate properties of asset returns. Therefore, we the methods described in Pewsey (2000). The results for the skewed normal consider the set of univariate distributions. skewed normal point is thedoes not distribution are rather disappointing since the The starting distribution normal distribution, fat tails.is the most popular in portfolio construction since state, as a account for which Therefore, it is not investigated any further. We may Markowitz (1952). For nance inthe inclusion normality assumption is found in most is more rule of thumb, that general, the of heavy-tails in return distributions models. The rst appearance of the normal distribution in nance dates back to Bachelier important than the skewness aspect. A fat-tailed extension of the normal distribution is the t distribution. The t (1900). Another reason for the popularity of the normal distribution is because of the use distribution converges to the normal distribution as the parameter tends to innity, of Brownian motion in nance. Although Brownian motion has been mathematically see (A.2) for details. Therefore, a large value of indicates that the considered rigorously introduced in 1923 by Norbert Wiener, the Brownian motion rst shows up random variable may also be considered normal. A further extension would be the in nance in Osborne (1959). A lot of continuous-time nance results have emerged skewed t distribution. We use the method of Fernandez and Steel (1998) to extend from Samuelson (1969) and Merton (1969). The central limit theorem makes the the t distribution to be skewed, see (A.1) for details. The skewness is measured by normal distribution the most important distribution in probability. Some similar the parameter (0, ). We have no skewness for = 1 which results in the phenomenon may also be observed for equity returns. The lower the frequency of ordinary t distribution. The skewed t distribution obviously has the properties of the returns, the more the distribution of the returns resembles a normal distribution. being leptokurtic and skewed. Note that Hansen (1994) also introduces a skewed This means that we may reasonably model yearly returns as normal. However, daily version of the t distribution. returns cannot be assumed normal, statistically. The generalized hyperbolic (GH) distribution is introduced by Barndor-Nielsen (1977), although not in a nancial context. Eberlein and Keller (1995) use the GH disOne of the rst published doubts about the normality assumption of asset returns tribution to describe nancial return data and also suggest a hyperbolic Levy are Mandelbrot (1963) and Fama (1965). Since then, many more publications on this motion. The GH distribution is a very exible distribution and is well suited for subject have appeared. Motivated by the fact that nancial returns are skewed and describing return data. It contains many important special and limiting cases. Among leptokurtic (fat-tailed), we want to investigate suitable extensions of the normal these are the hyperbolic, normal inverse Gaussian (NIG), a version of the skewed t, distribution. Figure variance gamma, t, and the normal distribution. assumption proposed so far. 3.1 shows promising extensions of the normality All these distributions are proposed as nancial return models

3.1 Statistical Properties of Asset Returns

in the literature. For more details on the GH distribution in nance see McNeil et al. (2005), Knight and Satchell (2000), Prause (1999), Raible (2000), Rydberg (1998), and Barndor-Nielsen and Shepard (2001). The density functions of the GH family are found in Appendix A.2.3. In his publication, Mandelbrot (1963) nds 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 innite second and higher moments. This is in contrast with empirical observations which have nite second moments. Madan and Seneta (1987) introduce the variance gamma distribution. A nancial 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 innitely divisible and therefore also suited for building a corresponding Levy 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 innite 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 Optimizationffd8ffe000104a46494600010201012c012c0000ffe20c584943435f50524f46494c4500010100000c48 4c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d534654000 0000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000 70 0000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000001163707274000001500000003364657363000001840000006c77747074000001f0000000146 26b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a000 60 0024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c00000 08676696577000003d4000000246c756d69000003f8000000146d6561730000040c000000247465636 50 8000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0 000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d50616 36b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d3 22e31000000000000000000000012735247422049454336313936362d322e310000000000000000000 00000000000000000000000000000000000

d e n si ty40 30 20 10 0

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

see Hardle (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 t in terms of the log-likelihood value. In this case, the deciency of the normal distribution is that it does not account for the fat tails and the thin middle.ffd8ffe000104a46494600010201012c012c0000ffe20c584943435f50524f46494c4500010100000c48 4c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d53465400 00000049454320735247420000000000000000000000000000f6d6000100000000d32d48502020000 000000000000000000000000000000000000000000000000000000000000000000000000000000000 0 00000000001163707274000001500000003364657363000001840000006c77747074000001f000000 014626b707400000204000000147258595a00000218000000146758595a0000022c00000014625859 5a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034 c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c000000247 5 4656368000004300000000c725452430000043c0000080c675452430000043c0000080c62545243000 0043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742 d5061636b61726420436f6d70616e7900006465736300000000000000127352474220494543363139 36362d322e31000000000000000000000012735247422049454336313936362d322e3100000000000 10 0000000000000000000000000000000000000000000

lo g( d e n si ty )

15

20

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

3.1 Statistical Properties of Asset Returns

In order to analyze the tails, the logarithmic density estimates are plotted in Figure 3.3. Obviously, the normal distribution ts 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 tted 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 the vector , the estimated parameters are denoted by . The log-likelihood value of the estimation is denoted by l(|y). The Akaike information criterion is dened as AIC = 2l(|y)+2q, (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 t 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 ts 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.

Equity index monthly min(AIC) ?1 30 3 Modeling of Financial Assets and Financial Optimization ?2 weekly min(AIC) ?1 ? Table 3.2 reports the results for dierent2 equity indices with data from 1990 to daily min(AIC) ?1 ? 2005. The GH, NIG, and the skewed t (s-t) t 2the data best in terms of the maximum S&P 500 likelihood value. In addition, the skewness NIG -0.36 the kurtosis 2 are given. The 1 and 3.43 NIG -0.41 normal distribution is the best model for monthly Nikkei 225 return data. This result 5.83 is supported by the values of 1 and 2 forGH -0.10 the monthly Nikkei 225 returns. The 6.89 Dow Jones considered stock indices, in general, have fat tails and are skewed to the left. This is s-t -0.27 3.71 seen from the values of 1, which are all negative, and from the values of 2, which s-t -0.40 6.34 are all larger than three. For daily returns, the GH and the NIG distribution are the GH -0.23 7.69 best models in terms of the AIC value. In terms of the maximum likelihood value, the Nasdaq GH -0.53 GH distribution ts best. 4.22 NIG -0.45 Table 3.2. Distributions for equity index returns. 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

Commodity (index) monthly min(AIC) ?1 ?2 weekly min(AIC) ?1 ?2 daily min(AIC) ?1 ?2 Gold

6.46 t 0.31 8.39 GH -0.08 14.34 Oil (West Texas Int.) 3.1 Statistical Properties of Asset Returns t 0.07 3.85 Table 3.3. Distributions for commodity returns. 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 Moodys Commodities Index N 0.30 3.64 t 0.12 FI index4.00 GH 0.04 monthly min(AIC) ?1 11.82 ?2 GSweekly min(AIC) ?1 considered so far. For monthly returnsCommodities Index of less than seven years, the with a maturity t 0.14 ?2 3.56 1 normal distribution is the best model indaily min(AIC)t?-0.59 value. We observe that the terms of the AIC ?2 8.67 US Govt. 1-3 the return distribution deviates from longer the maturity of the bond index, the moreyears-1.02 GH N 0.008 19.95 3.18 the normal distribution. The NIG distribution is particularly well suited for describing GS Energy Index t -0.04 t 0.26 3.88 daily bond index returns. Besides the normal and the NIG distribution, the skewed t NIG 0.063.80 t -0.04 7.05 distribution gives the best ts. 7.43 US Govt. 3-5 years GH -0.16 N -0.26 6.42 3.32 NIG -0.28 Table 3.4. Distributions for bond total return index returns. 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 Table 3.3 reports the results for some commodities. The results for the daily s-t -0.45 3.88 returns are similar to the ones in Table 3.2, the GH and the NIG distribution are the NIG -0.36 5.23 best models. For monthly and weekly returns, the t distribution is often the best US Govt. >10 years choice. Note that the t distribution is fat-taileds-t -0.49 whereas the GH distribution has semi3.83 s-t -0.41 heavy tails, see Prause (1999) for details. Commodity returns, in contrary to equity 3.84 index returns, may be significantly skewedNIG -0.33 right. The high values of the to the 4.67 US returns have fatter tails than the equity kurtosis give evidence that commodityGovt. all mat. s-t -0.40 3.56 indices in Table 3.2 and the bond indices in Table 3.4. Daily returns on oil are s-t -0.42 3.73 signicantly more non-normal than the returns on gold, indicated by the NIG -0.32 4.88 corresponding values of the skewness and kurtosis.

Table 3.4 reports the r