Life Tables and Selection

Transcription

1 Life Tables and Selection Lecture: Weeks 4-5 Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 1 / 29

2 Chapter summary Chapter summary What is a life table? also called a mortality table tabulation of basic mortality functions deriving probabilities/expectations from a life table Relationships to survival functions Assumptions for fractional (non-integral) ages Select and ultimate tables national life tables valuation or pricing tables Chapter 3, DHW Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 2 / 29

3 The life table What is the life table? A tabular presentation of the mortality evolution of a cohort group of lives. Begin with l 0 number of lives (e.g. 100,000) - called the radix of the life table. (Expected) number of lives who are age x: l x = l 0 S 0 (x) = l 0 p x 0 (Expected) number of deaths between ages x and x + 1: d x = l x l x+1. (Expected) number of deaths between ages x and x + n: d n x = l x l x+n. Conditional on survival to age x, the probability of dying within n years is: nqx = n d x /l x = (l x l x+n )/l x. Conditional on survival to age x, the probability of living to reach age x + n is: n p x = 1 nqx = l x+n /l x. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 3 / 29

4 The life table example of a life table Example of a life table x l x d x q x p x e x 0 100, , , , , , , , ,926 1, ,556 1, , Source: U.S. Life Table for the total population, 2004, Center for Disease Control and Prevention (CDC) Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 4 / 29

5 The life table Radix of the life table The radix of the life table does not have to start at age 0, e.g. start with age x 0, so that the table starts with radix l x0. The limiting age of the table is usually denoted by ω, in which case the table gives entries for only a period of ω x 0. All the formulas still work, e.g. conditional on survival to age x, the probability of surviving to reach age x + n is: npx = 1 nqx = l x+n. l x Note that among l x independent lives who have reached age x, the number of survivors L n within n years is a Binomial random variable with parameters l x and n p x so that E(L n ) = l x p n x. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 5 / 29

6 The life table Table 3.1 Revised example 3.1 Using Table 3.1, page 43 of DHW, calculate the following: the probability that (30) will survive another 5 years the probability that (39) will survive to reach age 40 the probability that (30) will die within 10 years the probability that (30) dies between ages 36 and 38 Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 6 / 29

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8 The life table examples Illustrative example 1 Complete the following life table: x l x d x p x q x 40 24, , , , , ,495 Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 7 / 29

9 The life table additional useful formulas Additional useful formulas From a life table, the following formulas can also easily be verified (or use your intuition): l x = k=0 d x+k: the number of survivors at age x should be equal to the number of deaths in each year of age for all the following years. ndx = l x l x+n = n 1 k=0 d x+k: the number of deaths within n years should be equal to the number of deaths in each year of age for the next n years. Finally, the probability that (x) survives the next n years but dies the following m years after that can be derived using: n mqx = n p x n+mpx = d m x+n l x = l x+n l x+n+m l x. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 8 / 29

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12 The life table force of mortality The force of mortality It is easy to show that the force of mortality can be expressed in terms of life table function as: Thus, in effect, we can also write µ x = 1 l x dl x dx. l x = l 0 exp ( x 0 ) µ z dz. With a simple change of variable, it is easy to see also that µ x+t = 1 dl x+t l x+t dt It follows immediately that: d dt p t x = p t x µ x+t. = 1 d t p x t p x dt. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 9 / 29

13 The life table curtate expectation of life Curtate expectation of life Recall the expected value of K x is called the curtate expectation of life. It can be expressed now as E[K x ] = e x = k=1 kpx = k=1 l x+k l x. The n-year temporary curtate expectation of life is e x : n = n k=1 kpx = n k=1 l x+k l x, which gives the average number of completed years lived over the interval (x, x + n] for a life (x). Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 10 / 29

14 The life table examples Illustrative example 2 Suppose you are given the following extract from a life table: x l x 94 16, , , , , , Calculate e Calculate the variance of K 95, the curtate future lifetime of (95). 3 Calculate e 95: 3. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 11 / 29

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17 The life table examples Illustrative example 3 For a life (x), you are given l x = 10, 000 and the following extract from a life table: Calculate q 2 x+1 k d x+k and interpret this probability. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 12 / 29

18 The life table typical human mortality curves mortality rates life expectancy age age Figure: Source: Life Tables, 2007 from the Social Security Administration - male (blue), female (red) Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 13 / 29

19 Fractional age assumptions Fractional age assumptions When adopting a life table (which may contain only integer ages), some assumptions are needed about the distribution between the integers. The two most common assumptions (or interpolations) used are (where 0 t 1): 1 linear interpolation (also called UDD assumption): l x+t = (1 t)l x + tl x+1 2 exponential interpolation (equivalent to constant force assumption): log l x+t = (1 t) log l x + t log l x+1 Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 14 / 29

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22 Fractional age assumptions summary of results Some results on the fractional age assumptions Linear Exponential Function (UDD) (constant force) tqx t q x 1 (1 q x ) t µ x+t q x 1 t q x µ = log p x tpxµ x+t q x µe µt Here we have 0 t 1. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 15 / 29

23 Fractional age assumptions examples Illustrative example 4 You are given the following extract from a life table: x l x 55 85, , , ,114 Estimate 1.4p55 and q non-integral ages: (a) UDD; and (b) constant force. Interpret these probabilities. under each of the following assumptions for Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 16 / 29

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26 Fractional age assumptions examples Illustrative example 5 Assume the Uniform Distribution of Death (UDD) assumption holds between integer ages. You are given: 0.5p65 = p66 = 0.92 Calculate the probability that (65) will survive the next two years. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 17 / 29

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43 Fractional age assumptions Fractional part of the year lived Denote by R x the fractional part of a year lived in the year of death. Then we have T x = K x + R x where T x is the time-until-death and K x is the curtate future lifetime of (x). We can describe the joint probability distribution of (K x, R x ) as Pr [(K x = k) (R x s)] = Pr[k < T x k + s] = p k x q s x+k, for k = 0, 1,... and for 0 < s < 1. The UDD assumption is equivalent to the assumption that the fractional part R x occurs uniformly during the year, i.e. R x U(0, 1). It can be demonstrated that K x and R x are independent in this case. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 18 / 29

44 Select and ultimate tables Select and ultimate tables Group of lives underwritten for insurance coverage usually has different mortality than the general population (some test required before insurance is offered). Mortality then becomes a function of age [x] at selection (e.g. policy issue, onset of disability) and duration t since selection. For select tables, notation such as t q [x], t p [x], and l [x]+t, are then used. However, impact of selection diminishes after some time - the select period (denoted by r). In effect, we have q [x]+j = q x+j, for j r. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 19 / 29

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46 Select and ultimate tables example of a select and ultimate table Example of a select and ultimate table [x] 1000q [x] 1000q [x] q x+2 l [x] l [x]+1 l x+2 x ,907 9,905 9, ,903 9,901 9, ,899 9,896 9, ,894 9,892 9, ,889 9,887 9, From this table, try to compute probabilities such as: (a) 2 p [30] ; (b) 5 p [30] ; (c) 1 q [31] ; and (d) 3 q [31]+1. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 20 / 29

47 Select and ultimate tables examples Illustrative example 6 A select and ultimate table with a three-year select period begins at selection age x. You are given the following information: l x+6 = 90, 000 q [x] = 1 6 p 5 [x+1] = p[x]+1 = p [x+1]. Evaluate l [x]. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 21 / 29

48 Select and ultimate tables examples Illustrative example 7 You are given the following extract from a select and ultimate life table: Calculate 1000 fractional ages. q [x] l [x] l [x]+1 l x+2 x ,616 29,418 29, ,131 28,920 28, ,601 28,378 28, [60]+0.8, assuming a constant force of mortality at Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 22 / 29

49 Select and ultimate tables examples Illustrative example 8 You are given the following extract from a select and ultimate life table: [x] l [x] l [x]+1 l x+2 x ,625 79,954 78, ,137 78,402 77, ,770 75, Approximate e [65]: 2 using the trapezium (trapezoidal) rule with h = 0.5 and assuming UDD for fractional ages. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 23 / 29

50 Select and ultimate tables examples Illustrative example 9 For a select-and-ultimate mortality table with a 3-year select period, you are given: (i) x q [x] q [x]+1 q [x]+2 q x+3 x (ii) Becky was a newly selected life on 01/01/2012. (iii) Becky s age on 01/01/2012 is 61. (iv) Q is the probability on 01/01/2012 that Becky will be dead by 01/01/2017. Calculate Q. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 24 / 29

51 Select and ultimate tables examples Illustrative example 10 - modified SOA MLC Spring 2012 Suppose you are given: p 50 = 0.98 p 51 = 0.96 e 51.5 = 22.4 The force of mortality is constant between ages 50 and 51. Deaths are uniformly distributed between ages 51 and 52. Calculate e Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 25 / 29

52 Select and ultimate tables examples Illustrative example 11 - modified SOA MLC Spring 2012 In a 2-year select and ultimate mortality table, you are given: q [x]+1 = 0.96 q x+1 l 65 = 82, 358 l 66 = 81, 284 Calculate l [64]+1. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 26 / 29

53 Select and ultimate tables examples Illustrative example 12 - SOA MLC Fall 2014 MC#20 For a mortality table with a select period of two years, you are given: (i) x q [x] q [x]+1 q x+2 x (ii) The force of mortality is constant between integral ages. Calculate q[50]+0.4. Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 27 / 29

54 Mortality trends Mortality projection factors Read Section 3.11 Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 28 / 29

55 Mortality trends other notation Only other symbol used in the MLC exam Expression number of lives SOA adopts the symbol l x Lecture: Weeks 4-5 (Math 3630) Life Tables and Selection Fall Valdez 29 / 29