Chapter 4 The International Parity Conditions

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Chapter 4The International Parity Conditions
Learning objectives ? The Law of One
Price Exchange rate equilibrium ? Internation
al parity conditions Relate exchange rates to
cross-currency interest rate and inflation
differentials Interest rate parity the most
important of these relations The
international parity conditions can be useful in
exchange rate forecasting ? The real exchange
rate
Butler, Multinational Finance, 4e
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• Though this be madness,
• yet there is method in it.
• William Shakespeare

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Prices
Prices appear as Upper Case Symbols Ptd
price of an asset at time t in currency d
Std/f spot exchange rate at time t in
currency d Ftd/f forward exchange rate
between currencies d and f also E expectatio
n operator (e.g. ESt/)
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Changes in prices(rates of change)
Changes in a price appear as lower case
symbols rtd an assets return in currency
d during period t ptd inflation in currency d
in period t étd real interest rate in currency
d in period t std/f change in the spot rate
during period t
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The law of one price

• Equivalent assets sell for the same price
• (also called purchasing power parity, or PPP)
• Seldom holds for nontraded assets
• Cant compare assets that vary in quality
• May not hold precisely when there are market
  frictions

The law of one price
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Purchasing power (dis)parityThe Big Mac Index
The law of one price
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An example of PPP The price of gold

• Suppose P 250/oz in London
• P 400/oz in Berlin
• The law of one price requires
• Pt Pt St/
• Þ 250/oz (400/oz) (0.6250/)
• or 1/(0.6250/) 1.6000/
• If this relation does not hold, then there is an
  opportunity to lock in a riskless arbitrage
  profit.

The law of one price
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An example with transactions costs

• Gold dealer A Gold dealer B

401.40/oz Offer
401.00/oz Bid
Sell high to B
FX dealer 1.599/ bid 1.601/ ask
Buy low from A
250.25/oz Offer
250.00/oz Bid
The law of one price
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Arbitrage profit
Pay 250.25 million to buy 1 million oz from A
(1 million oz)
-250,250,000
Sell 1 million oz to B for 401m
Arbitrage profit 218,500
401,000,000
-(1 million oz)
Buy s with s at the 1.601/ pound sterling ask
price
250,468,500
-401,000,000
The law of one price
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Cross exchange rate equilibrium
Sd/e Se/f Sf/d 1 If Sd/eSe/fSf/d either Sd/e, Se/f or Sf/d must rise Þ For each
spot rate, buy the currency in the denominator
with the currency in the numerator If
Sd/eSe/fSf/d 1, then either Sd/e, Se/f or Sf/d
must fall Þ For each spot rate, sell the currency
in the denominator for the currency in the
numerator
The law of one price
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A cross exchange rate table
Sd/f Sf/d 1 Û Sd/f 1 / Sf/d
C SFr CNY UK () 1.000 0.500 0.713 0.00
44 0.435 0.487 0.066 Canada (C) 1.998 1.000 1.4
26 0.0088 0.870 0.973 0.131 Euro-zone
() 1.400 0.700 1.000 0.0061 0.610 0.682 0.092
Japan () 228.1 114.0 162.8 1.0000 99.30 111.0 1
4.94 Swiss (SFr) 2.296 1.147 1.638 0.0101 1.000
1.117 0.150 US () 2.054 1.027 1.466 0.0090 0.894
1.000 0.135 China (CNY) 15.23 7.613 10.87 0.066
7 6.630 7.414 1.000
The law of one price
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Cross exchange rates and triangular arbitrage
Suppose SRbl/ Rbl 5.000/ Û S/Rbl
0.2000/Rbl S/ 0.01000/ Û S/
100.0/ S/Rbl 20.20/Rbl Û SRbl/ Rbl
0.04950/ SRbl/ S/ S/Rbl (Rbl
5/)(.01/)(20.20/Rbl) 1.01 1
The law of one price
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Cross exchange rates and triangular arbitrage

• SRbl/ S/ S/Rbl 1.01 1
• Þ Currencies in the denominators are too high
  relative to the numerators, so
• sell dollars and buy rubles
• sell yen and buy dollars
• sell rubles and buy yen

The law of one price
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An example of triangular arbitrage
SRbl/ S/ S/Rbl 1.01 1 Sell 1 million
and buy Rbl 5 million Sell 100 million yen and
buy 1 million Sell Rbl 4.950 million and buy
100 million Þ Profit of 50,000 rubles
10,000 at Rbls5.000/ or 1 of the initial
amount
The law of one price
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International parity conditionsthat span both
currencies and time

• Interest rate parity Less reliable
  linkages
• Ftd/f / S0d/f (1id)/(1if)t EStd/f /
  S0d/f
• (1Epd)/(1Epf)t
• where
• S0d/f todays spot exchange rate
• EStd/f expected future spot rate
• Ftd/f forward rate for time t exchange
• i a nominal interest rate
• p an expected inflation rate

The international parity conditions
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Interest rate parity

• Ftd/f / S0d/f (1id) / (1if) t
• Forward premiums and discounts are entirely
  determined by interest rate differentials.
• This is a parity condition that you can trust.

The international parity conditions Interest
rate parity
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Interest rate parityWhich way do you go?

• If Ftd/f/S0d/f (1id)/(1if)t
• then so...
• Ftd/f must fall Sell f at Ftd/f
• S0d/f must rise Buy f at S0d/f
• id must rise Borrow at id
• if must fall Lend at if

The international parity conditions Interest
rate parity
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Interest rate parityWhich way do you go?

• If Ftd/f/S0d/f
• then so...
• Ftd/f must rise Buy f at Ftd/f
• S0d/f must fall Sell f at S0d/f
• id must fall Lend at id
• if must rise Borrow at if

The international parity conditions Interest
rate parity
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Interest rate parity is enforced through covered
interest arbitrage

• An Example
• Given i 7 S0/ 1.20/
• i 3 F1/ 1.25/
• F1/ / S0/ (1i) / (1i)
• 1.041667 1.038835
• The fx and Eurocurrency markets are not in
  equilibrium.

The international parity conditions Interest
rate parity
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Covered interest arbitrage
1,000,000

• 1. Borrow 1,000,000
• at i 7
• 2. Convert s to s
• at S0/ 1.20/
• 3. Invest s
• at i 3
• 4. Convert s to s

-1,070,000
833,333
-1,000,000
858,333
-833,333
1,072,920
-858,333
The international parity conditions Interest
rate parity
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Forward rates as predictorsof future spot
ratesFtd/f EStd/f or Ftd/f / S0d/f
EStd/f / S0d/f

• That is, forward rates are unbiased estimates of
  future spot rates
• Speculators will force this relation to hold on
  average

The international parity conditions Less
reliable relations
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Forward rates as predictorsof future spot rates

• EStd/f / S0d/f Ftd/f / S0d/f
• Speculators will force this relation to hold on
  average
• For daily exchange rate changes, the best
  estimate of tomorrow's spot rate is the current
  spot rate
• As the sampling interval is lengthened, the
  performance of forward rates as predictors of
  future spot rates improves

The international parity conditions Less
reliable relations
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Yen-per- forward parity
S3//S0/ - 1
F3//S0/ - 1
Based on 3-month forward and spot exchange rates.
The international parity conditions Less
reliable relations
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Relative purchasing power parity (RPPP)

• Let Pt a consumer price index level at time t
• Then inflation pt (Pt - Pt-1) / Pt-1
• EStd/f / S0d/f (EPtd / EPtf) / (P0d
  /P0f)
• (EPtd/P0d) / (EPtf/P0f)
• (1Epd)t / (1Epf)t
• where pd and pf are geometric mean inflation
  rates.

The international parity conditions Less
reliable relations
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Relative purchasing power parity (RPPP)

• EStd/f / S0d/f (1Epd)t / (1Epf)t
• Speculators will force this relation to hold on
  average
• The expected change in a spot exchange rate
  should reflect the difference in inflation
  between the two currencies.
• This relation only holds over the long run.

The international parity conditions Less
reliable relations
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Relative purchasing power parityover monthly
intervals
S1//S0/ - 1
(1p)/(1p) - 1
Based on monthly spot exchange rate changes and
monthly inflation
The international parity conditions Less
reliable relations
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Relative purchasing power parityin the long run
(1995-2006)
S1f//S0f/ - 1
Venezuela
Iran
Australia, Canada, Denmark, India, Japan, Korea,
Malaysia, New Zealand, Norway, Pakistan,
Singapore, S. Africa, Sweden, Switzerland,
Thailand, United Kingdom
Indonesia
Columbia
Brazil
(1pf)/(1p) - 1
Based on exchange rates and monthly inflation
from IMF Statistics
The international parity conditions Less
reliable relations
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International Fisher relation (Fisher Open
hypothesis)
The Fisher equation provides a starting
point (1i) (1é)(1p) i nominal
interest rate é real interest rate p
inflation rate or i é p for small é
and p
The international parity conditions Less
reliable relations
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International Fisher relation (Fisher Open
hypothesis)

• (1id)/(1if)t (1pd)/(1pf)t
• Fisher relation (1i) (1é)(1p)
• If real rates of interest are equal across
  currencies (éd éf), then
• (1id)/(1if)t
• (1éd)(1pd)t / (1éf)(1pf)t
• (1pd)/(1pf)t

The international parity conditions Less
reliable relations
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International Fisher relation (Fisher Open
hypothesis)

• (1id)/(1if)t (1pd)/(1pf)t
• Speculators will force this relation to hold on
  average
• If real rates of interest are equal across
  countries, then interest rate differentials
  merely reflect inflation differentials
• This relation is unlikely to hold at any point in
  time, but should hold in the long run

The international parity conditions Less
reliable relations
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Summary Intl parity conditions
International Fisher relation
Interest rates (1id)/(1if)t
Inflation rates (1pd)/(1pf)t
Interest rate parity
Relative PPP
EStd/f / S0d/f Expected change in the spot rate
Ftd/f / S0d/f Forward-spot differential
Forward rates as predictors of future spot rates
The international parity conditions
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The parity conditions are usefuleven if they are
poor FX rate predictors

• Interest rate differentials reflect the
  difference in the opportunity cost of capital
  between two currencies
• Forward prices similarly reflect the relative
  opportunity cost of capital, and have some
  predictive ability over long horizons
• Inflation differentials reflect the rate of
  change of relative purchasing power between two
  currencies

The international parity conditions Forecasting
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The real exchange rate

• The real exchange rate adjusts the nominal
  exchange rate for differential inflation since an
  arbitrarily defined base period

The real exchange rate
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Change in the nominal exchange rate

• Example
• S0/ 100/
• S1/ 110/
• p 0
• p 10
• s1/ (S1/S0/)/S0/ 0.10,
• or a 10 percent nominal change

The real exchange rate
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The expected nominal exchange rate

• But RPPP implies
• ES1/ S0/ (1 p)/(1 p)
• 90.91/
• What is the change in the nominal exchange rate
  relative to the expectation of 90.91/?

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Actual versus expected change
St/
130/
120/
110/
Actual S1/ 110/
100/
ES1/ 90.91/
90/
time
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Change in the real exchange rate

• In real (or purchasing power) terms, the dollar
  has appreciated by
• (110/) / (90.91/) - 1 0.21
• or 21 percent more than expected

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Change in the real exchange rate

• (1xtd/f) (Std/f / St-1d/f) (1ptf)/(1ptd)
• where
• xtd/f percentage change in the real exchange
  rate
• Std/f the nominal spot rate at time t
• ptd inflation in currency d during period t
• ptf inflation in currency f during period t

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Change in the real exchange rate

• Example S0/ 100/ ? S1/ 110/
• Ep 0 and Ep 10
• (1xt/) (110/)/(100/)1.10/1.00
• 1.21
• or a 21 percent increase in relative purchasing
  power

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Behavior of real exchange rates

• Deviations from PPP
• can be substantial in the short run
• and can last for several years
• Both the level and variance of the real exchange
  rate are autoregressive

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Real value of the dollar (1970-2006)
Japan
Dollar over-valued relative to average
UK
Dollar under-valued relative to average

• Mean level 100 for each series

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Appendix 4-AContinuous time finance

• Most theoretical and empirical research in
  finance is conducted in continuously compounded
  returns

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Holding period returnsare asymmetric
r1 100
r2 50
200
100
100
(1rTOTAL) (1r1)(1r2) (11)(1½) (2)(½)
1 ? rTOTAL 0
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Continuous compounding

• Let
• r holding period (e.g. annual) return
• r continuously compounded return
• r ln (1r) Û (1 r) er
• where
• ln(.) is the natural logarithm
• with base e 2.718

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Properties of natural logarithms(for x 0)

• eln(x) ln(ex) x
• ln(AB) ln(A) ln(B)
• ln(At) (t) ln(A)
• ln(A/B) ln(AB-1)
• ln(A) - ln(B)

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Continuous returns are symmetric
r1 69.3
r2 -69.3
200
100
100
rTOTAL ln(1r1)(1r2) ln(1r1)
ln(1r2) r1r2 0.693-0.693 0.000
? rTOTAL 0
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Continuously compounded returns are additive

• ln (1r1) (1r2) ... (1rT)
• ln(1r1) ln(1r2) ... ln(1rT)
• r1 r2 ... rT

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The international parity conditionsin continuous
time

• Over a single period
• ln(F1d/f / S0d/f ) i d i f
• Ep d Ep f
• Es d/f
• where s d/f, p d, p f, i d, and i f are
  continuously compounded

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The international parity conditionsin continuous
time

• Over t periods
• ln(Ftd/f / S0d/f ) t (i d i f )
• t (Ep d Ep f )
• t Es d/f
• where s d/f, p d, p f, i d, and i f are in
  continuous returns