Question 1

On 1 January 2008, a life insurance company issued a number of without profit endowment

policies maturing at age 60 to lives then aged 40 exact. The sum assured is payable at the end of

year of death or on survival to the end of the term and level premiums are payable annually in

advance throughout the term of the contract.

Premiums and reserves on each policy are both calculated on the following basis:

Mortality: AM92 Select

Interest: 4% per annum

Initial commission: 60% of the first premium

Renewal commission: 6% of each annual premium excluding the first

(i) Calculate the annual office premium per 1,000 sum assured for each policy. [2]

(ii) Calculate the gross premium prospective reserve per 1,000 sum assured for each policy

in force at 31 December 2012. [2]

(iii) Calculate the profit or loss to the company in 2013 in respect of these policies given the

following information:

The total sums assured in force on 1 January 2013 were 15,500,000.

The company incurred expenses relating to these policies of 76,500 on 1 January

2013 (including renewal commission).

The total sums assured paid on 31 December 2013 in respect of deaths during

2013 were 295,000.

The total sums assured surrendered during 2013 were 625,000. The surrender

value on each policy (which was paid on 31 December 2013) was calculated as

85% of the gross premium prospective reserve applicable at the date of payment

of the surrender value.

The company earned interest of 3.5% per annum on its assets during 2013.

4 Under a policy issued by a life insurance company, the death benefit payable at the end of year of

death is a return of premiums paid without interest. A level premium of 3,000 is payable

annually in advance throughout the term of the policy.

For a policy in force at the start of the 12th policy year, you are given the following information:

Reserve at the start of the policy year 25,130

Reserve at the end of the policy year per survivor 28,950

Probability of death during the policy year 0.03

Expenses incurred at the start of the policy year 90

Rate of interest earned 4% per annum

Reserves given above are immediately before payment of the premium due.

Calculate the profit/loss expected to emerge at the end of the 12th policy year per policy in force

at the start of that year. [3]

Calculate 55:10 s .

Basis:

Mortality: PFA92C20

Interest: 4% per annum

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]