Homework's Mathematical Methods
Introduction to Exchange Rates and the Foreign Exchange Market
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Exchange Rate Essentials
Exchange Rates in Practice
The Market for Foreign Exchange
Arbitrage and Spot Exchange Rates
Arbitrage and Interest Rates
Conclusions
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Exchange rates affect large flows of international trade by influencing the prices in different currencies.
Foreign exchange also facilitates massive flows of international investment, which include direct investments as well as stock and bond trades.
In the foreign exchange market, trillions of dollars are traded each day and the economic implications of shifts in the market can be dramatic.
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Introduction
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1 Exchange Rate Essentials
An exchange rate (E) is the price of some foreign currency expressed in terms of a home (or domestic) currency.
Because an exchange rate is the relative price of two currencies, it may be quoted in either of two ways:
The number of home currency units that can be exchanged for one unit of foreign currency.
The number of foreign currency units that can be exchanged for one unit of home currency.
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To avoid confusion, we must specify which country is the home country and which is foreign.
When we refer to a particular country’s exchange rate, we will quote it in terms of units of home currency per units of foreign currency.
For example, Denmark’s exchange rate with the Eurozone is quoted as Danish krone per euro (or kr/€).
Defining the Exchange Rate
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1 Exchange Rate Essentials
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TABLE 2-1
Exchange Rate Quotations This table shows major exchange rates as they might appear in the financial media. Columns (1)to (3) show rates on December 31, 2015. For comparison, columns (4) to (6) show rates on December 31, 2014. For example, column (1) shows that at the end of 2015, one U.S. dollar was worth 1.501 Canadian dollars, 6.870 Danish krone, 0.921 euros, and so on. The euro–dollar rates appear in bold type.
E$/€ = 1.086 = U.S. exchange rate (American terms)
E€/$ = 0.921 = Eurozone exchange rate (European terms)
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1 Exchange Rate Essentials
| EXCHANGE RATES ON DECEMBER 31, 2015 | EXCHANGE RATES ON DECEMBER 31, 2014 ONE YEAR PREVIOUSLY | ||||||||
| (1) | (2) | (3) | (4) | (5) | (6) | ||||
| Country (currency) | Currency Symbol | Per $ | Per € | Per £ | Per $ | Per € | Per £ | ||
| Canada (dollar) | C$ | 1.501 | 1.389 | 2.047 | 1.158 | 1.402 | 1.806 | ||
| Denmark (krone) | DKr | 6.870 | 7.463 | 10.13 | 6.154 | 7.446 | 9.595 | ||
| Eurozone (euro) | € | 0.921 | — | 1.357 | 0.826 | — | 1.289 | ||
| Japan (yen) | ¥ | 120.3 | 130.7 | 177.3 | 119.9 | 145.1 | 187.0 | ||
| Norway (krone) | NKr | 8.851 | 9.612 | 13.05 | 7.498 | 9.072 | 11.69 | ||
| Sweden (krona) | SKr | 8.431 | 9.158 | 12.43 | 7.828 | 9.473 | 12.21 | ||
| Switzerland (franc) | SFr | 1.001 | 1.087 | 1.485 | 0.994 | 1.202 | 1.549 | ||
| United Kingdom (pound) | £ | 0.679 | 0.737 | — | 1.559 | 0.776 | — | ||
| United States (dollar) | $ | — | 1.086 | 1.474 | — | 1.210 | 1.559 |
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If one currency buys more of another currency, we say it has experienced an appreciation—its value has risen, appreciated, or strengthened.
If a currency buys less of another currency, we say it has experienced a depreciation—its value has fallen, depreciated, or weakened.
Appreciations and Depreciations
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1 Exchange Rate Essentials
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In U.S. terms, the following holds true:
When the U.S. exchange rate E$/€ rises, more dollars are needed to buy one euro. The price of one euro goes up in dollar terms, and the U.S. dollar experiences a depreciation.
When the U.S. exchange rate E$/€ falls, fewer dollars are needed to buy one euro. The price of one euro goes down in dollar terms, and the U.S. dollar experiences an appreciation.
Appreciations and Depreciations
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1 Exchange Rate Essentials
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To determine the size of an appreciation or depreciation, we compute the proportional change, as follows:
In 2011, at time t, the dollar value of the euro was
E$/€,t = $1.298.
In 2012, at time t + 1, the dollar value of the euro was
E$/€,t+1 = $1.318.
The change in the dollar value of the euro was
Δ E$/€,t = 1.318 − 1.298 = + $0.020.
The percentage change was
Δ E$/€,t/E$/€,t = +0.020/1.298 = +1.54%.
Thus, the euro appreciated against the dollar by 1.54%.
Appreciations and Depreciations
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1 Exchange Rate Essentials
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Similarly, over the same year:
In 2011, at time t, the euro value of the dollar was
E€ /$,t = €0.770.
In 2012, at time t + 1, the euro value of the dollar was
E€ /$,t+1 = €0.759.
The change in the dollar value of the euro was
ΔE€ /$,t = 0.759 − 0.770 = −€0.011.
The percentage change was
ΔE€/$,t / E€/$,t = −0.011/0.770 = −1.43%.
Thus, the dollar depreciated against the euro by 1.43%.
Appreciations and Depreciations
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1 Exchange Rate Essentials
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Economists calculate multilateral exchange rate changes by aggregating bilateral exchange rates using trade weights to construct an average over each currency in the basket. The resulting measure is called the change in the effective exchange rate. For example:
Suppose 40% of Home trade is with country 1 and 60% is with country 2. Home’s currency appreciates 10% against 1 but depreciates 30% against 2.
To find the change in Home’s effective exchange rate by multiply each exchange rate change by the trade share and sum:
(−10% * 40%) + (30% * 60%) = (−0.1 * 0.4) + (0.3 * 0.6) =
−0.04 + 0.18 = 0.14 = +14%.
Home’s effective exchange rate has depreciated by 14%.
Multilateral Exchange Rates
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1 Exchange Rate Essentials
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In general, suppose there are N currencies in the basket, and Home’s trade with all N partners is:
Trade = Trade1 + Trade2 + . . . + TradeN.
Applying trade weights to each bilateral exchange rate change, the home country’s effective exchange rate (Eeffective) will change according to the following weighted average:
Multilateral Exchange Rates
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1 Exchange Rate Essentials
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Multilateral Exchange Rates
FIGURE 2-1
Effective Exchange Rates: Change in the Value of the U.S. Dollar, 2002-2015 The chart shows the value of the dollar using two different baskets of foreign currencies. Against a basket of 7 major currencies, the dollar had depreciated by more than 25% by late 2004, and 35% by early 2008. Against a broad basket of 26 currencies, the dollar had lost only 15% by 2004 and 25% by 2008. This is because the dollar was floating against the major currencies, but the broad basket included important U.S. trading partners (such as China) that maintained fixed or tightly managed exchange rates against the dollar. These trends only briefly reversed during the global financial ciris of 2008 before continuing up to 2015.
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1 Exchange Rate Essentials
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