Question

You are managing a portfolio of $1.0 million. Your target duration is 21 years, and you can choose from two bonds: a zero-coupon bond with maturity five years, and a perpetuity, each currently yielding 2%.

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 twenty years?

Answer 1

Solution :-

Total Portfolio Amount = $1 million

Target Duration = 21 years

ZCB ( Zero Coupon Bond ) Maturity = 5 yrs

Yield of ZCB = 2%

Yield of Perpetuity = 2%

As we know Duration of ZCB is always equal to yield to maturity So Duration of ZCB = 5 yrs

and Duration of perpetuity Bond = (1 + y) / y where y is yield

= (1 + 0.02 ) / 0.02 = 51 years

let Share of ZCB be x and therefore perpetuity = (1 - x )

now x * 5 + ( 1 - x )*51 = 21

= 5x + 51 - 51x = 21

= 30 = 46x

x = 30 / 46 = 15 / 23

Bond = $1000000 * 15 / 23 = $652173.9 = 65.21%

now Therefore Perpetual bond = $1000000 - $652173.9 = $347826.1 = 34.78%

(b)

Now after 1 year

Duration of ZCB = 4 years as Time to maturity is 4 years now

But duration of Perpetual bond is same = 51 years ( As no change in its yield )

Now target Duration = 20 years

Now

x * 4 + ( 1 - x )*51 = 20

= 4x + 51 - 51x = 20

= 31 = 47x

x = 31 / 47

therefore share of ZCB in $1000000 = $1000000 * 31 / 47 = $659574.47 = 65.96%

now share of Perpetual Bond = $1000000 - $ 659574.47 = $340425.53 = 34.04%