Shapiro: Chapter 4 –
The Law of One Price
Chapter 4 Problems
Problem
4.7
The Law of One Price
“Exchange-adjusted
prices …
of identical, tradable goods and
financial assets …
The Law of One Price
“Exchange-adjusted
prices …
of identical, tradable goods and
financial assets …
must be within transaction costs of
equality worldwide.”
The Law of One Price
Enforced
by arbitrageurs - ???
Enforced by arbitrageurs - buy in
one market and sell in another
“Students Find $100 Textbooks
Cost $50, Purchased Overseas”
New York Times, 10-21-03
Biology Textbook
Amazon (USA)
Amount
$146.15
Amazon (UK)
Gross Margin
Shipping Cost
$ 63.49
$ 82.67
$ 8.05
Net Margin
$ 74.62
The Law of One Price
Enforced
by arbitrageurs - buy in
one market and sell in another
Risk-adjusted expected returns on
financial assets in different markets
should be equal
5 Key Economic Relationships





Purchasing Power Parity (PPP)
Fisher Effect (FE)
International Fisher Effect (IFE)
Interest Rate Parity (IRP)
Forward Rates as unbiased
predictors of future spot rates
(UFR)
Exhibit 4.1
Five Key Theoretical Relationships
Among Spot Rates, Forward Rates,
Inflation Rates, and Interest Rates
Expected %
Change of
Spot Rate
-3%
UFR
Forward
Discount or
Premium
-3%
PPP
IFE
Interest
Rate
Differential
+3%
IRP
FE
Expected
Inflation Rate
Differential
+3%
Purchasing Power Parity (PPP)
“The
ratio between domestic and
foreign price levels should equal the
equilibrium exchange rate between
domestic and foreign currencies.”
$1 = loaf of bread
$1 = £0.6475
£0.6475 = loaf of bread
Purchasing Power Parity (PPP)
Version: “Exchange
rate between the home currency
and any foreign currency will
adjust to reflect changes in the
price levels of the two
countries.”
Relative
Purchasing Power Parity (PPP)
 Inflation
= 5% (USA)
Inflation = 3% (Switzerland)
$ value of SFr should rise by 2%
 et
t
)
(1 + ih
--- = -----------t
e0
(1 + if)
2%
Purchasing Power Parity (PPP)
 Inflation
= 5% (USA)
 Inflation
= 3% (Switzerland)
 e0
= SFr 1 = $0.75 (spot)
 What
is e3?
Purchasing Power Parity
e3 (1  .05)

3
e0 (1  .03)
3
Purchasing Power Parity
e3 (1  .05)

3
e0 (1  .03)
3
e3  e0 (1.0194)
3
Purchasing Power Parity
e3 (1  .05)

3
e0 (1  .03)
3
e3  e0 (1.0194)
3
e3  .75(1.0594)  $0.7945
Purchasing Power Parity (PPP)
The
exchange rate change during a
period should equal the inflation
differential for that same period.
“Currencies with high rates of
inflation should devalue relative to
currencies with lower rates of
inflation.”
Purchasing Power Parity
(PPP)
Shapiro:
Exhibit 4.4
Fisher Effect
Nominal
interest rate (r) consists
of:
– real required rate of return, a
– an inflation premium, i
1 + r = (1 + a)(1 + i)
Real returns are equalized across
countries through arbitrage.
Fisher Effect
ah
= af
rh - rf = ih - if
Currencies with high rates of inflation
should bear higher interest rates than
currencies with lower rates of inflation
Exhibit 4.7
Fisher Effect: Empirical Data,
May 2007
International Fisher Effect
“Currencies
with low interest
rates are expected to appreciate
relative to currencies with high
interest rates.”
International Fisher Effect
1  rh e1

1  r f e0
where e1 is the expected
exchange rate.
International Fisher Effect
The
expected home country (HC)
returns from investing at home and
abroad should be equal.
1
+ rh = (1 + rf)(e1/e0)
Nominal vs. Real Rate of Interest
[The Economist, 2-5-16]
Country
Euro Area
Japan
Canada
USA
Britain
Australia
Mexico
Brazil
10 Yr Nominal 2015 Inflation
Real Rate (%)
Rate(%)
Rate (%)
Nominal vs. Real Rate of Interest
[The Economist, 2-5-16]
Country
10 Yr Nominal 2015 Inflation
Real Rate (%)
Rate(%)
Rate (%)
Euro Area
0.45
Japan
0.22
Canada
1.25
USA
2.01
Britain
1.75
Australia
2.68
Mexico
6.13
Brazil
16.52
Nominal vs. Real Rate of Interest
[The Economist, 2-5-16]
Country
10 Yr Nominal 2015 Inflation
Real Rate (%)
Rate(%)
Rate (%)
Euro Area
0.45
0.1
Japan
0.22
0.7
Canada
1.25
1.2
USA
2.01
0.2
Britain
1.75
0.1
Australia
2.68
1.6
Mexico
6.13
2.7
Brazil
16.52
9.6
Nominal vs. Real Rate of Interest
[The Economist, 2-5-16]
Country
10 Yr Nominal 2015 Inflation
Real Rate (%)
Rate(%)
Rate (%)
Euro Area
0.45
0.1
0.35
Japan
0.22
0.7
-0.48
Canada
1.25
1.2
0.05
USA
2.01
0.2
1.81
Britain
1.75
0.1
1.65
Australia
2.68
1.6
1.08
Mexico
6.13
2.7
3.43
Brazil
16.52
9.6
6.92
Interest Rate Parity (IRP)
[Covered Interest Arbitrage]
“A
condition where the interest
rate differential is approximately
equal to the forward differential
between two currencies.”
Interest Rate Parity (IRP)
[Covered Interest Arbitrage]
 Interest
rate (London) = 12%
 Interest rate (New York) = 7%
 £ spot rate: £1 = $1.95
 1-year forward rate: £1 = $1.88
 Forward differential:
•
($1.88 - $1.95) / $1.95 = -3.6%
 Rate
differential: 12% - 7% = 5%
Covered Interest Arbitrage
Spot Market:
– beginning of the year
 Borrow $1,000,000
(NY) at 7%
 Buy £ at £ = $1.95
 Receive £ 512,820.5
 Invest £ 512,820.5
at 12%
Spot Market:
– end of the year
Forward Market:
- beginning of the year
Forward Market:
– end of the year
Covered Interest Arbitrage
Spot Market:
– beginning of the year
 Borrow $1,000,000
(NY) at 7%
 Buy £ at £ = $1.95
 Receive £ 512,820.5
 Invest £ 512,820.5 at
12%
Spot Market:
– end of the year
Forward Market:
- beginning of the year
 Sell £ 574,358.9 @ £1
= $1.88
Forward Market:
– end of the year
Covered Interest Arbitrage
Spot Market:
– beginning of the year
 Borrow $1,000,000
(NY) at 7%
 Buy £ at £ = $1.95
 Receive £ 512,820.5
 Invest £ 512,820.5 at
12%
Spot Market:
– end of the year
Forward Market:
- beginning of the year
 Sell £ 574,358.9 @ £1
= $1.88
Forward Market:
– end of the year
 Collect £574,358.9
 Convert to $1,079,795
Covered Interest Arbitrage
Spot Market:
– beginning of the year
 Borrow $1,000,000
(NY) at 7%
 Buy £ at £ = $1.95
 Receive £ 512,820.5
 Invest £ 512,820.5 at
12%
Spot Market:
– end of the year
 Repay: $1,000,000
 Pay Interest: $70,000
 Net Profit: $9,795
Forward Market:
- beginning of the year
 Sell £ 574,358.9 @ £1
= $1.88
Forward Market:
– end of the year
 Collect £574,358.9
 Convert to $1,079,795
Forward Rates as Unbiased
Predictors of Future Spot Rates (UFR)
“The
forward rate should reflect
the expected future spot rate on the
date of settlement.”
f1
= e1
Forward Rates as Unbiased
Predictors of Future Spot Rates (UFR)
Example:
90-day forward rate: SFr1 = $0.75
– future spot rate: SFr1 = $0.75
Empirical evidence:
– forward rate appears biased
– bias caused by risk premium
– premium appears to average out
–
Currency Forecasting
Forecasting
requirements:
superior models
superior, consistent information
small, temporary deviations
predict government intervention
Currency Forecasting
Market-based
forecasts:
– forward rates
– interest rate differentials
Model-based forecasts
– fundamental analysis
– technical analysis
Shapiro: Problem 4-7a
iUSA
= 10%
iGDR = 4%
€1 = $0.95
Which currency will appreciate?
Exchange rates for next five years?
€t = $0.95(1.10/1.04)t
Shapiro: Problem 4-7a
Year (t)
Exchange Rate
1
e1 = 1.0048
2
e2 = 1.0628
3
e3 = 1.1241
4
e4 = 1.1889
5
e5 = 1.2575