B481/S98/NOTES/Grabbe-5 FORWARDS, SWAPS, AND INTEREST ...

COVERED INTEREST ARBITRAGE
The derivation here is slightly “reversed” from that given in the text.
Note that here we begin holding a foreign currency; in the text we begin by
borrowing US$.
Suppose we start with X units of the currency of a country C. We can
convert the C-currency into US$ at today’s spot rate S(t). This yields
S(t)X US$. These US$ can be deposited for a period of T days in a
eurocurrency account paying i. When mature, this deposit is worth
(1+i(T/360))S(t)X US$. Finally, we can convert this amount, without risk,
in currency-C at the forward rate for time t+T. Thus, leaving S(t) and
F(t+T) stated in the same terms, we have as the time t+T value, in
currency-C, as (1+i(T/360))S(t)X(1/F(t+T)). Alternatively, we could have
simply deposited the X units of C-currency at the interest rate i*. This
would yield at time t+T a value, in C-currency, of (1+i*(T/360))X. There
are no arbitrage profits if and only if these two values are equal, i.e., if
and only if
(1+i(T/360))S(t)X(1/F(t+T)) = (1+i*(T/360))X.
Eliminating X and rearranging terms, we have the famous (covered)
interest parity condition:
F ( t + T ) = S( t ) [ 1 + i ( T
)]
360
[ 1 + i ∗ ( T
)] .
360
Now note that if i > ( ( S(t), then it costs more US$
in the future to buy one unit of C-currency. Thus, if i > i*, so that F(t+T) >
S(t), then the US$ is at a discount and the C-currency is at a premium.
Similarly, if i < i*, so that F(t+T) < S(t), then the US$ is at a premium and
the C-currency is at a discount. Finally, if i* = i, so that F(t+T) = S(t),
then the US$ and the C-currency are at parity.
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