Question

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

Sol:

Portfolio value = $10 million

Zero coupon bond duration = 7 years

Perpetuity duration = 1.08/0.8 = 13.5 years

i) To determine weight of each bond Tony will hold in his portfolio is as follows,

Assume weight (W) invested in zero coupon bond

n = Zero coupon bond duration weight + (-weight) x (Perpetuity duration)

10 = 7W + (1-W) x (13.5)

10 = 7W + 13.5 - 13.5W

10 - 13.5 = 7W - 13.5W

-3.5 = -6.5W

W = -3.5/-6.5 = 0.5385 or 53.85%

Zero coupon bond weight = 0.5385 or 53.85%

Perpetuity weight = 1 - 0.5385 = 0.4615 or 46.15%

ii) Zero coupon bond duration = 6 years

Perpetuity duration = 1.06/0.6 = 17.67 years

Assume weight (W) invested in zero coupon bond

9 = 6W + (1-W) x (17.67)

9 = 6W + 17.67 - 17.67W

9 - 17.67 = 6W - 17.67W

-8.67 = -11.67W

W = -8.64/-11.67 = 0.7429 or 74.29%

Zero coupon bond weight = 0.7429 or 74.29%

Perpetuity weight = 1 - 0.7429 = 0.2571 or 25.71%