Question

You are managing a portfolio of $1.0 million. Your target duration is 29 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 eight years? zero-coupon bond? perpetuity bond? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

Answer:

Duration is a measure of the risk of a bond and it indicates the change in the price of a bond as a result of a change in the yield of that bond. Duration of a portfolio of bonds is the market value weighted average of the individual durations of the bond; i.e.

   Dp = Wm * Dm + Wn *Dn

Here; Dp- Duration of Portfolio

             Dm and Dn are duration of individual bond securities named m and n

             Wm and Wn are their respective weight in portfolio

Step-1: Calculate the Duration of Perpetuity Bond:

    Duration of Perpetuity Bond = (1 + Yield)/Yield

                                                             = (1 + 0.02) / 0.02 = 51 Years.

Step-2: Calculation of Zero Coupon Bond’s Duration:

    Maturity Period of zero coupon bond = 5 years.

    As, in the case of a zero-coupon bond, the bond's remaining time to its maturity date is equal to its duration, so duration = 5 years.

Part-A:

Target Duration of Portfolio = 29 Years

Let x be weight of zero coupon bond in portfolio, and

(1 – x) be weight of Perpetuity Bond

Then; (x* 5) + (1 – x)(51) = 29

             5x + 51 – 51x = 29

            46x = 22

            x = 0.478

Weight of Zero coupon bond = 0.4782 or 47.82%

Then Weight of Perpetuity Bond = (1 – 0.4782) = 0.5218 or 52.18%

Part-B:

After 1 years, duration of perpetuity will remain same (51 years) but duration of zero coupon bond becomes 4 years;

Target Duration = 28 years

Let x be weight of zero coupon bond in portfolio, and

(1 – x) be weight of Perpetuity Bond

Then; (x * 4) + (1 – x)(51) = 28

                4x + 51 – 51x = 28

                X = 0.4893

Weight of Zero coupon bond = 0.4893 or 48.93%

Then Weight of Perpetuity Bond = (1 – 0.4893) = 0.5107 or 51.07%