Chapter
1.3 Finance and investments
2 The basic deterministic model
2.2 An analogy with currencies
2.4 Calculating the discount function
2.5 Interest and discount rates
2.7 Values and actuarial equivalence
2.9 Regular pattern cash flows
2.10 Balances and reserves
2.10.2 Relation between balances and reserves
2.10.3 Prospective versus retrospective methods
2.10.4 Recursion formulas
2.11 Time shifting and the splitting identity
*2.11 Change of discount function
2.12 Internal rates of return
*2.13 Forward prices and term structure
2.14 Standard notation and terminology
2.14.1 Standard notation for cash flows discounted with interest
2.15 Spreadsheet calculations
3.3 Constructing the life table from the values of qx
3.5 Choice of life tables
3.6 Standard notation and terminology
4.2 Calculating annuity premiums
4.3 The interest and survivorship discount function
4.3.1 The basic definition
4.3.2 Relations between yx for various values of x
4.5 Deferred annuities with annual premiums
4.6 Some practical considerations
4.7 Standard notation and terminology
4.8 Spreadsheet calculations
5.2 Calculating life insurance premiums
5.3 Types of life insurance
5.4 Combined insurance–annuity benefits
5.5 Insurances viewed as annuities
5.7 A general insurance–annuity identity
5.7.1 The general identity
5.7.2 The endowment identity
5.8 Standard notation and terminology
5.8.1 Single-premium notation
5.8.2 Annual-premium notation
5.9 Spreadsheet applications
6 Insurance and annuity reserves
6.1 Introduction to reserves
6.2 The general pattern of reserves
6.4 Detailed analysis of an insurance or annuity contract
6.4.2 The risk–savings decomposition
6.7 Policies involving a return of the reserve
6.8 Premium difference and paid-up formulas
6.8.1 Premium difference formulas
6.8.3 Level endowment reserves
6.9 Standard notation and terminology
6.10 Spreadsheet applications
7.2 Cash flows discounted with interest only
7.3 Life annuities paid mthly
7.3.1 Uniform distribution of deaths
7.3.2 Present value formulas
7.5 Approximation and computation
*7.6 Fractional period premiums and reserves
7.7 Reserves at fractional durations
7.8 Standard notation and terminology
8.1 Introduction to continuous annuities
8.2 The force of discount
8.3 The constant interest case
8.4 Continuous life annuities
8.4.3 Life expectancy revisited
8.5 The force of mortality
8.6 Insurances payable at the moment of death
8.7 Premiums and reserves
8.8 The general insurance–annuity identity in the continuous case
8.9 Differential equations for reserves
8.10 Some examples of exact calculation
8.10.1 Constant force of mortality
8.10.3 An example of the splitting identity
8.11 Further approximations from the life table
8.12 Standard actuarial notation and terminology
9.2 Select and ultimate tables
9.4 Projections in annuity tables
10 Multiple-life contracts
10.2 The joint-life status
10.3 Joint-life annuities and insurances
10.4 Last-survivor annuities and insurances
10.4.2 Reserves on second-death insurances
10.5 Moment of death insurances
10.6 The general two-life annuity contract
10.7 The general two-life insurance contract
10.8 Contingent insurances
10.8.1 First-death contingent insurances
10.8.2 Second-death contingent insurances
10.8.3 Moment-of-death contingent insurances
10.8.4 General contingent probabilities
*10.10 Applications to annuity credit risk
10.11 Standard notation and terminology
10.12 Spreadsheet applications
11 Multiple-decrement theory
11.2.1 The multiple-decrement table
11.2.2 Quantities calculated from the multiple-decrement table
11.4 Determining the model from the forces of decrement
11.5 The analogy with joint-life statuses
11.7 Associated single-decrement tables
11.7.2 Forces of decrement in the associated single-decrement tables
11.7.3 Conditions justifying the two methods
12.3 Realistic reserve and balance calculations
12.4.1 Advanced gain and loss analysis
13.1.1 Description of the contract
13.1.2 Calculating account values
Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL
14 Survival distributions and failure times
14.1 Introduction to survival distributions
14.3.1 The basic functions
14.5 Shifted distributions
14.6 The standard approximation
14.7 The stochastic life table
14.8 Life expectancy in the stochastic model
14.9 Stochastic interest rates
15 The stochastic approach to insurance and annuities
15.2 The stochastic approach to insurance benefits
15.2.2 The continuous case
15.2.4 Endowment insurances
15.3 The stochastic approach to annuity benefits
15.3.1 Discrete annuities
15.3.2 Continuous annuities
15.5 The stochastic approach to reserves
15.6 The stochastic approach to premiums
15.6.1 The equivalence principle
15.6.2 Percentile premiums
15.6.3 Aggregate premiums
15.6.4 General premium principles
15.8 Standard notation and terminology
16 Simplifications under level benefit contracts
16.2 Variance calculations in the continuous case
16.2.3 Prospective losses
16.2.4 Using equivalence principle premiums
16.3 Variance calculations in the discrete case
16.4.1 The distribution of Z
16.4.2 The distribution of Y
16.4.3 The distribution of L
16.4.4 The case where T is exponentially distributed
16.5 Some non-level benefit examples
16.5.2 Deferred insurance
16.5.3 An annual premium policy
17 The minimum failure time
17.3 The distribution of T
17.3.2 The independent case
17.4 The joint distribution of (T, J)
17.4.1 The distribution function for (T, J)
17.4.2 Density and survival functions for (T, J)
17.4.3 The distribution of J
17.4.4 Hazard functions for (T, J)
17.4.5 The independent case
17.4.6 Nonidentifiability
17.4.7 Conditions for the independence of T and J
17.6 The common shock model
Part III ADVANCED STOCHASTIC MODELS
18 An introduction to stochastic processes
18.4 Finite-state Markov chains
18.4.1 The transition matrix
18.4.2 Multi-period transitions
*18.4.4 Limiting distributions
*18.4.5 Recurrent and transient states
18.5 Introduction to continuous time processes
18.6.2 Nonhomogeneous Poisson processes
18.7.1 The main definition
18.7.2 Connection with random walks
*18.7.4 Conditional distributions
18.7.5 Brownian motion with drift
18.7.6 Geometric Brownian motion
19.2 The discrete-time model
19.2.1 Non-stationary Markov Chains
19.2.2 Discrete-time multi-state insurances
19.2.3 Multi-state annuities
19.3 The continuous-time model
19.3.1 Forces of transition
19.3.2 Path-by-path analysis
19.3.3 Numerical approximation
19.3.4 Stationary continuous time processes
19.3.5 Some methods for non-stationary processes
19.3.6 Extension of the common shock model
19.3.7 Insurance and annuity applications in continuous time
19.4 Recursion and differential equations for multi-state reserves
19.5 Profit testing in multi-state models
20 Introduction to the Mathematics of Financial Markets
20.2 Modelling prices in financial markets
20.5 Option prices in the one-period binomial model
20.6 The multi-period binomial model
20.8 A general financial market
20.9 Arbitrage-free condition
20.10 Existence and uniqueness of risk-neutral measures
20.10.1 Linear algebra background
20.10.2 The space of contingent claims
20.10.3 The Fundamental theorem of asset pricing completed
20.11 Completeness of markets
20.12 The Black–Scholes–Merton formula
20.13.2 Extending the notion of conditional expectation
20.13.3 The arbitrage-free condition in the bond market
20.13.4 Short-rate modelling
20.13.5 Forward prices and rates
20.13.6 Observations on the continuous time bond market
21 Compound distributions
21.2 The mean and variance of S
21.3 Generating functions
21.4 Exact distribution of S
21.5 Choosing a frequency distribution
21.6 Choosing a severity distribution
21.7 Handling the point mass at 0
21.8 Counting claims of a particular type
21.8.2 Special classes in the Poisson case
21.9 The sum of two compound Poisson distributions
21.10 Deductibles and other modifications
21.10.1 The nature of a deductible
21.10.2 Some calculations in the discrete case
21.10.3 Some calculations in the continuous case
21.10.4 The effect on aggregate claims
21.10.5 Other modifications
21.11 A recursion formula for S
21.11.1 The positive-valued case
21.11.2 The case with claims of zero amount
22.3 Convex and concave functions: Jensen’s inequality
22.3.2 Jensen’s inequality
22.4 A general comparison method
22.5 Risk measures for capital adequacy
22.5.1 The general notion of a risk measure
22.5.3 Tail value-at-risk
22.5.4 Distortion risk measures
23.2 A functional equation approach
23.3 The martingale approach to ruin theory
23.3.2 The optional stopping theorem and its consequences
23.3.3 The adjustment coefficient
23.3.4 The main conclusions
23.4 Distribution of the deficit at ruin
23.5.1 Calculating ruin probabilities
23.5.2 The distribution of D(u)
23.6 The compound Poisson surplus process
23.6.1 Description of the process
23.6.2 The probability of eventual ruin
23.6.4 The distribution of D(0)
23.6.5 The case when X is exponentially distributed
23.7 The maximal aggregate loss
24.1 Introductory material
24.1.1 The nature of credibility theory
24.1.2 Information assessment
24.2 Conditional expectation and variance with respect to another random variable
24.2.1 The random variable E(X|Y)
24.2.2 Conditional variance
24.3 General framework for Bayesian credibility
24.5.3 Bühlman–Straub Model
24.6 Conditions for exactness
24.7.1 Unbiased estimators
24.7.2 Calculating Var(X) in the credibility model
24.7.3 Estimation of the Bülhman parameters
24.7.4 Estimation in the Bülhman–Straub model
Appendix A review of probability theory
A.1 Sample spaces and probability measures
A.2 Conditioning and independence
A.5 Expectations and moments
A.6 Expectation in terms of the distribution function
A.8 Conditioning and independence for random variables
A.9 Moment generating functions
A.10 Probability generating functions
A.11 Some standard distributions
A.11.1 The binomial distribution
A.11.2 The Poisson distribution
A.11.3 The negative binomial and geometric distributions
A.11.4 The continuous uniform distribution
A.11.5 The normal distribution
A.11.6 The gamma and exponential distributions
A.11.7 The lognormal distribution
A.11.8 The Pareto distribution
A.12.2 The continuous case
A.12.3 Notation and remarks
Fundamentals of Actuarial Mathematics
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