Download Parity Theory

1. The Foreign Exchange Market
Some currency rates as of May 21, 2004:
Per U.S. dollar:
Brazil (Real)
Mexico (Peso)
Japan (Yen)
3.1939
11.5754
112.2839
Indonesia (Rupiah)
89066
South Africa (Rand)
6.7295
United Kingdom (Pound)
0.5593
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The Foreign Exchange Market...
Some forward currency rates as of May 24, 2004:
U.S. dollars per Euro (bid prices):
Spot rate
1.2017
One-month forward
1.20062
3 months forward
1.19898
6 months forward
1.19789
12 months forward
1.19854
24 months forward
1.19804
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2. Some basic questions
Why aren’t FX rates all equal to one?
Why do FX rates change over time?
Why don’t all FX rates change in the same direction?
What drives forward rates – the rates at which you can
trade currencies at some future date?
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Definitions
r$ : dollar rate of interest (r¥, rHK$,…)
i$ : expected dollar inflation rate
f€/$ : forward rate of exchange
s€/$ : spot rate of exchange
“Indirect quote”:
s€/$ = 0.83215
 1 $ buys 0.83215 €
“Direct quote”:
s$/€ = 1.2017
 1 € buys $1.2017
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3. Four theories
.
Difference in
interest rates
1 + r€
1 + r$
Fisher
Theory
Interest
Rate
parity
Exp. difference in
inflation rates
1 + iSFr
1 + i$
Relative PPP
Difference between
forward & spot rates
F€/$
Exp. Theory
s€/$
of forward
rates
Expected change
in spot rate
E(s€/$)
S€/$
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Theory #1: Purchasing power parity
Law of One Price
Versions of
PURCHASING
POWER
PARITY
Absolute PPP
Relative PPP
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The Law of One Price
A commodity will have the same price in terms of
common currency in every country
In the absence of frictions (e.g. shipping costs, tariffs,..)
Example
Price of wheat in France (per bushel): P€
Price of wheat in U.S. (per bushel): P$
S€/$ = spot exchange rate
P€ = s€/$  P$
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The Law of One Price, continued
Example:
Price of wheat in France per bushel (p€) = 3.45 €
Price of wheat in U.S. per bushel (p$) = $4.15
S€/$ = 0.83215 (s$/€ = 1.2017)
Dollar equivalent price
of wheat in France
= s$/€ x p€
= 1.2017 $/€ x 3.45 € = $4.15
 When law of one price does not hold, supply and
demand forces help restore the equality
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Absolute PPP
Extension of law of one price to a basket of goods
Absolute PPP examines price levels
Apply the law of one price to a basket of goods with
price P€ and PUS (use upper-case P for the price of the
basket):
S€/$ = P€ / PUS
where
P€ = i (wFR,i  p€,i )
PUS = i (wUS,i  pUS,i )
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Absolute PPP
If the price of the basket in the U.S. rises relative to
the price in Euros, the U.S. dollar depreciates:
May 21 : s€/$ = P€ / PUS
= 1235.75 € / $1482.07 = 0.8338 €/$
May 24: s€/$ = 1235.75 € / $1485.01 = 0.83215 €/$
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Relative PPP
Absolute PPP:
P€ = s€/$  P$
For PPP to hold in one year:
P€ (1 + i€) = E(s€/$)  P$ (1 + i$),
or: P€ (1 + i€) = s€/$ [E(s€/$)/s€/$ )]  P$ (1 + i$)
Using absolute PPP to cancel terms and rearranging:
Relative PPP:
1 + i€ = E(s€/$)
1 + i$
s€/$
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Relative PPP
Main idea – The difference between (expected)
inflation rates equals the (expected) rate of change in
exchange rates:
1 + i€ = E(s€/$)
1 + i$
s€/$
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What is the evidence?
The Law of One Price frequently does not hold.
Absolute PPP does not hold, at least in the short run.
See The Economist’s Big McCurrencies
The data largely are consistent with Relative PPP, at
least over longer periods.
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Deviations from PPP
Simplistic model
Why does
PPP
not
hold?
Imperfect Markets
Statistical difficulties
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Deviations from PPP
Simplistic model
Imperfect Markets
Statistical difficulties
Transportation costs
Tariffs and taxes
Consumption patterns differ
Non-traded goods & services
Sticky prices
Markets don’t work well
Construction of price indexes
- Different goods
- Goods of different qualities
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Summary of theory #1:
.
Exp. difference in
inflation rates
1 + i€
1 + i$
Relative PPP
Expected change
in spot rate
E(s€/$)
S€/$
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Theory #2: Interest rate parity
Main idea: There is no fundamental advantage to
borrowing or lending in one currency over another
This establishes a relation between interest rates, spot
exchange rates, and forward exchange rates
Forward market: Transaction occurs at some point in future
BUY: Agree to purchase the underlying currency at a
predetermined exchange rate at a specific time in the future
SELL: Agree to deliver the underlying currency at a
predetermined exchange rate at a specific time in the future
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Example of a forward market transaction
Suppose you will need 100,000€ in one year
Through a forward contract, you can commit to lock in
the exchange rate
f$/€ : forward rate of exchange
Currently, f$/€ = 1.19854
 1 € buys $1.19854
 1 $ buys 0.83435 €
At this forward rate, you need to provide $119,854 in
12 months.
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Interest Rate Parity
START (today)
$117,228
r$=2.24%
END (in one year)
$117,228  1.0224 = $119,854
(Invest in $)
s€/$=0.83215
$117,228  0.83215 = 97,551€
One year
(Invest in €)
f€/$=0.83435
97,551€  1.0251 = 100,000€
r€=2.51%
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Interest rate parity
Main idea: Either strategy gets you the 100,000€
when you need it.
This implies that the difference in interest rates must
reflect the difference between forward and spot
exchange rates
Interest
Rate Parity:
1 + r€ = f€/$
1 + r$
s€/$
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Interest rate parity example
Suppose the following were true:
12 month interest
rate
U.S Dollar
Euro
2.24%
2.70%
Spot rate
1.2017 € / $
Forward rate
1.19854 € / $
Does interest rate parity hold?
Which way will funds flow?
How will this affect exchange rates?
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Evidence on interest rate parity
Generally, it holds
Why would interest rate parity hold better than PPP?
Lower transactions costs in moving currencies than real
goods
Financial markets are more efficient that real goods
markets
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Summary of theories #1 and #2:
.
Difference in
interest rates
1 + r€
1 + r$
Interest
Rate
parity
Difference between
forward & spot rates
f€r/$
s€/$
Exp. difference in
inflation rates
1 + i€
1 + i$
Relative PPP
Expected change
in spot rate
E(s€/$)
s€/$
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Theory #3: The Fisher condition
Main idea: Market forces tend to allocate resources to
their most productive uses
So all countries should have equal real rates of interest
Relation between real and nominal interest rates:
(1 + rNominal) = (1 + rReal)(1 + i )
(1 + rReal) = (1 + rNominal) / (1 + i )
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Example of capital market equilibrium
Fisher condition in U.S. and France:
(1 + r$(Real)) = (1 + r$) / (1 + i$)
(1 + r€(Real)) = (1 + r€) / (1 + i€)
If real rates are equal, then the Fisher condition
implies:
1 + r€ = 1 + i€
1 + r$
1 + i$
The difference in interest rates is equal to the expected
difference in inflation rates
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Summary of theories 1-3:
.
Difference in
interest rates
1 + r€
1 + r$
Interest
Rate
parity
Difference between
forward & spot rates
f€/$
s€/$
Fisher
Theory
Exp. difference in
inflation rates
1 + i€
1 + i$
Relative PPP
Expected change
in spot rate
E(s€/$)
s€/$
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Theory #4: Expectations theory of
forward rates
Main idea:
The forward rate equals expected spot exchange rate
f€/$ = E(s€/$)
Expectations theory
of forward rates:
f€/$ = E(s€/$ )
s€/$
s€/$
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Expectations theory of forward rates
With risk, the forward rate may not equal the spot rate
Group 1: Receive €
in six months, want $
Group 2: Contracted to
pay out € in six months
• Wait six months and
convert € to $
• Wait six months and
convert $ to €
• Sell € forward
• Buy € forward
or
or
If Group 1 predominates, then E(s€/$) < f€/$
If Group 2 predominates, then E(s€/$) > f€/$
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Takeaway: Summary of all four theories
.
Difference in
interest rates
1 + r€
1 + r$
Fisher
Theory
Interest
Rate
parity
Exp. difference in
inflation rates
1 + i€
1 + i$
Relative PPP
Difference between
forward & spot rates
f€/$
Exp. Theory
s€/$
of forward
rates
Expected change
in spot rate
E(s€/$)
s€/$
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