Thayer Watkins



Exchange Rate Determination Under Purchasing Power Parity

The purchasing power parity exchange rate (PPP) between a foreign currency and the dollar is defined as:


(Cost of a Market Basket of Goods and Services at Foreign Prices)
(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+$)]
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:

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:

= eet+1/et) + iFF(eet+1/et)
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:


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.

= 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.