Relative purchasing power parity relates the change in two countries' expected inflation rates to the change in their exchange rates. Inflation reduces the real purchasing power of a nation's currency. If a country has an annual inflation rate of 10%, that country's currency will be able to purchase 10% less real goods at the end of one year. Relative purchasing power parity examines the relative changes in price levels between two countries and maintains that exchange rates will change to compensate for inflation differentials.
The relationship can be expressed as follows, using indirect quotes:
Formula 5.6
S_{1} / S_{0} = (1 + I_{y}) Ã· (1 + I_{x})
Where,
S_{0} is the spot exchange rate at the beginning of the time period (measured as the "y" country price of one unit of currency x)
S_{1} is the spot exchange rate at the end of the time period.
I_{y} is the expected annualized inflation rate for country y, which is considered to be the foreign country.
I_{x} is the expected annualized inflation rate for country x, which is considered to be the domestic country.
Look Out! Note that the spot exchange rate used must be the quantity of currency y (the foreign currency) needed to purchase one unit of currency x (the domestic currency). If we want the spot value of the U.S. dollar in British pounds, the quote must be 0.6667 British pounds per dollar, not $1.50 per British pound. 
Example 1:
Suppose that Mexico's expected annual rate of inflation is equal to 6% per year, while the expected annual inflation rate for the U.S. is 3%. As an approximation, we would expect that the Mexican peso would depreciate at the rate of 3% per year (or we could say that the U.S. dollar should appreciate at the rate of 3% per year.
Example 2:
Suppose that the annual inflation rate is expected to be 8% in the Eurozone and 2% in the U.S. The current exchange rate is $1.20 per euro (Â¬1.00 = $1.20). What would the expected spot exchange rate be in six months for the euro?
Answer:
So the relevant equation is:
S_{0.5 }Ã· S_{0} = (( 1 + I_{us}) Ã· (1 + I_{eurozone}))^{0.5}
= S_{0.5} Ã· $1.20 per euro = (1.02 Ã· 1.08)^{0.5}
Which implies S_{0.5} = (1.20) Ã— 0.978125 = 1.1662
So the expected spot exchange rate at the end of six months would be $1.1662 per euro.
Example 3:
Assume that the U.S. is the foreign country and that Japan is the domestic country. The current spot exchange rate is S_{0} = 115 yen per dollar ($1 per Â¥115.00). The expected annual inflation rate for the U.S. is 4.89%, and the annual expected Japanese inflation rate is 6.23%. Compute the approximate expected spot rate and the expected spot rate one year from now.
Answer:
Because Japan is the domestic country we have:
S_{0} = 115 yen per dollar. (1 + Iy) is 1.0489, and (1 + Ix) is equal to 1.0623.
The approximation method would indicate that the yen should decline against the dollar by: 1.0623  1.0489 = 0.0134 = 1.34%
So the value of the yen relative to the dollar would be expected to decline to
(1 + 0.0134) Ã— 115 = Â¥116.54 per $
We can calculate the rate more exactly as:
S_{1} = (1.0623) / (1.0489) Ã— 115 = Â¥116.47 per $
Purchasing Power Parity and Real Return on Assets
The purchasing power parity principle also applies to the real returns on assets earned by various investors across the world. It holds that the real rate of return on assets should be the same for investors from any nation.
Suppose that a financial asset from Mexico has an annual rate of return of 10% in Mexican pesos. Assume that Mexico has an annual inflation rate of 4%, the U.S. has an annual inflation rate of 2% and that the U.S. dollar is appreciating by 2% a year, as predicted by purchasing power parity.
By the approximation method, U.S. investors would be earning about 8% per year in terms of Mexican pesos. Their real rate of return would be approximately 6% per year after U.S. inflation is taken into account. This is equal to what Mexican investors are earning, which is about 6% (10% nominal rate of return  4% inflation). All investors are getting the same real rate of return on specific assets.
Note that purchasing power parity is a theoretical concept that may not be true in the real world, especially in the short run.
Exam Tip!
Past exams have included questions that require purchasing power parity calculations, so it is a good idea to practice solving questions such as the examples given above. 

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