Mathematical Risk Analysis Dependence Risk Bounds Optimal Allocations And Portfolios 2013th Edition Ludger Rschendorf by Ludger Rüschendorf 9783642335891, 3642335896 instant download after payment.

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.
Table of Contents
Cover
Mathematical Risk Analysis - Dependence, Risk Bounds, Optimal Allocations and Portfolios
ISBN 9783642335891 ISBN 9783642335907
Preface
Contents
Part I Stochastic Dependence and Extremal Risk
Copulas, Sklar's Theorem, and Distributional Transform
1.1 Sklar's Theorem and the Distributional Transform
1.2 Copula Models and Copula Constructions
o 1.2.1 Some Classes of Copulas
o 1.2.2 Copulas and L2-Projections
1.3 Multivariate Distributional and Quantile Transform
1.4 Pair Copula Construction of Copula Models
1.5 Applications of the Distributional Transform 1.5.1 Application to Stochastic Ordering
o 1.5.2 Optimal Couplings
o 1.5.3 Identificatio and Goodness of Fit Tests
o 1.5.4 Empirical Copula Process and Empirical Dependence Function
1.6 Multivariate and Overlapping Marginals
o 1.6.1 Generalized Frec� het Class
o 1.6.2 Copulas with Given Independence Structure
o 1.6.3 Copulas, Overlapping Marginals, and L2-Projections
Fr�chet Classes, Risk Bounds, and Duality Theory
2.1 Dual Representation of Generalized Fr�chet Bounds
2.2 Fr�chet Bounds Comonotonicity and Extremal Risk
Convex Order, Excess of Loss, and Comonotonicity
3.1 Convex Order and Comonotonicity
3.2 Schur Order and Rearrangements
3.3 Rearrangements and Excess of Loss
3.4 Integral Orders and F -Diffusions
Bounds for the Distribution Function and Value at Risk of the Joint Portfolio
4.1 Standard Bounds
4.2 Conditional Moment Method
4.3 Dual Bounds
Restrictions on the Dependence Structure
5.1 Restriction to Positive Dependent Risk Vectors
5.2 Higher Order Marginals
o 5.2.1 A Reduction Principle and Bonferroni Type Bounds
o 5.2.2 The Conditioning Method
o 5.2.3 Reduction Bounds for the Joint Portfolio in General Marginal Systems
Dependence Orderings of Risk Vectors and Portfolios
6.1 Positive Orthant Dependence and Supermodular Ordering
6.2 Association, Conditional Increasing Vectors, and Positive Supermodular Dependence
6.3 Directionally Convex Order
o 6.3.1 Basic Properties of the Directionally Convex Order
o 6.3.2 Further Criteria for dcx
o 6.3.3 Directionally Convex Order in Functional Models
6.4 Dependence Orderings in Models with Multivariate Marginals
Part II Risk Measures and Worst Case Portfolios
Risk Measures for Real Risks
7.1 Some Classes of Risk Measures for Real Variables
o 7.1.1 Basic Properties of Risk Measures
o 7.1.2 Examples of Risk Measures
7.2 Representation and Continuity Properties of Convex Risk Measures on Lp-Spaces
o 7.2.1 Convex Duality and Continuity Results
o 7.2.2 Representation of Coherent and Convex Risk Measures on Lp
o 7.2.3 Continuity Results for Risk Measures on Lp
Risk Measures for Portfolio Vectors
8.1 Basic Properties of Portfolio Risk Measures
8.2 Classes of Examples of Portfolio Risk Measures
o 8.2.1 Aggregation Type Risk Measures
o 8.2.2 Multivariate Distortion and Quantile-Type Risk Measures
8.3 Representation and Continuity of Convex Risk Measures on Lp
Law Invariant Convex Risk Measures on Lp and Optimal Mass Transportation
9.1 Law Invariant Risk Measures and Optimal Mass Transportation
9.2 Multivariate Comonotonicity and the n-Coupling Problem
9.3 Worst Case Portfolio Vectors and Diversifcation Effects
9.4 Examples of Worst Case Risk Portfolios and Worst Case Diversifcation Effects
Part III Optimal Risk Allocation
Optimal Allocations and Pareto Equilibrium
10.1 Pareto Equilibrium and Related Risk Measures in the Coherent Case
10.2 Optimal Allocations Under Admissibility Restrictions
10.3 Pareto Equilibrium for Convex Risk Measures
10.4 Pareto Optimality, Comonotonicity, and Existence of Optimal Allocations
Characterization and Examples of Optimal Risk Allocations for Convex Risk Functionals
11.1 Inf-Convolution and Convex Conjugates
11.2 Characterization of Optimal Allocations
11.3 Examples of Optimal Risk Allocations
o 11.3.1 Expected Risk Functionals
o 11.3.2 Dilated Risk Functionals
o 11.3.3 Average Value at Risk and Stop-Loss Contracts
o 11.3.4 Mean Variance Versus Standard Deviation Risk Functionals
11.4 Optimal Allocation of Risk Vectors
o 11.4.1 Characterization of Optimal Allocations
o 11.4.2 Law Invariant Risk Measures and Comonotonicity
o 11.4.3 Existence of Minimal Risk Allocations
o 11.4.4 Uniqueness of Optimal Allocations
o 11.4.5 Examples of Optimal Allocations
11.5 The Capital Allocation Problem
Optimal Contingent Claims and (Re)insurance Contracts
12.1 Optimal Contingent Claims
o 12.1.1 Optimal Investment Problems
o 12.1.2 Minimal Demand Problem
12.2 Optimal (Re)insurance Contracts
o 12.2.1 Optimality of Stop-Loss Contracts
o 12.2.2 Optimal Worst Case (Re)insurance Contracts
Part IV Optimal Portfolios and Extreme Risks
Optimal Portfolio Diversificatio w.r.t. Extreme Risks
13.1 Heavy-Tailed Portfolios and Multivariate Regular Variation
13.2 Extreme Risk Index and Portfolio Diversificatio
13.3 Estimation of the Extreme Risk Index and the Optimal Portfolio
13.4 Asymptotic Normality of b
13.5 Application to Risk Minimization
Ordering of Multivariate Risk Models with Respect to Extreme Portfolio Losses
14.1 Asymptotic Portfolio Loss Ordering
14.2 Characterization of apl in Multivariate Regularly Varying Models
o 14.2.1 Multivariate Regular Variation
o 14.2.2 Ordering of Canonical Spectral Measures
o 14.2.3 Unbalanced Tails
14.3 Relations to the Convex and Supermodular Order
14.4 Examples of apl-Ordering
References
List of Symbols
Index