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International Parity Relationships
and Forecasting Exchange Rates
Chapter Six
Copyright © 2012 by the McGraw-Hill
Companies, Inc. All rights reserved.
Chapter Outline
 Interest Rate Parity
–
–
–
–
Covered Interest Arbitrage
IRP and Exchange Rate Determination
Currency Carry Trade
Reasons for Deviations from IRP
 Purchasing Power Parity
– PPP Deviations and the Real Exchange Rate
– Evidence on Purchasing Power Parity
 The Fisher Effects
 Forecasting Exchange Rates
–
–
–
–
Efficient Market Approach
Fundamental Approach
Technical Approach
Performance of the Forecasters
6-2
Interest Rate Parity Defined
 IRP is a “no arbitrage” condition.
 If IRP did not hold, then it would be
possible for an astute trader to make
unlimited amounts of money by
exploiting the arbitrage opportunity.
 Since we don’t typically observe persistent
arbitrage conditions, we can safely assume
that IRP holds.
…almost all of the time!
6-3
Interest Rate Parity Carefully Defined
Consider alternative one-year investments for $100,000:
1. Invest in the U.S. at i$. Future value = $100,000 × (1 + i$).
2. Trade your $ for £ at the spot rate and invest
$100,000/S$/£ in Britain at i£ while eliminating any
exchange rate risk by selling the future value of the
British investment forward.
F$/£
Future value = $100,000(1 + i£)×
S$/£
Since these investments have the same risk, they must
have the same future value (otherwise an arbitrage would
exist).
F$/£
(1 + i£) ×
= (1 +Fi$) = S × (1 + i$)
$/£
$/£
S$/£
(1 + i£) 6-4
$1,000
S$/£
Alternative 2:
Send your $ on
a round trip to
Britain
IRP
Step 2:
Invest those
pounds at i£
Future Value =
$1,000
$1,000
IRP
Alternative 1:
Invest $1,000 at i$
$1,000×(1 + i$) =
Step 3: Repatriate
future value to the
U.S.A.
$1,000
S$/£
S$/£
 (1+ i£)
 (1+ i£) × F$/£
Since both of these investments have the same risk, they must have the
same future value—otherwise an arbitrage would exist.
6-5
Interest Rate Parity Defined
 The scale of the project is unimportant.
$1,000
 (1+ i£) × F$/£
$1,000×(1 + i$) =
S$/£
F$/£
× (1+ i£)
(1 + i$) =
S$/£
 IRP is sometimes approximated as:
i$ – i£ ≈ F – S
S
6-6
Interest Rate Parity Carefully Defined
 No matter how you quote the exchange rate
($ per ¥ or ¥ per $) to find a forward rate,
increase the dollars by the dollar rate and the
foreign currency by the foreign currency
rate:
1 + i¥
F¥/$ = S¥/$ ×
1 + i$
or
1 + i$
F$/¥ = S$/¥ × 1 + i
¥
…be careful—it’s easy to get this wrong.
6-7
IRP and Covered Interest Arbitrage
 If IRP failed to hold, an arbitrage would
exist. It’s easiest to see this in the form of
an example.
 Consider the following set of foreign and
domestic interest rates and spot and
forward exchange rates.
Spot exchange rate
360-day forward rate
S($/£) = $2.0000/£
F360($/£) = $2.0100/£
U.S. discount rate
i$ = 3.00%
British discount rate
i£ = 2.49%
6-8
IRP and Covered Interest Arbitrage
 A trader with $1,000 could invest in the U.S. at
3.00%. In one year his investment will be worth:
$1,030 = $1,000  (1+ i$) = $1,000  (1.03)
 Alternatively, this trader could:
1. Exchange $1,000 for £500 at the prevailing spot rate.
2. Invest £500 for one year at i£ = 2.49%; earn £512.45.
3. Translate £512.45 back into dollars at the forward rate
F360($/£) = $2.01/£. The £512.45 will be worth $1,030.
6-9
Other Choice:
Buy £500 at $2/£.
£1
£500 = $1,000×
$2.00
$1,000
One Choice:
Invest $1,000 at 3%.
FV = $1,030
Arbitrage I
£500
Step 2:
Invest £500 at
i£ = 2.49%.
£512.45 In one year £500
will be worth
Step 3:
£512.45 =
Repatriate to the
£500  (1+ i£)
U.S. at F360($/£)
= $2.01/£
$1,030
F£(360)
$1,030 = £512.45 ×
£1
6-10
IRP& Exchange Rate Determination
 According to IRP only one 360-day
forward rate F360($/£) can exist. It must be
the case that
F360($/£) = $2.01/£
 Why?
 If F360($/£)  $2.01/£, an astute trader could
make money with one of the following
strategies.
6-11
Arbitrage Strategy I
 If F360($/£) > $2.01/£:
1. Borrow $1,000 at t = 0 at i$ = 3%.
2. Exchange $1,000 for £500 at the prevailing
spot rate (note that £500 = $1,000 ÷ $2/£.);
invest £500 at 2.49% (i£) for one year to
achieve £512.45.
3. Translate £512.45 back into dollars; if
F360($/£) > $2.01/£, then £512.45 will be more
than enough to repay your debt of $1,030.
6-12
Step 2:
Buy pounds
£1
£500 = $1,000×
$2.00
$1,000
£500
Arbitrage I
Step 3:
Invest £500 at
i£ = 2.49%.
£512.45 In one year £500
will be worth
£512.45 =
£500 (1+ i£)
Step 4: Repatriate
to the U.S.
Step 1:
Borrow $1,000. More
F£(360)
$1,030 < £512.45 ×
than $1,030
Step 5: Repay
£1
your dollar loan
with $1,030.
If F£(360) > $2.01/£, £512.45 will be more than enough to repay your
dollar obligation of $1,030. The excess is your profit.
6-13
Arbitrage Strategy II
 If F360($/£) < $2.01/£:
1. Borrow £500 at t = 0 at i£= 2.49%.
2. Exchange £500 for $1,000 at the prevailing
spot rate; invest $1,000 at 3% for one year to
achieve $1,030.
3. Translate $1,030 back into pounds; if
F360($/£) < $2.01/£, then $1,030 will be more
than enough to repay your debt of £512.45.
6-14
Step 2:
Buy dollars
$2.00
$1,000 = £500×
£1
£500
Arbitrage II
Step 1:
Borrow £500.
$1,000
Step 5: Repay
More
Step 3:
your pound loan
than
Invest $1,000
with £512.45.
£512.45
at i$ = 3%.
Step 4:
Repatriate to
the U.K.
In one year $1,000
F£(360)
will be worth
$1,030 > £512.45 ×
$1,030
£1
If F£(360) < $2.01/£, $1,030 will be more than enough to repay your
dollar obligation of £512.45. Keep the rest as profit.
6-15
Currency Carry Trade
 Currency carry trade involves buying a
currency that has a high rate of interest
and funding the purchase by borrowing in
a currency with low rates of interest,
without any hedging.
 The carry trade is profitable as long as the
interest rate differential is greater than the
appreciation of the funding currency
against the investment currency.
6-16
Currency Carry Trade Example







Suppose the 1-year borrowing rate in dollars is 1%.
The 1-year lending rate in pounds is 2½%.
The direct spot ask exchange rate is $1.60/£.
A trader who borrows $1m will owe $1,010,000 in one year.
Trading $1m for pounds today at the spot generates £625,000.
£625,000 invested for one year at 2½% yields £640,625.
The currency carry trade will be profitable if the spot bid rate
prevailing in one year is high enough that his £640,625 will
sell for at least $1,010,000 (enough to repay his debt).
 No less expensive than:
b
S360($/£)
$1,010,000
=
£640,625
=
$1.5766
£1.00
6-17
Reasons for Deviations from IRP
 Transactions Costs
– The interest rate available to an arbitrageur for
borrowing, ib, may exceed the rate he can lend
at, il.
– There may be bid-ask spreads to overcome, Fb/Sa
< F/S.
– Thus, (Fb/Sa)(1 + i¥l)  (1 + i¥ b)  0.
 Capital Controls
– Governments sometimes restrict import
and export of money through taxes or
outright bans.
6-18
Transactions Costs Example
 Will an arbitrageur facing the following
prices be able to make money?
Borrowing
Lending
$
5.0%
4.50%
€
5.5%
5.0%
Spot
F($/ €) = S($/ €) ×
Bid
$1.42 = €1.00
(1 + i$)
(1 + i€)
Ask
$1.45 = €1,00
Forward $1.415 = €1.00 $1.445 = €1.00
F1b($/€) =
S0a($/€)(1+i$b)
(1+i€l )
F1a($/€) =
S0b($/€)(1+i$l )
(1+i€b)
6-19
$1m
0
Borrow $1m at i$b
Step 1
$1m×(1+ib$)
IRP
1
Step 2
1
$1m × a
×(1+il€)×Fb1($/€) = $1m×(1+ib$)
Buy € at
S0($/€)
spot ask No arbitrage forward bid price (for
customer):
b
b
a
(1+i
)
S
($/€)
(1+i
$
0
$)
Step 4
b
F1($/€) =
=
1
l
l
Sell € at
×(1+i
)
(1+i
)
€
€
S0a($/€)
forward
bid
= $1.4431/€
$1m ×
1
1
l
l
Step
3
invest
€
at
i
$1m
×
×(1+i
€
€)
a
a
S0($/€)
S0($/€)
6-20
(All transactions at retail prices.)
b
€1m × S0($/€)
Step 3:
lend at
€1m × Sb0($/€) × (1+il$)
i$l
0
IRP
1
€1m × Sb0($/€) × (1+il$) ÷ Fa1($/€) = €1m×(1+ib€)
No arbitrage forward ask price:
Step 2:
sell €1m at
spot bid
€1m
F1a($/€) =
S0b($/€)(1+i$l )
(1+i€b)
= $1.4065/€
Step 1: borrow €1m at i€b
Step 4
buy € at
forward
ask
€1m×(1+i€b)
(All transactions at retail prices.)
6-21
Why This May Seem Confusing
 On the last two slides we found “no arbitrage.”
– Forward bid prices of $1.4431/€.
– Forward ask prices of $1.4065/€.
 Normally the dealer sets the ask price above the
bid—recall that this difference is his expected
profit.
 But the prices on the last two slides are the prices of
indifference for the customer, NOT the dealer.
– At these forward bid and ask prices the customer is
indifferent between a forward market hedge and a
money market hedge.
6-22
Setting Dealer Forward Bid and Ask
 Dealer stands ready to be on the opposite side of every trade.
– Dealer buys foreign currency at the bid price.
– Dealer sells foreign currency at the ask price.
– Dealer borrows (from customer) at the lending rates.
– Dealer lends to his customer at the posted borrowing rates.
il$ = 4.5% and i€l = 5.0%
Borrowing Lending
i$b = 5.0%, ib€ = 5.5%.
$
5.0%
4.50%
€
5.5%
5.0%
Bid
Spot
$1.42 = €1.00
Ask
$1.45 = €1.00
Forward $1.415 = €1.00 $1.445 = €1.00
6-23
Setting Dealer Forward Bid Price
Our dealer is indifferent between buying euros today
at the spot bid price and buying euros in 1 year at the
forward bid price.
$1m ×
Invest at
$1m×(1+ib$)
i$b
He is willing to spend He is also willing to buy at
$1m today and receive
b
1
$1m × b
S0($/€)
1
S0b($/€)
Invest at i€b
F1b($/€) =
S0b($/€)(1+i$)
(1+i€b)
forward bid
spot bid
$1m
1
b
$1m × b
×(1+i€)
6-24
S0($/€)
Setting Dealer Forward Ask Price
Our dealer is indifferent between selling euros today
at the spot ask price and selling euros in 1 year at the
forward ask price.
€1m
Invest at
i$b
€1m × S0b($/€) ×(1+ib$)
He is willing to spend He is also willing to buy at
€1m today and receive
b
€1m × S0b($/€)
Invest at i€b
Fa1($/€) =
S0a($/€)(1+i$)
(1+i€b)
forward ask
spot ask
€1m × S0b($/€)
€1m×(1+ib€)
6-25
PPP and Exchange Rate Determination
 The exchange rate between two currencies should
equal the ratio of the countries’ price levels:
P$
S($/£) =
P£
For example, if an ounce of gold costs $300 in the U.S.
and £150 in the U.K., then the price of one pound in
terms of dollars should be: S($/£) = P$ = $300 = $2/£
P£
£150
Suppose the spot exchange rate is $1.25 = €1.00. If the inflation
rate in the U.S. is expected to be 3% in the next year and 5% in
the euro zone, then the expected exchange rate in one year
should be $1.25×(1.03) = €1.00×(1.05).
6-26
PPP and Exchange Rate Determination
 The euro will trade at a 1.90% discount in the forward
market:
F($/€)
=
S($/€)
$1.25×(1.03)
€1.00×(1.05)
$1.25
€1.00
1.03
1 + $
=
=
1.05
1 + €
Relative PPP states that the rate of change in the
exchange rate is equal to differences in the rates of
inflation—roughly 2%.
6-27
PPP and IRP
 Notice that our two big equations equal
each other:
PPP
F($/€)
1 + $
=
S($/€)
1 + €
IRP
=
1 + i$
F($/€)
=
1 + i€
S($/€)
6-28
Expected Rate of Change in Exchange
Rate as Inflation Differential
 We could also reformulate
our equations as inflation or
interest rate differentials:
F($/€) 1 + $
=
S($/€) 1 + €
F($/€) – S($/€)
1 + $
1 + $ 1 + €
=
–1=
–
S($/€)
1 + €
1 + € 1 + €
F($/€) – S($/€)
 $ – €
E(e) =
≈ $ – €
=
S($/€)
1 + €
6-29
Expected Rate of Change in Exchange
Rate as Interest Rate Differential
E(e) =
F($/€) – S($/€)
=
S($/€)
i$ – i€
1 + i€
≈ i$ – i€
 Given the difficulty in measuring expected
inflation, managers often use a “quick and dirty”
shortcut:
$ – € ≈ i$ – i€
6-30
Evidence on PPP
 PPP probably doesn’t hold precisely in the
real world for a variety of reasons.
– Haircuts cost 10 times as much in the
developed world as in the developing world.
– Film, on the other hand, is a highly
standardized commodity that is actively
traded across borders.
– Shipping costs, as well as tariffs and quotas,
can lead to deviations from PPP.
 PPP-determined exchange rates still
provide a valuable benchmark.
6-31
Approximate Equilibrium Exchange
Rate Relationships
E(e)
≈ IFE
(i$ – i¥)
≈ FEP
≈ PPP
F–S
≈ IRP
S
≈ FE
≈ FRPPP
E($ – £)
6-32
The Exact Fisher Effects
 An increase (decrease) in the expected rate of inflation
will cause a proportionate increase (decrease) in the
interest rate in the country.
 For the U.S., 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.
6-33
International Fisher Effect
If the Fisher effect holds in the U.S.,
1 + i$ = (1 + $ ) × E(1 + $)
and the Fisher effect holds in Japan,
1 + i¥ = (1 + ¥ ) × E(1 + ¥)
and if the real rates are the same in each country,
$ = ¥
then we get the International Fisher Effect:
1 + i¥
E(1 + ¥)
=
1 + i$
E(1 + $)
6-34
International Fisher Effect
If the International Fisher Effect holds,
E(1 + ¥)
1 + i¥
=
1 + i$
E(1 + $)
and if IRP also holds,
1 + i¥
F¥/$
=
1 + i$
S¥/$
then forward rate PPP holds:
F¥/$
E(1 + ¥)
=
S¥/$
E(1 + $)
6-35
Exact Equilibrium Exchange Rate
Relationships
IFE
1 + i¥
1 + i$
E (S ¥ / $ )
S¥ /$
FEP
PPP
F¥ / $
S¥ /$
IRP
FE
FRPPP
E(1 + ¥)
E(1 + $)
6-36
Forecasting Exchange Rates:
Efficient Markets Approach
 Financial markets are efficient if prices reflect
all available and relevant information.
 If this is true, exchange rates will only change
when new information arrives, thus:
St = E[St+1]
and
Ft = E[St+1| It]
 Predicting exchange rates using the efficient
markets approach is affordable and is hard to
beat.
6-37
Forecasting Exchange Rates:
Fundamental Approach
 Involves econometrics to develop models
that use a variety of explanatory variables.
This involves three steps:
– Step 1: Estimate the structural model.
– Step 2: Estimate future parameter values.
– Step 3: Use the model to develop forecasts.
 The downside is that fundamental models
do not work any better than the forward
rate model or the random walk model.
6-38
Forecasting Exchange Rates:
Technical Approach
 Technical analysis looks for patterns in the
past behavior of exchange rates.
 Clearly it is based upon the premise that
history repeats itself.
 Thus, it is at odds with the EMH.
6-39
Performance of the Forecasters
 Forecasting is difficult, especially with
regard to the future.
 As a whole, forecasters cannot do a better
job of forecasting future exchange rates
than the forecast implied by the forward
rate.
 The founder of Forbes Magazine once said,
“You can make more money selling
financial advice than following it.”
6-40