International Parity
Relationships and Forecasting
Foreign Exchange Rates
6
Chapter Six
Chapter Objective:
This chapter examines several key international
parity relationships, such as interest rate parity and
purchasing power parity.
6-0
Chapter Outline
Interest
InterestRate
RateParity
Parity

Covered Interest
Arbitrage
Purchasing
Purchasing
Power
Power
Parity
Parity

IRP and Exchange Rate
Determination
the Real
Exchange Rate
Reasons
IRP
Evidencefor
onDeviations
Purchasingfrom
Power
Parity

PPP
Deviations
and
The
The
Fisher
Fisher
Effects
Effects


Forecasting
ForecastingExchange
ExchangeRates
Rates

PowerApproach
Parity
 Purchasing
The
Fisher Market
Effects
 Efficient



The
Fisher Effects
Forecasting
Exchange
 Fundamental
ApproachRates
Forecasting
Exchange Rates
 Technical Approach

6-1
Performance of the Forecasters
Interest Rate Parity




Interest Rate Parity Defined
Covered Interest Arbitrage
Interest Rate Parity & Exchange Rate
Determination
Reasons for Deviations from Interest Rate Parity
6-2
Interest Rate Parity Defined



IRP is an “no arbitrage” condition.
If IRP did not hold, then it would be possible for
an astute trader to make unlimited amounts of
money 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 = $100,000 × (1 + i$)
Trade your $ for £ at the spot rate, invest $100,000/S$/£ in
Britain at i£ while eliminating any exchange rate risk by
selling the future value of the British investment forward.
2.
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)
6-4
F$/£
(1 + i£) ×
= (1 + i$)
S$/£
(1 + i$)
F$/£ = S$/£ × (1 + i )
£
Alternative 2:
Send your $ on a
round trip to
Britain
$1,000
S$/£
Step 2:
Invest those
pounds at i£
Future Value =
$1,000
Step 3: repatriate
future value to the
U.S.A.
Alternative 1:
invest $1,000 at i$
IRP
$1,000
 (1+ i£)
S$/£
IRP
$1,000
 (1+ i£) × F$/£
$1,000×(1 + i$) =
S$/£
6-5
Since both of these investments have the same risk, they must
have the same future value—otherwise an arbitrage would exist
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$) =
S$/£
6-6
Interest Rate Parity Defined
Formally,
1 + i$
F$/¥
=
1 + i¥
S$/¥
IRP is sometimes approximated as
i$ – i ¥ ≈ F – S
S
6-7
Interest Rate Parity Carefully Defined

Depending upon how you quote the exchange rate
(as $ per ¥ or ¥ per $) we have:
1 + i¥
F¥/$
=
1 + i$
S¥/$
or
1 + i$
F$/¥
=
1 + i¥
S$/¥
…so be a bit careful about that.
6-8
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.
6-9
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
6-10
S($/£) = $2.0000/£
F360($/£) = $2.0100/£
U.S. discount rate
i$ = 3.00%
British discount rate
i£ =
2.49%
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.
6-11
Alternative 2:
Arbitrage I
buy pounds
£500
£1
£500 = $1,000×
$2.00
$1,000
Alternative 1:
invest $1,000
at 3%
FV = $1,030
6-12
Step 2:
Invest £500 at
i£ = 2.49%
£512.45 In one year £500
will be worth
Step 3: repatriate
£512.45 =
to the U.S.A. at
£500  (1+ i£)
F360($/£) =
$2.01/£
$1,030
F£(360)
$1,030 = £512.45 ×
£1
Interest Rate Parity
& Exchange Rate Determination
According to IRP only one 360-day forward rate,
F360($/£), can exist. It must be the case that
F360($/£) = $2.01/£
Why?
If F360($/£)  $2.01/£, an astute trader could make
money with one of the following strategies:
6-13
Arbitrage Strategy I
If F360($/£) > $2.01/£
i. Borrow $1,000 at t = 0 at i$ = 3%.
ii. Exchange $1,000 for £500 at the prevailing spot
rate, (note that £500 = $1,000 ÷ $2/£) invest £500 at
2.49% (i£) for one year to achieve £512.45
iii. Translate £512.45 back into dollars, if
F360($/£) > $2.01/£, then £512.45 will be more than
enough to repay your debt of $1,030.
6-14
Step 2:
buy pounds
Arbitrage I
£500
£1
£500 = $1,000×
$2.00
$1,000
Step 1:
borrow $1,000
Step 5: Repay
your dollar loan
with $1,030.
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.A.
More
than $1,030
F£(360)
$1,030 < £512.45 ×
£1
If F£(360) > $2.01/£ , £512.45 will be more than enough to repay
your dollar obligation of $1,030. The excess is your profit.
6-15
Arbitrage Strategy II
If F360($/£) < $2.01/£
i. Borrow £500 at t = 0 at i£= 2.49% .
ii. Exchange £500 for $1,000 at the prevailing spot
rate, invest $1,000 at 3% for one year to achieve
$1,030.
iii. 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.
6-16
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
6-17
If F£(360) < $2.01/£ , $1,030 will be more than enough to repay
your dollar obligation of £512.45. Keep the rest as profit.
IRP and Hedging Currency Risk
You are a U.S. importer of British woolens and have just ordered
next year’s inventory. Payment of £100M is due in one year.
Spot exchange rate
360-day forward rate
S($/£) = $2.00/£
F360($/£) = $2.01/£
U.S. discount rate
i$ = 3.00%
British discount rate
i£ =
2.49%
IRP implies that there are two ways that you fix the cash outflow to a
certain U.S. dollar amount:
a) Put yourself in a position that delivers £100M in one year—a long
forward contract on the pound.
You will pay (£100M)($2.01/£) = $201M in one year.
b) Form a money market hedge as shown below.
6-18
IRP and a Money Market Hedge
To form a money market hedge:
1. Borrow $195,140,989.36 in the U.S.
(in one year you will owe $200,995,219.05).
2. Translate $195,140,989.36 into pounds at the spot
rate S($/£) = $2/£ to receive £97,570,494.68
3. Invest £97,570,494.68 in the UK at i£ = 2.49% for
one year.
4. In one year your investment will be worth £100
million—exactly enough to pay your supplier.
6-19
Money Market Hedge
Where do the numbers come from? We owe our supplier £100
million in one year—so we know that we need to have an
investment with a future value of £100 million. Since i£ = 2.49%
we need to invest £97,570,494.68 at the start of the year.
£97,570,494.68 =
£100,000,000
1.0249
How many dollars will it take to acquire £97,570,494.68
at the start of the year if S($/£) = $2/£?
$2.00
$195,140,989.36 = £97,570,494.68 ×
£1.00
6-20
Money Market Hedge


This is the same idea as covered interest arbitrage.
To hedge a foreign currency payable, buy a bunch
of that foreign currency today and sit on it.



6-21
Buy the present value of the foreign currency payable
today.
Invest that amount at the foreign rate.
At maturity your investment will have grown enough to
cover your foreign currency payable.
Money Market Hedge: an Example
Suppose that the spot dollar-pound exchange rate is $2.00/£ and
i$ = 1%
i£ = 4%
0
1
Step 1
Step 5
Order Inventory; agree to
Redeem £-denominated
pay supplier £100 in 1 year.
investment receive £100
million
Step 2
Step 6
Borrow $192,307,692 at i$ = 1%
($192,307,692 = £96,153,846×$2/£) Pay supplier £100 million
Step 3
£100,000,000 Step 7
Buy £96,153,846 =
Repay dollar loan with
1.04
at spot exchange rate.
$194,230,769
Step 4 Invest £96,153,846 at i£ = 4%
6-22
Another Money Market Hedge
A U.S.–based importer of Italian bicycles



In one year owes €100,000 to an Italian supplier.
The spot exchange rate is $1.50 = €1.00
The one-year interest rate in Italy is i€ = 4%
€100,000
Can hedge this payable by buying €96,153.85 = 1.04
today and investing €96,153.85 at 4% in Italy for one year.
At maturity, he will have €100,000 = €96,153.85 × (1.04)
Dollar cost today = $144,230.77 = €96,153.85 × $1.50
€1.00
6-23
Another Money Market Hedge


With this money market hedge, we have
redenominated a one-year €100,000 payable into
a $144,230,77 payable due today.
If the U.S. interest rate is i$ = 3% we could borrow
the $144,230,77 today and owe in one year
$148,557.69 = $ 144,230,77 × (1.03)
€100,000
T
$148,557.69 = S($/€)×
×
(1+
i
)
$
(1+ i€)T
6-24
Generic Money Market Hedge: Step One
Suppose you want to hedge a payable in the amount
of £y with a maturity of T:
i. Borrow $x at t = 0 on a loan at a rate of i$ per year.
£y
$x = S($/£)× (1+ i )T
£
$x
0
6-25
Repay the loan in T years
–$x(1 + i$)T
T
Generic Money Market Hedge: Step Two
£y
ii. Exchange the borrowed $x for
(1+ i£)T
at the prevailing spot rate.
£y
Invest
at i£ for the maturity of the payable.
T
(1+ i£)
At maturity, you will owe a $x(1 + i$)T.
Your British investments will have grown to £y. This
amount will service your payable and you will have no
exposure to the pound.
6-26
Generic Money Market Hedge
£y
1. Calculate the present value of £y at i£
(1+ i£)T
2. Borrow the U.S. dollar value of receivable at the spot rate.
3. Exchange $x = S($/£)×
4. Invest
£y
(1+ i£)T
£y
at i£ for T years.
T
(1+ i£)
for
£y
(1+ i£)T
5. At maturity your pound sterling investment pays your
receivable.
6. Repay your dollar-denominated loan with $x(1 + i$)T.
6-27
Forward Premium



It’s just the interest rate differential implied by
forward premium or discount.
For example, suppose the € is appreciating from
S($/€) = 1.25 to F180($/€) = 1.30
The forward premium is given by:
f180,€v$
6-28
F180($/€) – S($/€) 360 $1.30 – $1.25
=
× 180 =
× 2 = 0.08
S($/€)
$1.25
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

6-29
Governments sometimes restrict import and export of
money through taxes or outright bans.
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
(1 + i$)
F($/ €) = S($/ €) ×
(1 + i€)
Bid
$1.42 = €1.00
Forward $1.415 = €1.00
6-30
a
b
S
($/€)
(1+i
0
$)
b
F1($/€) =
(1+i€l )
Ask
$1.45 = €1,00
$1.445 = €1.00
b
l
S
($/€)
(1+i
0
$)
a
F1($/€) =
(1+i€b)
Step 1
$1m
0
Borrow $1m at i$b
$1m×(1+ib$)
IRP
1
Step 2
1 ×(1+il )×Fb($/€) = $1m×(1+ib)
$1m
×
€
1
$
Buy € at
S0a($/€)
spot ask No arbitrage forward bid price (for customer):
b
b
a
(1+i
)
S
($/€)
(1+i
$
0
$)
Step 4
b
F1($/€) =
=
1
l
l
Sell € at
×(1+i
)
(1+i
)
€
€
S0a($/€)
forward
= $1.4431/€
bid
$1m × a 1
Step 3 invest € at i€l
$1m × a 1
×(1+i€l )
S0($/€)
S0($/€)
6-31
(All transactions at retail prices.)
Step 3
€1m ×
S0b($/€)
0
lend at
€1m × Sb0($/€) × (1+i$l )
i$l
IRP
1
€1m × Sb0($/€) × (1+i$l ) ÷ F1a($/€) = €1m×(1+ib€)
No arbitrage forward ask price:
Fa1($/€) =
Step 2
sell €1m at
spot bid
€1m
6-32
S0b($/€)(1+i$l )
(1+i€b)
= $1.4065/€
Step 1borrow €1m at i€b
(All transactions at retail prices.)
Step 4
buy € at
forward
ask
€1m×(1+ib€)
Why This Seems Confusing

On the last two slides we found “no arbitrage”


Forward bid prices of $1.4431/€ and
Forward ask prices of $1.4065/€
Normally the dealer sets the ask price above the
bid—recall that this difference is his expected profit.
 But the prices on the last two slides are the prices of
indifference for the customer NOT the dealer.
At these forward bid and ask prices the customer
is indifferent between a forward market hedge
and a money market hedge.
6-33

Setting Dealer Forward Bid and Ask

Dealer stands ready to be on opposite side of every trade




Dealer buys foreign currency at the bid price
Dealer sells foreign currency at the ask price
Dealer borrows (from customer) at the lending rates
i$l = 4.5% and i€l = 5.0%
Dealer lends to his customer at the borrowing rate i$b = 5.0%, i€b = 5.5%.
Borrowing
Lending
$
5.0%
4.50%
€
5.5%
5.0%
Spot
6-34
Bid
$1.42 = €1.00
Forward $1.415 = €1.00
Ask
$1.45 = €1.00
$1.445 = €1.00
Setting Dealer Forward Bid Price
Our dealer is indifferent between buying euro today at spot
bid price and buying euro in 1 year at forward bid price.
Invest at
i$b
$1m×(1+ib$)
spot bid
He is willing to spend He is also willing to buy at
$1m today and receive
b
b
S
($/€)
(1+i
0
$)
b
1
F1($/€) =
$1m × b
S0($/€)
(1+i€b)
$1m ×
6-35
1
S0b($/€)
Invest at i€b
$1m ×
forward bid
$1m
1
b
×(1+i
€)
b
S0($/€)
Setting Dealer Forward Ask Price
Our dealer is indifferent between selling euro today at spot
ask price and selling euro in 1 year at forward ask price.
Invest at
i$b
€1m × S0b($/€) ×(1+i$b)
spot ask
He is willing to spend He is also willing to buy at
€1m today and receive
b
a
S
($/€)
(1+i
0
$)
b
a
€1m × S0($/€)
F1($/€) =
(1+i€b)
€1m
6-36
Invest at i€b
forward ask
€1m × S0b($/€)
€1m×(1+i€b)
Purchasing Power Parity



Purchasing Power Parity and Exchange Rate
Determination
PPP Deviations and the Real Exchange Rate
Evidence on PPP
6-37
Purchasing Power Parity 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 $300 in
the U.S. and £150 in the U.K., then the price of
one pound in terms of dollars should be:
P$ $300
S($/£) =
=
= $2/£
P£ £150

6-38
Purchasing Power Parity and
Exchange Rate Determination



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 $1.25×(1.03) = €1.00×(1.05)
F($/€) = $1.25×(1.03) = $1.23
€1.00×(1.05)
€1.00
6-39
Purchasing Power Parity and
Exchange Rate Determination

The euro will trade at a 1.90% discount in the forward market:
F($/€)
=
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%
6-40
Purchasing Power Parity
and Interest Rate Parity

Notice that our two big equations today equal
each other:
PPP
IRP
F($/€)
1 + $
=
S($/€)
1 + €
1 + i$
F($/€)
=
1 + i€
S($/€)
6-41
=
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($/€)
1 + €
6-42
Expected Rate of Change in
Exchange Rate as Interest Rate
Differential
E(e) =
6-43
F($/€) – S($/€)
=
S($/€)
i$ – i€
1 + i€
≈ i$ – i€
Quick and Dirty Short Cut

Given the difficulty in measuring expected
inflation, managers often use
$ – € ≈ i$ – i€
6-44
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.
6-45
Approximate Equilibrium Exchange
Rate Relationships
E(e)
≈ IFE
(i$ – i¥)
≈ FEP
≈ PPP
≈ IRP
S
≈ FE
≈ FRPPP
E($ – £)
6-46
F–S
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

6-47
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 + $)
6-48
International Fisher Effect
If the International Fisher Effect holds,
E(1 + ¥)
1 + i¥
=
1 + i$
E(1 + $)
and if IRP also holds
1 + i¥
F¥/$
=
1 + i$
S¥/$
F¥/$
E(1 + ¥)
then forward rate PPP holds:
=
S¥/$
E(1 + $)
6-49
Exact Equilibrium Exchange Rate
Relationships
IFE
1 + i¥
1 + i$
E (S ¥ / $ )
S¥ /$
PPP
F¥ / $
S¥ /$
IRP
FE
FRPPP
E(1 + ¥)
E(1 + $)
6-50
FEP
Forecasting Exchange Rates




Efficient Markets Approach
Fundamental Approach
Technical Approach
Performance of the Forecasters
6-51
Efficient Markets Approach



Financial Markets are efficient if prices reflect all
available and relevant information.
If this is so, 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.
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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.
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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
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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 forward
rate.
The founder of Forbes Magazine once said:
“You can make more money selling financial
advice than following it.”
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End Chapter Six
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