Risk and Insurance

Preface
Contents
Notation
Internal Reference System
Special Typeface
Miscellaneous Mathematical Notation
Standard Distributions
Abbreviations
Chapter I: Basics
1 Actuarial Versus Financial Pricing
2 Utility
3 Premium Rules
4 Reinsurance
5 Poisson Modelling
5.1 The Poisson Distribution as a Binomial Limit
5.2 The Poisson Process
5.3 Superposition and Thinning
Chapter II: Experience Rating
1 Bayes and Empirical Bayes
1.1 The Bayes Premium
2 Exponential Families and Conjugate Priors
3 Credibility Premiums
3.1 The Simple Credibility Model
3.2 The General Credibility Model
3.3 The Bühlmann Model
3.4 The Bühlmann–Straub Model
3.5 Quadratic Loss
4 Bonus-Malus Systems
4.1 Introduction
4.2 Loimaranta Efficiency
4.3 Bayes Premiums
Chapter III: Sums and Aggregate Claims
1 Introduction
2 Heavy Tails. Subexponential Distributions
2.1 Definition of Subexponentiality and Sufficient Conditions
2.2 Further Mathematical Properties
3 Large Deviations of Sums of Light-Tailed Random Variables
3.1 The Cumulant Function
3.2 The Legendre–Fenchel Transform
3.3 Exponential Families and Change of Measure
3.4 The Chernoff Bound and the Saddlepoint Approximation
4 Tails of Sums of Light-Tailed Random Variables
5 Aggregate Claims and Compound Sums: Generalities
5.1 Poisson Compounding
6 Panjer's Recursion
7 Tails of Compound Sums
7.1 Heavy Tails: The Subexponential Approximation
7.2 Light Tails: The Saddlepoint Approximation
7.3 The NP Approximation
Chapter IV: Ruin Theory
1 The Cramér–Lundberg Model
2 First Results: Martingale Techniques
3 Ladder Heights. Heavy Tails
4 Proof of the Cramér–Lundberg Approximation
5 Finite Time Ruin Probabilities
5.1 Finite Horizon Ruin with Light Tails
5.2 Finite Horizon Ruin with Heavy Tails
6 Markov Regime Switching
6.1 The Averaged Model
6.2 The Matrix m.g.f.
6.3 Cramér–Lundberg Theory
7 Level-Dependent Premiums
8 The Diffusion Approximation
Chapter V: Markov Models in Life Insurance
1 The Contract Payments and the Probability Model
2 Canonical Models
2.1 Mortality Modelling
2.2 The Survival Model
2.3 The Disability Model
2.4 The Spouse Model
3 Valuation of the Payments
4 Valuation in Canonical Models
4.1 The Survival Model
4.2 The Disability Model
Chapter VI: Financial Mathematics in Life Insurance
1 Background and Simple Claims
2 Payment Streams
3 Unit-Link Insurance
4 With-Profit Insurance and the Dynamics of the Surplus
5 Cash Dividends and Market Reserve
6 The Pure Case of Cash Dividends
7 Bonus Payments and Market Reserve
8 The Pure Case of Bonus Payments
9 Comparison of Products
Chapter VII: Special Studies in Life Insurance
1 Duration-Dependent Intensities and Payments
2 Reserve-Dependent Payments and Intensities
3 Bonds and Forward Interest Rates
4 Survival Probabilities and Forward Mortality Rates
5 Dependent Interest and Mortality Rates
6 Stochastic Interest and Mortality Rate Models
7 Reserves Revisited
8 Incidental Policy Holder Behavior
9 Rational Policy Holder Behavior
10 Higher Order Moments. Hattendorf's Theorem
Chapter VIII: Orderings and Comparisons
1 Stochastic Ordering of Risks
2 Convex and Increasing Convex Ordering
2.1 Preliminaries on Stop-Loss Transforms and Convexity
2.2 Convex Ordering
2.3 Increasing Convex Ordering
3 Closure Properties of Orderings
4 Utility, Deductibles and Reinsurance
4.1 Underinsurance and Coinsurance
4.2 Utility and Reinsurance
4.3 Local Versus Global Reinsurance
4.4 Optimal Compensation Functions
5 Applications to Ruin Theory
6 Maximizing the Adjustment Coefficient
Chapter IX: Extreme Value Theory
1 Introduction
2 Elementary Examples and Considerations
2.1 The Exceedance Time
3 Convergence Results
3.1 MDA(Gumbel)
3.2 MDA(Fréchet)
3.3 MDA(Weibull)
3.4 Further Remarks and Examples
4 Proof of the Fisher–Tippett Theorem
5 Records
Chapter X: Dependence and Further Topics in Risk Management
1 Risk Measures
1.1 General Theory
1.2 Value-at-Risk and Expected Shortfall
1.3 Further Remarks
1.4 Quotes on Risk Diversification
2 The Fréchet–Höffding Bounds. Comonotonicity
2.1 Applications of Comonotonicity
3 Special Dependence Structures
3.1 The Multivariate Normal Distribution
3.2 Normal Mixtures
3.3 Spherical and Elliptical Distributions
3.4 Multivariate Regular Variation
4 Copulas
4.1 Archimedean Copulas
4.2 Extreme Value Copulas
4.3 Frailty Copulas
4.4 Further Examples
5 Pearson, Kendall and Spearman
5.1 Linear (Pearson) Correlation
5.2 Kendall's Tau and Spearman's Rho
5.3 Empirical Versions
6 Further Dependence Concepts
6.1 Association
6.2 Tail Dependence
7 Tails of Sums of Dependent Risks
7.1 Bounds
7.2 A Copula Representation
7.3 Asymptotics
8 Dependence Orderings
8.1 Orthant Dependencies and Orderings
8.2 The Supermodular Ordering
8.3 The Multivariate Normal Distribution
Chapter XI: Stochastic Control in Non-Life Insurance
1 Introduction
2 Minimizing the Ruin Probability
2.1 General Theory
2.2 Reinsurance Control
2.3 Investment Control
2.4 Premium Control
3 The Hamilton–Jacobi–Bellman Equation
4 Optimal Dividends
5 Control Problems for the Cramér–Lundberg Model
5.1 The Generator and the HJB Equation
5.2 Ruin Probability Minimization
5.3 Optimal Investment
5.4 Optimal Dividends
6 Examples Involving Game Theory
6.1 An Optimality Property of Stop-Loss Reinsurance
6.2 Premiums Based on Customer Preferences
Chapter XII: Stochastic Control in Life Insurance
1 The Diffusion Approximation
2 Finite-State Markov Process Linear Regulation
3 The Consumption-Investment Problem
4 Uncertain Lifetime
5 The Consumption-Investment-Insurance Problem
6 The Multi-State Problem
6.1 The Survival Model
7 The Pension Fund's Problem
7.1 The Survival Model
Chapter XIII: Selected Further Topics
1 Claims Reserving
2 Multivariate Extreme Value Theory
3 Statistical Methods for Tails and Extreme Values
3.1 The Mean Excess Plot
3.2 The Hill Estimator
3.3 Peaks-over-Threshold
3.4 Block Maxima
4 Large Deviations Theory in Function Spaces
4.1 The Large Deviations Principle. One-Dimensional Cramér
4.2 Schilder and Mogulski
4.3 Implementation and Examples
4.4 LD Results for SDEs
5 Gaussian Maxima
5.1 Borell's Inequality
5.2 Slepian's Inequality
Appendix
A.1 Integral Formulas
A.2 Differential Equations
A.2.1 The One-Dimensional Case
A.2.2 Systems of ODEs
A.3 Inhomogeneous Markov Processes
A.3.1 Feed-Forward Systems
A.4 Itô's Formula
A.5 Diffusion First Passage Probabilities
A.6 L2 Projections. Least Squares. Conditional Expectations
A.6.1 Least Squares
A.6.2 Conditional Expectations
A.7 Supplements on the Normal Distribution
A.7.1 Conditioning
A.7.2 Mill's Ratio
A.7.3 A PDE for the Multivariate Normal Density
A.8 Generalized Inverses
A.9 The Distributional Transform
A.10 Types of Distributions
A.11 Transforms
References
Index