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 5.6%. a. 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.) Zero-coupon bond Perpetuity bond b. 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.) Zero-coupon bond Perpetuity bond

Explanation / Answer

a.

Duration of perpetuity = (1+y)/y

y = yield = 5.6 %

Duration of perpetuity = (1 + 0.056)/0.056

                                = 1.056/0.056 = 18.8571 or 18.86 years

Portfolio duration = w1D1 + w2D2 +…. + wkDk

w1, w2… wk are the weight of bonds 1,2,…k and D1, D2…Dk are the durations.

Let w be the weight of zero coupon bond and (w-1) be the weight of perpetuity.

Substituting values in above formula, we get w as:

w x 5 + (1 – w) x 18.8571 = 10

5w +18.8571 - 18.8571 w = 10

18.8571 - 10 = 18.8571 w – 5w

8.8571 = 13.8571 w

w = 8.8571/13.8571 = 0.639175 or 63.92 %

1 – w = 1 - 0.639175 = 0.360825 or 36.08 %

Weight of Zero coupon bond 63.92 %

Weight of perpetuity 36.08 %

b.

Net year the duration of the zero coupon bond will have 4 years.

But duration of perpetuity will remain same.

Again substituting the values in above portfolio duration formula we get, w as:

w x 4 + (1 – w) x 18.8571 = 9

4w +18.8571 - 18.8571 w = 9

18.8571 - 9 = 18.8571 w – 4w

9.8571 = 14.8571 w

w = 9.8571/14.8571 = 0.663462 or 66.35 %

1 – w = 1 - 0.663462 = 0.336538 or 33.65 %

Weight of Zero coupon bond 66.35 %

Weight of perpetuity 33.65 %

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