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Computation and Modelling in Insurance and Finance

Book Description

Focusing on what actuaries need in practice, this introductory account provides readers with essential tools for handling complex problems and explains how simulation models can be created, used and re-used (with modifications) in related situations. The book begins by outlining the basic tools of modelling and simulation, including a discussion of the Monte Carlo method and its use. Part II deals with general insurance and Part III with life insurance and financial risk. Algorithms that can be implemented on any programming platform are spread throughout and a program library written in R is included. Numerous figures and experiments with R-code illustrate the text. The author's non-technical approach is ideal for graduate students, the only prerequisites being introductory courses in calculus and linear algebra, probability and statistics. The book will also be of value to actuaries and other analysts in the industry looking to update their skills.

Table of Contents

  1. Cover
  2. Half Title
  3. Series
  4. Title
  5. Copyright
  6. Contents
  7. Preface
  8. 1 Introduction
    1. 1.1 A view on the evaluation of risk
      1. 1.1.1 The role of mathematics
      2. 1.1.2 Risk methodology
      3. 1.1.3 The computer model
    2. 1.2 Insurance risk: Basic concepts
      1. 1.2.1 Introduction
      2. 1.2.2 Pricing insurance risk
      3. 1.2.3 Portfolios and solvency
      4. 1.2.4 Risk ceding and reinsurance
    3. 1.3 Financial risk: Basic concepts
      1. 1.3.1 Introduction
      2. 1.3.2 Rates of interest
      3. 1.3.3 Financial returns
      4. 1.3.4 Log-returns
      5. 1.3.5 Financial portfolios
    4. 1.4 Risk over time
      1. 1.4.1 Introduction
      2. 1.4.2 Accumulation of values
      3. 1.4.3 Forward rates of interest
      4. 1.4.4 Present and fair values
      5. 1.4.5 Bonds and yields
      6. 1.4.6 Duration
      7. 1.4.7 Investment strategies
    5. 1.5 Method: A unified beginning
      1. 1.5.1 Introduction
      2. 1.5.2 Monte Carlo algorithms and notation
      3. 1.5.3 Example: Term insurance
      4. 1.5.4 Example: Property insurance
      5. 1.5.5 Example: Reversion to mean
      6. 1.5.6 Example: Equity over time
    6. 1.6 How the book is planned
      1. 1.6.1 Mathematical level
      2. 1.6.2 Organization
      3. 1.6.3 Exercises and R
      4. 1.6.4 Notational rules and conventions
    7. 1.7 Bibliographical notes
      1. 1.7.1 General work
      2. 1.7.2 Monte Carlo
  9. PART I TOOLS FOR RISK ANALYSIS
    1. 2 Getting started the Monte Carlo way
      1. 2.1 Introduction
      2. 2.2 How simulations are used
        1. 2.2.1 Introduction
        2. 2.2.2 Mean and standard deviation
        3. 2.2.3 Example: Financial returns
        4. 2.2.4 Percentiles
        5. 2.2.5 Density estimation
        6. 2.2.6 Monte Carlo error and selection of m
      3. 2.3 How random variables are sampled
        1. 2.3.1 Introduction
        2. 2.3.2 Inversion
        3. 2.3.3 Acceptance–rejection
        4. 2.3.4 Ratio of uniforms
      4. 2.4 Making the Gaussian work
        1. 2.4.1 Introduction
        2. 2.4.2 The normal family
        3. 2.4.3 Modelling on logarithmic scale
        4. 2.4.4 Stochastic volatility
        5. 2.4.5 The t-family
        6. 2.4.6 Dependent normal pairs
        7. 2.4.7 Dependence and heavy tails
        8. 2.4.8 Equicorrelation models
      5. 2.5 Positive random variables
        1. 2.5.1 Introduction
        2. 2.5.2 The Gamma distribution
        3. 2.5.3 The exponential distribution
        4. 2.5.4 The Weibull distribution
        5. 2.5.5 The Pareto distribution
        6. 2.5.6 The Poisson distribution
      6. 2.6 Mathematical arguments
        1. 2.6.1 Monte Carlo error and tails of distributions
        2. 2.6.2 Algorithm 2.7 revisited
        3. 2.6.3 Algorithm 2.9 revisited
        4. 2.6.4 Algorithm 2.10 revisited
        5. 2.6.5 Algorithm 2.14 revisited
      7. 2.7 Bibliographical notes
        1. 2.7.1 Statistics and distributions
        2. 2.7.2 Sampling
        3. 2.7.3 Programming
      8. 2.8 Exercises
    2. 3 Evaluating risk: A primer
      1. 3.1 Introduction
      2. 3.2 General insurance: Opening look
        1. 3.2.1 Introduction
        2. 3.2.2 Enter contracts and their clauses
        3. 3.2.3 Stochastic modelling
        4. 3.2.4 Risk diversification
      3. 3.3 How Monte Carlo is put to work
        1. 3.3.1 Introduction
        2. 3.3.2 Skeleton algorithms
        3. 3.3.3 Checking the program
        4. 3.3.4 Computing the reserve
        5. 3.3.5 When responsibility is limited
        6. 3.3.6 Dealing with reinsurance
      4. 3.4 Life insurance: A different story
        1. 3.4.1 Introduction
        2. 3.4.2 Life insurance uncertainty
        3. 3.4.3 Life insurance mathematics
        4. 3.4.4 Simulating pension schemes
        5. 3.4.5 Numerical example
      5. 3.5 Financial risk: Derivatives as safety
        1. 3.5.1 Introduction
        2. 3.5.2 Equity puts and calls
        3. 3.5.3 How equity options are valued
        4. 3.5.4 The Black–Scholes formula
        5. 3.5.5 Options on portfolios
        6. 3.5.6 Are equity options expensive?
      6. 3.6 Risk over long terms
        1. 3.6.1 Introduction
        2. 3.6.2 The ruin problem
        3. 3.6.3 Cash flow simulations
        4. 3.6.4 Solving the ruin equation
        5. 3.6.5 An underwriter example
        6. 3.6.6 Financial income added
      7. 3.7 Mathematical arguments
        1. 3.7.1 The Black–Scholes formula
        2. 3.7.2 The derivative with respect to σ
        3. 3.7.3 Solvency without financial earning
        4. 3.7.4 A representation of net assets
      8. 3.8 Bibliographical notes
        1. 3.8.1 General work
        2. 3.8.2 Monte Carlo and implementation
        3. 3.8.3 Other numerical methods
      9. 3.9 Exercises
    3. 4 Monte Carlo II: Improving technique
      1. 4.1 Introduction
      2. 4.2 Table look-up methods
        1. 4.2.1 Introduction
        2. 4.2.2 Uniform sampling
        3. 4.2.3 General discrete sampling
        4. 4.2.4 Example: Poisson sampling
        5. 4.2.5 Making the continuous discrete
        6. 4.2.6 Example: Put options
      3. 4.3 Correlated sampling
        1. 4.3.1 Introduction
        2. 4.3.2 Common random numbers
        3. 4.3.3 Example from finance
        4. 4.3.4 Example from insurance
        5. 4.3.5 Negative correlation: Antithetic designs
        6. 4.3.6 Examples of designs
        7. 4.3.7 Antithetic design in property insurance
      4. 4.4 Importance sampling and rare events
        1. 4.4.1 Introduction
        2. 4.4.2 The sampling method
        3. 4.4.3 Choice of importance distribution
        4. 4.4.4 Importance sampling in property insurance
        5. 4.4.5 Application: Reserves and reinsurance premia
        6. 4.4.6 Example: A Pareto portfolio
      5. 4.5 Control variables
        1. 4.5.1 Introduction
        2. 4.5.2 The control method and reinsurance
        3. 4.5.3 Control scheme with equity options
        4. 4.5.4 Example: Put options
      6. 4.6 Random numbers: Pseudo- and quasi-
        1. 4.6.1 Introduction
        2. 4.6.2 Pseudo-random numbers
        3. 4.6.3 Quasi-random numbers: Preliminaries
        4. 4.6.4 Sobol sequences: Construction
        5. 4.6.5 Higher dimension and random shifts
        6. 4.6.6 Quasi-random numbers: Accuracy
      7. 4.7 Mathematical arguments
        1. 4.7.1 Efficiency of antithetic designs
        2. 4.7.2 Antithetic variables and property insurance
        3. 4.7.3 Importance sampling
        4. 4.7.4 Control scheme for equity options
      8. 4.8 Bibliographical notes
        1. 4.8.1 General work
        2. 4.8.2 Special techniques
        3. 4.8.3 Markov chain Monte Carlo
        4. 4.8.4 High-dimensional systems
      9. 4.9 Exercises
    4. 5 Modelling I: Linear dependence
      1. 5.1 Introduction
      2. 5.2 Descriptions of first and second order
        1. 5.2.1 Introduction
        2. 5.2.2 What a correlation tells us
        3. 5.2.3 Many correlated variables
        4. 5.2.4 Estimation using historical data
      3. 5.3 Financial portfolios and Markowitz theory
        1. 5.3.1 Introduction
        2. 5.3.2 The Markowitz problem
        3. 5.3.3 Solutions
        4. 5.3.4 Numerical illustration
        5. 5.3.5 Two risky assets
        6. 5.3.6 Example: The crash of a hedge fund
        7. 5.3.7 Diversification of financial risk I
        8. 5.3.8 Diversification under CAPM
      4. 5.4 Dependent Gaussian models once more
        1. 5.4.1 Introduction
        2. 5.4.2 Uniqueness
        3. 5.4.3 Properties
        4. 5.4.4 Simulation
        5. 5.4.5 Scale for modelling
        6. 5.4.6 Numerical example: Returns or log-returns?
        7. 5.4.7 Making tails heavier
      5. 5.5 The random walk
        1. 5.5.1 Introduction
        2. 5.5.2 Random walk and equity
        3. 5.5.3 Elementary properties
        4. 5.5.4 Several processes jointly
        5. 5.5.5 Simulating the random walk
        6. 5.5.6 Numerical illustration
      6. 5.6 Introducing stationary models
        1. 5.6.1 Introduction
        2. 5.6.2 Autocovariances and autocorrelations
        3. 5.6.3 Estimation from historical data
        4. 5.6.4 Autoregression of first order
        5. 5.6.5 The behaviour of first-order autoregressions
        6. 5.6.6 Non-linear change of scale
        7. 5.6.7 Monte Carlo implementation
        8. 5.6.8 Numerical illustration
      7. 5.7 Changing the time scale
        1. 5.7.1 Introduction
        2. 5.7.2 Historical data on short time scales
        3. 5.7.3 The random walk revisited
        4. 5.7.4 Continuous time: The Wiener process
        5. 5.7.5 First-order autoregression revisited
        6. 5.7.6 Continuous-time autoregression
      8. 5.8 Mathematical arguments
        1. 5.8.1 Markowitz optimality
        2. 5.8.2 Risk bound under CAPM
        3. 5.8.3 Covariances of first-order autoregressions
        4. 5.8.4 Volatility estimation and time scale
        5. 5.8.5 The accuracy of covariance estimates
      9. 5.9 Bibliographical notes
        1. 5.9.1 General work
        2. 5.9.2 Continuous-time processes
        3. 5.9.3 Historical data and the time scale
      10. 5.10 Exercises
    5. 6 Modelling II: Conditional and non-linear
      1. 6.1 Introduction
      2. 6.2 Conditional modelling
        1. 6.2.1 Introduction
        2. 6.2.2 The conditional Gaussian
        3. 6.2.3 Survival modelling
        4. 6.2.4 Over-threshold modelling
        5. 6.2.5 Stochastic parameters
        6. 6.2.6 Common factors
        7. 6.2.7 Monte Carlo distributions
      3. 6.3 Uncertainty on different levels
        1. 6.3.1 Introduction
        2. 6.3.2 The double rules
        3. 6.3.3 Financial risk under CAPM
        4. 6.3.4 Insurance risk
        5. 6.3.5 Impact of subordinate risk
        6. 6.3.6 Random claim intensity in general insurance
      4. 6.4 The role of the conditional mean
        1. 6.4.1 Introduction
        2. 6.4.2 Optimal prediction and interest rates
        3. 6.4.3 The conditional mean as a price
        4. 6.4.4 Modelling bond prices
        5. 6.4.5 Bond price schemes
        6. 6.4.6 Interest rate curves
      5. 6.5 Stochastic dependence: General
        1. 6.5.1 Introduction
        2. 6.5.2 Factorization of density functions
        3. 6.5.3 Types of dependence
        4. 6.5.4 Linear and normal processes
        5. 6.5.5 The multinomial situation
      6. 6.6 Markov chains and life insurance
        1. 6.6.1 Introduction
        2. 6.6.2 Markov modelling
        3. 6.6.3 A disability scheme
        4. 6.6.4 Numerical example
      7. 6.7 Introducing copulas
        1. 6.7.1 Introduction
        2. 6.7.2 Copula modelling
        3. 6.7.3 The Clayton copula
        4. 6.7.4 Conditional distributions under copulas
        5. 6.7.5 Many variables and the Archimedean class
        6. 6.7.6 The Marshall–Olkin representation
        7. 6.7.7 Copula sampling
        8. 6.7.8 Example: An equity portfolio
        9. 6.7.9 Example: Copula log-normals against pure log-normals
      8. 6.8 Mathematical arguments
        1. 6.8.1 Portfolio risk when claim intensities are random
        2. 6.8.2 Optimal prediction
        3. 6.8.3 Vasiček bond prices
        4. 6.8.4 The Marshall–Olkin representation
        5. 6.8.5 A general scheme for copula sampling
        6. 6.8.6 Justification of Algorithm 6.4
      9. 6.9 Bibliographical notes
        1. 6.9.1 General work
        2. 6.9.2 Applications
        3. 6.9.3 Copulas
      10. 6.10 Exercises
    6. 7 Historical estimation and error
      1. 7.1 Introduction
      2. 7.2 Error of different origin
        1. 7.2.1 Introduction
        2. 7.2.2 Quantifying error
        3. 7.2.3 Numerical illustration
        4. 7.2.4 Errors and the mean
        5. 7.2.5 Example: Option pricing
        6. 7.2.6 Example: Reserve in property insurance
        7. 7.2.7 Bias and model error
      3. 7.3 How parameters are estimated
        1. 7.3.1 Introduction
        2. 7.3.2 The quick way: Moment matching
        3. 7.3.3 Moment matching and time series
        4. 7.3.4 The usual way: Maximum likelihood
        5. 7.3.5 Example: Norwegian natural disasters
      4. 7.4 Evaluating error I
        1. 7.4.1 Introduction
        2. 7.4.2 Introducing the bootstrap
        3. 7.4.3 Introductory example: The Poisson bootstrap
        4. 7.4.4 Second example: The Pareto bootstrap
        5. 7.4.5 The pure premium bootstrap
        6. 7.4.6 Simplification: The Gaussian bootstrap
        7. 7.4.7 The old way: Delta approximations
      5. 7.5 Evaluating error II: Nested schemes
        1. 7.5.1 Introduction
        2. 7.5.2 The nested algorithm
        3. 7.5.3 Example: The reserve bootstrap
        4. 7.5.4 Numerical illustration
        5. 7.5.5 A second example: Interest-rate return
        6. 7.5.6 Numerical illustration
      6. 7.6 The Bayesian approach
        1. 7.6.1 Introduction
        2. 7.6.2 The posterior view
        3. 7.6.3 Example: Claim intensities
        4. 7.6.4 Example: Expected return on equity
        5. 7.6.5 Choosing the prior
        6. 7.6.6 Bayesian simulation
        7. 7.6.7 Example: Mean payment
        8. 7.6.8 Example: Pure premium
        9. 7.6.9 Summing up: Bayes or not?
      7. 7.7 Mathematical arguments
        1. 7.7.1 Bayesian means under Gaussian models
      8. 7.8 Bibliographical notes
        1. 7.8.1 Analysis of error
        2. 7.8.2 The bootstrap
        3. 7.8.3 Bayesian techniques
      9. 7.9 Exercises
  10. PART II GENERAL INSURANCE
    1. 8 Modelling claim frequency
      1. 8.1 Introduction
      2. 8.2 The world of Poisson
        1. 8.2.1 Introduction
        2. 8.2.2 An elementary look
        3. 8.2.3 Extending the argument
        4. 8.2.4 When the intensity varies over time
        5. 8.2.5 The Poisson distribution
        6. 8.2.6 Using historical data
        7. 8.2.7 Example: A Norwegian automobile portfolio
      3. 8.3 Random intensities
        1. 8.3.1 Introduction
        2. 8.3.2 A first look
        3. 8.3.3 Estimating the mean and variance of μ
        4. 8.3.4 The negative binomial model
        5. 8.3.5 Fitting the negative binomial
        6. 8.3.6 Automobile example continued
      4. 8.4 Intensities with explanatory variables
        1. 8.4.1 Introduction
        2. 8.4.2 The model
        3. 8.4.3 Data and likelihood function
        4. 8.4.4 A first interpretation
        5. 8.4.5 How variables are entered
        6. 8.4.6 Interaction and cross-classification
      5. 8.5 Modelling delay
        1. 8.5.1 Introduction
        2. 8.5.2 Multinomial delay
        3. 8.5.3 IBNR claim numbers
        4. 8.5.4 Fitting delay models
        5. 8.5.5 Syntetic example: Car crash injury
      6. 8.6 Mathematical arguments
        1. 8.6.1 The general Poisson argument
        2. 8.6.2 Estimating the mean and standard deviation of μ
        3. 8.6.3 Large-sample properties
        4. 8.6.4 The negative binomial density function
        5. 8.6.5 Skewness of the negative binomial
        6. 8.6.6 The convolution property of the negative binomial
        7. 8.6.7 IBNR: The delay model
      7. 8.7 Bibliographical notes
        1. 8.7.1 Poisson modelling
        2. 8.7.2 Generalized linear models
        3. 8.7.3 Reserving over long
      8. 8.8 Exercises
    2. 9 Modelling claim size
      1. 9.1 Introduction
      2. 9.2 Parametric and non-parametric modelling
        1. 9.2.1 Introduction
        2. 9.2.2 Scale families of distributions
        3. 9.2.3 Fitting a scale family
        4. 9.2.4 Shifted distributions
        5. 9.2.5 Skewness as a simple description of shape
        6. 9.2.6 Non-parametric estimation
      3. 9.3 The log-normal and Gamma families
        1. 9.3.1 Introduction
        2. 9.3.2 The log-normal: A quick summary
        3. 9.3.3 The Gamma model
        4. 9.3.4 Fitting the Gamma familiy
        5. 9.3.5 Regression for claims size
      4. 9.4 The Pareto families
        1. 9.4.1 Introduction
        2. 9.4.2 Elementary properties
        3. 9.4.3 Likelihood estimation
        4. 9.4.4 Over-threshold under Pareto
        5. 9.4.5 The extended Pareto family
      5. 9.5 Extreme value methods
        1. 9.5.1 Introduction
        2. 9.5.2 Over-threshold distributions in general
        3. 9.5.3 The Hill estimate
        4. 9.5.4 The entire distribution through mixtures
        5. 9.5.5 The empirical distribution mixed with Pareto
      6. 9.6 Searching for the model
        1. 9.6.1 Introduction
        2. 9.6.2 Using transformations
        3. 9.6.3 Example: The Danish fire claims
        4. 9.6.4 Pareto mixing
        5. 9.6.5 When data are scarce
        6. 9.6.6 When data are scarce II
      7. 9.7 Mathematical arguments
        1. 9.7.1 Extended Pareto: Moments
        2. 9.7.2 Extended Pareto: A limit
        3. 9.7.3 Extended Pareto: A representation
        4. 9.7.4 Extended Pareto: Additional sampler
        5. 9.7.5 Justification of the Hill estimate
      8. 9.8 Bibliographical notes
        1. 9.8.1 Families of distributions
        2. 9.8.2 Extreme value theory
        3. 9.8.3 Over thresholds
      9. 9.9 Exercises
    3. 10 Solvency and pricing
      1. 10.1 Introduction
      2. 10.2 Portfolio liabilities by simple approximation
        1. 10.2.1 Introduction
        2. 10.2.2 Normal approximations
        3. 10.2.3 Example: Motor insurance
        4. 10.2.4 The normal power approximation
        5. 10.2.5 Example: Danish fire claims
      3. 10.3 Portfolio liabilities by simulation
        1. 10.3.1 Introduction
        2. 10.3.2 A skeleton algorithm
        3. 10.3.3 Danish fire data: The impact of the claim size model
      4. 10.4 Differentiated pricing through regression
        1. 10.4.1 Introduction
        2. 10.4.2 Estimates of pure premia
        3. 10.4.3 Pure premia regression in practice
        4. 10.4.4 Example: The Norwegian automobile portfolio
      5. 10.5 Differentiated pricing through credibility
        1. 10.5.1 Introduction
        2. 10.5.2 Credibility: Approach
        3. 10.5.3 Linear credibility
        4. 10.5.4 How accurate is linear credibility?
        5. 10.5.5 Credibility at group level
        6. 10.5.6 Optimal credibility
        7. 10.5.7 Estimating the structural parameters
      6. 10.6 Reinsurance
        1. 10.6.1 Introduction
        2. 10.6.2 Traditional contracts
        3. 10.6.3 Pricing reinsurance
        4. 10.6.4 The effect of inflation
        5. 10.6.5 The effect of reinsurance on the reserve
      7. 10.7 Mathematical arguments
        1. 10.7.1 The normal power approximation
        2. 10.7.2 The third-order moment of X
        3. 10.7.3 Normal power under heterogeneity
        4. 10.7.4 Auxiliary for linear credibility
        5. 10.7.5 Linear credibility
        6. 10.7.6 Optimal credibility
        7. 10.7.7 Parameter estimation in linear credibility
      8. 10.8 Bibliographical notes
        1. 10.8.1 Computational methods
        2. 10.8.2 Credibility
        3. 10.8.3 Reinsurance
      9. 10.9 Exercises
    4. 11 Liabilities over long terms
      1. 11.1 Introduction
      2. 11.2 Simple situations
        1. 11.2.1 Introduction
        2. 11.2.2 Lower-order moments
        3. 11.2.3 When risk is constant
        4. 11.2.4 Underwriter results in the long run
        5. 11.2.5 Underwriter ruin by closed mathematics
        6. 11.2.6 Underwriter ruin under heavy-tailed losses
      3. 11.3 Time variation through regression
        1. 11.3.1 Introduction
        2. 11.3.2 Poisson regression with time effects
        3. 11.3.3 Example: An automobile portfolio
        4. 11.3.4 Regression with random background
        5. 11.3.5 The automobile portfolio: A second round
      4. 11.4 Claims as a stochastic process
        1. 11.4.1 Introduction
        2. 11.4.2 Claim intensity as a stationary process
        3. 11.4.3 A more general viewpoint
        4. 11.4.4 Model for the claim numbers
        5. 11.4.5 Example: The effect on underwriter risk
        6. 11.4.6 Utilizing historical data
        7. 11.4.7 Numerical experiment
      5. 11.5 Building simulation models
        1. 11.5.1 Introduction
        2. 11.5.2 Under the top level
        3. 11.5.3 Hidden, seasonal risk
        4. 11.5.4 Hidden risk with inflation
        5. 11.5.5 Example: Is inflation important?
        6. 11.5.6 Market fluctuations
        7. 11.5.7 Market fluctuations: Example
        8. 11.5.8 Taxes and dividend
      6. 11.6 Cash flow or book value?
        1. 11.6.1 Introduction
        2. 11.6.2 Mathematical formulation
        3. 11.6.3 Adding IBNR claims
        4. 11.6.4 Example: Runoff portfolios
      7. 11.7 Mathematical arguments
        1. 11.7.1 Lower-order moments of Y[sub(k)] under constant risk
        2. 11.7.2 Lundberg’s inequality
        3. 11.7.3 Moment-generating functions for underwriting
        4. 11.7.4 Negative binomial regression
      8. 11.8 Bibliographical notes
        1. 11.8.1 Negative binomial regression
        2. 11.8.2 Claims as stochastic processes
        3. 11.8.3 Ruin
      9. 11.9 Exercises
  11. PART III LIFE INSURANCE AND FINANCIAL RISK
    1. 12 Life and state-dependent insurance
      1. 12.1 Introduction
      2. 12.2 The anatomy of state-dependent insurance
        1. 12.2.1 Introduction
        2. 12.2.2 Cash flows determined by states
        3. 12.2.3 Equivalence pricing
        4. 12.2.4 The reserve
        5. 12.2.5 The portfolio viewpoint
      3. 12.3 Survival modelling
        1. 12.3.1 Introduction
        2. 12.3.2 Deductions from one-step transitions
        3. 12.3.3 Modelling through intensities
        4. 12.3.4 A standard model: Gomperz–Makeham
        5. 12.3.5 Expected survival
        6. 12.3.6 Using historical data
      4. 12.4 Single-life arrangements
        1. 12.4.1 Introduction
        2. 12.4.2 How mortality risk affects value
        3. 12.4.3 Life insurance notation
        4. 12.4.4 Computing mortality-adjusted annuities
        5. 12.4.5 Common insurance arrangements
        6. 12.4.6 A numerical example
      5. 12.5 Multi-state insurance I: Modelling
        1. 12.5.1 Introduction
        2. 12.5.2 From one-step to k-step transitions
        3. 12.5.3 Intensity modelling
        4. 12.5.4 Example: A Danish disability model
        5. 12.5.5 Numerical examples
        6. 12.5.6 From intensities to transition probabilities
        7. 12.5.7 Using historical data
      6. 12.6 Multi-state insurance II: Premia and liabilities
        1. 12.6.1 Introduction
        2. 12.6.2 Single policies
        3. 12.6.3 Example 1: A widow scheme
        4. 12.6.4 Example 2: Disability and retirement in combination
        5. 12.6.5 Portfolio liabilities
        6. 12.6.6 Example: A disability scheme
      7. 12.7 Mathematical arguments
        1. 12.7.1 Savings and mortality-adjusted value
        2. 12.7.2 The reserve formula (12.19)
        3. 12.7.3 The k-step transition probabilities
      8. 12.8 Bibliographical notes
        1. 12.8.1 Simple contracts and modelling
        2. 12.8.2 General work
      9. 12.9 Exercises
    2. 13 Stochastic asset models
      1. 13.1 Introduction
      2. 13.2 Volatility modelling I
        1. 13.2.1 Introduction
        2. 13.2.2 Multivariate stochastic volatility
        3. 13.2.3 The multivariate t-distribution
        4. 13.2.4 Dynamic volatility
        5. 13.2.5 Volatility as driver
        6. 13.2.6 Log-normal volatility
        7. 13.2.7 Numerical example
        8. 13.2.8 Several series in parallel
      3. 13.3 Volatility modelling II: The GARCH type
        1. 13.3.1 Introduction
        2. 13.3.2 How GARCH models are constructed
        3. 13.3.3 Volatilities under first-order GARCH
        4. 13.3.4 Properties of the original process
        5. 13.3.5 Fitting GARCH models
        6. 13.3.6 Simulating GARCH
        7. 13.3.7 Example: GARCH and the SP 500 index
      4. 13.4 Linear dynamic modelling
        1. 13.4.1 Introduction
        2. 13.4.2 ARMA models
        3. 13.4.3 Linear feedback
        4. 13.4.4 Enter transformations
      5. 13.5 The Wilkie model I: Twentieth-century financial risk
        1. 13.5.1 Introduction
        2. 13.5.2 Output variables and their building blocks
        3. 13.5.3 Non-linear transformations
        4. 13.5.4 The linear and stationary part
        5. 13.5.5 Parameter estimates
        6. 13.5.6 Annual inflation and returns
      6. 13.6 The Wilkie model II: Implementation issues
        1. 13.6.1 Introduction
        2. 13.6.2 How simulations are initialized
        3. 13.6.3 Simulation algorithms
        4. 13.6.4 Interest-rate interpolation
      7. 13.7 Mathematical arguments
        1. 13.7.1 Absolute deviations from the mean
        2. 13.7.2 Autocorrelations under log-normal volatilities
        3. 13.7.3 The error series for GARCH variances
        4. 13.7.4 Properties of GARCH variances
        5. 13.7.5 The original process squared
        6. 13.7.6 Verification of Table 13.5
      8. 13.8 Bibliographical notes
        1. 13.8.1 Linear processes
        2. 13.8.2 Heavy tails
        3. 13.8.3 Dynamic volatility
      9. 13.9 Exercises
    3. 14 Financial derivatives
      1. 14.1 Introduction
      2. 14.2 Arbitrage and risk neutrality
        1. 14.2.1 Introduction
        2. 14.2.2 Forward contracts
        3. 14.2.3 Binomial movements
        4. 14.2.4 Risk neutrality
      3. 14.3 Equity options I
        1. 14.3.1 Introduction
        2. 14.3.2 Types of contract
        3. 14.3.3 Valuation: A first look
        4. 14.3.4 The put–call parity
        5. 14.3.5 A first look at calls and puts
      4. 14.4 Equity options II: Hedging and valuation
        1. 14.4.1 Introduction
        2. 14.4.2 Actuarial and risk-neutral pricing
        3. 14.4.3 The hedge portfolio and its properties
        4. 14.4.4 The financial state over time
        5. 14.4.5 Numerical experiment
        6. 14.4.6 The situation at expiry revisited
        7. 14.4.7 Valuation
      5. 14.5 Interest-rate derivatives
        1. 14.5.1 Introduction
        2. 14.5.2 Risk-neutral pricing
        3. 14.5.3 Implied mean and forward prices
        4. 14.5.4 Interest-rate swaps
        5. 14.5.5 Floors and caps
        6. 14.5.6 Options on bonds
        7. 14.5.7 Options on interest-rate swaps
        8. 14.5.8 Numerical experimenting
      6. 14.6 Mathematical summing up
        1. 14.6.1 Introduction
        2. 14.6.2 How values of derivatives evolve
        3. 14.6.3 Hedging
        4. 14.6.4 The market price of risk
        5. 14.6.5 Martingale pricing
        6. 14.6.6 Closing mathematics
      7. 14.7 Bibliographical notes
        1. 14.7.1 Introductory work
        2. 14.7.2 Work with heavier mathematics
        3. 14.7.3 The numerical side
      8. 14.8 Exercises
    4. 15 Integrating risk of different origin
      1. 15.1 Introduction
      2. 15.2 Life-table risk
        1. 15.2.1 Introduction
        2. 15.2.2 Numerical example
        3. 15.2.3 The life-table bootstrap
        4. 15.2.4 The bootstrap in life insurance
        5. 15.2.5 Random error and pension evaluations
        6. 15.2.6 Bias against random error
        7. 15.2.7 Dealing with longer lives
        8. 15.2.8 Longevity bias: Numerical examples
      3. 15.3 Risk due to discounting and inflation
        1. 15.3.1 Introduction
        2. 15.3.2 Market-based valuation
        3. 15.3.3 Numerical example
        4. 15.3.4 Inflation: A first look
        5. 15.3.5 Simulating present values under stochastic discounts
        6. 15.3.6 Numerical examples
      4. 15.4 Simulating assets protected by derivatives
        1. 15.4.1 Introduction
        2. 15.4.2 Equity returns with options
        3. 15.4.3 Equity options over longer time horizons
        4. 15.4.4 Money-market investments with floors and caps
        5. 15.4.5 Money-market investments with swaps and swaptions
      5. 15.5 Simulating asset portfolios
        1. 15.5.1 Introduction
        2. 15.5.2 Defining the strategy
        3. 15.5.3 Expenses due to rebalancing
        4. 15.5.4 A skeleton algorithm
        5. 15.5.5 Example 1: Equity and cash
        6. 15.5.6 Example 2: Options added
        7. 15.5.7 Example 3: Bond portfolio and inflation
        8. 15.5.8 Example 4: Equity, cash and bonds
      6. 15.6 Assets and liabilities
        1. 15.6.1 Introduction
        2. 15.6.2 Auxiliary: Duration and spread of liabilities
        3. 15.6.3 Classical immunization
        4. 15.6.4 Net asset values
        5. 15.6.5 Immunization through bonds
        6. 15.6.6 Enter inflation
      7. 15.7 Mathematical arguments
        1. 15.7.1 Present values and duration
        2. 15.7.2 Reddington immunization
      8. 15.8 Bibliographical notes
        1. 15.8.1 Survival modelling
        2. 15.8.2 Fair values
        3. 15.8.3 Financial risk and ALM
        4. 15.8.4 Stochastic dynamic optimization
      9. 15.9 Exercises
    5. Appendix A Random variables: Principal tools
      1. A.1 Introduction
      2. A.2 Single random variables
        1. A.2.1 Introduction
        2. A.2.2 Probability distributions
        3. A.2.3 Simplified description of distributions
        4. A.2.4 Operating rules
        5. A.2.5 Transforms and cumulants
        6. A.2.6 Example: The mean
      3. A.3 Several random variables jointly
        1. A.3.1 Introduction
        2. A.3.2 Covariance and correlation
        3. A.3.3 Operating rules
        4. A.3.4 The conditional viewpoint
      4. A.4 Laws of large numbers
        1. A.4.1 Introduction
        2. A.4.2 The weak law of large numbers
        3. A.4.3 Central limit theorem
        4. A.4.4 Functions of averages
        5. A.4.5 Bias and standard deviation of estimates
        6. A.4.6 Likelihood estimates
    6. Appendix B Linear algebra and stochastic vectors
      1. B.1 Introduction
      2. B.2 Operations on matrices and vectors
        1. B.2.1 Introduction
        2. B.2.2 Addition and multiplication
        3. B.2.3 Quadratic matrices
        4. B.2.4 The geometric view
        5. B.2.5 Algebraic rules
        6. B.2.6 Stochastic vectors
        7. B.2.7 Linear operations on stochastic vectors
        8. B.2.8 Covariance matrices and Cholesky factors
      3. B.3 The Gaussian model: Simple theory
        1. B.3.1 Introduction
        2. B.3.2 Orthonormal operations
        3. B.3.3 Uniqueness
        4. B.3.4 Linear transformations
        5. B.3.5 Block representation
        6. B.3.6 Conditional distributions
        7. B.3.7 Verfication
    7. Appendix C Numerical algorithms: A third tool
      1. C.1 Introduction
      2. C.2 Cholesky computing
        1. C.2.1 Introduction
        2. C.2.2 The Cholesky decomposition
        3. C.2.3 Linear equations
        4. C.2.4 Matrix inversion
      3. C.3 Interpolation, integration, differentiation
        1. C.3.1 Introduction
        2. C.3.2 Numerical interpolation
        3. C.3.3 Numerical integration
        4. C.3.4 Numerical integration II: Gaussian quadrature
        5. C.3.5 Numerical differentiation
      4. C.4 Bracketing and bisection: Easy and safe
        1. C.4.1 Introduction
        2. C.4.2 Bisection: Bracketing as iteration
        3. C.4.3 Golden section: Bracketing for extrema
        4. C.4.4 Golden section: Justification
      5. C.5 Optimization: Advanced and useful
        1. C.5.1 Introduction
        2. C.5.2 The Newton–Raphson method
        3. C.5.3 Variable metric methods
      6. C.6 Bibliographical notes
        1. C.6.1 General numerical methods
        2. C.6.2 Optimization
  12. References
  13. Index