1       Calculate, showing all working, 3.5q56.25 , assuming a constant force of mortality between integer ages.

Basis:

Mortality        ELT15 (females) [4]

2       A non-tax-paying investor wishes to invest a lump sum for a period of 5 years. The   investor is considering depositing the lump sum with a bank. Interest will be credited to the initial lump sum at a variable market rate. Interest will only be added when the cash is withdrawn.

The investor has set up a generalised cashflow model for this investment.

Describe the cashflows with reference to the certainty and uncertainty of size and

timing of payments from the investor’s point of view.                                                  [4]

3       An individual aged 20 exact wishes to purchase a commercial property at age 45,        which is currently priced at $1,000,000. They plan to save a fixed amount, X, monthly in advance for the first 10 years, then 2Xmonthly in advance for the remainder of the term.

The investment return is expected to be 2.5% p.a. effective, and property price inflation is expected to be 0.5% effective per half year.

Calculate X, using annuity functions. You must show all working.                            [5]

4

(i)        Describe, in words, the meaning of the following: A 2                

Two lives, x and y, are independent with respect to mortality. Life x is subject to a constant force of mortality of 0.025, and life y is subject to a constant force of      mortality of 0.02.

(ii)       Calculate the probability that life y dies more than 3 years after life x. When

function and show all your working.                                                                  [5]

[Total 7]

When pricing fixed interest bonds, the capital gains test compares i(p ) to (1  t1 )   ,

where

p is the frequency of the coupon payment,

t1 is the income tax rate,

D is the annual coupon rate,

R is the redemption rate, and

i is the annual effective rate of return.

(i)        Show that if i(p ) > (1  t1 )   then there is a capital gain at redemption.      [4] Two situations where a capital gains test needs to be performed are:

1.   for an investor who pays capital gains tax.

2.   where the redemption date of the bond is variable.

(ii)       Explain, for these two situations, why the capital gains test is necessary.      [6] [Total 10]

8       A life insurance company issues 20-year term assurance policies to lives aged 45     exact. The sum assured of £300,000 is payable at the end of the year of death. Level premiums are payable annually in advance for the term of the policy ceasing on the death of the policyholder if earlier. The insurance company uses the following         assumptions:

Mortality Interest

Expenses Initial     Renewal Claim

AM92 (select)

4% p.a. effective

£250 incurred at the outset of the policy

£50 incurred at the start of the second and each subsequent policy year £75 incurred when the benefit is paid out

Commission

Initial               25% of the annual premium

Renewal          1.5% of each annual premium excluding the first

(i)        Write down the gross future loss random variable at outset for the policy in      terms of Kx, the curtate future lifetime of a life aged x exact.                          [4]

The company sets the premium such that the expected present value of the gross      future loss random variable is equal to 4% of the expected present value of premium income.

(ii)       Calculate the gross premium, showing all working.                                        [7]

[Total 11]

9       A unit-linked policy has the following profit vector:

(i)        Calculate, showing all working, the reserves required to zeroise the expected   negative future cashflows at the end ofyears 2 and 3, assuming the non-unit    fund accumulates at 3% p.a. effective and that qx = 0.02 for all ages.             [3]

(ii)       Determine the net present value of the profits, assuming a risk discount rate of

8% p.a.:

(a)       before zeroisation.

(b)       after zeroisation. [6]

(iii)      Comment on the results you obtained in part (ii).                                            [2]

[Total 11]

10     On 1 January 2006, a life insurance company issued a portfolio of identical 25-year endowment assurances with sums assured of £200,000 to lives then aged 40 exact.

Premiums of £3,885 are payable annually in advance and cease at the end of the           policy term or on earlier death. The sum assured is payable at maturity or at the end of the year of death if earlier. At the start of 2021, there were 520 policies in force.

1 January 2021, using the prospective method.                                                 [4]

Reserving basis

Interest                                      4% p.a. effective

Mortality                                 AM92 ultimate

Surrenders                               Ignore

Renewal expense                    £25 incurred at the beginning of each policy year

Claim expense                         £100 incurred at the same time as the benefit

payment is payable

Renewal commission             1% of the annual premium

The actual experience of the insurance company in respect of this portfolio of policies during 2021 was:

    actual investment return on assets backing the reserves was 4.3%.     2 deaths.

     10 surrenders, each receiving a surrender payment of £113,000 at the end of the year.

    actual renewal expenses attributed to the entire portfolio were £55,000.             total actual expenses incurred on handling the death claims were £200.             actual renewal commission payments were 1% of the annual gross premium.

The gross premium reserve per policy on 1 January 2022 is £113,238, calculated prospectively on the reserving basis above.

(ii)       Calculate, showing all working, the total profit earned from this portfolio of endowment assurances during 2021.                                                                  [8]

(iii)      Comment on the sources of profit, comparing the actual experience with the reserving basis.                    [3]

[Total 15]