Topics in Mathematics II - Actuarial Mathematics

Hand-Out 4: Life Annuities & Net Premiums

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

We present life annuities, both considering whole-life annuities and n-year temporary annuities,

again focussing on the net single premiums of such contracts. Thereafter, we consider net premiums,
where we aim to calculate the premiums required if a contract is not paid for by a single payment at
the start of the contract. Our presentation is closely linked to Chapters 4 and 5 of the textbook by
Gerber. Again, 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; available from the module
webpage).

Exercises

4-1. Prove that

n

p

x

d¨

a

n

+

n−1

X

k=0

(1 − v

k+1

)

k

p

x

q

x+k

= 1 − A

x:n

4-2. Let Y be the present value random variable of a whole life annuity due of 1 unit per year,
issued to (x). It is given that ¨

a

x

= 10 if evaluated with AER i =

1

24

= e

δ

− 1, and if we evaluate

it with AER set equal to e

2δ

− 1 we get ¨

a

x

= 6. Calculate the variance of Y , and comment on the

risk to the insurer.

4-3. The following function values have been calculated with i = 0.03:

x

¨

a

x

72

8.06

73

7.73

74

7.43

75

7.15

Calculate p

73

and

3

q

72

.

(Hint: Derive a recursive relation for ¨

a

x

.)

4-4. Calculate the following net single premiums, based on our life table and corresponding com-
mutation columns (so with AER i = 0.05), and give some brief comments:
(a) ¨

a

20

, a

20

, ¨

a

21

, ¨

a

80

, a

80

, ¨

a

81

;

(b) ¨

a

20:30

, ¨

a

20:31

, ¨

a

21:30

, ¨

a

80:10

, ¨

a

80:11

, ¨

a

81:10

;

(c)

30|

¨

a

20

,

30|

¨

a

21

,

10|

¨

a

20

,

10|

¨

a

80

,

11|

¨

a

80

.

4-5. Calculate ¨

a

20

and ¨

a

80

for a life with as lifetime probability distribution the De Moivre model

with maximum possible age 100, and AER 5%. Comment briefly on these net single premiums (also
compare them to net single premiums in Exercise 4-4).

4-6. On his 40th birthday, a man buys a deferred annuity, with constant payments per annum
starting at his 60th birthday, if he is then alive, and continuing until his death. The net single
premium of this contract is £100,000, and assume that this contract is based on our provided life
table and AER i = 0.05. Calculate the annual payments under this contract.

4-7. Consider an n-year temporary immediate annuity with annual unit payments, issued to (x),
and with AER i > 0. Give the definition of the present value random variable Y of this contract.
We define two further random variables:

Z

1

=

 (1 + i)v

K+1

for 0 ≤ K < n

0

for K ≥ n

and

Z

2

=

 0

for 0 ≤ K < n

v

n

for K ≥ n

Express Y in terms of Z

1

and Z

2

. Derive an expression for the net single premium of this contract,

denoted by a

x:n

in terms of i, and the net single premiums of Z

1

and Z

2

. Calculate a

20:30

, a

20:10

and a

80:10

based on the provided life tables, with AER 5%, and briefly comment on these net single

premiums.

4-8. Calculate the net single premium for a 30-year deferred immediate life annuity issued to (20),
denoted by

30|

a

20

, based on our life tables and commutation columns (AER 5%). Do the same

for

30|

a

21

,

10|

a

20

,

11|

a

20

,

10|

a

80

and

11|

a

80

, and comment on these values (also compare to net single

premiums in Ex. 4-4).

4-9. Prove that A

x:n

= v¨

a

x:n

− a

x:n−1

, and give an interpretation for this equality.

4-10. In this exercise, we consider a life annuity with varying annual payments.

Define the indicator random variable I

k

=

 1 if T (x) ≥ k

0 if T (x) < k

. Consider a life annuity issued to (x),

with annual payment b

k

on survival to age x + k, k = 0, 1, 2, . . ., and AER i > 0. (Note: you can

assume throughout this exercise, that the expected value of an infinite sum of random variables is
equal to the infinite sum of the expected values of these random variables, and similar as required
for computation of the variance.)

(a) Show that the present value random variable for this contract can be written as

∞

X

k=0

v

k

b

k

I

k

, and

give an expression for the net single premium of this contract.
(b) Show that, for j ≤ k, E(I

j

I

k

) =

k

p

x

and Cov(I

j

, I

k

) = (

k

p

x

)(

j

q

x

).

(c) Show that Var

∞

X

k=0

v

k

b

k

I

k

!

=

∞

X

k=0

v

2k

b

2
k

(

k

p

x

)(

k

q

x

) + 2

∞

X

k=0

X

j<k

v

j+k

b

j

b

k

(

k

p

x

)(

j

q

x

).

4-11. The following portfolio of annuities-due is currently paid from the assets of a pension fund:

Age

Number of annuitants

65

30

75

20

85

10

Each annuity has an annual payment of £10,000 as long as the annuitant survives (with one payment
‘now’). Assume an AER of 5% and mortality, for each of these people, according to our provided
life table. Calculate the expected value of the present value of these obligations of the pension fund.

4-12. Consider a 10-year term life insurance issued to (40), with sum insured C > 0 and death
benefit paid at the end of the year of death. The net annual premium Π is to be paid in advance
for the length of this contract while the insured is alive. Let the AER be i = 0.05, and assume that
the mortality of (40) follows De Moivre’s model with terminal age w = 100. Calculate Π. Compare
this with the corresponding net annual premiums in case the mortality is modelled via our provided
life tables, also with AER 5% (as derived in an example in the lectures).

4-13. Consider a whole life insurance of 1 unit, issued to (x), payable at the end of the year of
death, and financed by net annual premiums (for the whole length of the contract, and paid in
advance as usual), which we denote by P

x

.

(a) Derive an expression for the random loss to the insurer.

(b) Show that P

x

=

A

x

¨

a

x

.

(c) Show that L =



1 +

P

x

d



v

K+1

−

P

x

d

.

(d) Derive an expression for Var(L), and compare this to the variance of the loss to the insurer in
case this contract is paid by a net single premium.

4-14. In this exercise, we consider a general type of life insurance, with varying annual payments
and varying annual premiums.
Consider a whole life insurance, with the sum c

j

to be paid at the end of the jth year after policy

issue, if the insured dies during that year.

Assume that this insurance is financed by annual

premiums Π

i

, i = 0, 1, 2 . . ., to be paid while the insured is alive. Derive an expression for the

insurer’s loss, and an equation that these annual premiums must satisfy in order to be net premiums.

4-15. Given the following values, calculated at d = 0.08, for two whole life insurance policies issued
to (x):

Death Benefit

Premium

Variance of Loss

Policy A

4

0.18

3.25

Policy B

6

0.22

These premiums are paid at the beginning of each year, until death, and the death benefits are paid
at the end of the year of death. Calculate the variance of the loss for policy B.

4-16. A whole life insurance issued to (x) provides varying death benefits, with death benefit in
year j of the policy equal to 1, 000(1.06)

j

, payable at the end of the year of death. This contract is

financed by net annual premiums, payable for life. It is given that 1, 000 × P

x

= 10, with P

x

the net

annual premium of a whole life insurance with unit payment (see Exercise 4.13), and that the AER
is 6%. Calculate the net annual premium for this contract, and compare it with the similar whole
life insurance, but with constant death benefits, that could be financed by the same net annual
premiums.
(Hint: you may want to show first that

1

¨

a

x

= d + P

x

.)

4-17. Consider an m-year deferred life insurance issued to (x), with unit payment at the end of
the year of death if the insured survives m years after buying this contract. The mortality of (x) is
according to our provided life table, and AER is 5%.
(a) Show that the net annual premium, payable (in advance) throughout the duration of this

contract (so until death), is equal to

m|

A

x

¨

a

x

.

(b) Show how this net annual premium is calculated using commutation functions. Calculate this
net annual premium for x = 20 and m = 10, 20, and the same for x = 70, and comment on these
values.
(c) Now consider this same contract, but financed by constant annual payments (in advance) for
at most s years (so one payment at the start of each of the first s years, if the insured is still alive).
Derive the corresponding net annual premium, and show how it is calculated using commutation
functions. Calculate this net annual premium for x = 20, 70, with m = 10 and s = 5, 15, and
comment on these values.

4-18. (a) Derive the net annual premium P

x:n

for an n-year endowment, issued to (x), with unit

payment and death benefit paid at the end of the year of death, and financed by annual premiums
paid in advance throughout the term of this policy or until earlier death of the life assured.
(b) Express P

x:n

using commutation functions.

(c) Assume that such a 25-year endowment is issued to a life aged 40, with a sum assured of £50,000.
Calculate the net annual premium, based on our provided life table and AER 5%.
(d) Explain how P

x:n

can be derived from the general results for life insurances in Exercise 4-14.

4-19. For the setting of Exercise 4-13, we have d¨

a

x

+ A

x

= 1, which leads to the equalities:

1

¨

a

x

= P

x

+ d and P

x

=

dA

x

1 − A

x

. Give detailed interpretations for these two equalities.

4-20. (a) Calculate P

x

(see Exercise 4-13) if

k|

q

x

= c(0.96)

k+1

for k = 0, 1, 2, . . ., with c =

0.04
0.96

and

i = 0.06.
(b) Generalize part (a), by using

k|

q

x

= (1 − r)r

k

with r ∈ (0, 1), to derive A

x

, ¨

a

x

, and P

x

in terms

of r and i.

4-21. Consider the following contract, issued to a life aged 40, based on our provided life tables
and with i = 0.05. In case of death before age 60, a lump sum of £100,000 is paid at the end of
the year of death. Else, an annuity is provided, starting at age 60, with payments of £10,000 at
the end of each further year that the policy holder survives. Net annual premiums Π are to be paid
until the age of 60, or earlier death. Calculate Π.

4-22. A four-year loan issued to (25) is to be repaid with equal annual payments at the end of
each year. A four-year term life insurance has a death benefit which will pay off the loan at the
end of the year of death, including the payment then due. Given: (i) i = 0.06 is used both for the
calculations involved with the insurance and for the loan; (ii) ¨

a

25:4

= 3.667; (iii)

4

q

25

= 0.005.

(a) Express the insurer’s random loss in terms of the curtate future lifetime K(25), for a loan of 1
unit, assuming that the insurance is purchased with a single premium of G.
(b) Assume now that G is the net single premium of this term insurance providing cover for this
loan of 1 unit. Calculate G.
(c) (Challenging!) The loan is intended to buy a car for £10,000. However, the buyer can add the
additional amount required to buy the term insurance (costing its net single premium) to the loan,
with the same conditions. Calculate the total annual payment for this loan.