Topics in Mathematics II - Actuarial Mathematics

Hand-Out 3: Life Insurance

Frank Coolen (CM206 - [e-mail]), February 2008

We present life insurances, and related contracts such as endowments, with most attention to the present

values of such contracts. Our presentation is closely linked to Chapter 3 of the textbook by Gerber. The
life table and corresponding commutation columns, used in the exercises below, are those handed out in
the lectures (taken from the book by Gerber).

Exercises
3-1. Let

t

p

30

= 1 −

t

70

, for 0 ≤ t ≤ 70, and interest compounded continuously with force of interest

δ = 0.10. Calculate the net single premium of a whole life insurance paying £50,000 at the moment of
death. Give two different approximations for the corresponding net single premium if the payment is at
the end of the year of death, and briefly explain the justifications for these approximations.

3-2. Calculate the net single premium of a whole life insurance, with a unit payment at the moment
of death, issued to (x). The probability density function of the future lifetime T (x) is g(t) =

t

5000

for

0 ≤ t ≤ 100 (and g(t) = 0 elsewhere). Let interest be compounded continuously, with δ = 0.1.

3-3. A person aged 60 wishes to arrange for a payment to be made to a charity in 10 years time: if he
is still alive at that date, the payment will be £1,000; if he dies before the payment date, the amount
given will be £500. Assume that the future lifetime of this person is based on the provided life tables, and
that the AER is 6%. Specify the probability distribution of the actual payment in 10 years time, and its
expected value, and do the same for the present value of this payment.

3-4. Use the provided life tables and corresponding commutation columns, with AER 5%, to calculate the
following: (a) A

20

, A

50

, A

80

; (b) A

1

20:10

, A

1

20:30

, A

1

20:60

; (c) A

1

40:10

, A

1

60:10

, A

1

80:10

; (d)

10|

A

20

,

30|

A

20

,

10|

A

40

,

10|

A

80

. Briefly comment on some of these net single premiums.

3-5. A whole life insurance issued to (x) pays £10,000 at the end of the year of death if this person dies
during the first 20 years of this policy, and £20,000 at the end of the year of death if this person dies
thereafter. In addition, if this person dies during the first 20 years of the policy, the net single premium
is also returned, without interest. Express the net single premium of this contract using commutation
functions (assume constant AER i).

3-6. Assume that the probability density function of the future lifetime T (x) is g

x

(t) =

1

80

for t ∈ [0, 80]

(zero else). Assume that interest is compounded continuously, with constant δ > 0. Derive the net single
premium of a whole life insurance for (x), with payment at the moment of death, as a function of δ.

3-7. Consider an m-year deferred whole life insurance for (x), with a unit payment payable at the moment
of death. Assume that the individual is subject to a constant force of mortality µ, and that interest is
continuously compounded with force of interest δ. Derive the net single premium of this contract, denoted
by

m|

¯

A

x

, as function of m, µ and δ. Calculate this net single premium with µ = 0.04 and δ = 0.10, for the

values m = 1, 5, 10, and briefly comment on these net single premiums.

3-8. Consider a 25-year term insurance for a life aged 40, with payment at the moment of death. Assume
that this person belongs to a population, whose lifetimes have a probability distribution with

t

p

0

= (100 −

t)/100, for t ∈ [0, 100]. Assume that the force of interest is δ = 0.05. Calculate the net single premium of
this contract, denoted by ¯

A

1

40:25

.

3-9. Calculate ¯

A

1

30:10

, based on De Moivre’s lifetime model with maximum age w = 100 and i = 0.10.

3-10. Consider the following contract: an insurance, issued to (x), which pays £10,000 at the end of 20
years if (x) is still alive, and which returns the current net single premium Π at the end of the year of death
if (x) dies during the first 20 years. Express this net single premium Π using commutation functions.

3-11. Consider the following contract: a pure endowment of duration 20 years, with a sum insured of
£10,000, for a life aged 50 with future lifetime probability distribution based on the provided life tables,
and AER of 5%. Give the present value random variable for this contract, its probability distribution, and
calculate its net single premium.

3-12. Use the provided life tables and corresponding commutation columns, with AER 5%, to calculate
the following: (a) A

1

20:10

, A

1

20:60

, A

1

50:10

, A

1

80:10

; (b) A

20:10

, A

20:60

, A

50:10

, A

80:10

; (c) A

25:15

, A

25:30

,

A

25:45

, A

25:60

. Briefly comment on some of these net single premiums.

3-13. Let A

x:n

= u, A

1

x:n

= y and A

x+n

= z. Determine A

x

in terms of u, y and z.

3-14. Consider a 10-year endowment, with payment of £50,000, for a person aged 50. Calculate the net
single premiums of this contract, in the following two cases:
(a) Death benefit payment takes place at the moment of death, interest is compounded continuously with
force of interest δ = 0.05, and the future lifetime distribution of this person is given by

t

p

50

= 1 −

t

50

for

0 ≤ t ≤ 50.
(b) Death benefit payment takes place at the end of the year of death, the future lifetime probabilities of
this person are based on the provided life tables, and AER is 5%.
(c) Briefly compare the net single premiums for parts (a) and (b).

3-15. Calculate the net single premium of the following contract: a 30-year endowment with payment
of £10,000, providing death benefit at the moment of death, issued to a life aged 35 and based on the
provided life tables with constant AER i = 0.05. Use the assumption that death occurs at a uniformly
distributed moment during the year of death.
(Hint: use the relation ¯

A

x:n

=

i

δ

A

1

x:n

+ A

1

x:n

, which holds under these assumptions.)

3-16. Consider three independent lives, say persons A, B and C, all of identical age x. Let Z

A

be the

present value of an n-year endowment issued to A, Z

B

the present value of an n-year term insurance issued

to B, and Z

C

the present value of an n-year pure endowment issued to C. Assume that all these three

contracts involve 1-unit payments, and that the same constant AER i > 0 applies to all three contracts,
over the whole period considered. Prove that Var(Z

A

) < Var(Z

B

) + Var(Z

C

), and explain the relevance of

this result in terms of risk to an insurer.

3-17. Assume that each of 100 independent lives is of age x at time 0, is subject to a constant force of
mortality µ = 0.04, and has a whole life insurance for a payment of £100,000 payable at the moment of
death. The payments involved are to be withdrawn from an investment fund earning continuously com-
pounded interest with force of interest δ = 0.06. Calculate the minimum amount required in this fund, at
time 0, so that the probability is approximately 0.95 for the event that sufficient funds will be on hand to
withdraw the benefit payments at the deaths of all individuals. Briefly comment on this minimum required
initial amount.
(Hint: use the Central Limit Theorem (1H Probability), and the fact that, for a standard normally dis-
tributed random quantity X, we have P (X ≤ 1.645) = 0.95.)

3-18. A group of 100 independent lives, all aged 30, set up a fund to pay £1,000 at the end of the year
of death of each member, to a designated survivor. Their mutual agreement is to pay into the fund an
amount equal to the net single premium for the corresponding whole life insurance, based on the provided
life tables and with assumed constant AER i = 0.05. The actual experience of this fund is one death in
each of the second and fifth years, and the actual interest income for the fund is 5% in the first year, 5.5%
in each of years 2 and 3, and 6% in each of years 4 and 5. What is the difference, at the end of the fifth
year, between the expected size of the fund as determined in the inception of the plan, and the actual
fund?
(Hint: calculate the balance of the fund at the end of each of the years 1 to 5.)

3-19. Prove that A

x:n

= A

1

x:m

+ v

m

m

p

x

A

x+m:n−m

for m < n (assume constant AER i > 0), and give an

interpretation of this result.