104: Diploma – Practice Questions
Note: For questions 2 and 3, the mortality tables can be found from
www.actuaries.org.uk (then “actuarial education “ → “current students”
→“Formulae and tables for actuarial examinations (old version)”.
Q1.
The age at death, X , of an animal aged 0 is assumed to have a Weibull distribution
with parameters c = 0.75 and y = 0.25 for 0 < X, so the density for the age at death is

f  x   cx 1 exp cx 

0x
Find the survival function for the animal up to age 10, and hence find the probability
that the animal dies between ages 5 and 10, given that it survives to age 2.
[8]
Q2.
(a)
A group of lives experience mortality which can be represented between
ages 80 and 90 by the a(55) males ultimate table with a deduction from the
force of mortality of 0.05 at age 80, the deduction increasing linearly with age
to a deduction of 0.15 at age 90.
Find the probability of a life aged 80 dying within the next 10 years.
(b)
A life aged 60 has just retired due to ill health and is assumed to be subject to
a constant force of mortality of 0.04879 for two years and then will be subject
to the mortality of the a(55) males ultimate table.
Find the value of l60 if l65.  100,000 .
[10]
Q3.
A life aged 55 who has just retired owing to ill-health is assumed to be subject to a
constant force of mortality of 0.02 up to age 62, after which he is assumed to be
subject to the mortality of the ELT12 (Males) life table.
Calculate:
(a)
the probability that he will die before age 60,
(b)
the probability that he will die between ages 60 and 65
[8]
Q4.
The mortality in a certain aggregate life table is such that:
1
x 2

lx  l0  1 

 100 
(a)
determine  ( the limiting age).
(b)
obtain an exact expression for X.
(c)
obtain an exact value for q80 .
[6]
Q5.
Assume that, for a certain life, the survival function for age x, Sx  t  , (t  0), is of the
form:
  
Sx  t   

t

,   0.
Obtain and simplify expressions involving ,  and t
for:
(i)
Fx  t 
px
(ii)
n
(iii)
fx  t 
(iv)
 x n .
[7]
Q6.
A life table follows Makeham’s law for the force of mortality,  x  A  B.c x .
You are given that 5 p70  0.73, 5p75  0.62 and 5p80  0.50.
Find A, B and c.
[10]
Q7.
If x  3x at all ages, what is the relationship between qy and qy ?
[4]
Q8.
A mortality table is defined so that:
1
t 2

t p x  1 
 for x  70,
 70  x 
t  70  x and tp x  0 for t  70  x .
Find an expression for  x .
[3]
Q9.
Given that q63  0.019 and q64  0.021, and assuming a uniform distribution of
deaths between integer ages, calculate:
(a)
0.4
q64
(b)
0.4
q63.6
(c)
0.3
q63.6
[5]
Q10.
In a certain survival model, the force of mortality between ages 61 and 62 is 0.015
and the force of mortality between ages 62 and 63 is 0.017.
(a)
Show that 2 p61  0.96851
[2]
(b)
Calculate e63, given that e61  20
[6]