Topics in Mathematics II - Actuarial Mathematics

Solutions to Exercises on Hand-Out 5

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

5-1. Short solutions are provided where the main reasoning is similar to that of Exercise 4-21, with parts
of the solutions identical to the earlier ones not repeated.
(a)

Π

a

=

100, 000A

1

40:20

+ 10, 000

20|

a

40

¨

a

40:10

,

with ¨

a

40:10

=

N

40

−N

50

D

40

we get Π

a

= 5, 417.20.

(b) The extra benefit now has expected present value 20, 000 × (a

30

− a

40:30

), with a

30

=

1−v

30

i

and

a

40:30

=

N

40

−N

71

−D

40

D

40

. The annual premiums (paid for at most 20 years) are now Π

b

= 5, 117.38.

(c) Contract of (b), but at most 10 annual premiums (so same change as used in (a)), gives Π

c

= 8, 087.56.

Note that the proportional increase in annual premium, if the maximum number of years of premium
payment is reduced from 20 to 10 years, is not affected by the inclusion of the family income insurance
(

5,117.38
8,087.56

=

3,427.72
5,417.20

).

(d) By the equivalence principle, we get

Π

d

¨

a

40:20

= 100, 000A

1

40:20

+ 10, 000

20|

a

40

+ 1, 000 + 0.05Π

d

¨

a

40:20

,

which leads to Π

d

= 3, 691.42. It is easy to check that if only the acquisition expenses had been taken

into account (so not the collection expenses), then the corresponding annual premium would have been
3, 506.85, while without any expense-loading we had Π = 3, 427.72 (from 4-21), so from these figures we
can deduce the effects of the premium loadings involved (clearly, the collection expenses have most effect
in this case).

5-2. A quick way to solve such questions, if one knows all the present value terms of the components of
the contracts, is by listing the present values of all the benefits to the insured under the contract, and of all
the costs to the insured, assuming an annual premium Π, and to use the equivalence principle to compute
the corresponding premium Π. We briefly give the present values of the components of the contracts, and
a short solution.
(a) The benefits to the insured, and corresponding present values, are (1) family income insurance with
present value 20, 000(a

35

− a

30:35

), (2) deferred immediate life annuity with present value 15, 000

35|

a

30

(take care when the first payment is made), and (3) whole life insurance with present value 50, 000A

30

.

The costs to the insured are just the annual premiums, for at most 35 years, with present value Π¨

a

30:35

.

Using the equivalence principle, and calculating all these present values using the provided commutation
tables, leads to Π = 2, 516.16.
(b) Clearly, the present value of the total costs is now Π

b

¨

a

30:35

+ 20, 000, while the benefits are as before,

and the equivalence principle leads to Π

b

= 1, 306.59.

(c) Extending the contract considered in (a) in this manner, and denoting the new annual premium by ˜

Π

a

,

the ‘benefits-side’ of the equivalence principle equation now gets the expense-loading terms added, namely
additional expected present values of 400 and 0.015 ˜

Π

a

¨

a

30:35

, which leads to ˜

Π

a

= 2, 579.03. Extending (b)

similarly leads to ˜

Π

b

= 1, 351.05.

Remark: For complete answers, it must be clearly stated which expected present values relate to which
components of the contract (as briefly done above). If one does not know the formulae for the expected
present values (‘net single premiums’) involved, one can find the answer again by carefully deriving the
random loss to the insurer, as done in the lectures for a more basic contract (this is, of course, more time
consuming).