1. Why do insurers generally require evidence of health from a person applying for life insurance but not for an annuity.

(3 marks)

2. Derive (to the nearest integer) the median of the complete future lifetime of a person aged 30 exact who is subject to the force of mortality shown below:

( .01

| |

0  t  10

10  t  20

20  t

(6 marks)

3.       You are given the following survival function of a newborn:

S0 (x ) =〈 1(5)))|2

Calculate the probability that (30) dies within the next 20 years.

(5 marks)

4. For a fully discrete 3-year endowment insurance of £1000 on (x), you are given:

(a) µx+t = 0.08, for 0 ≤ t ≤ 3

(b) σ = 0.08

(c) The annual premium is 400.

(d) L0  is the present value random variable for the loss at issue for this insurance.

What is the meaning that a policy is fully discrete?

(1 marks)

Calculate the propability P r[L0  > 0].

(4 marks)

A term assurance contract for a life aged 50 exact for a term of 10 years provides a benefit of £10,000 payable at the end of the year of death.  Calculate the expected present value and      variance of benefits payable under this contract.

Basis:

Mortality:   AM92 Select

Interest:      4% per annum

(9 marks)

6.  On  1 January 2014 a life insurance company issued  1,000 identical special    endowment insurance policies to select lives then aged 50.  Each policy has a term  of  15  years  and  level  premiums  are  payable  annually  throughout  the policy term.  On  death within the policy term,  a  sum  insured  of £100,000  is paid.  On survival to the end of the term, a sum insured of £75,000 is paid.       The premium is calculated using the AM92 select tables and assuming

Interest: 4%

•  Initial expenses: 50% of the first gross premium

•  Renewal expenses: 5% of gross premium after the first

(i)  Show that the annual premium for each policy is £4,302.               [3 marks]

(ii)  Calculate the gross policy value per policy in force at the start of 2016,

just before the premium due is paid.                                               [3 marks]

(iii)  Calculate the asset share for each policy at the start of 2016.

[5 marks]

(iv)  Comment your answers to parts (ii) and (iii).                                 [3 marks]

(v) There were 995 policies in force on 1 January 2016. During 2016 there were

2 actual deaths, the actual interest rate earned by the company was 3.8% and the expenses were 7%. Calculate the profit or loss in 2016 for this group of policies. [5 marks]

7.     Assuming a uniform distribution of deaths between integral ages. Show that

i

(5 marks)

8. Calculate 0.5p45.75 using the Uniform Distribution of Deaths assumption.

Basis: AM92 Ultimate

(4 marks)

10.       On 1 January 2001, a life insurance company issued a number of 30-year endowment assurance policies that pay £100,000 at maturity, or £50,000 at the end of the year of earlier death to lives then aged 35 exact.  Premiums are payable annually in advance.

The company uses the following basis for calculating premiums and reserves:

Mortality                 AM92 Select

Interest                     4% per annum

Initial commission   50% of the premium payable in the first policy year

Initial expenses        £300 paid at policy commencement date

Renewal expenses    2.5% of each premium from the start of the second policy year

(i)         Write down the recursive relationship between the gross premium reserves at

successive durations of these policies, defining all symbols used.                           (6 marks)

(ii)       Show that the annual premium for each policy is approximately £1,803.              (7 marks)

There were 385 policies in force on 1 January 2010.  During 2010, there were three actual deaths, actual interest earned by the company was 5% and expenses were as expected.

(iii)      Calculate the profit or loss made by the company from both mortality and interest in respect of these policies for the year 2010 based on the formula

stated in (i) above.                                                                                                            (7 marks)