ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS

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

2 Contents Preface Introduction to life insurance 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 Survival models The future lifetime random variable 2.3 The force of mortality 2.4 Actuarial notation 2.5 Mean and standard deviation of T x 2.6 Curtate future lifetime K r and e r page xiv pensions Vll

3 viii Contents Lifei The complete and curtate expected future lifetimes, e x and e x tables and selection 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 Insurance benefits 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/rathly case, A^ Recursions Term insurance Pure endowment Endowment insurance Deferred insurance benefits Relating A x, A x and A x m) Using the uniform distribution of deaths assumption Using the claims acceleration approach Variable insurance benefits Functions for select lives Annuities Introduction

4 Contents 5.3 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 m times per year Introduction " Life annuities payable m times a year Term annuities payable m times a year 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 Premium calculation Preliminaries Assumptions The present value of future loss random variable The equivalence principle Net premiums Gross premium calculation Profit The portfolio percentile premium principle Extra risks Age rating Constant addition to \x x Constant multiple of mortality rates 167 ix

5 Contents Policy values 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, Asset shares ' Policy values for policies with cash flows at discrete intervals other than annually 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 value Negative policy values Multiple state models Examples of multiple state models The alive-dead model Term insurance with increased benefit on accidental death The permanent disability model The disability income insurance model The joint life and last survivor 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 model Thiele's differential equation - the general case 255

6 Contents Multiple decrement models Joint life and last survivor benefits The model and assumptions Joint life and last survivor probabilities Joint life and last survivor annuity and insurance functions An important special case: independent survival models Transitions at specified ages xi Pension mathematics , Introduction The salary scale function Setting the DC contribution The service table Valuation of benefits Final salary plans Career average earnings plans Funding plans Interest rate risk 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 Emerging costs for traditional life insurance Profit testing for traditional life insurance The net cash flows for a policy Reserves Profit measures A further example of a profit test

7 xii Contents Emerging costs for equity-linked insurance Equity-linked insurance 12.3 Deterministic profit testing for equity-linked insurance 12.4 Stochastic profit testing 12.5 Stochastic pricing 12.6 Stochastic reserving Reserving for policies wjth non-diversifiable risk Quantile reserving CTE reserving Comments on reserving Option pricing Introduction The 'no arbitrage' assumption Options The binomial option pricing model Assumptions Pricing over a single time period Pricing over two time periods of the binomial model option pricing technique The Black-Scholes-Merton model The model The Black-Scholes-Merton option pricing formula Embedded options Introduction Guaranteed minimum maturity benefit Pricing Reserving Guaranteed minimum death benefit Pricing Reserving

8 Contents 14.5 Pricing methods for embedded options Risk management Emerging costs A Probability theory 464 A. 1 Probability distributions 464 A. 1.1 Binomial distribution 464 A. 1.2 Uniform distribution 464 A. 1.3 Normal distribution v 465 A. 1.4 Lognormal distribution 466 A.2 The central limit theorem 469 A.3 Functions of a random variable 469 A.3.1 Discrete random variables. 470 A.3.2 Continuous random variables 470 A.3.3 Mixed random variables 471 A.4 Conditional expectation and conditional variance 472 A B Numerical techniques 474 B.I Numerical integration 474 B. 1.1 The trapezium rule 474 B.I.2 Repeated Simpson's rule 476 B.1.3 Integrals over an infinite interval 477 B.2 Woolhouse's formula 478 B C Simulation 480 C.I The inverse transform method 480 C.2 Simulation from a normal distribution 481 C.2.1 The Box-Muller method, 482 C.2.2 The polar method ' 482 C References 483 Author index 487 Index 488 xiii

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