Question

a. Explain which of the following bonds has higher interest rate sensitivity. Bond A is a 15-year, noncallable bond with a coupon rate of 7%, selling at par.

Bond B is a 15-year, callable bond with a coupon rate of 9%, also selling at par.

b. Tony, a fixed-income portfolio manager, is managing a portfolio of $10 million. His target duration is 7 years, and he can choose from two bonds: a zero-coupon bond with maturity of 3 years, and a perpetuity, each currently yielding 8%.

i. What is the weighting of each bond will Tony hold in his portfolio?

ii. Suppose that a year has passed and the yield has fallen to 6%. What will these weightings be if target duration is now 6 years?

Answer 1

a. Please find the excel screenshots for duration calculation of A and B. I have also provided one screenshot with formulas. As both the bonds are trading at par, ytm will be equal to coupon rate. Duration is calculated using the weighted cash flows. Please check the formulas to understand how its calculated. The bond with the higher duration will be more sensitive to interest rate changes. So Bond A will be more sensitive to interest rate changes.

b. i. Target Duration = 7 years

Duration of ZCB = 3 years

Duration of perpetual bond = (1+yield)/yield = 1.08/0.08 = 13.5

Let the weights of ZCB and perpetual bond be x, (1-x).

Weighted Duration = Weight of ZCB * Duration of ZCB + Weight of Perpetual Bond * Duration of Perpetual Bond

7 = x*3 + (1-x)*13.5

3x + 13.5 - 13.5x = 7

-10.5x = 7 - 13.5

Weight of ZCB or x = 0.62 or 62% and Weight of Perpetual Bond = (1-x) = (1-0.62) = 0.38 or 38%

ii. If the target duration is 6 years, using the formulas in answer above:

6 = x*3 + (1-x)*13.5

3x + 13.5 - 13.5x = 6

-10.5 x = 6 - 13.5

Weight of ZCB or x = 0.71 or 71% and Weight of Perpetual Bond = (1-x) = (1-0.71) = 0.29 or 29%