SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins
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FOREIGN EXCHANGE MARKETS
Exchange Rate Determination Under Purchasing Power Parity
The purchasing power parity exchange rate (PPP) between a foreign currency
and the dollar is defined as:
PPP =
(Cost of a Market Basket of Goods and Services at Foreign Prices)
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(Cost of the Same Market Basket of Goods and Services at U.S. Prices)
This gives the exchange rate in the indirect form, the units of foreign
currency per dollar. The direct form of the exchange rate, the dollars
per unit of foreign currency is just the reciprocal of the indirect form.
If the exchange rate is at PPP at time 0 and the foreign country experiences
a rate of inflation of (1+F) while the U.S.
experiences a rate of inflation of $ then the
cost of the market basket in foreign currency will increase in one year by a factor
of (1+F) while the cost of the market basket in
dollars will increase by a factor of (1+$).
Thus the exchange rate at time 1 year, E1 will equal to:
E1 = E0[(1+F/
(1+$)]
This means that the proportional change in the exchange rate is given by:
E1/E0 = 1 + g
= [(1+F)/(1+$)]
or
g = (F - $)/(1+$)
When $ is small then g is close to
(F - $).
If we consider the direct form of the exchange rate, dollars per
unit of foreign currency the growth rate g' would be:
g' = ($ - F)/(1+F)
Note that g' is approximately -g, but not quite. To see why this is so
consider the case in which the exchange rate for the French franc changes
from 4 ff/$ to 5. This is a 25 percent increase in the exchange rate
in the indirect form. But the direct forms of the exchange rates are
$0.25/ff and $0.20/ff. In this form the exchange rate has decreased by
0.05/0.25 = .2 or 20 percent. The results mean the same thing but the
numerical values depend upon which form is being considered.
Strictly speaking, the projection of the future exchange rate considered
above should be based upon the current PPP exchange rate. This would give
the expected future PPP exchange rate that would be a reasonable estimate
of the market exchange rate will be. Let E stand for the PPP exchange
as opposed to E for the market exchange rate. Then the theory is given
by the followiing equations:
Expected Value of ET = ET
ET = E0[(1+F)/(1+$)]T
where E0 is the PPP exchange rate at time 0, which may be
different from the market exchange rate at time 0.
Frederic Mishkin's Theory of the Determination of Exchange Rates
Mishkin's Interest Rate Parity Condition
Consider an American investor choosing between investing funds in the
U.S. market or the French market. Financial equilibrium requires that
the expected rate of return in dollar terms be the same for the two
markets. Let i$ be the nominal interest rate in the U.S. and
iFF in France. Let Et be the spot exchange rate of
francs per dollar at time t.
The expected rate of return for a dollar investment, RET$,
is just i$. The expected rate of return in dollar terms for
an investment in France, RETFF, is given by computing the
number of francs a dollar amount will be equivalent to, determineing
the number of francs that will be received after one year, and finding
the number of dollars those future francs can be expected to be
converted to on the basis of the expected exchange rate at
t+1, Eet+1 ; i.e.,
1+RETFF = Et(1+iFF)/
Eet+1
(Et/Eet+1)
+ iFF(Et/Eet+1)
= 1/(Eet+1/Et)
+ iFF/(Eet+1/Et)
This means that:
RETFF =
iFF/(Eet+1/Et)
- (Eet+1-Et)/Eet+1
Mishkin makes the assumption that Eet+1 is approximately
equal to Et and thus their ratio is approximately equal to one
so he asserts that:
RETFF = iFF
- (Eet+1-Et)/Et
and hence for equilibrium
i$ = iFF
- (Eet+1-Et)/Et
Let consider one numerical example in which Mishkin's assumptions
are reasonable and one in which they are not reasonable.
Suppose the exchange rate at t is 5 FF/$ and it is expected to
be 5.2 FF/$ at t+1. Furthermore, let the French interest rate be
10 percent. Then according to Mishkin's formula the equilibrium
interest rate in the U.S. must be:
i$ = 0.10 - (5.2 - 5.0)/5.0 = 0.10 - 0.04 = 0.06
The exact value is:
i$ = 0.10/(5.2/5.0) - (5.2 - 5.0)/5.2 = 0.0962 - 0.0385 = 0.0577
The error is about a quarter of a percentage point.
Now suppose next years exchange rate is expected to be 20. This
represents a devaluation of the franc by 75 percent. Suppose the
interest rate in France is 320 percent. According to Mishkin's
formula the equilibrium interest rate in the U.S. would have
to be:
i$ = 3.2 - (20 - 5)/5 = 3.2 - 3.0 = 0.2
This is an interest rate of 20 percent. The correct value
is:
i$ = 3.2/(20/5) - (20 - 5)/20
= 0.80 - 0.75
= 0.05
The error is 15 percentage points. Thus if the interest rate
in the U.S. were 5 percent the two financical markets would be
in equilibrium, but Mishkin's formula would indicate that the
interest rate in the U.S. is too low and investors should transfer
their funds to France. This would lead to an increasing value of
the franc compared to what it was expected to be in the future.
The Analysis in Terms of the Direct Quote Form of the
Exchange Rate
Let et the exchange rate in terms of units of domestic
currency per unit of foreign currency. In the previous example
this would be dollars per French franc. Then the rate of return
on an investment in the French market is:
1+RETFF =
(1/et)(1+iFF)eet+1
= eet+1/et)
+ iFF(eet+1/et)
Thus
RETFF=
(eet+1/et -1)
+ iFF(eet+1/et)
= (eet+1/et -et)
+ iFF(eet+1/et)
The first term on the right is the rate of change of the value
of the foreign currency with respect to the domestic currency.
Let g be this rate of change. Then the rate of return, RET, of
an investment in a foreign market is:
RETFF = g + iFF(1+g).
Note that in the case of a depreciating foreign currency the
value of g is negative so 1+g is less than one. Let us now consider
some numerical examples. Suppose the U.S. nominal interest rate is
7 percent and the Canadian nominal interest rate is 12 percent. If the
Canadian dollar is expected to depreciate 5 percent a year with respect to
the American dollar; i.e., g = -0.05, then the expected nominal rate of
return for Americans from a Canadian investment is
(0.12)(1-0.05) + (-0.05) = (0.12)(0.95)-0.05 = 0.064
which is not as good as the U.S. rate. Under these circumstances no
American investor would want to invest in the Canadian financial market.
It is intuitive that Canadians would under these circumstances prefer to
invest in the American market, but we should check the numbers. If the
direct quote exchange rate from the American perspective went from $1.00/C$
to $0.95/C$ (the 5 percent depreciation mentioned above) then the direct
quote (units of domestic currency per unit of foreign currency) exchange
rate from the Canadian perspective went from C$1.00/$ to C$1.05263/$, an
appreciation of the dollar of 5.263 percent. Thus for the Canadian investor
the rate of return from an investment in the U.S. market is:
(0.07)(1+0.05263)+(0.05263) = (0.07)(1.05263)+0.05263 = 0.12631.
This is significantly better than what the Canadian investor can get in the
Canadian financial market. Rationally all Canadian investors would want to
withdraw their funds from the Canadian markets and transfer them to the
U.S. The outflow would depress the value of the Canadian dollar further
and create a capital shortage in Canada which would drive up interest rates
there. The inflow of capital to the U.S. would depress interest rates in
the U.S. markets. The net effect of these consequences is to reduce the
difference between what Canandian and U.S. rates of return. The adjustment
process would continue until there was interest rate parity between the
two countries.
Covered Interest Arbitrage
This topic is essentially the same as the preceding one except
that the existence of a forward market is taken into account.
This means that when a domestic investor decides to convert
domestic currency into foreign currency for investment in the
foreign financial markets the uncertainty about the future
exchange rate can be eliminated by entering into a forward
contract to sell the future foreign currency. Thus the future
conversion rate is known at the time the decision is made to
invest in foreign financial markets.
Let et be the spot exchange rate (direct form)
at time t and let ft,t+1 be the forward rate at time
t for conversions at time t+1. Then the rate of return RET on
an investment in the foreign financial market is:
1+RETFF =
(1/et)(1+iFF)ft,t+1
This means that if there is equilibrium between the two
markets so that
1+RETFF = 1+i$
then it follows that
ft,t+1/et = (1+iFF)/(1+i$)
This is the same relationship found for the expected future spot
exchange rate compared to the current spot exchange rate. This
naturally follows from the forward rate being the current
expectation of what the future spot rate will be.
Mishkin's Theory
In Mishkin's interest rate parity condition he takes the expected future
exchange rate to be given. The current exchange rate has to adjust to
achieve interest rate parity. This is best viewed graphically.
RET$ = RETFF
= iFF
- (Eet+1-Et)/Et
= iFF
- (Eet+1/Et - 1)
The left-hand side (LHS) is a constant.
The right-hand side (RHS) of the above condition is a function of -1/Et
so the value of the RHS is as shown below:
and hence for equilibrium
i$ = iFF
- (Eet+1-Et)/Et
Modeling Exchange Rate Determination When There is a Difference
in Real Rates of Interest in Two Countries
It is clear that when a country has a higher real rate of interest
than other countries do, as was the case with the U.S. in the early
1980's, there is an increase in the value of its currency with respect
to the value of other countries' currencies. However it is also clear that
financial markets do not adjust instantaneously to achieve an equilibrium.
The difference in the real interest rates between the U.S. and other countries
persisted for years despite the substantial net inflows of capital to the
U.S.
Let us consider the particular case of the Deutsche Mark/Dollar exchange
rate. There are various components of demand for DM. One componenet stems from
the demand for the importation of German goods and services. Let q(P$)
be the demand function for a German product, say sport cars, as a function of
the dollar price of these cars. If E$/DM is the exchange rate
for the DM, then the dollar price for the cars to American buyers is
PDME$/DM, where . But the demand function is in terms of physical
units. The DM expenditure on German sports cars is then
PDMq(PDME$/DM). It should always be remembered
that although the demand function q() is downward sloping, the expenditure
function pq(p) can have any slope. However in this case if the German price
of the cars is independent of the exchange rate the relationship between the
quantity of DM demanded and the exchange rate will the same as the relationship
betweeen physical demand function and the price of the product.
Likewise the supply of DM from Germans who want to buy American goods
and services. Suppose the demand function of Germans for some American product,
say wheat, is h(PDM). The DM price for the wheat is then
P$/E$/DM so the supply of DM is
(Pa
$/E$/DM)h(P$/E$/DM). The
dependence of the quantity of DM supplied to the American currency market
from this source is a little more complicated than the case of the
demand for DM. Generally we expected that the supply of DM will be a
positive function of the exchange rate.
The supply of DM through capital market flows depends upon the expected
rate of return in the U.S. capital market compared to the rate of return in the
German capital market. Interestingly enough this does not depend directly upon the
rates of inflation in the U.S. and Germany. It depends upon the nominal
interest rates in the two countries and the expected rate of change of the
exchange rate, as was shown in the previous section; i.e.,
i$
- (Eet+1-Et)/Et - iDM
The difference in the expected rates of return in the two markets
then determines the amount of net transfer of funds between the
two markets.