Sommen van copula-afhankelijke toevalsveranderlijken en optimale Expected-Shortfall portfolio keuze

With this thesis we aim to accomplish two goals. First, we lay out in a general context how semiparametric methods, specifically the method of local likelihood, can be used to complement parametric time series models. Second, we study sums of dependent random variables, where we make use of the copula framework to introduce dependence between the summands. Chapter 1 reviews existing literature relevant to our approach and gives a detailed overview of the structure of this thesis. In Chapter 2 we introduce semiparametric modeling techniques to complement the well established parametric Copula-GARCH models. Within this time series setting we generalize existing approaches by providing a modeling framework, that uses an arbitrary order polynomial in the local likelihood formulation, as opposed to the available local constant fitting in a bivariate setting. A study on multivariate wind speed data concludes the chapter. With the work in Chapter 3 we contribute to the understanding of sums of dependent random variables, by providing a new viewpoint using copulas. This allows us to clearly separate the marginal information from the contributions linked to the involved dependence structure. Utilizing this novel point of view, we introduce and study new numerical approximations to the distribution function of such a sum, as well as the respective Value-at-Risk and Expected-Shortfall widely used in financial and actuarial applications. In contrast to competing methods, these approximations are also valid for sums of unbounded random variables, and thus contribute to the existing literature in this respect. We connect our results to the portfolio choice problem in Chapter 4. Here we exemplify how Expected-Shortfall optimal portfolios can be constructed utilizing the methods derived in Chapter 3. This allows us to clearly visualize how changing dependence structures impact the portfolio choice, thus adding to a better understanding of dependence uncertainty. Chapter 5 combines the methodologies introduced in the previous chapters. To this end we construct time dynamic Expected-Shortfall optimal portfolios based on return data of the Société Générale Group, the KBC Group NV and BNP Paribas SA. As anticipated from the well-known mean-variance optimal portfolios, we can report a clearly visible diversification effect. To investigate the validity of the proposed final model, we back-test both the Value-at-Risk and the Expected-Shortfall. Here we cannot detect a statistically significant model misspecification. Final conclusions and research perspectives are given in Chapter 6.Abstract iii Contents vii 1 Introduction . . 1 1.1 Computing & Software . . 4 1.2 Notation . . 4 1.3 Copulas and Sklar’s Theorem . . 5 1.3.1 Farlie–Gumbel–Morgenstern Copula . . 9 1.3.2 Gaussian Copula . . 9 1.3.3 Frank Copula . . 10 1.3.4 Gumbel Copula . . 11 1.4 Generalized Autoregressive Conditional Heteroscedasticity Models . . 11 2 Conditional Copulas and Local Likelihood . . 15 2.1 Local Likelihood . . 17 2.1.1 General Methodology . . 17 2.1.2 Illustrative example . . 21 2.2 Dynamic Modeling via Copulas: Application in Forecasting . . 23 3 Sums of Dependent Random Variables . . 34 3.1 Introduction . . 34 3.2 Theoretical Properties of Sums of Random Variables . . 36 3.3 Numerical Computation of $F_Z(z)$ . . 47 3.3.1 Monte Carlo Integration . . 48 3.3.2 Deterministic Numerical Integration . . 48 3.4 Numerical Computation of $F^{-1}_Z$ and $E[Z|Z ≤ z]$ . . 58 3.4.1 Computation of $F^{-1}_Z$ . . 58 3.4.2 Computation of $E[Z|Z ≤ z]$ . . 59 3.5 Numerical Examples . . 61 3.5.1 Random Variables with a Lower Bound . . 61 3.5.2 Unbounded Random Variables . . 69 3.6 Further Discussion . . 74 3.7 Application: Value-at-Risk Forecasting . . 76 3.8 Multivariate Distribution Refinement . . 77 3.8.1 Multivariate Distribution Refinement . . 77 3.8.2 Multivariate Distribution Refinement Extrapolation . . 82 4 Expected-Shortfall Based Portfolio Optimization . . 89 4.1 Introduction . . 89 4.2 Risk Measures and Portfolio Selection . . 91 4.3 Numerical Computation . . 96 4.3.1 Formulas for $C^{−j}$ . . 97 4.3.2 Numerical Integration . . 102 4.4 Applications . . 107 4.4.1 Illustrative Examples . . 107 4.4.2 Real Data Example: Apple, Microsoft and IBM . . 116 4.5 Conclusion . . 126 4.A Expected-Shortfall for the type I skew logistic distribution . . 128 4.B Expected-Shortfall for the student-t distribution . . 130 4.C Expected-Shortfall for the Laplace distribution . . 130 4.D Expected-Shortfall Optimization Gradient . . 131 5 Application of Methodologies to Bank Data . . 134 5.1 Data Pre-Processing . . 135 5.2 Marginal Time Series Modelling . . 136 5.3 Copula Parameter Estimation and Bandwidth Selection . . 147 5.4 Portfolio Selection . . 152 5.5 Back-Testing . . 160 6 General Conclusions and Research Perspectives . . 164nrpages: 174status: publishe