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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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