MTH120 Exercises 1
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MTH120 Questions
Exercises
1. An investor is considering two investments. One investment is a 91-day bond issued by a bank which pays a rate of interest of 4% per annum effective. The second is a 91-day treasury bill which pays out $100.
(a) Calculate the price of the treasury bill and the annual simple rate of discount from the treasury
bill if both investments are to provide the same effective rate of return. [3] (b) Suggest one factor, other than the rate of return, which might determine which investment is chosen. [1]
[Total 4]
2. The effective rate of discount per annum is 5%.
Calculate:
(a) the equivalent force of interest; [1] (b) the equivalent rate of interest per annum convertible monthly; [2] (c) the equivalent rate of discount per annum convertible monthly. [1]
[Total 4]
3. The force of interest, 6(t), is a function of time and at any time t, measured in years, is given by the formula:
6(t) = 
(b) Calculate the equivalent constant force of interest from t = 0 to t = 20. [2]
(c) Calculate the present value at time t = 0 of a continuous payment stream payable at a rate of e −女﹒女6t from time t = 4 to time t = 8. [4]
[Total 10]
4. The force of interest 6(t) is a function of time, and at any time t, measured in years is given by the formula:
6(t) = 
(a) Derive, and simplify as far as possible, expressions in terms of t for the present value of a unit investment made at any time, t. You should derive separate expressions for each time interval
0 < t ≤ 6 and 6 < t. [5]
(b) Determine the discounted value at time t = 4 of an investment of 1．000 due at time t = 10. [2] (c) Calculate the constant nominal annual interest rate convertible monthly implied by the trans- action in part (b). [2]
(d) Calculate the present value of a continuous payment stream invested from time t = 6 to t = 10 at a rate of β(t) = 20e女﹒36|女﹒32t per annum. [4]
[Total 13]
5. (a) Calculate the time in days for 戈6,000 to accumulate to 戈7,600 at:
i. a simple rate of interest of 3% per annum.
ii. a compound rate of interest of 3% per annum effective.
iii. a force of interest of 3% per annum. [6]
Note: You should assume there are 365 days in a year.
(b) Calculate the effective rate of interest per half year which is equivalent to a force of interest of 3% per annum. [1]
[Total 7]
6. The force of interest, 6(t), is a function of time and at any time t, measured in years, is given by the formula:
6(t) = 
(a) Calculate the corresponding constant effective annual rate of interest for the period from t = 0
to t = 10. [4]
(b) Express the rate of interest in part (a) as a nominal rate of discount per annum convertible half-yearly. [1]
(c) Calculate the accumulation at time t = 15 of 戈1,500 invested at time t = 5. [3]
(d) Calculate the corresponding constant effective annual rate of discount for the period t = 5 to t = 15. [1]
(e) Calculate the present value at time t = 0 of a continuous payment stream payable at a rate of
10e女﹒女1t from time t = 11 to time t = 15. [6]
[Total 15]
7. Calculate the nominal rate of discount per annum convertible monthly which is equivalent to:
(b) a force of interest of 5% per annum. [2]
(c) a nominal rate of discount of 4% per annum convertible every three months. [2]
[Total 6]
8. The nominal rate of interest per annum convertible quarterly is 5%. Calculate, giving all the answers as a percentage to three decimal places:
(b) the equivalent effective rate of interest per annum. [1]
(c) the equivalent nominal rate of discount per annum convertible monthly. [2]
[Total 4]
9. At the beginning of 2015 a 182–day commercial bill, redeemable at 戈100, was purchased for 戈96 at the time of issue and later sold to a second investor for 戈97.50. The initial purchaser obtained a simple rate of interest of 3．5% per annum before selling the bill.
(a) Calculate the annual simple rate of return which the initial purchaser would have received if they had held the bill to maturity. [2]
(b) Calculate the length of time in days for which the initial purchaser held the bill. [2]
The second investor held the bill to maturity.
(c) Calculate the annual effective rate of return achieved by the second investor. [2]
[Total 6]
10. The force of interest, 6(t), is a function of time and at any time t, measured in years, is given by the formula:
,．0．06 0 ≤ t ≤ 4
6(t) = ．0．10 − 0．01t 4 < t ≤ 7
．(0．01t − 0．04 7 < t
(a) Calculate, showing all working, the value at time t = 5 of 戈10,000 due for payment at time t = 10. [5]
(b) Calculate the constant rate of discount per annum convertible monthly which leads to the same result as in part (a). [2]
[Total 7]
11. An investor wishes to obtain a rate of interest of 3% per annum effective from a 91-day treasury bill. Calculate:
(a) the price that the investor must pay per 戈100 nominal.
[3]
12. The nominal rate of discount per annum convertible monthly is 5．5%.
(a) Calculate, giving all your answers as a percentage to three decimal places:
i. the equivalent force of interest.
ii. the equivalent effective rate of interest per annum.
iii. the equivalent nominal rate of interest per annum convertible monthly.
[3]
(b) Explain why the nominal rate of interest per annum convertible monthly calculated in part (a)(iii) is less than the equivalent annual effective rate of interest calculated in part (a)(ii) [1]
(c) Calculate, as a percentage to three decimal places, the effective annual rate of discount offered by an investment that pays 戈159 in eight years’ time in return for 戈100 invested now. [1]
(d) Calculate, as a percentage to three decimal places, the effective annual rate of interest from an investment that pays 12% interest at the end of each two-year period. [1]
[Total 6]
13. The force of interest, 6(t), is a function of time and at any time t (measured in years) is given by
,．0．08 for 0 ≤ t ≤ 4
6(t) = ．0．12 − 0．01t for 4 < t ≤ 9
．(0．05 for t > 9
(a) Determine the discount factor, 〇(t), that applies at time t for all t ≥ 0. [5]
(b) Calculate the present value at t = 0 of a payment stream, paid continuously from t = 10 to
t = 12, under which the rate of payment at time t is 100e女﹒女3t [4]
(c) Calculate the present value of an annuity of 戈1．000 paid at the end of each year for the first three years. [3]
[Total 12]
2022-07-27