Scribd
Upload
279 views12 pages

AM1 - Tutorial 1

This document is a tutorial on survival models for an actuarial mathematics course. It contains 12 questions about calculating survival probabilities, forces of mortality, and properties of survival models. The tutorial covers concepts like survival functions, life tables, distribution and density functions of lifetimes, and forces of mortality for mixed male-female populations. It also explores the relationships between survival probabilities, forces of mortality, and proportional hazards models.

Uploaded by

Nguyễn Quang Trường
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
279 views12 pages

AM1 - Tutorial 1

This document is a tutorial on survival models for an actuarial mathematics course. It contains 12 questions about calculating survival probabilities, forces of mortality, and properties of survival models. The tutorial covers concepts like survival functions, life tables, distribution and density functions of lifetimes, and forces of mortality for mixed male-female populations. It also explores the relationships between survival probabilities, forces of mortality, and proportional hazards models.

Uploaded by

Nguyễn Quang Trường
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 12

Tutorial 1 - Survival Models

Lecturer: Trần Minh Hoàng

Actuarial Mathematics 1

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 1 / 12


Question

(40) is subject to the survival function



1 − 0.005t if t < 20,
S40 (t) =
1.3 − 0.02t if 20 ≤ t < 65.

Calculate the probability that (50) survives at least 30 years.

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 2 / 12


Question

The following values are taken from the Australian Life Tables,

x px
55 0.99403
56 0.99336
57 0.99261
58 0.99177
59 0.99082

Calculate
a) the probability that a life aged 55 survives for at least three years,
b) the probability that a life aged 56 dies before age 60,
c) the probability that a life aged 55 dies aged 57 or 58.

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 3 / 12


Question

You are given the following mortality table:


x qx
60 0.001
61 0.002
62 0.003
63 0.004
64 0.005

Calculate the probability that a person age 60 will die sometime between 2 and 5
years from now.

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 4 / 12


Question

Let  2
x
S0 (x) = exp − for x ≥ 0.
10
Calculate the following to 4 decimal places:
a) S1 (1)
b) µ1
c) 1|0.5 q1

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 5 / 12


Question

Which of the following functions can serve as a force of mortality


a) µx = Bc x where B > 0,c > 1
a
b) µx = b+cx where a, b > 0
c) µx = (1 + x)−3 .

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 6 / 12


Question

Let Tx denote the future lifetime of a life aged x. Suppose


3
µx = for x > 0.
1+x
Find expressions for the distribution function, the density function, the mean and
variance of Tx .

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 7 / 12


Question

For a population which contains equal numbers of males and females at birth:
(i) For males, µM
x = 0.10 for x ≥ 0,
(ii) For females, µFx = 0.08 for x ≥ 0.
Calculate q60 for this population.

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 8 / 12


Question

Lives A and B are both aged exactly 40.


Life A is subject to a force of mortality that is k% greater than the force of
mortality for life B at all ages.
The probability that life A survives to age 50 is 0.97247.
The probability that life B survives to age 50 is 0.973.
Find k.

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 9 / 12


Question

You are given:


(i) 3 p70 = 0.95
(ii) 2 p71 = 0.96
R 75
(iii) 71 µx dx = 0.107
Calculate 5 p70 .

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 10 / 12


Question

a) Show that
∂
t px = t px (µx − µx+t ).
∂x
b) Suppose
x
t px = for x > 0.
x +t
Find µx .

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 11 / 12


Question

If µx is given by
1
µx = for x > 0.
(a0 + a1 x)(b0 + b1 x)

Show that px is directly proportional to


 1/r
b0 + b1 x
,
a0 + a1 x

where r = a1 b0 − a0 b1 .

Lecturer: Trần Minh Hoàng Tutorial 1 - Survival Models Actuarial Mathematics 1 12 / 12

You might also like