parity conditions

The Basic International Parity Conditions

2

Introduction Managers of multinational firms, international

investors, importers and exporters, and government officials must deal with these fundamental issues: Are changes in exchange rates predictable? How are exchange rates related to interest rates? What, at least theoretically, is the “proper”

exchange rate?

To answer these questions we need to first understand the economic fundamentals of international finance, known as parity conditions.

3

Parity Conditions Parity Conditions provide an intuitive explanation of the

movement of prices and interest rates in different markets in relation to exchange rates.

The derivation of these conditions requires the assumption of Perfect Capital Markets (PCM). no transaction costs no taxes complete certainty

NOTE – Parity Conditions are expected to hold in the long-run, but not always in the short term.

4

Purchasing Power Parity (PPP)

PPP is based on the notion of arbitrage across goods markets and the basic building block of PPP is the Law of One Price (LOP).

LOP states that the price of an identical good should be the same in all markets (assuming no transactions costs). Otherwise, one could make profits by buying the

good in the cheap market and reselling it in the expensive market.

May 2, 2023 ASHOK PATIL 5

Arbitrage• Arbitrage is used in determining many of the relationships between

exchange rate (spot and forward), interest rates and inflation rates• Five key theoretical economic relationships between two countries

result from these arbitrage activities.– Purchasing power parity (PPP):

• Reln betn Inflation and expected change in spot price in future.– Fisher effect (FE):

• Reln betn interest and inflation – International Fisher Effect (IFE): Uncovered Interest Rate Parity (UIP)

• Interest rate and expected change in spot price in future– Interest rate parity (IRP):

• Interest rate and difference between forward and spot rate.– Forward rates as unbiased predictors of future spot rates (UFR)

• Difference between forward -spot rate and expected change in spot price in future


The law of one price• The law of one price states that in the absence

of frictions such as differential shipping costs and tariffs, the price of a product when converted into a common currency such as the US dollar, using the spot exchange rate, is the same in every country.

• E.g. the dollar equivalent of the price of wheat in Britain [i.e. S($/£)times the price of wheat in UK] is equal to the price of wheat in US.

wheatd

wheatn

wheatIN

wheatUS

wheatUK

wheatUS

pp

pporpp

*S(n/d)

*S($/Rs)__*S($/£)

7

LOP Example Commodity market: Suppose Pwheat, Aust = A$4/bushel and

Pwheat, UK = £2.5/bushel

Currency market: But if spot rate (A$/£) is 1.70

Using LOP equation, A$ equivalent price of wheat in UK is according to spot rate is A$1.70/£ *£2.5

= A$4.25/bushel Implication: The demand for Australian wheat will increase

forcing up its price. The price of UK wheat will drop.

wheatUKA

wheatAust PSP £/$


Absolute (or static) form of the PPP condition

– If the law of one price were to hold for each and every good and service, and we computed the cost of the same basket of goods and services in UK and US, the relationship is called absolute form of PPP condition, and it is written as: where PUS is price index in US, etc.

– This indicates that exchange rate is the equal to the ratio of their price indexes.

d

n

UK

US

dnUKUS

PP

PP

elyalternativorPPPP

S(n/d)or ;S($/£)

_*S(n/d)or ;*S($/£)

If price increases of ‘d’ country in relation to ‘n’ country, the currency d depreciates.


Problems associated with using price indexes to determine exchange rate

• Inflation vs relative price changes– Inflation results in increase in all prices, but relative price

changes indicate that not all prices move together. (analogy of lift with tennis balls)• If two lifts (and balls are stationary) move together then the

exchange rate would be constant • If we look at long intervals, then we can ignore the bouncing

balls inside. Lift movement is prominent and it determines the exchange rate.

• If we look at very short intervals, bouncing balls become prominent; and they would largely determine the exchange rate.


The relative (or dynamic) form of PPP

• Calculating the rate of basket of goods and services in every country is difficult.

• The needs and tastes are varied across countries and weight given to a particular commodity within a basket will differ.

• This means that even if the law of one price holds for each commodity, price indexes, which depend on the weights attached to each commodity, will not conform to the law of one price.

• Therefore, the relative form takes care of this situation by taking inflation into account.


• If the PPP condition holds in its absolute form at some moment in time, then for PPP to continue to hold, say at the end of one year, it is necessary that (Here Inf1 =change in P1 and so on) the following equation holds. Relative PPP talks about % changes in S, P of n and P of d.

(n/d)1S

(n/d)]

1S1S-

[1)1(

)1(

)1(*(n/d)][1)1( case, dmultiperiofor

or

(n/d)1S

(n/d)2]1S

1S-2[1)1(

)1(

),1(*(n/d)][1)1(

__*S(n/d)

_,_____)1(*(n/d)][1 S(n/d))1(

'

'

'

NSNS

NdInf

NnInf

InfSInf

SS

dInf

nInf

ISI

getweDividingPP

knowwePPPofformabsolutethefromInfPSInfP

NdN

Nn

dn

dn

ddnn

Expected exchange rate

If inflation increases of ‘d’ country in relation to ‘n’ country, the currency d depreciates.


Expected change in spot exchange rate equals difference in expected inflation

)/(1S]1

)[/(1S)/(2

11S1S-2

1(n/d)

111

(n/d)

11

(n/d)1

'

'

'

dnInfInfInf

dndnS

InfInfInfS

InfInfInf

S

InfInf

S

InfInf

S

d

dn

d

dn

d

dn

d

n

d

n

Expected change in exchange rate


• Several researchers have found that PPP holds better for high-inflation countries.

• More the time frame, better is the PPP relationship

14

Relative PPP Example Given inflation rates of 5% and 10% in Australia and the UK

respectively, what is the prediction of PPP with regards to $A/GBP exchange rate?

= (0.05 – 0.10)/(1 + 0.10) = - 0.045 = - 4.5%The general implication of relative PPP is that countries with high rates of inflation will see their currencies depreciate against those with low rates of inflation.

d

dn

III

SSS

11

12

Relative PPP

15

Relative Purchasing Power Parity

Applications of Relative PPP:

1. Forecasting future spot exchange rates.

2. Calculating appreciation in “real” exchange rates. This will provide a measure of how expensive a country’s goods have become (relative to another country’s).

16

Forecasting Future Spot Rates Suppose the spot exchange rate and expected inflation

rates are:

What is the expected ¥/$ exchange rate if relative PPP holds?

2% 5%;$;/¥90S ..t0¥/$, JapanSU II

$/¥43.8705.102.1$/¥90

11

$

¥0,/¥1$,/¥

II

SS tSPPP

t


• Suppose the price of a good is INR 121.6 and price of the same good in foreign currency is 2 USD.

• Then according to PPP, the exchange rate should be – 121.6/2=INR60.8/USD

• But, if the spot exchange rate is – INR 61.2/USD,

• then the price of this foreign good in INR will be 2*61.2 = 122.4 – i.e. if I import it will cost me 122.4

• Real exchange rate = Actual rate/PPP rate• =

pricedomesticrateactualbypriceimport

ratePPPrateActualSpotrateexchangeal

_____

6.1212*2.61

2*8.604.122

2*8.602*2.61

8.602.61

____Re

18

Real Exchange Rate By definition, the real exchange rate measures

deviations from PPP. That is, changes in the spot exchange rate that do not

reflect differences in inflation rates between the two currencies in question.

Real Exchange Rate: PPP1t

1t

SSE

0065.1hcin goods domestic of price

hcin goodsforeign of price6.12120.61*2

_2/6.121_/20.61_

/80.60_/20.61_

USDINRUSDINRE

USDINRUSDINRE

19

Real Exchange Rate By definition, the real exchange rate measures

deviations from PPP. That is, changes in the spot exchange rate that do not

reflect differences in inflation rates between the two currencies in question.

Real Exchange Rate: PPP1t

1t

SSE

hcin goods domestic of pricehcin goodsforeign of price/*

//

1

1

PhcfchcPfc

PfcPhcfchcE

SS

E pppt

t


• Therefore, if nominal exchange rate appreciates (hc/fc) equal to domestic inflation increase compared to foreign inflation, there is no real exchange rate change.

• However, if nominal exchange rate appreciates more than domestic inflation increase compared to foreign inflation, real exchange rate also appreciates.

• If nominal exchange rate appreciates less than domestic inflation increase compared to foreign inflation, real exchange rate depreciates.

flationDomesticInlationForeignInf

SS

SS

EE

ngInSubstitutiflationDomesticInlationForeignInf

SS

PP

PPPP

PP

PPPP

SSbut

SS

SS

EE

SSandE

SSE

pppt

pppt

t

t

t

t

pppt

pppt

fct

fct

hct

hct

fct

fct

hct

hct

fct

hct

fct

hct

pppt

pppt

pppt

pppt

t

t

t

t

pppt

ttppp

t

tt

*change rate exchange Nominal*

,

*/1*

//,...

*

Then,

1

11

1

1

1

1

1111

1

11

1

11


99997.095238.0*00493.195710.0*Re

*fc)change(hc/ rate exchange Nominal*)/(___Re

99997.095710.095238.0*

90.602.61

*fc)change(hc/ rate exchange Nominal*)/(__Re

,95710.095238.0

1.22*

05.127/6.1211*

/1

1.22*

6.12105.127

2/6.1211.2/05.127

//

80.6050.60,...

80.6050.60*

90.602.61___Re

Then,50.6090.60_

80.602.61

1

11

1

1

1111

1

1

11

REalal

lationForeignInfflationDomesticInfchcchangerateexchangeal

REal

flationDomesticInlationForeignInf

SS

SSfchcrateexchangeal

EE

ngInSubstitutiflationDomesticInlationForeignInf

SS

PP

PPPPPP

SSbut

changerateexchangealEE

SSEand

SSE

pppt

pppt

t

t

t

t

pppt

pppt

fct

fct

hct

hct

fct

hct

fct

hct

pppt

pppt

t

t

pppt

ttppp

t

tt


• Therefore, if nominal exchange rate appreciates (hc/fc) equal to domestic inflation increase compared to foreign inflation, there is no real exchange rate change.

• However, if nominal exchange rate appreciates more than domestic inflation increase compared to foreign inflation, real exchange rate also appreciates.

• If nominal exchange rate appreciates less than domestic inflation increase compared to foreign inflation, real exchange rate depreciates.


Summary

yr 1an greater th is N and

yr 1for is I where,muliperiodfor PPP of form Relative ;)1(

)1(

(n/d)1S

(n/d)

1)-(2 periodgiven afor is I wherePPP of form Relative;11S

1S-2

1)-(2 periodgiven afor is I wherePPP, of form Relative ;)1(

)1(

(n/d)1S

(n/d)2

PPP of form Absolute ;S(n/d)

price one of Law ;S(n/d)

NdI

NnINS

IIISdI

nIS

PP

PP

d

dn

d

n

dn


Derivation of covered interest parity

• To derive– First we will see how to make short-term

investment and borrowing decisions in the international context.

– Secondly, we will see how checking for highest investment yield or lowest borrowing costs causes the costs and yields of different currencies toward equality. (an interest parity condition)


Background and definitions

• This phenomenon occurs in money market, where short-term borrowing and investment takes place.

• Some forward contract have a short-term maturity and hence allow money market borrowers and investors to avoid foreign exchange risk and exposure.– Foreign exchange risk is the result of uncertainty in asset

or liability values or income flows due to unexpected changes in exchange rates.

– Foreign exchange exposure is the amount that is subject to foreign exchange risk.


Interest Parity: determining the currency of investment

(1+r$/4)

1/S(

$/ £

) F1/4 ($/ £)

(1+r£ /4)

$0 $n

£n£0

)4

1)(d)/(

d)/(-d)/((4

_

)4

1)(£)/($

£)/($-£)/($(4

_

)4

1(£)/($£)/($)

41(

d4/1d

£4/1£$

£4/1$

rnS

nSnFrr

generalin

rS

SFrr

grearranginby

rSFr

n


Interest Parity: determining the currency of investment

04985.0)01015.0(06.0

)406.01)(

2.0002.000-9950.1(406.0

)4

1)(d)/(

d)/(-d)/((4

_

)4

1)(£)/($

£)/($-£)/($(4

_

)4

1(£)/($£)/($)

41(

d4/1d

£4/1£$

£4/1$

n

n

n

r

r

rnS

nSnFrr

generalin

rS

SFrr

grearranginby

rSFr

(1+?/4)

1/S(

$/ £

) F1/4 ($/ £)

(1+.06/4)

100 101.24625

50.

75

50

S($/ £)=2 F1/4($/ £)= 1.9950 (hedged today)

Ri= 4.985%


Interest Parity: determining the currency in which to borrow

)4

1)(d)/(

d)/(-d)/((4

_

)4

1)(£)/($

£)/($-£)/($(4

_

)4

1(£)/($£)/($)

41(

___

d4/1d

£4/1£$

£4/1$

rnS

nSnFrr

generalin

rS

SFrr

grearranginby

rSFr

ifswapviapoundsborrow

n

(1+r$/4)

1/S(

$/ £

) F1/4 ($/ £)

(1+r£ /4)

$0 $n

£n£0


Interest Parity: determining the currency in which to borrow

04985.0)406.01)(

2.000)000.2-9950.1(406.006.0

)406.01)(

2.000)000.2-9950.1(406.0

_

)4

1)(£)/($

£)/($-£)/($(4

_

)4

1(£)/($£)/($)

41(

___

£4/1£$

£4/1$

n

n

r

r

generalin

rS

SFrr

grearranginby

rSFr

ifswapviapoundsborrow

(1+0.06/4)

1/S(

$/ £

) F1/4 ($/ £)

(1+.06 /4)

100 101.2

4625

50.7

5

50

S($/ £)=2 F1/4($/ £)= 1.9950 (hedged today)


Borrowing and investing for arbitrage profit

(1+r$/4)

1/S(

$/ £

) F1/4 ($/ £)

(1+r£ /4)

$0 $n

£n£0

Borrow, let us say, 1$ for 3 months; so you have to pay (1+r$/4). At the same time engage into selling £ forward

Use this borrowed $ to buy £;So you will receive 1/S($/£) pounds.

Invest these pounds for 3 months;so you will receive 1/S($/£)*(1+r£ /4)pounds at the end of 3 months.

Honor the forward contract by sellingthese pounds received toreceive F1/4($/ £)/S($/£)*(1+r£ /4) dollars

Naturally this amount should be equal to (1+r$/4) or else arbitrage is possible

Borrowing and investing in this way withexchange rate risk hedged in the forward market is known as covered interest arbitrage(CIA).


• The previous equations were applicable for short term investment and borrowing. If we allow for long-term investing and borrowing and consider compound interest, then the short-term equation

could be written as )1()1(

)/()/(

periodgiven afor isr where,)1()1(

)/()/(

periodgiven afor isr where),1()/()/()1(

d

d

d

rr

dnSdnF

rr

dnSdnF

rdnSdnFr

n

n

n

yr 1for isr andyr 1n where,)1()1(

n/d)()/(

d

N

NnN

rr

SdnF


)1()1()/()/(or )/()/(

)1(n/d)(

)///(

1

1n/d)(

)/(1)1()1(

n/d)()/(

)1()1(

year one of case for the ),1(n/d)(

)/()1(

)1(n/d)(

)/()1(

dd

d

d

d

d

d

rrdnSdnFdnSdnS

rrrF

ordS(n

d)d)-S(nnF)r(

rrS

dnFrr

SdnF

rr

rS

dnFr

rS

dnFr

nn

d

dn

n

n

Nn

NNNn

Interest rate parity:Difference in interest rates is equal to difference in spot and forward

If interest increases of ‘d’ country in relation to ‘n’ country, the currency d sells at discount in future.


Summary (IRP)

)1(n/d)()/(-)/(

yr 1for isr andyr 1N where,)1()1(

n/d)()/(

periodgiven afor isr where,)1()1(

)/()/(

periodgiven afor isr where),1()/()/()1(

d

d

d

d

d

rrr

SdnSdnFrr

SdnF

rr

dnSdnF

rdnSdnFr

n

N

NnN

n

n


Combining PPP and interest parity The uncovered interest parity condition: Approximation to International

Fisher effect

• We know that speculation will make forward exchange rate approximately equal to the expected future spot exchange rate.

• If this does not hold, then arbitrage is possible. Substituting this in our covered interest parity condition

• We get,

)/(F)/( N* dndnSN

NNNn r

SdnF

r )1(n/d)(

)/()1( d

NNNn r

SdnS

r )1(n/d)(

)/()1( d

*


• But we know, S*N= S(n/d) (1+S*’)N

• Where S*’ is the average annual expected rate of change of the exchange rate. Substituting this in above equation gives us

)1()(

11*2

1)1()1(1]

11*2[1

)1()1())/(S(1

)1))(/(S(1)1(

sidesboth ofroot nth taking)1()S(1)1(

)1(n/d)(

)S)(1/()1(

d

d

d

d

*'

d*'

d*'

d

*'

rrr

SSS

rr

SSS

rrdn

rdnr

rr

rSdnSr

n

n

n

n

NNNn

NN

Nn

The uncovered interest parity condition

If interest increases of ‘d’ country in relation to ‘n’ country, the currency d Is expected to depreciate.


ConditionFisherOpenIII

rrr

II

rr

Ir

Ir

DividingUIPffectnalFisherEInternatiorr

PPPISI

d

dnn

d

nn

dn

n

n

dn

/;)1()1(

1)1()1(1

)1()1(

)1()1(

)1()1(

)(:)1)(S(1)1(

:)1(*][1)1(

*

**

d

d

*

*

d

*d

*

d*'

*'**

We know


Summary


rrr

UIPrrr

SSS

d

dnn

n

/;)1()1(

;)1()(

11*2

*

**

d

d

d

d


Difference in interest rates

d

d

1 rrrn

Difference in expected inflation

*

**

1 d

dn

III

Fisher Open Condition (Effect)

Expected change in spot

1

1*

'* )/(S

SSdnS N

RelativePPP

*

**

d

d

11 d

dnn

III

rrr

*'

d

d

d*'

S)1(

)1)(S(1)1(

rrr

orrr

n

n

International Fisher effectOR Uncovered Interest RateParity (UIP)

'**

**

11

1

*'**

)1(

(n/d)S(n/d)

]S

S-[1

)1()1(

)1(*][1)1(

SIII

or

SSII

ISI

d

dn

NNN

d

Nn

dn

Difference between spot and forward

n/d)()/(-)/(

SdnSdnF

n/d)()/(-)/(

)1(

_____

)1(n/d)(

)/()1(

d

d

d

SdnSdnF

rrr

yearoneofcasethefor

rS

dnFr

n

NNNn

Expectations theoryUnbiased forward rate

Covered Interest Rate Parity (CIP)

)/()/( * dnSdnF N



d

d

1 rrrn


*

**

1 d

dn

III



1

1*

'* )/(S

SSdnS N

RelativePPP

)1()1(

)1()1(

*

*

d d

nn

II

rr

1

*

d )1()1(

SS

rr Nn

International Fisher effectOR Uncovered Interest RateParity (UIP)

(n/d)S(n/d)

)1()1(

1

NN

d

Nn SII


n/d)()/(-)/(

SdnSdnF

n/d)()/(

)1()1(

d SdnF

rr N

N

Nn

Expectations theoryUnbiased forward rate

Covered Interest Rate Parity (CIP)

)/()/( * dnSdnF N

Simplified version



rrr

UIPrrr

SSS

IRPrrr

SdnSdnFrr

SdnF

rr

dnSdnF

NdI

NnINS

IIISdI

nIS

PP

PP

d

dnn

n

n

N

NnN

n

d

dn

d

n

dn

/;)1()1(

;)1()(

11*2

;)1(n/d)(

)/(-)/(

yr 1for isr andyr 1N where,)1()1(

n/d)()/(

IRP period;given afor isr where,)1()1(

)/()/(

yr 1an greater th is N and

yr 1for is I where,muliperiodfor PPP of form Relative ;)1(

)1(

(n/d)1S

(n/d)

1)-(2 periodgiven afor is I wherePPP of form Relative;11S

1S-2

1)-(2 periodgiven afor is I wherePPP, of form Relative ;)1(

)1(

(n/d)1S

(n/d)2

PPP of form Absolute ;S(n/d)

price one of Law ;S(n/d)

*

**

d

d

d

d

d

d

d

d


The interrelationship of parity conditions• The expectations model of PPP,

• Now, we can compare uncovered interest parity condition with the above equation

•

'***

'**

'***'**

*'**

,

1)1(

)1(*][1)1(

SII

xSignoringI

xSIISI

ISI

dn

d

ddn

dn

*d

*

**d

'***

*'d

,

,

S

dnn

dnn

dn

n

IrIr

grearranginIIrr

WehaveSII

rr

Fisher-open condition

Approximations



drrn


**dn II



1

1*

'* )/(S

SSdnS N

PPP

*d

*

**d

dnn

dnn

IrIr

orIIrr

*'d

d*'

S

)1)(S(1)1(

rr

orrr

n

n

International Fisher effect (uncovered interest rateparity

'***

11

1

*'**

(n/d)S(n/d)

]S

S-[1

)1()1(

)1(*][1)1(

SII

or

SSII

ISI

dn

NNN

d

Nn

dn


n/d)()/(-)/(

SdnSdnF

n/d)()/(-)/(_____

)1(n/d)(

)/()1(

d

d

SdnSdnFrr

yearoneofcasethefor

rS

dnFr

n

NNNn

Expectations theoryOr Unbiased forward rate

Interest Rate Parity

)/()/( * dnSdnF N

Approximate version

parity conditions

Economy & Finance