advertisement

Chapter 3. Life tables. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. c 2008. Miguel A. Arcones. All rights reserved. Extract from: ”Arcones’ Manual for SOA Exam MLC. Fall 2009 Edition”, available at http://www.actexmadriver.com/ 1/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Continuous computations Although, a life table does not show values of `x for non integers numbers, we will assume that `x is known for each x ≥ 0. In the next section, we will discuss how to estimate `x for non integers. Knowing `x , for each x ≥ 0, we can get `x , `0 d `0 µ(x) = − (log(`x )) = − x , dx `x Z ∞ `x ◦ e0 = dx, ` Z0 ∞ 0 `x+t ◦ ex = dt. `x 0 s(x) = 2/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Definition 1 Tx is the expected number of years lived beyond age x by the cohort group with l0 members. TheoremR1 (i) Tx = (ii) ∞ 0 `x+t ◦ dt and e x = E [T (x)] = 2 E [(T (x)) ] = 2 Tx `x . R∞ x Ty dy . `x Notice that the expected number of years lived beyond age x by an ◦ individual alive at age x is e x . The expected number of individuals ◦ alive at age x is `x . Hence, Tx = `x e x . 3/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Proof. (i) We have that Tx = `0 E [(X − x)I (X > x)] = `0 P{X > x}E [X − x|X > x] Z ∞ Z ∞ =`x E [T (x)] = `x `x+t dt. t px dt = 0 (ii) Using that Tx = Z 0 ∞ 2 0 R∞ `x+t dt = Z `t dt, we get that Z ∞Z t `t dt dy = 2 x y ∞ Z (t − x)`t dt = 2 =2 x ∞Z ∞ Z Ty dy = 2 x R∞ x `t dy dt x x ∞ u`x+u du. 0 So, 2 R∞ x Ty dy = `x Z ∞ 2u · u px du = E [(T (x))2 ]. 0 4/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Example 1 2 ◦ Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ] and Var(T (x)) using Theorem 1. 5/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Example 1 2 ◦ Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ] and Var(T (x)) using Theorem 1. Solution: We have that Z ∞ Z Tx = `x+t dt = ω−x (x + t)2 `0 1 − dt ω2 0 0 2 2 3ω (ω − x) − ω 3 + x 3 3ω t − (x + t)3 ω−x = `0 = =`0 3ω 2 3ω 2 0 3 2ω − 3ω 2 x + x 3 =`0 ,0 ≤ x ≤ ω 3ω 2 Hence, ◦ ex = 2ω 3 − 3ω 2 x + x 3 2ω 2 − ωx − x 2 (ω − x)(2ω + x) Tx = = = . `x 3(ω 2 − x 2 ) 3(ω + x) 3(ω + x) 6/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Example 1 2 ◦ Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ] and Var(T (x)) using Theorem 1. Solution: We also have that Z ∞ Z ω 3 2ω − 3ω 2 y + y 3 Ty dy = `0 dy 3ω 2 x x 3 8ω y − 6ω 2 y 2 + y 4 ω = 12ω 2 x 8ω 4 − 6ω 4 + ω 4 − 8ω 3 x + 6ω 2 x 2 − x 4 12ω 2 3ω 4 − 8ω 3 x + 6ω 2 x 2 − x 4 = . 12ω 2 = 7/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Example 1 2 ◦ Suppose `x = `0 1 − ωx 2 , for 0 ≤ x ≤ ω. Find e x , E [(T (x))2 ] and Var(T (x)) using Theorem 1. Solution: R∞ Ty dy 3ω 4 − 8ω 3 x + 6ω 2 x 2 − x 4 = `x 6(ω 2 − x 2 ) 3ω 3 − 5ω 2 x + ωx 2 + x 3 (ω − x)2 (3ω + x) = = , 6(ω + x) 6(ω + x) (ω − x)(2ω + x) 2 (ω − x)2 (3ω + x) Var(T (x)) = − 6(ω + x) 3(ω + x) 2 (ω − x) 2 = 3(ω + x)(3ω + x) − 2(2ω + x) 18(ω + x)2 (ω − x)2 = ω 2 + 4ωx + x 2 . 2 18(ω + x) 2 E [(T (x)) ] = 2 x 8/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Definition 2 n Lx is the expected number of years lived between age x and age x + n by the `x survivors at age x. Theorem 2 ◦ Z n Lx = Tx − Tx+n = `x e x:n| = n `x+t dt. 0 9/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Definition 2 n Lx is the expected number of years lived between age x and age x + n by the `x survivors at age x. Theorem 2 ◦ Z n Lx = Tx − Tx+n = `x e x:n| = n `x+t dt. 0 Proof: (i) Since the deceased at age x do not live between age x and age x + n, n Lx is the expected number of years lived between age x and age x + n by the initial `0 lives. So, n Lx = Tx − Tx+n . 10/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Definition 2 n Lx is the expected number of years lived between age x and age x + n by the `x survivors at age x. Theorem 2 ◦ Z n Lx = Tx − Tx+n = `x e x:n| = n `x+t dt. 0 Proof: (i) Since the deceased at age x do not live between age x and age x + n, n Lx is the expected number of years lived between age x and age x + n by the initial `0 lives. So, n Lx = Tx − Tx+n . ◦ (ii) Since e x:n| is the expected number of years lived between age x and age x + n by a live aged x, n Lx ◦ = `x e x:n| . 11/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Definition 2 n Lx is the expected number of years lived between age x and age x + n by the `x survivors at age x. Theorem 2 ◦ Z n Lx = Tx − Tx+n = `x e x:n| = n `x+t dt. 0 Proof: (i) Since the deceased at age x do not live between age x and age x + n, n Lx is the expected number of years lived between age x and age x + n by the initial `0 lives. So, n Lx = Tx − Tx+n . ◦ (ii) Since e x:n| is the expected number of years lived between age x and age x + n by a live aged x, n Lx ◦ (iii) `x e x:n| = `x Rn 0 t px dt = ◦ = `x e x:n| . Rn 0 c 2008. Miguel A. Arcones. All rights reserved. `x+t dt. Manual for SOA Exam MLC. 12/15 Chapter 3. Life tables. Section 3.3. Continuous computations. R1 We will abbreviate Lx = 1 Lx , i.e. Lx = 0 `x+t dt is the expected number of years lived between age x and age x + 1 by the `x survivors at age x. Previous equation display implies that Z 1 Lx = `x+t dt = Tx − Tx+1 . 0 13/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Corollary 1 (i) Tx = ◦ (ii) e x = ◦ P∞ k=x Lk . P∞ k=x Lk . `x P (iii) e x:n| = x+n−1 k=x `x Lk . 14/15 c 2008. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC. Chapter 3. Life tables. Section 3.3. Continuous computations. Corollary 1 (i) Tx = ◦ (ii) e x = P∞ k=x Lk . P∞ k=x Lk . `x P x+n−1 ◦ L (iii) e x:n| = k=x`x k . Proof: (i) Z ∞ Z Tx = `x+t dt = 0 = ∞ X ∞ `t dt = x ∞ Z X k=x k+1 `t dt = k ∞ Z X k=x 1 `k+t dt 0 Lk . k=x ◦ (ii) e x = (iii) Rn ◦ ex = 0 Tx `x = P∞ k=x `x `x+t dt = `x Lk . R x+n x `t dt `x Px+n−1 R k+1 = c 2008. Miguel A. Arcones. All rights reserved. k=x k `t dt `x Manual for SOA Exam MLC. Px+n−1 = k=x `x Lk . 15/15