You need to understand how the PPP is derived. Understanding the relationship between inflation differentials and changes in the exchange rate enables you to attach a number to the change in the exchange rate, such as 2 percent depreciation. Then because spot exchange rates are observable, you can apply the expected change in the exchange rate to the spot rate, to predict the future spot rate.

## Derivation of the PPP

Suppose that πH and πF indicate the home and foreign country’s inflation rates, respectively. In the following equations, you work with inflation factors of home and foreign countries, (1+ πH) and (1+ πF), respectively.

Remember, the relative PPP implies that changes in an exchange rate follow the changes in both countries’ price levels. You can express this relationship first by realizing that you cannot compare home and foreign country’s inflation factors:

One way to express the relationship between the home and foreign countries’ inflation factors is to adjust the foreign inflation factor with the expected change in the exchange rate. In this case, you have the following:

Here, e indicates the percent change in the exchange rate and is defined as the number of home currency per foreign currency. You can solve this equation for e by dividing both sides of the equation by (1+ πF) and moving 1 to the other side:

This equation indicates that if the home inflation rate is larger than the foreign inflation rate, the ratio of the two inflation factors becomes larger than 1, making e a positive percent change in the exchange rate. This scenario implies a depreciation in the home currency.

On the other hand, if the home inflation rate is smaller than the foreign inflation rate, the ratio of the two inflation factors becomes smaller than 1, making e a negative percent change in the exchange rate. It implies an appreciation in the home currency.

The idea behind the relationship between the change in the exchange rate and the inflation differential is related to the exchange rate determination. For example, when home inflation rate is higher than foreign inflation rate, you are inclined to buy foreign goods, which leads to exchanging domestic currency for foreign currency. Therefore, domestic currency depreciates.

As an approximation, for a smaller inflation differential between home and foreign country, you can use this formula as the difference between the home and foreign inflation rates:

The previous equation means that the percent change in the exchange rate should approximately equal the difference between the home and foreign inflation rates.

In 2010, the inflation rates based on the Consumer Price Indices of the U.S. and Turkey were 1.64 and 8.52 percent, respectively. Based on these inflation rates, the PPP indicates an expected change in the exchange rate of:

The U.S. and Turkish inflation rates imply a 6.34 percent appreciation in the U.S. dollar. If you use the approximation (1.64 – 8.52 = –6.88), the appreciation in the U.S. dollar becomes 6.88 percent. These two rates of appreciation are considered as similar for most purposes.

## Application of the PPP

If you know the current spot rate, you can use the expected change in the exchange rate given in the previous example to predict the next period’s future spot rate. The dollar–Turkish lira exchange rate in 2009 was \$0.67. Look at this rate as the spot rate in 2009, and suppose you want to guess the spot rate in 2010.

For simplicity, further assume that the mentioned U.S. and Turkish inflation rates for 2010 reflect your expectations for 2010 in 2009.

The following equation connects the current and expected spot rate next period:

Here, the components of the equation imply the expected spot rate at time t+1, the current spot rate at time t, and the change in the exchange rate. Using this formula, calculate your expected exchange rate for 2010:

Based on the inflation differential between the U.S. and Turkey, your expected dollar–Turkish lira exchange rate for 2010 is \$0.63, which implies appreciation in the dollar. (By the way, the actual dollar–Turkish lira exchange rate in 2010 was \$0.65.)

This numerical example doesn’t constitute a test for the PPP. It aims to show the interpretation of the relevant equations. Clearly, you need to collect decades of data and apply suitable econometric techniques to test for the PPP or use the PPP for forecasting. Also, this example is closer to the future spot rate than what is often observed.