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 St

    • Spot currency exchange rate previous period, St-1

    • Random disturbance, et

    • Assumed normally distributed

    • Mean is zero

    • Constant variance

Random walk

  • Random walk appears to drift in a particular direction

    • Forecast - today's forecast is the value of the previous period because we do not know what the random disturbance is.

  • We can do a trick called the first difference

Random walk

  • Example - U.S. dollar per euro exchange rate in months

Random walk-U.S. dollar per euro exchange rate

  • We do a first difference of U.S. dollar per euro, and it appears random

Random Walk-First Difference of .S. dollar-euro exchange rate

Purchasing Power Parity (PPP)

1. Purchasing Power Parity (PPP) - goods denominated in differrent currencies should have the same price

  1. 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

  2. Purchasing Power Parity = the exchange rate is the ratio of a product's price in two countries

    • Notation

    • Domestic price is Pd

    • Foreign price is Pf

    • S is the spot exchange rate

Purchasing Power Parity equation

  1. Example 1: Petroleum

    • PUSA = $90 per barrel

    • PMexico = 850 Mexican pesos

    • Implied exchange rate: St = 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

  2. Example 2: Big Mac Index

    • The Economist

    • Most ingredients are locally produced

    • Peuro = 3.06 € per Big Mac

    • PUSA = $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

Euro is overvalued relative to the U.S.

  • Note - Starbucks index exists

  1. 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

      • Inflation rate - percent change in a price level

    • Example: Law of One Price for CPIs

      • CPIUSA = $755.3

      • CPISwit = 1,241.2 Swiss francs

      • The implied exchange rate:

Calculating the PPP for an example

  • 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.

  1. 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

  • The Inflation Rule

  • The relationship works for large inflation rate differences

  • Notation

    • The percent change in the exchange rate, e

    • Foreign inflation rate is pf

    • Domestic inflation rate is pd

    • ST and S0 are the domestic exchange rates measured at times 0 and T

  • The exchange rates are:

Implied exchange rate at Time 0       Implied exchange rate at Time 0

Define the percent change in the exchange rate as:

The Relative PPP

We can use the approximation if the inflation rates are low:

The Relative PPP

  • Example 1: The U.S. inflation rate is 3% while Mexico is 25%

    • The U.S. dollar should appreciate approximately 25% - 3% = 22% per year, while the peso depreciates

    • Mexicans and Americans prefer to hold U.S. dollars

  • Example 2: The U.S. inflation rate is 3% and Turkey is 10%

    • The U.S. dollar should appreciate approximately 7% per year, while the lira depreciates

  • Consequences

    • Investors hold currencies with low inflation rates

    • Inflation erodes value of currency

    • Countries with high inflation rates tend to have currencies that depreciate

3. Competitiveness Ratio - we can modify the relative PPP to determine if a country has competitive exports

We start with the equation below:

The Relative PPP

Add one to both sides and divide by (1 + e), to yield:

The Relative PPP

  • 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

The Relative PPP

  • 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 Relative PPP

 

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 MD

    • 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

Quantity Theory of Money

  • Demand for money must equal the supply of money, or MD = MS

    • Central bank issues money that the public uses

  • Example

    • Nominal GDP = P Y = $15 trillion for U.S.

    • Money supply is $1 trillion

    • Velocity of money is 15, which means every one-dollar bill is circulated in economy on average 15 times each year

Quantity Theory of Money

  • Apply the Quantity Theory of Money to the PPP Theory

  • United States is in the numerator

  • Eurozone is in the denominator

Quantity Theory of Money and PPP Theory

  • Use mathematical trick to convert equation into percent changes

    • Upper case letters switched to lower case letters

    • Dot represents percentage change

  • Note - if a variable does not change, then it is zero

  • Euro is defined as the base currency

currency exchange rate percent change

  • 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

  • Real interest rate is r

  • Nominal interest rate is i

  • Expected inflation rate is p

The equation is:

The Fisher Effect

Solve for i:

Fisher Effect

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

    • id is the domestic nominal interest rate in APR and rd is the rate of return of the investment in T days

    • if 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 T-day return on a $1 foreign bank deposit is:

Effective return on a bank deposit

The effective T-day return on a bank deposit in the domestic country is:

Return on home bank deposit

Arbitrage causes the rates of return to converge for both the foreign and domestic investment. Set both equations equal to each other:

Set home return equal to foreign bank return

Solve for the change in the exchange rate, e:

Solve for exchange rate

Use the linear approximation if numbers are small:

Linear approximation for change in exchange rate

  • Example 1

    • The time is T = 90 days

    • Domestic interest rate is for United States, id = 3%

    • Foreign interest rate for Japan is, if = 12%

  • Using the International Fisher Effect, the U.S. dollar should appreciate approx. 2.25% while the yen would depreciate

An example calculating the change in exchange rate

  • Example 2

  • Mexican peso depreciated by 5% a year during the early 90s

  • Interest rate differential between U.S. and Mexico ranged between 7% and 16%

  • Strategy

    1. Borrow funds from the United States at id = 5%

    2. Convert dollars to Mexcian Pesos on the spot market, S

    3. Invest in Mexican funds and earn the higher interest rate at if= 12%

    4. Convert pesos back into dollars in one year, T = 1, with spot exchange rate

    5. Expected foreign exchange loss, e = -5%, because Mexican peso is depreciating

Expected return from Mexican investment:

Expected return from a Mexican investment

Investors repay U.S. bank the following:

Investors pay U.S. bank interest

  • The expected profit is 6.7% - 5% = 1.7% per year

    • In reality, profits ranged from 1.5% to 11%

    • Fidelity used this uncovered strategy during the early 90s

    • Fidelity lost all its profits in December 1994 after the peso devaluation

    • Uncovered position - company uses the future spot exchange rate to transfer profit back

    • Covered position - company uses a derivative to fixed the future exchange rate, when profits are converted back to home currency

  • Notes

    • International Fisher Effect is an equilibrium exchange rate; investors will profit from arbitrage

    • Equilibrium is no capital flows from one country to another to take advantage of interest rate differentials

    • A country with a high interest rate is expected to see its currency depreciate, because that country is experiencing greater inflation

 

Interest Rate Parity Theorem

 

  • Interest Rate Parity Theorem is a theory to price forward contracts

  • Uses Law of One Price and Arbitrage

  • Notation:

    • Domestic nominal interest rate in APR is id, while domestic rate of return is rd

    • Foreign nominal interest rate in APR is if, while foreign rate of return is rf

    • The spot currency exchange rate at time t is S

      • Written as a ratio, such as USD / ringgit

    • Forward contract that expires at time T is F

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:

Rate of return to invest in domestic currency

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.

Convert funds and invest in foreign country

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:

Investor buys a forward to lock into an exchange 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.

Use arbitrage to set both equations equal to each other

5. Solve for F to obtain the following equation:

Solve for the price of the forward contract

We could use the approximation:

A linear approximation to estimate forward contract's price

Another approximation estimates the percent change of the forward price over the spot price. The percent change is proportional to the interest rate differential:

Percent change of forward over the spot price

 

  • 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 six-month 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

Calculate forward price for an example

Percent change of forward over the spot price

  • 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, id = 3.4

    • Foreign interest rate is U.S., if = 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

An example calculating a foreign return

 

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