Risk Measures and Insurance Solvency Benchmarks Fixed Probability Levels in Renewal Risk Models Chapman and Hall CRC Financial Mathematics Series 1st Edition by Vsevolod Malinovskii ISBN 1000411095 9781000411096

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ISBN 10: 1000411095
ISBN 13: 9781000411096
Author: Vsevolod K. Malinovskii

Risk Measures and Insurance Solvency Benchmarks: Fixed-Probability Levels in Renewal Risk Models is written for academics and practitioners who are concerned about potential weaknesses of the Solvency II regulatory system. It is also intended for readers who are interested in pure and applied probability, have a taste for classical and asymptotic analysis, and are motivated to delve into rather intensive calculations. The formal prerequisite for this book is a good background in analysis. The desired prerequisite is some degree of probability training, but someone with knowledge of the classical real-variable theory, including asymptotic methods, will also find this book interesting. For those who find the proofs too complicated, it may be reassuring that most results in this book are formulated in rather elementary terms. This book can also be used as reading material for basic courses in risk measures, insurance mathematics, and applied probability. The material of this book was partly used by the author for his courses in several universities in Moscow, Copenhagen University, and in the University of Montreal. Features Requires only minimal mathematical prerequisites in analysis and probability Suitable for researchers and postgraduate students in related fields Could be used as a supplement to courses in risk measures, insurance mathematics and applied probability.

Risk Measures and Insurance Solvency Benchmarks Fixed Probability Levels in Renewal Risk Models Chapman and Hall CRC Financial Mathematics Series 1st Table of contents:

Risk Measures in Finance and Insurance
1.1 Risk Measures in Finance and Portfolio Management
1.1.1 Value-at-Risk (VaR) Quantile Method and Related Concepts
1.1.2 Tail Value-at-Risk (TVaR) and Expected Shortfall
1.1.3 Value-at-Risk and Its Variants: A Blessing or a Curse?
1.1.4 Economic Capital: A Comprehensive Approach
1.1.5 Simulation in Finance and Insurance
1.1.6 Analytical Methods and Simulation Techniques
1.2 Risk Measures in the Solvency II System
1.2.1 Capital Requirements in the Solvency II Framework
1.2.2 Risk Measurement in the Solvency II System
1.2.3 Criteria for Selecting Loss Distributions
1.2.4 Concerns and Challenges Regarding the Solvency II System
1.3 Risk Measures in Risk Theory
1.3.1 Integral Model of Multi-Year Insurance Processes
1.3.2 Annual Models and the Risk of Ruin
1.3.3 Annual Solvency Provisions: Two Approaches
1.3.4 Non-Loss and Non-Ruin Capital in Exceptional Risk Models
1.3.5 Non-Loss Capital in General Risk Models
1.3.6 Probability of Ruin in General Risk Models
1.3.7 Non-Ruin Capital in General Risk Models
1.4 Aim and Structure of the Book
1.5 Target Audience: Who This Book is Intended For
Problems

Fixed-Probability Level in a Diffusion Model
2.1 Diffusion Model: An Auxiliary Tool
2.1.1 Model Overview and Equivalent Terminology
2.1.2 Roadmap Outline
2.2 Direct Level-Crossing Problem
2.3 Inverse Level-Crossing Problem
2.3.1 Analytical Structure of Fixed-Probability Level
2.3.2 Monotonicity of the Fixed-Probability Level
2.3.3 Convexity of the Fixed-Probability Level
2.4 Asymptotic Behavior of Fixed-Probability Level
2.5 Primary Upper Bounds on Fixed-Probability Level
2.6 Refined Upper Bounds on Fixed-Probability Level
2.6.1 Bounds Based on Monotonicity
2.6.2 Bounds Based on Convexity
2.7 Conclusions and Perspectives
Problems

Fixed-Probability Level in an Exceptional Renewal Model
3.1 Exponential Renewal Model: A Special Case
3.1.1 Model Overview and Equivalent Terminology
3.1.2 Roadmap Outline
3.2 Direct Level-Crossing Problem
3.2.1 Closed-Form Expressions for the Distribution of First Level-Crossing Time
3.2.2 Numerical Comparison
3.2.3 Connections to a Random Walk with Random Displacements
3.2.4 Pollaczek-Khinchin Formula
3.2.5 An Upper Bound Independent of the Distribution
3.3 Inverse Level-Crossing Problem
3.3.1 Analytical Structure of Fixed-Probability Level
3.3.2 Monotonicity of the Fixed-Probability Level
3.3.3 Convexity of the Fixed-Probability Level
3.4 Asymptotic Behavior of Fixed-Probability Level
3.5 Primary Upper Bounds on Fixed-Probability Level
3.6 Refined Upper Bounds on Fixed-Probability Level
3.6.1 Bounds Based on Monotonicity
3.6.2 Bounds Based on Convexity
3.7 Conclusions
Problems

Implicit Function Defined by M-Equation
4.1 Analytical Properties of Core Integral Expression
4.1.1 Equivalent Expressions for μu,c(t)mu_{u,c}(t)μu,c​(t) and μu,cmu_{u,c}μu,c​
4.1.2 Analytical Structure of μu,c(t)mu_{u,c}(t)μu,c​(t) and μu,cmu_{u,c}μu,c​
4.1.3 Asymptotic Behavior of μu,c(t)mu_{u,c}(t)μu,c​(t) as $t\to \infty$
4.2 Proximity Between μu,c(t)mu_{u,c}(t)μu,c​(t) and μu,c(t∣v)mu_{u,c}(t | v)μu,c​(t∣v)
4.3 Analytical Properties of M-Level
4.3.1 Analytical Structure of M-Level
4.3.2 Monotonicity and Convexity of M-Level
4.3.3 Asymptotic Behavior of M-Level as $t\to \infty$
Problems

Fixed-Probability Level in General Renewal Models
5.1 General Renewal Model: Main Framework
5.1.1 Notations and Terminology
5.1.2 Roadmap Adjustments
5.2 Direct Level-Crossing Problem
5.2.1 Kendall’s Identities: Key Results
5.2.2 Inverse Gaussian Approximation
5.2.3 Derivatives of the First Level-Crossing Time Distribution
5.2.4 Upper Bounds on $P\{Yu,c\le t\}$ when $c>c\ast$
5.2.5 Approach Based on Laplace Transforms
5.3 Inverse Level-Crossing Problem
5.3.1 Analytical Structure of Fixed-Probability Level
5.4 Primary Upper Bounds on Fixed-Probability Level
5.5 Proximity to M-Level
5.6 Conclusion
Problems

Case Study: Numerical Evaluation of Fixed-Probability Level
6.1 Distributions of T and Y Selected for Numerical Calculations
6.1.1 Rationale for Selection
6.1.2 Models and Examples for Numerical Calculations
6.2 Simulation in Level-Crossing Problems
6.2.1 Monte Carlo Simulation Algorithm
6.2.2 Simulated Values vs. Inverse Gaussian Approximation and Solutions to M-Equation
6.2.3 Simulated Values in Direct Level-Crossing Problem
6.2.4 Simulated Values in Inverse Level-Crossing Problem
6.3 Numerically Calculated Bounds on Fixed-Probability Level
6.3.1 Elementary Upper Bounds
6.3.2 Refined Upper Bounds: Price of Elegance
6.4 Conclusion
Problems

Probability Mechanism of Insurance with Migration and ERS-Analysis
7.1 Structural Model of Insurance Business: The Origin and Purpose of ERS-Analysis
7.2 Price Competition, Migration, and Market Price
7.2.1 Migration and Market Price Dynamics
7.2.2 Ultimate Migration Rate Function: The M-Function
7.2.3 Migration Rate Function Over Finite Time Intervals
7.2.4 Cumulative Migration Rate Functions
7.2.5 Stochastic Migration Rate Functions
7.3 Compound Poisson Risk Model with Migration
7.3.1 Basic Inhomogeneous Processes
7.3.2 Technical Tools for ERS-Analysis
7.4 ERS-Analysis When Y is Exponentially Distributed
7.4.1 Claim Amount Distribution
7.4.2 Probability of Ruin
7.4.3 Non-Ruin Capital
7.4.4 Conservative Intrinsic Value
7.5 ERS-Analysis When Y is Generally Distributed
7.5.1 Non-Ruin Capital
7.5.2 Conservative Intrinsic Value
7.5.3 Simulation Algorithms and Numerical Illustrations
7.6 Conclusions
Problems

Auxiliary Results from Analysis
8.1 Elementary Concepts and Results
8.1.1 Criteria of Monotonicity, Convexity, and Concavity
8.1.2 Chain Rules of Differentiation
8.1.3 Tangent Lines Passing Through a Point
8.1.4 Inverse and Implicit Functions and Their Derivatives
8.1.5 Remarks on Asymptotic Methods
8.1.6 Remarks on Analytical and Numerical Methods
8.2 Advanced Concepts and Results
8.2.1 Bessel Functions
8.2.2 Stirling Numbers of the Second Kind
8.2.3 A Formula from Combinatorics
8.2.4 Series Sk(u∣p)S_k(u | p)Sk​(u∣p) and Integrals Hk(t∣p)H_k(t | p)Hk​(t∣p)
8.2.5 Auxiliary Function of the Second Kind

Auxiliary Results from Probability
9.1 Distributions on the Positive Half-Line
9.2 Gaussian Distribution and Related Issues
9.3 Auxiliary Function of the First Kind
9.4 Random Walks and Ladder Random Variables
9.5 Wiener and Diffusion Processes

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