Question

Suppose that you are responsible for managing a $100 million bond portfolio at a large pension fund. Because the pension fund has obligations to retirees in 2029, you want your portfolio to have a duration of 10 years. Suppose that you can only use 2-year and 12-year Zeros to immunize your portfolio. Assume that new 2-year Zeros and 12-year Zeros are issued only in 2019, 2021, etc. (i.e., odds years). Assume that interest rates are the same for all these securities.

a. In 2019, how much of your $100 million do you need to invest in each of the two bonds to immunize your portfolio from interest rate risk?
b. Suppose that one year has passed (i.e., it’s 2020) and your portfolio now has a target duration of 9 years. What are the portfolio weights on the 2-year and 12-year Zeros (issued in 2019) required to immunize your portfolio? (Hint: consider what happens to the maturity of the Zeros in your portfolio as time passes.) If the yield curve has been flat and constant, do you need to rebalance your portfolio in 2020?

Answer 1

Part a:

We are standing in 2019 and our liabilities will mature in 2029 i.e. 10 years form now the target duration of the asset portfolio to immunise the liabilities is given 10 years i.e. duration of assets D(a) =10

We have two assets in our portfolio

2 year Zero coupon bond(2 years ZCB) and 12 years zero coupon bond(12 year ZCB)

Now according to the properties of duration ,the duration of a ZCB is=its maturity term(n)

So duration of 2 years ZCB (D1)=2

AND duration of 12 years ZCB (D2)=12

let the proportions be

W1 which is the proportion of 2 year ZCB

W2 which is the proportion of 12 year ZCB

And W2=1-W1

Hence

Duration of asset portfolio =weighted average duration of all assets in the portfolio

I.e. D(a)=W1*D1+W2*D2

10=W1*2+W2*12

10=W1*2+(1-W1)*12

10=W1*2+12-W1*12

10-12=W1(2-12)

W1=2/10=.20

W2=1-.2=.80

These are the required proportions

Out of $100 million we should invest $100*.2 =$20 million in 2 year ZCB and rest $80 million in 12 year ZCB

Part b :

Now we have

D(a)=9

Since 1 year has paased new duration of assets will be

D1=2-1=1

D2=12-1=11

Now the new proportions are given by

9=W1*1+W2*11

9=W1*1+(1-W1)*11

9-11=W1(1-11)

W1=2/10=.20

W2=.80

NOTE:is this a coincidence or these had to be remain unchanged? well it depends upon the type of assets an immunised portfolio has to rebalanced with passage of time and change of interest rates if assets are

Coupon bearing bond

Perpetual bond or

Any fixed income security other than ZCB because in case of ZCB passing of time doesn't impact the portfolio hance it doesn't need to be rebalanced that is why in our case where both the assets were ZCBs the proportions didn't change at all

Now as we discussed we don't need to rebalance our portfolio because of passage of time but we need to do this if the interest rate changes since in the qustion we are given that yield curve is flat i.e. interest rate are expected to be constant both in short as well as in long term hence we don't need to rebalance it in this case as well.