Chapter 04
International Parity Conditions
1
International Parity Conditions
• Foreign exchange rate quotations
• Measuring a change in spot rates
• Prices and exchange rates
– Law of one price
– Absolute Purchasing Power Parity (PPP)
– Relative Purchasing Power Parity
• Interest rates and exchange rates
–
–
–
–
The Fisher effect
International Fisher effect
Forward rate
Interest Rate Parity (IRP)
• Covered/Uncovered Interest Arbitrage
• Forward rates as an unbiased predictor of future spot rates
2
Foreign Exchange Rates & Quotations
• Direct and Indirect Quotes
• A direct quote is a home currency price of a
unit of a foreign currency
– Sfr1.6000/$ is a direct quote in Switzerland
• An indirect quote is a foreign currency price of
a unit of the home currency
– Sfr1.6000/$ is an indirect quote in the US,
– $0.6250/Sfr is a direct quote in the US and an
indirect quote in Switzerland
3
Measuring a Change in Spot Rates
• Assume that Swiss franc is quoted at Sfr1.6000/$ (same as
$0.6250/Sfr). Suddenly it strengthens to Sfr1.2800/$ (same
as $0.78125/$). What is the percentage change in the
dollar value of the franc? (Home currency is dollar)
• Using Direct Quotes:
%ΔDQ 
Ending Rat e-Beginnin g Rate
$0.78125Sfr-$0.6250 /Sfr
x100 
x100=+25%
Beginning Rate
$0.6250 /Sfr
S2$ / FC -S1$ / FC
S1$ / FC
SAME
• Using Indirect Quotes:
%ΔIQ 
Beginning Rate-Endin g Rate
Sfr1.6000 /$-Sfr1.2800 /$
x100 
x100=+25%
Ending Rat e
Sfr1.2800 /$
S1FC / $ -S 2FC / $
S 2FC / $
4
Prices and Exchange Rates
• The Law of one price states that all else being equal (no
transaction costs) a product’s price should be the same in
all markets
• The law of one price states that
$
P xS
¥/$
=P
¥
• Where the price of the product in US dollars (P$),
multiplied by the spot exchange rate (S, yen per dollar),
equals the price of the product in Japanese yen (P¥)
• Conversely, if the prices were stated in local currencies, and
markets were efficient, the exchange rate could be
calculated from the relative local product prices
S
¥ /$
P¥
= $
P
5
Prices and Exchange Rates
• What if you have the following scenario:
• You live in Switzerland and observe that a product costs Sfr6.20 in
Switzerland while the same product costs $2.65 in the US
• Also assume that Homer trades Sfr and $ from his couch and that
he stands ready to sell you one dollar for Sfr1.36 – This is also to say
he is going to buy Sfr1.36 for a buck
• Can you make any arbitrage profits?
– Buy the product in the US and sell it in Switzerland
– Buy: need to spend $2.65 for the purchase. Since you are a Swiss
person ask Homer to sell you dollars for Sfr. You need to pay him
$2.65 x Sfr1.36/$ = Sfr3.604
– Sell: receive Sfr6.20 for the sale of the product that you just spent
Sfr3.604 for
– What is your profit (Sfr6.20/Sfr3.604) – 1 = + 72%
– Thanks to Homer!!!
6
The Big Mac Hamburger Standard
• The Economist developed the Big Mac
Standard to track PPP:
– Assuming that the Big Mac is identical in all
countries, it serves as a comparison point as to
whether or not currencies are trading at market
prices
– Big Mac in Switzerland costs Sfr6.20 while the
same Big Mac in the US costs $2.65
– The implied PPP of this exchange rate is
Sfr6.20
= Sfr2.34/$
$2.65
7
The Big Mac Hamburger Standard
(Continued)
• If the actual exchange rate is Sfr1.36/$, then we can
determine relative valuation
Indirect Quote (FC/$) over (  ) under (  ) valuation :
Theorethical Exchange Rate

1
Actual Spot Exchange Rate
• The Swiss franc is overvalued against dollar by
Sfr 2.34 /$

 1  72%
Sfr1.36 /$
• Above equation applies when we have an “Indirect
Quote” (FC/$) to measure Sfr value against the dollar
8
The Big Mac Hamburger Standard
(Continued)
• If Direct Quotes ($/FC) are used the equation would
become
Direct Quote ($/FC) over (  ) under (  ) valuation :
Actual Spot Exchange Rate

1
Theorethical Exchange Rate
• Since home currency is $, Direct Quotes are:
• Theoretical = 1 / Sfr2.34/$ = $0.4274/Sfr and
• Actual = 1 / Sfr1.36/$ = $0.7353/Sfr
$0.7353 /Sfr

 1  72%
$0.4274 /Sfr
• Again we are measuring the value of Sfr against the dollar
9
Homer’s Solution
• Homer should change his conversion rate, Spot
Exchange Rate, to Sfr2.34/$ = (Sfr6.20/$2.65)
– Buy the product in the US and sell it in Switzerland
– Buy = need to spend $2.65 for the purchase. Since you
are a Swiss person ask Homer to sell you dollars for
Sfr. You need to pay him $2.65 x Sfr2.34/$ = Sfr6.20
– Sell = receive Sfr6.20 for the sale of the product that
you just spent Sfr6.20 for????
– What is your profit (Sfr6.20/Sfr6.20) – 1 = 0%
10
Purchasing Power Parity & The Law of
One Price
• If the Law of One Price were true for all goods,
the purchasing power parity (PPP) exchange rate
could be found from any set of prices
• Prices of individual products can be determined
through the PPP exchange rate
• This is the absolute theory of purchasing power
parity
• Absolute PPP states that the spot exchange rate is
determined by the relative prices of similar
basket of goods
11
Relative Purchasing Power Parity
• If the assumptions of absolute PPP theory are relaxed, we
observe relative purchasing power parity
– The idea is that the relative change in prices between countries
over a period of time determines the change in exchange rates
– Moreover, if the spot rate between 2 countries starts in
equilibrium, any change in the differential rate of inflation
between them tends to be offset over the long run by an equal
change in the spot rate. Higher inflation currency should
depreciate
– Using the direct quotes mathematically the following
relationship should hold:
$
S 2$/FC - S1$/FC
(1

π
)
$
FC
$/FC
$/FC
=
π
π
and
S

S
2
1
S1$/FC
(1  π FC )
– Using indirect quotes, we have:
FC
S1FC / $ -S 2FC / $
(
1
+
π
)
$
FC
FC / $
FC / $
= π - π and S 2
= S1
FC / $
S2
( 1+ π$ )
12
Relative Purchasing Power Parity
• High inflation currency should depreciate
PPP Line
π$ – πFC
1
4
Increased purchasing
power of foreign goods.
Cheaper to buy foreign
goods.
3
Home inflation rate is higher,
but home currency does not
depreciate as much (direct
quote goes up). Foreign goods
become relatively cheaper to
US consumers.
2
S2$ / FC -S1$ / FC
S1$ / FC
or
1
0
-4
-3
-2
-1
0
1
2
3
-1
-2
2
-3
Decreased purchasing
power of foreign goods.
Cheaper to buy
domestic goods.
4
S1FC / $ -S 2FC / $
S 2FC / $
-4
13
Does PPP Hold?
• Empirical tests of both relative and absolute purchasing
power parity show that for the most part, PPP is not
accurate in predicting future exchange rates
• Reasons:
–
–
–
–
Price indexes include non tradable goods
Goods baskets are not similar
Differential inflation rates might be inaccurate
Test periods are subject to government interventions
• Two general conclusions can be drawn from the tests:
– PPP holds up well over the very long term but is poor for short
term estimates
– The theory holds better for countries with relatively high rates
of inflation and underdeveloped capital markets
14
Exchange Rate Indices: Real and
Nominal
• In order to evaluate a single currency’s value
against all other currencies in terms of whether
or not the currency is “over” or “undervalued,”
exchange rate indices were created
– These indices are formed by trade-weighting the
bilateral exchange rates between the home country
and its trading partners
• The nominal exchange rate index uses actual
exchange rates to create an index on a weighted
average basis of the value of the subject currency
over a period of time
15
Exchange Rate Indices: Real and
Nominal
• The real effective exchange rate index indicates how the weighted
average purchasing power of the currency has changed relative to
some arbitrarily selected base period
$
C
ER$ = E N$ x FC
C
$
E
•
R The real effective rate for the US dollar
$
• E N The nominal rate index
• C$ US dollar costs
FC
• C
Foreign currency costs (CFC)
– If changes in exchange rates just offset differential inflation rates – if
purchasing power parity holds – all the real effective exchange rate
indices would stay at 100
– If a rate strengthened (overvalued) or weakened (undervalue) then
the index value would be above or below 100, respectively
16
Real Effective Exchange Rate Indices
Undervalued
Overvalued
IMF’s Real Effective Exchange Rate Indexes for the United States & Japan (2000 = 100)
17
Exchange Rate Pass-Through
• Incomplete exchange rate pass-through is one reason
that country’s real effective exchange rate index can
deviate from it’s PPP equilibrium point
• The degree to which the prices of imported & exported
goods change as a result of exchange rate changes is
termed pass-through
• Example: assume BMW produces a car in Germany and
all costs are incurred in euros. When the car is
exported to the US, the price of the BMW should be
the euro value converted to dollars at the spot rate
• Where P$ is the BMW price in dollars, P€ is the BMW
price in euros and S is the spot rate
$
Pr oduct
P
P
FC
Pr oduct
$/FC
xS
18
Exchange Rate Pass-Through
• If the euro appreciated 20% against the dollar, but the price
of the BMW in the US market rose to only $40,000 from
$35,000, and not $42,000 as is the case under complete
pass-through, the pass-through is partial
• The degree of pass-through is measured by the proportion
of the exchange rate change reflected in dollar prices
$
PBMW,
2
$
PBMW,
1
$40,000

 1.1429, or  14.29%
$35,000
• The degree of pass-through in this case is partial, 14.29% ÷
20.00% or approximately 0.71. Only 71.0% of the change
has been passed through to the US dollar price
19
Fisher Effect
• Prices between countries are related by exchange rates
and now we discuss how exchange rates are linked to
interest rates
• The Fisher Effect states that nominal interest rates in
each country are equal to the required real rate of
return plus compensation for expected inflation. As a
formula, The Fisher Effect is
(1  i)  (1  r)(1  π)
i  r  π  rπ
• Where i is the nominal rate, r is the real rate of
interest, and π is the expected rate of inflation over
the period of time
• The cross-product term, rπ , is usually dropped due to
its relatively minor value
20
Fisher Effect
• Applied to two different countries, like the US
and Japan, The Fisher Effect would be stated as
i  r π ; i  r π
$
$
$
¥
¥
¥
• It should be noted that this requires a forecast of
the future rate of inflation, not what inflation has
been, and predicting the future can be difficult!
• International Fisher Effect assumes that the real
rates across countries are the same.
21
International Fisher Effect (IFE)
• The International Fisher Effect, or Fisher-open,
states that the spot exchange rate should change
in an amount equal to the difference in interest
rates between countries
– if we were to use the US dollar and the Japanese yen,
the expected change in the spot exchange rate
between the dollar and yen should be
– Using direct quotes:
$
S 2$ / FC -S1$ / FC
(
1
+
i
)
$
FC
$ / FC
$ / FC
= i - i and S 2
= S1
$ / FC
S1
( 1 + i FC )
– Using indirect quotes:
FC
S1FC / $ -S 2FC / $
(
1
+
i
)
$
FC
FC / $
FC / $
= i - i and S 2
= S1
FC / $
S2
( 1 + i$ )
22
International Fisher Effect (IFE)
• Justification for the international Fisher effect is
that investors must be rewarded or penalized to
offset the expected change in exchange rates
• The international Fisher effect predicts that
currencies with low interest rates are expected to
appreciate relative to currencies with high
interest rates
• This reasoning is due to the fact that low interest
rates imply low expected inflation rates
• Similar to PPP except that expected inflation
rather than past inflation figures are incorporated
23
International Fisher Effect (IFE)
• High interest rate currency should depreciate
International Fisher Effect
i$ – iFC
4
3
Lower returns from
investing in foreign
deposits. Invest in the US.
Home interest rate is higher, but
home currency does not
depreciate (direct quote goes up)
as much. It is better to invest in
the US. Foreigners will do the
same.
1
2
S2$ / FC -S1$ / FC
S1$ / FC
or
1
0
-4
-3
-2
-1
0
1
2
3
-1
-2
Higher returns from
investing in foreign
deposits. Invest abroad.
4
S1FC / $ -S 2FC / $
S 2FC / $
-3
-4
24
Does International Fisher Effect Hold?
• Empirical studies indicate that the IFE theory
holds during some time frames. However, there is
also evidence that it does not consistently hold
• Since the IFE is based on PPP, it will not hold
when PPP does not hold
• For example, if there are factors other than
inflation that affect exchange rates, the rates will
not adjust in accordance with the inflation
differential
25
Interest Rates and Exchange Rates
• The Forward Exchange Rate
– A forward rate is an exchange rate quoted today for
settlement at some future date
– The forward exchange agreement between currencies
states the rate of exchange at which a foreign
currency will be bought or sold forward at a specific
date in the future (typically 30, 60, 90, 180, 270 or 360
days)
– The forward rate is calculated by adjusting the current
spot rate by the ratio of euro currency interest rates of
the same maturity for the two subject currencies
26
Interest Rates and Exchange Rates
• The Forward Rate (using direct quotes)
F
$ / FC
=S
$ / FC
( 1  i$ )
( 1  i FC )
FC
)
The equation for determining the
FC/$
FC/$ ( 1  i
1.F
=S
Forward Rate
( 1  i$ )
2. Equilibrium condition where the
1
investment return in $ is the same
2.( 1+i $ )=S FC/$( 1+i FC ) FC/$
regardless where the investment is
F
made after taking into account the
FC/$
FC
F
(
1

i
)
change in exchange rate
3. FC/$ 
1
$
S
(1 i )
3. Another way of stating equilibrium
condition
• Note each one of these equations can
be driven from one another
1.
27
Interest Rates and Exchange Rates
• The Forward Rate example with spot rate of
Sfr1.4800/$, a 90-day euro Swiss franc deposit
rate of 4.00% p.a. and a 90-day euro-dollar
deposit rate of 8.00% p.a. Home currency is
US $.
Sfr/$
F90
90 
 
 1   0.04 x 360 


 Sfr1.4800/$ x
90 
 
 1   0.08 x 360 


 Sfr1.4800/$ x
1.01
 Sfr1.4655/$
1.02
28
Interest Rates and Exchange Rates
• The forward premium or discount is the
percentage difference between the spot and
forward rates stated in annual percentage terms
– When stated in direct terms (home currency per
foreign currency units, $/FC) then formula is
Foward-Spot
360
$/FC
f

x
x 100
Spot
days
– For indirect quotes (FC/$), then (S – F) / F should be
applied
Spot - Foward
360
FC/$
f

x
x 100
Foward
days
29
Interest Rates and Exchange Rates
• Using the previous Sfr example, the forward discount
or premium would be as follows:
Spot - Foward
360
FC/$
f

x
x 100
Foward
days
f
Sfr/$
Sfr1.4800 /$ - Sfr1.4655 /$ 360

x
x 100   3.96% p.a.
Sfr1.4655 /$
90
• The positive sign indicates that the Swiss franc is selling
forward at a premium of 3.96% per annum (it takes
3.96% more dollars to get a franc at the 90-day forward
rate – it would be the same if direct quotes are used)
30
Interest Rates and Exchange Rates
• Same example with direct quotes:
1
Spot RateDirect Quote 
 $0.6757 / Sfr
Sfr1.4800/$
1
Forward RateDirect Quote 
 $0.6824 / Sfr
Sfr1.4655/$
f
$ / FC
f $ / Sfr
Foward-Spot
360

x
x 100
Spot
days
$0.6824 /Sfr-$ 0.6757 /Sfr 360

x
x 100  3.96%
$0.6757 /Sfr
90
31
Interest Rate Parity (IRP)
• Interest rate parity theory provides the linkage
between foreign exchange markets and
international money markets
• The theory states that the difference in the
national interest rates for securities of similar risk
and maturity should be equal to the forward rate
discount or premium for the foreign currency,
except for transaction costs
• If not true then there exists risk-free profit
opportunities (arbitrage)
32
Interest Rate Parity (IRP)
• In the diagram in the following slide, a US
dollar-based investor with $1 million to invest,
is shown indifferent between dollardenominated securities for 90 days earning
8.00% per annum, or Swiss francdenominated securities of similar risk and
maturity earning 4.00% per annum, when
“cover” against currency risk is obtained with
a forward contract
33
Interest Rate Parity (IRP)
i $ = 8.00 % per annum
(2.00 % per 90 days)
Start
$1,000,000
S = SF 1.4800/$
End
 1.02
$1,020,000
Dollar money market
$1,019,993*
90 days
F90 = SF 1.4655/$
Swiss franc money market
SF 1,480,000
 1.01
SF 1,494,800
i SF = 4.00 % per annum
(1.00 % per 90 days)
Note that the Swiss franc investment yields $1,019,993, $7 less on a $1 million investment.
34
Covered Interest Arbitrage (CIA)
• Because the spot and forward markets are not
always in a state of equilibrium as described by
IRP, the opportunity for arbitrage exists
• The arbitrageur who recognizes this imbalance
can invest in the currency that offers the higher
return on a covered basis
• This is known as covered interest arbitrage (CIA)
• The following slide describes a CIA transaction in
much the same way as IRP was transacted
35
CIA – Example
•
•
•
•
Assume that you observe the following (HC is $):
i$ = 8% p.a. and i¥ = 4% p.a.
S¥/$ = ¥106.00/$ and six-month F¥/$ = ¥103.50/$
Compare interest differential to forward
premium/discount
• Interest differential = i$ – i¥ = 8% – 4% = 4%
• Forward premium/discount:
f
¥ /$
¥106.00 /$ - ¥103.50 /$ 360

x
x 100  4.83% p.a.
¥103.50 /$
180
• This observation would be below the IRP line,
therefore we borrow $ and invest in ¥
36
CIA – Example
Eurodollar rate = 8.00 % per annum
Start
End
 1.04
$1,000,000
$1,040,000
$1,044,638
Arbitrage
Potential
Dollar money market
S = ¥ 106.00/$
F180 = ¥ 103.50/$
180 days
Yen money market
¥ 106,000,000
 1.02
¥ 108,120,000
Euroyen rate = 4.00 % per annum
37
Uncovered Interest Arbitrage (CIA)
• A deviation from CIA is uncovered interest
arbitrage, UIA, wherein investors borrow in
currencies exhibiting relatively low interest
rates and convert the proceeds into currencies
which offer higher interest rates
• The transaction is “uncovered” because the
investor does not sell the currency forward,
thus remaining uncovered to any risk of the
currency volatility
38
Uncovered Interest Arbitrage (UIA):
The Yen Carry Trade
Investors borrow yen at 0.40% per annum
End
Start
¥ 10,000,000
Then exchanges
 1.004
Japanese yen money market
¥ 10,040,000 Repay
¥ 10,500,000 Earn
¥ 460,000 Profit
the yen proceeds
for US dollars,
S = ¥ 120.00/$
investing in US
360 days
S360 = ¥ 120.00/$
dollar money
US dollar money market
markets for
one year
$ 83,333.33
 1.05
$ 87,500.00
Invest dollars at 5.00% per annum
Note that the final outcome is determined by future spot rate. It is likely to test some nerves!!!
¥ appreciates to below ¥114.74/$ you face losses.
39
Does Interest Rate Parity Hold?
• Various empirical studies indicate that IRP
generally holds
• While there are deviations from IRP, they are
often not large enough to make covered
interest arbitrage worthwhile
• This is due to the characteristics of foreign
investments, including transaction costs,
political risk, and differential tax laws
40
Forward Rates as an Unbiased
Predictor
• If the foreign exchange markets are thought to be
“efficient” then the forward rate should be an unbiased
predictor of the future spot rate
• This is roughly equivalent to saying that the forward
rate can act as a prediction of the future spot exchange
rate, and it will often “miss” the actual future spot rate,
but it will miss with equal probabilities (directions) and
magnitudes (distances)
• Evidence indicate that forward rates are biased
predictors of future spot rates
• Forecasting services should have value otherwise
would not exist
41
Forward Rates as an Unbiased
Predictor
Exchange rate
F2
S2
Error
Error
S1
F1
F3
Error
S3
S4
Time
t1
t2
t3
t4
The forward rate available today (Ft,t+1), time t, for delivery at future time t+1, is used as a “predictor” of the
spot rate that will exist at that day in the future. Therefore, the forecast spot rate for time St2 is F1; the actual spot
rate turns out to be S2. The vertical distance between the prediction and the actual spot rate is the forecast error.
When the forward rate is termed an “unbiased predictor,” it means that the forward rate over or underestimates
the future spot rate with relatively equal frequency and amount, therefore it misses the mark in a regular and
orderly manner. The sum of the errors equals zero.
42