You are managing a portfolio of $1.9 million. Your target duration is 13 years,

Question

You are managing a portfolio of $1.9 million. Your target duration is 13 years, and you can choose from two bonds: a zero-coupon bond with maturity 6 years, and a perpetuity, each currently yielding 8%. a. How much of each bond will you hold in your portfolio? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Zero-coupon bond % Perpetuity bond % b. How will these fractions change next year if target duration is now thirteen years? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Zero-coupon bond % Perpetuity bond %

Explanation / Answer

Duration of a zero cpupon bond with 6 years maturity = 6 years

Duration of a perpetual bond = (1 + Yield)/Yield = 1.08/0.08 = 13.5 years

Let x be the weight of zero coupon bond and 1-x be the weight of perpetuity bond

6x + (1-x)13.5 = 13

6x -13.5x = -0.5

-7.5x = -0.5

x = 0.5/7.5 =6.67%

1- x = 93.33%

Zero Coupon Bond % = 6.67%

Perpetuity Bond % = 93.33%

Thus value of zero coupon bond = 6.67% * 1.9 = $0.13 million

Thus value of zero coupon bond = 93.33% * 1.9 = $1.77 million

Next year, the target duration becomes 13 years.

5x + (1-x)13.5 = 13

5x -13.5x = -0.5

-8.5x = -0.5

x = 0.5/8.5 = 5.88%

1- x = 94.12%

Zero Coupon Bond % = 5.88%

Perpetuity Bond % = 94.12%

Thus value of zero coupon bond = 5.88% * 1.9 = $0.11 million

Thus value of zero coupon bond = 94.12% * 1.9 = $1.79 million

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