FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 99
3.5 Suppose the spot rates of interest for investment horizons of 1, 2, 3 and 4
years are, respectively, 4%, 4.3%, i S 3 and 4.8%. You are given that (Da) 4⌉ is
5% large than (Ia) 4⌉
.
(a) Calculate the spot rate of interest i S 3 .
(b) Calculate the values of (Da) 4⌉ and (Ia) 4⌉ .
3.6 You are given the following information about various annuities available in
the market. All the annuities pay $100 at the end of each year over the period
of the investment. It is also given that the spot rate of interest for 1-year
maturity is 3%.
Period of investment (in years) Price (in $)
2 190.89
3 281.96
4 369.53
Calculate the forward rates of interest i F t for t =2, 3 and 4. You want to
accumulate $10,000 after four years by putting a single payment into a bank
account. Based on the information above, calculate the deposit required.
3.7 Let δ(t) =0.01t, fort ≥ 0. Findi S 2 and iF 3 .
3.8 The following table shows the spot rates of interest over a number of horizons:
Period of investment (in years) Spot rate (in %)
1 6.00
2 7.00
3 7.75
4 8.25
(a) Find the present value of payments of $100 at the end of each year for
4 years.
(b) What is the effective rate of interest earned by this annuity-immediate
if it were constant through the whole investment period? [Hint: Use
the Excel function Rate to compute the rate of interest.]
3.9 Suppose that the spot rates of interest over a number of horizons (in quarters)
with continuous compounding are as follows: