6.4. HEDGING WITH FUTURES CONTRACTS 239
as your exposure. The convergence property means thatf ̃T(e 1 ),T 2 =ST(e 1 ): on the last
day of trading asekfutures price exactly equals the spot rate at the same moment
because both stipulate delivery att+2. Thus, in this special case of a perfect match,
Equation [6.3] tells us, we should regress the variable upon itself. There of course
is no need to actually do so: in that regression, the slope coefficient (and theR^2 )
can only be unity. So you sell forward one for one: if the exposure isBunits of
forex, you sellBunits. In short, this is standard hedging where nothing needs to
be estimated.
But usually one is not that lucky:6.4.3 Case 2: The Currency-Mismatch Hedge or Cross-Hedge
We now consider a case where the futures contract matches the maturity of the
foreign-currency inflow but not the currency (h 6 = e). For instance, theusexporter’s
sekinflow is hedged using aeurfuture. We can use the convergence property
fT 1 ,T 1 =ST 1 to specify the hedge ratio as
β =cov(S ̃(Te 1 ),S ̃(Th 1 ))
var(S ̃(Th 1 )), (6.4)
= the slope coefficient inS ̃T(e 1 )=α+βS ̃T(h 1 )+ ̃. (6.5)This measure of linear exposure will come up again and again in this book, most
prominently in Chapter 9 on option pricing and hedging, or in Chapter 13 when we
quantify operating exposure, so reading on is useful even if we get technical, initially.
Actually, another reason for getting technical is that it helps us understand the pros
and cons of the relative-value hedging rule that we introduced before, like hedging
everysekby 0.10eur, the current cross-rate.
Recall that in the definition of exposure we hold the time constant, and we instead
compare possible future scenarios. Similarly, our regression is, in principle, forward
looking: it should be run across a representative number of (probability-weighted)
possible future scenarios. This is not easy, so you may want to run the regression
on past data instead. One assumption then is thatβis constant, so that the past
is a good guide to the future. For technical and statistical reasons that are beyond
the scope of this chapter, one should not regress levels of exchange rates on levels if
the data are time series. A regression between changes of the variables, in contrast,
would be statistically more acceptable:
regress ∆S(te)=α′+β∆St(h)+ ̃′t, (6.6)where, this time, deltas refer to changes over time. Many careful researchers would
still be unhappy with this, and actually prefer to work with a regression in percentage
changes: in a long time series with much variation in the level ofS, it is hard to