180 CHAPTER 5. USING FORWARDS FOR INTERNATIONAL FINANCIAL MANAGEMENT
net transaction can be hedged. Of course, a firm typically has many contracts
denominated in a given foreign currency and these contracts may have different
maturity dates. In such a case, it is sometimes inefficient to hedge individually
the transactions for each particular date. In Section 17.3, we show how one can
define an aggregate measure of the firm’s exposure to foreign-currency-denominated
contracts that have different maturity dates, and how one can hedge this exposure
with a single transaction.
5.3.1 Measuring Exposure from Transactions on a Particular Date
By exposure we usually mean a number that tells us by what multiple thehcvalue
of an asset or cash flow changes when the exchange rate moves by ∆S, everything
else being the same. We denote this multiple byB∗t,T:
Bt,T∗ =∆V ̃T
∆S ̃T
. (5.6)
Note that the delta’s are for constantT—and remember thatT is a known future
date. That is, we are not relating a change inSover time to a change inVover time;
rather, we compare two possible situations or scenarios for a future timeTthat differ
as far asSis concerned. In continuous-math terms, we might have in mind a partial
derivative. In SciFi terms, we’re comparing two closely related parallel universes,
each having its ownST. Economists, more grandly, talk about comparative statics.
This is the general definition, and it may look rather other-wordly. To reassure
you, in the case of contractual exposureB∗t,Tis simply thefcvalue of the contract
at maturity:
Example 5.5
Assume that your firm (located in theus) has anA/Rnext month ofjpy1m. Then,
for a given change in theusd/jpyexchange rate, the impact on theusdvalue of
the cash flows from thisA/Ris 1m times larger. For example, if the future exchange
rate turns out to be fromusd/jpy0.0103 instead of the expected 0.0100, then the
usdvalue of theA/Rchanges fromusd10,000 to 10,300. Thus, the exposure of the
firm is
Bt,T∗ =10 , 300 − 10 , 000
0. 0103 − 0. 0100
= 1, 000 , 000. (5.7)
To the mathematically gifted, this must have been obvious all along: if the cash flow
amounts to a known number offcunitsC∗, then itshcvalue equalsVT=C∗×ST,
implying that the derivative ∂VT/∂ST or the relative difference ∆VT/∆ST both
equalC∗, thefccash flow. A point to remember, though, is that while exposure
might be a number described in a contract or found in an accounting system, it
generally is not. We’ll get back to this when we talk about option pricing and
hedging, or operations exposure, or hedging with futures.