Actuarial mathematics

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1 Actuarial mathematics Multiple life tables Edward Furman Department of Mathematics and Statistics York University November 2, 2011 Edward Furman Actuarial mathematics MATH / 15

2 Life table Random survivorship group Definition 1.1 (Life table.) We shall call the distribution of (u) that describes its life time with respect to all existing sources of decrement a multiple decrement life table. Example 1.1 [x, x + 1) q (1) x q (2) x qx τ lx τ d (1) x d (2) x [0, 1) [1, 2) [2, 3) [3, 4) [4, 5) Looks like we shall need even more notations... Edward Furman Actuarial mathematics MATH / 15

3 Definition 1.2 (Random number of survivors to age x.) Let us have a group of l τ 0 new born children. Then, for 1τ {j} indicating the survival of the new born child number j to age x, L τ (x) := l0 τ 1 τ {j} j=1 denotes the number of children alive at age x. L τ (x) is an r.v. Definition 1.3 (Expected number of survivors to age x.) For x p0 τ = P[1τ {j} = 1] for every j = 1,...,l 0, the expectation of L τ (x) is l0 τ lx τ := E[Lτ (x)] = E 1 τ {j} = l0 τ x p0 τ. j=1 Edward Furman Actuarial mathematics MATH / 15

4 More generally The same can be done for any person or life status age y for y 0. Namely, the number of people surviving from age y to age x, where x y is l τ x := l τ y x y p τ y. Definition 1.4 Let n D τ (x) := L τ (x) L τ (x + n) denote the group of deaths between ages x and x + n. We then define the expected number of deaths (out of l0 τ and between the aforementioned ages) ndx τ := E[ nd τ (x)] = l0 τ ( xp0 τ x+np0 τ ) = lτ x lτ x+n. Edward Furman Actuarial mathematics MATH / 15

5 Definition 1.5 We shall define by n D i (x), the random number of people age 0 that leave the group between the ages x and x + n. due to source of decrement i. Then the expected number of such people is for any y x. x+n nd x (i) := E[ n D i (x)] = l0 τ = l τ 0 = l τ y x+n x x+n y x y x tp τ 0 µ(i) (t)dt yp τ 0 t y p τ y µ (i) (t)dt sp τ y µ (i) (s+y)ds, Edward Furman Actuarial mathematics MATH / 15

6 Remark. We should have that nd τ (x) = nd i (x), for m being the number of sources of decrement and fixed x and n. Thus ndx τ = nd x (i), Definition 1.6 Divide the previous identity by l τ x throughout, and have that nq x (i) := n d x (i) /lx τ and nq x (i) = n q τ x Edward Furman Actuarial mathematics MATH / 15

7 Remember. For, e.g., two decrements only we have that t qx τ = tq x (1) + t q x (2) as we it has been just shown. However, same additive relation does not hold for the p-related functions. Namely, Definition 1.7 tpx τ = 1 t qx τ = 1 t q x (1) t q x (2) 1 t q x (1) + 1 t q x (2) = t p x (1) + t p x (2). The force of decrement due to the i-th source of decrement is defined for a new born child as µ i (t) := 1 tp 0 d dt t q (i) 0. Edward Furman Actuarial mathematics MATH / 15

8 Proposition 1.1 We have that µ τ (t) = µ i (t). Proof. Recall that tp τ 0 µτ (t) = d dt qτ (t) = d dt tq (i) m 0 = d dt t q (i) 0 = t p0 τ µ i (t), as needed. Note that for µ τ (t) to be a legitimate force of mortality, there is no need that lim t µi (t) = for every i = 1,...,m. Edward Furman Actuarial mathematics MATH / 15

9 Definition 1.8 Let (T(u), I(u)) := (T, I) be a random vector with T(u) standing for the future life time of a status age u, and I(u) denoting the source of decrement. The expression f T, I (t, i) t := P[t < T t + t, I = i], t R +, i = 1,...,m is then interpreted as the probability that a life status will leave in the interval (u + t, u + t + t] due to the i-th source of decrement. Also, 1 m f T (t) := lim t 0 dt P[t < T t + t] = f T, I (t, i) and p I (i) := P[I = i] = 0 f T, I (t, i) := q (i) u. Edward Furman Actuarial mathematics MATH / 15

10 Remark. Note that the density f T (t) is exactly the one we have encountered in the single decrement life tables, and it is interpreted using t as f T (t) t is the probability that (u) leaves the life table (all possible decrements are included). Also, p I (i) is a new object, and it is seen as the probability (u) leaves due to the i-th source of decrement any time. Of course f T (t)dt = 1 and p I (i) = 1. Definition Similarly to Definition 1.8. we define the probability that (u) leaves due to the i-th source of decrement during the following t years as t tq u (i) := f T, I (s, i)ds, i = 1,...,m. 0 Edward Furman Actuarial mathematics MATH / 15

11 Remark. As when discussing the random survivorship group, we redenote tq τ u := P[T(u) < t] = t o f T (s)ds tpu τ := P[T(u) t] = 1 t qu τ µ τ (u + t) := 1 d tqu τ dt t qu τ µ (i) (u + t) := lim P[t < T t + h, I = i, T > t] h = f T, I(t, i) P[T t] = f T, I(t, i), from where h 0 1 f T, I (t, i) := t p τ u µ (i) (u + t) tp τ u Edward Furman Actuarial mathematics MATH / 15

12 Proposition 1.2 We have that and µ τ (u + t) = µ i (u + t) tq τ u = m tq (i) u. Proof. Indeed tq τ u = = t 0 f T (s)ds = t 0 t 0 f T, I (s, i)ds f T, I (s, i)ds = tq (i) u, Edward Furman Actuarial mathematics MATH / 15

13 Proof. that proves the second expression. Differentiating it throughout completes the proof. Proposition 1.3 The conditional density of I T is f I T (i t) = µi (u + t) µ τ (u + t). Edward Furman Actuarial mathematics MATH / 15

14 Proposition 1.4 The p.m.f. of (K, I) is P[K = k, I = i] = k pu τ q (i) u 0, k 0, and i = 1,...,m. Proof. u+k, for fixed P[K = k, I = i] = P[k T < k + 1, I = i] = P[k < T k + 1, I = i] = = = k+1 k 1 tp τ u µ i (u + t)dt 0 s+k p τ u µ i (u + s + k)ds 1 0 = k p τ u q (i) u+k, sp τ u+s kp τ u µ i (u + s+k)ds Edward Furman Actuarial mathematics MATH / 15

15 Proof. that is true if there is no select periods in the table and thus completes the proof. Edward Furman Actuarial mathematics MATH / 15