144 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY
−Ft 0 ,T. We shall use this result repeatedly in Chapter 5 (on uses of forward con-
tracts), in Chapter 9 when we discuss option pricing, and in Chapters 13 where we
analyse exposure and risk management.
Make sure you realize that the hedged liability may make you worse off,ex post,
than the unhedged one. Buying at a pre-set rateFt,Tgives that great warm feeling,
ex post, if the spot rateSTturns out to be quite high; but it hurts if the spot rate
turns out to be quite cheap. The same conclusion was already implicit in [4.17]: the
value of the contract at expiry can be either sign. This rises the question whether
hedging is really so good as it is sometimes cracked up to be. We return to the
economics of hedging in Chapter 12.
4.4.3 Corollary 2: The Value of a Forward Contract at Inception
The value at expiry, above, probably was so obvious that it is, in a way, just a
means of proving that the general valuation formula [4.15] makes sense. The same
holds for the next special case: the value at inception,i.e. the time the contract
is initiated or signed. At inception, the market valuemustbe zero. We know this
because (a) when we sign a forward contract, we have to pay nothing; and (b) hard-
nosed bankers would never give away a positive-value contract for free, nor accept
a negative-value contract at a zero price. To show the (initial) zero-value property
formally, we use the general value formula [4.16] and consider the special case where
t 0 =t, implying thatFt 0 ,T =Ft,T (That is, the contract we are valuing is new.)
Obviously,
[Initial value of a
forward contract
with rateFt,T
]
=
Ft,T−Ft,T
1 +rt,T= 0. (4.19)
The value of a forward contract is zero at the moment it is signed because the
contract can be replicated at zero cost. Notably, if a bank tried to charge you
money for a contract at the equilibrium (Covered Interest Parity) forward rate, you
would refuse, and create a synthetic forward contract through the spot and money
markets:
Example 4.14
LetSt = 100, r∗t,T = 0. 10 , rt,T = 0. 21 , Ft,T = 110; but a bank wants to charge
you a commission of 3 for a forward purchase. You would shrug dismissively and
immediately construct a synthetic forward contract at 110 at a zero cost:
- write apnadhc110, discount it;
- convert the proceeds, 110/1.21 = 90.909090, intofc: you getfc0.90909.
- invest at 10 percent, to gethc1 atT.
Thus, you can replicate a forward purchase contract under which your payment at
T amounts to 110, just like in the genuine, direct forward contract, but it does not
cost you anything now.