FM for Actuaries

100 CHAPTER 3
Period of investment (in quarters) Spot rate (% per annum)
1 3.80
2 3.90
3 4.00
4 4.25
5 4.30
6 4.50
Calculate the forward rates of interest for the second, third, fourth, fifth, and
sixth quarters.
3.10 Fill in the following table of forward rates of interest:
i F t,τ
t i S t i F t τ = 1 2 3 4 5 6 7
1 4.00%
2 4.20% –
3 4.40% – –
4 4.60% – – –
5 4.70% – – – –
6 4.50% – – – – –
7 4.30% – – – – – –
8 4.10% – – – – – – –
3.11 Denote the spot rates of interest for investment horizons of 1 to n years by i S t
for t =1, 2, ··· ,n. Derive an expression (in terms of i S t ) of the future value
of the $1-payment annuity-immediate at the end of year n, assuming future
payments earn the spot rates of interest as at time 0.
3.12 Suppose the spot rates of interest for investment horizons of 1 to 5 years are
4%, and for 6 to 10 years are 5%.
(a) Compute the forward rates of interest i F t for t =1, 2, ··· , 10.
(b) Calculate the present value of an annuity-immediate of $100 over 10
years.
(c) Compute the future value of the annuity-immediate at the end of year
10, assuming future payments earn the forward rates of interest, using
equation (3.18).