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

Question

You are managing a portfolio of $1 million. 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 5.2%.

What weight of each bond will you hold to immunize your portfolio? (Round your answers to 2 decimal places. Omit the "%" sign in your response.)

How will these weights change next year if target duration is now 9 years? (Round your answers to 2 decimal places. Omit the "%" sign in your response.)

You are managing a portfolio of $1 million. 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 5.2%.

Explanation / Answer

Part a)

We need to calculate the weight of each type of bond in the portfolio. We will take the weight of zero coupon bond as X, therefore, the weight of perpetuities would be 1-X. The formula for calculating duration of a perpetuity is:

Duration of a Perpetuity = (1+y)/y where y is the yield

We will use the formula for total duration of the portfolio to calculate the weight of each type. The formula would be:

Target Duration = Duration of Zero Coupon Bond*Weight of Zero Coupon Bond + Duration of Perpetuity*Weight of Zero Coupon Bond

______________

Solution:

Using the formula for duration of perpetuity, we get,

Duration of Perpetuity = (1+.052)/.052 = 20.23 years

Substituting duration of perpetuity and and other information in the equation for target duration we get,

10 = 5X + (1-X)*20.23

10 = 5X + 20.23 - 20.23X

X (Weight of Zero-Coupon Bond)= (20.23 - 10)/(15.23) = .6717 or 67.17%

Weight of Perpetuity = 1 - .6717 = .3283 or 32.83%

________________

Part b)

Using the equation for target duration, we get,

9 = 4X + (1-X)*20.23 [We will take 4 for Zero Coupon Bond because 1 year has passed]

9 = 4X + 20.23 - 20.23X

X (Weight of Zero-Coupon Bond) = (20.23 - 9)/(16.23) =.6919 or 69.19%

Weight of Perpetuity = 1 - .6919 = 30.81%

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