Question

The yield of a 1-year zero-coupon bond is 5% per annum (continuously compounded). A 2-year coupon bond with a coupon rate of 7% per year (paid annually) has a price of £98 and a yield of 7.81% per annum (continuously compounded). The bonds have principal value of £100. i) What is the forward rate over the 2nd year? Provide the answer as a semi-annually compounded forward rate, as well as a continuously compounded forward rate.

Answer 1

Continuously Compounded 1-Year Rate = 5 % and Continuously Compounded 2-Year Rate = 7.8 %

Let the continuously compounded one-year forward rate be r

Therefore, [e^(0.078 x 2)] / [e^(0.05 x 1)] = e^(r x 1)

e^(r x 1) = 1.11182

r = = 0.106 or 10.6 %

Let the 1-Year Equivalent Semi-Annually Compounded rate be r1

Therefore, (1+r1)^(2) = e^(0.05 x 1)

r1 = [(1.05127)^(1/2)] - 1 = 0.02532 or 2.532 %

Let the 2-year Equivalent Semi-Annually Compounded Rate be r2

Therefore, (1+r2)^(4) = e^(0.078 x 2)

(1+r2)^(4) = 1.16883

r2 = [(1.16883)^(1/4)] - 1 = 0.03977or 3.977%

Let the semi-annually compounded forward rate be r

Therefore, (1+r)^(2) = [(1.03977)^(4)/(1.02532)^(2)] = 1.11181

r = [(1.11181)^(1/2)] - 1 = 0.05442 or 5.442 %

Annualized One Year Forward Rate = 2 x r = 2 x 5.442 = 10.884 %