Question 1

A life insurance company is considering selling with-profit endowment policies with a term of

twenty years and initial sum assured of 100,000. Death benefits are payable at the end of the

policy year of death. Bonuses will be added at the end of each policy year.

The company is considering three different bonus structures:

(1) simple reversionary bonuses of 4.5% per annum

(2) compound reversionary bonuses of 3.84615% per annum

(3) super compound bonuses where the original sum assured receives a bonus of 3%

each year and all previous bonuses receive an additional bonus of 6% each year.

(i) Calculate the amount payable at maturity under the three structures. [4]

(ii) Calculate the expected present value of benefits under structure (2) for an individual aged

45 exact at the start, using the following basis:

Interest 8% per annum

Mortality AM92 Select

Expenses ignore [4]

(iii) Calculate the expected present value of benefits, using the same policy and basis as in (ii)

but reflecting the following changes:

(a) Bonuses are added at the start of each policy year (the death benefit is payable at

the end of the policy year of death).

(b) The death benefit is payable immediately on death (bonuses are added at the end

of each policy year).

(c) The death benefit is payable immediately on death, and bonuses are added

continuous

The following model for the force of mortality for a life insurance company's annuitants has been proposed: u(t, i) = (0.015 -0.0001).exp [a(x; -70) + B.y; + y.z; ] where: u(t, i) = force of mortality for the ith life, in calendar year 2000 + t ; x; = age of the ith life; y; = 1 if the ith life is a smoker, or y; = 0 if a non-smoker; z, = 1 if the ith life is male, or z; =0 if female; and a, B, y are the parameters of the model. The following data have been observed over the calendar year 2003: Risk characteristics Number of annuitants Number dying Male non-smoker, average age 65 800 o Male smoker, average age 60 200 Female non-smoker, average age 70 450 Female smoker, average age 65 150 You can assume that the numbers of annuitants in each class remained constant throughout the investigation period, and that the average age for each class can be treated as representing the value of x, for each individual in that class. (i) Explain why this model is a proportional hazards model. [2] (ii) Explain the importance of subdividing the data by age, sex and smoking status, and explain whether you think each of the parameters o, B and y would be likely to be positive or negative. [3] (iii) Calculate the force of mortality for female non-smokers with average age 70 in 2007, according to this model. [1] (iv) (a) Obtain an expression for the partial likelihood based on the given data, expressing your answer in terms of o, B and y only. (b) State how you would estimate the parameters of the model using the partial likelihood. [6] [Total 12]