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Module Code: MATH352001
Module Title: Actuarial Mathematics 2
Calculator instructions:
• You are allowed to use a calculator.
Exam information:
• There are 10 pages to this exam.
• Answer all questions.
• The numbers in brackets indicate the marks available for each question.
• Actuarial tables are included from page 7.
• You must submit your solutions through Minerva (Submit My Work)
Page 1 of 10 Turn the page over
Module Code: MATH352001
1. (a) Suppose we have two independent lives aged 40 and 50 respectively. Using the
actuarial tables provided, calculate the following assuming that A40:50 = 0.13,
A60:70 = 0.35 and i =5% p.a. Clearly state any approximations you make.
(i) 20q40:50
(ii) A
(12)
40:50:20
[6 marks]
(b) Consider two independent lives initially aged x and y respectively. Explain the
meaning of nq
1
xy and nq
2
xy, and prove that
nq
1
xy + nq
2
xy = nqx.
[4 marks]
(c) Suppose that µx denotes the force of mortality for a life aged x, and δ denotes the
force of interest.
(i) Show that:
a¯x =
∫ ∞
0
e−δttpx dt ≈ a¨x − 1
2
− 1
12
(δ + µx).
Hint: Use the Woolhouse formula (stated below) with g(t) = e−δttpx and
h = 1. ∫ ∞
0
g(t) dt = h
∞∑
k=0
g(kh)− h
2
g(0) +
h2
12
g′(0)− · · ·
(ii) Use the Woolhouse formula to show that:
a¨(2)x − a¨(4)x ≈
1
8
+
1
64
(δ + µx).
[5 marks]
(d) Consider a whole life assurance issued to a select life aged [30]. Premiums are
paid annually in advance and a death benefit B = £100, 000 is paid at the end of
year of death. You may assume an interest rate of 5% p.a. applies throughout the
period. The select period has a duration of two years.
Policy expenses are incurred annually as detailed below:
- Initial expenses are E1 =£500.
- Renewal expenses are i1 = 2% of the annual gross premium. Renewal expenses
are only due at the start of the second and subsequent policy years.
(i) Write down the future gross loss random variable LGt at t = 0. [3 marks]
(ii) Determine the minimum future lifetime for the policyholder in order that the
insurance company makes a profit on the policy. [7 marks]
[ Total 25 marks]
Page 2 of 10 Turn the page over
Module Code: MATH352001
2. Consider a 45-year term insurance issued to a select life aged [35]. Premiums are paid
annually in advance and benefits due are paid at the end of year of death. You may
assume an interest rate of 5% p.a. applies throughout the period. The select period has
a duration of two years.
The death benefits for the policy are as set out below:
- If death occurs within the first 10 years of the policy (i.e. at time t < 10), then a
benefit of B1 = £10, 000 is paid.
- If death occurs at time t ≥ 10 and within the 45-year term, then a benefit of
B2 = £300, 000 is paid.
Policy expenses are incurred annually as detailed below:
- Initial expenses are E1 =£200 and i1 = 5% of the annual gross premium at the
start of the first policy year.
- Renewal expenses are E2 =£50 and i2 = 1% of the annual gross premium. Renewal
expenses are only due at the start of the second and subsequent policy years.
- A terminal expense of 2% of the benefits is due at the time of payment of those
benefits.
(a) (i) Write down the future gross loss random variable LGt for t = 0.
(ii) Write down recursive relationships between the gross policy values tV
G for
each year of the policy.
Hint: You may wish to group similar years into a single relationship. [9 marks]
(b) Use the equivalence principle to calculate the gross premium G and compute the
gross policy value at time t = 9. [12 marks]
(c) Use the gross policy value calculated in part (b) to calculate the death strain at
risk at time t = 9. [2 marks]
Now suppose that the insurer issued 2,000 identical policies to independent select lives
aged [35].
(d) Calculate the mortality profit to the insurer on this portfolio arising during the ninth
year, knowing that there was 1 death during the ninth year of the policy and there
had been only 1 death up to time t = 8. [2 marks]
[ Total 25 marks]
Page 3 of 10 Turn the page over
Module Code: MATH352001
3. (a) You are given the following population data.
Age Leeds Population Leeds Deaths UK Population UK Population Deaths
34 35, 000 150 1, 800, 000 5, 500
35 42, 000 182 1, 600, 000 3, 100
36 40, 000 120 980, 000 2, 300
(i) Calculate the crude mortality rate for Leeds.
(ii) Calculate the directly standardised mortality rate for Leeds.
(iii) Calculate the area compatibility factor for Leeds.
[7 marks]
(b) Assume the following 4 state Markov model for a life initially aged x. The possible
transitions and the corresponding intensities are shown in the graph below: