Question

Consider the following set of interest rates on zero-coupon bonds.

1 year: 6.0% p.a.

2 years: 6.1% p.a.

3 years: 6.2% p.a.

4 years: 6.3% p.a.

(a) How many forward rates are associated with these zero rates?

(b) Calculate the forward rates and discount factors.

(c) What is the relation between the zero rate curve and the forward rate curve?

Answer 1

(a) and (b)

Forward Rates are expected one-year interest rates at certain points of time in the future such as one-year forward rate one year from now is the expected one-year spot rate at t=1 year with current time being t=0

Therefore, f1 ={ [(1.061)^(2) / 1.06] - 1} x 100 = 6.20009 % approximately. (One-Year Forward Rate at t=1)

f2 = {[(1.062)^(3) / (1.061)^(2)] - 1} x 100 = 6.40028 %approximately (One-Year Forward Rate at t=2)

f3 = {[(1.063)^(4) / (1.062)^(3)] - 1} x 100 = 6.6006 % approximately (One-Year Forward Rate at t=3)

Expected Discount Factor:

At t=2 would be 1 / 1+f1 = 0.9416

At t=3 would be 1 / 1+f2 = 0.9398

At t=4 would be 1/ 1+f3 = 0.9381

(c) According to the pure expectations hypothesis, short-term spot rates combined with forward rates give an indication of the long-term spot rates. This essentially implies that an investment if invested for 1 year at 6% (the 1 year spot rate) and then rolled over into another investment at f1 for another year would give investment proceeds at t=2, equal to a single investment made for two years at 6.1% (the 2 year spot rate). Representing this mathematically we get:

K x (1.06) x (1+f1) = K x (1.061)^(2)