Question 1

A life insurance company issues a with profit whole life assurance policy to a life aged 55 exact.

The sum assured is 75,000 together with any attaching bonuses and is payable immediately on

death. Level premiums are payable monthly in advance ceasing on the policyholder's death or on

reaching age 85 if earlier.

Simple annual bonuses are added at the end of each policy year (ie the death benefit does not

include any bonus relating to the policy year of death).

The company calculates the premium on the following basis:

Mortality AM92 Select

Interest 4% per annum

Expenses

Initial 275

Renewal 65 at the start of the second and subsequent policy years and payable until

death

Claim 200 on death

Commission

Initial 75% of the total premium payable in the first policy year

Renewal 2.5% of the second and subsequent monthly premiums

Bonuses Simple bonus of 2.0% of basic sum assured per annum

(i) Calculate the monthly premium for this policy.

(ii) Calculate the gross prospective policy value at the end of the 30th policy year given that

the total actual past bonus additions to the policy have followed the assumptions stated

in the premium basis above (including the bonus just vested).

Policy value basis:

Mortality AM92 Ultimate

Interest 4% per annum

Expenses

Renewal 80 at the start of each policy year and payable until death

Claim 250 on death

Commission

Renewal 2.5% of the monthly premiums

Bonuses Simple bonus of 2.5% of basic sum assured per ann

The Cox proportional hazards model is to be used to model the rate at which students leave a certain profession before qualification. Assuming they stay in the profession, students will qualify three years after joining the profession. In the fitted model, the hazard depends on the time, t, since joining the profession and three covariates. The covariates, their categories and the fitted parameters for each category are shown in the table below: Covariate Possibility Parameter Size of employer large 0 small 0.4 Degree studied none 0.3 Science -0.1 Arts 0.2 other 0 Location London 0 other UK -0.3 overseas 0.4 (i) Defining clearly all the terms you use, write down an expression for the hazard function in this model. [3] (ii) State the class of students that is most likely to proceed to qualification under this model, and that which is least likely. [2] (iii) A student who has been in the profession for one year moves from a "small" employer to a "large" employer. Express the probability that he will qualify with the "large" employer Pr in terms of the probability that he would have qualified if he had stayed with the "small" employer Ps, all other factors being equal. [2] [Total 7]