Question

2. Carter Bank makes a 3-year loan for $1,000 with an annual coupon rate of 8%.

a. Compute the annual coupon bond payment of a U.S. Treasury bond with a risk-free rate of 8%.

b. Suppose the spot rate for 3-year Treasury bonds increases to 9%, find the duration and use it to explain what happens to the present value of the bond in part a. SHOW YOUR WORK.

Answer 1

2. a.

Annual coupon payment = coupon rate x face value of the bond = 8% x 1,000 = $80

b.

When the spot rate is 9%,

Present value of the bond PV = PMT x [1 - (1 +y)^-n ] / y + FV / (1 + y)ⁿ = 80 x (1 - 1.09^-3) / 0.09 + 1000 / 1.09³ = $974.69

When spot rate is 8%, PV^- = 80 x (1 - 1.08^-3) / 0.08 + 1000 / 1.08³ = $1,000

When spot rate is 10%, PV⁺ = 80 x (1 - 1.1^-3) / 0.1 + 1000 / 1.1³ = 950.26

Duration of bond = (PV^- - PV⁺) / (2 x PV x (y⁺ - y^-)) = 49.74 / ( 2 x 974.69 x 0.02) = 1.26

This means that a 1% change in yields would inversely change the price of the bond by 1.26% (1% increase in rates will cause a 1.26% decrease in prices)