Faculty of Actuaries Institute of Actuaries
EXAMINATIONS
April 1999
Subject D — Actuarial Mathematics
Paper One
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Write your surname in full, the initials of your other names and your
Candidate’s Number on the front of the answer booklet.
2. Begin your answers to Parts One, Two and Three on a separate sheet.
3. Mark allocations are shown in brackets.
4. Attempt all 14 questions.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet and this question paper.
In addition to this paper you should have available
Actuarial Tables and an electronic calculator.
ã Faculty of Actuaries
D1—A99 ã Institute of Actuaries
D1(A)—2
PART ONE
For questions 1–7 indicate in your answer booklet which one of the answers A, B, C or D
is correct.
1 Which of the following is equal to 3 2 3
q 2 : ?
A 1 3 23 1 - q :
B 3 3 3 2 3
q - q1:
C 3 3 3 2 3
q - q 1 :
D 3 2 3 23 1 q - q : [2]
2 Two lives, (x) and (y), are assumed to be independent with respect to mortality
and are each assumed to be subject to a constant force of mortality of 0.015.
Which of the following is the probability that (x) dies more than 5 years after
(y)?
A 0.0750
B 0.4291
C 0.4304
D 0.4639 [3]
3 Tx and Ty are random variables denoting the complete future lifetimes of (x)
and (y) respectively. The following statements have been made:
I E a Tx Ty
( ) max( , ) = ax + ay - axy
II P(max(Tx , Ty) < t) = 1 -tpxy
III E v A T T
xy
x y min , d i FH
IK
=
A I and II only are correct
B II and III only are correct
C I only is correct
D III only is correct [3]
D1(A)—3 PLEASE TURN OVER
4 Under the Manchester Unity model of sickness, you are given the following
values:
tpx zx +t dt z0
1
= 0.1
vt p z dt
t x x +t z0
1
= 0.08
Which of the following is the value of sx ?
A 0.08
B 0.1
C 4.1744
D 5.218 [2]
5 A life insurance company issues a number of 3-year unit-linked policies to
lives each aged 50 exact. (PRO)t is the expected profit at the end of policy year
t per policy in force at the start of the year. (SCF)t is the year-end sterling
fund cash flow per policy in force at the start of year t. The values of (SCF)t
are as follows (in £’s):
year(t) 1 2 3
(SCF)t +100 -40 -40
(SCF)t equals (PRO)t since there is no allowance for reserves. Sterling
reserves are to be set up at each year-end for each policy then in force to
zeroise future negative cash flows. Assuming that the sterling fund earns
interest at 5% per annum and that mortality is that of A1967–70 Ultimate and
that sterling reserves are set up using these assumptions, which of the
following gives the adjusted value of (PRO)1?
A +25.07
B +25.62
C +26.17
D +26.24 [3]
D1(A)—4
6 In a stationary population in which the only mode of decrement is death,
which of the following gives a correct formula for the average age at death of
individuals who die between the ages of 30 and 35?
A 30
30 35 5 35
30 35
+ - -
-
T T l
l l
B 30
30 35 5 30 35
30 35
+ - - -
-
T T l l
l l
( )
C 30
30 35 5 30
30 35
+ - -
-
T T l
l l
D 30 30 35
30 35
+ -
-
T T
l l
[3]
7 On the basis of A1967–70 mortality and 4% p[1] [2] [3] [4] 下一页 |
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