MTH 120 Financial and Actuarial Mathematics 2021/22
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MTH 120
Financial and Actuarial Mathematics
2st SEMESTER 2021/22 FINAL EXAMINATIONS
BACHELOR DEGREE - Year 2
Theory of Interest
Questions
Q 1. Describe the characteristics of
。an interest-only loan (or mortgage); and
。a repayment loan (or mortgage).
Q 2. (a) The rate of interest is 4.5% per annum effective. Calculate the equivalent
i. annual effective rate of discount.
ii. nominal rate of discount per annum convertible monthly.
iii. nominal rate of interest per annum convertible quarterly.
iv. effective rate of interest over a five year period.
(b) Prove that d(p) < d(p+1) mathematically, for p e N.
Q 3. 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 戈1, 000 at time 10 .
(b) Calculate the annual effective rate of interest that is equivalent to this variable force of interest from
time 2 to time 10.
(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]
Q 4. Uncle Stephen has decided to buy a new car. The car’s face value is 戈10, 000. He adds some accessories
which cost 戈250. The salesman agrees to a discount of 戈300 and Stephen part-exchanges his old car for 戈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 戈175, plus a final lump sum payment of 戈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.
(a) Calculate the monthly repayment amount under the bank loan.
(b) Estimate the rate of interest charged by the dealership’s finance package.
Q 5. 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.
(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.
Q 6. Mr Cooper has 戈5, 000 and he is looking to invest in Company ABC. Company ABC has shares which are
expected to provide regular dividends annually. The next dividend is expected to be 戈4.50 and will be paid in 2 months time and every 12 months thereafter. Dividends are expected to increase by 2% per annum.
(a) Describe the characteristics of equities.
Shares can only be purchased in blocks of 10.
(b) Suppose share price is evaluated through the discounted model with an effective interest rate of 6% per
annum, how many shares can Mr Cooper buy with his 5, 000?
Q 7. One of your best friends, Miss Hearland, has inherited E1m and is asking your opinion on two investment
opportunities. Opportunity A requires the investment of the whole sum in properties that are to be let out to tenants:
。Miss Hearland expects to receive rents at an annual rate of E60,000 a year for four years after an initial
period of one year in which no income will be received.
。Rents are expected to rise thereafter at the start of each year at a rate of 0 .5% per annum. 。The income will be received monthly in advance.
。There are costs of E10,000 per annum in the first year, rising at a constant rate of 0 .5% per annum. 。The costs will be incurred at the beginning of each year.
。At the end of 20 years, Miss Hearland expects to sell the properties for E2m after which there will be
no further revenue or costs.
Opportunity B involves the investment of the whole sum in an investment fund:
。The fund is expected to pay an income of E60,000 per annum annually in advance and return the whole
invested sum at the end of 20 years.
(a) i. Define the payback period.
ii. Calculate the payback period for opportunity B.
iii. Discuss, without any further calculation, whether the payback period from opportunity A is longer or shorter than that from opportunity B.
(b) i. Define the discounted payback period.
ii. Determine the discounted payback period from opportunity B at a rate of interest of 1% per annum effective.
iii. Discuss, without any further calculation, whether the discounted payback period from opportunity A is longer or shorter than that from opportunity B.
(c) Determine the internal rate of return from opportunity B expressed as an annual effective return.
(d) Show that the internal rate of return from opportunity A is higher than that from opportunity B.
(e) Discuss which opportunity is better given your answers above.
2022-07-26