504 CHAPTER 13. MEASURING EXPOSURE TO EXCHANGE RATES
DoItYourself problem 13.1
Verify that if the forward rate had been different, the level of the hedged cash flow
would be affected but not the fact that the investment is hedged. For instance, with
a forward rate of 57 instead of 58 the hedged asset would have been 1.2m higher, at
96.6m. Show it.
Remember two things from the example. First, exposure is computed from a
comparison of alternative future outcomes, not from one single number found in a
balance sheet or a pro forma cash flow statement for next year. Second, the size and
(here) even the sign of exposure can be very different from what gut feeling would
suggest. Here, an accounting-tiedCFOwould have taken for granted that exposure
is positive: we talk about agbpasset, don’t we? Wrong; the position behaves like
a 1.2m liability.
DoItYourself problem 13.2
We just showed that exposure defined as a slope of the line linking the two points
does work: in this (overly simple) example, all risk is gone. Show that if you would
have followed your intuition and had hedged (sold forward)gbp 1.55m, orgbp
1.8m, or in fact any positive number, the uncertainty after such “hedging” would
have been higher than before.
A Problem with two Possible Exchange Rates and Noise
Let us generalize. The fact that the regression hedge always succeeds in taking away
all exchange-related risk can be proven in just two lines:
V ̃hedged
T = At,T+Bt,T
S ̃T+ ̃t,T
︸ ︷︷ ︸
value unhedged+ [−Bt,T]
︸ ︷︷ ︸
size of
hedge[
S ̃T−Ft,T]
︸ ︷︷ ︸
expiry value
pergbp,
= At,T+ ̃t,T
︸ ︷︷ ︸
uncorrelated
withS ̃+ Bt,TFt,T
︸ ︷︷ ︸
risk-free. (13.5)
Thus, what regression-based hedging generally achieves is eliminating all uncer-
tainty that is linearly related with the exchange rate:Bt,TFt,Thas taken the place
ofBt,TS ̃T. The uncertainty that isnotcorrelated with the exchange rate, in con-
trast, cannot be picked up by the forward contract, so it remains there. It can be
shown that this regression hedge ratio is also the one that reduces the variance of
the remaining risk to the lowest possible level. This is why this section is called
Minimum-Variance hedging and why ordinary regression is called Least Squares
(=minimal residual variance).
In the Android example there was assumed to be no residual risk, which is hard to
believe. Our earlier Freedonia example, in contrast, does have this feature: the state