ACTSC232-Ch3

ACTSC232-Ch3 - Chapter 3 Life tables and selection Chapter...

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Unformatted text preview: Chapter 3 Life tables and selection Chapter 3 Life tables and selection ACTSC 232 Introduction to Actuarial Mathematics Tianxiang Shi Department of Statistics and Actuarial Science University of Waterloo Winter 2012 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Outline 1 Life tables 2 Fractional age assumptions UDD assumption CFM assumption 3 Select and ultimate survival models Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Life tables Heuristic analysis on a life table What a life table looks like? Age x 0 1 2 . . . lx 100,000 99,329 99,285 . . . dx 671 44 30 . . . qx 0.006713 0.000444 0.000300 . . . ex 77.7 77.2 76.3 . . . l0 : initial number (say 100,000 newborns). TEMPORARILY consider lx : # of survivors up to age x dx : # of death between age x and age x + 1 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Life tables US/Canada life tables 2006 US life tables http: //www.cdc.gov/nchs/data/nvsr/nvsr58/nvsr58_21.pdf 2002 Canada life tables http://www.statcan.gc.ca/pub/84-537-x/ 84-537-x2006001-eng.htm Some comments tabulated at integer ages only, and containing lx , dx , qx and other functions. l0 : radix of the table. Arbitrary positive number, usually large. the maximum age is NOT necessarily the limiting age. Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Life tables Survivors to age x (Lx ) Expected number of survivors Denote Lx as the number of survivors up to age x among l0 newborns. Let Ik = 1, if the kth life survives to age x (with prob. x p0 ) 0, otherwise (with prob. x q0 ) l0 Then, Lx = Ik . Define k=1 lx := E[Lx ] If Ik are independent, Lx ∼ Bin(l0 , x p0 ) Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Life tables t px and lx The mean of Lx l0 E[Ik ] = l0 · x p0 lx = E[Lx ] = k=1 In general t px = x+t p0 x p0 = lx+t /l0 lx+t = lx /l0 lx or lx+t = lx · t px In particular lx+1 = lx · px , and px = Tianxiang Shi([email protected]) lx+1 lx Chapter 3 Life tables and selection Life tables Death between ages x and x + n (n Dx ) Expected number of death between ages x and x + n Denote n Dx as the number of death between ages x and x + n among l0 newborns. Let uk = 1, kth life dies b/t ages x & x + n (w.p. 0, otherwise (w.p. 1 − x|n q0 ) l0 Then, n Dx = Lx − Lx+n = uk . Define k=1 n dx := E[n Dx ] and denote: dx = 1 dx If uk are independent, Lx ∼ Bin(l0 , x|n q0 ) Tianxiang Shi([email protected]) x|n q0 ) Chapter 3 Life tables and selection Life tables n qx , n dx and lx The mean of n Dx n dx = E[n Dx ] = E[Lx − Lx+n ] = lx − lx+n or l0 n dx E[uk ] = l0 · x|n q0 = lx · n qx = k=1 In particular lx = lx+1 + dx , and qx = dx lx Other property n dx Tianxiang Shi([email protected]) = dx + dx+1 + · · · + dx+n−1 Chapter 3 Life tables and selection Life tables Other functions k|n qx in terms of lx k|n qx = k px − k+n px = lx+k − lx+k+n lx Total expected # of years lived beyond age x by those survivors of l0 newborns ∞ ∗ Tx = lx+t dt = lx · ˚x e 0 ∗ Note: Tx corresponds to Tx in the US life table. Total expected # of years lived b/t ages x and x + n of l0 newborns n n Lx = 0 Tianxiang Shi([email protected]) ∗ ∗ lx+t dt = lx · ˚x:n = Tx − Tx+n e Chapter 3 Life tables and selection Life tables Example Example x lx dx 30 1500 50 31 1450 (1) 32 1390 65 33 1025 75 34 950 90 35 (2) You are given the incomplete life table, (i) Fill in the blanks (1) and (2) (ii) Calculate 2 p30 , q34 and 3 q31 (iii) Calculate the probability that a life (31) dies b/t ages 33 and 35 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Life tables Life table under De Moivre’s law Under the De Moivre’s law, Tx ∼ U (0, ω − x) µx = 1 t , and t px = 1 − ω−x ω−x for 0 ≤ t ≤ ω − x (x ≤ ω) Life table under De Moivre’s law lx = c · (ω − x), where c is a constant. For the generalized De Moivre’s law with µx = lx = c · (ω x)α , α > 0 Tianxiang Shi([email protected]) α ω−x Chapter 3 Life tables and selection Fractional age assumptions Fractional age assumptions (FAA) Life table tabulated at integer ages, which is not sufficient to calculate probabilities involving fractional ages, e.g., 0.6 px or 7.2 q10 . Additional assumption on the distribution of the future lifetime r.v. b/t integer ages is needed. We use the term fractional age assumption” (FAA) refer to these assumptions. Common FAAs (based on different interpolations) Uniform distribution of deaths (UDD) Constant force of Mortality (CFM) Balducci (also known as Hyperbolic) Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Fractional age assumptions UDD assumption Two equivalent UDD assumptions UDD 1 For x ∈ N and 0 ≤ s < 1, assume that s qx = s · qx UDD 2 Let Rx = Tx − Kx be the fractional part of Tx . For x ∈ N assume that Rx ∼ U (0, 1), and Rx is independent of Kx Immediate consequences lx = lx+s + s · dx Linear interpolation: lx+s = (1 − s) · lx + s · lx+1 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Fractional age assumptions UDD assumption Probabilities under UDD assumption For x ∈ N and 0 ≤ s < 1, we have s qx = s · qx , and s px = 1 − s · qx = (1 − s)0 px + s · px We also have fx (s) = qx , and µx+s = (Question: What about fx (n + s) and For t px+s , qx 1 − s · qx n+s px ?) Transfer to an integer age and use prev. formula t+s px t px+s = s px Using life table function t px+s Tianxiang Shi([email protected]) = lx+t+s lx+s Chapter 3 Life tables and selection Fractional age assumptions UDD assumption Example Example Under UDD assumption, express following terms using life table functions, i.e. lx and/or dx , for x ∈ N (i) 0.6 p50 (ii) 0.3 q31.6 (iii) 6.7 q8.5 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Fractional age assumptions CFM assumption CFM assumptions CFM b/t ages Recall that 1 0 px = e− µx+s ds for x ∈ N and 0 ≤ s < 1. Now we assume that µx+s = c(x) for 0 ≤ s < 1 (Note that c(x) depends on x), then px = e−c(x) Immediate consequences s px Tianxiang Shi([email protected]) = e− s 0 µx+t dt = (px )s Chapter 3 Life tables and selection Fractional age assumptions CFM assumption Probabilities under CFM assumption Exponential (Log linear) interpolation: ln lx+s = (1 − s) ln lx + s ln lx+1 or ln n+s px = (1 − s) ln n px + s ln n+1 px for x ∈ N and 0 ≤ s < 1. We also have s qx = 1 − (px )s For t px+s , transfer to an integer age t px+s Tianxiang Shi([email protected]) = t+s px s px Chapter 3 Life tables and selection Fractional age assumptions CFM assumption Example Example Under CFM assumption, express following terms using life table functions, i.e. lx and/or dx , for x ∈ N (i) 0.6 p50 (ii) 0.3 q31.6 (iii) 6.7 q8.5 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Fractional age assumptions CFM assumption Balducci assumption and others Harmonic interpolation: 1 1−s s = + lx+s lx lx+1 or 1 1−s = + n+s px n px s n+1 px for x ∈ N and 0 ≤ s < 1. UDD v.s. CFM: s px under UDD is larger than one under CFM Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Select process Underwriting Consider an applicant who wants to buy a life insurance product. Underwriting is the process of collecting the applicant’s relevant information and evaluating related factors, such as age, gender, smoking habits, occupation, dangerous hobbies, personal/family health history, etc.. These factors are called rating factors. The insurers may also require the applicant to do medical examinations. Purpose of underwriting to classify potential policyholders into broadly homogeneous risk categories to assess appropriate premiums for applicants Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Risk categories Preferred lives: very low mortality risk based on the standard information; no undesirable indicators and evidence. Normal lives: have some higher rated risk factors than preferred lives but still insurable at standard rates. Most applicants fall into this category. (The above 2 categories account for 95% for traditional life insurance) Rated lives: one or more risk factors at raised levels. Can be insured at a higher premium. e.g. a family history of heart disease. (2-3%) Uninsurable lives: significant risk that insurer won’t sell insurance to them. (2-3%) Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Underwriting effect Adverse selection: If underwriting process is not strict, very high-risk individuals will buy insurance in disproportionate numbers, leading to excessive losses. When insurers increase their rates, they tends to further lose preferred lives. Selection effect Accepted applicants are expected to have lower mortality rates than the general population (national life tables). Selection effect will wear off eventually. This leads us to study the select and ultimate survival models. Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Select and ultimate survival models Model description An individual who enters the group (e.g. who buys a term insurance) at, say, age x, is said to be selected (or select) at age x. A select and ultimate survival model follows: future survival probabilities for a life in the group depends on the current age and on the age at which s/he joined the group. The initial selection effect is assumed to have worn off after d years, where d is called the selection period (generally a positive integer). After d years, the mortality is called ultimate mortality. Remark: A select and ultimate survival model is also shorten for select survival model. Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Notations under select survival models [x]: a life selected at age x ([x] + t): a life selected at age x and currently aged at x + t t p[x]+s := S[x]+s (t): Pr(a life ([x] + s) will survive t years) t q[x]+s = 1 − t p[x]+s : Pr(a life ([x] + s) will die within t years) u|t q[x]+s : Pr(a life ([x] + s) will survive u years and then die within the next t years) the force of mortality at age x + t for a life selected at age x µ[x]+t := lim h→0+ 1 Pr(T[x]+t ≤ h), h where T[x]+t is the future lifetime of ([x] + t) Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Formulas under select survival models Under a select survival model, T[x]+t = T[x] − t|T[x] ≥ t From this assumption, we have t+u p[x]+s = u p[x]+s · t p[x]+s+u , or t p[x]+s+u = In particular, t p[x]+u = t+u p[x]+s u p[x]+s t+u p[x] u p[x] BUT, the select age can not be broken in the formula t p[x+u] Tianxiang Shi([email protected]) = t+u p[x] u p[x] Chapter 3 Life tables and selection Select and ultimate survival models Aggregate/Select and ultimate survival models Aggregate (Ch. 2) v.s. Select and ultimate Aggregate survival models (the model in Ch. 2) all functions such as t px , t qx , and lx depend only on current age x no selection effect assume Tx = T0 − x|T0 ≥ x Select and ultimate survival models all functions such as s p[x]+t , s q[x]+t , and l[x]+t depends on current age x + t and select age x assume T[x]+t = T[x] − t|T[x] ≥ t All the formulas regarding t px , t qx , u|t qx and µx can be transplanted from the aggregate survival models to the select and ultimate survival models, as long as we DO NOT break the select age x. Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Select periods Select period d and probabilities Select part: within the selection period e.g., for t < d, s p[x]+t = s px+t , and s q[x]+t = s qx+t Ultimate part: selection effect will wear off after d years, that is T[x]+t and Tx+t are identically distributed for t ≥ d. e.g., s p[x]+t Tianxiang Shi([email protected]) = s px+t , and s q[x]+t = s qx+t Chapter 3 Life tables and selection Select and ultimate survival models Example Example Assume that the selection period is 2 years. Compare (i.e. >, =, or <) (i) s q[x] v.s. s qx (ii) s q[x]+1 v.s. s q[x+1] (iii) s q[x]+2 v.s. s q[x+1]+1 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Example Example x 30 31 32 33 34 1000q[x] 0.222 0.234 0.250 0.269 0.291 1000q[x]+1 0.330 0.352 0.377 0.407 0.441 1000qx+2 0.422 0.459 0.500 0.545 0.596 x+2 32 33 34 35 36 From the above 2-year select and ultimate life table, find out and 3 q[31]+1 5 p[30] , 1|2 q[31] Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Select and ultimate survival models Select and ultimate life tables Construct d-year select life tables l[x]+t : expected number of survivors at age x + t for those individuals selected at age x. Similarly, we can define n d[x]+t . For t ≥ d, the selection effect has worn off l[x]+t = lx+t For 0 ≤ t < d, we define in a backward way l[x]+t = Tianxiang Shi([email protected]) lx+d d−t p[x]+t Chapter 3 Life tables and selection Select and ultimate survival models Probabilities: l[x]+t and n d[x]+t For t p[x]+s l[x]+t = l[x] · t p[x] , or t p[x] = l[x]+t l[x] and in general t p[x]+s = l[x]+s+t l[x]+s For t q[x]+s t q[x]+s For = t d[x]+s l[x]+s , and t q[x] = u|t q[x]+s u|t q[x]+s Tianxiang Shi([email protected]) = t d[x]+s+u l[x]+s t d[x] l[x] Chapter 3 Life tables and selection Select and ultimate survival models Example Example x l[x] l[x]+1 20 21 . . . 995 970 . . . 973 948 . . . lx+2 1000 975 949 923 . . . x+2 20 21 22 23 . . . From the above 2-year select and ultimate life table, find out (i) 3 p[20] (ii) 1|2 q[20] (iii) 2 q[20]+1 (iv) 0.9 q[20]+0.6 Tianxiang Shi([email protected]) ...
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This note was uploaded on 03/30/2012 for the course ACTSC 232 taught by Professor Matthewtill during the Winter '08 term at Waterloo.

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