HERIOT-WATT UNIVERSITY
DEPARTMENT OF ACTUARIAL MATHEMATICS & STATISTICS
F70LA1/F70LB2 — LIFE INSURANCE MATHEMATICS A/B
7 May 2010, 09.30 – 12.30
Please answer ALL NINE questions.
Total marks: 100
Approved electronic calculators only can be used.
1. Calculate the following on the basis of AM92 Mortality:
(a) ä32 at 4% interest per annum effective.
[1 mark]
(b) A[46] at 6% interest per annum effective.
[1 mark]
(c) (IA)30 at 4% interest per annum effective.
[1 mark]
(d) A30:15 at 4% interest per annum effective.
[2 marks]
[Total 5 marks]
2. Throughout this question, mortality is that of the AM92 tables.
(a) Calculate q[22] , 1| q[22] , 2| q[22] and 3 p[22] .
[4 marks]
(b) A pure endowment with term 3 years and sum assured £1 is sold to a select
life aged 22. Calculate the expected present value of this benefit, at effective
interest of 1% per annum.
[2 marks]
(c) Calculate the variance of the present value of the benefit in (b) above.
[2 marks]
(d) An endowment with term 3 years and sum assured £1, payable at the end of
the year of death or at the end of the term if that is sooner, is sold to a select
life aged 22. Calculate the expected present value of this benefit, at effective
interest of 1% per annum.
[2 marks]
(e) For the endowment in (d) above, what is the probability that the sum assured
is paid to the policyholder between time 0 and time 3 (0 and 3 included)?
[2 marks]
(f) What is the variance of the present value of the benefit in (d) above when the
effective rate of interest is 0% per annum?
[2 marks]
[Total 14 marks]
continued on next page . . .
3. A life insurance company sells endowment assurances with a term of 35 years to
persons age 30. The sum assured is £200,000, payable at the end of the year of
death or on maturity. Level premiums are payable annually in advance throughout
the term. Use AM92 Ultimate mortality and 4% interest per annum effective as the
basis of your calculations.
(a) What is the standard deviation of the present value of the future loss on a
single policy at outset? You are given that A30:35 at 8.16% is 0.07475.
[4 marks]
(b) What loss will be exceeded with probability 1% if the insurer sells (i) 20, or
(ii) 2,000 contracts? Express your answers as a percentage of one year’s total
premiums.
[4 marks]
(c) What happens to the percentage in (b) when the number of policies grows to
infinity?
[2 marks]
[Total 10 marks]
4. A life office is to issue a with-profits term assurance to a life aged 40. The basic sum
assured is £250,000, the term 25 years. The sum assured plus bonuses is payable at
the end of the year of death. The policyholder will pay level premiums monthly in
advance throughout the term.
Calculate the monthly premium for the following premium basis:
Mortality
Interest
Expenses
Bonus
AM92 Select
4% per annum effective
none
simple reversionary bonus of 4% per year vesting
at the start of each policy year
Profit criterion E[P V profit] = £2, 500.
[Total 10 marks]
continued on next page . . .
5. A deferred annuity contract is issued to a man aged 35. The policyholder will pay
level premiums monthly in advance ceasing after 30 years or on earlier death. On
survival to age 65 an annuity of £25,000 per annum is paid monthly in advance for
life. The premium basis is given by
Mortality
Interest
Initial Expenses
Renewal Expenses
Claims Expenses
Profit criterion
AM92 Select
4% per annum effective
£150
£1.50 at the time of payment of
the second and each subsequent premium
£1.50 at the time of each annuity payment
E[P V profit] = 0.
(a) Calculate the monthly premium.
[7 marks]
V30P
(b) Calculate the prospective policy value
at time 30 under the assumption
that the valuation and premium bases are the same.
[2 marks]
(c) Assume now that the prospective policy value V30P at time 30 is calculated as
in (b), except that the interest rate is 6% per annum effective. State, giving
reasons, whether this policy value would be higher or lower than that calculated
in (b). (No calculation needed.)
[2 marks]
[Total 11 marks]
6. Two different approaches to specifying a model of a person’s life history are as
follows:
• Random future lifetimes: specify the distributions of the random times between successive events.
• Multiple state model: specify the transition intensities governing movements
between states.
Describe briefly, giving reasons, which approach you would prefer when required
to obtain premium rates and policy values for the following insurance contracts:
(a) Disability insurance, paying an annuity-type benefit during any period when
the policyholder is ill and unable to work.
[3 marks]
(b) A reversionary annuity, payable annually in advance to a person (y) provided
another person (x) is dead.
[3 marks]
[Total 6 marks]
continued on next page . . .
7. A multinational company has large workforces in two countries, Country A and
Country B. It has in place a group life insurance contract covering all employees so
it wishes to analyse the mortality experienced in each country. The available data,
in respect of a single calendar year, are denoted as follows:
Country A Country B Combined
A
B
S
dx
dx
dx
A c
B c
S c
Ex
Ex
Ex
S
B
A
µx
µx
µx
Deaths in age group x
Exposure to risk in age group x
Crude force of mortality in age group x
and are shown in the table below:
Age Group x
20–29
30–39
40–49
50–59
Total
A
dx
1
4
8
30
43
A
Exc
2,000
2,000
2,000
2,000
8,000
B
dx
1
4
9
15
29
B
Exc
2,000
1,500
1,000
500
5,000
S
S c
dx
Ex
2 4,000
8 3,500
17 3,000
45 2,500
72 13,000
Based on these, the following crude rates (forces) of mortality have been obtained:
Age Group x
20–29
30–39
40–49
50–59
A
µx
0.000500
0.002000
0.004000
0.015000
B
µx
0.000500
0.002667
0.009000
0.030000
S
µx
0.000500
0.002286
0.005667
0.018000
(a) Calculate the following single-figure indices:
(i) The crude mortality rates in Country A and Country B.
(ii) The standardised mortality rates in Country A and Country B, using the
combined workforce as the standard population.
[4 marks]
(b) Comment on your results.
[3 marks]
[Total 7 marks]
continued on next page . . .
8. A life office uses the following Markov model for calculating premiums and policy
values for joint-life assurances and annuities.
State 0
(x) alive, (y) alive
-
µ01
y+t
µ02
x+t
State 1
(x) alive, (y) dead
µ13
x+t
?
State 2
(x) dead, (y) alive
µ23
y+t
?
State
3
(x) dead, (y) dead
The life office sells a contingent assurance with term ten years to a man (x) age
60 and a woman (y) age 55. The benefit is £100,000, payable immediately on the
death of the woman, provided that event occurs within ten years, and the man is
already dead. The policy is purchased by a single premium payable at outset.
Let V i (t) be the policy value if the joint life history is in state i at time t years after
issue. The valuation basis is as follows:
Force of interest:
Male mortality:
Female mortality:
Expenses:
δ per annum
13
µ02
x+t , µx+t as in the diagram
01
23
µy+t , µy+t as in the diagram
None.
(a) Show from first principles that:
d 02
00 02
02 23
t ps = t ps µx+t − t ps µy+t .
dt
[5 marks]
(b) Write down Thiele’s equations for:
d 0
d
d
d
V (t), V 1 (t), V 2 (t) and V 3 (t)
dt
dt
dt
dt
and state what boundary conditions you would use in solving them.
[4 marks]
continued on next page . . .
(c) Write down the first step of an Euler scheme, with step-size h years, to solve
Thiele’s equations for the system V 0 (t), V 1 (t), V 2 (t) and V 3 (t).
[4 marks]
(d) By considering the joint distribution of the random lifetimes Tx and Ty , assuming these two random variables to be independent, show that:
2
n qxy
=
Z
n
(1 − t px ) t py µ01
y+t dt.
0
[4 marks]
µ01
y+t
µ23
y+t ,
(e) Suppose that
6=
in which case the random lifetimes Tx and Ty are
not independent. Write down an integral expression for n qxy2 in terms of the
transition intensities. You should describe your reasoning but you need not
give a formal proof. [HINT: You need to consider the death or survival of (y)
separately before and after the exact time at which (x) dies.]
[3 marks]
[Total 20 marks]
continued on next page . . .
9. A life insurer issues three-year unit-linked savings policies. Each policy has a level
premium of £2,000 per year. 90% of the first premium, and 103% of the second and
third premiums, is invested in units at the offer price. There is a bid/offer spread
in unit prices, the bid price being 95% of the offer price.
A fund management charge of 0.5% of the bid value of the policyholder’s fund is
deducted at the end of each policy year.
At the end of the policy term, the policyholder receives the larger of:
• 100% of the bid value of their unit fund, after the deduction of the fund management charge; or
• 90% of the total of the premiums they have paid, that is, £5,400.
The policy may be terminated early by either death or surrender. In either case, the
policyholder (or their estate) receives the bid value of the unit fund, after deduction
of the fund management charge. The insurer expects that 0.2% of policies in force
at the start of each year will terminate by death during the year; and that 5% of
policies in force at the start of each of the first and second years will terminate by
surrender during the year.
The insurer incurs expenses of £80 at the beginning of year one and £20 at the
beginning of each of years two and three.
It is assumed that the growth in the unit price will be 8% per annum, and that
sterling funds will earn interest at 5% per annum. Zero sterling reserves will be held
by the insurer at the end of each year.
(a) Calculate the value of the unit fund for this contract at the end of each year.
[4 marks]
(b) Calculate the profit signature for this policy.
[5 marks]
(c) Suppose the risk discount rate is 20% per annum. Calculate the expected value
of the profit under this policy.
[3 marks]
(d) (i) What would the unit growth rate have to be in the final year to make the
guaranteed minimum maturity value of £5,400 exceed the bid value of the
unit fund?
(ii) What are the good and bad features of including a guaranteed minimum
maturity value in this policy from the point of view of the insurer and of
the policyholder?
[5 marks]
[Total 17 marks]
Total 100 marks
END OF PAPER
COMMUTATION FUNCTIONS FORMULAE SHEET
Commutation functions:
Dx = v x lx
Nx =
∞
X
Dx+k
∞
X
Nx+k
k=0
Sx =
k=0
Cx = v x+1 dx
Mx =
∞
X
Cx+k
∞
X
Mx+k
k=0
Rx =
k=0
Insurances:
Ax = Mx /Dx
A1x:n
= (Mx − Mx+n )/Dx
Ax:n1
= Dx+n /Dx
Ax:n
= (Mx − Mx+n + Dx+n )/Dx
(IA)x = Rx /Dx
(IA)1x:n
= (Rx − Rx+n − nMx+n )/Dx
(IA)x:n
= (Rx − Rx+n − nMx+n + nDx+n )/Dx
Annuities:
äx = Nx /Dx
ax = Nx+1 /Dx
äx:n
= (Nx − Nx+n )/Dx
(Iä)x = Sx /Dx
(Ia)x = Sx+1 /Dx
(Iä)x:n
= (Sx − Sx+n − nNx+n )/Dx
Select functions:
D[x]+t = v x l[x]+t
N[x]+t =
∞
X
D[x]+t+k
∞
X
N[x]+t+k
k=0
S[x]+t =
k=0
C[x]+t = v x+1 d[x]+t
M[x]+t =
∞
X
C[x]+t+k
∞
X
M[x]+t+k
k=0
R[x]+t =
k=0
A[x]+t = M[x]+t /D[x]+t
A1[x]+t:n
= (M[x]+t − M[x]+t+n )/D[x]+t
ä[x]+t = N[x]+t /D[x]+t
a[x]+t = N[x]+t+1 /D[x]+t