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### INBU 4200INTERNATIONAL FINANCIAL MANAGEMENT

### Appendix 1: Formulas for the Relative PPP

### Appendix 2: Formulas for the IFE

Lecture 6: Part 2

Topic: Forecasting Exchange Rates With Parity Models

(1) Purchasing Power Parity

(2) International Fisher Effect Model

What are Parity Models all About?

- Parity can be defined as a state of equilibrium.
- Foreign exchange parity models are attempts to “estimate” what the equilibrium spot exchange rate should be at some date.
- The date could be today (i.e., is the current spot rate realistic?).
- Or the date could be some time in the future (what might the equilibrium spot rate be then).
- Parity models have an economic basis (or theory) for their spot rate determination.

Two Major Spot FX Parity Models

- Two important spot foreign exchange parity models in use today are:
- Purchasing Power Parity (PPP)
- Assumes that relative rates of inflation between two countries determines the equilibrium spot exchange rate.
- International Fisher Effect (IFE)
- Assumes that relative rates of interest (long term interest rates) between two countries determines the equilibrium spot exchange rate.

Long Term Spot FX Parity Models

- Both the Purchasing Power Parity Model and the International Fisher Effect are regarded as longer term forecasting models.
- Thus they would appear to be helpful for companies involved in intermediate and long term location decisions (i.e., capital budgeting) and intermediate and long term financing decisions.
- Probably not that useful in assess very short term foreign currency exposure.
- And only limited usefulness as a short term trading strategy.

Purchasing Power Parity Theory

- The Purchasing Power Parity (PPP) explains and quantifies the relationship between inflation and spot exchange rates.
- The theory states that the spot exchange rate between two currencies should be equal to the ratio of the two countries’ price levels.
- Idea was first proposed by the classical economist, David Ricardo, in the 19th century.
- But the concept was fully developed by the Swedish economist, Gustav Cassel, during the years after WW1 when countries in Europe were experiencing hyperinflation.

Two Popular Forms of PPP

- There are two popular forms of the Purchasing Power Parity:
- (1) The Absolute PPP Model and
- (2) The Relative PPP Model.
- Absolute PPP:
- After adjusting for exchange rates, the prices of similar goods in two different countries should be equal.
- Relative PPP:
- The change in the exchange rate between two currencies should be equal to the rate of change in the prices of similar goods between the two countries.

Rationale Behind the PPP: The Law of One Price

- Both Purchasing Power Parity models are based on the Law of One Price:
- The Law of One Price states that all else equal (i.e., no transaction costs or other frictions, like tariffs) a product’s price should be the same in all markets.
- Question: Why should the product’s price be the same?
- Answer: The principle of competitive markets assumes that prices will equalize as consumers and business firms shift their purchases to those markets (or countries) where prices are the lowest.

Law of One Price and Exchange Rates

- Law of One Price: In equilibrium, after adjusting for exchange rates, prices should be the same for similar products in different countries.
- Using the U.S. and Japan as an example:
- In equilibrium the price of product “X” in the U.S. in US dollars (P$), adjusted by the spot exchange rate (S = Yen per dollar), will equal the price of product “X” in Japan in Japanese yen (P¥), or:
- P$ S = P¥
- The formula above is the Law of One Price formula

Law of One Price Example

- Assume a Big Mac hamburger costs $2.00 in the United States and the current yen spot exchange rate is ¥120.00
- According to the Law of One Price, the “equilibrium” Big Mac hamburger price in Japan is:
- P$ S = P¥
- $2.00 x ¥120.00 = ¥240.00

The Absolute PPP Spot Exchange Rate

- The Law of One Price formula can be re-arranged to calculate the “Absolute” PPP Spot Exchange Rate:
- P$ x S = P¥
- Sppp = P¥ / P$
- Where Sppp is the Absolute PPP Spot Exchange Rate
- Example: If a Big Mac cost $2.00 in the U.S. and 300 yen in Japan, the Absolute PPP spot rate is:
- ¥300/$2 = ¥150 (in European terms)
- Note: The Absolute PPP exchange rate of ¥150 is the equilibrium spot exchange rate that would produce “similar prices.”
- How do we calculate the Absolute PPP Spot rate?
- We do so from local currency prices for similar goods in two different countries.

Absolute PPP Spot Exchange Rate: European Terms and American Terms

- Since the absolute PPP spot rate is simply the ratio of the two prices of similar goods in two local currencies, we can calculate this equilibrium exchange rate for either a European terms or an American terms quoted currency as follow:
- For European terms, the Absolute PPP Spot is
- Absolute PPP = Foreign price/U.S. price
- For American terms, the Absolute PPP Spot is
- Absolute PPP = U.S. price/Foreign price.

European Terms Example

- Big Mac: United States : $3.08 (excluding taxes)
- Big Mac: Japan: ¥250 (excluding taxes)
- Calculate Absolute PPP European Terms as follows:
- Absolute PPP Spot Exchange Rate = Yen Price/Dollar Price
- Absolute PPP Spot Exchange Rate = ¥250/$3.08 = ¥81.17
- Question: How do we use this number?
- Answer: Compare the Absolute PPP Spot rate to the actual spot rate, to determine if the current spot rate is overvalued or undervalued.
- Rate on June 20, 2007 was 123.61
- Question: What is this model telling us about the yen’s current spot rate (i.e., is it overvalued or undervalued?)
- ANSWER: Undervalued (by about 34%).
- 123.61 – 81.17/123.61 = 34.33%

American Terms Example

- Tall Starbucks Latte: United States: $2.80 (excluding taxes)
- Tall Starbucks Latte: France: €2.90 (excluding taxes)
- Calculate Absolute PPP American Terms as follows:
- Absolute PPP Spot Exchange Rate = Dollar Price/Euro Price
- Absolute PPP Spot Exchange Rate = $2.80 / €2.90 = $0.9655
- Compare this Absolute PPP Spot rate to the actual rate:
- Rate on June 20, 2007 was 1.3409
- Question: What is this model telling us about the euro’s spot rate (i.e., is it overvalued or undervalued?)
- ANSWER: Overvalued (by about 28%)
- 1.3409 - .9655/1.3409 = 27.99%

Rules for the Absolute PPP

- As noted, the Absolute PPP can be used to “estimate” whether a foreign currency’s spot rate is overvalued or undervalued and by how much.
- Absolute PPP European Terms:
- If PPP Spot < Current Spot, then the currency is undervalued.
- E.g.: PPP = 100; Current Spot = 110
- If PPP Spot > Current Spot, then the currency is overvalued.
- E.g.: PPP = 100; Current Spot = 90
- Absolute PPP American Terms:
- If PPP Spot > Current Spot, then the currency is undervalued.
- E.g.: PPP = $1.20; Current Spot = $1.00
- If PPP Spot < Current Spot, then the currency is overvalued.
- E.g.: PPP = $1.20 Current Spot = $1.40

Absolute PPP in Practice

- In practice, the “absolute” PPP Spot exchange rate is used to assess the “correctness” of a current spot rate on the basis of similar goods in different countries.
- It examines the possibility that a currency is overvalued or undervalued, and by how much?
- Where can we get data for the Absolute PPP model?
- The Economist magazine’s "Big Mac” index.
- http://www.economist.com/markets/Bigmac/Index.cfm
- Go to this web-site an examine specific currencies.

One Test of the Big-Mac: The Introduction of the Euro

- The Euro was introduced on January 1, 1999. The first day trading price was $1.1874.
- According to the Big-Mac data, at the time of the euro’s introduction the Absolute PPP Spot rate could be calculated as follows:
- Average price of a Big-Mac in the euro zone = €2.53
- Average price of a Big-Mac in the U.S. = $2.63
- Absolute PPP Spot rate = $2.63/€2.53 = $1.04
- Comparing the actual spot ($1.1874) to the Absolute PPP Spot ($1.04) suggested the euro was overvalued by about 12.5% at the time it began trading.
- This would suggest the currency should weaken in the period ahead.

What Happened to the Euro? The Euro: January 1, 1999 – December 31, 1999

Relative Purchasing Power Parity

- The second PPP model, the relative Purchasing Power Parity model is concerned with the “rate of change” in the exchange rate.
- It is not assessing the “correctness” of the current spot rate.
- It is concerned with forecasting a future equilibrium rate.
- The relative PPP model suggests that spot exchange rates change(i.e., move) in a manner opposite to the inflation differential between the two countries.
- The Relative PPP model suggests that the percent change in a spot exchange rate over time should be equal to, but opposite in direction to, the difference in the rates of inflation between countries.

Relative PPP Example

- Assume the following:
- Annual rate of inflation in U.S. = 2.0%
- Annual rate of inflation in U.K. = 3.0%
- Question: What should the pound do over time according to the Relative PPP?
- Answer: The British pound should depreciate (i.e., weaken) 1% per year against the U.S. dollar.
- So, if the current spot rate is $1.80, then
- 1 year from now the spot rate should be: $1.7820
- $1.80 – (1.80 x. 01) = $1.7820
- Note: This represents a depreciation of 1% over the current spot rate, An amount which is equal to the inflation differential.
- Note: See Appendix 1 for specific Relative PPP formulas for both American terms and European terms currencies.

Where can we get Inflation Data?

- Historical and Current Data:
- Visit Central Bank Web sites at:
- http://www.bis.org/cbanks.htm
- For Forecasts of Inflation:
- Visit: The Economist Magazine.
- http://www.economist.com/index.html

The International Fisher Effect

- The second foreign exchange parity model is the International Fisher Effect (IFE).
- This model uses long term interest rates to explain why exchange rates change over time.
- The model consists of two parts:
- (1) Fisher Effect which is an explanation of the market interest rate, and
- (2) The International Fisher Effect which is an explanation of the relationship of market interest rates to exchange rate changes.
- The model is attributed to the American

economist, Irving Fisher

(1895 - 1935).

Part 1: The Fisher Effect

- The IFE model begins with the Fisher interest rate model:
- According to Irving Fisher the market interest rate is made up of two components:
- Real rate requirement; which relates to the real growth rate (e.g., real GDP) in the economy.
- U.S. Data: Last 125 years: 1.8%; 1973-2006: 2.3%
- Inflationary expectations premium; which relates to the markets’ expectations regarding future rates of inflation.
- Or, simply put:
- Market rate of interest = real rate + expected inflation
- Real rate requirement is assumed to be relatively stable.
- Changes only occur slowly in response to technology changes, population growth, population skills, etc.
- Inflationary expectations, however, are subject to potentially wide variations over short periods of time.

Fisher Effect: International Assumptions

- On an international level, the Fisher Model assumes that the real rate requirement is similar across major industrial countries.
- If this is true, any observed market interest rate differences between counties will be accounted for on the basis of differences in inflation expectations.
- Example:
- If the United States 1 year market interest rate is 5% and the United Kingdom 1 year market interest rate is 7%, then:
- The expected rate of inflation over the next 12 months must be 2% higher in the U.K. compared to the U.S.

Part 2: International Fisher Effect

- The second part of the Fisher model, the International Fisher (IFE) effect assumes that:
- Changes in spot exchange rates are determined by differences in market interest rates between countries.
- Why this assumption?
- Because differences in interest rates reflect differences in expected inflation, which in turn affect exchange rates.
- IFE relationship to Exchange Rates
- Currencies of high interest rate countries will weaken.
- Why: These countries have high inflationary expectations
- Currencies of low interest rate countries will strengthen.
- Why: These countries have low inflationary expectations.
- Note: The IFE is a longer term model and its conclusions differ (they are opposite) from the short term asset choice model.

IFE Example

- Assume the following:
- I year Government bond rate in U.S. = 5.00%
- 1 year Government bond rate Japan = 2.00%
- According to the IFE, the yen should appreciate 3.0% per year against the U.S. dollar.
- Thus, if the current spot rate is 120, then
- 1 year from now the spot rate should be,
- 120 - (120 x .03) = 116.40
- Note: This represents a appreciation of 3% over the current spot rate.
- An amount which is equal to the interest rate differential.
- Note: See Appendix 2 for specific IFE formulas.

Problematic Issues Regarding the Two Parity Models

- PPP model issues:
- User needs to “forecast” the future rates of inflation.
- How does one do this for very long periods of time?
- Perhaps it is easier for short time periods.
- IFE model issues:
- User relies on market interest rate data to “proxy” for future inflation.
- However, are real rates similar across countries?
- And, do real rates change over time?
- And how stable are inflationary expectations over the period of the forecast?.

The following slides cover the specific formulas to be used in calculated the Relative PPP spot rate for some future date. Note the formula for an American Terms quoted currency and for an European Terms quoted currency.

Relative PPP Formula: American Terms

- For an American Term quoted currency:
- PPP Spot Rate = Current Spot Rate x (1 + infhome)n/(1 + infforeign)n)
- Where:
- PPP Spot Rate is the expected spot rate sometime in the future.
- Current spot rate is expressed in American terms.
- Infhome is the expected annual rate of inflation in the United States.
- Infforeign is the expected annual rate of inflation in the foreign country.
- N is the number of years in the future.

Relative PPP Formula: American Terms

- Example:
- Current spot rate for British pounds = $1.80
- Expected annual rate of inflation in the U.S. = 2.0%
- Expected annual rate of inflation in the U.K. = 3.0%
- Then, the spot pound 2 years from now is equal to:
- PPP Spot Rate = Current Spot Rate x (1 + infhome)n/(1 + infforeign)n)
- Spot rate in 2 years = 1.80 (1+.02)2/(1+.03)2
- Spot rate in 2 years = 1.80 (1.0404/1.0609)
- Spot rate in 2 years = 1.80 (.9807)
- Spot rate in 2 years = $1.7653

Relative PPP Formula: European Terms

- For European Term quoted currency:
- PPP Spot Rate = Current Spot Rate x (1 + infhome)n/(1 + infforeign)n)
- Where:
- PPP spot rate is the expected spot rate sometime in the future.
- Current spot rate is expressed in European terms.
- Infhome is the expected annual rate of inflation in the foreign country.
- Infforeign is the expected annual rate of inflation in the United States.
- N is the number of years in the future.

Relative PPP Formula: European Terms

- Example:
- Current spot rate for Japanese yen = 111.00
- Expected annual rate of inflation in the U.S. = 2.0%
- Expected annual rate of inflation in Japan = 1.0%
- Then, the spot yen 2 years from now is equal to:
- PPP Spot Rate = Current Spot Rate x (1 + infhome)n/(1 + infforeign)n)
- Spot rate in 2 years = 111 (1+.01)2/(1+.02)2
- Spot rate in 2 years = 111 (1.0201/1.0404)
- Spot rate in 2 years = 111 (.9805)
- Spot rate in 2 years = 108.84

The following slides cover the specific formulas to be used in calculated the IFE spot rate for some future date. Note the formula for an American Terms quoted currency and for an European Terms quoted currency.

IFE Formula: American Terms

- For American Term quoted currency:
- IFE Spot Rate = Current Spot Rate x (1 + inthome)n/(1 + intforeign)n)
- Note the similarity to the Relative PPP formula
- Where:
- IFE spot rate is the expected spot rate sometime in the future.
- Current spot rate is expressed in American terms.
- Inthome is the current annual market interest rate in the United States.
- Intforeign is the current annual market interest rate in the foreign country.
- N is the number of years in the future.

IFE Formula: American Terms

- Example:
- Current spot rate for British pounds = $1.80
- Annual rate of interest in the U.S. = 5.0%
- Annual rate of interest in the U.K. = 6.0%
- Then, the spot pound 2 years from now is equal to:
- PPP Spot Rate = Current Spot Rate x (1 + infhome)n/(1 + infforeign)n)
- Spot rate in 2 years = 1.80 (1+.05)2/(1+.06)2
- Spot rate in 2 years = 1.80 (1.1025/1.1236)
- Spot rate in 2 years = 1.80 (.9812)
- Spot rate in 2 years = $1.7679

IFE Formula: European Terms

- For European Term quoted currency:
- IFE Spot Rate = Current Spot Rate x (1 + inthome)n/(1 + intforeign)n)
- Again, note the similarity to the Relative PPP formula
- Where:
- IFE spot rate is the expected spot rate sometime in the future.
- Current spot rate is expressed in European terms quote.
- Inthome is the current annual market interest rate in the foreign country.
- Intforeign is the current annual market interest rate in the United States.
- N is the number of years in the future.

IFE Formula: European Terms

- Example:
- Current spot rate for Japanese yen = 120.00
- Annual rate of interest in the U.S. = 5.0%
- Annual rate of interest in Japan = 2.0%
- Then, the spot yen 2 years from now is equal to:
- PPP Spot Rate = Current Spot Rate x (1 + infhome)n/(1 + infforeign)n)
- Spot rate in 2 years = 120 (1+.02)2/(1+.05)2
- Spot rate in 2 years = 120 (1.0404/1.1025)
- Spot rate in 2 years = 120 (.9436)
- Spot rate in 2 years = 113.24

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