Question

You are managing a portfolio of $1 million. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with maturity five years, and a perpetuity, each currently yielding 5%. a. How much of (i) the zero-coupon bond and (ii) the perpetuity will you hold in your portfolio? (Do not round intermediate calculations. Round your answers to 2 decimal places.) b. How will these fractions change next year if target duration is now nine years? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

Ai.Zero coupon bonds = 68.75%

Aii. Perpetuity= 0.3125

Bi. Zero coupon bonds = 70.58%

Bii. Perpetuity= 0.2942

Step-by-step explanation:

ai. Calculation for How much of the zero-coupon bond

Duration of Zero Coupon Bond = 5 years

Duration of Perpetuity = 1.05/0.05 = 21 years

Using this formula

Dp=W×D1 +(1-W)×D2

Let plug in the formula

10 = 5w + (1 - w)21

10 = 5w + 21 - 21w

21w-10w=21-5

11w=16

w=11/16

w = 0.6875*100

w=68.75%

Zero coupon bonds = 68.75%

Therefore How much of the zero-coupon bond

you hold in your portfolio will be 68.75%

aii. Calculation for the perpetuity you will hold in your portfolio

Perpetuity= 1 - 0.6875

Perpetuity= 0.3125

Therefore the perpetuity you will hold in your portfolio will be 0.3125

bi. Calculation for how will these fractions change next year if target duration is now nine years

Duration of Zero Coupon Bond = (5 -1) =4years

Duration of Perpetuity = 1.05/0.05 = 21 years

Using this formula

Dp=W×D1 +(1-W)×D2

Let plug in the formula

9 = 4w + (1 - w)21

9 = 4w + 21 - 21w

21w-9w=21-4

12w=17

w=12/17

w = 0.7058×100

w=70.58%

Zero coupon bonds = 70.58%

Therefore How much of the zero-coupon bond

you hold in your portfolio will be 70.58%

bii. Calculation for the perpetuity you will hold in your portfolio

Perpetuity= 1 - 0.7058

Perpetuity= 0.2942

Therefore the perpetuity you will hold in your portfolio will be 0.2942