I’m trying to better understand the concepts behind this question:

27. A portfolio management firm manages the fixed-rate corporate bond portfolio owned by a
defined-benefit pension fund. The duration of the bond portfolio is 5 years; the duration of the
pension fund’s liabilities is 7 years. Assume that the fund sponsor strongly believes that rates will
decline over the next six months and is concerned about the duration mismatch between portfolio
assets and pension liabilities. Which of the following strategies would be the best way to
eliminate the duration mismatch?

a. Enter into a swap transaction in which the firm pays fixed and receives floating.
b. Enter into a swap transaction in which the firm receives fixed and pays floating.
c. Purchase an interest rate cap expiring in six months.
d. Sell Eurodollar futures contracts.

My thinking process:

As yields go down, the price of the portfolio and liabilities go down. However, the price of the liabilities will go up faster since the liabilities have a higher sensitivity to interest rate changes. So we want to increase the portfolio duration in order to match with the liability duration. How is this fulfilled with answer B? Why wouldn’t answer A also increase duration?

Also...the test mentions that answer D would not resolve the problem because selling Eurodollar futures contracts would reduce portfolio duration. But isn’t this the case for any security that has positive duration?

ANSWER: B
‘A’ is incorrect. Entering into a pay fixed swap will further reduce the duration of the assets. This will exacerbate the interest rate risk arising from the duration mismatch.

‘B’ is correct. Because the duration of the pension liabilities exceeds the duration of the bond portfolio (assets), the pension plan is at risk if interest rates fall. Specifically, in a falling rate environment, the value of the liabilities will increase by more than will the value of the assets, thereby eroding the pension surplus.
The duration of a swap equals the difference between the duration of the fixed leg and the duration of the floating leg. By entering into a receive-fixed swap, the pension plan can increase the portfolio duration to equal that of the pension liabilities and consequently reduce interest rate risk.

‘C’ is incorrect. The duration of the option is only six months and would therefore not offset the duration mismatch.

‘D’ is incorrect. Selling futures will reduce the portfolio duration and augment the duration mismatch.

“the price of the liabilities will go up faster since the liabilities have a higher sensitivity to interest rate changes” - that’s correct. Rates go down, both assets and liabilities go up, but liabilities go up faster year.

Recall we can value the swap as two bonds. Receive fixed = long fixed coupon bond and short a floating coupon bond. Duration of floater will be near zero, so receive fixed is pretty close to going long (buying) a bond with fixed coupons. As long as it’s duration > 5, it’s positive duration will add to the portfolio. Or, put another way, lower rates will increase the value of the swap to us as we pay lower but receive fixed.

if we instead went with (a), if we payed-fixed in the swap, we actually have a negative dollar duration for that position; lower rates benefit our counterparty (who we sold the bond to, and who benefits from the price increase associated with rate decline), but they hurt us.

Re “selling Eurodollar futures contracts would reduce portfolio duration. But isn’t this the case for any secrity that has positive duration?” Well, yes, and that’s exactly why (a) is wrong: to pay-fixed and receive floating (the leg with negligible duration) is like shorting a (+ duration) bond.

David