Engineering and Mathematical Sciences

01 / 10 / 2021 Assignment

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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]

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