International Finance: Putting Theory Into Practice

142 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY

gotten away with 110 if we had signed right now. This “loss”, however, is dated 4
years from now, so itsPVis discounted at the risk-free rate.

The sceptical reader may object that this “loss” is very fleeting: its value changes
every second; how comes, then, that we can discount at the risk-free rate? One
answer is that the value changes continuously because interest rates and (especially)
the spot rate are in constant motion, But that does not invalidate the claim that
we can always value eachpnusing the risk-free rates and the spot exchange rate
prevailing at that moment. Relatedly, the future loss relative to market conditions
attcan effectively be locked in at no cost, by selling forward for the same date:

Example 4.12
Consider a contract that has four years to go, signed in the past at a historic forward
price of 115, for speculative purposes. Right now you see there is a loss, and you
want to close out to avoid any further red ink. One way is to sell forwardhc1 at
the current forward rate, 110. On the common expiry date of old and new contract
we then just net the loss of 115–110:

hcflows atT fcflows atT

old contract: buy atFt 0 ,T=115 –115 1

new contract: sell atFt 0 ,T=110 110 –1

net flow –5 0

But because this loss is realized within four years only, itsPVis found by discounting.
Discounting can be at the risk-free rate since, as we see, the locked-in loss is risk
free.

We can now use the result in Equation [4.15] to determine the value of a forward
contract in two special cases: at its inception and at maturity.

4.4.2 Corollary 1: The Value of a Forward Contract at Expiration

At its expiration time, the market value of a purchase contract equals the difference
between the spot rate that prevails at timeT—the value of what you get—and the
forward rateFt 0 ,Tthat you agreed to pay:
[Expiration value of
a forward contract
with rateFt 0 ,T

]

=ST−Ft 0 ,T. (4.17)

Equation [4.17] can be derived formally from Equation [4.15], using the fact that
the effective return on a deposit or loan with zero time to maturity is zero (that
is,rT,T = 0 =r∗T,T. The result in [4.17] is quite obvious, as the following example
shows:

Example 4.13