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Questions

MTH120

Assignment 1

1.   (a) The rate of interest is 4.5% per annum effective. Calculate the equivalent

ii. nominal rate of discount per annum convertible monthly.                                               [2]

iii. nominal rate of interest per annum convertible quarterly.                                               [2]

iv. effective rate of interest over a five year period.                                                             [2]

(b) Prove that d(p)  < d(p+1)  mathematically, for p ∈ N.                                                               [5]

[Total 13]

2. The force of interest at time t is given by

  0.08 − 0.001t   0 ≤ t < 3

6(t) =〈   0.025t − 0.04   3 ≤ t < 5

(         0.03              5 ≤ t

(a) Calculate the present value at time 2 of a payment of £1, 000 at time 10 .                           [5]

(b) Calculate the annual effective rate of interest that is equivalent to this variable force of interest from time 2 to time 10.                [2]

(c) Denote the A(t1 ,t2 ) be the accumulation factor from t1  to t2 . Find an expression for 6(t) in terms of A(0,t), A(0,t) and 6(t).                                           [7]

[Total 14]

3. Uncle Stephen has decided to buy a new car.   The car’s face value is £10, 000.   He adds some accessories which cost £250. The salesman agrees to a discount of £300 and Stephen part-exchanges his old car for £2, 000 which also acts as a deposit. For the remaining finance Stephen must take out a loan. The dealership offer a finance package which requires 36 payments paid monthly in advance of £175, plus a final lump sum payment of £4, 000. As an alternative Stephen could borrow the money from his bank who will charge him an effective interest rate of 6.5% p.a.  for monthly repayments made monthly in advance over 3 years.

(b) Estimate the rate of interest charged by the dealership’s finance package.                             [7]

[Total 13]

4. An investor is considering purchasing loan stock which pays annual coupons at a rate of 7.5% per annum per 100 nominal and is redeemed at 110% at any time between time t = 5 and time t = 10 at the choice of the borrower. The investor pays income tax at a rate of 25% and capital gains tax at a rate of 30%.

(a) Calculate the price the investor should pay for 100 nominal of the loan stock if he wishes to  achieve a net return of 6% per annum.                                                                                 [6] 

(b) After 3 years another investor wishes to purchase the loan stock from the original investor. This second investor wants to achieve a return of just 4%p.a. What price should he offer to pay the original investor if this second investor is not subject to tax.                                                [6]

[Total 12]