# Currency

### Currency [viewable here in Excel]

Fixed-rate bonds are only risk-free in terms of its own unit; a US 10-year Treasury is risk-free in the US but not in the UK or Canada or Brazil. Let's say the interest rate on a UK 10-year risk-free government bond is 9% and on a Japanese 10-year risk-free government bond, the interest rate is 3%. Free money you say. Just move you Japanese money to the UK and make an additional 6% with no added risk. If it was so easy… Certainly, you can do that, and many investors do, but there's a little math involved involving currency exchange. Japan currency trades in Yen, UK trades in Pounds. If you're a Japanese investor you can safely buy the Japanese 10-year Treasury and earn 3%. If you elect to buy the UK 10-year Treasury the exchange rate is 150 yen to the pound, your rate of return in yen depends on the yen/pound exchange rate a year from now.

Invest 100 Pounds in UK bond.

- Convert 150 yen x 100 pounds = 15,000 yen into 100 pounds

- In a year your 100 pounds will equal 100 * 1.09 = 109 pounds

then

Exchange 150 yen to 1 Pound Investment

 Currency yen/lb Pounds Lbs x Yen LB (yr end) New Exchange Yen (yr end) Conversion Return yen 150 0.006667 100 0.666667 109 0.726666667 140 101.7333 1.73% pound 1 150 100 15000 109 16350 140 15260 1.73% 100 109 140 150 109 140

 This is tricky, you need to convert the change from 150 to 140: 10/140 = 1.07143 Yen Future Pound Convert Lb pound (100 + 9 interest = 109) after yr 1 1+(150 - 140)/140 LB Equivalent (Yr End) Yen\$ Converted after drop in Yen (150 to 140) This is what you earned playing the currency conversion game. You should have left you money in Japan and earned 3% Yen to Pound 15,000 1.09 16,350 1.07143 15,260 1.73%

Yen rate of return = 109 pounds x future yen price of the pound - 15,000 yen / 15,000 yen
Yen price at the end of year 1 = 140 yen / UK pound

Yen rate of return = 109 pounds x 140 yen - 15,000 yen / 15,000 yen

Yen realized rate of return is 1.73%, lower than the 3% Japanese 10-year bond. You should have invested in the Japanese bond.

 1-yr Yen Bond Yen Rate 3% year 0.03 Investment Currency yen/\$'s Total \$'s \$'s x Yen \$USD (yr end) New Exchange Yen\$ (yr end) Conversion Return yen 100 0.01 10,000 100 103 1 10,000 10,300 3.00% \$'s 1 100 10,000 1,000,000 103 10,300 10,000 1,030,000 3.00% 10,000 103 10,000 100 103 10,000

### Dollar Bonds vs. Yen Bonds. What's Better?

You have an opportunity to invest \$10,000 in dollar denominated bonds at 10% per year or in yen-dominated bonds offering 3% per year. What is the better investment?

 1-yr Yen Bond Yen Rate 3% year 0.03 Investment Currency yen/\$'s Total \$'s \$'s x Yen Yen (yr end) New Exchange \$ (yr end) Conversion Return yen 100 0.01 10,000 100 1,030,000 10,300 1 10,300 3.00% \$'s 1 100 10,000 1,000,000 1,030,000 103,000,000 1 1,030,000 3.00% 10,000 1,030,000 1 100 1,030,000 1

Keep in mind, you have invested in the 3% Yen bonds (hence the product is 1,030,000 * change in yen, not 1,000,000)

At the one year maturity date the USD \$10,000 that was converted to Yen \$1,000,000 you receive Yen \$1,030,000, regardless if the Yen currency increased or decreased.

Typically an increase in dollar represents higher GDP and higher interest rates, so if the Yen increases to .0108, interest rates may move higher, again this has no effect on the 1 year Yen bond. What the US investors need to be concerned with is if the US dollar increased in value or decreased in value.

 1-yr US Bond Investing \$10,000 0.1 Interest Earned US Bond \$1,000 Total \$11,000 Yen: US Dollar Exchange Rate (Start) \$0.0100 Yen: US Dollar Exchange Rate (End) \$0.0100 Yen Equivalent (\$10,000 US) 1,000,000 Interest Earned Yen Bond 30,000 Total 1,030,000 US Equivalent (US \$ : Yen) \$10,300 money losing proposition What US Dollar increase is required to equal \$11,000 USD \$0.01068 6.80% \$11,000 US Dollar with 8% appreciation \$0.0108 8.00% \$11,124 US Dollar with 6% appreciation \$0.0106 6.00% \$10,918

 1-yr Corporate Bond Yen Bond US Dollar Bond 5% 0.08 Yen = \$.01 \$10,000 100 Interest \$500 8 Expected Earnings \$10,500 108 Yen appreciated 4% \$0.0104 Cost to convert yen to dollar is now 109.2 -1.2 loss 0

 3% year 0.05 Investment Currency yen/\$'s Total \$'s \$'s x Yen Yen (yr end) New Exchange \$ (yr end) Conversion Return yen 100 0.01 10000 100 1050000 10500 1.000 10,500 5.00% \$'s 1 100 10000 1,000,000 1050000 105,000,000 1.000 1050000 5.00% 10000 1050000 1 100 1050000 1

 If the investor intends to keep the money in Yen then the loss would be \$1.20, but the currency, at some point in time, will need to return to the US. The US dollar loss against a 4% Yen appreciation is: Convert mature bond and interest to US dollar immediately US dollar weakens 1 to .996 US dollar straightens 1 to 1.02 2.00% (\$403.85) .01 to .0102 Yen weakens .01 to .009902 -0.98% \$10,500 \$10,500 \$10,500 Yen converted to US dollar (beginning yr) 100 100 100 Yen converted to US dollar (end yr) 96.15384615 97.64705882 103.009493 US dollar after change in Yen \$10,096.15 \$10,252.94 \$10,816.00 Gain or Loss (\$403.85) (\$247.06) \$316.00 Gain or loss as % -3.85% -2.35% 3.01%

### Currency Conversion [viewable here in Excel]

Calculating Foreign Exchange (FX) rates. When you're dealing with money from all over the world, exchange rates are important. In order to find them, all you have to do is compare the value of two different currencies. Seems simple enough, right?

Well, not particularly. If you often switch between two or more currencies, you know that calculating an exchange rate is not so straightforward.

First, the rates are in constant fluctuation. Just like the stock market, exchange rates move according to supply and demand; global money is traded around the clock. Second, you will receive different rates than people trading on the open market. Even then, the exchange rates you're offered will vary from bank to bank and service to service.

With so many variables, how do you know which exchange rate is fair to ensure you get the best deal available?

Let’s start with a basic example. Maybe you’re a US retiree who lives in Mexico but earns your \$1,000 retirement from the U.S. In that case, you’d need to change your U.S. dollars to Mexican Pesos every month.

Searching online, the exchange rate would look like this:

1US = 22.08 Mexican Peso

That would mean 1 US dollar buys 22.08 Mexican Pesos. Since your monthly retirement is US \$1,000, you would calculate it like this:

US \$1,000 X Mexican Peso 22.08 = Mexican Peso 22,080.

A US \$1,000 retirement here would equal Mexican Peso 22,080.

On the other hand, perhaps you are renting a condo in Canada and need to pay exactly Canadian \$500. How many US dollars would you need?

In this case, the exchange rate would be written the opposite way:

1\$US = 0.757 Canadian Dollar

To find the Canadian Dollar cost of US \$500, repeat the earlier calculation: \$500 X CD 0.757 = US \$378.50. Here, to pay CD \$500 rent, you’d need US \$378.50.

 Pound USD Conversion Day 1 to USD USD (Day 2) Lb Converted Gain/Loss Simple Pound to USD Conversion 5500 1.62 8910 1.61 5,534 0.62% Simple Pound to USD Conversion 5500 1.62 8910 1.65 5,400 -1.82% USD Yen Conversion Day 1 to Yen Yen (Day 2) US\$ Converted Gain/Loss Simple USD to Yen Conversion 10,000 0.01 100 0.010400 9,615 -3.85% Simple USD to Yen Conversion 10,000 0.01 100 0.009902 10,099 0.99% Yen USD Conversion Day 1 to Yen Dollar (Day 2) US\$ Converted Gain/Loss Simple Yen to USD Conversion 1,000,000 1 1,000,000 1.02 980,392 -1.96% Simple Yen to USD Conversion 1,000,000 1.04 1,040,000 1.00 1,000,000 4.00%

### Exchange Rates and Triangular Arbitrage

The Law of One Price applies to the foreign exchange market as well as to other financial markets. Arbitrage ensures that for any three currencies that are freely convertible in competitive markets, it is enough to know the exchange rates between any two in order to determine the third. Thus, as we show, if you know that the yen price of the U.S. dollar is ¥100 and the yen price of the U.K pound is ¥200, if follows by the Law of One Price that the dollar price of the pound is \$2.

To understand how arbitrage works in the foreign exchange market, it is helpful to start by considering the price of gold in different currencies. Suppose you know that the current dollar price of gold is \$100 per ounce and its price in yen is ¥10,000 per ounce. What would you expect the exchange rate to be between the dollar and yen?

The Law of One Price implies that it should not matter which currency you use to pay for gold. Thus, the ¥10,000 price should be equivalent to \$100, which implies that the dollar price of the yen must be \$0.01 or 1 cent per yen.

Suppose that the Law of One Price was violated, and the dollar price of the yen \$.009 rather than \$0.01. Suppose you currently have \$10,000 in cash in the bank. Since, by assumption, you can buy or sell gold either for ¥10,000 per ounce or for \$100 per ounce, you would convert your \$10,000 into \$10,000/\$0.009 = ¥1,111,111. You would use the yen to buy 111.111 ounces of gold (¥1,111,111/¥10,000 per ounce) and sell the gold for dollars to receive \$11,111 (111.11 x \$100 per ounce). You would now have \$11,111 less the transaction costs of buying and selling the gold and the yen. As long as these transaction costs are less than \$1,111, it would pay you to engage in arbitrage.

 Yen vs. Gold price Mismatch Gold per oz. US Yen (1 cent per yen) Yen - Gold by oz \$100 \$100 10,000 0.01 1 Yen gold mispriced \$10,000 1,111,111.11 0.009 111 \$11,111 Gain \$1,111

This type of transaction is called triangular arbitrage because it involves three assets: gold, dollars and yen.

Now let's look at the relations among the prices of three different currencies: yen, dollars, and pounds. Suppose the U.S. dollar price of the yen is \$0.01 per yen (or equivalent ¥100 yen to the dollar), and the price of the yen in terms of the British pound is a half pence (£0.005) to the yen (or equivalent, ¥200 to the pound). From these two exchange rates, we can determine that the U.S. dollar price of the pound is \$2.

Although it may not be immediately obvious, there are two ways to buy pounds for dollar. One way is indirectly through the yen market - by first buying yen for dollars and then using the yen to buy pounds. Since, by assumption, one pound costs ¥200, and ¥200 costs \$2.00, this indirect way costs \$2.00 per pound. Another way to buy pounds for dollars is to just do it directly.

The direct purchase of pounds for dollars must cost the same as the indirect purchase of pounds for dollars because of the Law of One Price. If it is violated, there will be an arbitrage opportunity that cannot persist for very long.

To see how the force of arbitrage works to uphold the Law of One Price in this example, let's look at what would happen if the price of the pound were \$2.10 rather than \$2. Suppose you walk into a bank in New York City, and you observe the following three exchange rates - \$0.01 per yen, ¥200 per pound, and \$2.10 per pound. Suppose that there is a window for exchanging dollars and yen, another for exchanging yen and pounds and a third window for exchanging dollars and pounds.

US dollar to the Yen is \$0.01 per yen (or equivalently 100 Yen to US dollar). Japanese yen to British pound is a half pence (Pound 0.005) or Yen 200 to the pound. With these two exchange rates, the computed US dollar to pound is \$2.00 (.01/.005 = 2).

You can possibly arbitrage currencies if a mismatch occurs. You can buy yen for US dollars and US dollars to buy pounds. Since one pound costs Yen 200, and Yen 200 costs \$2.00, pound cost is \$2.00.

If this method is mismatched an arbitrage opportunity exists. Let's say the pound is selling for \$2.10 rather than \$2.00.

Here is how you could make an immediate risk free \$10 profit:

 US Dollar Yen:USD Yen: Pound USD:Pound \$0.01 \$2.00 \$2.00 \$100.00 \$10,000.00 \$200.00 \$50.00 \$0.00500 \$20,000 \$100 \$210

• Convert (Dollar/Yen exchange) \$200 USD into Y20,000
• Convert (Yen/Pound exchange) Y\$20,000 to \$100 Pound
• Convert (Dollar/Pound exchange) \$100 Pound into \$2.10 USD for an immediate risk-free trade of \$10.

Congratulations, you have converted \$200 into \$210!

Three exchange rates:

Yen
Pound
Dollar

(Dollar/Yen) / (Pound/Yen) = Dollar/Pound
(Dollar/Yen) = Dollar/Pound x (Pound/Yen)
(Pound/Yen) = (Dollar Yen) / (Dollar/Pound)

Yen/Pound = US Dollar
Yen = Dollar x Pound
Pound = Yen/Dollar

Currency One-Year Future Price

One-year interest rate in the US
One-year interest rate in Germany
Spot rate (deutsche mark)

Future price/spot rate = 1 yr US rate/1yr German rate
One-year German Duestche Mark Future Price

### Forward and futures Prices for Foreign Exchange [viewable here in Excel]

A distinction between forward and futures prices emerges from the daily settlement feature of futures contracts. Consider forward and futures contracts on foreign exchange that have the same expiration. Both the futures and the forward will have the same profit in the end, exclusive of interest earned on the resettlement payments. If the futures position is likely to have more favorable interim cash flows due to its positive correlation with interest rates, the futures price should exceed the forward price. By the same token, if the futures price is negatively correlated with interest rates then the futures price should be lower than the forward price. This conclusion follows because the futures trader will then tend to experience losses just as interest rates rise. Finally, if the price of a commodity is uncorrelated with interest rates, then the forward and futures prices should be equal. Notice that all these conclusions arise strictly from economic reasoning and hold if investors are risk neutral.

While futures and forward prices differ in theory, the magnitude and practical significance of that difference is an empirical question. In general, studies of this issue find very little difference between foreign exchange forward and futures prices. As one study concluded, "The foreign exchange data reveal that mean differences between forward and futures prices are insignificantly different from zero, both in statistical and economic sense." In view of these finding, results based on research in the forward market will be regarded as holding for the futures as well.

### Futures Price Parity Relationships

Earlier, we noted the geographic or cross-rate arbitrage opportunities that occur when foreign exchange rates are improperly aligned among single contracts. The arbitrage example of Tables 9.1 and 9.2 arose from a pricing discrepancy in the foreign exchange rates for a single maturity of 90 days forward. Other price relationships are equally important and determine the permissible price differences that may exist between foreign exchange rates for delivery at different times. These relationships are expressed as the interest rate parity (IRP) theorem and the purchasing power parity (PPP) theorem. As we will see, the IRP theorem is simply the cost-of-carry model in a very thin disguise.

This is an arbitrage transaction since it yields a certain profit with no investment. Notice that the arbitrage is not complete until the transactions at t = 90 are completed.

Table 9.1 Geographic arbitrage

 t = 0 Buy €1 in New York 90 days forward for \$1.25 Sell €1 in Frankfurt 90 days forward for \$1.35 t = 90 Deliver €1 in Frankfurt: collect \$1.35 Pay \$1.25 in New York: collect €1 Profit: \$1.33 (\$1.25) \$0.08

Table 9.2 Cross-rates arbitrage transactions

 Key Currency Cross Rates Late New York Trading Thursday, September 2, 2020 Dollar Euro Pound Sfranc Peso Yen CDN Canada 1.16 1.41 2.03 0.89 0.11 0.01 0.00 Japan 117.34 142.63 204.96 90.44 10.74 0.00 101.08 Mexico 10.93 13.29 19.06 8.42 0.00 0.09 9.42 Switzerland 1.30 1.58 2.27 0.00 0.12 0.01 1.12 U.K. 0.57 0.70 0.00 0.44 0.05 0.00 0.49 Euro 0.82 0.00 1.44 0.63 0.08 0.01 0.71 U.S. 0.00 1.22 1.75 0.77 0.09 0.01 0.86

### The Interest Rate Parity Theorem [viewable here in Excel]

The interest rate parity theorem asserts that interest rates and exchange rates form one system. According to the IRP theorem, foreign exchange rates will adjust to ensure that a trader earns the same return by investing in risk-free instruments of any currency, assuming that the proceeds from investing are repatriated into the home currency by a forward contract initiated at the outset of the holding period. We can use the rates of Table 9.4 to illustrate interest rate parity. Faced with the rates in Table 9.4 and assuming interest rate parity holds, a trader must earn the same return by following either of the following strategies:

Strategy 1: Invest in the United States for 180 days.

Strategy 2:

(a) Sell dollar for euro at the spot rate.
(b) Invest euro proceeds for 180 days in Germany.
(c) Sell the proceeds of the German investment for dollars through a forward contract initiated at the outset of the investment horizon.

t = 0
Sell SF 1.0 90 days in Frankfurt for €0.63
Sell €0.63 90 days in New York for \$0.788
Sell \$0.788 90 days in New York for SF 1.023

t = 90
Deliver SF 1.0 90 days in Frankfurt; collect €0.63
Deliver €0.63 90 days in New York; collect \$0.788
Deliver \$0.788 90 days in New York; collect SF 1.023
Profit:
SF 1.023
(\$1.00)
SF 0.023

With our sample data, the following equation expresses the same equivalence:
\$1 x (1.20)0.5 = [(\$1/1.25) x (1.3697)0.5] x 1.17

Table 9.4 Interest rates and exchange rates to illustrate interest rate parity

 Exchange rates (\$/€) Interest Rates US Germany Spot 1.25 0 0.00 30-day 1.23 0.18 0.5760 90-day 1.20 0.19 0.3300 180-day 1.17 0.20 0.3697

In the equation, Strategy 1 is on the left-hand side. There, \$1 is invested at the 20 percent U.S. rate for six months (0.5 years). For strategy 2 on the right side, the dollar is first converted into euros at the spot rate of \$1.25 euro. The trader invests these proceeds at the Euribor rate, which represents the German interest rate, for 0.5 years. This 180-day rate is 36.97 percent. Investment of the euro will pay 0.920174 in 180 days. The investment proceeds are sold for dollars using the 180-day forward rate of 1.17. For this 180-day horizon, the equivalence between the two strategies holds, no arbitrage opportunity is available. In this example, the IRP theorem holds.

### Interest Rate Parity and the Cost-Of-Carry Model

In essence, the IRP theorem is simply the exchange rate equivalent of the cost-of-carry model. To see this equivalence, consider the cash-and-carry strategy for the interest rate market. In a cash-and-carry transaction a trader follows the following steps: borrow funds and buy a bond, carry the bond to the futures/forward expiration, and sell the good through a futures/forward contract arranged at the initial date. The cost of carry is the difference between the rate paid on the borrowed funds and the rate earned by holding the bond. Our familiar cash-and-carry strategy is known as covered interest arbitrage in the foreign exchange market. In covered interest arbitrage, a trader borrows domestic funds and buys foreign funds at the spot rate. The trader then invests these funds at the foreign interest rate until expiration of the forward/futures foreign investment back into the domestic currency. The cost of carry is the difference between the interest rate paid to borrow funds and the interest earned on the investment in foreign funds.

Thus, a trader borrows the domestic currency, DC, at the domestic rate of interest, DC, and exchanges these funds for foreign currency, FC, at the spot exchange rate. The trader receives DC/FC units of the foreign currency and invests at the foreign interest rate, FC. This rate, rFC is the interest rate applicable to the time from the present to the expiration of the forward or futures. At the outset of these transactions, t = 0, the trader also sells the forward or futures contract at F0,t for the amount of funds (DC/FC)(1+FC). With these transactions the trader has no net cash flow at t = 0. At expiration, the trader receives (DC/FC)(1+FC) units currency against the forward or futures contract and receives F0,t in the domestic currency. The trader must then pay the debt on the original borrowing, which is DC(1+DC). If the IRP theorem, or equivalent, the cost-of-carry model holds, the trader must be left with zero funds. Otherwise, an arbitrage opportunity exists.

Applying this notation to our previous example of the cost-and-carry transaction for the 180-day horizon, we can generalize this example to write an equation for the IRP theorem or the cost-of-carry model as it applies to foreign exchange. For convenience, we begin with \$1 as the amount of the domestic currency, DC. Earlier, for out example, we wrote:

\$1 x (1.20)0.5 = [(\$1/1.25) x (1.3697)0.5] x 1.17

In the new notation, this translate as follows:

DC(1 + DC) = (DC/FC)(1 + FC)F0,t

Remember that DC and FC are the interest rates for the specific period between the present, t = 0, and the expiration of the futures at time t.

Isolating the futures price on the left-hand side gives:

Equation (9.1) F0,t = DC(1 + DC) / ((DC/FC) x (1 + FC)) = FC x ((1+ DC)/(1+FC))

Equation 9.1 says that, for a unit of foreign currency, the futures price equals the spot rate of the foreign currency time the quantity:

Equation (9.2) ((1+ DC) / (1+FC))

This quantity is the ratio of the interest factor for the domestic currency to the interest factor for the foreign currency. We can compare this to our familiar Equation 3.3 for the cost-of-carry model in perfect markets with unrestricted short selling:

Equation (3.3) F0,t = S0 x (1 + C)

where F0,t is the futures or forward price at t = 0 for a foreign exchange contract to expire at time t, S0 is the spot price of the good at t = 0, and C is the percentage cost of carrying the good from t = 0 to time t. Equation 3.3 and 9.1 have the same form. Therefore, the quantity in Equation 9.2 equals one plus the cost of carry, (1 + C). The cash-and-carry strategy requires borrowing at the domestic rate, DC, so this is an element of the carrying cost. However, the borrowed domestic funds are converted to foreign currency and earn at the foreign interest rate FC. Therefore, the foreign earnings offset the cost being incurred through the domestic interest rate. The net result is that the quantity of Equation 9.2 gives the value for one plus the carrying cost. As a simpler approximation, we note that

Equation (9.3) 1 + cost of carry = ((1+ DC) / (1+FC)) =1 + ((1+ DC) / (1+FC))

Therefore, the cost of carry approximately equals the difference between the domestic and foreign interest rates for the period from t = 0 to the futures expiration. To complete this discussion, let us apply this equation for the 180-day horizon using the rates in Table 9.4. We have already seen that there is no arbitrage possible for this horizon. For this example, we have the following data: F0,t = 1.17; S0 = 1.25; DC = 0.095445 for the half-year; and FC = .17347 for the half-year.

Applying Equation 9.1 to this data, we have:

1.17 = 1.25 x 1.095445 ÷ 1.170347

This equation holds exactly. The cost of carry -.064. For this example, the approximate cost of carry for the half-year is:

DC - FC = 0.095445 - 0.170347 = -0.0749

Thus, the cost of carry for the half-year is approximately -0.075. The cost of carry is negative because the cash-and-carry trader pays at the domestic rate but earns interest at the higher foreign rate. For the same reason, the futures price of the foreign currency must exceed the spot price. If the foreign rate of interest had been lower, the futures price of the foreign currency would have to be lower than the spot price to avoid arbitrage.

Table 9.4 Interest rates and exchange rates to illustrate interest rate parity

 Exchange rates (\$/€) Interest Rates US Germany Spot 1.25 0 0.00 30-day 1.23 0.18 0.5760 90-day 1.20 0.19 0.3300 180-day 1.17 0.20 0.3697

### Exploiting Deviations from Interest Rate Parity

The analysis of the values in Table 9.4 shows that there is not an arbitrage opportunity in the 180-day contact. If the IRP theorem is to hold in general, there cannot be an arbitrage opportunity for any investment horizon. In Table 9.4, the rates allow an arbitrage opportunity in the 90-day contract. This is apparent when one realizes that the strategy of holding the US dollar and euro investment does not yield the same 90-day terminal wealth in US dollars when the euros are converted into dollars by issuing a forward contract. The following computation illustrates the different terminal dollar values earned by the two strategies:

Strategy 1
Hold in the US:
\$1 x (1.19)0.25 = \$1.0444

Strategy 2
Convert to euros, invest, and use a forward contract:
(\$1 ÷ \$1.25) x (1.33).025 x 1.20 = \$1.030942

Strategy 1, investing in the US, gives a higher payoff than converting dollars to euros and investing in Germany. This difference implies that an arbitrage opportunity exists.

This is also evident by applying the cost-of-carry model for foreign exchange to the 90-day values in Table 9.4. For this horizon, the values in Table 9.4 imply the following data: F0,t = 1.20; S0 = 1.25; DC = 0.-444448 for the quarter-year; and FC = 0.073898 for the quarter-year. With these values, the futures price should be 1.215721:

= FC x ((1+ DC)/(1+FC)) = 1.25 x 1.044448 ÷ 1.073898 = 1.215721

Because the futures price is less than this amount, an arbitrage opportunity exists. With our example data, it is clearly better to invest funds in the US rather than Germany. Table 9.5 shows the transaction that will exploit this discrepancy, assuming that the transactions begin with \$1.00.

This kind of arbitrage in foreign exchange is covered interest arbitrage. With these transactions, the trader uses a forward contract to cover the proceeds from the euro investment. The proceeds are covered, because the trader arranges through the forward contract to convert the euro proceeds into dollars as soon as the proceeds are received. The IRP theorem asserts that such opportunities should not exist. The section on market efficiency explores whether the IRP theorem actually holds.

Table 9.5 Covered interest arbitrage

t = 0 (present)
Borrow €0.80 in Germany for 90 days at 33%
Sell €0.80 spot for \$1.00
Invest \$1.0 in the US for 90 days at 19%
Sell \$1.030942 90 days forward for €0.804764

t = 90 (delivery)
Collect \$1.0444 on investment in the US
Deliver \$1.030943 on forward contract: collect €0.84764
Pay €0.84764 on €0.8 that was borrowed
Profit:
\$1.0444
(\$1.0342)
\$0.0140

### The Purchasing Power Parity Theorem (PPP)

The purchasing power parity theorem asserts that the exchange rates between two currencies must be proportional to the price level of traded goods in the two currencies. The PPP theorem is intimately tied to interest rate parity, as we discuss later. Violations of the PPP theorem can lead to arbitrage opportunities, such as the following example of "tortilla arbitrage."

For tortilla arbitrage, we assume that transportation and transaction cost are zero and that there are no trade barriers, such as quotas or tariffs. These assumptions are essentially equivalent to our usual assumptions of perfect markets. The spot rate value of the Mexican peso (MP) is \$0.10 and the cost of a tortilla in Mexico City is MP 1, as Table 9.6 shows. In New York a tortilla sells for \$0.15, so this price creates an arbitrage opportunity. A trader can exploit this opportunity by transacting as shown in the bottom portion of Table 9.6. Given the other values, the price of a tortilla in New York must be \$0.10 to exclude arbitrage.

Table 9.6 Tortilla arbitrage

 Table 9.6 Tortilla arbitrage Mexican Peso(MP) Cost of one tortilla Mexico City 10 MP 1 New York City 10 \$0.15 Arbitrage transaction Sell \$1 for MP 10 in the spot market Buy ten tortillas in Mexico City Ship the tortillas to New York Sell 10 tortillas in New York at 0.15 for \$1.50 \$1.50 -\$1.00 Profit: \$0.50

Over time, exchange rates must also conform to the PPP theorem. The left column of Table 9.7 presents prices and exchange rates consistent with the PPP theorem at t = 0. The right column shows values one year later at t = 1, after a year of inflation in Mexico and the US. During this year, Mexican inflation was 20 percent, so a tortilla now sells for MP 1.2. In the US, inflation was 10 percent, so a tortilla is now \$0.11. To be consistent with the PPP theorem, the exchange rates must also have adjusted to keep the relative value of the euro and dollar consistent with the relative purchasing power of the two currencies. As a consequence, the dollar must now be worth MP 10.91. Any other exchange rate would create an arbitrage opportunity. The requirement that the PPP theorem holds at all times means that the exchange rate must change proportionately to the relative price levels in the two currencies.

### Table 9.7 Purchasing Power Parity Over Time

 Expected inflation rates from t = 0 to t = 1: US dollar 0.10 Mexican Peso 0.20 Exchange rates (MP/\$) 10.00 10.91 Tortilla prices: Mexico City MP 1.00 MP 1.20 New York \$0.10 \$0.11

### Purchasing Power and Interest Rate Parity

The intimate relationship that exists between the purchasing power parity theorem and the interest rate parity theorem originates from the link between interest rates and inflation rates. According to the analysis of Irving Fisher, the nominal, or market, rate of interest consists of two elements, the real rate of interest and the expected inflation rate. This relationship can be expressed mathematically as follows:

1 + n = (1 + r*)[1 + E(I)]

where n is the nominal interest rate, r* is the real rate of interest, and E(I) is the expected inflation rate over the period in question. Since the expected inflation is the expected change in purchasing power, the PPP theorem expresses the linkage between exchange rates and relative inflation rates. A difference in nominal interest rates between two countries is most likely due to differences in expected inflation. This means that interest rates, exchange rates, price levels, and foreign exchange rates form an integrated system.

### Hedging with Foreign Exchange Futures

Many firms, and some individuals, find themselves exposed to foreign exchange risk. Importers and exporters, for example, often need to make commitments to buy or sell goods for delivery at some future time, with the payment to be made in foreign currency. Likewise, multinational firms operating foreign subsidiaries receive payments from their subsidiaries that may be denominated in a foreign currency. A wealthy individual may plan an extended trip abroad and may be concerned about the chance that the price of a particular foreign currency might rise unexpectedly. All of these different parties are potential candidates for hedging unwanted currency risk by using foreign exchange futures market.

If a trader faces the actual exchange of one currency for another, the risk is called transaction exposure, because the trader will transact in the market to exchange one currency for another. Firms often face translation exposure, the need to restate one currency in terms of another. For example, a firm may have a foreign subsidiary that earns profits in a foreign currency. However, the parent company prepares its accounting statements in the domestic currency. For accounting purposes, the firm must translate the foreign earnings into the domestic currency. While this procedure does not involve an actual transaction in the foreign exchange market, the reported earnings of the firm expressed in the domestic currency can be volatile due to the uncertain exchange rate at which the subsidiary's foreign earnings will be translated into the domestic currency. In the example that follow, we consider hedges of both transaction and translation exposure.

### Hedging Transaction Exposure

The simplest kind of example arises in the case of someone like John Moncrief, who is planning a six-month trip to Switzerland. Moncrief plans to spend a considerable sum during this trip, enough to make it worthwhile to attend to exchange rates, shown in Table 9.16. With the more distant rates lying above nearby rates. Moncrief fears that spot rates may rise even higher, so he decides to lock in the existing rates by buying Swiss franc futures. Because he plans to depart for Switzerland in June, he buys two JUN SF futures contracts at the current price of 0.5134. He anticipates that SF 250,000 will be enough to cover his six-month stay, as Table 9.17 shows. By June 6 Moncrief's fears have been realized, and the spot rate for SF is 0.5211. Consequently, he delivers \$128,350 and collects SF 250,000. Had he waited and transacted in the spot market on June 6, the SF 250,000 would have cost \$130,275. Hedging his foreign exchange risk, Moncrief has saved \$1,925, which is enough to finance a few extra days in Switzerland.

Table 9.16 Swiss exchange rates, January 12

 Table 9.16 Swiss exchange rates, January 12 \$/SF Spot 0.4935 MAR 0.5034 JUN 0.5134 SEP 0.5237 DEC 0.5342

Table 9.17 Moncrief's Swiss franc hedge

 Date Cash Market Futures Market January 12 Moncrief plans to take a six-month vacation in Switzerland, to begin June; the trip will cost about SF 250,000 Moncrief buys two June SF futures contracts at 0.5134 \$/SF for a total cost of \$128,350 June 6 The \$/SF spot rate is now 0.5211, giving a dollar cost of \$130,275 for SF 250,000 Moncrief delivers \$128,350 and collects SF 250,000 The \$/SF spot rate is now 0.5211, giving a dollar cost of \$130,275 for SF 250,000. Savings on the hedge = \$130,275 - \$128,350 = \$1,925

In this example, Moncrief had a preexisting risk in the foreign exchange market, since it was already determined that he would acquire the Swiss francs. By trading futures, he guaranteed a price of \$0.5134 per Swiss franc. Of course, the futures market can be used for purposes that are even more serious than reducing the risk surrounding Moncrief's Swiss vacation.

### Hedging Import/Export Transactions [viewable here in Excel]

Consider a small import/export firm that is negotiating a large purchase of Japanese watches from a firm in Japan. The Japanese firm, being a very tough negotiator, has demanded that payment be made in yen upon delivery of the watches. (If the contract had called for payment in dollars, rather than yen, the Japanese firm would bear the exchange risk.) Delivery will take place in seven months, but the price of the watches is agreed today to be ¥2,850 per watch for 15,000 watches. This means that the purchaser will have to pay ¥42.75 million in about seven months. Table 9.18 shows the current exchange rates on April 11. With the current spot rate of 0.004173 dollars per yen, the purchase price for the 15,000 watches would be \$178,396. If the futures prices on April 11 are treated as a forecast of future exchange rates, it seems that the dollar is expected to lose ground against the yen. With the DEC futures trading at 0.004265, the actual dollar cost might be closer to \$182,329. If delivery and payment are to occur in December, the importer might reasonably estimate dollar outlay to be about \$182,000 instead of \$178,000.

Table 9.18 \$/¥ foreign exchange rates, April 11

 \$/¥ Spot 0.004173 JUN 0.004200 SEP 0.004237 DEC 0.004265

To avoid any worsening of his exchange position, the importer decides to hedge the transaction by trading foreign exchange futures. Delivery is expected in November, so the importer decides to trade the DEC futures. By selecting this expiration, the hedger avoids having to roll over a near-by contract, thereby reducing transaction costs. Also, the DEC contract has the advantage of being the first contract to mature after the hedge horizon, so the DEC futures exchange rate should be close to the spot exchange rate prevailing in November when the yen are needed.

The importer's next difficulty stems from the fact that the futures contract is written for ¥12.5 million. If he trades three contracts, his transaction will be ¥37.50 million. If he trades four contracts, however, he would be trading ¥50 million, when he really only needs coverage for ¥42.75 million. No matter which way he trades, the importer will be left with some unhedged exchange risk. Finally, he decides to trade three contracts. Table 9.19 shows his transactions. On April 11, he anticipates that he will need ¥42.75 million, with current dollar value of \$178,396 and an expected future value of \$182,329, where the expected future worth of the yen is measured by the DEC future price. This expected future price is the most relevant price for measuring by the DEC futures price. This expected future price is the most relevant price for measuring the success of the hedge. In the futures market, the importer buys three DEC yen contracts at 0.004265 dollars per yen.

### Table 9.19 The Importer's Hedge

 Date Cash Market Futures Market April 11 The importer anticipates a need for ¥42.75 million in November, the current value of which is \$178,396, and which has an expected value in November of \$182,329 The importer buys three DEC yen futures contracts at 0.004265 for a total commitment of \$159,938 futures contracts at 0.004265 for a total commitment of \$159,938 November 1 Receives watches; buys ¥42.75 million at the spot market rate of 0.004273 for a total of \$182,671 Sells three DEC yen futures contracts at 0.004270 for a total value of \$160,125 Spot market results: Anticipated cost \$182,329 Total commitment of \$159,938 -\$182,671.00 (\$342) Net loss: (Futures market result: Profit = \$187) (\$155)

On November 18, the watches arrive, and the importer purchases the yen on the spot market at 0.004273. Relative to his anticipated cost of yen, he pays \$342 more than expected. Having acquired the yen, the importer offsets his futures position. Since the futures has moved only at 0.000005, the futures profit is only \$187. This gives a total loss on the entire transaction of \$155. Had there been no hedge, the loss would have been the full change of the price in the cash market, or \$342. This hedge was only partially effective for two reasons. First, the futures price did not move as much as the cash price. The cash price changed by 0.000008 dollars per yen, but the futures price changed by only 0.00005 dollars per yen. Second, the importer was not able to fully hedge his position, due to the fact that his needs fell between two contract amounts. Since he needed ¥42.75 million and only traded futures for ¥37.5 million, he was left with an unhedged exposure of ¥5.25 million.

### Hedging Translation Exposure [viewable here in Excel]

Many corporation in international business have subsidiaries that earn revenue in foreign currencies and remit their profits to a US parent company. The US parent reports its income in dollars, so the parent's reported earnings fluctuate with the exchange rate between the dollar and the currency of the foreign country in which the subsidiary operate. This necessity to restate foreign currency earnings in the domestic currency is translation exposure. For many firms, fluctuating earnings are an anathema. To avoid variability in earnings stemming from exchange rate fluctuations, firms can hedge with foreign exchange futures.

Table 9.20 shows euro exchange rates for January 2 and December 15. Faced with these exchange rates is Schropp Trading Company of Neckarsulm, a subsidiary of an American firm. Schropp Trading expects to earn €12.8 million this year and plans to remit those funds to its American parent. With the DEC futures trading at 1.2533 dollars per euro on January 2, the expected dollar value of those earnings is \$16,042,240. If the euro falls, however, the actual dollar contribution to the earnings of the parent will be lower.

Table 9.20 Exchange rates for the euro

 January 2 December 15 Spot 1.2598 1.1955 DEC futures 1.2533 1.1955

The firm can either hedge or leave unhedged the value of the earnings in euros, as Table 9.21 shows. With the rates in Table 9.20, the €12.8 million will be worth only \$15,302,400 on December 15. This shortfall could have been avoided by selling the expected earnings in euros in the futures market in January at the DEC futures price of 1.2533. Table 9.21 shows this possibility. With a contract size of €125,000, the firm could have sold 103 contracts at the January 2 price. This strategy would have generated a futures profit of \$774,175 (103 contracts x €125,000 x \$0.0578 profit per euro). This futures profit would have almost exactly offset the loss in the value of the euro, and Schropp Trading could successfully make its needed contribution to the American parent by remitting \$16,046,575.

Table 9.21 The Schropp Trading Company of Neckarsulm

 January 2 Expected earnings in Germany for the year €12.8 million Anticipated value in US dollars (computed at 1,2533 \$/€) \$16,042,240 Schropp Trading Company's contribution to its parent's income Unhedged Hedged Contribution to parent's income in US dollars from €12.8 million earnings (assumes spot rate of 1.1955) \$15,302,400 \$15,302,400 Futures profit or loss (closed at the spot rate of 1.1955) 0 \$774,175 Total \$15,302,400 \$16,076,575

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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