On Asymptotic Diversification Effects for Heavy-Tailed Risks Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨atFreiburg im Breisgau vorgelegt von Georg Mainik Februar 2010 Dekan: Prof. Dr. Kay K¨onigsmann Referenten: Prof. Dr. Ludger R¨uschendorf Prof. Dr. Paul Embrechts Datum der Promotion: 29. April 2010 Contents 1 Introduction1 1.1 Motivation.............................1 1.2 Overview of related theory....................2 1.3 Central results and structure of the thesis...........3 1.4 Acknowledgements........................7 2 Multivariate regular variation9 2.1 Basic notation and model assumptions.............9 2.2 Canonical forms of exponent and spectral measures...... 14 2.3 Dependence functions....................... 18 2.4 Copulas.............................. 20 2.5 Spectral densities of Gumbel copulas.............. 24 2.6 Spectral densities of elliptical distributions........... 27 3 Extreme risk index 35 3.1 Basic approach.......................... 35 3.2 Representations in terms of spectral measures......... 39 3.3 Portfolio optimization and diversification effects........ 42 3.4 Minimization of risk measures.................. 49 4 Estimation 55 4.1 Basic approach.......................... 55 4.2 Main results............................ 59 4.3 Empirical processes with functional index............ 63 4.4 Proofs of the main results.................... 76 4.5 Examples and comments..................... 78 5 Stochastic order relations 87 5.1 Introduction............................ 87 5.2 Ordering of extreme portfolio losses............... 90 5.3 Ordering of spectral measures.................. 94 i ii CONTENTS 5.4 Convex and supermodular orders................ 106 5.5 Examples............................. 111 6 Modelling and simulation 117 6.1 Objectives and design....................... 117 6.2 Estimation of the tail index................... 118 6.3 Models............................... 121 6.4 Simulation results......................... 126 6.5 Conclusions............................ 147 A Auxiliary results 149 A.1 Regular variation......................... 149 A.2 Empirical processes........................ 149 Bibliography 155 Index 163 Chapter 1 Introduction 1.1 Motivation Management of financial risks is one of the central challenges in the area of finance and insurance. In particular, any market agent has a natural inter- est in protection against extreme losses arising from market crashes, natural catastrophes, or political turbulences. Furthermore, financial crashes have proved to be a serious danger not only to the prosperity of share holders and financial corporations, but also to the general economic and political sta- bility of entire societies. Thus, additionally to the microeconomic interest, management of extreme financial risks is an issue of macroeconomic, politi- cal, and social importance. This issue is even greater in a rapidly changing world with advancing globalization of financial markets and acceleration of communication networks. In consequence of these developments, permanent validation and adjustment of models used in the risk management is essential to the market agents and the regulating authorities as well. Portfolio diversification is one of the basic microeconomic approaches to the reduction of non-systemic risk. The first probabilistic concept of diversi- fication was given by Markowitz(1952). Quantifying risk and dependence by variance and correlation, respectively, Markowitz gave arguments explaining why diversification decreases the portfolio risk. Designed for the multivariate Gaussian model assumption, this approach shaped the intuition of portfolio diversification in economics (cf. Sharpe, 1964; Lintner, 1965) and has proved to be appropriate to many application areas. However, the mean-variance portfolio theory cannot be applied to all diversification problems. First, this approach assumes existence of second moments, which is questionable in some applications. There are even real- world risk data sets suggesting infinite first moments (cf. Moscadelli, 2004; 1 2 Chapter 1. Introduction Neˇslehov´aet al., 2006). Another limitation is the two-sided view of risk, as it automatically follows from measuring risk by variance. This may be misleading for asymmetric distributions and extreme risks related to rare events. Moreover, measuring dependence by correlation is also problematic in many applications (cf. Embrechts et al., 2002). Indeed, since moments depend on the entire distribution of a random variable in its value domain, they may be inappropriate to the quantification of risk and dependence in the tail region. The limitations of the Markowitz approach are well known and addressed by recent developments in financial risk theory. In particular, the two-sided view of risk is commonly replaced by the notion of downside risk and re- lated risk measures such as the value-at-risk or the expected shortfall. The theory of coherent risk measures founded by Artzner et al.(1999) provides an axiomatic characterization of risk measures that favours portfolio diver- sification. This objective can be considered as an extension of the original motivation of Markowitz, who wanted to find a mathematical reasoning for portfolio diversification (cf. Markowitz, 1991). Talking about rare events and extreme losses, one has to mention the heavy-tailed distributions. The evidence of heavy tails in finance and in- surance is manifold and the application area is so vast that an outline of the relevant literature would go far beyond the scope of this introduction. To mention only the basic facts, it should be said that heavy-tailed models are generally accepted in the realm of insurance (cf. Hogg and Klugman, 1984; Embrechts et al., 1997) and vividly discussed in mathematical finance since its earliest days (cf. Mandelbrot, 1963; Fama, 1965). The equivocality concerning heavy-tailed modelling of financial returns is a result of the high complexity of financial markets. On the one hand, extremal behaviour of the market data seems to suggest heavy-tailed distributions (cf., among others, Longin, 1996; Beirlant et al., 2004b). On the other hand, reliable estima- tion of tail parameters in non-stationary time series with clustering effects is known to be difficult (cf. Malevergne et al., 2006). This implies that indica- tion for heavy tails in financial return data should be handled very carefully. Still, despite the ongoing scientific discussion, it is commonly agreed that heavy tails are worth considering. 1.2 Overview of related theory Extremal behaviour of random variables is a subject of vital research since the very beginnings of probability theory. The core basis of extreme value theory is the Fisher{Tippett theorem, which characterizes the limit distribu- 1.3. Central results and structure of the thesis 3 tions for maxima of i.i.d. random variables (Fisher and Tippett, 1928; Gne- denko, 1943). The Fisher{Tippett theorem states that there are only three possible limit distribution types: the Fr´echet, the Gumbel, and the Weibull distributions. This result was applied to the estimation of high quantiles and probabilities of extremal events in many application areas, such as hydrol- ogy, network modelling, and risk management in finance and insurance. An elaborate overview of the latter application area is given by Embrechts et al. (1997). The probabilistic structure of multivariate extremes was characterized by de Haan and Resnick(1977). This seminal result provided a sound theoretical basis for various approaches to the modelling and the estimation of extremal dependence. In particular, modelling concepts based on multivariate extreme value theory and copulas have been developed for an adequate description and analysis of risks and risk portfolios. Comprehensive elaborations on this topic are given in McNeil et al.(2005) and Malevergne and Sornette(2006). Further developments in modelling extremal dependence are based on the notions of tail dependence and tail copulas, multivariate excess distributions, and the empirical distribution of excess directions. See, among others, Falk et al.(1994); Schmidt and Stadtm¨uller(2006); Kl¨uppelberg and Resnick (2008); Hauksson et al.(2000). The current state of extreme value theory in both univariate and multivariate settings with particular emphasis on statistical applications is presented in de Haan and Ferreira(2006), whereas heavy-tailed models are specially treated by Resnick(2007). There are also various applications of multivariate extreme value theory related to sums of random variables (cf. W¨uthrich, 2003; Barbe et al., 2006; B¨ocker and Kl¨uppelberg, 2008; Kortschak and Albrecher, 2009a). However, the focus of these results is put on loss aggregation rather than on portfolio diversification. Although tightly related, these problems are obviously not identical. In particular, characterization of risk aggregated in the sum of random variables does not provide a direct link to the location of the optimal portfolio. Furthermore, as highlighted in Remark 5.31, the influence of the tail index on diversification effects turns out to be different from the influence on the aggregated risk. 1.3 Central results and structure of the thesis This thesis is dedicated to the diversification of extreme risks. In contrast to the Markowitz approach, the problem is reduced to the comparison and minimization of the extreme portfolio risks, while the consideration of average gains is omitted. Thus application of these results in practical portfolio