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mkThis page intentionally left blank Actuarial Mathematics for Life Contingent Risks How can actuaries best equip themselves for the products and risk structures of the future? In this new textbook, three leaders in actuarial science give a modern perspective on life contingencies. The book begins traditionally, covering actuarial models and theory, and emphasizing practical applications using computational techniques. The authors then develop a more contemporary outlook, introducing multiple state models, emerging cash ? ws and embedded options. Using spreadsheet-style software, the book presents large-scale, realistic examples. Over 150 exercises and solutions teach skills in simulation and projection through computational practice. Balancing rigour with intuition, and emphasizing applications, this textbook is ideal not only for university courses, but also for individuals preparing for professional actuarial examinations and quali? ed actuaries wishing to renew and update their skills.International Series on Actuarial Science Christopher Daykin, Independent Consultant and Actuary Angus Macdonald, Heriot-Watt University The International Series on Actuarial Science, published by Cambridge University Press in conjunction with the Institute of Actuaries and the Faculty of Actuaries, contains textbooks for students taking courses in or related to actuarial science, as well as more advanced works designed for continuing professional development or for describing and synthesizing research.The series is a vehicle for publishing books that re? ect changes and developments in the curriculum, that encourage the introduction of courses on actuarial science in universities, and that show how actuarial science can be used in all areas where there is long-term ? nancial risk. 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 Univ ersity, 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 United States of America by Cambridge University Press, New York www. cambridge. org Information on this title: www. cambridge. org/9780521118255  © D. C. M. Dickson, M. R. Hardy and H. R. Waters 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.First published in print format 2009 ISBN-13 ISBN-13 978-0-511-65169-4 978-0-521-11825-5 eBook (NetLibrary) Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To Carolann, Vivien and Phelim Contents Preface page xiv 1 Introduction to life insurance 1 1. 1 Summary 1 1. 2 Background 1 1. 3 Life insurance and annuity contracts 3 1. 3. 1 Introduction 3 1. 3. Traditional insurance contracts 4 1. 3. 3 Modern insurance contracts 6 1. 3. 4 Distribution methods 8 1. 3. 5 Underwriting 8 1. 3. 6 Premiums 10 1. 3. 7 Life annuities 11 1. 4 Other insurance contracts 12 1. 5 Pension bene? ts 12 1. 5. 1 De? ned bene? t and de? ned contribution pensions 13 1. 5. 2 De? ned bene? t pension design 13 1. 6 Mutual and proprietary insurers 14 1. 7 Typical problems 14 1. 8 Notes and further reading 15 1. 9 Exercises 15 2 Survival models 17 2. 1 Summary 17 2. 2 The future lifetime random variable 17 2. 3 The force of mortality 21 2. 4 Actuarial notation 26 2. Mean and standard deviation of Tx 29 2. 6 Curtate future lifetime 32 2. 6. 1 Kx and ex 32 vii viii 2. 6. 2 Contents The complete and curtate expected fu ture ? lifetimes, ex and ex 2. 7 Notes and further reading 2. 8 Exercises Life tables and selection 3. 1 Summary 3. 2 Life tables 3. 3 Fractional age assumptions 3. 3. 1 Uniform distribution of deaths 3. 3. 2 Constant force of mortality 3. 4 National life tables 3. 5 Survival models for life insurance policyholders 3. 6 Life insurance underwriting 3. 7 Select and ultimate survival models 3. 8 Notation and formulae for select survival models 3. Select life tables 3. 10 Notes and further reading 3. 11 Exercises Insurance bene? ts 4. 1 Summary 4. 2 Introduction 4. 3 Assumptions 4. 4 Valuation of insurance bene? ts ? 4. 4. 1 Whole life insurance: the continuous case, Ax 4. 4. 2 Whole life insurance: the annual case, Ax (m) 4. 4. 3 Whole life insurance: the 1/mthly case, Ax 4. 4. 4 Recursions 4. 4. 5 Term insurance 4. 4. 6 Pure endowment 4. 4. 7 Endowment insurance 4. 4. 8 Deferred insurance bene? ts (m) ? 4. 5 Relating Ax , Ax and Ax 4. 5. 1 Using the uniform distribution of deaths assu mption 4. 5. 2 Using the claims acceleration approach 4. Variable insurance bene? ts 4. 7 Functions for select lives 4. 8 Notes and further 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 96 101 101 102 107 107 107 5 Contents 5. 3 5. 4 Review of annuities-certain Annual life annuities 5. 4. 1 Whole life annuity-due 5. 4. 2 Term annuity-due 5. 4. 3 Whole life immediate annuity 5. 4. 4 Term immediate annuity 5. 5 Annuities payable continuously 5. 5. 1 Whole life continuous annuity 5. 5. 2 Term continuous annuity 5. 6 Annuities payable m times per year 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 Comparison of annuities by payment frequency 5. 8 Deferred annuities 5. 9 Guaranteed annuities 5. 10 Increasing annuities 5. 10. 1 Arithmetically increasing annuities 5. 10. 2 Geometrically increasing annu ities 5. 11 Evaluating annuity functions 5. 11. 1 Recursions 5. 11. 2 Applying the UDD assumption 5. 11. 3 Woolhouse’s formula 5. 12 Numerical illustrations 5. 13 Functions for select lives 5. 14 Notes and further reading 5. 15 Exercises Premium calculation 6. 1 Summary 6. 2 Preliminaries 6. Assumptions 6. 4 The present value of future loss random variable 6. 5 The equivalence principle 6. 5. 1 Net premiums 6. 6 Gross premium calculation 6. 7 Pro? t 6. 8 The portfolio percentile premium principle 6. 9 Extra risks 6. 9. 1 Age rating 6. 9. 2 Constant addition to  µx 6. 9. 3 Constant multiple of mortality rates ix 108 108 109 112 113 114 115 115 117 118 118 119 120 121 123 125 127 127 129 130 130 131 132 135 136 137 137 142 142 142 143 145 146 146 150 154 162 165 165 165 167 6 x Contents 6. 10 Notes and further reading 6. 11 Exercises Policy values 7. 1 Summary 7. 2 Assumptions 7. Policies with annual cash ? ows 7. 3. 1 The future loss random variable 7. 3. 2 Policy values for policies with annual cash ? ows 7. 3. 3 Recursive formulae for policy values 7. 3. 4 Annual pro? t 7. 3. 5 Asset shares 7. 4 Policy values for policies with cash ? ows at discrete intervals other than annually 7. 4. 1 Recursions 7. 4. 2 Valuation between premium dates 7. 5 Policy values with continuous cash ? ows 7. 5. 1 Thiele’s differential equation 7. 5. 2 Numerical solution of Thiele’s differential equation 7. 6 Policy alterations 7. 7 Retrospective policy value 7. 8 Negative policy values 7. Notes and further reading 7. 10 Exercises Multiple state models 8. 1 Summary 8. 2 Examples of multiple state models 8. 2. 1 The alive–dead model 8. 2. 2 Term insurance with increased bene? t on accidental death 8. 2. 3 The permanent disability model 8. 2. 4 The disability income insurance model 8. 2. 5 The joint life and last survivor model 8. 3 Assumptions and notation 8. 4 Formulae for probabilities 8. 4. 1 Kolmogorov’s forward equations 8. 5 Numerical evaluat ion of probabilities 8. 6 Premiums 8. 7 Policy values and Thiele’s differential equation 8. 7. 1 The disability income model 8. 7. Thiele’s 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 Multiple decrement models Joint life and last survivor bene? ts 8. 9. 1 The model and assumptions 8. 9. 2 Joint life and last survivor probabilities 8. 9. 3 Joint life and last survivor annuity and insurance functions 8. 9. 4 An important special case: independent survival models 8. 10 Transitions at speci? ed ages 8. 11 Notes and further reading 8. 12 Exercises Pension mathematics 9. Summary 9. 2 Introduction 9. 3 The salary scale function 9. 4 Setting the DC contribution 9. 5 The service table 9. 6 Valuation of bene? ts 9. 6. 1 Final salary plans 9. 6. 2 Career average earnings plans 9. 7 Funding plans 9. 8 Not es and further reading 9. 9 Exercises Interest rate risk 10. 1 Summary 10. 2 The yield curve 10. 3 Valuation of insurances and life annuities 10. 3. 1 Replicating the cash ? ows of a traditional non-participating product 10. 4 Diversi? able and non-diversi? able risk 10. 4. 1 Diversi? able mortality risk 10. 4. 2 Non-diversi? able risk 10. 5 Monte Carlo simulation 10. Notes and further reading 10. 7 Exercises Emerging costs for traditional life insurance 11. 1 Summary 11. 2 Pro? t testing for traditional life insurance 11. 2. 1 The net cash ? ows for a policy 11. 2. 2 Reserves 11. 3 Pro? t measures 11. 4 A further example of a pro? t test 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 costs for equity-linked insurance 12. 1 Summary 12. 2 Equity-linked insurance 12. 3 Deterministic pro? testing fo r equity-linked insurance 12. 4 Stochastic pro? t testing 12. 5 Stochastic pricing 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’ assumption 13. 4 Options 13. 5 The binomial option pricing model 13. 5. 1 Assumptions 13. 5. 2 Pricing over a single time period 13. 5. 3 Pricing over two time periods 13. 5. 4 Summary of the binomial model option pricing technique 13. The Black–Scholes–Merton model 13. 6. 1 The model 13. 6. 2 The Black–Scholes–Merton option pricing formula 13. 7 Notes and further reading 13. 8 Exercises Embedded options 14. 1 Summary 14. 2 Introduction 14. 3 Guaranteed minimum maturity bene? t 14. 3. 1 Pricing 14. 3. 2 Reserving 14. 4 Guaranteed minimum death 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 management 14. 7 Emerging costs 14. Notes and further reading 14. 9 Exercises A Probability theory A. 1 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 B Numerical techniques B. 1 Numerical integration B. 1. 1 The trapezium rule B. 1. 2 Repeated Simpson’s rule B. 1. 3 Integrals over an in? nite interval B. Woolhouse’s formula B. 3 Notes and further reading C Simulation C. 1 The inverse transf orm method C. 2 Simulation from a normal distribution C. 2. 1 The Box–Muller method C. 2. 2 The polar method C. 3 Notes and further reading References Author index Index 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 insurance has undergone enormous change in the last two to three decades. New and innovative products have been developed at the same time as we have seen vast increases in computational power.In addition, the ? eld of ? nance has experienced a revolution in the development of a mathematical theory of options and ? nancial guarantees, ? rst pioneered in the work of Black, Scholes and Merton, and actuaries have come to realize the importance of that work to risk management in actuarial contexts. Given the changes occurring in the interconnected worlds of ? nance and life insurance, we believe that this is a good time to recast the mathematics of life continge nt risk to be better adapted to the products, science and technology that are relevant to current and future actuaries.In this book we have developed the theory to measure and manage risks that are contingent on demographic experience as well as on ? nancial variables. The material is presented with a certain level of mathematical rigour; we intend for readers to understand the principles involved, rather than to memorize methods or formulae. The reason is that a rigorous approach will prove more useful in the long run than a short-term utilitarian outlook, as theory can be adapted to changing products and technology in ways that techniques, without scienti? c support, cannot.We start from a traditional approach, and then develop a more contemporary perspective. The ? rst seven chapters set the context for the material, and cover traditional actuarial models and theory of life contingencies, with modern computational techniques integrated throughout, and with an emphasis on the prac tical context for the survival models and valuation methods presented. Through the focus on realistic contracts and assumptions, we aim to foster a general business awareness in the life insurance context, at the same time as we develop the mathematical tools for risk management in that context. iv Preface xv In Chapter 8 we introduce multiple state models, which generalize the life– death contingency structure of previous chapters. Using multiple state models allows a single framework for a wide range of insurance, including bene? ts which depend on health status, on cause of death bene? ts, or on two or more lives. In Chapter 9 we apply the theory developed in the earlier chapters to problems involving pension bene? ts. Pension mathematics has some specialized concepts, particularly in funding principles, but in general this chapter is an application of the theory in the preceding chapters.In Chapter 10 we move to a more sophisticated view of interest rate models and intere st rate risk. In this chapter we explore the crucially important difference between diversi? able and non-diversi? able risk. Investment 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 emerging cash ? ows for traditional insurance products. One of the liberating aspects of the computer revolution for actuaries is that we are no longer required to summarize complex bene? s in a single actuarial value; we can go much further in projecting the cash ? ows to see how and when surplus will emerge. This is much richer information that the actuary can use to assess pro? tability and to better manage portfolio assets and liabilities. In Chapter 12 we repeat the emerging cash ? ow approach, but here we look at equity-linked contracts, where a ? nancial guarantee is commonly part of the contingent bene? t. The real risks for such products can only be assessed taking the random variation in potential outcomes into consideration, and we demonstrate this with Monte Carlo simulation of the emerging cash ? ws. The products that are explored in Chapter 12 contain ? nancial guarantees embedded in the life contingent bene? ts. Option theory is the mathematics of valuation and risk management of ? nancial guarantees. In Chapter 13 we introduce the fundamental assumptions and results of option theory. In Chapter 14 we apply option theory to the embedded options of ? nancial guarantees in insurance products. The theory can be used for pricing and for determining appropriate reserves, as well as for assessing pro? tability.The material in this book is designed for undergraduate and graduate programmes in actuarial science, and for those self-studying for professional actuarial exams. Students should have suf? cient background in probability to be able to calculate moments of functions of one or two random vari ables, and to handle conditional expectations and variances. We also assume familiarity with the binomial, uniform, exponential, normal and lognormal distributions. Some of the more important results are reviewed in Appendix A. We also assume xvi Preface that readers have completed an introductory level course in the mathematics of ? ance, and are aware of the actuarial notation for annuities-certain. Throughout, we have opted to use examples that liberally call on spreadsheetstyle software. Spreadsheets are ubiquitous tools in actuarial practice, and it is natural to use them throughout, allowing us to use more realistic examples, rather than having to simplify for the sake of mathematical tractability. Other software could be used equally effectively, but spreadsheets represent a fairly universal language that is easily accessible. To keep the computation requirements reasonable, we have ensured hat every example and exercise can be completed in Microsoft Excel, without needing an y VBA code or macros. Readers who have suf? cient familiarity to write their own code may ? nd more ef? cient solutions than those that we have presented, but our principle was that no reader should need to know more than the basic Excel functions and applications. It will be very useful for anyone working through the material of this book to construct their own spreadsheet tables as they work through the ? rst seven chapters, to generate mortality and actuarial functions for a range of mortality models and interest rates.In the worked examples in the text, we have worked with greater accuracy than we record, so there will be some differences from rounding when working with intermediate ? 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 give some of the most useful algorithms in Appendix B. The material in this book is appropriate for tw o one-semester courses. The ? rst seven chapters form a fairly traditional basis, and would reasonably constitute a ? st course. Chapters 8–14 introduce more contemporary material. Chapter 13 may be omitted by readers who have studied an introductory course covering pricing and delta hedging in a Black–Scholes–Merton model. Chapter 9, on pension mathematics, is not required for subsequent chapters, and could be omitted if a single focus on life insurance is preferred. Acknowledgements Many of our students and colleagues have made valuable comments on earlier drafts of parts of the book. Particular thanks go to Carole Bernard, Phelim Boyle, Johnny Li, Ana Maria Mera, Kok Keng Siaw and Matthew Till.The authors gratefully 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 w ork on this book. These visits signi? cantly shortened the time it has taken to write the book (to only one year beyond the original 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 range of other disciplines to problems involving risk, uncertainty and ? nance. In this chapter we set the context for the mathematics of later chapters, by describing some of the background to modern actuarial practice in life insurance, followed by a brief description of the major types of life insurance products that are sold in developed insurance markets. Because pension liabilities are similar in many ways to life insurance liabilities, we also describe 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 result ing problems, is the subject of the following chapters. 1. 2 Background The ? rst actuaries were employed by life insurance companies in the early eighteenth century to provide a scienti? c basis for managing the companies’ assets and liabilities. The liabilities depended on the number of deaths occurring amongst the insured lives each year. The modelling of mortality became a topic of both commercial and general scienti? interest, and it attracted many signi? cant scientists and mathematicians to actuarial problems, with the result that much of the early work in the ? eld of probability was closely connected with the development of solutions to actuarial problems. The earliest life insurance policies provided that the policyholder would pay an amount, called the premium, to the insurer. If the named life insured died during the year that the contract was in force, the insurer would pay a predetermined lump sum, the sum insured, to the policyholder or his or her estate. So, t he ? st life insurance contracts were annual contracts. Each year the premium would increase as the probability of death increased. If the insured life became very ill at the renewal date, the insurance might not be renewed, in which case 1 2 Introduction to life insurance no bene? t would be paid on the life’s subsequent death. Over a large number of contracts, the premium income each year should approximately match the claims outgo. This method of matching income and outgo annually, with no attempt to smooth or balance the premiums over the years, is called assessmentism.This method is still 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 later eighteenth century was the level premium contract. The problem with assessmentism was that the annual increases in premiums discouraged policyholders from renewing their contracts. The level premium policy offered the policyholde r the option to lock-in a regular premium, payable perhaps weekly, monthly, quarterly or annually, for a number of years.This was much more popular with policyholders, as they would not be priced out of the insurance contract just when it might be most needed. For the insurer, the attraction of the longer contract was a greater likelihood 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 modelling of mortality probabilities. In particular, it became necessary to incorporate ? nancial considerations into the modelling of income and outgo.Over a one-year contract, the time value of money is not a critical aspect. Over, say, a 30-year contract, it becomes a very important part of the modelling and management of risk. Another development in life insurance in the nineteent h century was the concept of insurable interest. This was a requirement in law that the person contracting to pay the life insurance premiums should face a ? nancial loss on the death of the insured life that was no less than the sum insured 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 incentive for a policyholder to hasten the death of the named insured life. Subsequently, insurance policies tended to be purchased by the insured 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 include life tables constructed by John Graunt and Edmund Halley. A life table summarizes a survival model by specifying the proportion of lives that are expected to survive to each age.Using London mortality data from the early seventeenth century, Graunt proposed, for example, that each new life had a probability of 40% of surviving 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 Graunt’s, was constructed by 1. 3 Life insurance and annuity contracts 3 proposing the average (‘medium’ in Halley’s 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 about the conditional survival probabilities at intermediate ages. That is, given the probability that a newborn life survives to each subsequent age, it is possible to infer the probability that a life aged, say, 20, will survive to each subsequent age, using the condition that a life ag ed zero survives to age 20. The second major innovation was that Halley combined 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 regular intervals while the named life (the annuitant) is still alive. The calculations in Halley’s 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 same techniques are relevant also in pensions mathematics. However, there have been many changes since the ? st long-term policies of the late eighteenth century. 1. 3 Life insurance and annuity contracts 1. 3. 1 Introduction The life insurance and annuity contracts that were the object of study of the early actuaries w ere very similar to the contracts written 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 sophisticated 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 increasingly complex products in order to attract more busines s.The risk management techniques in ? nancial products have also become increasingly complex, and insurers have offered some bene? ts, particularly 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 will later be used as examples in the book. Different countries have different names and types of contracts; we have tried to cover the major contract types in North America, the United Kingdom and Australia.The basic transaction of life insurance is an exchange; the policyholder pays premiums in return 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 usually use the term ‘insurance’ when the bene? t is paid as a single lump sum, either on the de ath of the policyholder or on survival to a predetermined maturity 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 regular series of payments, usually 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 maturity, usually with predetermined premium and bene? t amounts. We describe each in a little more detail here. Term insurance pays a lump sum bene? t on the death of the policyholder, provided death occurs before 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 policyholder’s death.Level term insurance indicates a level sum insured and regular, level premiums. Decreasing term insurance indicates tha t 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 mortgage; if the policyholder dies, the remaining mortgage is paid 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 evidence of the policyholder’s health status.In North America, Yearly 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 evidence of the policyholder’s health status. Whole life insurance pays a lump sum bene? t on the death of the policyholder whenever it occ urs.For regular premium contracts, the premium is often payable only up to some maximum age, such as 80. This avoids the problem that older lives may be less able to pay the premiums. Endowment insurance offers a lump sum bene? t paid either on the death of the policyholder or at the end of a speci? ed term, whichever occurs ? rst. This is a mixture of a term insurance bene? t and a savings element. If the policyholder dies, the sum insured is paid just as under term insurance; if the policyholder survives, the sum insured is treated as a maturing investment. Endowment insurance is obsolete in many jurisdictions.Traditional endowment insurance policies are not currently sold in the UK, but there are large portfolios of policies on the books of UK insurers, because until the late 1990s, endowment insurance policies were often used to repay home mortgages. The policyholder (who is the home owner) paid interest on the mortgage loan, and the principal was paid from the proceeds on the e ndowment insurance, either on the death of the policyholder or at the ? nal mortgage repayment date. Endowment insurance policies are becoming popular in developing nations, particularly for ‘micro-insurance’ where the amounts involved are small.It is hard for small investors to achieve good rates of return on investments, because of heavy expense charges. By pooling the death and survival bene? ts under the endowment contract, the policyholder gains on the investment side from the resulting economies of scale, and from the investment expertise of the insurer. With-pro? t insurance Also part of the traditional design of insurance is the division of business into ‘with-pro? t’ (also known, especially 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, the pro? s earned on the invested pr emiums are shared with the policyholders. In North America, the with-pro? t arrangement often takes the form of cash dividends or reduced premiums. In the UK and in Australia the traditional approach is to use the pro? ts to increase the sum insured, through bonuses called ‘reversionary bonuses’and ‘terminal bonuses’. Reversionary bonuses are awarded during the term of the contract; once a reversionary bonus is awarded it is guaranteed. Terminal 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 percentage of the total of the previous sum insured plus bonus, or as a percentage of the original sum insured plus a different percentage of the pr eviously declared bonuses. Reversionary and terminal bonuses are determined by the insurer based on the investment performance 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%. Then 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 earned 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 primarily for the life insurance cover, with the savings aspect a secondary feature. 1. 3. 3 Modern insurance contracts I n recent 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 policyholders’savings with other institutions, for example, banks or open-ended investment companies (e. g. mutual 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 money available. In this section we describe some examples of modern, ? exible insurance contracts. Universal 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 the designated sum insured under the ter m insurance part of the contract.Under variable universal life, there 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 will 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 publicity 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 will remain i mportant for many years as many companies carry 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. There are two different forms. The ? rst is where the policyholder’s premiums are invested in an open-ended investment company style account; at maturity, the bene? t is the accumulated value of the premiums. There is a guaranteed minimum death bene? t payable if the policyholder dies before 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 USA these contracts are called variable annuity contracts; maturity guarantees are increasingly common for these pol icies. (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 generally 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 Most people ? d insurance dauntingly complex. Brokers who connect individuals to an appropriate insurance product ha ve, since the earliest times, played 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 potential policyholders to take out an insurance policy is crucial in maintaining an adequate volume of new business. Brokers, or other ? nancial advisors, are often remunerated through a commission system. The commission would be speci? ed as a percentage of the premium paid.Typically, there is a higher 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 salary basis. An alternative to the broker method of selling insurance is direct marketing. Insurers may use television advertising or other telemarketing methods to sell direct to the public. The nature of the business sold by direct marketing meth ods tends to differ from the broker sold business. For example, often the sum insured is smaller.The policy may be aimed at a niche 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 credit insurance, where an insurer might cover loan or credit card payments in the event of the borrower’s death, disability or unemployment. 1. 3. 5 Underwriting It is important in modelling life insurance liabilities to consider what happens when a life insurance policy is purchased. Selling life insurance policies is a competitive business and life insurance companies (also known as life of? es) are constantly 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 necessaril y a simpli? ed version of what actually happens. For a given type of policy, say a 10-year term insurance, the life of? ce will have a schedule of premium rates. These rates will depend on the size of the policy and some other factors known as rating factors.An applicant’s risk level is assessed by asking them to complete a proposal form giving information on 1. 3 Life insurance and annuity contracts 9 relevant rating factors, generally including their age, gender, smoking habits, occupation, any dangerous hobbies, and personal and family health history. The life insurer may ask for permission to contact the applicant’s doctor to enquire about their medical history. In some cases, particularly for very large sums insured, the life insurer may require that the applicant’s health be checked by a doctor employed by the insurer.The process of collecting and evaluating this information is called underwriting. The purpose of underwriting is, ? rst, to classify potenti al policyholders into broadly homogeneous risk categories, and secondly to assess what additional premium would be appropriate for applicants whose risk factors indicate that standard premium rates would be too low. On the basis of the application and supporting medical information, potential life insurance policyholders will generally be categorized into one of the following groups: †¢ Preferred lives have very low mortality risk based on the standard infor- mation.The preferred applicant would have no recent record of smoking; no evidence of drug or alcohol abuse; no high-risk hobbies or occupations; no family history of disease known to have a strong genetic component; no adverse medical indicators such as high blood pressure or cholesterol level or body mass index. The preferred life category is common in North America, but has not yet caught on elsewhere. In other areas there is no separation of preferred and normal lives. †¢ Normal lives may have some higher rated ri sk factors than preferred lives (where this category exists), but are still insurable at standard 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 fundamental 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 2–3% may be accepted at non-standard rates 10 Introduction to life insurance because of an impairment, or a dangerous occupation, leaving around 2–3% who ill be refused insurance. The rigour of the underwriting process will depend on the type of insurance being purchased, on the sum insured and on the distribution process of the insurance company. 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 majority 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 disproportionate numbers, leading to excessive 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 relatively low bene? t levels, and with the attraction that no medical evidence will be sought beyond a standard questionnaire.The insurer may assume relatively 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 note 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 insurer and policyholder may assume ‘utmost good faith’ or ‘uberrima ? es’ on the part of the other side of the contract. In practi ce, 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 policyholder’s discretion for investment products such as equity-linked insurance, or at the insurer’s 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 policyho lders to have their outgoings payable with approximately the same frequency as their income. . 3 Life insurance and annuity contracts 11 An important feature of all premiums is that they are paid at the start 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 annui ty 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 retirement. Buying 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, the re may be some minimum payment period, called the guarantee period, and the balance would be paid to the policyholder’s estate. Single Premium Immediate Annuity (SPIA) This contract is the same as the SPDA, except 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 similar to the joint life annuity, except that payment continues while at least 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 described 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 continu es 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 replace at least some income during periods of sickness. They usually cease at retirement age. Critical illness insurance Some serious 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 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, valua tion 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 pensionab le salary 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 accrual rate, and is usually around 1%–2%. The formula may be interpreted 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 must be enough, on average, to pay the pensions when they become due. De? ned Contribution (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 must 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 company. 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 employee’s 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 employee’s spouse. . 6 Mutual and proprietary insurers A mutual insurance company 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 must be shared in some predetermined proportion, between the shareholders and the with-pro? t policyholders.Many early life insurance companies were formed 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 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 needs 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 covered 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 responsible 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 actu ary is responsible for assessing the cost and impact. When one company with a pension plan takes over another, the actuary must assist with determining 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 found 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 interested 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 policyholder’s health status for a te rm insurance 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 year’s 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) Describe how this term insurance might differ from the standard term insurance described in Section 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 $500 000. 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 calculated under this framework.We then de? ne an important quantity known as the force of mortality, introduce some actuarial notation, and discuss some properties of the distribution of future lifetime. W e introduce 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 policyholder’s date of death is unknown, so the insurer does not know exactly when the death bene? t will be payable. In order 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. Let (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 whic h we denote 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) = Pr[Tx ? 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) = Pr[Tx > t]. Thus, Sx (t) represents the probability that (x) survives for at least t years, and Sx is known as the survival function. Given 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