


International Parity Conditions Lecture 6


Random Walk 
Random Walk  a variable is determined by the previous period plus a random disturbance
Notation
Spot currency exchange rate is S_{t}
Spot currency exchange rate previous period, S_{t1}
Random disturbance, e_{t}
Assumed normally distributed
Mean is zero
Constant variance
 We do a first difference of U.S. dollar per euro, and it appears random

Purchasing Power Parity (PPP) 
1. Purchasing Power Parity (PPP)  goods denominated in differrent currencies should have the same price
Law of One Price  the price of a product should be the same between two separate markets

Transportation and transaction costs would cause price difference between markets
Arbitrage is possible if markets have different prices
Purchasing Power Parity  predict changes in exchange rates

Purchasing Power Parity = the exchange rate is the ratio of a product's price in two countries
Example 1: Petroleum
P_{USA} = $90 per barrel
P_{Mexico} = 850 Mexican pesos
Implied exchange rate: S_{t} = pesos 850 / $90 = pesos 9.444 / $1
The actual exchange rate is $1 = 10 Mexican pesos
The dollar is overvalued and the peso is undervalued
Traders can buy petroleum from Mexico and sell petroleum to United States
Petroleum price increases in Mexico (lower supply)
Petroleum price decreases in United States (higher supply)
Trade stops when prices become equal between countries
Example 2: Big Mac Index
The Economist

Most ingredients are locally produced
P_{euro} = 3.06 € per Big Mac
P_{USA} = $3.41 per Big Mac
Implied exchange rate: S = $3.41 / 3.06 € = $1.114 / 1 €
The actual exchange rate is $1.35 / 1 €

The euro is overvalued while the U.S. dollar is undervalued

In theory, we could ship Big Macs from the U.S. to Europe; a Big Mac costs $3.41 in the U.S. and $4.13 in the Eurozone
The euro is overvalued approx. 21.1%; the euro should depreciate
Absolute PPP  the exchange rate is the ratio of two countries price levels

Purchasing Power Parity includes all products in a society
Price level is the average price for a basket of goods
Consumer Price Index (CPI)  price index of basket of goods
Example: Law of One Price for CPIs
The actual exchange rate is S = 1.43 francs per $1
The Swiss franc is overvalued while the dollar is undervalued. Using arbitrage, a trader buys a basket of goods in the U.S. for $755.3 and sells it for $867.97 ( 1,241.2 / 1.43) in Switzerland for a profit.
PPP tends to hold in the long run and not in the short run

PPP emphasizes trade and price levels
Countries have trade restrictions and trade barriers like quotas and tariffs
PPP does not account for transactions costs and transportation costs
Countries place different goods in the basket for the Consumer Price Index (CPI)

Some goods and services cannot be traded, such as haircuts, medical services, and real estate. Real estate prices could cause wide differences in prices
2. Relative Purchasing Power Parity  the change in the CPI's price level influences the exchange rate
Define the percent change in the exchange rate as:
We can use the approximation if the inflation rates are low:
3. Competitiveness Ratio  we can modify the relative PPP to determine if a country has competitive exports
We start with the equation below:
Add one to both sides and divide by (1 + e), to yield:
 Set the one into a variable k
 k should equal one in the long run but can change in the short run
 k > 1, a country's exports are not competitive
 k < 1, a country's exports are competitive
 Example
 U.S. is domestic country with 3% inflation rate
 Eurozone has 2% inflation rate
 U.S. dollar depreciates 2% per year against the euro
 k = 1.011
 U.S. industries are not competitive

The Quantity Theory of Money 
Quantity Theory of Money  relate the demand for money to a country's GDP
Notation
The demand for money is M^{D}
The price level is denoted by P
A country's income is Y; measure by a country's real GDP
The velocity of money, V
Apply the Quantity Theory of Money to the PPP Theory
United States is in the numerator
Eurozone is in the denominator

Use mathematical trick to convert equation into percent changes
Note  if a variable does not change, then it is zero
Euro is defined as the base currency
Example 1
Europe's real GDP grows at 3% per year
United States real GDP increases at 2% per year
U.S. money supply grows at 3% per year
Europe's money supply grows at 1% per year
Thus, the euro appreciates approx. 3% per year
Example 2
Malaysia's real GDP grows at 7% per year
U.S. grows at 3% per year
U.S. money supply grows at 2% per year
Malaysia's money supply grows at 5% per year
Define the ringgit as the base currency
Thus, ringgit appreciates approx. 1% per year

International Fisher Effect 
1. The Fisher Effect
The equation is:
Solve for i:
Use the approximation if numbers are small, because the cross produc, r p, would be small
Example
If expected inflation, p, is 10% and nominal interest rate, i, is 5%, then the real interest rate is 5%
The cross product, r p, is 0.5%
All prices in society rose 10% while your nominal money only grew by 5%
Consequently, you lost 5% of your purchasing power
2. International Fisher Effect builds upon the law of one price for financial transactions
International investors should earn comparable returns in foreign countries as their home country
Exchange rates reflect interest rate differentials
Interest rate is not a commodty that can be bought or sold
Notation
i_{d} is the domestic nominal interest rate in APR and r_{d} is the rate of return of the investment in T days
i_{f} is the foreign nominal interest rate for T days
e is the percent change in exchange rate from beginning of investment to time T
The expected Tday return on a $1 foreign bank deposit is:
The effective Tday return on a bank deposit in the domestic country is:
Arbitrage causes the rates of return to converge for both the foreign and domestic investment. Set both equations equal to each other:
Solve for the change in the exchange rate, e:
Use the linear approximation if numbers are small:
Expected return from Mexican investment:
Investors repay U.S. bank the following:

Interest Rate Parity Theorem 
To price a forward contract, investors use the arbitrage strategies:
1. An investor can invest $1 in the United States for T days. Consequently, he or she earns the rate of return, which is defined below:
2. An investor could invest into a foreign country to earn the foreign interest rate for T days. Consequently, the investor must convert his currency at the spot exchange rate before investing into foreign financial institution. The spot exchange rate is the domestic currency over the foreign currency.
3. The investor locks in his foreign rate of return by covering his investment with a forward contract. Thus, the investor converts his investment back into the home currency at a known future rate:
4. The investor is indifferent between investing in foreign country or domestic country. Thus, arbitrage allows us to set the rates of return equal to each other.
5. Solve for F to obtain the following equation:
We could use the approximation:
Another approximation estimates the percent change of the forward price over the spot price. The percent change is proportional to the interest rate differential:
Example 1
Malaysia has a 6% APR interest rate
United States has a 2% APR interest rate
If the spot exchange rate is 3 ringgits per $1 (or $0.3333 per ringgit), what is price of a sixmonth forward?
Remember, the exchange rate is the domestic currency over the foreign currency.
The price is $0.33266 per ringgit. Thus, the ringgit depreciates while the U.S dollar appreciates
We could percent change in the exchange rate, which is 2%
Note  the country with a greater interest rate experiences a depreciating currency, because inflation would be higher in that country
Example 2
The spot exchange rate is S = 79 yen per $1 (Remember to take reciprocal of exchange rate, because U.S. is defined as home country)
The forward price is F = 81 yen per $1 (Take reciprocal of exchange rate)
Domestic interest rate is for Japan, i_{d} = 3.4
Foreign interest rate is U.S., i_{f} = 5.0
Where would an investor invest his funds for one year, if he or she is indifferent about which country to invest in?
Invetor could earn 5% in the United States, which is greater, because the U.S. dollar is appreciating and the U.S. interest rate is greater
Investor could earn 0.85% in Japan

