Question

You are managing a portfolio of $1 million. Your target duration is 10 years, and you can invest in two bonds, a zero-coupon bond with maturity of five years and a perpetuity, each currently yielding 7.5%.

You are managing a portfolio of $1 million. Your target duration is 10 years, and you can invest in two bonds, a zero-coupon bond with maturity of five years and a perpetuity, each currently yielding 7.5%. a. What weight of each bond will you hold to immunize your portfolio? (Round your answers to 2 decimal places.) Answer is complete and correct. 46.41 Zero-coupon bond Perpetuity bond % % 53.59 b. How will these weights change next year if target duration is now nine years? (Round your answers to 2 decimal places.) Answer is complete but not entirely correct. 57.13 % Zero-coupon bond Perpetuity bond 42.87 %

Answer 1

a.

Portfolio duration is the weighted average of the duration of the individual bonds in the portfolio

Now we can immunizre the portfolio we need to invest in zcb and the perpeyuity bon din a way such that the weighted duration is equal to duration of portfolio of 10

Duration of Zero coupoon bond = maturity of ZCB = 5

Duration of Perpetual bond = ( 1 + yield ) / yeild = 1.075 / 0.075 = 14.33

let the weights of ZCB be w and perpetual bond be ( 1 - w)

w*5 + ( 1 - w)* 14.33 = 10

5w + 14.33 -14.33w = 10

9.33w = 4.33

w = 46.41

1 - w = 53.59

weight of ZCB = 46.41%

weight of perpetual bond = 53.59%

b.

after 1 year ,

duration of ZCB = 4

duration for perpetual bond = 14.33

now 4w + ( 1 - w)*14.33 = 9

4w + 14.33 -14.33w = 9

10.33w = 5.33

w = 51.60

1-w = 48.40

weight of ZCB = 51.60%

weight of Perpetual bond = 48.40%