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### The Foreign Exchange Market

### Five-Country Example of Cross Rates

The foreign exchange market is a market in national moneys; the exchange rate is the price.

(Robert Aliber)

The Goals of This Chapter

- Review the historical development of foreign exchange markets.
- Explain how the foreign exchange rate reflects the demand and supply of goods, services, and assets, and the other flows that make up the balance of payments.
- Explain geographic arbitrage, triangular arbitrage, and intertemporal arbitrage.
- Introduce the spot and forward foreign exchange markets and derive the interest parity condition.
- Explain foreign exchange risk and how to hedge risk.
- Describe effective exchange rates.

The Foreign Exchange Market

- Most international transactions require the exchange of national currencies.
- Foreign Exchange Markets are the markets where the many different national currencies are exchanged.
- Changing foreign exchange rates add to the risk of many foreign transactions.
- In markets where the forces of supply and demand are free to drive the prices of currencies, the exchange rates are said to float.
- Some countries try to fix the value of their currencies at some target rate, often by by selling or buying currencies in the foreign exchange markets to neutralize shifts in supply and demand.

The Evolution of the Foreign Exchange Market

- Markets for foreign exchange have operated for over two thousand years, ever since there have been distinct national moneys.
- Early money changers carefully weighed and examined coins in order to determine their true gold or silver content.
- The development of modern banking brought the exchange of bills rather than actual coins made of precious metals.
- With the exchange of paper, money changers had to consider the reputation of the banks that issued the paper.

The Evolution of the Foreign Exchange Market

- The advent of paper or fiat money made the job of the money changer much more difficult and increased the risk of holding different national moneys.
- The relative value of each fiat money depends on what it, and all other currencies, can buy currently and is expected to buy in terms of real goods and services in the future.
- Expectations depend on a variety of information about current policies and about likely political and economic developments in the future.
- Expectations are subject to constant revision as news about political and economic events becomes known.
- Whenever expectations change, the exchange rate changes.

Foreign Exchange Markets Today

- Nearly all of the trillion dollars worth of currencies that are exchanged every working day are traded in the over-the-counter market.
- Worldwide, 2,772 dealer institutions reported that they were active in the foreign exchange markets in 2000.
- According to the Federal Reserve Bank of New York, there were 93 major foreign exchange dealers operating in the United States in 1998, 82 of which were large commercial banks.
- There were 213 major dealers in London, the most important center for foreign exchange transactions.

Table 12.2Geographical Location of Foreign Exchange Trading(Daily Averages on April 1, percentages)

1989 1992 1995 1998 2001

United Kingdom 25.6% 27.0% 29.5% 32.5% 31.1%

United States 16.0 15.5 15.5 17.9 15.7

Japan 15.5 11.2 10.2 6.9 9.1

Singapore 7.7 6.9 6.7 7.1 6.2

Germany - 5.1 4.8 4.8 5.4

Switzerland 7.8 6.1 5.5 4.2 4.4

Hong Kong 6.8 5.6 5.7 4.0 4.1

Source: BIS (2001), Central Bank Survey of Foreign Exchange and Derivatives Market Activity, Basle: BIS

Table 12.3Currencies Involved in Foreign Exchange Market Trading

Currency Percent in: 1989 1992 1995 1998 2001

U.S. dollar 90% 82.0% 83.3% 87.3% 90.4%

Deusche mark 27 39.6 36.1 30.1

Japanese yen 27 23.4 24.1 20.2 22.7

British pound 15 13.6 9.4 11.0 13.2

Swiss franc 10 8.4 7.3 7.1 6.1

French franc 2 3.8 7.9 5.1

Canadian dollar 1 3.3 3.4 3.6 4.5

Australian dollar 2 2.5 2.7 3.1 4.2

Euro - - - - 37.6

All others 26 23.4 25.8 32.5 21.3

Total 200% 200% 200% 200% 200%

Source: BIS (2001), Central Bank Survey of Foreign Exchange and Derivatives Market Activity, Basle: BIS

The Process of Arbitrage

- Arbitrage effectively combines distinct markets into a single integrated market because profit-seeking arbitrageurs buy where prices are low and sell where prices are high.
- Suppose the two isolated markets for violins are as shown by the supply and demand curves in Figure 12.4.

The Process of Arbitrage

- Arbitrage by consumers reduces demand for violins in New York and increases demand in San Francisco.
- If there are no costs to moving violins from San Francisco to New York, then prices will equalize at a single national price of pUS.

An Example: The Market for Mexican Pesos

- The demand curve intersects the supply curve at the price $.10.
- That is, one peso costs ten U.S. cents.
- We use the letter e to represent the foreign exchange rate, so that the equilibrium can be written as e = $.10.

An Example: An Increase in Demand for Pesos

- If holders of dollars want to engage in more foreign transactions that require Mexican pesos, the demand for pesos will increase.
- Such an increase in demand for pesos will cause the dollar to depreciate and the exchange rate e to rise, all other things equal.
- In the example shown, e rises from $.10 to $.125

The Foreign Exchange Market: The Mexican Perspective

- The supply curve for dollars is seen as the demand curve for pesos from the Mexican perspective.
- Similarly, the U.S. demand curve for dollars is the viewed in Mexico as the supply curve of pesos.
- Thus the equilibrium exchange rate from the Mexican perspective is 1/e = 1/.10 = 10 pesos.

The Foreign Exchange Market: A Shift in the Supply of Pesos

- The shift in demand for dollars from the U.S. perspective is a shift in supply of pesos from the Mexican perspective.
- The shift in supply causes the exchange rate to decline from 10 pesos, or 1/e = 1/$.10, to 1/e = 1/$.125 = 8 pesos.
- 37.5 million dollars are exchanged for 300 million (8x37.5) pesos.

How Many Foreign Exchange Rates Are There?

- There were 216 currencies in the world at the start of 2002.
- This seems to suggest that there are 216 x 216 = 46,656 different exchange rates.
- There are actually fewer than 46,656 exchange rates.
- First, subtract the 216 diagonal values of 1.
- Then, subtract half of the remaining exchange rates, which are just reciprocals of the other half, which leaves 23,220 different exchange rates.
- In general, for n different currencies, there are [n(n – 1)]/2 different foreign exchange markets.
- In the real world of 216 countries and moneys, there are therefore [216(215)]/2 = 23,220 foreign exchange rates.

Table 12.6Example: Five Countries with Five Currencies

Currencies:a b c d e

Countries:

Country A1a/b a/c a/d a/e

Country Bb/a 1 b/c b/d b/e

Country Cc/a c/b 1 c/d c/e

Country Dd/a d/b d/c 1 d/e

Country Ee/a e/b e/c e/d 1

The Example Illustrated in Table 12.6

- There must be markets for exchanging a for b, a for c, a for d and a for e.
- there must be markets to trade b for c, b for d, b for e, c for d, c for e, and d for e.
- the exchange rate between currencies a and b can be expressed as either the amount of b per unit of a, that is b/a, or as the amount of a per unit of b, a/b.
- The diagonal set of 1's are not foreign exchange rates because two different currencies are not being compared.
- In the case of n currencies there are [n(n – 1)]/2 different foreign exchange markets; in the case of 5 currencies there are (5x4)/2 = 10 exchange rates.

A foreign exchange trader would detect arbitrage opportunities among the exchange rates shown.

There will be an increase in supply of a to demand c, an increase in supply of c to demand b, and an increase in the supply of b to demand a.

Triangular arbitrage will result in a price of c > .25a, a price of b > 1c, and the price of a > 2b.

Incompatible Exchange Rates between Currencies a, b, and c

Price of:a b c

in:

Country A1a .5a .25a

Country B2b 1b 1b

Country C4c 1c 1c

Triangular ArbitrageArbitrage will continue until the exchange rates have moved to where there is no longer any way of making a profit by exchanging several currencies.

On possible outcome of the arbitrage and the exchange rate changes that it caused is given in the table on the right.

Compatible Exchange Rates After Arbitrage

Price ofa= b= c=

in:

Country A1a .4a .29a

Country B2.5b 1b .71b

Country C3.5c 1.4c 1c

Triangular ArbitrageUS$ ¥ DM FFr SFr

US$ per1 .01 .40 .20 .50

¥ per _ _ _ _ _

DM per _ _ _ _ _

FFr per _ _ _ _ _

SFr per _ _ _ _ _

The Set of Cross Rates with Triangular Arbitrage

US$ ¥ DM Fr SFr

US$ per 1 .01 .40 .20 .50

¥ per 100 1 40 20 50

DM per 2.50 .025 1 .50 1.25

Fr per 5.00 .05 2.00 1 2.50

SFr per 2.00 .02 .80 .40 1

Forward Exchange Markets

- Forward exchange markets are where future foreign exchange transactions are contracted today.
- The forward exchange rate is the price of one currency in terms of another currency for an exchange that is contractually agreed on today but will not be carried out until some future date.
- Over half of all transactions on the world’s foreign exchange markets are forward transactions.
- The forward markets are operated by the same dealers who operate the spot markets.

Intertemporal Arbitrage

- Suppose that you can store your wealth of $100 in assets denominated in U.S. dollars or British pounds.
- If you purchase dollar-denominated assets in the U.S., the $100 will earn a return of r, which means that over the period of one year your wealth would grow to

w($)t+1 = $100(1 + r),

given that r is the return on U.S. assets.

- To decideing where to invest your $100, you will have to compare the U.S. returns to what you would end up with after one year if you invested the $100 in Britain.
- The British investment is somewhat more complex because you will have to pay attention not only to the British rate of return, but also the spot and future exchange rates.

Intertemporal Arbitrage

- If you invest in the U.K., you must first convert $’s to £’s.
- Your wealth in terms of pounds is w(£)t = $100/et, where et is the spot exchange rate.
- If r* is the rate of return on British assets, then after one year your wealth in British pounds will grow to be:

w(£)t+1 = ($100/et)(1 + r* ).

- Before you can compare the value of your British investment to the U.S. investment, you must convert the pound value of your investment back to your home currency, dollars, at next year’s exchange rate, et+1.
- If you use the forward market to contract the sale of your pounds one year from now at the forward exchange rate ftt+1, then the dollar value of British investment is:

w($)t+1 = w(£)t+1(ftt+1) = ($100/et)(1 + r*)(ftt+1 ).

Intertemporal Arbitrage

- Now you can compare this dollar value of the British investment to the dollar value of the U.S. investment:

$100(1 + r) ? ($100/et)(1 + r*)(ftt+1).

- With unrestricted international asset trade, there will be international investment arbitrage until the inequality becomes an equality, or when

$100(1 + r) = ($100/et)(1 + r*)(ftt+1).

The Interest Parity Condition

The relationship

$100(1 + r) = ($100/et)(1 + r*)(ftt+1)

can be rearranged to yield the interest parity condition. Dividing each side by $100 gives us:

(1 + r) = (1/et)(1 + r*)(ftt+1).

Dividing each side by (1 + r) and multiplying each side by es results in:

et = [(1 + r*)/(1 + r)](ftt+1).

The Interest Parity Condition

- The equation

et = [(1 + r*)/(1 + r)](ftt+1)

is known as the the covered interest parity condition

- It is “covered” because the use of the forward market eliminates exchange risk.
- This equation says that the spot exchange rate is related to the forward exchange rate by the factor [(1 + r*)/(1 + r)].

The Interest Parity Condition

- When there is no forward exchange market, investors must compare returns across countries using their expectations of the spot rate one year from now, denoted as Et(et+1).
- The choice is thus:

$100(1 + r) ? ($100/et)(1 + r*)(Etet+1)

- Intertemporal arbitrage will still occur if the difference between the right-hand and left-hand sides of the relationship is big enough to overcome exchange rate risk.
- There will be a tendency for the following relationship to hold approximately:

$100(1 + r) ≈ ($100/et)(1 + r*)(Etet+1).

The Interest Parity Condition

- Manipulating the relationship

$100(1 + r) ≈ ($100/et)(1 + r*)(Etet+1)

just as was done for the covered interest parity condition shows that the spot exchange rate is directly related to the expected future exchange rate:

et≈ [(1 + r*)/(1 + r)](Etet+1).

- This is known as the uncovered interest parity condition or simply as the interest parity condition.
- The interest parity condition is one of the most important relationships in international economics.

A Simplified Version of the Interest Parity Condition

- First, divide each side of

et≈ [(1 + r*)/(1 + r)](Etet+1)

by (1 + r*), then divide both sides by et, which leaves:

(1 + r)/(1 + r*) = (Etet+1/et).

- Subtract 1 from each side of the equal sign using

1 = (1 + r*)/(1 + r*) on the left hand side and 1 = es/es on the right hand side, which leaves

(1 + r)/(1 + r*) – (1 + r*)/(1 + r*) = Etef/et – et/et.

- This latter relationship can be written as

(r – r*)/(1 + r*) = (Etef – et)/et .

A Simplified Version of the Interest Parity Condition

- (1 + r*) is very close to 1; assuming that (1 + r*) = 1, the equation simplifies further to

(r – r*) ≈ (Etef – et)/et = Et(Δe)/et,

where the “Δ” stands for “the change in.”

- That is, the proportional change expected in the exchange rate is roughly equal to the difference in the interest rates of the two countries.
- Thus, when arbitrage has equalized the overall returns for domestic and foreign assets, the difference between the rates of return on the assets in the two countries is exactly offset by the expected percentage change in the exchange rate over the period that investors expect to hold the assets.

A Numerical Example

- Suppose that the interest rate in the United States is higher at 12 percent per year than the interest rate on British assets at 7 percent.
- Suppose also that economic conditions and policies in the two countries lead investors to expect that the exchange rate will be 2 dollars = 1 pound one year from now, so that Etet+1 = $2.
- In the case of complete intertemporal arbitrage, the spot exchange rate must then be:

es = [(1 + r*)/(1 + r)]Etet+1 =[(1.07/1.12)]$2 = $1.91.

A Numerical Example

- Applying the simplified equation,

(r – r*) ≈ (Etef – et)/et = Et(Δe)/et,

an interest differential of 5 percent (.12 - .07 = .05) implies that the dollar is expected to fall by 5 percent over the coming year.

- In the case of perfect intertemporal arbitrage, if the spot rate is expected to be $2.00 one year from now, a depreciation of 5 percent implies that the current spot rate must be about $1.90.
- Note that this answer is very similar to the more precise $1.91 given by the full interest parity condition.

There Are Many Future Exchange Rates

- Intertemporal arbitrage links all future exchange rates according to the interest parity condition.
- For example, if the rates of return in the United States and Britain are expected to be r and r*, respectively, for the next two periods, then in the case of perfect arbitrage, the following two-period interest parity condition will hold:

$100(1 + r)2 = ($100/et)(1 + r*)2(Etet+2)

- The spot rate is thus a function of the expected exchange rate two periods from now:

et = Etet+2[(1 + r*)/(1 + r)]2.

- In general, for n periods into the future:

et = Etet+n[(1 + r*)/(1 + r)]n

Predicting Exchange Rate Changes

- The multi-period interest parity condition predicts that a country’s exchange rate will either appreciate or depreciate as time passes, depending on whether r* > r or r > r*.
- Exchange rates actually fluctuate much more than interest rate differences across countries suggest.
- Only fundamental shifts in long-run expectations, which imply shifts in the whole long-run time path of exchange rates, can explain the large changes in spot exchange rates that we often observe.
- Predictions of exchange rate changes therefore must predict changes in long-run expectations, which is an impossible task!

Predicting Exchange Rate Changes

If people set their expectations rationally they will make use of all relevant information to set their expectations, which consists of:

1. Their understanding of how foreign exchange rates are determined, that is, their model of exchange rate determination;

2. All available information that helps to put values on the variables in their model of exchange rate determination.

Economists define items 1 and 2 as the information set.

Predicting Exchange Rate Changes

- The expected exchange rate at time t = n (n years in the future), given the current information set Ωt is written as:

Et=0(et=n | Ωt ).

- The information set Ω of course keeps changing as time passes as news arrives.
- The expected exchange rate for the period t = n at time t = 0 will generally not be the same as the expected exchange rate for period t = n at t = 1 because Ωt≠Ωt+1.
- That is, in general: Et[(et+n) ׀Ωt] ≠ Et+1[(et+n) ׀Ωt+1].

Predicting Exchange Rate Changes

- The spot rate will deviate from its long-run time path whenever news arrives.
- News is, by definition, unpredictable
- Thus, the logical conclusion is that:

In general, when expectations are rationally set and the interest parity condition holds (international investment is not restricted), future changes in the exchange rates are unpredictable.

Effective Exchange Rates

- The exchange rate between just two currencies tells a firm little about an economy’s competitive position in the global economy.
- Many government agencies and private financial firms compile broader exchange rate measures that attempt to capture the overall value of a country’s currency vis-a-vis many countries.
- Effective exchange rates are weighted averages of sets of foreign exchange rates.
- The U.S. Federal Reserve Bank compiles several effective exchange rates, including the Broad Dollar Index, the Major Currencies Dollar Index, and the Other Important Trading Partners Dollar Index.

Effective Exchange Rates

- Each of the Fed’s three effective exchange rates behaved very differently over the past 25 years.
- The Figure also shows how volatile exchange rates have been over the past 25 years.
- Notice that even the effective exchange rates of the United States dollar have fluctuated widely, changing by more than 10 percent in many years.

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