Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2014 Mixture models in financial risk modeling Krause, Lars Jochen Abstract: A new model class for univariate asset returns is proposed which involves the use of mixtures of stable Paretian distributions, and readily lends itself to use in a multivariate context for portfolio selection. The model nests numerous ones currently in use, and is shown to outperform all its special cases. In particular, an extensive out-of-sample risk forecasting exercise for seven major FX and equity indices confirms the superiority of the general model compared to its special cases and other competitors. An improved method (in terms of speed and accuracy) is developed for the computation of the stable Paretian density. Estimation issues related to problems associated with mixture models are discussed, and a new, general, method is proposed to successfully circumvent these. The model is straightforwardly extended to the multivariate setting by using an independent component analysis framework. The tractability of the relevant characteristic function then facilitates portfolio optimization using expected shortfall as the downside risk measure. Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-164486 Dissertation Published Version Originally published at: Krause, Lars Jochen. Mixture models in financial risk modeling. 2014, University of Zurich, Faculty of Economics. MIXTURE MODELS IN FINANCIAL RISK MODELING Dissertation submitted to the Faculty of Economics, Business Administration and Information Technology of the University of Zurich to obtain the degree of Doktor der Wirtschaftswissenschaften, Dr. oec. (corresponds to Doctor of Philosophy, PhD) presented by Lars Jochen Krause from Germany approved in February 2014 at the request of Prof. Dr. Marc S. Paolella Prof. Dr. Erich W. Farkas The Faculty of Economics, Business Administration and Information Technology of the University of Zurich hereby authorizes the printing of this dissertation, without indicating an opinion of the views expressed in the work. Zurich,¨ 12.02.2014 Chairman of the Doctoral Board: Prof. Dr. Josef Zweimuller¨ To my family. IV Contents I Introduction 1 II Research Papers 9 1 Stable Mixture GARCH 11 1.1 Introduction . 13 1.2 Stable Mixture GARCH . 16 1.3 Estimation of Mixture GARCH Models . 24 1.4 Univariate Empirical Results . 28 1.5 ICA-MixStable-GARCH . 33 1.6 Conclusions and Future Research . 36 Appendix . 37 2 Time-varying Mixture GARCH 55 2.1 Introduction . 57 2.2 Mixed Normal GARCH . 59 2.3 Time-varying weights . 60 2.4 Empirical Results . 69 2.5 Conclusions and Further Extensions . 72 Appendix . 73 3 Portfolio optimization using the noncentral t 85 3.1 Introduction . 87 3.2 Saddlepoint approximation of expected shortfall for transformed means . 88 3.3 Portfolio optimization using the noncentral t ........................ 90 3.4 Conclusion . 97 Appendix . 98 III Curriculum Vitae 113 V VI CONTENTS List of Tables 1.1 Small sample performance analysis of the studied estimators . 40 1.2 Profile log likelihood study of the GARCH power parameter . 41 1.3 Test statistics on the optimal choice of the GARCH power parameter . 42 1.4 Estimated parameter values and standard errors . 43 1.5 Estimated log likelihood, AIC and BIC values . 44 1.6 Out-of-sample forecast performance results I . 45 1.7 Out-of-sample forecast performance results II . 46 1.8 Predicted VaR coverage percentages . 47 1.9 Integrated root mean squared errors . 48 1.10 Christoffersen test statistics . 49 2.1 Estimated log likelihood values . 76 2.2 Estimated BIC values . 76 2.3 Out-of-sample performance comparison results . 77 2.4 Predicted VaR coverage percentages . 78 2.5 Integrated root mean squared errors . 78 2.6 Christoffersen test statistics . 79 2.7 Comparison of two variants of the new model . 80 3.1 Annualized portfolio performance measures . 103 3.2 Average realized predictive log likelihood values . 103 3.3 Out-of-sample forecast performance results for the min-ES portfolio . 104 3.4 Out-of-sample forecast performance results for the equally-weighted portfolio . 104 VII VIII LIST OF TABLES List of Figures 1.1 The Stable Paretian and Nolan’s density approximation . 22 1.2 The impact of mixture degeneracy for the estimators under study . 39 2.1 The news impact curve under different models . 74 2.2 Estimated news impact curves . 75 3.1 Average computation times for the min-ES portfolio . 93 3.2 Illustration of the behavior of terms included in the infinite sum in (3.17) . 101 3.3 Annual performance results for the min-ES portfolio . 105 3.4 Cumulative returns and predictions of mean and variance for the min-ES portfolio . 106 3.5 Predictions of median, value-at-risk and expected shortfall for the min-ES portfolio . 107 3.6 Evolution of the degrees of freedom parameter over time . 108 3.7 Evolution of the noncentrality parameters over time . 108 3.8 Predictions of mean and variance for the equally-weighted portfolio . 109 3.9 Predictions of median, value-at-risk and expected shortfall for the 1/K portfolio . 110 IX Part I Introduction 1 3 Acknowledgments My thanks go to my family, friends and colleagues who have contributed to this thesis in numerous ways throughout the years. Especially to Marc Paolella, Stefan Mittnik, Markus Haas, Simon Broda, Sven Steude, Pawel Polak, Kerstin Kehrle, Tatjana Puhan, Ingo Goetze and, last but not least, Katharina Schonau.¨ Of course, the biggest thank-you goes to my doctoral adviser Prof. Marc Paolella! Marc, this work would not have been possible without your guidance, patience and continued support. My special thanks! Finally, I also wish to express my gratitude to the Department of Banking and Finance of the University of Zurich (highlighting our IT team), as well as to the Swiss National Science Foundation! Introduction Risk management is a crucial ingredient of sustainable development in a world that is becoming increas- ingly quantitative while being continuously shaken by financial bubbles and crashes of varying mag- nitude. The recent and enduring hype on big data, including the colossal amounts of high-frequency financial returns data available today, stresses a trend towards a new form of high-level technical (risk) analysis that is not only present in (empirical) finance but basically in every discipline in academia and industry. Concerning finance and the financial sector at large, it is no overstatement to say that an elabo- rated quantitative risk management incorporating such data could have prevented, or at least attenuated, some of the recent financial crises. However, for a profound risk management, adequate measures of risk are indispensable comprising models, methods and techniques that characterize and capture the particular type of risk involved in a certain business activity; e.g., the risk to default, or the risk of loosing a certain amount of an investment for a given probability of occurrence (downside risk). The latter, the downside risk, is a cornerstone of asset allocation, and as such, of high value for virtually all financial institutions such as hedge funds, pension funds, or insurance companies where risk evaluation and management is a critical element in business. For the insurance sector, for example, according to CEIOPS (2010) and EIOPA (2011), one-quarter of the overall risk belongs to financial equity risk whose accurate assessment is one of the key aspects of this thesis. On the other hand, regulatory agencies specify procedures and parameters for companies for determining the downside risk in terms of capital requirements to, among other targets, reduce the likelihood of firms defaulting; see the regulatory frameworks Basel III and Sol- vency II of the European Union. Roughly summarized, quantifying the downsize risk has evolved into daily practice and has inevitably become one of the most important factors in determining the level of resources expended in order to mitigate risk. Stressing that the best risk measure is worth nothing if the underlying statistical model and the em- 4 ployed estimation methods and numerical techniques are not appropriate, this thesis contributes to the improvement of risk measurement by introducing new models, methods and techniques dedicated to pre- dict the density of tomorrow’s return more accurately. The goal is to produce accurate return density forecasts for financial assets from which the distribution and, thus, risk measures can be easily derived. Special emphasis is thereby set on (computational) feasibility, accuracy and numerical reliability through- out the chapters. Considering modeling the predictive density, i.e., the evolution of asset returns over time, the class of mixture generalized autoregressive conditional heteroscedasticity (mixture GARCH) models is chosen; either based on finite mixture distributions (see Haas et al., 2004; and Alexander and Lazar, 2006; and also Haas et al., 2009), or location scale mixtures of normals (see, e.g., Menc´ıa and Sentana, 2009; and Jondeau, 2010). Mixture distributions possess a successful and long history in the literature (for finite mixtures see, e.g., Pearson, 1894) and have been shown to be well suited for capturing styl- ized facts of financial asset returns, which refer to empirically observed phenomena common to all such data sets. Finite mixtures, in particular, being convex combinations of density functions, can flexibly mimic all kinds of functional shapes and therefore lend themselves to account for stylized facts such as non-normality, heavy-tails, asymmetry, and non-ellipticity; while GARCH filter (see Bollerslev, 1986; and for ARCH Engle, 1982) are well-known to excellently capture time-varying volatilities, volatility clusters and volatility persistence. For judging about the quality of the approaches devised, extensive out- of-sample forecast comparisons with respect to one-day ahead density and risk forecasts are conducted which demonstrate the applicability and usefulness on historic market data.