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Chapter 4 The International Parity Conditions

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Title: Chapter 4 The International Parity Conditions


1
Chapter 4 The 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
2
  • Though this be madness,
  • yet there is method in it.
  • William Shakespeare

3
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/)
4
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
5
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
6
Purchasing power (dis)parity The Big Mac Index
The law of one price
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
International parity conditions that 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
16
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
17
Interest rate parity Which 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
18
Interest rate parity Which 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
19
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
20
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
21
Forward rates as predictors of future spot
rates Ftd/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
22
Forward rates as predictors of 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
23
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
24
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
25
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
26
Relative purchasing power parity over 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
27
Relative purchasing power parity in 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
28
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
29
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
30
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
31
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
32
The parity conditions are useful even 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
33
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
34
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
35
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/?

36
Actual versus expected change
St/
130/
120/
110/
Actual S1/ 110/
100/
ES1/ 90.91/
90/
time
37
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

38
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

39
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

40
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

41
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

42
Appendix 4-A Continuous time finance
  • Most theoretical and empirical research in
    finance is conducted in continuously compounded
    returns

43
Holding period returns are asymmetric
r1 100
r2 50
200
100
100
(1rTOTAL) (1r1)(1r2) (11)(1½) (2)(½)
1 ? rTOTAL 0
44
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

45
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)

46
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
47
Continuously compounded returns are additive
  • ln (1r1) (1r2) ... (1rT)
  • ln(1r1) ln(1r2) ... ln(1rT)
  • r1 r2 ... rT

48
The international parity conditions in 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

49
The international parity conditions in 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
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