### THEORIES OF FOREIGN EXCHANGE

```THEORIES OF
FOREIGN EXCHANGE
International Parity Conditions
Exchange Rate Determination



What determines equilibrium relationships among
exchange rates?
International arbitrageur and the "Law of One-price"
insure that risk adjusted expected rates of returns
are approximately equal across countries.
There are five key relationships between:





Spot rate
Forward rate
Inflation rate
Interest rate
Exchange rate
Purchasing Power Parity


A unit of domestic currency should purchase
the same amount of goods in the home
country as it would of identical goods in a
foreign country.
Absolute form of PPP:
“law of one price”: price of similar products to
two countries should be equal when
measured in a common currency.
PPP Example



A bottle of wine costs €8 in Paris and \$10 in New
York. The exchange rate must reflect this price
relationship:
e0 = Pd/Pf = \$10/€8 = \$1.25 per €
(e0 = \$ per FC; direct or American quote)
Equivalent to €0.80 per \$ in indirect quote,
European terms.
The strictest version of PPP is not supported
empirically, but changes in relative inflation rates are
related to changes in exchange rates.
Relative form of PPP

Acknowledges market imperfections such as:



transport costs
tariffs and quotas.
Rate of change in the prices of products should
be similar when measured in a common
currency.
Relative PPP - Example

Example: Suppose the price of wine in Paris increases to
€ 9 in one year implying an inflation of 12.5%, while in
the U.S. the price of wine increases to \$10.50 indicating
an inflation rate of 5%. The new exchange rate:
e1 = Pd,1/Pf,1 = \$10.5/€9 = \$1.1667 per €
or €0.8571 per \$

What is the depreciation in the value of €?
1.1667/1.2500 -1 = -6.67%
PPP
Relative PPP:
e1 =
OR
OR
Pd ,1
P f ,1
=
Pd ,0
x
P f ,0
(1+  d )
(1+  f )
(1+  d )
e1 = e0 x
(1+  f )
e1 (1+ d )
=
e0 (1+ f )
Approximately: %Δe0 = πd - πf
PPP Implication


According to PPP, the currency of countries with
high inflation rates should devalue relative to
countries with low inflations rates.
Rationale:
if πd > πf, then:
 domestic imports increase; domestic exports decrease
 foreign imports decrease; foreign exports increase
 demand for FC increases; supply decreases
 demand for LC decreases; supply increases
 FC appreciates; LC depreciates
Relative PPP Example


Suppose the U.S. inflation rate is expected to
be 3 percent for the coming year, while the
Britain's expected rate of inflation is 5
percent.
The current exchange rate is \$1.50 per £.
What should be the £ spot rate in one year?
1.50 × 1.03/1.05 = \$1.47 per £
II: Fisher Effect



Recall the relationship between nominal and
real rates of interest, as expressed in the
Fisher theorem:
1 + i = (1 + r*) (1 + π)
Or
(1+ i)
1+ r* =

(1+  )
Approximately:
i = r* + π and r* = i - π
Generalized Fisher Effect




Real rates of interest are equalized across
countries through arbitrage.
Otherwise funds would flows from countries
with low expected real rates of interest to
countries with high expected real rates of
interests (in the absence of segmented
markets)
Therefore: r*f = r*d
OR
if - πf = id - πd
Generalized Fisher Effect

More precisely:
(1+ i d ) (1+ i f )
=
(1+ d ) (1+ f )

OR

Approximately:
(1+ i d ) (1+ d )
=
(1+ i f ) (1+ f )
id  i f   d   f
III. International Fisher Effect
Combines the generalized Fisher effect to show the
relationship between nominal interest rates and currency
exchange rates.
From PPP:
e1 (1+ d )
=
e0 (1+ f )
From GFE:
(1+ i d ) (1+ d )
=
(1+ i f ) (1+ f )
Therefore:
e1 (1+ i d )
=
e0 (1+ i f )
IFE Implications



Currencies with low interest rates would appreciate
with respect to currencies with high interest rates.
A long-run tendency for interest rates differentials to
offset exchange rate changes has been
demonstrated empirically.
Example: Interest rate in U.S. is 4%, while interest
rate in Switzerland is 10%. If the current SF spot
rate is \$0.80, what should be the SF spot rate one
year from now?
\$0.80 × 1.04/1.10 = \$0.7564 per SF
Or SF depreciates about 5.5%
IV. Forward Rates and Expected
Future Spot Rates




Early studies indicated forward exchange rates to be
unbiased predictor of future spot exchange rates.
f1 = E[e1]
Then the forward rate premium or discount
unbiasedly reflects potential gains to be realized
from the purchase or sale of forward currencies.
This equality captures the relationship between
forward and spot rates.
Recent work has demonstrated the existence of a
slight risk premium. The premium, however,
changes signs. Therefore, it is fair to assume that
the future spot rates would equal forwards rates.
V. Interest Rate Parity

Substituting f1 = E[e1] in the IFE equation:
e1 (1+ i d )
=
e0 (1+ i f )
f1
(1+ id )
=
e0 (1+ i f )
Covered interest arbitrage

Suppose the interest rates are 4% in the U.S.
and 10% in Switzerland. The Swiss Franc spot
rate is \$0.8000 and 180-day forward rate is
\$0.7800. Is covered arbitrage possible?
Forward discount on SF
= (.78 -.80)/0.80 = -2.5% for 180 days
or -5% per year.
Id = 4% while If + discount = 10% - 5% = 5%

Therefore, arbitrage is possible.

Covered Arbitrage
1. Borrow \$1,000,000 in US @ 4% per year or 2% for half year.
Loan plus Interest to be paid in 180 days = \$1,020,000
2. Convert \$ to SF at the spot rate:
\$1,000,000/0.80 = SF 1,250,000
3. Invest SF 1,250,000 @ 10% for 180 days:
Will receive SF 1,250,000 × (1+10%/2) = SF 1,312,500 in 180 days
4. Sell SF 1,312,500 in forward market @180 forward rate \$0.78/SF
Will receive 1,312,500 × \$0.78/SF = \$1,023,750 in 180 days
5. After 180 days receive \$1,023,750 from forward contract, and pay-off
loan
Net profit form arbitrage: \$1,023,750 -1,020,000 = \$3,750
Prices, Interest Rates and
Exchange Rates in Equilibrium
Forward rate
as an unbiased
predictor
(E)
Forward premium
on foreign currency
+4%
Forecast change in
spot exchange rate
+4%
(yen strengthens)
International
Fisher
Effect
(C)
(yen strengthens)
Interest
rate parity
(D)
Purchasing power
parity
(A)
Forecast difference
in rates of inflation
-4%
(less in Japan)
Difference in nominal
interest rates
-4%
(less in Japan)
Fisher effect
(B)
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