Topics in Mathematics II - Actuarial Mathematics

Hand-Out 5: Family Income Insurance & Expense-Loaded Premiums

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

In the final lectures of this module, we discuss Family Income Insurance as an example of a type of

contract for which we can also calculate required premiums, using similar methods as seen earlier in the
module. We also discuss Expense-Loading of premiums, considering three general types to enable insurers
to take costs of contracts into account. These topics are part of the examinable material of this course.
The examples in the lectures are of course good indications of possible questions. Such contracts (and
quite similar variations) can also be included in more extensive contracts. As an example exercise, we
suggest some additions to Exercise 4-21 (given again below), and we include a further exercise. Clearly,
such analyses of contracts with all sorts of variations, both in terms of benefits and premiums, combine all
aspects of the course, and hence are important exercises.

4-21. (From Hand-Out 4) 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.

Exercises

5-1. Calculate the net annual premiums (including expense-loading where appropriate) for each of the
following:
(a) Same contract as in 4-21, but with net annual premiums paid only until the age of 50, or earlier death.
(b) Same contract as in 4-21, but now with a 30-year family income insurance added, with payments of
£20,000 at the end of the year for this period (starting in the year of death), if the insured dies before
reaching age 70.
(c) Same contract as in (b), but now (as in (a)) with premiums paid only until the age of 50, or earlier
death.
(d) Same contract as in 4-21, but now include the following forms and amounts of expense-loading: Col-
lection expenses - 5% of the premium; Acquisition expenses - £1,000.

5-2. Consider the following policy, issued to a life aged 30, and based on the life and commutation tables
provided, with i = 0.05. If the policy holder dies before age 65, a family income insurance becomes available
consisting of an annual payment of 20,000 paid at the end of each year, starting the year in which this
person died and with the final payment in the year the insured would have reached age 65. If the policy
holder reaches the age of 65, then the policy provides a life annuity with annual payments of 15,000 and
the first payment at the start of the following year, until the moment the insured dies. As a further benefit
of this policy, there is a death benefit of 50,000 at the end of the year of death of the insured, no matter
at what age death occurs. Payment of the premiums is offered as follows: constant net annual premiums
Π are to be paid at the start of each year until the age of 65, or death if earlier.
(a) Calculate Π.
(b) Suppose that the insured has 20,000 available which he can use to buy the policy, in addition to annual
premiums as described above. Determine how the annual premiums are affected by the use of this 20,000
up front.
(c) Suppose that this policy can only be bought at a price that includes expense loading of the premiums,
with a one-off cost of 400 to cover acquisition expenses, and collection expenses at 1.5% of the annual
premium. Answer the questions (a) and (b) again, taking these expenses into account.