Arbitrage, money and purchasing power

Eco 328Arbitrage, money and purchasing power

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Contracts to exchange euros for dollars in one year’s time carry an exchange rate of F$/ € dollars per euro. This is known as the forward exchange rate.

If you invest in a dollar deposit, your $1 placed in a U.S. bank account will be worth (1 + i$) dollars in one year’s time. The dollar value of principal and interest for the U.S. dollar bank deposit is called the dollar return.

If you invest in a euro deposit, you first need to convert the dollar to euros. Using the spot exchange rate, $1 buys 1/E $/€ euros today.

These 1/E $/€ euros would be placed in a euro account earning i €, so in a year’s time they would be worth (1 + i €)/E$/€ euros.

These euros must then be exchanged for dollars. But at what rate?

Riskless Arbitrage: Covered Interest Parity

Arbitrage and Interest Rates

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To avoid that risk, you engage in a forward contract today to make the future transaction at a forward rate F$/€.

The (1 + i €)/E$/€ euros you will have in one year’s time can then be exchanged for (1 + i €)F$/€/E$/€ dollars, or the dollar return on the euro bank deposit.

Riskless Arbitrage: Covered Interest Parity

deposits euroon return Dollar

€/$

€/$

€

depositsdollar on return Dollar

$ 11E

Fii

This is called covered interest parity (CIP) because all exchange rate risk on the euro side has been “covered” by use of the forward contract.


In equilibrium,

Arbitrage and Covered Interest Parity Under CIP, returns to holding dollar deposits accruing interest going along the path AB must equal the returns from investing in euros going along the path ACDB with risk removed by use of a forward contract. Hence, at B, the riskless payoff must be the same on both paths:

1+ i$( ) =F$/€

E$/€

1+ i€( )4

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Evidence on Covered Interests Parity

Financial Liberalization and Covered Interest Parity: Arbitrage Between the United Kingdom and Germany

The chart shows the difference in monthly pound returns on deposits in British pounds and German marks using forward cover from 1970 to 1995. In the 1970s, the difference was positive and often large: traders would have profited from arbitrage by movingmoney from pound deposits to mark deposits, but capital controls prevented them from freely doing so.

Financial Liberalization and Covered Interest Parity

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Evidence on Covered Interests Parity

Financial Liberalization and Covered Interest Parity: Arbitrage Between the United Kingdom and Germany

After financial liberalization, these profits essentially vanished, and no arbitrage opportunities remained. The CIP condition held, aside from small deviations resulting from transactions costs and measurement errors.

Financial Liberalization and Covered Interest Parity

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• In this case, traders face exchange rate risk and must make a forecast of the future spot rate. We refer to the forecast as , which we call the expected exchange rate.

• Based on the forecast, you expect that the euros you will have in one year’s time will be worth when converted into dollars; this is the expected dollar return on euro deposits.

• The expression for uncovered interest parity (UIP) is:

Risky Arbitrage: Uncovered Interest Parity

eE $/€

$/€€ /)1( Ei

$/€ $/€€ /)1( EEi e

deposits euroon returndollar Expected

€/$

€/$

€

depositsdollar onreturn Dollar

$ 11E

Eii

e


Arbitrage and Uncovered Interest Parity

Under UIP, returns to holding dollar deposits accruing interest going along the path AB must equal returns from investing in euros going along the risky path ACDB. Hence, at B, the expected payoff must be the same on both paths:

€€/$

€/$

$ 11 iE

Ei

e

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What Determines the Spot Rate?

Uncovered interest parity is a no-arbitrage condition that describes an equilibrium in which investors are indifferent between the returns on unhedged interest-bearing bank deposits in two currencies.

We can rearrange the terms in the uncovered interest parity expression to solve for the spot rate:

Risky Arbitrage: Uncovered Interest Parity

$

€

€/$€/$1

1

i

iEE e


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Evidence on Uncovered Interest Parity

• Dividing the UIP by the CIP, we obtain , or

• Although the expected future spot rate and the forward rate are used in two different forms of arbitrage—risky and riskless, in equilibrium they should be exactly the same.

• If both covered interest parity and uncovered interest parity hold, the forward must equal the expected future spot rate.

• Investors have no reason to prefer to avoid risk by using the forward rate, or to embrace risk by awaiting the future spot rate.

€/$€/$ /1 FE e €/$€/$ FE e

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Evidence on Uncovered Interest Parity

If the forward rate equals the expected spot rate, the expected rate of depreciation equals the forward premium (the proportional difference between the forward and spot rates):

While the left-hand side is easily observed, the expectations on the right-hand side are typically unobserved.

ondepreciati of rate Expected

€/$

€/$

premium Forward

€/$

€/$ 11 E

E

E

F e

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Evidence on Uncovered Interest Parity Evidence on Interest Parity

When UIP and CIP hold, the 12-month forward premium should equal the 12-month expected rate of depreciation. A scatterplot showing these two variables should be close to the diagonal 45-degree line.

Using evidence from surveys of individual forex traders’ expectations over the period 1988 to 1993, UIP finds some support, as the line of best fit is close to the diagonal.

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The UIP approximation equation says that the home interest rate equals the foreign interest rate plus the expected rate of depreciation of the home currency.

Suppose the dollar interest rate is 4% per year and the euro 3%. If UIP is to hold, the expected rate of dollar depreciation over a year must be 1%. The total dollar return on the euro deposit is approximately equal to the 4% that is offered by dollar deposits.

Uncovered Interest Parity: A Useful Approximation


𝑖$ = 𝑖€ +%Δ𝐸$€

𝑒

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How Interest Parity Relationships Explain Spot and Forward Rates

In the spot market, UIP provides a model of how the spot exchange rate is determined. To use UIP to find the spot rate, we need to know the expected future spot rate and the prevailing interest rates for the two currencies.

Arbitrage and Interest Rates Summary

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How Interest Parity Relationships Explain Spot and Forward Rates

In the forward market, CIP provides a model of how the forward exchange rate is determined. When we use CIP, we derive the forward rate from the current spot rate (from UIP) and the interest rates for the two currencies.

Arbitrage and Interest Rates Summary

The Law of One Price

What if Reebok hockey sticks were selling for $150 in New York and $350 in Montreal?

Arbitrage occurs in the international goods markets just as in the international financial markets.

Therefore, there is an equalizing for acting on prices of goods in different countries expressed in a common currency.

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The law of one price (LOOP) states that in the absence of trade frictions and under free competition and price flexibility, identical goods sold in different locations must sell for the same price when expressed in a common currency.

We can state the law of one price as follows, for the case of any good g sold in two locations:


$in good of

price U.S.

$in good ofpriceEuropean

€/$

U.S. versusEuropein good of price Relative

/ /)(

g

g

US

g

g

EUR

g

g

EURUS PPEq

€/$EWhere expresses the rate at which currencies can be exchanged.

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We can rearrange the equation for price equality


to suggest that the exchange rate ought to equal the ratio of the goods’ prices expressed in the two currencies:

g

US

g

EUR PPE €/$

prices goods’of Ratio

rateExchange

€/$ / g

EUR

g

US PPE

Purchasing Power Parity

Suppose now one put together a basket of goods, including not just hockey sticks, but milk, eggs and gasoline too.

What would happen if the same two baskets were selling for different amounts of $$ in the US and Canada?

Same thing – goods arbitrage until the same basket of goods sells for the same price in both countries when expressed in a common currency.

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The principle of purchasing power parity (PPP) is the macroeconomic counterpart to the microeconomic law of one price (LOOP). To express PPP algebraically, we can compute the relative price of the two baskets of goods in each location:

Purchasing Power Parity

$in expressed

basket ofprice U.S.

$in expressed

basket ofpriceEuropean

€/$

U.S.versusEuropein basket of

price Relat ive

/ /)( USEUREURUS PPEq

There is no arbitrage when the basket is the same price in both locations qUS/EUR = 1.

PPP holds when price levels in two countries are equal when expressed in a common currency. This is called absolute PPP.

For example

Suppose the European basket costs €100, and the exchange rate is $1.20 per euro. For PPP to hold, the U.S. basket would have to cost 1.20 × 100 = $120.

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The real exchange rate is the relative price of the baskets.

• The U.S. real exchange rate qUS/EUR = E$/€ PEUR/PUS tells us how many U.S. baskets are needed to purchase one European basket.

• The exchange rate for currencies is a nominal concept. The real exchange rate is a real concept.

The real exchange rate has terminology similar to the nominal exchange rate:

• If the real exchange rate rises (more Home goods are needed in exchange for Foreign goods), Home has experienced a real depreciation.

• If the real exchange rate falls, Home has experienced a real appreciation

The Real Exchange Rate

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Purchasing power parity states that the real exchange rate is equal to 1.

• If the real exchange rate qUS/EUR is below 1 then Foreign goods are relatively cheap.

o In this case, the Home currency is said to be strong, the euro is weak, and we say the euro is undervalued.

• If the real exchange rate qUS/EUR is above 1, then Foreign goods are relatively expensive.

o In this case, the Home currency is said to be weak, the euro is strong, and we say the euro is overvalued.

Absolute PPP and the Real Exchange Rate

For example

If a European basket costs E$/€PEUR = $550 in dollar terms, and a U.S. basket costs only PUS = $500, then qUS/EUR = E$/€PEUR /PUS = $550/$500 = 1.10, the euro is strong, and the euro is 10% overvalued against the dollar.

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We can rearrange the no-arbitrage equation for the equality of price levels, to allow us to solve for the exchange rate that would be implied by absolute PPP:

Absolute PPP:

Absolute PPP, Prices, and the Nominal Exchange Rate

USEUR PPE €/$

levels price of Ratiorate Exchange

€/$ / EURUS PPE

Purchasing power parity implies that the exchange rate at which two currencies trade should equal the relative price levels of the two countries.

For example

If a basket of goods costs $460 in the United States and the same basket costs €400 in Europe, the theory of PPP predicts an exchange rate of…

$460/€400 = $1.15 per euro.

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Absolute PPP, Prices, and the Nominal Exchange Rate

We now have a model that takes as inputs the prices levels in the respective countries and outputs an exchange rate.

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How does our exchange rate model relate to inflation (the rate of change of the price level)?

Let’s evaluate both sides of the equation as the variables change.

Relative PPP, Inflation, and Exchange Rate Depreciation

rate exchange nominal theofon depreciat i of Rate

,€/$

,€/$1,€/$

,€/$

, €/$

t

tt

t

t

E

EE

E

E


€/$ / EURUS PPE

For example

If the price level today is 100, and one year from now it is 103.5, then the rate of inflation is…

(103.5 – 100) / 100 =

3.5% (for the year).

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How does our exchange rate model relate to inflation (the rate of change of the price level)?



€/$ / EURUS PPE

On the right, the rate of change of the ratio of two price levels equals the rate of change of the numerator minus that of the denominator:

EURUS

tEUR

tEURtEUR

tUS

tUStUS

tEUR

tEUR

tUS

tUS

EURUS

EURUS

tEURtUS

P

PP

P

PP

P

P

P

P

PP

PP

,,

Europein inflation of Rate

,

,1,

in U.S.inflation of Rate

,

,1,

,

,

,

,

)/(

)/(

For example

If the price level in the US today is 100, and one year from now it is 103.5, then the US rate of inflation is 3.5%.

If the price level in Europe today is 100, and one year from now it is 105, then the Eurozone rate of inflation is 5%.

The inflation differential between the two is

3.5% - 5% = -1.5%

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Combining the changes on both sides we obtain:


This way of expressing PPP is called relative PPP, and it implies that the rate of depreciation of the nominal exchange rate equals the difference between

the inflation rates of two countries.

Unlike absolute PPP, relative PPP predicts a relationship between changes in prices and changes in exchange rates, rather than a relationship between their levels.

aldifferentiInflation

,,

rate exchange nominal theofondepreciati of Rate

,€/$

,€/$

tEURtUS

t

t

E

E


Relative PPP is derived from absolute PPP. Hence, the latter always implies the former: if absolute PPP holds, this implies that relative PPP must hold also.

But the converse need not be true: relative PPP does not necessarily imply absolute PPP (if relative PPP holds, absolute PPP can hold or fail).

For example, imagine that all goods consistently cost 20% more in country A than in country B, so absolute PPP fails; however, it still can be the case that the inflation differential between A and B (say, 5%) is always equal to the rate of depreciation (5%).

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Evidence for PPP in the Long Run

Inflation Differentials and the Exchange Rate, 1975-2005

This scatterplot shows the relationship between the rate of exchange rate depreciation against the U.S. dollar and the inflation differential against the United States over the long run, for a sample of 82 countries. The correlation between thetwo variables is strong and bears a close resemblance to the prediction of PPP that all data points would appear on the 45-degree line.

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Evidence for PPP in the Short Run

Exchange Rates and Relative Price Levels

Data for the U.S. and the UK for 1975 to 2010 show that the exchange rate and relative price levels do not always move together in the short run. Relative price levels tend to change slowly and have a small range of movement; exchange rates move quickly and experience large fluctuations. Therefore, relative PPP does not hold in the short run. It is a better guide to the long run, and we can see that the two series do tend to drift together over the decades.

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Research shows that price differences—the deviations from PPP—can be quite persistent.

Estimates suggest that these deviations may die out at a rate of about 15% per year. This kind of measure is often called a speed of convergence.

Approximately half of any PPP deviation still remains after four years: economists would refer to this as a four-year half-life.

Such estimates provide a rule of thumb that is useful as a guide to forecasting real exchange rates.

How Slow Is Convergence to PPP?

For exampleSuppose the home basket costs $100 and the foreign basket $90, in home currency.

Home’s real exchange rate is 0.900, and the home currency is overvalued, with foreign goods less expensive than home goods.

The deviation of the real exchange rate from the PPP-implied level of 1 is −10% (or −0.1).

Our rule of thumb tells us that next year 15% of this deviation will have disappeared (i.e., 0.015), so the new deviation will be only −0.085,

meaning that Home’s real exchange rate would be forecast to be 0.915 after one year and thus end up a little bit closer to 1, after a small depreciation.

Similarly, after four years, all else being equal, 52% of the deviation (or 0.052) would have been erased, and the real exchange rate would by then be 0.952, only −5% from PPP.

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Forecasting When the Real Exchange Rate Is Undervalued or Overvalued

When relative PPP holds, forecasting exchange rate changes is simple: just compute the inflation differential.

But how do we forecast when PPP doesn’t hold, as is often the case? Knowing the real exchange rate and the convergence speed may still allow us to construct a forecast of real and nominal exchange rates.

The rate of change of the nominal exchange rate equals the rate of change of the real exchange rate plus home inflation minus foreign inflation:

aldifferentiInflation

,,

rate exchange real theofon depreciati of Rate

,/

,/

rate exchange nominal theofon depreciati of Rate

,€/$

,€/$

tEURtUS

tEURUS

tEURUS

t

t

q

q

E

E

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Transaction costs. Include costs of transportation, tariffs, duties, and other costs due to shipping and delays associated with developing distribution networks and satisfying legal and regulatory requirements in foreign markets. On average, they are more than 20% of the price of goods traded internationally.

Non-traded goods. Some goods are inherently non-traded; they have infinitely high transaction costs. Most goods and services fall somewhere between traded and non-traded.

What Explains Deviations from PPP?

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Imperfect competition and legal obstacles. Many goods are not simple undifferentiated commodities, as LOOP and PPP assume. Differentiated goods create conditions of imperfect competition because firms have some power to set the price of their good, allowing firms to charge different prices not just across brands but also across countries.

Price stickiness. Prices do not or cannot adjust quickly and flexibly to changes in market conditions.

What Explains Deviations from PPP?

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Home of the undervalued burger?

The Big Mac Index

For more than 20 years, The Economist has gauged the over- or undervaluation of a currency against the U.S. dollar by comparing the relative prices of a Big Mac in a common currency, and expressing the difference as a percentage deviation from one:

11Index Mac BigMac Big

US

Mac Big

localcurrency $/localMac Big

P

PEq

For example

The average price of a Big Mac in the US in 2012 was $4.33.

In Buenos Aires it was 19 pesos, which, at an actual exchange rate of 4.57 pesos per dollar, worked out to be $4.16 in U.S. currency, or 4% less than the U.S. price.

So the peso was 4% undervalued against the U.S. dollar according to this measure, and

Argentina’s exchange rate would have had to appreciate to 4.39 pesos per dollar to attain the level implied by a burger-based PPP theory.

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Similarly In Rio de Janeiro a Big Mac cost 10.08 reais, or $4.94 in U.S. currency at the prevailing exchange rate of 2.04 reais per dollar,

making the Brazilian burgers 14% more expensive than their U.S. counterparts.

To get to its PPP-implied level, and put the burgers at parity, Brazil’s currency would have needed to depreciate to 2.33 reais per dollar.

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The Big Mac Index

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The Big Mac Index

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The Big Mac Index

Economy & Finance

Arbitrage, money and purchasing power