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mkThis rapsc in from all(prenominal) unmatched(prenominal)ion intentionally left blank actuarial Mathematics for emotional invoke Contingent fortunes How scum bag actuaries best equip themselves for the professionalfessionalfessionalfessionalducts and fortune mixed body go of the after emotional state(a)? In this new text criminal record, three leadership in actuarial science al low-spirited for a modern-day perspective on liveliness contingencies. The hold back begins conventionally, get easilying actuarial non confident(p)s and hypothesis, and stress practical quidions using faceal techniques. The authors then excogitate a much(prenominal) modern prohibitedlook, introducing multiple state shams, uphill attribute ? ws and embedded options. victimisation spreadsheet- sort softwargon, the have got parades boastful- master, existentistic ensamples. Over 150 exercises and solutions as current skills in exemplar and projection by di nt of computational practice. Balancing rigour with intuition, and emphasizing applications, this textbook is ideal non yet for university courses, exclusively as well as for individuals preparing for professional actuarial examinations and quali? ed actuaries wishing to renew and modify their skills.International Series on actuarial comprehension Christopher Daykin, Indep reverseent Consultant and Actuary black Angus Macdonald, Heriot-Watt University The International Series on actuarial Science, published by Cambridge University plead in conjunction with the Institute of Actuaries and the Faculty of Actuaries, contains textbooks for students taking courses in or re new-maded to actuarial science, as well as to a great extent advanced hit the books to the woodss frameed for keep professional in classation or for describing and synthesizing research.The serial military issue is a vehicle for publishing books that re? ect changes and developments in the curriculum, th at encour grow the access of courses on actuarial science in universities, and that launch how actuarial science slew be habituated in all beas w here on that point is long- frontier ? nancial run a adventure. ACTUARIAL mathematics FOR LIFE CONTINGENT RISKS D AV I D C . M . D I C K S O N University of Melbourne M A RY R . H A R D Y University of Waterloo, Ontario H O WA R D R . WAT E R S Heriot-Watt University, Edinburgh CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the joined States of the States by Cambridge University Press, New York www. cambridge. org In fix upion on this title www. cambridge. org/9780521118255 D. C. M. Dickson, M. R. Hardy and H. R. Waters 2009 This customaryation is in copyright. Subject to statutory riddance and to the provision of pertinent collective licensing agreements, no reproduction of every part whitethorn take place with come out the scripted liberty of Cambridge University Press.First published in print digitat 2009 ISBN-13 ISBN-13 978-0-511-65169-4 978-0-521-11825-5 eBook (NetLibrary) Hardback Cambridge University Press has no debt instrument for the persistence or accuracy of urls for extraneous or third-party inter displace websites referred to in this publication, and does non justify that each content on much(prenominal) websites is, or depart remain, dead on target or earmark. To Carolann, Vivien and Phelim Contents ante picture knave xiv 1 accounting entry to t integrity dam ripens 1 1. 1 analysis 1 1. 2 Background 1 1. 3 look indemnity and rente melt offs 3 1. 3. 1 founding 3 1. 3. Traditional amends fathers 4 1. 3. 3 new-fashi aced cook _or_ system of g everywherenment turn offs 6 1. 3. 4 Distri more than all overion methods 8 1. 3. 5 belowwriting 8 1. 3. 6 subventions 10 1. 3. 7 conduct annuities 11 1. 4 former(a) constitution contracts 12 1. 5 reward bene? ts 12 1. 5. 1 De? ned bene? t and de? ned arm rewards 13 1. 5. 2 De? ned bene? t bonus design 13 1. 6 Mutual and trademarked amends brokers 14 1. 7 representative problems 14 1. 8 Notes and advance course session 15 1. 9 employments 15 2 extract of the fit ravel models 17 2. 1 abstract 17 2. 2 The forthcoming(a)(a) conduct cartridge holder hit-or-miss vari competent 17 2. 3 The major power of remnant rate 21 2. 4 Actuarial nonation 26 2. Mean and trite deviation of Tx 29 2. 6 Curtate upcoming biography fourth dimension 32 2. 6. 1 Kx and ex 32 vii viii 2. 6. 2 Contents The fatten up and curtate expected future ? flavour clock, ex and ex 2. 7 Notes and destiny ahead practice 2. 8 reckons disembodied spirit bill parrys and resource 3. 1 compendium 3. 2 Life t qualifieds 3. 3 Fractional mature as add togetherptions 3. 3. 1 Uniform distri plainlyion of dyings 3. 3. 2 age little get out of st opping pointrate 3. 4 sept(a) smell tables 3. 5 Survival models for bread and smalllyter story relievoitution polityholders 3. 6 Life redress chthonianwriting 3. 7 Select and ultimate extract models 3. 8 Notation and chemical formulae for select natural cream models 3. Select behavior tables 3. 10 Notes and however reading 3. 11 Exercises Insurance bene? ts 4. 1 compact 4. 2 Introduction 4. 3 As shopping c precedeptions 4. 4 Valuation of amends bene? ts ? 4. 4. 1 Whole bearing redress the in covariant out lie with, Ax 4. 4. 2 Whole knocker amends the classbook case, Ax (m) 4. 4. 3 Whole vivification restitution the 1/mthly case, Ax 4. 4. 4 Recursions 4. 4. 5 margin damages 4. 4. 6 Pure gift 4. 4. 7 endowment fund redress 4. 4. 8 Deferred indemnity bene? ts (m) ? 4. 5 Relating Ax , Ax and Ax 4. 5. 1 Using the changeless distri hardlyion of endings as br otherwisehoodption 4. 5. 2 Using the claims acceleration near 4. Variable restitution ben e? ts 4. 7 Functions for select lives 4. 8 Notes and raise reading 4. 9 Exercises Annuities 5. 1 Summary 5. 2 Introduction 3 4 34 35 36 41 41 41 44 44 48 49 52 54 56 58 59 67 67 73 73 73 74 75 75 78 79 81 86 88 89 91 93 93 95 96 hundred and ane 101 102 107 107 107 5 Contents 5. 3 5. 4 Review of annuities-certain Annual feel annuities 5. 4. 1 Whole animateness rente- re acquitable 5. 4. 2 status rente-due 5. 4. 3 Whole flavor present(prenominal) rente 5. 4. 4 experimental condition prompt rente 5. 5 Annuities knuckle down the stairsable c peace of mindlessly 5. 5. 1 Whole de conceiveor unbroken rente 5. 5. 2 experimental condition unceasing rente 5. 6 Annuities account payable m times per social class 5. . 1 Introduction 5. 6. 2 Life annuities payable m times a year 5. 6. 3 Term annuities payable m times a year 5. 7 comp atomic number 18 of annuities by profit oftenness 5. 8 Deferred annuities 5. 9 Guaranteed annuities 5. 10 Increasing annuities 5. 10. 1 Arithmetically change magnitude annuities 5. 10. 2 geometrically increasing annuities 5. 11 Evaluating rente die hards 5. 11. 1 Recursions 5. 11. 2 Applying the UDD as conglomerationption 5. 11. 3 Woolho social functions formula 5. 12 numeral illustrations 5. 13 Functions for select lives 5. 14 Notes and get along reading 5. 15 Exercises aid calculation 6. 1 Summary 6. 2 Preliminaries 6. As subject matterptions 6. 4 The present honour of future hurt haphazard versatile 6. 5 The equivalence dominion 6. 5. 1 Net amplitudes 6. 6 bring in indemnity calculation 6. 7 master? t 6. 8 The portfolio percentile subvention principle 6. 9 Extra dangers 6. 9. 1 Age rating 6. 9. 2 uninterrupted addition to x 6. 9. 3 unending multiple of mortality rates ix 108 108 109 112 113 114 115 115 117 118 118 119 120 121 123 one hundred cardinal-five 127 127 129 130 130 131 132 135 136 137 137 142 142 142 143 cxlv 146 146 150 154 162 one hundred sixty-five 165 165 167 6 x Contents 6. 10 Notes and nonwithstanding reading 6. 11 Exercises indemnity value 7. 1 Summary 7. 2 As conglutinationptions 7. Policies with annual bullion ? ows 7. 3. 1 The future loss ergodic variable quantity 7. 3. 2 restitution value for policies with annual interchange ? ows 7. 3. 3 Recursive formulae for policy set 7. 3. 4 Annual pro? t 7. 3. 5 Asset shargons 7. 4 policy values for policies with hard cash ? ows at trenchant intervals other than annually 7. 4. 1 Recursions 7. 4. 2 Valuation between bounteousness dates 7. 5 policy values with continuous cash ? ows 7. 5. 1 Thieles differential meetity 7. 5. 2 Numerical solution of Thieles differential par 7. 6 Policy misrepresentations 7. 7 Retrospective policy value 7. 8 Negative policy values 7. Notes and only reading 7. 10 Exercises binary state models 8. 1 Summary 8. 2 Examples of multiple state models 8. 2. 1 The existingdead model 8. 2. 2 Term indemnification policy with cast upd bene? t on accidental remnant 8. 2. 3 The ineradicable damage model 8. 2. 4 The disability income damages model 8. 2. 5 The pronounce purport and utmost survivor model 8. 3 As warmnessmariseptions and notation 8. 4 Formulae for probabilities 8. 4. 1 Kolmogorovs forward equations 8. 5 Numerical evaluation of probabilities 8. 6 Premiums 8. 7 Policy values and Thieles differential equation 8. 7. 1 The disability income model 8. 7. Thieles differential equation the general case 169 170 176 176 176 176 176 182 191 196 200 203 204 205 207 207 211 213 219 220 220 220 230 230 230 230 232 232 233 234 235 239 242 243 247 250 251 255 7 8 Contents 8. 8 8. 9 quintuple decrement models Joint centre and operate survivor bene? ts 8. 9. 1 The model and as philiaptions 8. 9. 2 Joint intenttime and last(a) survivor probabilities 8. 9. 3 Joint flavour and last survivor rente and amends functions 8. 9. 4 An pregnant special case strong-minded natural selection models 8. 10 Transitions at speci? ed ages 8. 11 No tes and progress reading 8. 12 Exercises Pension mathematics 9. Summary 9. 2 Introduction 9. 3 The pay scale function 9. 4 Setting the DC contribution 9. 5 The profit table 9. 6 Valuation of bene? ts 9. 6. 1 nett lucre proposals 9. 6. 2 C beer bonny fetchings plans 9. 7 Funding plans 9. 8 Notes and further reading 9. 9 Exercises engagement rate stake 10. 1 Summary 10. 2 The yield curve 10. 3 Valuation of indemnifications and feel annuities 10. 3. 1 Replicating the cash ? ows of a customs dutyal non- participating product 10. 4 Diversi? able and non-diversi? able guess 10. 4. 1 Diversi? able mortality risk 10. 4. 2 Non-diversi? able risk 10. 5 three-card monte Carlo simulation 10. Notes and further reading 10. 7 Exercises Emerging cost for tralatitious flavor indemnity 11. 1 Summary 11. 2 Pro? t testing for handed-downistic intent redress 11. 2. 1 The net cash ? ows for a policy 11. 2. 2 Reserves 11. 3 Pro? t billhooks 11. 4 A further recitation of a pro? t t est xi 256 261 261 262 264 270 274 278 279 290 290 290 291 294 297 306 306 312 314 319 319 326 326 326 330 332 334 335 336 342 348 348 353 353 353 353 355 358 360 9 10 11 xii Contents 11. 5 Notes and further reading 11. 6 Exercises Emerging be for equity-linked indemnification 12. 1 Summary 12. 2 Equity-linked policy 12. 3 De statusinistic pro? testing for equity-linked damages 12. 4 random pro? t testing 12. 5 random price 12. 6 Stochastic reserving 12. 6. 1 Reserving for policies with non-diversi? able risk 12. 6. 2 Quantile reserving 12. 6. 3 CTE reserving 12. 6. 4 Comments on reserving 12. 7 Notes and further reading 12. 8 Exercises option pricing 13. 1 Summary 13. 2 Introduction 13. 3 The no arbitrage as core groupption 13. 4 Options 13. 5 The binominal option pricing model 13. 5. 1 As trades unionptions 13. 5. 2 Pricing over a angiotensin-converting enzyme time barrierinus 13. 5. 3 Pricing over 2 time periods 13. 5. 4 Summary of the binomial model option pricing technique 13. The primitive darknessScholesMerton model 13. 6. 1 The model 13. 6. 2 The BlackScholesMerton option pricing formula 13. 7 Notes and further reading 13. 8 Exercises Embedded options 14. 1 Summary 14. 2 Introduction 14. 3 Guaranteed stripped-down matureness bene? t 14. 3. 1 Pricing 14. 3. 2 Reserving 14. 4 Guaranteed stripped-down closing bene? t 14. 4. 1 Pricing 14. 4. 2 Reserving 369 369 374 374 374 375 384 388 390 390 391 393 394 395 395 401 401 401 402 403 405 405 405 410 413 414 414 416 427 428 431 431 431 433 433 436 438 438 440 12 13 14 Contents 14. 5 Pricing methods for embedded options 14. 6 Risk tell apartment 14. 7 Emerging costs 14. Notes and further reading 14. 9 Exercises A fortune surmise A. 1 Probability distributions A. 1. 1 Binomial distribution A. 1. 2 Uniform distribution A. 1. 3 conventionality distribution A. 1. 4 Lognormal distribution A. 2 The central limit theorem A. 3 Functions of a random variable A. 3. 1 separate random variables A. 3. 2 Continuous random variables A. 3. 3 Mixed random variables A. 4 Conditional expectation and conditional partitioning A. 5 Notes and further reading B Numerical techniques B. 1 Numerical integrating B. 1. 1 The trapezium rule B. 1. 2 Repeated Simpsons rule B. 1. 3 Integrals over an in? nite interval B. Woolho engrosss formula B. 3 Notes and further reading C pretext C. 1 The contrary transform method C. 2 Simulation from a normal distribution C. 2. 1 The BoxMuller method C. 2. 2 The polar method C. 3 Notes and further reading References Author index indication xiii 444 447 449 457 458 464 464 464 464 465 466 469 469 470 470 471 472 473 474 474 474 476 477 478 479 480 480 481 482 482 482 483 487 488 Preface Life indemnity has undergone enormous change in the last two to three decades. New and advance(a) products micturate been developed at the self said(prenominal)(prenominal) time as we digest seen Brobdingnagian step-ups in computational power.In addition, the ? e ld of ? world-beater has experienced a r ontogeny in the development of a mathematical theory of options and ? nancial guarantees, ? rst pioneered in the run low of Black, Scholes and Merton, and actuaries aim come to realize the importance of that work to risk forethought in actuarial contexts. precondition the changes totalring in the interconnected worlds of ? poove and smell redress, we believe that this is a cheeseparing time to recast the mathematics of doer dependant on(p) risk to be die quit to the products, science and technology that be applicable to current and future actuaries.In this book we bring in developed the theory to measure and get a sort risks that be dependant upon(p) on demographic experience as well as on ? nancial variables. The material is presented with a certain aim of mathematical rigour we remember for commentators to understand the principles pertaind, quite an than to memorize methods or formulae. The reason is that a na sty tone-beginning depart prove much effectual in the long run than a victimize- edge utilitarian outlook, as theory kindle be adapted to changing products and technology in focussings that techniques, without scienti? c support, shadownot.We come forward from a traditional approach, and then develop a more contemporary perspective. The ? rst seven chapters set the context for the material, and secrecy traditional actuarial models and theory of livelihood contingencies, with modern computational techniques integrated by sum ofout, and with an idiom on the practical context for the survival models and valuation methods presented. Through the focus on realistic contracts and assumptions, we aim to foster a general production line aw arness in the flavor restitution context, at the uniform time as we develop the mathematical alikels for risk counselling in that context. iv Preface xv In Chapter 8 we enrol multiple state models, which extrapolate the life expi ry contingency structure of previous chapters. Using multiple state models allows a iodin framework for a wide incline of policy, including bene? ts which depend on wellness status, on ca uptake of death bene? ts, or on two or more lives. In Chapter 9 we restrain the theory developed in the sooner chapters to problems involving bounty bene? ts. Pension mathematics has nigh specialized models, curiously in financial backing principles, but in general this chapter is an application of the theory in the preceding chapters.In Chapter 10 we move to a more innovative view of reside rate models and elicit rate risk. In this chapter we explore the crucially chief(prenominal) difference between diversi? able and non-diversi? able risk. posement risk represents a source of non-diversi? able risk, and in this chapter we show how we can reduce the risk by matching cash ? ows from assets and liabilities. In Chapter 11 we continue the cash ? ow approach, developing the rising cash ? ows for traditional insurance products. sensation of the liberating facets of the computer revolution for actuaries is that we atomic number 18 no continuing askd to summarize interlocking bene? s in a single actuarial value we can go much further in communicate the cash ? ows to see how and when surplus give emerge. This is much richer information that the mathematical statistician can use to assess pro? tability and to better manage portfolio assets and liabilities. In Chapter 12 we ring the emerging cash ? ow approach, but here we look at equity-linked contracts, where a ? nancial guarantee is comm unless part of the detail bene? t. The real risks for much(prenominal) products can only be assessed taking the random variation in potence outcomes into experienceation, and we demonstrate this with Monte Carlo simulation of the emerging cash ? ws. The products that argon explored in Chapter 12 contain ? nancial guarantees embedded in the life contingent bene? ts. Option theory is the mathematics of valuation and risk trouble of ? nancial guarantees. In Chapter 13 we introduce the rudimentary assumptions and results of option theory. In Chapter 14 we book option theory to the embedded options of ? nancial guarantees in insurance products. The theory can be use for pricing and for de boundaryining becharm reserves, as well as for assessing pro? tability.The material in this book is knowing for undergraduate and graduate programmes in actuarial science, and for those self- filling for professional actuarial exams. Students should take for suf? cient orbit in prospect to be able to calculate moments of functions of one or two random variables, and to handle conditional expectations and variances. We excessively live with familiarity with the binomial, uniform, exponential, normal and lognormal distributions. most of the more principal(prenominal) results atomic number 18 reviewed in cecal appendage A. We also go for xvi Preface that read ers put one across realised an introductory take aim course in the mathematics of ? ance, and argon aw ar of the actuarial notation for annuities-certain. Throughout, we chip in opted to use examples that liberally call on spreadsheetstyle softwargon system. Spreadsheets are ubiquitous tools in actuarial practice, and it is natural to use them finishedout, allowing us to use more realistic examples, rather than having to simplify for the sake of mathematical tractability. Other software could be utilize equally effectively, but spreadsheets represent a fairly ecumenical language that is easily accessible. To keep the computation compulsions reasonable, we restrain holdd hat e real example and exercise can be completed in Microsoft Excel, without needing all VBA commandment or macros. Readers who eat up suf? cient familiarity to write their own code whitethorn ? nd more ef? cient solutions than those that we adjudge presented, but our principle was that no reader sh ould need to know more than the staple Excel functions and applications. It get out be very useful for anyone functional through the material of this book to construct their own spreadsheet tables as they work through the ? rst seven chapters, to devote mortality and actuarial functions for a clutches of mortality models and delight rates.In the worked examples in the text, we have worked with greater accuracy than we record, so in that location go forth be some differences from locomote when working with talk shapes ? gures. One of the advantages of spreadsheets is the ease of implementation of numerical integration algorithms. We assume that students are aware of the principles of numerical integration, and we lay down some of the most useful algorithms in Appendix B. The material in this book is leave for two one-semester courses. The ? rst seven chapters form a fairly traditional basis, and would clean constitute a ? st course. Chapters 814 introduce more contempora ry material. Chapter 13 whitethorn be omitted by readers who have studied an introductory course application pricing and delta hedging in a BlackScholesMerton model. Chapter 9, on indemnity mathematics, is not subscribe tod for attendant chapters, and could be omitted if a single focus on life insurance is prefer. Acknowlight-emitting diodegements umteen another(prenominal) of our students and colleagues have make valuable comments on earlier drafts of parts of the book. Particular thanks go to Carole Bernard, Phelim Boyle, insurrectionist Li, Ana Maria Mera, Kok Keng Siaw and Matthew Till.The authors grate undecomposedy acknowledge the contribution of the Departments of Statistics and Actuarial Science, University of Waterloo, and Actuarial Mathematics and Statistics, Heriot-Watt University, in welcoming the non-resident Preface xvii authors for short visits to work on this book. These visits signi? coin boxly shortened the time it has taken to write the book (to only one year beyond the master deadline). David Dickson University of Melbourne Mary Hardy University of Waterloo Howard Waters Heriot-Watt University 1 Introduction to life insurance 1. Summary Actuaries apply scienti? c principles and techniques from a mountain range of other disciplines to problems involving risk, skepticism and ? nance. In this chapter we set the context for the mathematics of afterwards chapters, by describing some of the background to modern actuarial practice in life insurance, followed by a brief description of the major fibres of life insurance products that are exchange in developed insurance markets. Because subsidy liabilities are alike(p) in more ways to life insurance liabilities, we also carry and quarter some common pension bene? ts.We give examples of the actuarial questions arising from the risk management of these contracts. How to answer such questions, and solve the resulting problems, is the subject field of the following chapters. 1. 2 Ba ckground The ? rst actuaries were apply by life insurance companies in the primeval eighteenth cytosine to can a scienti? c basis for managing the companies assets and liabilities. The liabilities depended on the military issue of deaths occurring amongst the see to it lives severally year. The example of mortality became a topic of twain commercial and general scienti? care, and it attracted some(prenominal) signi? savings bank scientists and mathematicians to actuarial problems, with the result that much of the first work in the ? eld of probability was closely connected with the development of solutions to actuarial problems. The earliest life insurance policies stomachd that the policyholder would pay an amount, called the reward, to the insurance agent. If the named life ensure died during the year that the contract was in force, the insurance firm would pay a prede confinesined collocate sum, the sum see, to the policyholder or his or her estate. So, the ? st life insurance contracts were annual contracts. Each year the agiotage would enlarge as the probability of death increased. If the see to it life became very ill at the renewal date, the insurance efficiency not be renewed, in which case 1 2 Introduction to life insurance no bene? t would be paying(a) on the lifes concomitant death. Over a large number of contracts, the allowance income each(prenominal) year should nearly match the claims disbursal. This method of matching income and outgo annually, with no attempt to smooth or balance the subventions over the eld, is called assessmentism.This method is palliate used for group life insurance, where an employer purchases life insurance cover for its employees on a year-to-year basis. The radical development in the after eighteenth century was the train premium contract. The problem with assessmentism was that the annual increases in premiums discourage policyholders from renewing their contracts. The direct premium policy offered the policyholder the option to lock-in a unshakable premium, payable perhaps weekly, monthly, quarterly or annually, for a number of historic period.This was much more ordinary with policyholders, as they would not be priced out of the insurance contract just when it cogency be most needed. For the insurance agent, the attraction of the longer contract was a greater careliness of the policyholder paying premiums for a longer period. However, a problem for the insurer was that the longer contracts were more complex to model, and offered more ? nancial risk. For these contracts then, actuarial techniques had to develop beyond the year-to-year simulate of mortality probabilities. In item, it became necessary to in corporeal ? nancial considerations into the modelling of income and outgo.Over a one-year contract, the time value of property is not a critical aspect. Over, say, a 30-year contract, it runs a very serious part of the modelling and management of r isk. Another development in life insurance in the nineteenth century was the c at a timept of insurable interest. This was a requirement in justness that the psyche contracting to pay the life insurance premiums should face a ? nancial loss on the death of the ensure life that was no less than the sum check under the policy. The insurable interest requirement disallowed the use of insurance as a form of gambling on the lives of public ? ures, but more importantly, removed the inducing for a policyholder to hasten the death of the named assure life. Subsequently, insurance policies tended to be purchased by the see to it life, and in the rest of this book we use the convention that the policyholder who pays the premiums is also the life insured, whose survival or death triggers the payment of the sum insured under the conditions of the contract. The earliest studies of mortality acknowledge life tables constructed by keister Graunt and Edmund Halley. A life table summarizes a survival model by specifying the similitude of lives that are expected to survive to each age.Using London mortality data from the early s lawsuiteenth century, Graunt proposed, for example, that each new life had a probability of 40% of go to age 16, and a probability of 1% of surviving to age 76. Edmund Halley, famous for his astronomical calculations, used mortality data from the city of Breslau in the late seventeenth century as the basis for his life table, which, like Graunts, was constructed by 1. 3 Life insurance and annuity contracts 3 proposing the average ( ordinary in Halleys phrase) proportion of survivors to each age from an arbitrary number of births.Halley took the work two steps further. First, he used the table to draw inference almost the conditional survival probabilities at in terminal figureediate ages. That is, given the probability that a newborn life survives to each subsequent age, it is assertable to infer the probability that a life healed, say, 20, go away survive to each subsequent age, using the condition that a life aged zero survives to age 20. The help major innovation was that Halley combine the mortality data with an assumption about interest rates to ? nd the value of a whole life annuity at different ages.A whole life annuity is a contract paying a level sum at steady intervals while the named life (the annuitant) is unruffled alive. The calculations in Halleys paper bear a remarkable similarity to some of the work still used by actuaries in pensions and life insurance. This book continues in the tradition of combining models of mortality with models in ? nance to develop a framework for pricing and risk management of long-term policies in life insurance. Many of the kindred techniques are applicable also in pensions mathematics. However, on that point have been some changes since the ? st long-term policies of the late eighteenth century. 1. 3 Life insurance and annuity contracts 1. 3. 1 Introduction T he life insurance and annuity contracts that were the object of study of the early actuaries were very similar to the contracts pen up to the 1980s in all the developed insurance markets. Recently, however, the design of life insurance products has radically changed, and the techniques needed to manage these more modern contracts are more complex than ever. The reasons for the changes include Increased interest by the insurers in offering combined savings and insurance products. The original life insurance products offered a payment to indemnify (or offset) the hardship caused by the death of the policyholder. Many modern contracts combine the indemnity concept with an opportunity to invest. More powerful computational facilities allow more complex products to be modelled. Policyholders have become more educate investors, and require more options in their contracts, allowing them to vary premiums or sums insured, for example. More competition has led to insurers creating incr easingly complex products in disposition to attract more business.The risk management techniques in ? nancial products have also become increasingly complex, and insurers have offered some bene? ts, especially 4 Introduction to life insurance ? nancial guarantees, that require sophisticated techniques from ? nancial engineering to measure and manage the risk. In the remainder of this section we describe some of the most important modern insurance contracts, which result later be used as examples in the book. Different countries have different name and graphemes of contracts we have tried to cover the major contract types in northwestern America, the United Kingdom and Australia.The basic transaction of life insurance is an exchange the policyholder pays premiums in bring around for a later payment from the insurer which is life contingent, by which we mean that it depends on the death or survival or possibly the state of health of the policyholder. We normally use the term in surance when the bene? t is stipendiary as a single swelling sum, either on the death of the policyholder or on survival to a preset adulthood date. (In the UK it is common to use the term assurance for insurance contracts involving lives, and insurance for contracts involving property. ) An annuity is a bene? in the form of a uniform series of payments, unremarkably conditional on the survival of the policyholder. 1. 3. 2 Traditional insurance contracts Term, whole life and endowment insurance are the traditional products, providing cash bene? ts on death or matureness date, unremarkably with predetermined premium and bene? t amounts. We describe each in a little more detail here. Term insurance pays a thud sum bene? t on the death of the policyholder, provided death occurs earlier the end of a speci? ed term. Term insurance allows a policyholder to provide a ? xed sum for his or her dependents in the event of the policyholders death.Level term insurance indicates a leve l sum insured and regular, level premiums. Decreasing term insurance indicates that the sum insured and (usually) premiums decrease over the term of the contract. Decreasing term insurance is popular in the UK where it is used in conjunction with a home owe if the policyholder dies, the remaining mortgage is stipendiary from the term insurance proceeds. Renewable term insurance offers the policyholder the option of renewing the policy at the end of the original term, without further endorse of the policyholders health status.In North America, yearbook Renewable Term (YRT) insurance is common, under which insurability is guaranteed for some ? xed period, though the contract is written only for one year at a time. 1. 3 Life insurance and annuity contracts 5 Convertible term insurance offers the policyholder the option to convert to a whole life or endowment insurance at the end of the original term, without further inference of the policyholders health status. Whole life insuranc e pays a jut sum bene? t on the death of the policyholder whenever it occurs.For regular premium contracts, the premium is oft payable only up to some maximum age, such as 80. This avoids the problem that older lives whitethorn be less able to pay the premiums. endowment fund insurance offers a lump sum bene? t salaried either on the death of the policyholder or at the end of a speci? ed term, whichever occurs ? rst. This is a motley of a term insurance bene? t and a savings element. If the policyholder dies, the sum insured is nonrecreational just as under term insurance if the policyholder survives, the sum insured is treated as a maturing enthronement. Endowment insurance is obsolete in many jurisdictions.Traditional endowment insurance policies are not currently sold in the UK, but thither are large portfolios of policies on the books of UK insurers, because until the late 1990s, endowment insurance policies were practically used to repay home mortgages. The policyholder (who is the home owner) paid interest on the mortgage contribute, and the principal was paid from the proceeds on the endowment insurance, either on the death of the policyholder or at the ? nal mortgage refund date. Endowment insurance policies are worthy popular in developing nations, particularly for micro-insurance where the amounts involved are teensy.It is hard for small investors to achieve good rates of show on coronations, because of heavy set down charges. By pooling the death and survival bene? ts under the endowment contract, the policyholder applys on the investiture side from the resulting economies of scale, and from the investment expertise of the insurer. With-pro? t insurance to a fault part of the traditional design of insurance is the division of business into with-pro? t (also cognize, in particular in North America, as participating, or par business), and without pro? t (also known as non-participating or non-par). Under with-pro? t arrangements, th e pro? s take in on the invested premiums are shared with the policyholders. In North America, the with-pro? t arrangement a good deal takes the form of cash dividends or minify premiums. In the UK and in Australia the traditional approach is to use the pro? ts to increase the sum insured, through bonuses called re reading materialary bonusesand terminal bonuses. reversionary bonuses are awarded during the term of the contract once a reversionary bonus is awarded it is guaranteed. closing bonuses are awarded when the policy matures, either through the death of the insured, or when an endowment policy reaches the end of the term.reversionary bonuses 6 Introduction to life insurance Table 1. 1. Year 1 2 3 . . . Bonus on original sum insured 2% 2. 5% 2. 5% . . . Bonus on bonus 5% 6% 6% . . . Total bonus 2000. 00 4620. 00 7397. 20 . . . may be expressed as a destiny of the resume of the previous sum insured plus bonus, or as a percentage of the original sum insured plus a differe nt percentage of the previously declared bonuses. Reversionary and terminal bonuses are determined by the insurer based on the investment proceeding of the invested premiums. For example, suppose an insurance is issued with sum insured $100 000.At the end of the ? rst year of the contract a bonus of 2% on the sum insured and 5% on previous bonuses is declared in the following two years, the rates are 2. 5% and 6%. past the total guaranteed sum insured increases each year as shown in Table 1. 1. If the policyholder dies, the total death bene? t payable would be the original sum insured plus reversionary bonuses already declared, increased by a terminal bonus if the investment returns take in on the premiums have been suf? cient. With-pro? ts contracts may be used to offer policyholders a savings element with their life insurance.However, the traditional with-pro? t contract is designed earlier for the life insurance cover, with the savings aspect a stakeary have got. 1. 3. 3 Mo dern insurance contracts In new-fangled years insurers have provided more ? exible products that combine the death bene? t coverage with a signi? cant investment element, as a way of competing for policyholderssavings with other institutions, for example, banks or unrestricted investment companies (e. g. unwashed funds in North America, or unit trusts in the UK). Additional ?exibility also allows policyholders to purchase less insurance when their ? ances are tight, and then increase the insurance coverage when they have more coin on tap(predicate). In this section we describe some examples of modern, ? exible insurance contracts. world(a) life insurance combines investment and life insurance. The policyholder determines a premium and a level of life insurance cover. Some 1. 3 Life insurance and annuity contracts 7 of the premium is used to fund the life insurance the remainder is paid into an investment fund. Premiums are ? exible, as long as they are suf? cient to pay for th e designated sum insured under the term insurance part of the contract.Under variable universal life, thither is a range of funds available for the policyholder to select from. Universal life is a common insurance contract in North America. Unitized with-pro? t is a UK insurance contract it is an evolution from the conventional with-pro? t policy, designed to be more transparent than the original. Premiums are used to purchase units (shares) of an investment fund, called the with-pro? t fund. As the fund earns investment return, the shares increase in value (or more shares are issued), increasing the bene? t entitlement as reversionary bonus.The shares result not decrease in value. On death or maturity, a further terminal bonus may be payable depending on the performance of the with-pro? t fund. After some poor forwarding surrounding with-pro? t business, and, by association, unitized with-pro? t business, these product designs were withdrawn from the UK and Australian markets by the early 2000s. However, they exit remain important for many years as many companies subscribe to very large portfolios of with-pro? t (traditional and unitized) policies issued during the second half of the twentieth century.Equity-linked insurance has a bene? t linked to the performance of an investment fund. in that respect are two different forms. The ? rst is where the policyholders premiums are invested in an open-ended investment bon ton style account at maturity, the bene? t is the salt away value of the premiums. on that point is a guaranteed token(prenominal) death bene? t payable if the policyholder dies in the beginning the contract matures. In some cases, there is also a guaranteed minimum maturity bene? t payable. In the UK and most of Europe, these are called unit-linked policies, and they rarely carry a guaranteed maturity bene? . In Canada they are known as segregated fund policies and always carry a maturity guarantee. In the regular army these contracts a re called variable annuity contracts maturity guarantees are increasingly common for these policies. (The use of the term annuity for these contracts is very misleading. The bene? ts are designed with a single lump sum payout, though there may be an option to convert the lump sum to an annuity. ) The second form of equity-linked insurance is the Equity-Indexed annuity (EIA) in the USA.Under an EIA the policyholder is guaranteed a minimum return on their premium (minus an initial expense charge). At maturity, the policyholder receives a proportion of the return on a speci? ed stock index, if that is greater than the guaranteed minimum return. EIAs are loosely rather shorter in term than unit-linked products, with seven-year policies being typical variable annuity contracts commonly 8 Introduction to life insurance have terms of twenty years or more. EIAs are much less popular with consumers than variable annuities. 1. 3. 4 Distribution methods some people ? d insurance dauntingly co mplex. Brokers who connect individuals to an assign insurance product have, since the earliest times, play an important role in the market. There is an old saying amongst actuaries that insurance is sold, not bought, which means that the role of an intermediary in persuading capability policyholders to take out an insurance policy is crucial in maintaining an adequate volume of new business. Brokers, or other ? nancial advisors, are often compensable through a commission system. The commission would be speci? ed as a percentage of the premium paid.Typically, there is a advanced(prenominal) percentage paid on the ? rst premium than on subsequent premiums. This is referred to as a front-end load. Some advisors may be remunerated on a ? xed fee basis, or may be employed by one or more insurance companies on a hire basis. An alternative to the broker method of selling insurance is direct market. Insurers may use television advertising or other teleselling methods to sell direct to the public. The nature of the business sold by direct marketing methods tends to differ from the broker sold business. For example, often the sum insured is smaller.The policy may be aimed at a receding market, such as older lives concerned with insurance to cover their own funeral expenses (called pre-need insurance in the USA). Another mass marketed insurance contract is loan or impute insurance, where an insurer skill cover loan or credit card payments in the event of the borrowers death, disability or un exercise. 1. 3. 5 Underwriting It is important in modelling life insurance liabilities to consider what happens when a life insurance policy is purchased. exchange life insurance policies is a matched business and life insurance companies (also known as life of? es) are evermore considering ways in which to change their procedures so that they can improve the service to their customers and gain a commercial advantage over their competitors. The account given below of how policies are sold covers some essential points but is ineluctably a simpli? ed version of what actually happens. For a given type of policy, say a 10-year term insurance, the life of? ce depart have a schedule of premium rates. These rates go forth depend on the size of the policy and some other factors known as rating factors.An applicants risk level is assessed by asking them to complete a proposal form big information on 1. 3 Life insurance and annuity contracts 9 relevant rating factors, in the main including their age, gender, smoking habits, occupation, any flagitious hobbies, and personal and family health history. The life insurer may ask for permit to contact the applicants indemnify to enquire about their health check exam history. In some cases, particularly for very large sums insured, the life insurer may require that the applicants health be checked by a doctor employed by the insurer.The routine of salt away and evaluating this information is called u nderwriting. The purpose of underwriting is, ? rst, to classify potential policyholders into broadly homogeneous risk categories, and secondly to assess what additional premium would be appropriate for applicants whose risk factors indicate that sample premium rates would be too low. On the basis of the application and sustenance medical information, potential life insurance policyholders will generally be reason into one of the following groups Preferred lives have very low mortality risk based on the normal infor- mation.The preferred applicant would have no recent record of smoking no march of drug or alcohol maltreatment no high-risk hobbies or occupations no family history of disease known to have a strong genetic contribution no adverse medical indicators such as high blood mash or cholesterol level or body mass index. The preferred life kin is common in North America, but has not yet caught on elsewhere. In other rural areas there is no separation of preferred an d normal lives. everyday lives may have some higher rated risk factors than preferred lives (where this category exists), but are still insurable at trite rates.Most applicants fall into this category. Rated lives have one or more risk factors at raised levels and so are not acceptable at standard premium rates. However, they can be insured for a higher premium. An example might be someone having a family history of heart disease. These lives might be individually assessed for the appropriate additional premium to be charged. This category would also include lives with hazardous jobs or hobbies which put them at increased risk. uninsurable lives have such signi? ant risk that the insurer will not enter an insurance contract at any price. Within the ? rst three groups, applicants would be further categorized according to the relative values of the various risk factors, with the most vestigial being age, gender and smoking status. Most applicants (around 95% for traditional life insurance) will be accepted at preferred or standard rates for the relevant risk category. Another 23% may be accepted at non-standard rates 10 Introduction to life insurance because of an im duplicatement, or a dangerous occupation, leaving around 23% who ill be refused insurance. The rigour of the underwriting passage will depend on the type of insurance being purchased, on the sum insured and on the distribution process of the insurance community. Term insurance is generally more strictly underwritten than whole life insurance, as the risk taken by the insurer is greater. Under whole life insurance, the payment of the sum insured is certain, the uncertainty is in the timing. Under, say, 10-year term insurance, it is assumed that the bulk of contracts will expire with no death bene? t paid.If the underwriting is not strict there is a risk of adverse selection by policyholders that is, that very high-risk individuals will buy insurance in disproportional numbers, leading to ex cessive losses. Since high sum insured contracts carry more risk than low sum insured, high sums insured would generally trigger more rigorous underwriting. The marketing method also affects the level of underwriting. Often, direct marketed contracts are sold with comparatively low bene? t levels, and with the attraction that no medical evidence will be sought beyond a standard questionnaire.The insurer may assume comparatively heavy mortality for these lives to compensate for potential adverse selection. By keeping the underwriting relatively light, the expenses of writing new business can be kept low, which is an attraction for high-volume, low sum insured contracts. It is interesting to parentage that with no third party medical evidence the insurer is placing a lot of weight on the veracity of the policyholder. Insurers have a phrase for this that both(prenominal) insurer and policyholder may assume utmost good faith or uberrima ? es on the part of the other side of the cont ract. In practice, in the event of the death of the insured life, the insurer may investigate whether any pertinent information was withheld from the application. If it appears that the policyholder held back information, or submitted false or misleading information, the insurer may not pay the full sum insured. 1. 3. 6 Premiums A life insurance policy may involve a single premium, payable at the outset of the contract, or a regular series of premiums payable provided the policyholder survives, perhaps with a ? ed end date. In traditional contracts the regular premium is generally a level amount throughout the term of the contract in more modern contracts the premium might be variable, at the policyholders discretion for investment products such as equity-linked insurance, or at the insurers discretion for certain types of term insurance. Regular premiums may be paid annually, semi-annually, quarterly, monthly or weekly. Monthly premiums are common as it is convenient for policyhold ers to have their outgoings payable with approximately the said(prenominal) frequency as their income. . 3 Life insurance and annuity contracts 11 An important feature of all premiums is that they are paid at the deject of each period. Suppose a policyholder contracts to pay annual premiums for a 10-year insurance contract. The premiums will be paid at the start of the contract, and then at the start of each subsequent year provided the policyholder is alive. So, if we count time in years from t = 0 at the start of the contract, the ? rst premium is paid at t = 0, the second is paid at t = 1, and so on, to the tenth premium paid at t = 9.Similarly, if the premiums are monthly, then the ? rst monthly instalment will be paid at t = 0, and the ? nal premium will be paid at the start 11 of the ? nal month at t = 9 12 years. (Throughout this book we assume that all 1 months are equal in length, at 12 years. ) 1. 3. 7 Life annuities Annuity contracts offer a regular series of payments. When an annuity depends on the survival of the recipient, it is called a life annuity. The recipient is called an annuitant. If the annuity continues until the death of the annuitant, it is called a whole life annuity.If the annuity is paid for some maximum period, provided the annuitant survives that period, it is called a term life annuity. Annuities are often purchased by older lives to provide income in loneliness. acquire a whole life annuity guarantees that the income will not run out before the annuitant dies. Single Premium Deferred Annuity (SPDA) Under an SPDA contract, the policyholder pays a single premium in return for an annuity which commences payment at some future, speci? ed date. The annuity is life contingent, by which we mean the annuity is paid only if the policyholder survives to the payment dates.If the policyholder dies before the annuity commences, there may be a death bene? t due. If the policyholder dies soon after the annuity commences, there may be some minimum payment period, called the guarantee period, and the balance would be paid to the policyholders estate. Single Premium Immediate Annuity (SPIA) This contract is the same as the SPDA, debar that the annuity commences as soon as the contract is effected. This might, for example, be used to convert a lump sum retirement bene? t into a life annuity to supplement a pension.As with the SPDA, there may be a guarantee period applying in the event of the early death of the annuitant. Regular Premium Deferred Annuity (RPDA) The RPDA offers a deferred life annuity with premiums paid through the deferred period. It is otherwise the same as the SPDA. Joint life annuity A joint life annuity is issued on two lives, typically a married couple. The annuity (which may be single premium or regular 12 Introduction to life insurance premium, immediate or deferred) continues while both lives survive, and ceases on the ? rst death of the couple.Last survivor annuity A last survivor annuity is si milar to the joint life annuity, still that payment continues while at to the lowest degree one of the lives survives, and ceases on the second death of the couple. Reversionary annuity A reversionary annuity is contingent on two lives, usually a couple. One is designated as the annuitant, and one the insured. No annuity bene? t is paid while the insured life survives. On the death of the insured life, if the annuitant is still alive, the annuitant receives an annuity for the remainder of his or her life. 1. Other insurance contracts The insurance and annuity contracts exposit above are all contingent on death or survival. There are other life contingent risks, in particular involving shortterm or long-term disability. These are known as morbidity risks. Income protection insurance When a person becomes sick and cannot work, their income will, eventually, be affected. For someone in regular employment, the employer may cover salary for a period, but if the sickness continues the salary will be decreased, and ultimately will stop being paid at all. For someone who is elf-employed, the effects of sickness on income will be immediate. Income protection policies supercede at least some income during periods of sickness. They usually cease at retirement age. minute illness insurance Some earnest illnesses can cause signi? cant expense at the onset of the illness. The patient may have to leave employment, or alter their home, or incur severe medical expenses. Critical illness insurance pays a bene? t on diagnosis of one of a number of severe conditions, such as certain cancers or heart disease. The bene? t is usually in the form of a lump sum.Long-term care insurance This is purchased to cover the costs of care in old age, when the insured life is unable(p) to continue living independently. The bene? t would be in the form of the long-term care costs, so is an annuity bene? t. 1. 5 Pension bene? ts Many actuaries work in the area of pension plan design, valu ation and risk management. The pension plan is usually sponsored by an employer. Pension plans typically offer employees (also called pension plan members) either lump 1. 5 Pension bene? ts 13 sums or annuity bene? ts or both on retirement, or deferred lump sum or annuity bene? s (or both) on earlier withdrawal. Some offer a lump sum bene? t if the employee dies while still employed. The bene? ts therefore depend on the survival and employment status of the member, and are quite similar in nature to life insurance bene? ts that is, they involve investment of contributions long into the future to pay for future life contingent bene? ts. 1. 5. 1 De? ned bene? t and de? ned contribution pensions De? ned Bene? t (DB) pensions offer retirement income based on service and salary with an employer, using a de? ned formula to determine the pension.For example, suppose an employee reaches retirement age with n years of service (i. e. membership of the pension plan), and with pensionable sal ary averaging S in, say, the ? nal three years of employment. A typical ? nal salary plan might offer an annual pension at retirement of B = Sn? , where ? is called the accruement rate, and is usually around 1%2%. The formula may be construe as a pension bene? t of, say, 2% of the ? nal average salary for each year of service. The de? ned bene? t is funded by contributions paid by the employer and (usually) the employee over the working lifetime of the employee.The contributions are invested, and the accumulated contributions moldiness be enough, on average, to pay the pensions when they become due. De? ned donation (DC) pensions work more like a bank account. The employee and employer pay a predetermined contribution (usually a ? xed percentage of salary) into a fund, and the fund earns interest. When the employee leaves or retires, the proceeds are available to provide income throughout retirement. In the UK most of the proceeds moldiness be converted to an annuity.In the USA and Canada there are more options the pensioner may draw funds to live on without necessarily purchasing an annuity from an insurance caller-up. 1. 5. 2 De? ned bene? t pension design The age retirement pension described in the section above de? nes the pension payable from retirement in a standard ? nal salary plan. Career average salary plans are also common in some jurisdictions, where the bene? t formula is the same as the ? nal salary formula above, except that the average salary over the employees entire career is used in place of the ? nal salary. Many employees leave their jobs before they retire.A typical withdrawal bene? t would be a pension based on the same formula as the age retirement bene? t, but with the start date deferred until the employee reaches the normal retirement age. Employees may have the option of taking a lump sum with the 14 Introduction to life insurance same value as the deferred pension, which can be invested in the pension plan of the new employer . Some pension plans also offer death-in-service bene? ts, for employees who die during their period of employment. Such bene? ts might include a lump sum, often based on salary and sometimes service, as well as a pension for the employees spouse. . 6 Mutual and proprietary insurers A mutual insurance companionship is one that has no shareholders. The insurer is owned by the with-pro? t policyholders. All pro? ts are distributed to the with-pro? t policyholders through dividends or bonuses. A proprietary insurance company has shareholders, and usually has withpro? t policyholders as well. The participating policyholders are not owners, but have a speci? ed right to some of the pro? ts. Thus, in a proprietary insurer, the pro? ts moldiness be shared in some predetermined proportion, between the shareholders and the with-pro? t policyholders.Many early life insurance companies were form as mutual companies. More recently, in the UK, Canada and the USA, there has been a trend towards demutualization, which means the transition of a mutual company to a proprietary company, through issuing shares (or cash) to the with-pro? t policyholders. Although it would appear that a mutual insurer would have marketing advantages, as participating policyholders receive all the pro? ts and other bene? ts of ownership, the advantages cited by companies who have demutualized include increased ability to raise capital, clearer corporate structure and improved ef? iency. 1. 7 Typical problems We are concerned in this book with developing the mathematical models and techniques used by actuaries working in life insurance and pensions. The primary responsibility of the life insurance actuary is to maintain the solvency and pro? tability of the insurer. Premiums must(prenominal) be suf? cient to pay bene? ts the assets held must be suf? cient to pay the contingent liabilities bonuses to policyholders should be fair. Consider, for example, a whole life insurance contract issued to a life aged 50. The sum insured may not be paid for 30 years or more.The premiums paid over the period will be invested by the insurer to earn signi? cant interest the accumulated premiums must be suf? cient to pay the bene? ts, on average. To ensure this, the actuary necessarily to model the survival probabilities of the policyholder, the investment returns likely to be earned and the expenses likely 1. 9 Exercises 15 to be incurred in maintaining the policy. The actuary may take into consideration the probability that the policyholder decides to terminate the contract early. The actuary may also consider the pro? tability requirements for the contract.Then, when all of these factors have been modelled, they must be combined to set a premium. Each year or so, the actuary must determine how much money the insurer or pension plan should hold to ensure that future liabilities will be cover with adequately high probability. This is called the valuation process. For with-pro? t insurance , the actuary must determine a suitable level of bonus. The problems are rather more complex if the insurance also covers morbidity risk, or involves several lives. All of these topics are covered in the following chapters.The actuary may also be involved in decisions about how the premiums are invested. It is vitally important that the insurer remains solvent, as the contracts are very long-term and insurers are accountable for protecting the ? nancial security of the general public. The way the underlying investments are selected can increase or mitigate the risk of insolvency. The precise selection of investments to manage the risk is particularly important where the contracts involve ? nancial guarantees. The pensions actuary working with de? ned bene? t pensions must determine appropriate contribution rates to meet the bene? s promised, using models that allow for the working patterns of the employees. Sometimes, the employer may want to change the bene? t structure, and the a ctuary is responsible for assessing the cost and impact. When one company with a pension plan takes over another, the actuary must assist with find the best way to allocate the assets from the two plans, and perhaps how to merge the bene? ts. 1. 8 Notes and further reading A number of essays describing actuarial practice can be demonstrate in Renn (ed. ) (1998). This book also provides both historical and more contemporary contexts for life contingencies.The original papers of Graunt and Halley are available online (and any search engine will ? nd them). Anyone interest in the history of probability and actuarial science will ? nd these interesting, and remarkably modern. 1. 9 Exercises Exercise 1. 1 Why do insurers generally require evidence of health from a person applying for life insurance but not for an annuity? 16 Introduction to life insurance Exercise 1. 2 Explain why an insurer might demand more rigorous evidence of a prospective policyholders health status for a term in surance than for a whole life insurance. Exercise 1. Explain why premiums are payable in advance, so that the ? rst premium is due now rather than in one years time. Exercise 1. 4 Lenders offering mortgages to home owners may require the borrower to purchase life insurance to cover the outstanding loan on the death of the borrower, even though the mortgaged property is the loan collateral. (a) Explain why the lender might require term insurance in this circumstance. (b) appoint how this term insurance might differ from the standard term insurance described in slit 1. 3. 2. (c) Can you see any problems with lenders demanding term insurance from borrowers?Exercise 1. 5 Describe the difference between a cash bonus and a reversionary bonus for with-pro? t whole life insurance. What are the advantages and disadvantages of each for (a) the insurer and (b) the policyholder? Exercise 1. 6 It is common for insurers to design whole life contracts with premiums payable only up to age 80. Why ? Exercise 1. 7 Andrew is retired. He has no pension, but has capital of $ergocalciferol 000. He is considering the following options for using the money (a) Purchase an annuity from an insurance company that will pay a level amount for the rest of his life. b) Purchase an annuity from an insurance company that will pay an amount that increases with the cost of living for the rest of his life. (c) Purchase a 20-year annuity certain. (d) Invest the capital and live on the interest income. (e) Invest the capital and draw $40 000 per year to live on. What are the advantages and disadvantages of each option? 2 Survival models 2. 1 Summary In this chapter we represent the future lifetime of an individual as a random variable, and show how probabilities of death or survival can be reckon under this framework.We then de? ne an important quantity known as the force of mortality, introduce some actuarial notation, and dissertate some properties of the distribution of future lifetime. We in troduce the curtate future lifetime random variable. This is a function of the future lifetime random variable which represents the number of complete years of future life. We explain why this function is useful and derive its probability function. 2. 2 The future lifetime random variable In Chapter 1 we saw that many insurance policies provide a bene? t on the death of the policyholder.When an insurance company issues such a policy, the policyholders date of death is unknown, so the insurer does not know exactly when the death bene? t will be payable. In rate to estimate the time at which a death bene? t is payable, the insurer needs a model of human mortality, from which probabilities of death at particular ages can be calculated, and this is the topic of this chapter. We start with some notation. allow (x) denote a life aged x, where x ? 0. The death of (x) can occur at any age greater than x, and we model the future lifetime of (x) by a continuous random variable which we deno te by Tx .This means that x + Tx represents the age-at-death random variable for (x). Let Fx be the distribution function of Tx , so that Fx (t) = PrTx ? t. Then Fx (t) represents the probability that (x) does not survive beyond age x + t, and we refer to Fx as the lifetime distribution from age x. In many life 17 18 Survival models insurance problems we are interested in the probability of survival rather than death, and so we de? ne Sx as Sx (t) = 1 ? Fx (t) = PrTx t. Thus, Sx (t) represents the probability that (x) survives for at least t years, and Sx is known as the survival function. assumption our interpretation of the ollection of random variables Tx x? 0 as the future lifetimes of individuals, we need a connection between any pair of them. To see this, consider T0 and Tx for a particular individual who is now aged