Question

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

> Formula

• Duration Zero Coupen Bond = Maturity of bond
• Duration perpetual bond = [ 1 + yield ] / Yield

> Calculation

• Duration of ZCB = 3 years
• Duration of perpetual bond = [ 1 + 0.08 ] / 0.08

                                                 = 13.5 years

(i) Weights of each bond

Let us assume weight of Zero coupen Bond be "w". Then weight of perpetual bond be "1-w"

=> Target Duration = w * Duration of ZCB + (1-w) * Duration of perpetual bond

=> 7 = w * 3 + (1-w) * 13.5

=> 7 = 3w + 13.5 - 13.5w

=> w = 6.5 / 10.5

=> w = 0.62

=> (1-w) = 0.38

• Weight of ZCB = 0.62
• Weight of Perpetual bond = 0.38

(ii)

• Revised duration of perpetual bond = ( 1.06 ) / 0.06

                                                  = 17.67 years

• Revised duration of ZCB = 2 years

Let us assume weight of Zero coupen Bond be "w". Then weight of perpetual bond be "1-w"

=> Target Duration = w * Duration of ZCB + (1-w) * Duration of perpetual bond

=> 6 = w * 2 + (1-w) * 17.67

=> 6 = 2w + 17.67 - 17.67w

=> w = 11.67 / 15.67

=> w = 0.75

=> (1-w) = 0.25

• Weight of ZCB = 0.75
• Weight of Perpetual bond = 0.25

Hope you understand the solution.