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, δ(t), is a function of time and at any time t, measured in years, is given by the formula:

δ(t) = 

(a)  Calculate the present value of a unit sum of money due at time t = 20.                                  [4] (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 −4.46t  from time t = 4 to time t = 8.                                                                                         [4]

[Total 10]

4. The force of interest δ(t) is a function of time, and at any time t, measured in years is given by the formula:

δ(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 ρ(t) = 20e4.36|4.31t  per annum.

[4]

[Total 13]

5.   (a)  Calculate the time in days for e6,000 to accumulate to e7,600 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, δ(t), is a function of time and at any time t, measured in years, is given by the formula:

δ(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 e1,500 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

10e4.4dt  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:

(a) an effective rate of interest of 1% per quarter.                                                                          [2] (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:

(a) the equivalent annual force of interest.

(b) the equivalent effective rate of interest per annum.

[1] [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 e100, was purchased for e96 at the time of issue and later sold to a second investor for e97.50.  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.

The second investor held the bill to maturity.

(c)  Calculate the annual effective rate of return achieved by the second investor.

[2]

[2]

[Total 6]

10. The force of interest, δ(t), is a function of time and at any time t, measured in years, is given by the formula:

,．0.06                0 ≤ t ≤ 4

δ(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 e10,000 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 e100 nominal.

(b) the annual simple rate of discount from the treasury bill.

[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 e159 in eight years’ time in return for e100 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, δ(t), is a function of time and at any time t (measured in years) is given by

,．0.08                 for 0 ≤ t ≤ 4

δ(t) = ．0.12 − 0.01t     for 4 < t ≤ 9

．．0.05                 for t > 9

(a) Determine the discount factor, v(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 100e4.43t                                                                             [4]

(c)  Calculate the present value of an annuity of e1, 000 paid at the end of each year for the first