Question

a. You wish to create a bond portfolio with a duration of 6 years. At your disposal is a 2 year zero-coupon bond and a perpetual bond, both with a YTM of 5%. How much of each bond should you buy? b. What will be the new weights in part A after rebalancing one year later?

Answer 1

Modified Duration of perpetual bond=1/ytm+ytm=1/5%+1=21

Duration of zero coupon bond=maturity=2 years

1.

Let w be the proportion invested in perpetual bond and 1-w in zero coupon bond

Hence,

w*21+(1-w)*2=6

=>w=0.210526

So 21% in perpetual bond and 79% in zero coupon bond

2.

Case 1: After 1 year, the duration to be matched will become 5 years

After 1 year, duration of zero coupon bond=1 year

So, w*21+(1-w)*1=5

=>w=0.2

So 20% in perpetual bond and 80% in zero coupon bond

Case 2: After 1 year, the duration to be matched will remain 6 years

After 1 year, duration of zero coupon bond=1 year

So, w*21+(1-w)*1=6

=>w=0.25

So 25% in perpetual bond and 75% in zero coupon bond