Question

1. You are managing a portfolio of $5 million. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with a maturity of 5 years, and perpetuity, each currently yielding 8.00%.

1. What weight of each bond will you hold to immunize your portfolio? (10 points)
2. How will these weights change next year if the target duration is now 9 years? (15 points)
3. If you do not rebalance your portfolio of immunizing assets over the second year of the investment period. What will be the difference in duration between the portfolio and the obligation it seeks to immunize? (10 points)
4. How far apart approximately will the value of the obligation and the value of the portfolio be if, under the conditions of part c., interest rates increase by 50 basis points? (10 points)

(Please show how to do it and excel if possible) thank you.

Answer 1

First of all we will calculate the duration of perpetual bond using the formulae 1+y / y

1.08 / 0.08 = 13.5

Duration of ZCB is its maturity ie 5 years

Calculation of weight

Let the weight of ZCB be X, then weight of perpetail bond is 1-X

(X * 5) + (1-X)*13.5 = 10

X = 41.17 (weight of ZCB)

(1-X) = 58.83 (weight of Perpetual Bond)

b) after 1 year duration of ZCB = 4, duration of perpetaul bond = 13.5

Calculation of wiight for target duration of 9

(X*4) + (1-X)*13.5 = 9

X = 47.36 weight of ZCB

52.63 = weight of perpetual bond

c) Calculation of duration of portfolio using orignal weights after 1 year

(0.4117 * 4) + (0.5883 * 13.5) = 9.58

Since target duration after 1 year is 9 difference in duration is 0.58