1. id = if +
Eπ + ρ
where id and if are the domestic
and foreign interest rates, ρ is the combined country-specific
and foreign exchange risk premium and Eπ is the
expected rate of change in the domestic currency price of foreign currency.
We now turn to the implications of this relationship for the determination
of domestic real interest rates. In the Lesson entitled Interest
Rates and Asset Values it was noted that real interest rates
equal nominal rates minus the expected rate of inflation as follows:
2. r = i - Ep
where Ep is the rate of inflation expected during the
term of the loan, and r is the real interest rate on which people
base their decisions. The basis for this relationship is the fact that
inflation reduces the amount of real goods and services the principal and
interest on bonds and other assets that are fixed in nominal terms will buy
when they come due. The real interest rate that the contracting parties
expect to pay and receive is therefore the nominal rate minus an allowance
for the expected erosion of the real value of the principal and interest due
to inflation.
Using Equation 2 to characterize the real interest rates in both the domestic
economy and the rest of the world, we can express the domestic and foreign
real interest rates as
2a. rd = id - Epd
The domestic/foreign real interest rate differential can be obtained by
subtracting Equation 2b from Equation 2a to yield
3. rd - rf
= id - if - Epd + Epf
If we obtain the nominal interest rate differential by subtracting
if from both sides of Equation 1 and substitute the
resulting equation into Equation 3 we obtain
4. rd = rf
+ ρ - Epd + Eπ + Epf
Two conditions must hold for domestic and foreign real interest
rates to be equal. The combined risk premium, ρ , must be
zero and so must be the
term Epd - Eπ - Epf . Common sense
tells us that the risk premium must be zero for domestic and foreign interest
rates to be the same. The condition that
Epd - Eπ - Epf = 0
also has a straight-forward interpretation. The first term on the
left Epd is the expected rate of increase in the
level of prices of domestic output in domestic currency. The second
term Eπ is the expected rate of increase in the
price of foreign currency in terms of domestic currency and the third
term Epf is the expected rate of increase in the
level of prices of foreign output in foreign currency. The sum of these
latter two terms Eπ + Epf is the
expected rate of increase in the domestic currency price of foreign output.
Suppose, for example, that the price of foreign currency in units of
domestic currency goes up by 3% and the price of foreign output in units
of foreign currency goes up by 2%. Then it follows that the price of foreign
output in units of domestic currency will go up by approximately 5%. The
term
Epd - Eπ - Epf = 0
thus equals the expected rate of increase in the price level of domestic
output minus the expected rate of increase in the price level of foreign
output when both price levels are measured in domestic currency.
The ratio of the price level of domestic output to the domestic currency
price level of foreign output,
5. Pd / (Π Pf )
is called the real exchange rate. The real exchange rate is the
relative price of domestic output in terms of foreign output or,
in other words, the rate at which domestic output can be
exchanged for foreign output. The expression
Epd - Eπ - Epf
can thus be interpreted as the expected percentage change in the domestic
real exchange rate. This interpretation and others in the preceding screens
follow from two elementary mathematical relationships:
1) that the percentage change in a ratio equals the percentage change in the
numerator minus the percentage change in the denominator, and
2) that the percentage change in the product of two numbers equals the sum
of their percentage changes.
Equation 4 can thus be rewritten as
6. rd = rf
+ ρ - Eq
where Eq is the expected rate of change in the domestic real
exchange rate as defined above. The domestic real interest rate equals the
foreign real interest rate plus a risk premium minus the expected rate of
increase in the domestic real exchange rate. The role of the risk premium
in determining the domestic real interest rate is obvious. But why does the
domestic real interest rate fall in response to an expected increase in the
relative price of domestic output in terms of foreign output?
An increase in the relative price of domestic in terms of foreign goods
represents an increase in the value of the income flow from capital employed
in the domestic economy relative to the value of the income flow from capital
employed abroad. A positive value of Eq thus represents
an expected future capital gain on domestic relative to foreign capital which
makes it more profitable to hold domestic capital at equal risk-adjusted
interest rates. The prices of the sources of real income from capital
employed in the domestic economy will thus be bid up, and the interest rate
on that capital bid down, until net yield on capital, taking everything into
account, is the same in the domestic economy as abroad.
Equation 6 imposes a constraint on the domestic real interest rate similar to
the constraint on the domestic nominal interest rate imposed by Equation 1.
It says that, regardless of the mechanics by which government policy affects
the economy, the authorities can only bring about a change in the domestic
real interest rate by either inducing a change in the risk of holding
domestic assets or inducing a change in the expected rate of change of the
domestic real exchange rate. This, of course, assumes that the domestic
government cannot influence the foreign real interest rate. To affect the
real interest rate the government must change either the risk premium or the
expected rate of change in the country's real exchange rate---to affect the
nominal interest rate, it must change either the risk premium or the expected
rate of change in the country's nominal exchange rate.
Finally, from the relationships between the nominal and real interest rates
in the domestic and foreign economies, given by the two country's Fisher
Equations
7a. id = rd + Epd
which are simply reorganizations of equations 2a and 2b, it is clear that
the domestic/foreign nominal interest rate differential equals the
domestic/foreign real interest rate differential plus the difference between
the domestic and foreign expected inflation rates:
8. id - if =
rd - rf + Epd - Epf
A change in the domestic relative to the foreign expected inflation rate will
lead to an equal change in the domestic nominal interest rate relative to
the nominal interest rate abroad---assuming, of course, that real interest
rates are unaffected. If real interest rates are approximately the same
in different countries---that is, if risk differences are not too
great---nominal interest rates will differ across countries by the
differences in their expected inflation rates.
It's time for a test. Be sure to think up your own answers before looking at
the ones provided.
The last topic developed the following relationship between
the domestic and foreign nominal interest rates, the combined
foreign exchange and country-specific risk premiums, and the
expected rate of change (depreciation) of the domestic exchange
rate:
2b. rf = if - Epf
7b. if = rf + Epf