Measuring the risk of Bond futures

Most bond futures contracts are based on a hypothetical benchmark bond. The Chicago Board of Trade’s U.S. Treasury bond futures contract is based on a 6 percent bond with at least 15 years from the futures expiration to maturity or the first call date. Even though the benchmark bond has a 6 percent coupon, any bond meeting the maturity requirement can be delivered. At any time, a single bond exists that the holder of the short position would find optimal to deliver if current conditions continued. That bond is called the cheapest to deliver and can be thought of as the bond on which the futures contract is based. In other words, the cheapest to deliver bond is the underlying. The responsiveness of the futures contract to an interest rate change is equivalent to the responsiveness of that bond on the futures expiration day to an interest rate change.
We can think of this concept as the responsiveness of the underlying bond in a forward context. This responsiveness can be measured as that bond’s modified duration on the futures expiration and, as such, we can use the price sensitivity formula to capture the sensitivity of the futures contract to a yield change. Accordingly, we shall, somewhat loosely, refer to this as the implied duration of the futures contract, keeping in mind that what we mean is the duration of the underlying bond calculated as of the futures expiration. Moreover, we also mean that the underlying bond has been identified as the cheapest bond to deliver and that if another bond takes its place, the duration of that bond must be used. We use the term implied to emphasize that a futures contract does not itself have a duration but that its duration is implied by the underlying bond. In addition to the duration, we also require an implied yield on the futures, which reflects the yield on the underlying bond implied by pricing it as though it were delivered at the futures contract expiration.
Hence, we can express the sensitivity of the futures price to a yield change as
where MDURf is the implied modified duration of the futures, f is the futures price, and Ayf is the basis point change in the implied yield on the futures.
Now that we have a measure of the responsiveness of a bond portfolio and the responsiveness of a bond futures contract to interest rate changes, we should be able to find a way to balance the two to offset the risk.