Question

An investor wishes to purchase a 1-year forward contract on a risk-free bond which has a current market price of £97 per £100 nominal. The bond will pay coupons at a rate of 7% per annum half-yearly. The next coupon payment is due in exactly 6 months, and the following coupon payment is due just before the forward contract matures. The 6-month risk-free spot interest rate is 5% per annum effective and the 12-month risk-free spot interest rate is 6% per annum effective. Stating all necessary assumptions

(a) Calculate the forward price of the bond.

(b) Calculate the 6-month forward rate for an investment made in 6 months’ time.

(c) Calculate the purchase price of a risk-free bond with exactly 1 year to maturity which is redeemed at par and which pays coupons of 4% per annum half-yearly in arrears.

(d) Calculate the gross redemption yield from the bond in (c).

(e) Comment on why your answer in (d) is close to the 1-year spot rate.

Answer 1

(i) A future is a contract which obliges the parties to deliver/take delivery of a particular quantity of a particular asset at a particular time at a fixed price.

An option is the right to buy or sell a particular quantity of a particular asset at (or before) a particular time at a given price.

(ii) Assume no arbitrage

a. Buying the forward is exactly the same as buying the bond except that the forward will not pay coupons and the forward does not require immediate settlement.

Let the forward price = F. The equation of value is:

F = (97*1.06) – 3.5*(1.06/ 1.05^1/2) – 3.5
=> 102.82 – (3.5* 1.034454077) – 3.5
= 95.6994

b. Let six month forward interest rate be f_0.5,0.5

f_0.5,0.5 = (1.06/ 1.05^0.5) – 1 = 0.034454 or 3.4454%

Note: It is not required to be expressed as annual interest rate.

c. P = (2 * 1.05^-0.5) + (102 *1.06) = 1.9518 + 96.2264 = 98.1782

d. Gross redemption yield is i such that:

98.1782 = 2*(1 + i)^-0.5+ 102* (1+ i)^-1

Using the formula for solving a quadratic (interpolation will do):

(1 + i)^-0.5 = 0.97133. Therefore, i ? 6% (in fact 5.99%).

e) Answer is very close to 6% (the one-year spot rate) because the payments from the bond are so heavily weighted towards the redemption time in one year.