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We now make the simple assumption that a single interest rate exists that drives all interest rates in the market. We assume that a 1 basis point change in this interest rate will cause a 1 basis point change in the yield on the bond portfolio and a 1 basis point change in the implied yield on the futures. We will relax that assumption later. For now, consider a money manager who holds a bond portfolio of a particular market value and will not be adding to it or removing some of it to balance the risk. In other words, the manager will not make any transactions in the actual bonds themselves. The manager can, however, trade any number of futures contracts to adjust the risk. Let Nf be the number of futures contracts traded. To balance the risk, suppose we combine the change in the value of the bond portfolio and the change in the value of Nf futures and set these equal to zero: AB + NfAf = 0.Solving for Nf produces Nf = -ABlAf. Substituting our formulas for AB and Af, we obtain
where we assume that AyB/Ayf= 1; or in other words, the bond portfolio yield changes one-for-one with the implied yield on the futures.’
Now let us go back to the major simplifying assumption we made. We assumed that an interest rate change occurs in the market and drives the yield on the bond and the implied yield on the futures one-for-one. In reality, this assumption is unlikely to hold true. Suppose, for example, the rate driving all rates in the United States is the overnight Fed funds rate.9 If this rate changes by 1 basis point, not all rates along the term structure are likely to change by 1 basis point. What actually matters, however, is not that all rates change by the same amount but that the yield on the bond portfolio and the implied yield on the futures change by the same amount for a 1 basis point change in this rate. If that is not the case, we need to make an adjustment.
Suppose the yield on the bond portfolio changes by a multiple of the implied yield on the futures in the following manner:
We refer to the symbol Py as the yield beta. It can be more or less than 1, depending on whether the bond yield is more sensitive or less sensitive than the implied futures yield. If we take the formula we previously obtained for AB, substitute PyA yf where we previously hadAyB,and use this new variation of the formula in the formula Nf=-AB/Af ,we obtain ^&3$
This is the more general formula, because Py = 1.0 is just the special case we assumed at the start.