ed5 04 the international parity conditions(1)

Butler / Multinational Finance Chapter 4 The International Parity Conditions 4-1

Chapter 4The International Parity Conditions

Learning objectives

The Law of One Price– Exchange rate equilibrium

International parity conditions– These relate exchange rates to cross-currency interest rate and inflation differentials– Interest rate parity is the most important relation

The real exchange rate– Measuring inflation-adjusted currency values


“Though this be madness, yet there is method in it.”

William Shakespeare


Notation

Upper Case Symbols = Prices

lower case symbols = changes in a price

Ptd = price of an asset in currency d at time

t

pd = inflation rate in currency d (% change in CPI)

id = nominal interest rate in currency d

ʀd = real interest rate in currency d

Std/f = spot exchange rate at time t between

d and f

std/f = change in the spot rate during period t

Ftd/f = forward exchange rate between d and

f

E[…] = expectations operator (e.g., E[St$/£])

✓ The law of one price and interest rate parityLess reliable parity conditions

The real exchange rateAppendix: Continuous compounding

Notation and definition ✓Parity in commodities marketsCross exchange rate equilibriumInterest rate parity and covered interest arbitrage


The law of one price

“Equivalent assets sell for the same price”

(also called purchasing power parity, or PPP)

- Seldom holds for nontraded assets

- Can’t compare assets that vary in quality

- May not be precise when there are market

frictions



Notation and definition ✓Parity in commodities marketsCross exchange rate equilibriumInterest rate parity and covered interest arbitrage


As of October 14, 2010 Relative

P$ = Pf / Sf/$ Pf Sf/$ P$to the $

USA ($) 3.71 1.000 3.71 100%Brazil (BRL) 8.74 1.661 5.26 142%Britain (£) 2.29 0.631 3.63 98%Canada (C$) 4.20 1.005 4.18 113%Euro-zone (€) 3.43 0.716 4.79 129%China 14.51 6.657 2.18 59%J apan (¥) 319.9 81.82 3.91 105%S. Korea (Won) 3392 1119 3.03 82%Switzerland (SFr) 6.49 0.9575 6.78 183%

Purchasing power (dis)parity: The Big Mac Index

Notation and definition ✓Parity in commodities markets Cross exchange rate equilibriumInterest rate parity and covered interest arbitrage



Sources: The Economist ( “The Big Mac Index,” Oct. 14, 2010) and www.oanda.com


An example of PPP: The price of gold

Suppose P£ = £940/oz in LondonP$ = $1504/oz in New York

The law of one price requires:

Pt£ = Pt

$ St£/$

Þ St£/$ = Pt

£/Pt$ = (£940/oz) / ($1504/oz)

= £0.6250/$

or St$/£ = 1 / St

£/$ = 1 / (£0.6250/$) = $1.6000/£

If this relation does not hold, then there may be an opportunity to lock in a riskless arbitrage profit.

Notation and definitionParity in commodities markets ✓Cross exchange rate equilibriumInterest rate parity and covered interest arbitrage




London gold dealer New York gold dealer $1522/oz Ask

$1510/oz Bid

£940/oz Ask

£930/oz Bid

Buy low from London

Sell high to New York

FX dealer$1.599/£ bid$1.601/£ ask

An example with transactions costs





+£943,161

-$1,510,000

+$1,510,000

+(1000 oz)

Cover your £ position at the $1.601/£ ask price for pounds

Buy 1000 oz at London’s £940/oz offer price

Sell 1000 oz at NY’s $1510/oz bid price

-£940,000

-(1000 oz)

Arbitrage profit

+£3,161

Arbitrage profit in gold


= ($1,510,000)/($1.601/£)




A cross exchange rate table

Sd/f Sf/d = 1 Sd/f = 1 / Sf/d

$ € ¥ £ SFr INR BRL CNYUS ($) 1 1.481 0.012 1.671 1.157 0.023 0.637 0.154Euro (€) 0.675 1 0.008 1.128 0.781 0.015 0.430 0.104Japan (¥) 81.23 120.3 1 135.7 93.99 1.836 51.72 12.51

UK (£) 0.598 0.886 0.007 1 0.693 0.014 0.381 0.092Swiss (SFr) 0.864 1.278 0.011 1.444 1 0.020 0.550 0.133India (INR) 44.25 65.53 0.545 73.94 51.20 1 28.18 6.817Brazil (BRL) 1.570 2.326 0.019 2.624 1.817 0.036 1 0.242China (CNY)6.491 9.613 0.080 10.85 7.511 0.147 4.133 1



Notation and definitionParity in commodities markets Cross exchange rate equilibrium ✓Interest rate parity and covered interest arbitrage


Cross exchange rate equilibrium

Sd/e Se/f Sf/d = 1

If Sd/eSe/fSf/d < 1, then 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





Cross exchange rates and triangular arbitrage

An example with U.S. dollars ($), Japanese yen (¥), & Egyptian pounds (E£)

SE£/$ = E£5.000/$ Û S$/E£ = $0.2000/E£

S$/¥ = $0.01000/¥ Û S¥/$ = ¥100.0/$

S¥/E£ = ¥20.20/E£ Û SE£/¥ » E£0.04950/¥

SE£/$ S$/¥ S¥/E£

= (E£5/$)($0.01/¥)(¥20.20/E£)

= 1.01 > 1





Cross exchange rates and triangular arbitrage

SE£/$ S$/¥ S¥/E£ = 1.01 > 1

Þ Currencies in the denominators are too high relative to the numerators, so…

sell dollars and buy Egyptian pounds

sell yen and buy dollars

sell Egyptian pounds and buy yen








+¥100m

-E£4.95m

+$1m

+E£5m

Sell E£4.95m and buy ¥100m at ¥20.20/E£

Sell $1m and buy E£5m at E£5/$

Sell ¥100m and buy $1m at $0.01/¥

-$1m

-¥100m

Arbitrage profit

= +E£50,000

or about 1% of the initial amount

An example of triangular arbitrage


Interest rate parity Ft

d/f / S0d/f = [(1+id)/(1+if)]t

where S0

d/f = today’s spot exchange rate

E[Std/f] = expected future spot rate

Ftd/f = forward rate for time t

exchange i = a nominal interest rate p = an expected inflation rate

Less reliable linkages=E[St

d/f] / S0d/f

=[(1+E[pd])/(1+E[pf])]t

International parity conditions that span both currencies and time



Notation and definitionParity in commodities markets Cross exchange rate equilibriumInterest rate parity and covered interest arbitrage ✓


Interest rate parity

Ftd/f / S0

d/f = [ (1+id) / (1+if) ]t

Forward premiums and discounts are entirely determined by interest rate differentials.

This is a parity condition that you can trust.





Invest in eurosFVt

€ = PV0€

(1+i€)t

Invest in dollarsFVt

$ = PV0$

(1+i$)t

Moving $s today into €s tomorrow

Time 0 Time t

Convert to euros at Ft

$/€

Convert to euros at S0

$/€

V0$

Vt€

Interest rate parity





Interest rate parity: Which way do you go?

If Ftd/f/S0

d/f > [(1+id)/(1+if)]t

then so...

Ftd/f must fall Sell f at Ft

d/f

S0d/f must rise Buy f at S0

d/f

id must rise Borrow at id if must fall Lend at if





Interest rate parity: Which way do you go?

If Ftd/f/S0

d/f < [(1+id)/(1+if)]t

then so...

Ftd/f must rise Buy f at Ft

d/f

S0d/f must fall Sell f at S0

d/f

id must fall Lend at id if must rise Borrow at if





Interest rate parity is enforced through “covered interest arbitrage”

An Example:

Given: i$ = 7% S0$/£ = $1.20/£

i£ = 3% F1$/£ = $1.25/£

F1$/£ / S0

$/£ > (1+i$) / (1+i£) 1.041667 > 1.038835

The currency and Eurocurrency markets are not in equilibrium





-£833,333

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 at F1

$/£ = $1.25/£

5. Arbitrage profit = $2,920

-$1,070,000

+$1,000,000

+£833,333-

$1,000,000+£858,333

+$1,072,920-

£858,333

Covered interest arbitrage





Relative purchasing power parity (RPPP)

Let Pt = a consumer price index level at time t

Then inflation pt = (Pt - Pt-1) / Pt-1

E[Std/f] / S0

d/f = (E[Ptd] / E[Pt

f]) / (P0d / P0

f)

= (E[Ptd] / P0

d) / (E[Ptf] / P0

f)

= (1+E[pd])t / (1+E[pf])t

where pd and pf are geometric mean inflation rates.

Relative purchasing power parity ✓Forward parityThe international Fisher relationSummary

The law of one price and interest rate parity✓ Less reliable parity conditions



Relative purchasing power parity (RPPP)

E[Std/f] / S0

d/f = (1+E[pd])t / (1+E[pf])t

Speculators will force this relation to hold on average

- The expected change in a spot rate should reflect the difference in inflation between the two currencies

- This relation only holds over the long run





-15%

-10%

-5%

0%

5%

10%

15%

-2% -1% 0% 1% 2%

S1¥/$/S0

¥/$ - 1

(1+p¥)/(1+p$) - 1

RPPP over monthly intervals





S1f/$/S0

f/$ - 1

(1+pf)/(1+p$) - 1

RPPP over a 5-year horizon (2006-2010)









S1f/$/S0

f/$ - 1

(1+pf)/(1+p$) - 1


Forward rates as predictors of spot rates

E[Std/f ] / S0

d/f = Ftd/f / S0

d/f


- 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

Relative purchasing power parityForward parity ✓The international Fisher relationSummary




-15%

-10%

-5%

0%

5%

10%

15%

-1% 0% 1%

S1¥/$/S0

¥/$ - 1

F1¥/$/S0

¥/$ - 1

Yen-per-$ forward parity over monthly intervals



Relative purchasing power parityForward parity ✓The international Fisher relationSummary


International Fisher relation(Fisher Open hypothesis)

The Fisher equation provides a starting point…

(1+i) = (1+ʀ)(1+p)

i = nominal interest rate ʀ = real interest rate p = inflation rate

or i ≈ ʀ + p for small ʀ and p

Relative purchasing power parityForward parityThe international Fisher relation ✓Summary





[(1+id)/(1+if)]t = [(1+pd)/(1+pf)]t

where (1+i) = (1+ʀ)(1+p) from the Fisher relation

If real rates of interest are equal across currencies (ʀd = ʀf), then…

[(1+id)/(1+if)]t

= [(1+ʀd)(1+pd)]t / [(1+ʀf)(1+pf)]t

= [(1+pd)/(1+pf)]t






[(1+id)/(1+if)]t = [(1+pd)/(1+pf)]t


- If real rates of interest are equal across countries, then interest rate differentials merely reflect inflation differentials

- This relation is unlikely to hold at a point in time, but should hold in the long run





Interest rates[(1+id)/(1+if)]t

Inflation rates[(1+pd)/(1+pf)]t

E[Std/f] / S0

d/f

Expected changein the spot rate

Ftd/f / S0

d/f

Forward-spotdifferential

Interestrate parity

RelativePPP

International Fisher relation

Forward rates as predictors of future spot rates

Summary: The international parity conditions

Relative purchasing power parityForward parityThe international Fisher relationSummary ✓




The parity conditions are useful even if they are poor short-run predictors of the FX rate

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 predictive ability over long horizons

Inflation reflects change in a currency’s purchasing power and inflation differentials are a key element of the real exchange rate

Relative purchasing power parityForward parityThe international Fisher relationSummary ✓




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 ✓Change in the real exchange rateThe behavior of real exchange rates

The law of one price and interest rate parityLess reliable parity conditions

✓ The real exchange rateAppendix: Continuous compounding


Change in the nominal exchange rate

ExampleS0

¥/$ = ¥100/$

S1¥/$ = ¥110/$

p¥ = 0%p$ = 10%

s1¥/$ = (S1

¥/$–S0¥/$)/S0

¥/$ = 0.10,

or a 10 percent nominal change





The expected nominal exchange rate

But RPPP impliesE[S1

¥/$] = S0¥/$ (1+ p¥)/(1+ p$)

= ¥90.91/$

What is the change in the nominal exchange rate relative to the expectation of ¥90.91/$?





St¥/$

Actual S1¥/$ = ¥110/$

E[S1¥/$] = ¥90.91/$

¥130/$

¥120/$

¥100/$

¥110/$

¥90/$

time

Actual versus expected change





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

The real exchange rateChange in the real exchange rate ✓The behavior of real exchange rates





(1+xtd/f) = (St

d/f / St-1d/f)

[(1+ptf)/(1+pt

d)]

where

xtd/f = percentage change in the real

exchange rateSt

d/f = the nominal spot rate at time t

ptd = inflation in currency d during period t

ptf = inflation in currency f during period t






Example S0¥/$ = ¥100/$ S1

¥/$ = ¥110/$

p¥ = 0% and p$ = 10%

(1+xt¥/$) = [(¥110/$)/(¥100/$)][1.10/1.00]

= 1.21

= a 21 percent increase in relative purchasing power





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 the variance of the real exchange rate are autoregressive

The real exchange rateChange in the real exchange rateThe behavior of real exchange rates ✓




0%

50%

100%

150%

1970 1980 1990 2000 2010

Euro areaJapanUnited KingdomUnited States

Real exchange rates (Xf/d)

The real exchange rateChange in the real exchange rateThe behavior of real exchange rates ✓



Source: www.bis.org/statistics/eer/


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

Appendix 4-AContinuous time finance

Continuously compounded returns ✓The international parity conditions in continuous time


The real exchange rate✓ Appendix: Continuous compounding


100

200

100

r1 = +100% r2 = –50%

(1+rTOTAL) = (1+r1)(1+r2)

= (1+1)(1–½) = (2)(½) = 1

rTOTAL = 0%

Holding period returns are asymmetric





Continuous compounding

Let

r = holding period (e.g. annual) return

r = continuously compounded return

r = ln (1+r) Û (1 + r) = er

where ln(.) is the natural logarithm

with base e » 2.718





Properties of natural logarithms (for x > 0)

eln(x) = ln(ex) = x

ln(AB) = ln(A) + ln(B)for positive values A and B

ln(At) = (t) ln(A)

ln(A/B) = ln(AB-1)= ln(A) - ln(B)





100

200

100

r1 = ln(1+1) = +69.3%

r2 = ln(1-½) = -69.3%

rTOTAL = ln [ (1+r1)(1+r2) ]

= ln(1+r1) + ln(1+r2)

= r1 + r2= +0.693 -0.693 = 0.000

rTOTAL = 0%

Continuous returns are symmetric





Continuously compounded returns are additive

ln [ (1+r1) (1+r2) ... (1+rT) ]

= ln(1+r1) + ln(1+r2) ... + ln(1+rT)

= r1 + r2 +... + rT





The international parity conditionsin continuous time

Over a single period

ln(F1d/f / S0

d/f ) = i d – i f

= E[p d ] – E[p f ]= E[s d/f ]

where s d/f, p d, p f, i d, and i f are continuously compounded returns

Continuously compounded returnsThe international parity conditions in continuous time ✓




The international parity conditionsin continuous time

Over t periods

ln(Ftd/f / S0

d/f ) = t (i d – i f)

= t (E[p d ] – E[p f ])= t (E[s d/f ])

where s d/f, p d, p f, i d, and i f are continuously compounded returns

Continuously compounded returnsThe international parity conditions in continuous time ✓



ed5 04 the international parity conditions(1)

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