138 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY
spot forward swap rate
level 0 100.0 100.0×^11 ..^003333002500 = 100.0831 0.0831
level 1 100.5 100.5×^11 ..^003333002500 = 100.5835 0.0835
change 0.5 0.5004 0.0004The rule of thumb of not updating the swap rate all the time used to work reason-
ably well because, in olden days, interest rates were low^8 and rather similar across
currencies (the gold standard, remember?), and maturities short. This makes the
fraction on the right hand side of [4.12] a very small number. In addition, interest
rates used to vary far less often than spot exchange rates. Nowadays, of course,
computers make it very easy to adjust all rates simultaneously without creating
arbitrage opportunities, so we no longer need the trick with the swap rates. But
while the motivation for using swap rates is gone, the habit has stuck.
DoItYourself problem 4.4
Use the numbers of Example 4.9 to numerically evaluate the partial derivative in
Equation [4.13],
∂(Ft,T−St)
∂St=
[
rt,T−r∗t,T
1 +rt,T∗]
≈rt,T−rt,T∗.Check whether this is a small number, when interest rates are low (and rather similar
across currencies) and maturities short. (If so, it means that the swap rate hardly
changes when the spot rate moves.) Also check that the analytical result matches
the calculations in the Example.
We now bring up an issue we have been utterly silent about thus far: taxes.CIP: Capital GainsvInterest Income, and Taxes
When comparing the direct and synthetichcdeposits, in Example 4.7, we ignored
taxes. This, we now show, is fine as long as the tax law does not discriminate
between interest income and capital gains.
(^8) During the Napoleonic Wars, for instance, theukissued perpetual (!) debt (the consolidated
war debt, orconsol) with an interest rate of 3.25 percent. Toward the end of the nineteenth century,
Belgium issued perpetual debt with a 2.75 percent coupon (to pay off a Dutch toll on ships plying
for Antwerp). Rates crept up in the inflationary 70s to, in some countries, 20 percent short-term
or 15 percent long-term around 1982. They then fell slowly to quite low levels, as a result of falling
inflation, lower government deficits and, in the first years of the 21st century, high uncertainty and
a recession—the “flight for safety” effect.