SECOND EDITION. MARY R. HARDY University of Waterloo, Ontario. HOWARD R. WATERS Heriot-Watt University, Edinburgh

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1 ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS SECOND EDITION DAVID C. M. DICKSON University of Melbourne MARY R. HARDY University of Waterloo, Ontario HOWARD R. WATERS Heriot-Watt University, Edinburgh RSI CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface to the second edition page xvii 1 Introduction to life insurance Summary Background Life insurance and annuity contracts Introduction Traditional insurance contracts Modern insurance contracts Distribution methods Underwriting Premiums Life annuities Other insurance contracts Pension benefits Defined benefit and defined contribution Defined benefit pension design Mutual and proprietary insurers Typical problems Notes and further reading Exercises 15 2 Survival models Summary The future lifetime random variable The force of mortality Actuarial notation Mean and standard deviation of T x 28 VÜ

3 viii Contents 2.6 Curtate future lifetime A, and The complete and curtate expected future lifetimes, e x and e x Notes and further reading Exercises 3 Life tables and selection Summary Life tables Fractional age assumptions Uniform distribution of deaths Constant force of mortality National life tables Survival models for life insurance policyholders Life insurance underwriting Select and ultimate survival models Notation and formulae for select survival models Select life tables Some comments on heterogeneity in mortality Mortality trends Notes and further reading Exercises 70 4 Insurance benefits Summary Introduction Assumptions Valuation of insurance benefits Whole life insurance: the continuous case, A x Whole life insurance: the annual case, A x Whole life insurance: the 1 /mthly case, A 1 "'' Recursions Term insurance Pure endowment Endowment insurance Deferred insurance benefits Relating Ä x, A x and A n) x Using the uniform distribution of deaths assumption Using the claims acceleration approach Variable insurance benefits 9%

4 Contents ix 4.7 Functions for select lives Notes and further reading Exercises Annuities Summary Introduction Review of annuities-certain Annual life annuities Whole life annuity-due Term annuity-due Whole life immediate annuity Term immediate annuity Annuities payable continuously Whole life continuous annuity Term continuous annuity Annuities payable 1/rathly Introduction Whole life annuities payable 1/mthly Term annuities payable 1/mthly Comparison of annuities by payment frequency Deferred annuities Guaranteed annuities Increasing annuities Arithmetically increasing annuities Geometrically increasing annuities Evaluating annuity functions Recursions Applying the UDD assumption Woolhouse's formula Numerical illustrations Functions for select lives Notes and further reading Exercises Premium calculation Summary Preliminaries Assumptions The present value of future loss random variable The equivalence principle Net premiums 147

5 X Contents 6.6 Gross premiums 6.7 Profit 6.8 The portfolio percentile premium principle Extra risks Age rating Constant addition to (i x Constant multiple of mortality rates Notes and further reading Exercises ' 7 ' 7 Policy values ' 7 * 7.1 Summary ' 7^ 7.2 Assumptions Policies with annual cash flows The future loss random variable Policy values for policies with annual cash flows Recursive formulae for policy values Annual profit by source Asset shares Policy values for policies with cash flows at 1 /mthly intervals Recursions Valuation between premium dates Policy values with continuous cash flows Thiele's differential equation Numerical solution of Thiele's differential equation Policy alterations Retrospective policy values Prospective and retrospective valuation Defining the retrospective net premium policy value Negative policy values Deferred acquisition expenses and modified premium reserves Notes and further reading 23 I 7.11 Exercises Multiple state models Summary Examples of multiple state models The alive-dead model 243

6 Contents xi Term insurance with increased benefit on accidental death The permanent disability model The disability income insurance model Assumptions and notation Formulae for probabilities Kolmogorov's forward equations Numerical evaluation of probabilities Premiums Policy values and Thiele's differential equation The disability income insurance model Thiele's differential equation - the general case Multiple decrement models Multiple decrement tables Fractional age assumptions for decrements Constructing a multiple decrement table Deriving independent rates from dependent rates Deriving dependent rates from independent rates Comments on multiple decrement notation Transitions at exact ages Markov multiple state models in discrete time The Chapman-Kolmogorov equations Transition matrices Notes and further reading Exercises Joint life and last survivor benefits Summary Joint life and last survivor benefits Joint life notation Independent future lifetimes A multiple state model for independent future lifetimes A model with dependent future lifetimes The common shock model Notes and further reading Exercises Pension mathematics Summary Introduction The salary scale function 335

7 xii Contents 10.4 Setting the DC contribution 10.5 The service table Valuation of benefits Final salary plans Career average earnings plans Funding the benefits Notes and further reading Exercises Yield curves and non-diversifiable risk Summary The yield curve Valuation of insurances and life annuities Replicating the cash flows of a traditional non-participating product Diversifiable and non-diversifiable risk Diversifiable mortality risk Non-diversifiable risk Monte Carlo simulation Notes and further reading Exercises Emerging costs for traditional life insurance Summary Introduction Profit testing a term insurance policy Time step Profit test basis Incorporating reserves Profit signature Profit testing principles Assumptions The profit vector The profit signature The net present value Notes on the profit testing method Profit measures Using the profit test to calculate the premium Using the profit test to calculate reserves Profit testing for multiple state models Notes Exercises 423

8 Contents xiii 13 Participating and Universal Life insurance Summary Introduction Participating insurance Introduction Examples Notes on profit distribution methods Universal Life insurance Introduction Key design features Projecting account values Profit testing Universal Life policies Universal Life Type B Universal Life Type A No-lapse guarantees Comments on UL profit testing Comparison of UL and whole life insurance policies Notes and further reading Exercises Emerging costs for equity-linked insurance Summary Equity-linked insurance Deterministic profit testing for equity-linked insurance Stochastic profit testing Stochastic pricing Stochastic reserving Reserving for policies with non-diversifiable risk Quantile reserving CTE reserving Comments on reserving Notes and further reading Exercises Option pricing Summary Introduction The 'no-arbitrage' assumption Options The binomial option pricing model Assumptions Pricing over a single time period 507

9 xiv CoMfeMü Pricing over two time periods Summary of the binomial model option pricing technique 15.6 The Black-Scholes-Merton model The model The Black-Scholes-Merton option pricing formula 15.7 Notes and further reading 15.8 Exercises 16 Embedded options 16.1 Summary 16.2 Introduction 16.3 Guaranteed minimum maturity benefit Pricing Reserving 16.4 Guaranteed minimum death benefit Pricing Reserving 16.5 Pricing methods for embedded options 16.6 Risk management 16.7 Emerging costs 16.8 Notes and further reading 16.9 Exercises A B Probability theory A.l Probability distributions A. 1.1 Binomial distribution A.1.2 Uniform distribution A. 1.3 Normal distribution A. 1.4 Lognormal distribution A.2 The central limit theorem A.3 Functions of a random variable A.3.1 Discrete random variables A.3.2 Continuous random variables A.3.3 Mixed random variables A.4 Conditional expectation and conditional variance A.5 Notes and further reading Numerical techniques B.l Numerical integration B.l.l The trapezium rule B.1.2 Repeated Simpson's rule »

10 Contents XV B. 1.3 Integrals over an infinite interval 576 B.2 Woolhouse's formula 577 B.3 Notes and further reading 578 C Simulation 579 C.l The inverse transform method 579 C.2 Simulation from a normal distribution 580 C.2.1 The Box-Muller method 580 C.2.2 The polar method 581 C.3 Notes and further reading 581 D Tables 582 References 589 Index 592

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