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:
𝑎̈𝑛̅| − 𝑛𝜈
Hint: You have to show your full workings from basic principles of valuing a constant payment
[Total 8 marks]
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.
(ii) The amount of loan outstanding immediately after the instalment now due is paid.
(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.
(b) The amount of the amended final payment.
(c) The interest component of the amended final payment.
[Total 21 marks]
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-
(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
[Total 20 marks]
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]
The force of interest
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
(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.
(iii) Calculate, to the nearest 0.01%, the constant nominal annual rate of interest convertible
half-yearly implied by the transaction in part (ii).
(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
[Total 28 marks]
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;
(ii) The annual effective rate of discount equivalent to this transaction, to the nearest 0.01%;
(iii) The annual nominal rate of discount convertible monthly equivalent to this transaction, to
the nearest 0.01%.
[Total 10 marks]