Question

You are managing a portfolio of $1.0 million. Your target duration is 15 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?

Zero-coupon bond ___ %

Perpetuity bond ____ %

How will these fractions change next year if target duration is now fourteen years?

Zero-coupon bond ___ %

Perpetuity bond ____ %

Answer 1

Solution :-

Total Portfolio Amount = $1.0 million

Target Duration = 15 Years

Bond I

Maturity of Zero coupon Bond = 5 Years

Yield of Zero Coupon Bond = 5%

Bond II

Yield of Perpetuity = 5%

Duration of Zero Coupon Bond = 5 Years

Duration of Perpetual Bond = (1 + Y) / Y = ( 1 + 0.05 ) / 0.05 = 21 Years

Target Duration = 15 Years

Proportion of Zero Coupon Bond = X

Duration of Zero Coupon Bond = D1 = 5 years

Proportion of Perpetual Bond = ( 1 - X )

Duration of Perpetual Bond = D2 = 21 years

Now Target Duration = X * 5 + ( 1 - X ) * 21 = 15

5X + 21 - 21X = 15

6 = 16X

X = 6 / 16

Therefore Share of Zero Coupon Bond = 6 /16 = 37.50%

= $1,000,000 * 6 / 16 = $375,000

Share of Perpetual Bond = 10 / 16 = 62.50%

= $1,000,000 * 10 / 16 = $625,000

(B)

Target Duration = 14 Years

Bond I

Maturity of Zero coupon Bond = 5 Years

Yield of Zero Coupon Bond = 5%

Bond II

Yield of Perpetuity = 5%

Duration of Zero Coupon Bond = 5 Years

Duration of Perpetual Bond = (1 + Y) / Y = ( 1 + 0.05 ) / 0.05 = 21 Years

Target Duration = 14 Years

Proportion of Zero Coupon Bond = X

Duration of Zero Coupon Bond = D1 = 5 years

Proportion of Perpetual Bond = ( 1 - X )

Duration of Perpetual Bond = D2 = 21 years

Now Target Duration = X * 5 + ( 1 - X ) * 21 = 14

5X + 21 - 21X = 14

7 = 16X

X = 7 / 16

Therefore Share of Zero Coupon Bond = 7 /16 = 43.75%

= $1,000,000 * 7 / 16 = $437,500

Share of Perpetual Bond = 9 / 16 = 56.25%

= $1,000,000 * 9 / 16 = $562,500

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