Question

Suppose the price of a 1-year 5%-coupon bond is $1,000 and the price of a 2-year 5%-coupon bond is $1027.81. Using bootstrapping, what is the two-year zero-coupon spot rate? Each bond has a face value of $1,000 and makes annual coupon payments.

Answer 1

1. 1 year Spot Rate = 5% because the 1 year bond is trading at par value

2. Forward rate for year 2 =

1027.81 = 50 / 1.05 + 1050 / (1 + Forward rate)

1027.81 = 47.62 + 1050 / (1 + Forward rate)

980.19 = 1050 / (1 + Forward rate)

1 + Forward Rate = 0.93351

Forward Rate = 6.65%

2. Two-year zero-coupon spot rate = ((1 + 1 year spot rate) * (1 + Forward rate))^(1/2) - 1

Two-year zero-coupon spot rate = (1.05 * 1.0665)^(1/2) - 1

Two-year zero-coupon spot rate = (1.1198)^(1/2) - 1

Two-year zero-coupon spot rate = 1.0582 - 1

Two-year zero-coupon spot rate = 5.82%