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Question 1
An annuity is payable continuously for 𝑛 years. The rate of payment is constant through each year
and is as follows:
1 unit per annum during the first year;
2 units per annum during the second year;
3 units per annum during the third year; and so on.
Show that the present value of the annuity is given by:
(𝐼𝑎)𝑛| =
𝑎̈𝑛̅| − 𝑛𝜈
𝑛
𝛿
Hint: You have to show your full workings from basic principles of valuing a constant payment
stream.
[Total 8 marks]
Question 2
Exactly 5 years ago, a loan was taken out that was to be repaid by level annual instalments made in
arrears over a 15-year contract. Given that the instalments (of capital and interest) were set to
£883 per annum based on a 8% p.a. effective interest rate on the borrowing, calculate the following:
(i) The initial amount of loan taken out on this contract.
[3 marks]
(ii) The amount of loan outstanding immediately after the instalment now due is paid.
[3 marks]
(iii) It is agreed that, immediately after the instalment now due, the rate of interest charged
on the outstanding loan is reduced to 5.5% p.a. effective. Consequently, the same annual
instalments will be payable for a revised remaining term and also with an amended final
payment. Thus, find the following:
(a) The revised remaining term of the loan outstanding in whole years.
[8 marks]
(b) The amount of the amended final payment.
[4 marks]
(c) The interest component of the amended final payment.
[3 marks]
[Total 21 marks]
Question 3
An individual deposits £10,500 each year into a tax-free savings plan over a 20-year period. The
payments are made monthly in arrears during the first 5 years and thereafter quarterly in arrears for
the remaining 15 years.
The savings plan pays compound interest at the rates of:
6% p.a. nominal convertible monthly for the first 10 years, and
7.5% p.a. nominal convertible quarterly for the remaining 10 years.
(i) Calculate the total amount of fund accumulated in the savings plan at the end of the 20-
year period.
[13 marks]
(ii) At the end of the 20-year period, the individual intends to invest the total savings into a
level fixed term annuity product that provides a future retirement income. Calculate the
monthly income that the individual can obtain by investing the sum calculated in part (i)
into a 25-year term annuity payable monthly in arrears at an effective interest rate of
4.5% p.a.
[7 marks]
[Total 20 marks]
Question 4
A manufacturer is considering investing in a new production line that requires an initial capital
investment of £79,000. Once in operation, the production line is expected to generate the following
net earnings at the end of the years stated:
Furthermore, exactly mid-year during the second year there will be a further maintenance cost
amounting to £1,600. Then at the end of the planned 4-year service it is expected that the company
will be able to sell the machinery for £41,000.
Calculate, to the nearest 0.01%, the annual internal rate of return the manufacturer can expect to
earn from this investment.
[Total 13 marks]
Question 5
The force of interest
t
at any time t, measured in years, is given by:
Derive, and simplify as far as possible, expressions in terms of t for the discount factor of
a unit investment made at time 𝑡. You should derive separate expressions for the three
sub-intervals.
[14 marks]
(ii) Hence, making use of the result in part (i), calculate the value at time t = 3 of a payment of
£2,500 made at time t = 15.
[4 marks]
(iii) Calculate, to the nearest 0.01%, the constant nominal annual rate of interest convertible
half-yearly implied by the transaction in part (ii).
[3 marks]
(iv) Making use of the result in part (i), calculate the present value of a payment stream 𝜌(𝑡)
paid continuously from time t = 15 to t = 20 at a rate of payment at time t given by:
𝜌(𝑡) = 300𝑒
0.02𝑡
.
[7 marks]
[Total 28 marks]
Question 6
A short term loan of £5,000 is repayable in 25 days at a simple rate of interest of 6% p.a.
Assuming that 1 year is equivalent to exactly 365 days, calculate the following:
(i) The amount of interest, to the nearest £0.01, accrued on the loan in 25 days;
[3 marks]
(ii) The annual effective rate of discount equivalent to this transaction, to the nearest 0.01%;
[4 marks]
(iii) The annual nominal rate of discount convertible monthly equivalent to this transaction, to
the nearest 0.01%.
[3 marks]
[Total 10 marks]
