International Parity Relationships and Forecasting Foreign Exchange Rates
[1]
Chapter
International Parity Relationships and
Forecasting Foreign Exchange Rates
Chapter Outline
6.1 Interest Rate Parity
6.2 Purchasing Power Parity
6.3 The Fisher Effects
6.4 Forecasting Exchange Rates
6.0 INTRODUCTION
It is important to understand the forces driving exchange rate changes as these changes
would affect investment and financing opportunities. This chapter examines several key
international parity relationships, such as interest rate parity and purchasing power
parity that have profound implications for international financial management. Some of
these are, in fact, manifestations of the law of one price that must hold in arbitrage
equilibrium. An understanding of these parity relationships provides insights into (l)
how foreign exchange rates are determined, and (2) how to forecast foreign exchange
rates.
The term arbitrage can be defined as the act of simultaneously buying and selling the
same or equivalent assets or commodities for the purpose of making certain,
guaranteed profits. As long as there are profitable arbitrage opportunities, the market
cannot be in equilibrium.
The market can be said to be in equilibrium when no profitable arbitrage opportunities
exist. Such well-known parity relationships as interest rate parity and purchasing
power parity, in fact, represent arbitrage equilibrium conditions.
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International Parity Relationships and Forecasting Foreign Exchange Rates
6.1 INTEREST RATE PARITY
Interest rate parity (IRP) is an arbitrage condition that must hold when international
financial markets are in equilibrium. Interest rate parity (IRP) holds forward premium
or discount should be equal to the interest rate differential between two countries.IRP
represents an arbitrage equilibrium that should hold in the absence of barriers to
international capital flows.
If IRP is violated, one can make profit by borrowing in one currency and lending in
another with exchange risk hedged via forward contract.IRP implies that in short run,
the exchange rate depends on (a) relative interest rates between two countries (b) the
expected future exchange rate. Arbitrage equilibrium then would dictate that the future
dollar proceeds (or, equivalently, the dollar interest rates) from investing in the two
equivalent investments must be the same, implying that (1 +
) =
(1 + ) or
F=
Example 1 Covered Interest Arbitrage (Page 141-142)
Suppose that the annual interest rate is 5 percent in the United States and 8 percent in
the U.K., and that the spot exchange rate is $1.80/£ and the forward exchange rate, with
one-year maturity, is $1.78/£. In terms of our notation, i$ = 5%, i£ = 8%, S = $1.80, and
F = $1.78. Assume that the arbitrager can borrow up to $1,000,000 or £555,556, which
is equivalent to $1,000,000 at the current spot exchange rate.
Solution
Step 1: Checking if IRP is holding under current market conditions F=
$1.78 > 1.80
.
.
and $1.78 > $1.75 or (1 +
) = (1 + )in which 1.05 < 1.068
Clearly, IRP is not holding, implying that a profitable arbitrage opportunity exists.
Since the interest rate is lower in the United States, an arbitrage transaction should
involve borrowing in the United States and lending in the U.K.
The arbitrager can carry out the following transactions:
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1. In the United States, borrow $1,000,000. Repayment in one year will be $1,050,000
= $1,000,000 x 1.05.
2. Buy £555,556 spot using $1,000,000.
3. Invest £555,556 in the U.K. The maturity value will be £600,000 = £555,556 x
1.08.
4. Sell £600,000 forward in exchange for $1, 068,000 = (£600,000)($1.78/£).
5. The arbitrage profit $1,068,000 - $1,050,000=$18,000
How long will this arbitrage opportunity last? A simple answer is: only for a short
while. As soon as deviations from IRP are detected, informed traders will immediately
carry out CIA transactions. As a result of these arbitrage activities, IRP will be restored
quite quickly. To see this, let's get back to our numerical example, which induced
covered interest arbitrage activities. Since every trader will (1) borrow in the United
States as much as possible, (2) lend in the U.K., (3) buy the pound spot, and, at the
same time, (4) sell the pound forward, the following adjustments will occur to the
initial market condition:
1. The interest rate will rise in the United States (i$ ↑).
2. The interest rate will fall in the U.K. (i£↓ )
3. The pound will appreciate in the spot market (S↑).
4. The pound will depreciate in the forward market (F ↓).
Example 2
Consider the following set of foreign and domestic interest rates and spot and forward
exchange rates and the amount of money you may borrow $1,000.
Spot exchange rate(S)
$2.00
360-day forward rate(F)
$2.01
U.S. discount rate(Ih)
British discount rate (If)
3%
2.49%
Solution
Step One: A trader with $1,000 could invest in the U.S. at 3%, in one year and his
investment will be worth = $1,000 (1.03) = $1,030
Step Two: Exchange $1,000 for £500 at the prevailing spot rate of $2
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£1= $2
X= $1000 therefore, x= £500
Step Three: Invest £500 for one year at If = 2.49%,
a. The amount that will be received= 500(1+2.49%) = £512.45
b. Sign a forward contract at rate of $2.01
Step Four: Reconvert £512.45 back into dollars at the forward rate F= $2.01, the
£512.45 will be worth of $1,030.
Step Five: therefore, IRP holds and CIA is not possible
Example 3: Covered Interest Arbitrage
The spot exchange rate is $1.50/£ and the three-month forward exchange rate is
$1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per
annum in the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000.
a. Determine whether the interest rate parity is currently holding.
b. If the IRP is not holding, how would you carry out covered interest arbitrage?
c. Show all the steps and determine the arbitrage profit.
d. Explain how the IRP will be restored as a result of covered arbitrage activities.
Solution
S = $1.5/£
F = $1.52/£
Ih = 2.0%
If = 1.45%
Proceed = $1,500,000 or
£1,000,000.
a. (1+Ih) = (1+If)(F/S) = (1.0145)(1.52/1.50)
1.02 <1.0280, thus, IRP is not holding exactly.
b. Strategy
Step (1): Borrow $1,500,000 and repayment will be $1,530,000.
Repayment= 1,500,000 (1+0.02) = $1,530,000.
Step (2): convert $1,500,000, at S = $1.5
£1= $1.5
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International Parity Relationships and Forecasting Foreign Exchange Rates
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X= 1,500,000
Then x= £1,000,000
Step (3): Invest this amount £1,000,000 at the interest rate of 1.45%.
a. The amount that you will receive= £1,000,000 (1+0.0145) = £1,014,500.
b. Sign a forward contract with F = $1.52
Step (4): Reconvert this amount at £1,014,500 at rate of F = $1.52
£1= $1.52
£1,014,500 = x
Then x= $1,542,040
Step 5: Calculation of Arbitrage Profit
Arbitrage= $1,542,040-$1,530,000= $12040
Therefore the arbitrage profit will be $12,040
c. Followings are the arbitrage transactions described below.
i.
The dollar interest rate will rise.
ii.
The pound interest rate will fall.
iii.
The spot exchange rate will rise.
iv.
The forward exchange rate will fall.
These adjustments will continue until IRP holds.
Interest Rate Parity and Exchange Rate Determination
IRP has an immediate implication for exchange rate determination. An arbitrage in
equilibrium condition involves the (spot) exchange rate. The forward exchange rate is
an important factor in spot exchange rate determination. Under certain conditions the
forward exchange rate can be viewed as the expected future spot exchange rate.
Where St+1 is the future spot rate when the forward contract matures, and It denotes the
set of information currently available.
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Currency Carry Trade
Unlike IRP, the uncovered interest rate parity often doesn't hold, giving rise to
uncovered interest arbitrage opportunities. A popular example of such trade is provided
by currency carry trade. Currency carry trade involves buying a high-yielding currency
and funding it with a low-yielding currency, without any hedging.
6.2 PURCHASING POWER PARITY
When the law of one price is applied internationally to a standard commodity basket,
we obtain the theory of purchasing power parity (PPP). This theory states that the
exchange rate between currencies of two countries should be equal to the ratio of the
countries' price levels.
The exchange rate between two currencies should equal the ratio of the countries’ price
levels.
=
When the country’s inflation rises, the demand of its currency and exports decline and
consumers and firms increase imports. Both of these forces place downward pressure
on high inflation country’s currency. The purchasing power parity (PPP) theory
attempts to quantify the inflation exchange rate relationship.
There are two popular techniques:
1. Absolute PPP that states that similar products in different countries should be
priced equally when measured in common currency.
2. Relative PPP that accounts for imperfections like transportation costs, tariffs and
quotas. It states that the rate of price changes should be similar.
=
−
+
=
( +
)
ef = % ∆ of foreign currency value, IH = home inflation If = foreign inflation
IH > If , ef > 0 ⇒ FC appreciates ,IH < If , ef < 0 ⇒ FC depreciates
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Example 4
Ih= 5%
=
If = 3%
+
+
−
Interpretation
=
ef =?
+ .
+ .
−
= .
%
The foreign currency should appreciate 1.94% to high inflation of home country with
respect to foreign country. Price index of both countries should rise 5% according to
home country.
Example 5
IH = 4%
=
+
+
If = 7%
−
Interpretation
=
+ .
+ .
ef =?
−
=− . %
The foreign currency should depreciate 2.8% to high inflation of home country with
respect to foreign country. Price index of both countries rises 4% according to home
country. Price index of both countries rises 4% according to home country.
Purchasing power parity doesn’t hold.
Purchasing power parity doesn’t consistently occur because of confounding effect and
lack of substitutes for some traded goods.
i.
Confounding Effect
PPP theory presumes that exchange rates are completely driven by inflation
differentials between two countries but there are some other factors, such as:
a. Interest rate
b. Income level
c. Government controls
d. Change in expectation of future exchange
ii.
Lack of substitutes for traded goods
The idea behind PPP theory is as soon as prices become high in one country. The
consumers in another country buying imported goods and shift to domestic goods. This
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shift influences exchange rate. If substitute goods are not available domestically, the
consumers may not stop buying imported goods.
Evidence on PPP
1. PPP probably doesn’t hold precisely in the real world for a variety of reasons.
a. Haircuts cost 10 times as much in the developed world as in the developing world.
b. Film, on the other hand, is a highly standardized commodity that is actively traded
across borders.
c. Shipping costs, as well as tariffs and quotas can lead to deviations from PPP.
2. PPP-determined exchange rates still provide a valuable benchmark.
Implications of Purchasing Power Parity
If exchange rate changes satisfy PPP, competitive positions of countries will remain
unaffected following exchange rate changes. Otherwise, exchange rate changes will
affect relative competitiveness of countries. If a country’s currency appreciates
(depreciates) by more than is warranted by PPP that will hurt (strengthen) the country’s
competitive position in the world market.
6.3 FISHER EFFECTS
Fisher’s effect states an increase (decrease) in the expected rate of inflation will cause a
proportionate increase (decrease) in the interest rate in the country. The Fisher effect is
written as:
1 + i$ = (1 + $ ) × E(1 + $)
Where
$ is the equilibrium expected “real” U.S. interest rate
E($) is the expected rate of U.S. inflation
i$ is the equilibrium expected nominal U.S. interest rate
International Fisher Effect
International fisher effect (IFE) uses interest rates rather than inflation rates but it is
closely related with PPP theory because interest rates are highly correlated with
inflation rates. According to IFE theory, nominal interest rates contain real rate of
return and anticipated inflation.
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IFE theory states that currencies with high interest rates will depreciate because higher
interest rates reflect higher expected inflation. Investors who hope to capitalize on
higher foreign interest rates should earn return no better than what they have earned
domestically.
+
+
=
=
−
IH > If , ef > 0 ⇒ FC appreciates
IH < If , ef < 0 ⇒ FC depreciates ,
−
IH = home interest , If = foreign interest
Example 6
Ih = 11%
=
ef = ?
If = 12%
+
+
Interpretation
−
+ .
+ .
=
−
=− .
%
The foreign currency depreciates by 0.89% to make actual return on foreign deposit
equal to the 11 percent in the home country.
Example 7: International Fishers Effect and exchange rate
St = $0.85
=
=
+
+
( +
Interpretation
,
−
If = 4%,
=
)
∴
+ .
+ .
IH = 3% ,
−
=− .
St+1=?
%≈− .
= $0.85[1 + (−0.0196)] = $0.8415
The Canadian dollar should depreciate by 0.01 and the expected future spot rate of
CD$ is $0.8415
Example 8: International Fishers Effect and exchange rate
P#5: St = $0.90
=
1+
1+
=
IH=6%
−1=
(1 +
Interpretation
)
If=5%+6%=11%
1 + 0.06
− 1 = +0.045
1 + 0.11
⟹ $0.90[1 − 0.045] = $0.8595
The foreign currency depreciates by 0.045 and new spot rate is $0.8595
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6.4 FORECASTING EXCHANGE RATES
1. Efficient Markets Approach
Financial Markets are efficient if prices reflect all available and relevant
information.
2. Fundamental Approach involves econometrics to develop models that use a
variety of explanatory variables. This involves three steps:
a. step 1: Estimate the structural model.
b. step 2: Estimate future parameter values.
c. step 3: Use the model to develop forecasts.
3. Technical Approach
Technical analysis looks for patterns in the past behavior of exchange rates.
4. Performance of the Forecasters
a. Forecasting is difficult, especially with regard to the future.
b. As a whole, forecasters cannot do a better job of forecasting future exchange
rates than the forward rate.
c. The founder of Forbes Magazine once said: “You can make more money selling
financial advice than following it.”
Comparison among IRP, PPP and IFE
key variables of
Summarized
Theory
theory
theory
Interest rate parity (IRP)
Forward rate
Premium ( discount)
Interest rate
differential
Percentage change
of spot rate
Inflation rate
differential
The forward rate of one currency to another
will contain premium or discount that is
determined by differential in interest rates
between two countries. As a result, the
covered interest arbitrage will provide return
that in not higher than domestic return.
Purchasing power parity (PPP)
The spot rate of one currency with respect to
another currency will change in reaction to
differential in inflation rates between two
countries. Consequently, the purchasing
power for consumers when purchasing
goods in their own country should be similar
when importing goods from foreign country.
International Fisher Effect (IFE)
Percentage change
of Spot rate
The spot rate of one currency with respect to
another currency will change in reaction
differential in interest rates between two
countries consequently. The return on
foreign money market will be no higher than
return on domestic money market.
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PROBLEMS
Question One
Suppose that the treasurer of IBM has an extra cash reserve of $1,000,000 to invest for
six months. The six-month interest rate is 8% per annum in the U.S. and 6% per annum
in Germany. Currently, the spot exchange rate is DM1.60 per dollar and the six-month
forward exchange rate is DM1.56 per dollar. The treasurer of IBM does not wish to
bear any exchange risk. Where should he/she invest to maximize the return?
Ans: $16,410 and it is better to invest $1,000,000 in Germany with exchange risk hedging
Question Two
While you were visiting London, you purchased a Jaguar for £35,000, payable in three
months. You have enough cash at your bank in New York City, which pays 0.35%
interest per month, compounding monthly, to pay for the car. Currently, the spot
exchange rate is $1.45/£ and the three-month forward exchange rate is $1.40/£. In
London, the money market interest rate is 2.0% for a three-month investment. There are
two alternative ways of paying for your Jaguar.
a) Keep the funds at your bank in the U.S. and buy £35,000 forward.
b) Buy a certain pound amount spot today and invest the amount in the U.K. for
three months so that the maturity value becomes equal to £35,000.Evaluate each
payment method. Which method would you prefer? Why?
Ans:
a. The cost of Jaguar as of today is $48,489
b. $1,266
Question Three
Suppose that the current spot exchange rate is FF6.25/$ and the three-month forward
exchange rate is FF6.28/$. The three-month interest rate is 5.6% per annum in the U.S.
and 8.8% per annum in France. Assume that you can borrow up to $1,000,000 or FF
6,250,000.
a. Show how to realize a certain profit via covered interest arbitrage, assuming that
you want to realize profit in terms of U.S. dollars. Also determine the magnitude of
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arbitrage profit.
b. Assume that you want to realize profit in terms of French francs. Show the covered
arbitrage process and determine the arbitrage profit in French francs.
Ans: a) Arbitrage profit will be $3,118 b) Arbitrage profit will be FF19, 580
Question Four
In the issue of October 23, 1999, the Economist reports that the interest rate per annum
is 5.93% in the United States and 70.0% in Turkey. Why do you think the interest rate
is so high in Turkey? Based on the reported interest rates, how would you predict the
change of the exchange rate between the U.S. dollar and the Turkish lira?
Ans: The Turkish lira thus is expected to depreciate against the U.S. dollar by about 64%.
Question Five
As of November 1, 1999, the exchange rate between the Brazilian real and U.S. dollar
is R$1.95/$. The consensus forecast for the U.S. and Brazil inflation rates for the next
1-year period is 2.6% and 20.0%, respectively. How would you forecast the exchange
rate to be at around November 1, 2000?
Ans: R$2.29/$
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