International Finance

1. Eco 328
Arbitrage, money and purchasing power

2. 2
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

3. 3
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
   


depositseuroonreturnDollar
€/$
€/$
€
depositsdollaronreturnDollar
$ 11
E
F
ii 
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.
Arbitrage and Interest Rates
In equilibrium,

4. 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

5. 5
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 moving
money from pound deposits to mark deposits, but capital controls prevented them from freely doing so.
Financial Liberalization and Covered Interest Parity

6. 6
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

7. 7
• 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
e
E$/€
$/€€ /)1( Ei
$/€$/€€ /)1( EEi e

   


depositseuroon
returndollarExpected
€/$
€/$
€
depositsdollar
onreturnDollar
$ 11
E
E
ii
e

Arbitrage and Interest Rates

8. 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 i
E
E
i
e
 8

9. 9
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
i
EE e



Arbitrage and Interest Rates

10. 10
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 FEe
 €/$€/$ FEe


11. 11
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.

ondepreciatiofrateExpected
€/$
€/$
premiumForward
€/$
€/$
11 
E
E
E
F e

12. 12
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.

13. 13
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
Arbitrage and Interest Rates
𝑖$ = 𝑖€ + %Δ𝐸$
€
𝑒

14. 14
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

15. 15
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

16. 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.
16

17. 17
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:
The Law of One Price

$ingoodof
priceU.S.
$ingoodof
priceEuropean
€/$
U.S.versusEuropein
goodofpriceRelative
/ /)(
g
g
US
g
g
EUR
g
g
EURUS PPEq


€/$EWhere expresses the rate at which currencies can be
exchanged.

18. 18
We can rearrange the equation for price equality
The Law of One Price
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 €/$
 
pricesgoods’
ofRatio
rate
Exchange
€/$ / g
EUR
g
US PPE 

19. 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.
19

20. 20
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
basketof
priceU.S.
$in
expressed
basketof
priceEuropean
€/$
U.S.versus
Europein
basketof
priceRelative
/ /)( 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.

21. 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.
21

22. 22
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

23. 23
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

24. 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.
24

25. 25
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 €/$
 
levelspriceofRatiorateExchange
€/$ / 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.

26. 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.
26

27. 27
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.

28. 28
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
  
rateexchangenominaltheofondepreciatiofRate
,€/$
,€/$1,€/$
,€/$
,€/$
t
tt
t
t
E
EE
E
E 

 
 
levelspriceofRatiorateExchange
€/$ / EURUS PPE 

29. 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).
29

30. 30
How does our exchange rate model relate to inflation (the rate of
change of the price level)?
Relative PPP, Inflation, and Exchange Rate Depreciation
 
levelspriceofRatiorateExchange
€/$ / 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







 







 










    
,,
EuropeininflationofRate
,
,1,
in U.S.inflationofRate
,
,1,
,
,
,
,
)/(
)/(

31. 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%
31

32. 32
Combining the changes on both sides we obtain:
Relative PPP, Inflation, and Exchange Rate Depreciation
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
,,
rateexchangenominaltheof
ondepreciatiofRate
,€/$
,€/$
tEURtUS
t
t
E
E



33. Relative PPP, Inflation, and Exchange Rate Depreciation
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%).
33

34. 34
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 the
two variables is strong and bears a close resemblance to the prediction of PPP that all data points would appear on the 45-
degree line.

35. 35
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.

36. 36
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?

37. For example
Suppose 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.
37

38. 38
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
,,
rateexchangereal
theofondepreciatiofRate
,/
,/
rateexchangenominal
theofondepreciatiofRate
,€/$
,€/$
tEURtUS
tEURUS
tEURUS
t
t
q
q
E
E





39. 39
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?

40. 40
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?

41. 41
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:
11IndexMacBig MacBig
US
MacBig
localcurrency$/localMacBig










P
PE
q

42. 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.
42

43. 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.
43

44. 44
The Big Mac Index

45. 45
The Big Mac Index

46. 46
The Big Mac Index