The Foreign Exchange Market

Interest parity and efficient markets require that forward exchange rates equal the expected future spot rates on the dates at which the forward contracts mature. True or False: Explain your answer.

False! Neither require it.

We define the interest parity condition as

(1) --- id - if = FD + PRP

where id = the domestic nominal interest rate, if = the foreign nominal interest rate, FD = the forward discount and PRP = the country specific or political risk premium.

The forward discount is defined as

(2) --- FD = (F - S)/S

where S is the spot rate (domestic currency price of foreign currency) and F is the forward rate (domestic currency forward price of foreign currency).

Finally, the efficient markets condition is defined as

(3) --- FD = Es + FXRP

where Es is the expected rate of change of S and FXRP is the foreign exchange risk premium.

Foreign exchange markets, and markets in general, are efficient if market participants use all the information available to them in making transactions decisions.

The efficient markets condition will always hold, as will the interest parity condition, if investors are rational (i.e., use all information available to them in making investment decisions). Thus (1) and (3) define the conditions under which the relevant markets will be efficient.

For the forward exchange rate to equal the future spot rate expected on the day that contracts mature (or, what is the same thing, for the forward discount to equal the expected rate of change in the exchange rate) the foreign exchange risk premium FXRP must be zero (and, of course, the foreign exchange market must be efficient so that (3) holds).

The interest parity condition, as we have defined it, says nothing about the relationship between the forward and expected future spot exchange rates.

Our statements of the efficient markets and interest parity conditions in (3) and (1) above are true by definition. Sometimes in the literature you will see the efficient markets condition referred to as, simply,

FD = Es.

This is incorrect because markets can be efficient when this equality does not hold if there is a risk premium. Moreover, even if this equality holds, markets could be inefficient if there is were a non-zero risk premium (because equation (3) would be violated).

Quite apart from the definition of interest rate parity in equation (1) above, there are two common types of interest rate parity appearing frequently in the literature:

• Covered Interest Parity --- which says that the nominal interest rate differential will equal the forward discount (and hence that the political risk premium PRP is zero).

• Uncovered Interest Parity --- which says that the nominal interest rate differential will equal the expected rate of change in the exchange rate (and hence that both the political and foreign exchange risk premia, PRP and FXRP, are zero).

It turns out that covered interest rate parity seems to occur to a reasonable approximation (i.e., political risk premia seem to be small) while uncovered interest rate parity is almost always rejected by a correct interpretation of the data except where there are big differences in the rates of inflation in the two countries in question.

Tests of whether forward rates are unbiased predictors of future spot rates are not sufficient to enable us to determine whether or not foreign exchange markets are efficient, regardless of the outcomes of those tests. True or False: Explain your answer.

A test of whether the forward rate equals the expected future spot rate is really a joint test of three hypotheses; the efficient markets hypothesis, the hypothesis that the risk premium is zero (or, depending on the test, constant) and and an hypothesis about how expectations of the future spot rate are formed. If the test fails, there is no way of knowing whether it failed because the efficient markets hypothesis is not true, because a risk premium exists (or varies through time), or because the wrong theory of how expectations about future spot rates are formed is being used. It is thus impossible to test whether markets are efficient, because there is no independent way of determining whether or not a risk premium exists, or whether agents form unbiased expectations.

To put things more formally, we can write the forward rate as

(1) F = EFS + RP

where F is the forward rate (in, say, Pounds per Canadian Dollar), EFS is the expected future spot rate (Pounds per Canadian Dollar) and RP is the risk premium (measured in Pounds). This equation will hold if the market is efficient. If it is inefficient the equality will not hold. Instead, we would have

(2) F = EFS + RP + ERR

where ERR is the error in the forward prediction due to the failure of market efficiency.

Even if markets are efficient the expected future spot rate can differ from the actual as a result of bias in the agents' formation of their expectations or as a result of random error. We will then have

(3) EFS = MAFS + BIAS

where MAFS is the mean of the actual future spot rate and BIAS is the bias in the prediction of the actual future spot rate resulting from the way agents form their expectations.

Then, substituting (2) into (3) we obtain

(4) F = MAFS + BIAS + RP + ERR.

To establish that markets are efficient we must establish that ERR = 0.

Suppose we do a statistical analysis and find that on average

F - MAFS > 0

---i.e., that the forward rate differs from the mean expected future spot rate and hence does not predict it well. There could be three possible reasons why this forward-prediction test can fail.

• The market is inefficient --- i.e., ERR is not zero.

• The market is efficient (ERR = 0) but agents' expectations formation is biased --- i.e., BIAS is not zero.

• The market is efficient and agents form unbiased expectations but there is a positive risk premium --- i.e., RP is not zero.

The failure of our test does not necessarily imply that the market is inefficient.

Alternatively, suppose we do a statistical analysis and find that on average

F - MAFS = 0

---i.e., that the forward rate is an unbiased predictor of the future spot rate. From equation (4), this only means that

BIAS + ERR + RP = 0.

There are a number of possible reasons why this could be true, among them:

• The market is efficient, agents form unbiased expectations and the risk premium is zero --- i.e., BIAS = ERR = RP = 0.

• The market is inefficient so that ERR is not zero but BIAS = - ERR and RP = 0 so that BIAS + ERR + RP = 0.

• The market is inefficient so that ERR is not zero but RP = - ERR and BIAS = 0 so that, again, BIAS + ERR + RP = 0.

We can never show empirically that the market is efficient by examining the ability of the forward rate to predict the future spot rate because the test will necessarily be a joint test of three hypotheses:

(1) Markets are efficient (ERR = 0).

(2) Agents form unbiased expectations (BIAS = 0).

(3) There is no risk premium (RP = 0).

The efficient markets hypothesis is simply a working hypothesis that enables us to conduct a rational analysis of the problem. It is a necessary condition for any useful analysis to be possible. Exactly the same can be said for utility maximization! If people do not exhibit coherent behavior, analysis of the consequences of that behavior is impossible.

The first point that has to be brought into the discussion is that observed real exchange rate series are, for practical purposes, indistinguishable from random walks. Some mean reversion is evident when the series are viewed over long periods but from the point of view of forecasting, unless agents have information about the innovation or error term, their best forecast of next period's real exchange rate will typically be the observed real exchange rate this period. Next period's nominal exchange rate will, of course, differ from the real exchange rate by an adjustment for ongoing inflation.

If the real exchange rate is a random walk, it can be viewed as having been generated by a time-series process of the form

(1) --- rex(t) = rex(t-1) + v(t)

where rex(t) is the logarithm of the real exchange rate and v(t) is a random innovation process (error term) with mean zero and constant variance.

The nominal exchange rate, of course, will be a random walk if the real exchange rate is a random walk. But it will nevertheless move up and down in relation to the real exchange rate in accordance with increases and decreases in agents' expectations of the difference between the domestic and foreign inflation rates.

To illustrate the consequence of the real exchange rate being a random walk, let us imagine a world where the domestic and foreign inflation rates exhibit zero trends. Then let us generate a simulated series representing the logarithm of the deviation of the nominal exchange rate around some constant value (which we choose to be zero) using (1) above with the innovation (renamed e(t) because the exchange rate is now the nominal rate) specified as having a normal distribution with variance equal to 1.0. In this case, equation (1) becomes

(2) --- s(t) = s(t-1) + e(t)

Then let us suppose that agents know that the real and nominal exchange rates are random walks but have no information about the innovation process e(t). Their best prediction of next period's real exchange rate is the current period's value so, under the assumption that all agents are risk neutral (so the risk premium will be equal to 0), the forward rate f(t) (in logarithms) will equal s(t) and the forward discount will then be zero. Let us suppose that agents make guesses about the actual inflation rate so that the the forward real exchange rate is chosen with an error u(t) having a variance that is small relative to the variance of e(t). Then the forward rate in period (t), which will be the forward forecast of the spot exchange rate in period (t+1), will equal

(3) --- f(t) = s(t) + u(t)

and the forward discount will be

(4) --- fd(t) = f(t) - s(t) = u(t)

If the forward price is not in line with the interest rate differential, an arbitrage opportunity exists. In other words, forward prices do not contain expectations about future movements of spot price. Doesn't this contradict with what I am learning in your class?"

No! It is true that, ignoring risk (an important assumption), the log of the forward price will equal the log of the spot price plus the interest rate differential---an expression virtually identical to the one you note above. Letting id(t) and if(t) be domestic and foreign interest rates and f(t) and s(t) be the logarithms of the forward and spot exchange rates respectively, we have the interest parity condition

(1) id(t) - if(t) = f(t) - s(t) + risk premium

which when we assume that the risk premium is zero reverts to

(2) f(t) = s(t) + id(t) - if(t).

This expression is otherwise known as covered interest parity.

Once we take into account the length of the forward contract and adjust the interest rates to apply to that length of time, this is exactly what you are learning in your options and futures course. And, as long as we ignore the country specific risk, violation of this condition will present an arbitrage opportunity---an investor can switch funds from one country to the other, covering himself with an appropriate forward contract, and make a sure profit as long as the two countries' securities are perfect substitutes (i.e., there is no country-risk differential).

But market efficiency (again assuming no risk) implies that

(3) f(t) - s(t) = E{s(t+1)} - s(t).

If we substitute (2) into (3) to eliminate f(t) - s(t) we obtain the uncovered interest parity condition

(4) id(t) - if(t) = E{s(t+1) - s(t)

which says that the interest rate differential depends on the expected future spot rate.

So the forward exchange rate depends on the expected future spot rate because the interest rate differential depends on it. Or alternatively, the interest rate differential depends on the expected future spot rate because the forward exchange rate depends upon it (equation (3)). Which ever way we look at it the model being used in my course is the same as the one in your Options and Futures course.

Actually, if you go into a bank and want to buy forward exchange, the bank will apply the interest rate differential to the current spot price to obtain an appropriate forward price to charge you. They do this because they assume that covered interest parity holds to a reasonable approximation (i.e., there is no country-risk premium) in which case the observed interest rate differential will reflect the markets expectations as to the movement of the spot rate over the life of the forward contract after appropriate adjustment for foreign exchange risk.