CHAPTER 6 - INTERNATIONAL PARITY RELATIONSHIPS
We now look at Int'l. Parity Relationships, starting with the Law of One Price (LOP), extended to:
Purchasing Power Parity (PPP) and Interest Rate Parity (IRP). These parity relationships help us to understand: 1) how ex-rates are determined, and 2) how to forecast ex-rates.
Int'l. Parity is based on EMH (Efficient Market Hypothesis). FX/securities markets are efficient when:
1) securities/FX are priced efficiently reflecting all currently available information, and 2) no arbitrage
Arbitrage: Riskless, certain profit opportunities by exploiting price discrepancies. Simultaneously
buying and selling mispriced securities/FX to make a guaranteed, riskless profit without any
investment. "Picking up dimes with a bulldozer." Example: triangular arbitrage. Int'l. parity conditions exist when there are no arbitrage opportunities and markets are in
equilibrium. "No $100 bills lying on the sidewalk."
Law of One Price (LOP): P
= S ($/?) P, where D F
P = Domestic Price ($) D
P = Foreign Price (?) F
S ($/?) = spot ex-rate.
Example: Gold in U.S. is $579.50/oz., gold in U.K. = ?305 and S= $1.9000/?
In USD: ?305 x $1.9000/? = $579.50, Gold is selling in both countries for the same price in USD
In BP: $579.50/oz. ? $1.9000 = ?305/oz, Gold is selling in both countries for the same price in BP
If Law of One Price (Price Equalization Principle) did not hold, arbitrage would be possible, and would
quickly restore parity. For example, what if gold in U.K was $575? What if gold in US was ?300?
INTEREST RATE PARITY (IRP)
IRP: “No Arbitrage condition” when int'l. financial markets (FX and money markets) are in
equilibrium. Assuming free movement of capital, int'l. financial markets should be efficient. "Smell of
profits" eliminates any discrepancies. Covered Interest Rate Parity = Parity conditions in fin. mkts., when forward markets are used to eliminate or "cover" any FX risk.
Example: U.S. investor has $1 to invest for one year. You consider two strategies: 1) Invest in U.S.
treasury securities at
i, the domestic interest rate, for one year; or 2) Invest in foreign U.K. treasury $
securities at i, and hedge FX risk by selling maturity value of ?s forward one year. ?
In U.S., your payoff (maturity value) in one year will be: $1(1 + i) $
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BUS 466/566: International Finance – CH 6 Professor Mark J. Perry
In equilibrium this should be the same as your payoff in U.K.
In U.K., your investment strategy involves:
1. Sell $1 for ?s to get $1 ? S($/?) pounds. (We assume that S = S($/?)).
2. Invest ?s at U.K. int. rate ( i) with payoff = $1/S x (1 + i) ??
3. Sell ?s forward at F ($/?) for the maturity value of the UK investment, to get a guaranteed amount 360
For either investment, you start and end with U.S. dollars. For Strategy #2, you have completely
hedged ("covered") FX risk with the forward contract.
The Interest Rate Parity (IRP) condition would be:
(1 + i) = (F / S) (1 + i) $?
IRP is an application of the Law of One Price (LOP) to financial securities, says that two identical
securities (e.g. Treasury securities or bank CDs) should have the same return, after accounting for the
ex-rates (S and F). We need the F rate here because we have added the time dimension, in this case
one year into the future.
Example: i= 5%; i = 8%; F = $1.4583/? and S = $1.50/? $ ?
IRP Holds: (1.05) = ($1.4583 / $1.50) x 1.08
Invest $1000 in U.S.: $1000 x 1.05 = $1050 in 1 year.
Invest $1000 in U.K.: $1000 ? $1.50/? = ?666.6667 x 1.08 = ?720 x $1.4583333/? = $1050 in 1 year.
One of the reasons IRP should hold is because of Covered Interest Arbitrage (CIA), no risk, no net
investment arbitrage when IRP does not hold. Covered Interest Arbitrage (CIA) involves: 1) Borrow
$s in U.S. at
i, and buy UK pounds at S in spot market, 2) Invest (lend) in UK at i, 3) Sell pounds $?
forward at F, to cover ex-rate risk. No investment, no risk arbitrage opportunities if IRP does not hold.
See Example 6.1 (p. 135). i = 8% and i = 5%. S = $1.50/? and F = $1.48/?. IRP does not hold and ?$can be exploited by CIA.
Logic: Nominal interest rates are 3% higher in U.K. (8%) than U.S.(5%). If IRP holds, what would we
expect will happen to the ?? BP should depreciate by approx. 3% if IRP holds.
%CHG = (F - S) / S x 100.
($1.50 - 1.48) / $1.50 x 100 = -1.333% (or use %CHG function on calculator)
British Pound is expected to depreciate by only -1.33% instead of 3%, and is selling at a 1.33%
discount in the forward market. Therefore, expected covered return in U.K. to a U.S. investor would be
8% - 1.33% ? 6.667% (U.K.) vs. 5% (U.S.).
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BUS 466/566: International Finance – CH 6 Professor Mark J. Perry
Effective Return to U.S. Investor = i + % Appreciation Foreign Currency F
Effective Return to U.S. Investor = i - % Depreciation Foreign Currency F
Logic: When investing in a foreign market you are making 2 simultaneous investments: 1) the foreign security, and 2) the foreign currency.
We can also check IRP formula:
1.05 ?=? (1.48/1.50) (1.08) = 1.0656
1.05 < 1.0656
5% < 6.56%
Effective one-year return to a U.S. investor in U.K. (6.56%) is higher than return in U.S. (5%) by more than 1.5%.
1. Borrow $1m in U.S. at 5%, promise to pay $1.05m back in one year.
2) Buy $1m worth of BP in spot market at S($1.50/?) for ?666,667 ($1m ? $1.50/?).
3. Invest ?666,667 in U.K. at 8% to get guaranteed ?720,000 payoff in one year (?666,667 x 1.08).
4. Enter into a 1 yr. forward contract to sell ?720,000s forward at $1.48/?, for $1,065,600 guaranteed in one year (?720,000 x $1.48/?).
5. Pay back $1,050,000 on the loan in U.S., and make $15,600 arbitrage profit.
No risk, no investment, arbitrage strategy, see CF diagram, p. 136, Exhibit 6.2.
What will happen over time?
1. Int. rates will ____ in U.S. due to borrowing pressure. Demand for Credit goes up.
2. Int. rates will ____ in U.K. due to buying pressure for bonds. Bond prices rise, int. rates fall. 3. ? will _________ in spot market due to buying pressure, S will rise.
4. ? will _________ in the forward market, due to selling pressure, F will fall.
The difference between the two int. rates (3%) will narrow, and the difference between the S and F will widen (the forward discount for ? will increase from 1.33%), until IRP is restored, possibly at a
forward discount of 2% for the ?, until the int. rate spread is EXACTLY equal to the %CHG in ?. For example, suppose interest rates end up around 5.5% in U.S. and 7.5% in U.K., and the ? sells at a
forward discount of 2% in the Forward Mkt. (S = $1.505, F = $1.4749). In that case, your effective
return is about 5.5% in EITHER country, and IRP is restored, partly by: a) a decrease in the interest rate differential and partly by: b) an increase in the forward premium.
Another way to view IRP:
? i + (F - S) / S $?
(i - i) ? (F - S) / S (Forward discount or premium for the ?) $?
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Shows that int. rates (bond prices) are directly linked to S and F ex-rates, and says that the difference in
interest rates should be equal to the forward discount or premium for FX.
The above equality can be represented graphically, IRP line on page 136 (Exhibit 6.3). Note: Units for
both axes are %.
Point A represents the previous example. Int. rates are 3% higher in U.K. than U.S., so that the
? should depreciate by 3% and be selling at a 3% forward discount according to IRP, however it is
actually selling at a 1.33% forward discount, representing profit opportunities in U.K. Anything above
the IRP line represents profits by either: a) investing in U.K. instead of U.S., or by b) borrowing in U.S.
and lending in UK (arbitrage).
Anything below the IRP line represents profit opportunities by either: a) investing in U.S., or b)
borrowing in U.K. and lending in U.S. Point B: U.S. interest rates are 4% higher than U.K., so U.S.
dollar should depreciate by 4% and pound appreciate by 4%. However, if the dollar was actually
selling at a 2% discount (pound at 2% premium) in the forward market, U.S. investment would be very
attractive. You could borrow in U.K., invest in U.S. and make money.
See CIA Example 6.2 (p. 137) for a 3-month period using interest rates in Germany and the ?. Interest
rates are usually quoted at annual rates, so an adjustment must be made. Also, IRP formula (6.1) is
based on American terms ($1.50/?), and the ? is quoted here in European terms (?/$).
3-month Interest Rates: i
= 2% (U.S. based on 8% annual) and i = 1.25% (Germany based on 5%) $?
Convert ex-rates on p. 137 to American terms to check IRP:
S = $1.250/? and F = $1.2510/?, dollar selling at 3
1.02 ?=? (1.2510/1.250) (1.0125)
IRP: 1.02 > 1.0133, 3-month return is higher in U.S. than Germany. Strategy: Invest in US, or use
CIA by borrowing in Germany, and invest in U.S. to make money.
According to the difference in interest rates (0.75%), the $ should be selling at a 3-month 0.75%
forward discount for IRP to hold. However, the dollar is selling in the forward market at only about a
0.08% discount [(F - S) / S]. A German investor can get 1.25% yield in Germany vs. a 2% - .08% =
1.92% in the U.S., starting by converting euros for dollars, investing in U.S. at 2%, and selling the $s
forward at F
to get ?s back in 3 months (start and end with euros). 3
1. Borrow ?800,000 (equal to $1m) in Germany @ 1.25%, promise to pay ?810,000 in 3 months.
2. Sell ?800,000 for $ at $1.2500/? = $1m
3. Invest $1m for three months in U.S. @ 2% to get $1,020,000 in three months.
4. Sell $1,020,000 forward @ $1.2510/? = ?815,347.7218 ($1,020,000 ? $1.2510) 5. Pay back loan of ?810,000 and end up with ?5,347.7218 arbitrage profit (or $6690 @ F = $1.2510, ?5,347.7218 x $1.2510/?).
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Note: This would represent a point below the line on Exhibit 5.3. Adjustment would take place by a
decrease in the interest rate differential and an increase in the forward discount for the dollar. Example:
i = 1.75% and i = 1.50%, and $ (?) sells at forward discount (premium) of .25%. German investor $?
gets effective return of 1.5% in either country, U.S. investor gets 1.75% effective return in either
US: 1.75% - .25% = 1.50%, same as 1.50% in Germany
IRP and EX-RATE DETERMINATION
IRP helps explain ex-rate determination, by linking interest rates and ex-rates. We can rearrange the
S ($/?) = (1 + i
) x F ?
(1 + i) $
Spot ex-rates ($/?) are partly determined by relative interest rates, i.e., the interest rate in U.K. (and
other countries) relative to interest rates in U.S. From the formula above, we can see that if interest
rates increase in the U.K. (U.S.), the ? ($) will appreciate (holding F constant) as capital flows to the
countries where interest rates are increasing, especially if the real interest rate is increasing.
Forward rates also influence ex-rates. We can express F as: t+1
F= E (S | I ) t+1 t+1t
which says that the Forward Rate (F) is the Expected (E) future Spot Rate when the forward contract t+1matures in the future at time t+1, conditional upon all current and relevant information now, at time t
(I). What relevant information?? t
Combining the two equations above, we have:
S = (1 + i) x E (S | I ) t?t+1t
(1 + i) $
We conclude that: 1) Expectations about the future spot rate, influence today's spot rate. If the
expected spot rate goes up, the current spot rate (S) goes up now. Expectations drive the spot and
2) Information (I) drives ex-rates. Information changes daily, and forward and spot rates change daily
as information changes. Ex-rates are dynamic and volatile. EMH says that prices (ex-rates) reflect
ALL currently available information, and the only thing that changes prices is NEW information.
Conclusion: Expectations and Information are important determinants of ex-rates, Spot and Forward
markets, which are extremely dynamic markets.
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We can also say that: (i - i) = E(e), $?where E(e) is the expected rate of change in S, or the expected percentage (%) change. The relative
interest rate differential between two countries should reflect the expected percentage change in S. If
one-year interest rates are 3% higher (lower) in UK, we expect the Pound to depreciate (appreciate) by
(5% - 8%) = -3% Interest rates are higher in U.K. than U.S., ? is expected to depreciate by 3% over yr.
(7% - 4%) = +3% Interest rates are higher in U.S., ? is expected to appreciate by 3% over the year.
Current example using current market data (9/24/06): Does IRP currently hold?
S = $1.2788 / ?
= $1.2984 / ? 360
LIBOR = 5.31% (1-year rate for USD)
EURIBOR = 3.725% (1-year rate for Euros)
Calculation for IRP:
WHY MIGHT IRP NOT ALWAYS HOLD?
Although it should and does hold most of the time, why might IRP not always hold exactly, i.e. how to
explain deviations from IRP.
1. Transaction costs. We have so far assumed that transaction costs = 0. Deviations from IRP could be explained by transaction costs. For example, a) the borrowing and lending interest rate are NOT
usually the same, as we assumed in the previous examples of CIA. Interest rates are usually higher for
borrowing than for lending, e.g., commercial banks. Therefore, higher interest rates for borrowing may
eliminate or exceed any potential arbitrage profits.
b) Also, there are transaction costs (commissions, bid-ask spreads, fees, etc.) to buy/sell currency and
forward contracts and securities (bank CDs), which we have ignored, these may reduce or eliminate
Therefore, the IRP line should have a band around it to reflect transactions costs, see Exhibit 5.4, p.
139. Deviations within the shaded area (Point D) do not represent arbitrage opportunities, only deviations outside the shaded area (Point C).
2. Capital controls that limit, restrict or ban cross border capital flows (in or out of a country), can create significant barriers to intl. arbitrage, resulting in possible deviations from IRP. See Japan story
on pages 140-141. Between 1978-1979, Japan restricted capital inflows to prevent Yen from
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appreciating. Why? Deviations from IRP resulted in 1978 and 1979, but did not reflect unexploited
arbitrage opportunities. See Exhibit 6.5 on p. 140.
PURCHASING POWER PARITY (PPP) PPP is the Law of One Price (LOP) applied to a standard commodity basket.
P= S x P $ ?
where Pis the domestic price level (CPI), Pis the foreign price level (CPI), and S = $/? ex-rate. PPP $ ?
says that the dollar price of the commodity basket in the U.S. should be equal to the dollar price of the
same commodity basket purchased in U.K. PPP is the LOP applied to a standard commodity basket,
which should be priced the same in all countries, when measured in a common currency. What if
prices were higher on average in U.S. than in the U.K.?:
P > S x P $ ?
Commodities would be purchased in ____ and sold in the _____, which would put upward pressure on:
a) the _____ (S goes up) and b) _____ prices (go up), restoring PPP.
Or we can also say:
S = P $
where the spot rate (S) is the ratio between the two country's price levels. If domestic U.S. prices rise
(P goes up), S should go up, meaning that the _____ is appreciating and the ____ is depreciating. . $
Q: What would cause U.S. prices to be rising?
Absolute PPP links ex-rates to the relative price levels (P) in two countries
Relative PPP looks at Relative Inflation Rates (%): e = %INF - %INF , usF
where e is the % change in the ex-rate.
If INF > INF, then the dollar will depreciate and the pound will appreciate (S will get bigger and e usF
will be positive).
If INF < INF, then the dollar will appreciate and the pound will depreciate (S will get smaller, and usF
e will be negative).
Therefore, ex-rates are linked to relative prices and relative inflation rates. If annual inflation in the
U.S. is 6% and only 4% in U.K., then the dollar should depreciate by 2% annually and the pound will BUS 466/566: International Finance – CH 6 Professor Mark J. Perry
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appreciate by 2%. If inflation in the U.S. is 6% and 9% in U.K., the pound should depreciate by 3%
and the dollar should appreciate by 3%.
Whichever country has the higher (lower) inflation rate will experience a currency depreciation
BIG MAC PPP, see p. 142-143, annual report by The Economist to test for PPP/LOP. If LOP holds,
the price of a Big Mac should sell for the same price around the world, measured in dollars using the
actual spot ex-rate. Procedure: Big Mac sells for $2.49 in U.S., take $2.49 to countries around the
world, convert $2.49 to local currency, and go buy a Big Mac. If Big Macs are cheap (expensive), the
dollar is overvalued (undervalued), according to PPP or LOP.
Example: Take $2.49 to Argentina, convert to 7.79 pesos ($2.49 x pesos3.13/$), compare 7.79 pesos to
Big Mac price in pesos: only 2.50 pesos, so you could buy 3 Big Macs, dollar (peso) is overvalued
(undervalued) by 68%. Take $2.49 to Switzerland, convert to SF4.134 (2.49 x 1.66), compare to Big
Mac price of SF6.30, dollar is undervalued by +53% (4.134 vs. 6.30).
EVIDENCE on PPP and the REAL EXCHANGE RATE:
e = %INF
- %INF, or usF
(INF - INF) - e = 0 if PPP holds USF
When PPP does NOT hold, the real, effective exchange rate changes, and affects a country’s
international competitiveness and trade balance.
The real, effective exchange-rate for the USD rises IF US inflation (5%) exceeds inflation abroad
(3.5%), and the dollar doesn’t depreciate by the full inflation differential of 1.5%. In that case, the real,
effective ex-rate value of the USD has risen, making the U.S. exports less competitive.
The real, effective exchange-rate for the USD falls IF US inflation (5%) exceeds inflation abroad
(3.5%), and the dollar depreciates by more than the full inflation differential of 1.5%. In that case, the
real, effective ex-rate value of the USD has fallen, making the U.S. more competitive.
Example: Inflation in U.S. is 5%, inflation in U.K. is 3.5%, we would expect e = +1.5%. The dollar
should depreciate by 1.5% and the pound should appreciate by 1.5%, according to PPP. If the dollar
actually depreciates by more than 1.5%, i.e., by more than predicted by PPP, it strengthens the
competitiveness of U.S. industries in world markets (below line in Exhibit 6.6 on p. 145) because the
“real, effective exchange rate” has fallen. On the other hand, if the dollar depreciates by less than 1.5%,
the real, effective ex-rate rises, making the US less competitive.
Or, assume that U.S. inflation is 2% and U.K. inflation is 5%, and the dollar is expected to appreciate
by +3%. If dollar actually appreciates by more than 3%, it weakens our competitiveness in world
markets (mid-1980s), because the real, effective exchange rate has increased. If the dollar appreciates
by less than the full 3%, the real, effective ex-rate decreases, making the US more competitive.
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Exhibit 6.6 shows that ex-rates (indexes) often deviate from PPP, and are often above and below the
rate predicted by PPP. Example: US dollar was much stronger in mid-1980s than justified by PPP. Our inflation rate was not that much lower than other countries, and the $ still got much stronger
than PPP would explain. (Appreciation of the dollar was not explained by low U.S. inflation compared
to inflation in other countries, see example above.) During this period, U.S. manufacturers and
exporters were complaining that the dollar was too strong, their products were not competitive globally.
Why doesn’t PPP always hold?
PPP is just the LOP extended to a standard commodity basket. For PPP to hold, the commodity basket
would have to be: a) consistent across countries, b) all goods in basket would have to be tradable so
that arbitrage would be possible, and c) prices would have to be perfectly flexible (no price controls).
Not all of these conditions are met, so PPP does not always hold. For example, see world prices in
Exhibit 6.7 on p. 146. The standard CPI commodity basket is not the same in U.S. vs. Japan vs. U.K.,
and not all goods are tradable (e.g. haircuts) and not all prices are perfectly flexible. Also, there are
transportation costs, transactions and possible trade barriers (tariffs, quotas) that would prevent PPP
from strictly holding.
PPP would hold with "frictionless trading." Holds most closely with homogenous, traded commodities,
(Treasury bills, gold, oil, wheat, steel, bank CDs, etc.)
SUMMARY OF PPP:
1. More of a long run condition than short run.
2. Can be used to tell if a currency is over or under-valued.
3. Int'l. comparisons are more accurate using PPP-ex-rates vs. mkt ex-rates. See Exhibit 6.8 page 147.
To compare GDP across countries, we have to convert to a common currency, like dollars. However
using market rates can distort the comparison. If the dollar (pound) is under/over valued, then
converting into dollars may be distorted.
For example, according to PPP, China's currency is artificially undervalued (and $ is artificially
overvalued) at the market ex-rate, lowering the value of China's GDP in the first column ($1.41T),
ranking 7th. Adjusting for PPP value of China's currency, they are second ($6.43T). India and Brazil
also rise in the rankings because their currencies are undervalued according to PPP, and Japan,
Germany, France, U.K. and Italy fall. The ? and ? are OVERVALUED according to PPP. Also, prices
are cheaper in China and India than in U.S. for the same goods and services, which distorts
international comparisons at market ex-rates.
FISHER EFFECTS (PARITY CONDITION)
Nominal Int Rate
ei = r (real rate) + Π (Expected inflation) Fisher Equation (p. 147) us
Example: One-year nominal interest rates in U.S. are 5%, the one-year real rate (ρ) is 2%, and the one-
year expected rate of U.S. inflation is 3%. This would hold for other countries as well.
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If the real interest rate (r) is constant, changes in nominal interest rates reflect changes in expected einflation (Δi = ΔП). Fisher Effect: One-to-one relationship between changes in inflation expectations and changes in nominal interest rates. Assume that the real interest rate is the same internationally, due
to free capital flows. In that case the relative difference in nominal interest rates reflects relative
differences in expected inflation:
(i - i) = E(П - П) $?$?
Assuming a constant real interest rate (r), the differences in nominal interest rates between US and UK
reflect differences in expected inflation between U.S. and U.K.
With unrestricted capital flows, we can assume that the real rate of interest is constant internationally.
We would have the following additional parity conditions (see page 148):
International Fisher Effect (IFE):
E (e) ? ( i
- i ) $?
The difference in nominal interest rates (i - i) between U.S. and U.K. reflects the expected change in $?the ex-rate. For a one year period, if i = 5% and i= 7%, the dollar (pound) is expected to appreciate $? (depreciate) by 2% over the next year. If i = 5% and i= 4%, the pound (dollar) is expected to $? appreciate (depreciate) by 1% over the next year.
Forward Expectations Parity (FEP): (F - S) / S = E (e)
Any forward premium or discount will be equal to the expected change in the ex-rate. If the pound is
expected to depreciate (appreciate) by 2% over the next year, it will be priced at a 2% forward discount
(premium) in the forward market.
SUMMARY OF INTERNATIONAL PARITY CONDITIONS
See Exhibit 6.9, p. 148: The difference in nominal interest rates (%) reflects the difference in inflation
rates (%), which is also equal to the expected change in S (%) or E (e), which is also reflected in the
forward premium or discount (%) or (F – S) / S.
EXAMPLE 1: If one-year T-bills = 8% in U.S. and 6% in U.K., the +2% difference reflects the
expectation of 2% higher inflation in U.S. In addition, the BP (USD) is expected to appreciate
(depreciate) by +2%, which is reflected in a forward premium (discount) of +2% for the BP (USD).
EXAMPLE 2: Assume that real rate r = 2%, Expected inflation is 2% in U.S. and 4% in
U.K. Nominal interest rates will be 6% in U.K. and 4% in the U.S. for one-year T-bills, and (i
- i) = $?
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