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Topic 4: International ParityRelationship (Shapiro, Chapter 4)
Law of one price
Purchasing power parity (PPP)
Fisher effect International Fisher effect
Interest rate parity (IRP)
Unbiased forward rates
Covered interest arbitrage (CIA)
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The law of one price: in competitivemarkets identical commodity (say, FX) sold
in different countries must sell for the sameprice when their price is expressed in termsof the same currency
Purchasing Power Parity: Money supplyand price inflation: when growth in acountrys money supply is faster thangrowth in output, price inflation is fueled,
which will lead to depreciation in currency ofthat country.
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(A)Purchasing power parity(PPP):
Money supply and price inflation: when growth in acountrys money supply is faster than growth inoutput, price inflation is fueled, which will lead todepreciation in currency of that country.
PPP holds that the expected change in the exchangerate is due to the difference in expected inflationrates in the respective countries.
tfi
thieet)1(
)1(0
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where e0is the spot exchange rate and etisexpected future spot rate in period t, ihand iftheexpected home and foreign inflation rates,respectively.
PPP states that the exchange rate between twocurrencies will adjust to reflect the relative
inflation rates in the two countries. It is assumedthat the law of one price is valid.
Eg., the expected inflation rates for the US and
Australia in one year are 5% and 3%, respectively,current spot exchange rate beingUSD0.6000/AUD (or AUD1.6667/USD), what isthe expected future spot exchange rate in oneyear?
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(1) When AUD is the commodity currency:
(2) When USD is the commodity currency:
AUDUSD
ee tf
t
h
i
i
t
/6117.0
6.0 11
)03.01(
)05.01(
)1(
)1(
0
USDAUD
ee tf
t
h
i
i
t
/6349.1
1
1
)05.01(
)03.01(
6.01
)1(
)1(
0
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Given that the U.S.s relative expected inflationrate is higher, USD is expected to depreciatefrom AUD1.6667/USD to AUD1.6349/USD.Why?
Because American goods are relatively moreexpensive, American will convert their USD to
get AUD so that they can buy goods inAustralia. The increases in the supply ofUSD and the demand for AUD will drive upthe price of AUD. USD will continue todepreciate until an equilibrium point isreached (ie., AUD1.6349/USD in this case).
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Therefore, PPP states that the currency of
the country with higher inflation rate will
depreciate.PPP is simple and easy to understand.
Empirical research shows that it does
not hold in reality, especially in the
short-run. If it does hold, it holds only in
the long-run
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According to PPP, given the previous example,AUD should appreciate against USD by 2%roughly (ie.,e = 2% roughly).
Mathematically:
0194.1
)1(
)1()1(
%)31(
%)51(f
i
ie
h
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Given
Let
Let
1
'
11 eee
)1(
)1(
)1(
)1(
1
'1
T
C
h
f
i
i
i
i
e
e
or
)1)(1(
)1(
h
f
ie
iq
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If PPP holds, (1 + e) = (1 + if)/(1 + ih), then q=
1. Competitiveness of domestic country in
international trade remains unchangedIf q< 1, competitiveness of domestic country
improves with currency depreciations
because domestic currency depreciatedmore than required by PPP.
If q> 1, competitiveness of domestic
country deteriorates with currencydepreciations because domestic currency
depreciated less than required by PPP.
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(B) Fisher Theorem
Fisher equation maintains that, for any given
currency, the nominal interest rate (r) will beset by the market such that it coversexpected inflation rate (E(i)) and provides arequired real rate of return (real interest rate,
r*).
Roughly, nominal interest rate
= Real interest rate + Expected inflation rate
Mathematically,
1 + r = (1+r*)(1 + E(i))
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International Fisher Theorem
If Fisher equation holds for both countries, A
and B, then
We have
)](1*)[1(1
)](1*)[1(1
BB
AA
iErr
andiErr
)](1[)](1[
*)1(*)1(
11
B
A
B
A
iEiE
rr
rr
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Nominal interest rates are rates quotedin the market. Real interest rates are
nominal rates adjusted for expectedinflation
(C) International Fisher Effect
Tthe expected change in spot exchangerate between two currencies is thedifference in the interest rates between
the two currencies.
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If real interest rates are equal across
countries, then
Given PPP,
)](1[
)](1[
1
1
B
A
B
A
iE
iE
r
r
B
A
B
A
B
A
r
r
iE
iE
S
SEiE
iE
S
SE
1
1
)(1
)(1)()(1
)(1)(have, we
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(D) Unbiased Forward Rates
Since forward rate is an unbiased predictor
for future spot rate, we have
And interest rate parity is derived, ie.,s
sE
s
f )(
C
T
B
A
r
r
i
i
s
f
1
1
1
1
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(E) Interest rate parity(IRP):
The IRP states that relative interest rates
between two countries, say UK and USA,determine the relativity between the forward
exchange rate (f) and spot exchange rate
(s):
f
h
C
T
r
r
r
r
s
f
or
1
1
1
1
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The IRP simply states that at the equilibriumexchange rates, returns from investing in thetwo currencies must be equal, otherwisethere will be arbitrage opportunity.
Arbitrage activity will ensure that the exchangerates will go back to the equilibrium. This is
the so-called law of one price.The IRP relationship: the higher (lower)
interest rate currency will sell at a discount
(premium) in the forward markets.
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IRP relationship is derived from PPP and
works through the international Fisher
theorem.PPP: The currency with higher inflation
will depreciate
Fisher Theorem: The currency with
higher inflation rate will have higher
nominal interest rate. Therefore:
IRP: The currency with higher interest
rate will have a lower forward rate.
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IRP is derived from PPP through the Fisher theorem.
It is a very simple model that assumes exchange
rate is only determined by relative inflation rates ofthe two countries. There are many other factors
that help to determine exchange rate.
IRP is useful because it provides an equation to
allow forward rates to be calculated. All banksuse IRP to calculate forward rates.
For the forward rates that banks quote, banks can
hedge their FX position using IRP.
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Integrating Parity & Equilibrium ConditionsDirect FX rates: d/f (domestic/foreign currency)
E[S] = S(1 + rh)
(1 + rf
)
=(1 + r*h)
(1 + r*f
)
(1 + ih)
(1 + if
)
S
DFE
=(1 + ih)
(1 + if)S
IFEC
IFE
IFE: International Fisher EffectDFE: Domestic Fisher Effect
IRP: Interest Rate Parity
RPPP: Relative Purchasing Power Parity
IRP
FRP: Forward Rate Parity
FRP
IFEC: International Fisher Relation Corollary (equality of real returns
RPPP
F =
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Integrating Parity & Equilibrium ConditionsIndirect FX rates: f/d (foreign/domestic currency)
F = E[S] = S(1 + if)(1 + id) =
(1 +i*f)
(1 +i*d
)
(1 + pf)
(1 +p
d
)
S
IFE DFE
=(1 + pf)
(1 + pd)S
IFEC
FRP: Forward Rate Parity
IFE: International Fisher EffectDFE: Domestic Fisher Effect
IRP: Interest Rate Parity
RPPP: Relative Purchasing Power Parity
IRP
FRP
IFEC: International Fisher Effect Corollary (equality of real returns)
RPPP
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From IRP:
Example: Given that interest rates for USD andGBP are 7.1% and 11.56% p.a. respectively
and the spot exchange rate isUSD1.2500/GBP, calculate the 1-year forwardrate.
The equilibrium forward rate is USD1.2000/GBP.
)1(
)1(
C
T
r
rsf
2000.125.1 )1156.01()071.01(f
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If IRP does not hold, the situation gives rise to
a covered interest arbitrage (CIA)
opportunity.
For example, what if the bank quotes you the
1-year forward rate as USD1.1950/GBP in
the above example?What would you do? Conduct a secret CIA
operation!!!
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Covered Interest Arbitrage
The calculated f is higher than the quoted f. To
conduct the CIA operation, one would buy
low and sell high in order to make anarbitrage profit, ie., sell GBP spot and buy
GBP forward.
1950.12000.1
2500.1)1156.01()071.01(
f
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Three steps in the CIA operation:
1. Borrow GBP at 11.56% pa for one yearand sell it (ie., buy USD) at
USD1.2500/GBP
2. Invest the USD to earn 7.1% for one year
3. Sell the USD 1-year forward (ie., buy
GBP forward) at USD1.1950.
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Calculation:
Borrow GBP1 at 11.56% for one year and you
payback GBP1(1+0.1156) = GBP1.1156 Convert into USD at 1.2500, invest the sum toearn 7.1% pa for one year, you get
USD1.2500(1+ 0.071) = USD1.33875;
Sell USD1.33875 1-year forward at 1.1950,you get GBP1.1203.
Arbitrage profit = 1.12031.1156 =
GBP0.0047. Arbitrage profit is GBP0.0047 per GBP in one
year.
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Another example on CIA: Given that USD
interest rate: 5% pa, GBP interest rate 8% pa,
Spot exchange rate: USD1.5000/GBP, One-year forward: USD1.4800/GBP
(a) First, calculate the effective
0656.1)08.1(50.1
48.1
)1()1(
GBPUSD rS
Fr
USDr
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Since 1.0656 > 1.05, IRP is violated.
The effective interest rate of USD is still lower that thatof GBP at 8%, one should borrow USD at 5% and
lend in GBP to arbitrage.This approach is easier, but it is better if you
understand the following approach in (b)
(b) Alternately, calculate the forward rate:
So sell GBP forward (ie., buy GBP spot). To buy GBP,one must borrow USD.
The two approaches would give the same answer.
48.14583.1)08.1/)05.1(50.1
)1/()1(
GBPUSD rrSF
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Effective Exchange Rate (called Trade
Weighted Index in Australia, TWI)
Effective exchange rate, is a multilateral
exchange rate which is a weighted average of
exchange rates of homeand foreign
currencies, with the weight for each foreigncountry equal to its share in trade. It measures
the average price of a homegood relative to
the average price of goods of trading partners,
using the share of trade with each country as
the weight for that country.
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The trade weighted index is an economic
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The trade weighted index is an economic
instrument used by economies to compare their
exchange rate against those of their major trading
partners. Those trading partners that constitute alarger portion of an economy's exports and
imports receives a higher index.
The trade weighted index is used to make a
complete comparison between one economy's
currency and other currencies it interacts with. It is
a much more comprehensive analysis thancomparing two currencies, for example, the
Australian dollar and the U.S. dollar.
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The interpretation of the effective exchange
rate is that if the index increases, the
purchasing power of that currency is higher(the currency strengthened against those of
the country's or area's trading partners).
A lower index means that the currencydepreciated (devaluation) so that you need
more of that currency to pay for imports.
(See Wikipedia:
http://en.wikipedia.org/wiki/Exchange_rate)
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