1. Chapter 4 The International Parity Conditions and their Consequences
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

2. yet there is method in it.
4.1 The Law of One Price
Though this be madness,
yet there is method in it.
William Shakespeare

3. 4.1 The Law of One Price Notation Upper Case Symbols = Prices
lower case symbols = changes in a price
Vtd = value of an asset in currency d at time t
rtd = return (% change) in currency d during period t
id = nominal rate of interest in currency d
pd = inflation rate in currency d
ʀd = real (inflation-adjusted) interest rate in currency d
Std/f = spot exchange rate between currencies d and f at time t
std/f = percentage change in the spot rate during period t
Ftd/f = forward rate between d and f for exchange at time t
ftd/f = percentage change in the forward rate during period t
E[…] = expectations operator (e.g., E[Std/f])

4. 4.1 The Law of One Price
The law of one price (also called purchasing power parity, or PPP) is the principle that equivalent assets sell for the same price:
Seldom holds for nontraded assets.
Can’t compare assets that vary in quality.
May not be precise when there are market frictions.
Market participants promote the law of one price through speculation and through riskless arbitrage.

5. 4.1 The Law of One Price Arbitrage
Riskless arbitrage is a profitable position obtained with no net investment and no risk.
The no-arbitrage condition refers to an absence of arbitrage opportunities, so that the law of one price holds within the bounds of transaction costs.

6. 4.1 The Law of One Price A purchasing power parity (PPP) example
Suppose V£ = £940/oz in London
V$ = $1504/oz in New York
The law of one price requires that…
St£/$ = Vt£ / Vt$ = (£940/oz) / ($1504/oz) = £0.6250/$
Alternatively,
St$/£ = 1 / St£/$ = 1 / (£0.6250/$) = $1.6000/£
If this relation does not hold within the bounds of transactions costs, then there may be an opportunity to lock in a riskless arbitrage profit.

7. London gold dealer New York gold dealer
4.1 The Law of One Price
A purchasing power parity (PPP) example
London gold dealer New York gold dealer
Equilibrium prices in New York
should be set such that
V$ = V£ S$/£
= (£940/oz)($1.6/oz)
= $1504/oz
S$/£
$1.6000/£
£940/oz
$1504/oz

8. London gold dealer New York gold dealer
4.1 The Law of One Price
A PPP example with transactions costs
London gold dealer New York gold dealer
$1522/oz Ask
FX dealer
$1.599/£ bid
$1.601/£ ask
$1510/oz Bid
Sell high to New York
£940/oz Ask
Buy low from London
£930/oz Bid

9. 4.1 The Law of One Price Arbitrage profit in gold Arbitrage profit
Buy 1000 oz at London’s £940/oz offer price
+(1000 oz)
-£940,000
Arbitrage profit
+£3,161
Sell 1000 oz at NY’s $1510/oz bid price
+$1,510,000
-(1000 oz)
Cover your £ position at the $1.601/£ ask price for pounds
+£943,161
= ($1,510,000)/($1.601/£)
-$1,510,000

10. 4.2 Exchange Rate Equilibrium
A cross exchange rate table
Sd/f Sf/d = 1  Sd/f = 1 / Sf/d
USD EUR JPY GBP CHF CAD AUD CNY
USD U.S.A
EUR Euro
JPY Japan
GBP U.K
CHF Swiss
CAD Canada
AUD Australia
CNY China

11. 4.2 Exchange Rate Equilibrium
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.

12. 4.2 Exchange Rate Equilibrium
Cross exchange rates and triangular arbitrage
An example with U.S. dollars ($), Japanese yen (¥), and Egyptian pounds (E£):
SE£/$ = E£5.000/$ Û S$/E£ = $0.2000/E£
S$/¥ = $ /¥ Û S¥/$ = ¥100.0/$
S¥/E£ = ¥20.20/E£ Û SE£/¥ » E£ /¥
SE£/$ S$/¥ S¥/E£
= (E£5/$)($0.01/¥)(¥20.20/E£)
= 1.01 > 1

13. 4.2 Exchange Rate Equilibrium
Cross exchange rates & 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.

14. 4.2 Exchange Rate Equilibrium
An example of triangular arbitrage
Arbitrage profit
= +E£50,000
or about 1% of the initial amount
Sell $1m and buy E£5m at E£5/$
+E£5m
-$1m
Sell ¥100m and buy $1m at $0.01/¥
+$1m
-¥100m
Sell E£4.95m and buy ¥100m at ¥20.20/E£
+¥100m
-E£4.95m

15. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
International parity conditions that span both currencies and time
Less reliable linkages
= E[Std/f] / S0d/f
= [(1+E[pd])/(1+E[pf])]t
Interest rate parity
Ftd/f / S0d/f = [(1+id)/(1+if)]t
where
S0d/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

16. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
Interest rate parity
Ftd/f / S0d/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.

17. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
Suppose yield curves are flat
S0$/€ = $1.20/€
i$ = 7% and i€ = 3%
What are forward rates F1$/€, F2$/€, and F3$/€?
Ft$/€ = S0$/€ [(1+i$)/(1+i€)]t
F1$/€ = ($1.20/€) [(1.07)/(1.03)]1 ≈ $1.2466/€
F2$/€ = ($1.20/€) [(1.07)/(1.03)]2 ≈ $1.2950/€
F3$/€ = ($1.20/€) [(1.07)/(1.03)]3 ≈ $1.3453/€

18. Moving dollars today into euros tomorrow
4.3 Interest Rate Parity
and Covered Interest Arbitrage
Interest rate parity
Invest in dollars
FVt$ = PV0$ (1+i$)t
V0$
Dollars
Moving dollars today into euros tomorrow
Convert to euros at S0$/€
Convert to euros at Ft$/€
Euros
Vt€
Invest in euros
FVt€ = PV0€ (1+i€)t
Time 0 Time t

19. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
Interest rate parity: Which way do you go?
If Ftd/f/S0d/f > [(1+id)/(1+if)]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

20. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
Interest rate parity: Which way do you go?
If Ftd/f/S0d/f < [(1+id)/(1+if)]t
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

21. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
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£)
>
The currency and Eurocurrency markets are not in equilibrium.

22. and Covered Interest Arbitrage
4.3 Interest Rate Parity
and Covered Interest Arbitrage
Covered interest arbitrage
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
+$1,000,000
-$1,070,000
+£833,333
-$1,000,000
+£858,333
-£833,333
+$1,072,920
-£858,333

23. 4.4 Less Reliable Parity Conditions
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] / S0d/f = (E[Ptd] / E[Ptf]) / (P0d / P0f)
= (E[Ptd] / P0d) / (E[Ptf] / P0f)
= (1+E[pd])t / (1+E[pf])t
where pd and pf are geometric mean inflation rates.

24. 4.4 Less Reliable Parity Conditions
Relative purchasing power parity (RPPP)
E[Std/f] / S0d/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.

25. 4.4 Less Reliable Parity Conditions
RPPP over monthly intervals
S1¥/$/S0¥/$ - 1
(1+p¥)/(1+p$) - 1

26. 4.4 Less Reliable Parity Conditions

27. 4.4 Less Reliable Parity Conditions

28. 4.4 Less Reliable Parity Conditions

29. 4.4 Less Reliable Parity Conditions
Forward rates as predictors of spot rates
E[Std/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.

30. 4.4 Less Reliable Parity Conditions
Yen-per-$ forward parity over monthly intervals
S1¥/$/S0¥/$ - 1
F1¥/$/S0¥/$ - 1

31. 4.4 Less Reliable Parity Conditions
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

32. 4.4 Less Reliable Parity Conditions
International Fisher relation
(Fisher Open hypothesis)
[(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

33. 4.4 Less Reliable Parity Conditions
International Fisher relation
(Fisher Open hypothesis)
[(1+id)/(1+if)]t = [(1+pd)/(1+pf)]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 a point in time, but should hold in the long run.

34. 4.4 Less Reliable Parity Conditions
Summary: The international parity conditions
International Fisher relation
Interest rates
[(1+id)/(1+if)]t
Inflation rates
[(1+pd)/(1+pf)]t
Interest
rate parity
Relative
PPP
Ftd/f / S0d/f
Forward-spot
differential
E[Std/f] / S0d/f
Expected change
in the spot rate
Forward rates as predictors of future spot rates

35. 4.4 Less Reliable Parity Conditions
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.

36. 4.5 The Real Exchange Rate
The real exchange rate adjusts the nominal exchange rate for differential inflation since an arbitrarily defined base period.
Example
S0¥/$ = ¥100/$
S1¥/$ = ¥110/$
p¥ = 0%
p$ = 10%
s1¥/$ = (S1¥/$–S0¥/$)/S0¥/$ = 0.10,
or a 10 percent nominal change

37. 4.5 The Real Exchange Rate The expected nominal exchange rate
But RPPP implies
E[S1¥/$] = S0¥/$ (1+ p¥)/(1+ p$)
= ¥90.91/$
What is the change in the nominal exchange rate relative to the expectation of ¥90.91/$?

38. 4.5 The Real Exchange Rate Actual versus expected change St¥/$ ¥130/$
¥120/$
¥110/$
Actual S1¥/$ = ¥110/$
¥100/$
E[S1¥/$] = ¥90.91/$
¥90/$
time

39. 4.5 The Real Exchange Rate 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.

40. 4.5 The Real Exchange Rate Change in the real exchange rate
(1+xtd/f) = (Std/f / St-1d/f) / [(1+ptd)/(1+ptf)]
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

41. 4.5 The Real Exchange Rate Change in the real exchange rate
Example S0¥/$ = ¥100/$  S1¥/$ = ¥110/$
p¥ = 0% and p$ = 10%
(1+xt¥/$) = [(¥110/$)/(¥100/$)] / [1.00/1.10]
= 1.21
= a 21 percent increase in relative purchasing power

42. 4.5 The Real Exchange Rate Consequences of real exchange rate changes
A real appreciation of the domestic currency increases the purchasing power of domestic residents:
Domestic companies and consumers benefit from lower prices Pd on foreign goods; Pd = Pf Sd/f.
This helps importers and hurts exporters.
A real depreciation of the domestic currency decreases the purchasing power of domestic residents:
Domestic companies and consumers benefit from higher prices Pd on foreign goods; Pd = Pf Sd/f.
This hurts importers and helps exporters.

43. 4.5 The Real Exchange Rate 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.

44. 4.5 The Real Exchange Rate Real exchange rates (Xf/d) 2010 base year
Source:
bis.org/statistics/eer/
The Bank for International Settlements uses 2010 as the base year in their “effective exchange rates”

45. 4.5 The Real Exchange Rate
Real exchange rates (Xf/d) rescaled average = 100
Source:
bis.org/statistics/eer/

46. Appendix 4-A Continuous Compounding
Most theoretical and empirical research in finance is conducted in continuously compounded returns.

47. Appendix 4-A Continuous Compounding
Holding period returns are asymmetric
r1 = +100%
r2 = –50%
200
100
100
(1+rTOTAL) = (1+r1)(1+r2)
= (1+1)(1–½) = (2)(½) = 1
 rTOTAL = 0%

48. Appendix 4-A 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

49. Appendix 4-A Continuous Compounding
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)

50. Appendix 4-A Continuous Compounding
Continuous returns are symmetric
r1 = ln(1+1) = +69.3%
r2 = ln(1-½) = -69.3%
200
100
100
rTOTAL = ln [ (1+r1)(1+r2) ]
= ln(1+r1) + ln(1+r2)
= r1 + r2 = = 0.000
 rTOTAL = 0%

51. Appendix 4-A Continuous Compounding
Continuously compounded returns are additive
ln [ (1+r1) (1+r2) ... (1+rT) ]
= ln(1+r1) + ln(1+r2) ln(1+rT)
= r1 + r rT

52. Appendix 4-A Continuous Compounding
The international parity conditions in continuous time
Over a single period
ln(F1d/f / S0d/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

53. Appendix 4-A Continuous Compounding
The international parity conditions in continuous time
Over t periods
ln(Ftd/f / S0d/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