parity conditions
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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
May 2, 2023 ASHOK PATIL 6
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$4.25/bushel Implication: The demand for Australian wheat will increase
forcing up its price. The price of UK wheat will drop.
wheatUKA
wheatAust PSP £/$
May 2, 2023 ASHOK PATIL 8
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.
May 2, 2023 ASHOK PATIL 9
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.
May 2, 2023 ASHOK PATIL 10
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.
May 2, 2023 ASHOK PATIL 11
• 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.
May 2, 2023 ASHOK PATIL 12
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
May 2, 2023 ASHOK PATIL 13
• 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
May 2, 2023 ASHOK PATIL 17
• 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
May 2, 2023 ASHOK PATIL 20
• 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
May 2, 2023 ASHOK PATIL 21
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
May 2, 2023 ASHOK PATIL 22
• 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.
May 2, 2023 ASHOK PATIL 23
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
May 2, 2023 ASHOK PATIL 24
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)
May 2, 2023 ASHOK PATIL 25
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.
May 2, 2023 ASHOK PATIL 26
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
May 2, 2023 ASHOK PATIL 27
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%
May 2, 2023 ASHOK PATIL 28
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 $n
£n£0
May 2, 2023 ASHOK PATIL 29
Interest Parity: determining the currency in which to borrow
04985.0)406.01)(
2.000)000.2-9950.1(406.006.0
)406.01)(
2.000)000.2-9950.1(406.0
_
)4
1)(£)/($
£)/($-£)/($(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)
May 2, 2023 ASHOK PATIL 30
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).
May 2, 2023 ASHOK PATIL 31
• 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
May 2, 2023 ASHOK PATIL 32
)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.
May 2, 2023 ASHOK PATIL 33
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
May 2, 2023 ASHOK PATIL 34
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
*
May 2, 2023 ASHOK PATIL 35
• 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.
May 2, 2023 ASHOK PATIL 36
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
May 2, 2023 ASHOK PATIL 37
Summary
ConditionFisherOpenIII
rrr
UIPrrr
SSS
d
dnn
n
/;)1()1(
;)1()(
11*2
*
**
d
d
d
d
May 2, 2023 ASHOK PATIL 38
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
May 2, 2023 ASHOK PATIL 39
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
)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
Difference between spot and forward
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
May 2, 2023 ASHOK PATIL 40
ConditionFisherOpenIII
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
May 2, 2023 ASHOK PATIL 41
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
May 2, 2023 ASHOK PATIL 42
Difference in interest rates
drrn
Difference in expected inflation
**dn II
Fisher Open Condition (Effect)
Expected change in spot
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
Difference between spot and forward
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
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