You are managing a portfolio of $1,000,000. Your target duration is 10 years and

Question

You are managing a portfolio of $1,000,000. Your target duration is 10 years and you can choose from two bonds: a zero coupon bond with maturity of 5 years, and a perpetuity, each currently yielding 5%. How much of each bond will you hold in your portfolio? How will these fractions change next year if target duration is now 9 years? What is the value of perpetuity bond today, assuming it pays $1000 per year? What is the value of the zero coupon bond today? How many of the perpetual and zero bonds would you be purchasing today?

Explanation / Answer

a) duration of perpetual bond = (1 + yield)/yield = 1.05/.05 = 21

duration of zero coupon bond = maturity = 5 years

portfolio duration = weight of perptual bond*duration of perpetual bond + weight of zero coupon bond*duration of zero coupon bond

10 = Wp*21+(1-Wp)*5

Wp=0.3125

Perpetual bond holdings = 0.3125*1000000 = 312500

Zero coupon bond holding = 1000000-312500=687500

b) after 1 year duration of perpetual bond will be same and that of zero coupon bond will be 4 years

portfolio duration = weight of perptual bond*duration of perpetual bond + weight of zero coupon bond*duration of zero coupon bond

9 = Wp*21+(1-Wp)*4

Wp=0.29411

Wz=1-Wp=0.70588

c) value of perpetuity = payment/yield = 1000/0.05 = 20000

d) value of ZCB = parvalue/(1 + YTM)^n = 1000/(1+0.05)^5 = 783.526

e) number of perpetual bonds to be held = weight in portfolio/value of each perpetual bond = 312500/20000=15.625

number of zero coupon bonds to be held = weight in portfolio/value of each Zero coupon bond =

687500/ 783.526 = 877.4437

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