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Actuarial Mathematicsand Life-Table StatisticsEric V. SludMathematics DepartmentUniversity of Maryland, College Parkc 2001

c 2001Eric V. SludStatistics ProgramMathematics DepartmentUniversity of MarylandCollege Park, MD 20742

Contents0.1Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Basics of Probability & Interestvi11.1Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Theory of Interest . . . . . . . . . . . . . . . . . . . . . . . . .71.2.1Variable Interest Rates . . . . . . . . . . . . . . . . . . 101.2.2Continuous-time Payment Streams . . . . . . . . . . . 151.3Exercise Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 181.5Useful Formulas from Chapter 1 . . . . . . . . . . . . . . . . . 212 Interest & Force of Mortality2.12.223More on Theory of Interest . . . . . . . . . . . . . . . . . . . . 232.1.1Annuities & Actuarial Notation . . . . . . . . . . . . . 242.1.2Loan Amortization & Mortgage Refinancing . . . . . . 292.1.3Illustration on Mortgage Refinancing . . . . . . . . . . 302.1.4Computational illustration in Splus . . . . . . . . . . . 322.1.5Coupon & Zero-coupon Bonds . . . . . . . . . . . . . . 35Force of Mortality & Analytical Models . . . . . . . . . . . . . 37i

iiCONTENTS2.2.1Comparison of Forces of Mortality . . . . . . . . . . . . 452.3Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.4Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 542.5Useful Formulas from Chapter 2 . . . . . . . . . . . . . . . . . 583 Probability & Life Tables613.1Interpreting Force of Mortality . . . . . . . . . . . . . . . . . . 613.2Interpolation Between Integer Ages . . . . . . . . . . . . . . . 623.3Binomial Variables &Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 663.3.13.4Exact Probabilities, Bounds & Approximations . . . . 71Simulation of Life Table Data . . . . . . . . . . . . . . . . . . 743.4.1Expectation for Discrete Random Variables . . . . . . 763.4.2Rules for Manipulating Expectations . . . . . . . . . . 783.5Some Special Integrals . . . . . . . . . . . . . . . . . . . . . . 813.6Exercise Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.7Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 873.8Useful Formulas from Chapter 3 . . . . . . . . . . . . . . . . . 934 Expected Present Values of Payments4.14.295Expected Payment Values . . . . . . . . . . . . . . . . . . . . 964.1.1Types of Insurance & Life Annuity Contracts . . . . . 964.1.2Formal Relations among Net Single Premiums . . . . . 1024.1.3Formulas for Net Single Premiums . . . . . . . . . . . 1034.1.4Expected Present Values for m 1 . . . . . . . . . . . 104Continuous Contracts & Residual Life. . . . . . . . . . . . . 106

CONTENTSiii4.2.1Numerical Calculations of Life Expectancies . . . . . . 1114.3Exercise Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.4Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1184.5Useful Formulas from Chapter 4 . . . . . . . . . . . . . . . . . 1215 Premium Calculation5.1123m-Payment Net Single Premiums . . . . . . . . . . . . . . . . 1245.1.1Dependence Between Integer & Fractional Ages at Death1245.1.2Net Single Premium Formulas — Case (i) . . . . . . . 1265.1.3Net Single Premium Formulas — Case (ii) . . . . . . . 1295.2Approximate Formulas via Case(i) . . . . . . . . . . . . . . . . 1325.3Net Level Premiums . . . . . . . . . . . . . . . . . . . . . . . 1345.4Benefits Involving Fractional Premiums . . . . . . . . . . . . . 1365.5Exercise Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.6Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1425.7Useful Formulas from Chapter 5 . . . . . . . . . . . . . . . . . 1456 Commutation & Reserves6.16.2147Idea of Commutation Functions . . . . . . . . . . . . . . . . . 1476.1.1Variable-benefit Commutation Formulas . . . . . . . . 1506.1.2Secular Trends in Mortality . . . . . . . . . . . . . . . 152Reserve & Cash Value of a Single Policy . . . . . . . . . . . . 1536.2.1Retrospective Formulas & Identities . . . . . . . . . . . 1556.2.2Relating Insurance & Endowment Reserves . . . . . . . 1586.2.3Reserves under Constant Force of Mortality . . . . . . 1586.2.4Reserves under Increasing Force of Mortality . . . . . . 160

ivCONTENTS6.2.5Recursive Calculation of Reserves . . . . . . . . . . . . 1626.2.6Paid-Up Insurance . . . . . . . . . . . . . . . . . . . . 1636.3Select Mortality Tables & Insurance . . . . . . . . . . . . . . . 1646.4Exercise Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.5Illustration of Commutation Columns . . . . . . . . . . . . . . 1686.6Examples on Paid-up Insurance . . . . . . . . . . . . . . . . . 1696.7Useful formulas from Chapter 6 . . . . . . . . . . . . . . . . . 1717 Population Theory7.1161Population Functions & Indicator Notation . . . . . . . . . . . 1617.1.1Expectation & Variance of Residual Life . . . . . . . . 1647.2Stationary-Population Concepts . . . . . . . . . . . . . . . . . 1677.3Estimation Using Life-Table Data . . . . . . . . . . . . . . . . 1707.4Nonstationary Population Dynamics . . . . . . . . . . . . . . 1747.4.1Appendix: Large-time Limit of λ(t, x) . . . . . . . . . 1767.5Exercise Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.6Population Word Problems . . . . . . . . . . . . . . . . . . . . 1798 Estimation from Life-Table Data1858.1General Life-Table Data . . . . . . . . . . . . . . . . . . . . . 1868.2ML Estimation for Exponential Data . . . . . . . . . . . . . . 1888.3MLE for Age Specific Force of Mortality . . . . . . . . . . . . 1918.3.1Extension to Random Entry & Censoring Times . . . . 1938.4Kaplan-Meier Survival Function Estimator . . . . . . . . . . . 1948.5Exercise Set 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.6Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 195

CONTENTS9 Risk Models & Select Mortalityv1979.1Proportional Hazard Models . . . . . . . . . . . . . . . . . . . 1989.2Excess Risk Models . . . . . . . . . . . . . . . . . . . . . . . . 2019.3Select Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . 2029.4Exercise Set 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.5Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 20410 Multiple Decrement Models20510.1 Multiple Decrement Tables . . . . . . . . . . . . . . . . . . . . 20610.2 Death-Rate Estimators . . . . . . . . . . . . . . . . . . . . . . 20910.2.1 Deaths Uniform within Year of Age . . . . . . . . . . . 20910.2.2 Force of Mortality Constant within Year of Age . . . . 21010.2.3 Cause-Specific Death Rate Estimators . . . . . . . . . 21010.3 Single-Decrement Tables and Net Hazards of Mortality . . . . 21210.4 Cause-Specific Life Insurance Premiums . . . . . . . . . . . . 21310.5 Exercise Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 21310.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 21411 Central Limit Theorem & Portfolio Risks21513 Bibliography217Solutions & Hints219

vi0.1CONTENTSPrefaceThis book is a course of lectures on the mathematics of actuarial science. Theidea behind the lectures is as far as possible to deduce interesting material oncontingent present values and life tables directly from calculus and commonsense notions, illustrated through word problems. Both the Interest Theoryand Probability related to life tables are treated as wonderful concrete applications of the calculus. The lectures require no background beyond a thirdsemester of calculus, but the prerequisite calculus courses must have beensolidly understood. It is a truism of pre-actuarial advising that students whohave not done really well in and digested the calculus ought not to consideractuarial studies.It is not assumed that the student has seen a formal introduction to probability. Notions of relative frequency and average are introduced first withreference to the ensemble of a cohort life-table, the underlying formal randomexperiment being random selection from the cohort life-table population (or,in the context of probabilities and expectations for ‘lives aged x’, from thesubset of lx members of the population who survive to age x). The calculation of expectations of functions of a time-to-death random variables isrooted on the one hand in the concrete notion of life-table average, which isthen approximated by suitable idealized failure densities and integrals. Later,in discussing Binomial random variables and the Law of Large Numbers, thecombinatorial and probabilistic interpretation of binomial coefficients are derived from the Binomial Theorem, which the student the is assumed to knowas a topic in calculus (Taylor series identification of coefficients of a polynomial.) The general notions of expectation and probability are introduced,but for example the Law of Large Numbers for binomial variables is treated(rigorously) as a topic involving calculus inequalities and summation of finiteseries. This approach allows introduction of the numerically and conceptuallyuseful large-deviation inequalities for binomial random variables to explainjust how unlikely it is for binomial (e.g., life-table) counts to deviate muchpercentage-wise from expectations when the underlying population of trialsis large.The reader is also not assumed to have worked previously with the Theory of Interest. These lectures present Theory of Interest as a mathematicalproblem-topic, which is rather unlike what is done in typical finance courses.

0.1. PREFACEviiGetting the typical Interest problems — such as the exercises on mortgage refinancing and present values of various payoff schemes — into correct formatfor numerical answers is often not easy even for good mathematics students.The main goal of these lectures is to reach — by a conceptual route —mathematical topics in Life Contingencies, Premium Calculation and Demography not usually seen until rather late in the trajectory of quantitativeActuarial Examinations. Such an approach can allow undergraduates withsolid preparation in calculus (not necessarily mathematics or statistics majors) to explore their possible interests in business and actuarial science. Italso allows the majority of such students — who will choose some other avenue, from economics to operations research to statistics, for the exercise oftheir quantitative talents — to know something concrete and mathematicallycoherent about the topics and ideas actually useful in Insurance.A secondary goal of the lectures has been to introduce varied topics ofapplied mathematics as part of a reasoned development of ideas related tosurvival data. As a result, material is included on statistics of biomedicalstudies and on reliability which would not ordinarily find its way into anactuarial course. A further result is that mathematical topics, from differential equations to maximum likelihood estimators based on complex life-tabledata, which seldom fit coherently into undergraduate programs of study, are‘vertically integrated’ into a single course.While the material in these lectures is presented systematically, it is notseparated by chapters into unified topics such as Interest Theory, ProbabilityTheory, Premium Calculation, etc. Instead the introductory material fromprobability and interest theory are interleaved, and later, various mathematical ideas are introduced as needed to advance the discussion. No book atthis level can claim to be fully self-contained, but every attempt has beenmade to develop the mathematics to fit the actuarial applications as theyarise logically.The coverage of the main body of each chapter is primarily ‘theoretical’.At the end of each chapter is an Exercise Set and a short section of WorkedExamples to illustrate the kinds of word problems which can be solved bythe techniques of the chapter. The Worked Examples sections show howthe ideas and formulas work smoothly together, and they highlight the mostimportant and frequently used formulas.

viiiCONTENTS

Chapter 1Basics of Probability and theTheory of InterestThe first lectures supply some background on elementary Probability Theoryand basic Theory of Interest. The reader who has not previously studied thesesubjects may get a brief overview here, but will likely want to supplementthis Chapter with reading in any of a number of calculus-based introductionsto probability and statistics, such as Larson (1982), Larsen and Marx (1985),or Hogg and Tanis (1997) and the basics of the Theory of Interest as coveredin the text of Kellison (1970) or Chapter 1 of Gerber (1997).1.1Probability, Lifetimes, and ExpectationIn the cohort life-table model, imagine a number l0 of individuals bornsimultaneously and followed until death, resulting in data dx , lx for eachage x 0, 1, 2, . . ., wherelx number of lives aged x (i.e. alive at birthday x )anddx lx lx 1 number dying between ages x, x 1Now, allowing the age-variable x to take all real values, not just wholenumbers, treat S(x) lx /l0 as a piecewise continuously differentiable non1

2CHAPTER 1. BASICS OF PROBABILITY & INTERESTincreasing function called the “survivor” or “survival” function. Then for allpositive real x, S(x) S(x t) is the fraction of the initial cohort whichfails between time x and x t, andS(x) S(x t)lx lx t S(x)lxdenotes the fraction of those alive at exact age x who fail before x t.Question: what do probabilities have to do with the life tableand survival function ?To answer this, we first introduce probability as simply a relative frequency, using numbers from a cohort life-table like that of the accompanyingIllustrative Life Table. In response to a probability question, we supply thefraction of the relevant life-table population, to obtain identities likeP r(life aged 29 dies between exact ages 35 and 41 or between 52 and 60 )no. S(35) S(41) S(52) S(60) (l35 l41 ) (l52 l60 )l29where our convention is that a life aged 29 is one of the cohort surviving tothe 29th birthday.The idea here is that all of the lifetimes covered by the life table areunderstood to be governed by an identical “mechanism” of failure, and thatany probability question about a single lifetime is really a question concerningthe fraction of those lives about which the question is asked (e.g., those aliveat age x) whose lifetimes will satisfy the stated property (e.g., die eitherbetween 35 and 41 or between 52 and 60). This “frequentist” notion ofprobability of an event as the relative frequency with which the event occursin a large population of (independent) identical units is associated with thephrase “law of large numbers”, which will bediscussed later. For now, remarkonly that the life table population should be large for the ideas presented sofar to make good sense. See Table 1.1 for an illustration of a cohort life-tablewith realistic numbers.Note: see any basic probability textbook, such as Larson (1982), Larsenand Marx (1985), or Hogg and Tanis (1997) for formal definitions of thenotions of sample space, event, probability, and conditional probability. Themain ideas which are necessary to understand the discussion so far are really

1.1. PROBABILITY3Table 1.1: Illustrative Life-Table, simulated to resemble realistic US (Male)life-table. For details of simulation, see Section 3.4 below.Age 52803

4CHAPTER 1. BASICS OF PROBABILITY & INTERESTmatters of common sense when applied to relative frequency but requireformal axioms when used more generally: Probabilities are numbers between 0 and 1 assigned to subsets of theentire range of possible outcomes (in the examples, subsets of the interval of possible human lifetimes measured in years). The probability P (A B) of the union A B of disjoint (i.e.,nonoverlapping) sets A and B is necessarily the sum of the separateprobabilities P (A) and P (B). When probabilities are requested with reference to a smaller universe ofpossible outcomes, such as B lives aged 29, rather than all membersof a cohort population, the resulting conditional probabilities of eventsA are written P (A B) and calculated as P (A B)/P (B), whereA B denotes the intersection or overlap of the two events A, B. Two events A, B are defined to be independent when P (A B) P (A)·P (B) or — equivalently, as long as P (B) 0 — the conditionalprobability P (A B) expressing the probability of A if B were knownto have occurred, is the same as the (unconditional) probability P (A).The life-table data, and the mechanism by which members of the population die, are summarized first through the survivor function S(x) which atinteger values of x agrees with the ratios lx /l0 . Note that S(x) has valuesbetween 0 and 1, and can be interpreted as the probability for a single individual to survive at least x time units. Since fewer people are alive at largerages, S(x) is a decreasing function of x, and in applications S(x) shouldbe piecewise continuously differentiable (largely for convenience, and becauseany analytical expression which would be chosen for S(x) in practice willbe piecewise smooth). In addition, by definition, S(0) 1. Another way ofsummarizing the probabilities of survival given by this function is to definethe density functionf (x) dS(x) S 0 (x)dxas the (absolute) rate of decrease of the function S. Then, by the fundamental theorem of calculus, for any ages a b,

1.1. PROBABILITY5P (life aged 0 dies between ages a and b) (la lb )/l0 S(a) S(b) Zb0( S (x)) dx aZbf (x) dx(1.1)awhich has the very helpful geometric interpretation that the probability ofdying within the interval [a, b] is equal to the area under the curve y f (x)over the x-interval[a, b]. Note also that the ‘probability’ rule which assignsRthe integral A f (x) dx to the set A (which may be an interval, a unionof intervals, or a still more complicated set) obviously satisfies the first twoof the bulleted axioms displayed above.The terminal age ω of a life table is an integer value large enough thatS(ω) is negligibly small, but no value S(t) for t ω is zero. For practicalpurposes, no individual lives to the ω birthday. While ω is finite in reallife-tables and in some analytical survival models, most theoretical forms forS(x) have no finite age ω at which S(ω) 0, and in those forms ω by convention.Now we are ready to define some terms and motivate the notion of expectation. Think of the age T at which a specified newly born member ofthe population will die as a random variable, which for present purposesmeans a variable which takes various values x with probabilities governedby the life table data lx and the survivor function S(x) or density functionf (x) in a formula like the one just given in equation (1.1). Suppose there is acontractual amount Y which must be paid (say, to the heirs of that individual) at the time T of death of the individual, and suppose that the contractprovides a specific function Y g(T ) according to which this paymentdepends on (the whole-number part of) the age T at which death

Actuarial Mathematics and Life-Table Statistics Eric V. Slud Mathematics Department University of Maryland, College Park c 2001

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