Chapter 2. Survival models.
Manual for SOA Exam MLC.
Chapter 2. Survival models. Section 2.7 Common Analytical Survival Models
c 2009. Miguel A. Arcones. All rights reserved.
Extract from: Arcones Manual for SOA Exam MLC. Fall 2009 Edition, available at http://www.actexmadriver.com/
1/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Common Analytical Survival Models
Sometimes it is of interest to assume that the survival function follows a parametric model, i.e. it is of the form S (x , ), where is an unknown parameter. There are several reasons to make this assumption: 1. Data supports this assumption. Actuaries realized that the models observed in real life follow this assumption. 2. Computations are simpler using a parametric model. 3. There are valid scientic reasons to justify the use of a particular parametric model.
2/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
The most common approach is not to use parametric models due to the following reasons: 1. Modern computers allow to handle the computations needed using the collected data. 2. It is dicult to justify that a parametric model applies. 3. Knowing that a particular model applies we can get more accurate estimates. But, this increase in accuracy is not much. If a parametric model does not apply, using the parametric approach we can get much worse estimates than the nonparametric estimates.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
De Moivre model.
De Moivres law (1729) assumes that deaths happen uniformly over the interval of deaths, i.e. the density of the ageatfailure is 1 , for 0 x . Therefore, fX (x ) = SX (x ) = x , for 0 x < , x FX (x ) = , for 0 x < , 1 (x ) = , for 0 x < , x s (x + t ) x t = , for 0 t x , t px = s (x ) x t , for 0 t x . t qx = x
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 1
Find the median of an ageatdeath subject to de Moivres law if 1 the probability that a life aged 20 years survives 40 years is 3 .
5/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 1
Find the median of an ageatdeath subject to de Moivres law if 1 the probability that a life aged 20 years survives 40 years is 3 . Solution: We know that 40 p20 = 2 3 . For the uniform distribution, 1 2040 x t t px = x . Hence, 3 = 20 , 20 = 3 180 and = 80. Let m be the median of the ageatdeath. Then, 1 m 80 m = SX (m) = = 2 80 and m = 40.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Under the De Moivres law, T (x ) has a uniform distribution on the interval [0, x ]. Hence,
Theorem 1
For the De Moivres law, ex =
( x )2 x and Var(T (x )) = . 2 12
7/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 2
Suppose that the survival of a cohort follows the De Moivres law. Suppose that the expected ageatdeath of a new born is 70 years. Find the expected future lifetime of a 50year old.
8/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 2
Suppose that the survival of a cohort follows the De Moivres law. Suppose that the expected ageatdeath of a new born is 70 years. Find the expected future lifetime of a 50year old. Solution: Since 70 = e0 = lifetime of a 50year old is
2,
= 140. The expected future
e 50 =
140 50 = 45. 2
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 2
For the de Moivres law, for 0 x , x, integers, ex = x 1 . 2
10/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 2
For the de Moivres law, for 0 x , x, integers, ex = Proof: We have that
x
x 1 . 2
ex =
k =1
x k = x
1
k =1 k =1
k x
x )(( x ) + 1) x) + 1 =( x ) = ( x ) 2( x ) 2 x 1 . = 2
11/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 3
For the de Moivres law with terminal age , where is a positive integer, and each for 0 x , where x is an integer, P{K (x ) = k } = and
k 1 P{Kx = k } = p1 (1 p1 ), k = 1, 2, . . . , x 1.
1 , k = 0, 1, 2, . . . , x . x
Proof:
k +1
P{K (x ) = k } = P{k < Tx k + 1} =
k
1 1 dt = x x
12/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Exponential model.
An exponential model assumes that the ageatdeath has an exponential distribution, i.e. SX (x ) = e x , x 0 where > 0. In this case, FX (x ) = 1 e x , for 0 x , fX (x ) = e x , for 0 x , (x ) = , for 0 x , s (x + t ) t . = e t = px t px = s (x ) For an exponential model, the force of mortality is constant. The exponential model is also called the constant force model.
13/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 3
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the probability that a 40yearold will survive to age 50. (ii) the probability that a 30yearold will survive to age 50. (iii) the probability that a 30yearold will die between ages 40 and 50.
14/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 3
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the probability that a 40yearold will survive to age 50. (ii) the probability that a 30yearold will survive to age 50. (iii) the probability that a 30yearold will die between ages 40 and 50. Solution: (i) Since the force of mortality is constant, 10 p40 = 10 p30 = 0.95.
15/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 3
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the probability that a 40yearold will survive to age 50. (ii) the probability that a 30yearold will survive to age 50. (iii) the probability that a 30yearold will die between ages 40 and 50. Solution: (i) Since the force of mortality is constant, 10 p40 = 10 p30 = 0.95. (ii) 20 p30 = 10 p30 10 p40 = (0.95)(0.95) = 0.9025.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 3
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the probability that a 40yearold will survive to age 50. (ii) the probability that a 30yearold will survive to age 50. (iii) the probability that a 30yearold will die between ages 40 and 50. Solution: (i) Since the force of mortality is constant, 10 p40 = 10 p30 = 0.95. (ii) 20 p30 = 10 p30 10 p40 = (0.95)(0.95) = 0.9025. (iii) We can do either
10 |10 p30
= 10 p30 20 p30 = 0.95 0.9025 = 0.0475. = 10 p30 10 q40 = (0.95)(1 0.95) = 0.0475.
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or
10 |10 p30
c 2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 4
Suppose that the lifetime random variable of a new born has constant mortality force . Then, ex =
1 1 and Var(X ) = 2 .
18/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 4
Suppose that the lifetime random variable of a new born has constant mortality force . Then, ex =
1 1 and Var(X ) = 2 .
Proof: Since t px = e t , T (x ) has an exponential distribution 1 with mean .
19/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 5
Suppose that the lifetime random variable of a new born has constant mortality force . Then, the curtate future lifetime of (x ) is ex = 1 . e 1
20/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Theorem 5
Suppose that the lifetime random variable of a new born has constant mortality force . Then, the curtate future lifetime of (x ) is ex = Proof: We have that
k px = k =1 k =1
1 . e 1
ex =
e k =
e 1 = . 1e e 1
21/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 4
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the future lifetime of a 40yearold. (ii) the future curtate lifetime of a 40yearold.
22/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 4
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the future lifetime of a 40yearold. (ii) the future curtate lifetime of a 40yearold. Solution: (i) We know that 10 p30 = 0.95 = e (10) . Hence, .95) 1 10 = ln(0 and e 40 = = ln(0 10 .95) = 194.9572575.
23/27
c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Example 4
Suppose that: (i) the force of mortality is constant. (ii) the probability that a 30yearold will survive to age 40 is 0.95. Calculate: (i) the future lifetime of a 40yearold. (ii) the future curtate lifetime of a 40yearold. Solution: (i) We know that 10 p30 = 0.95 = e (10) . Hence, .95) 1 10 = ln(0 and e 40 = = ln(0 10 .95) = 194.9572575. 0 . 1 (ii) Since e = (0.95) , 1 e40 = e 1 1 = (0.95)0.1 1 = 194.4576849.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Gompertz model.
Gompertzs model (1825) says that x = Bc x , where B > 0 and x B c > 1. Hence, s (x ) = e m(c 1) for x 0, where m = log c.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Makeham model.
Makehan (1860) introduced the model x = A + Bc x , where x A B , B > 0 and c > 1. Hence, s (x ) = e Ax m(c 1) for B x 0, where m = log c . We also have that fX (x ) = s (x )(x ) = (A + Bc x )e Ax m(c
x 1)
, x 0.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
Chapter 2. Survival models.
Section 2.7 Common Analytical Survival Models.
Weibull model.
Weibull (1939) introduced the model (x ) = kx n , for x 0, where k > 0 and n > 1. Then, s (x ) = e fX (x ) = s (x )(x ) = kx n e
kx n+1 n+1
,x 0
n+1 kx n+1
, x 0.
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c 2009. Miguel A. Arcones. All rights reserved.
Manual for SOA Exam MLC.
