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Ludger Rüschendorf

Mathematical Risk Analysis

Dependence, Risk Bounds, Optimal Allocations and Portfolios
2013. 2015. xii, 408 S. 12 SW-Abb. 235 mm
ISBN: 3-642-43016-3 (3642430163)
Neue ISBN: 978-3-642-43016-9 (9783642430169)

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The up-to-date material and logical structure of this volume provides the clarity and orientation needed to gain a solid working knowledge of mathematical risk analysis. It includes a specialized focus on the risk theory deployed in finance and insurance.
The author´s particular interest in the area of risk measures is to combine this theory with the analysis of dependence properties. The present volume gives an introduction of basic concepts and methods in mathematical risk analysis, in particular of those parts of risk theory that are of special relevance to finance and insurance. Describing the influence of dependence in multivariate stochastic models on risk vectors is the main focus of the text that presents main ideas and methods as well as their relevance to practical applications. The first part introduces basic probabilistic tools and methods of distributional analysis, and describes their use to the modeling of dependence and to the derivation of risk bounds in these models. In the second, part risk measures with a particular focus on those in the financial and insurance context are presented. The final parts are then devoted to applications relevant to optimal risk allocation, optimal portfolio problems as well as to the optimization of insurance contracts. Good knowledge of basic probability and statistics as well as of basic general mathematics is a prerequisite for comfortably reading and working with the present volume, which is intended for graduate students, practitioners and researchers and can serve as a reference resource for the main concepts and techniques.
Preface.-Part I: Stochastic Dependence and Extremal Risk.-1 Copulas, Sklar´s Theorem, and Distributional Transform.- 2 Fréchet Classes, Risk Bounds, and Duality Theory.- 3 Convex Order, Excess of Loss, and Comonotonicity.- 4 Bounds for the Distribution Function and Value at Risk of the Joint Portfolio.- 5 Restrictions on the Dependence Structure.- 6 Dependence Orderings of Risk Vectors and Portfolios.- Part II: Risk Measures and Worst Case Portfolios.- 7 Risk Measures for Real Risks.- 8 Risk Measures for Portfolio Vectors.- 9 Law Invariant Convex Risk Measures on L_d^p and Optimal Mass Transportation.- Part III: Optimal Risk Allocation.- 10 Optimal Allocations and Pareto Equilibrium.- 11 Characterization and Examples of Optimal Risk Allocations for Convex Risk Functionals.- 12 Optimal Contingent Claims and (Re)Insurance Contracts.- Part IV: Optimal Portfolios and Extreme Risks.- 13 Optimal Portfolio Diversification w.r.t. Extreme Risks.- 14 Ordering of Multivariate Risk Models with Respect to Extreme Portfolio Losses.- References.- List of Symbols.- Index.