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Part IV RISK THEORY
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Part IV RISK THEORY
by S. David Promislow
Fundamentals of Actuarial Mathematics, 3rd Edition
Preface
Acknowledgements
About the companion website
Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL
1 Introduction and motivation
1.1 Risk and insurance
1.2 Deterministic versus stochastic models
1.3 Finance and investments
1.4 Adequacy and equity
1.5 Reassessment
1.6 Conclusion
2 The basic deterministic model
2.1 Cash flows
2.2 An analogy with currencies
2.3 Discount functions
2.4 Calculating the discount function
2.5 Interest and discount rates
2.6 Constant interest
2.7 Values and actuarial equivalence
2.8 Vector notation
2.9 Regular pattern cash flows
2.10 Balances and reserves
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
2.15 Spreadsheet calculations
Notes and references
Exercises
3 The life table
3.1 Basic definitions
3.2 Probabilities
3.3 Constructing the life table from the values of qx
3.4 Life expectancy
3.5 Choice of life tables
3.6 Standard notation and terminology
3.7 A sample table
Notes and references
Exercises
4 Life annuities
4.1 Introduction
4.2 Calculating annuity premiums
4.3 The interest and survivorship discount function
4.4 Guaranteed payments
4.5 Deferred annuities with annual premiums
4.6 Some practical considerations
4.7 Standard notation and terminology
4.8 Spreadsheet calculations
Exercises
5 Life insurance
5.1 Introduction
5.2 Calculating life insurance premiums
5.3 Types of life insurance
5.4 Combined insurance–annuity benefits
5.5 Insurances viewed as annuities
5.6 Summary of formulas
5.7 A general insurance–annuity identity
5.8 Standard notation and terminology
5.9 Spreadsheet applications
Exercises
6 Insurance and annuity reserves
6.1 Introduction to reserves
6.2 The general pattern of reserves
6.3 Recursion
6.4 Detailed analysis of an insurance or annuity contract
6.5 Bases for reserves
6.6 Nonforfeiture values
6.7 Policies involving a return of the reserve
6.8 Premium difference and paid-up formulas
6.9 Standard notation and terminology
6.10 Spreadsheet applications
Exercises
7 Fractional durations
7.1 Introduction
7.2 Cash flows discounted with interest only
7.3 Life annuities paid mthly
7.4 Immediate annuities
7.5 Approximation and computation
*7.6 Fractional period premiums and reserves
7.7 Reserves at fractional durations
7.8 Standard notation and terminology
Exercises
8 Continuous payments
8.1 Introduction to continuous annuities
8.2 The force of discount
8.3 The constant interest case
8.4 Continuous life annuities
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
8.11 Further approximations from the life table
8.12 Standard actuarial notation and terminology
Notes and references
Exercises
9 Select mortality
9.1 Introduction
9.2 Select and ultimate tables
9.3 Changes in formulas
9.4 Projections in annuity tables
9.5 Further remarks
Exercises
10 Multiple-life contracts
10.1 Introduction
10.2 The joint-life status
10.3 Joint-life annuities and insurances
10.4 Last-survivor annuities and 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
10.9 Duration problems
*10.10 Applications to annuity credit risk
10.11 Standard notation and terminology
10.12 Spreadsheet applications
Notes and references
Exercises
11 Multiple-decrement theory
11.1 Introduction
11.2 The basic model
11.3 Insurances
11.4 Determining the model from the forces of decrement
11.5 The analogy with joint-life statuses
11.6 A machine analogy
11.7 Associated single-decrement tables
Notes and references
Exercises
12 Expenses and profits
12.1 Introduction
12.2 Effect on reserves
12.3 Realistic reserve and balance calculations
12.4 Profit measurement
Notes and references
Exercises
*13 Specialized topics
13.1 Universal life
13.2 Variable annuities
13.3 Pension plans
Exercises
Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL
14 Survival distributions and failure times
14.1 Introduction to survival distributions
14.2 The discrete case
14.3 The continuous case
14.4 Examples
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
Notes and references
Exercises
15 The stochastic approach to insurance and annuities
15.1 Introduction
15.2 The stochastic approach to insurance benefits
15.3 The stochastic approach to annuity benefits
*15.4 Deferred contracts
15.5 The stochastic approach to reserves
15.6 The stochastic approach to premiums
15.7 The variance of r  L
15.8 Standard notation and terminology
Notes and references
Exercises
16 Simplifications under level benefit contracts
16.1 Introduction
16.2 Variance calculations in the continuous case
16.3 Variance calculations in the discrete case
16.4 Exact distributions
16.5 Some non-level benefit examples
Exercises
17 The minimum failure time
17.1 Introduction
17.2 Joint distributions
17.3 The distribution of T
17.4 The joint distribution of (T, J)
17.5 Other problems
17.6 The common shock model
17.7 Copulas
Notes and references
Exercises
Part III ADVANCED STOCHASTIC MODELS
18 An introduction to stochastic processes
18.1 Introduction
18.2 Markov chains
18.3 Martingales
18.4 Finite-state Markov chains
18.5 Introduction to continuous time processes
18.6 Poisson processes
18.7 Brownian motion
Notes and references
Exercises
19 Multi-state models
19.1 Introduction
19.2 The discrete-time model
19.3 The continuous-time model
19.4 Recursion and differential equations for multi-state reserves
19.5 Profit testing in multi-state models
19.6 Semi-Markov models
Notes and references
Exercises
20 Introduction to the Mathematics of Financial Markets
20.1 Introduction
20.2 Modelling prices in financial markets
20.3 Arbitrage
20.4 Option contracts
20.5 Option prices in the one-period binomial model
20.6 The multi-period binomial model
20.7 American options
20.8 A general financial market
20.9 Arbitrage-free condition
20.10 Existence and uniqueness of risk-neutral measures
20.11 Completeness of markets
20.12 The Black–Scholes–Merton formula
20.13 Bond markets
Notes and references
Exercises
Part IV RISK THEORY
21 Compound distributions
21.1 Introduction
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
21.9 The sum of two compound Poisson distributions
21.10 Deductibles and other modifications
21.11 A recursion formula for S
Notes and references
Exercises
22 Risk assessment
22.1 Introduction
22.2 Utility theory
22.3 Convex and concave functions: Jensen’s inequality
22.4 A general comparison method
22.5 Risk measures for capital adequacy
Notes and references
Exercises
23 Ruin models:
23.1 Introduction
23.2 A functional equation approach
23.3 The martingale approach to ruin theory
23.4 Distribution of the deficit at ruin
23.5 Recursion formulas
23.6 The compound Poisson surplus process
23.7 The maximal aggregate loss
Notes and references
Exercises
24 Credibility theory:
24.1 Introductory material
24.2 Conditional expectation and variance with respect to another random variable
24.3 General framework for Bayesian credibility
24.4 Classical examples
24.5 Approximations
24.6 Conditions for exactness
24.7 Estimation
Notes and References
Exercises
Answers to exercises
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Appendix A review of probability theory
A.1 Sample spaces and probability measures
A.2 Conditioning and independence
A.3 Random variables
A.4 Distributions
A.5 Expectations and moments
A.6 Expectation in terms of the distribution function
A.7 Joint distributions
A.8 Conditioning and independence for random variables
A.9 Moment generating functions
A.10 Probability generating functions
A.11 Some standard distributions
A.12 Convolution
A.13 Mixtures
References
Notation index
Index
End User License Agreement
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20 Introduction to the Mathematics of Financial Markets
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21 Compound distributions
Part IV
RISK THEORY
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