You can compare the real returns on dollar-denominated securities with the real returns on euro-denominated securities. The concept that makes this comparison possible is the expected change in the exchange rate.

For example, when you invest in a euro-denominated security, you hope to make money in two ways. First, you expect to earn a return on this security you are planning to hold. Suppose that ris the real return on euro-denominated securities.

Second, you, the American investor, convert your dollars into euros, buy euro-denominated securities, earn returns in euros, and convert your euro earnings into dollars. Therefore, you care about the future exchange rate that you observe when you convert your earnings in euros into dollars.

However, you are investing in euro-denominated securities now, and you don’t know for sure what the dollar–euro exchange rate is going to be in the future — say, a year from now. Still, you need to have an expected dollar–euro exchange rate in mind.

Clearly, you can observe the current exchange rate. You can call the current time t. Suppose you have an expectation of what the dollar–euro exchange rate is going to be at a future date. You use the current and expected exchange rate to calculate the expected percent change in the exchange rate using the following formula:

where:

ER = Exchange rate

Note the line over the expected exchange rate, which indicates that you are holding its value constant.

Now you, the investor, can make the appropriate comparison between the real return on a risk- and maturity-comparable dollar- and euro-denominated security. You are indifferent between these securities if the following parity condition holds:

Note that the previous formula drops the time subscript for simplicity.

where:

r\$ = Real return on the dollar-denominated security (in dollars)
r = Real return on the euro-denominated security (in euros)

The right side of the equation expresses the expected real returns on the euro-denominated security in dollars. By adding the expected change in the exchange rate to the real return on the euro-denominated security, you account for two possible sources of your return from a euro-denominated security: the interest rate on the euro-denominated security (r) and whether you will enjoy additional returns when you convert your earnings in euros into dollars.

Now you can compare your earnings on a dollar-denominated security to your earnings on the euro-denominated security. In other words, you can compare the number of dollars you will have in the future from holding the dollar-denominated security to the number of dollars you will have in the future from holding the euro-denominated security.

Another way of expressing the parity condition is that the difference between the real return on a dollar-denominated security and that of a euro-denominated security in dollars must be zero:

The following figure shows four numerical examples to which you can apply the parity condition and decide which security gives you a higher real return. The second and third columns indicate the real return on the dollar- and euro-denominated securities, respectively. The fourth column shows the expected change in the exchange rate.

The last column indicates the parity condition as the difference among these three variables, and the difference is zero. In other words, after adjusting for the expected changes in the exchange rate, a security denominated in euros offers the same (expected) real rate of return as the dollar-denominated security.

For simplicity, the real returns on the dollar-denominated security don’t change. However, you can see changes in the real returns on the euro-denominated security and the expected change in the exchange rate.

Take a look at the figure and consider each example:

• Example #1: The real return on the dollar- and euro-denominated security is 10 percent and 6 percent, respectively. Additionally, the expected change in the exchange rate indicates 4 percent depreciation in the dollar.

In other words, if you invest in the euro-denominated security, in addition to earning 6 percent interest on the security, you earn 4 percent by holding a security whose currency is expected to appreciate. In this case, you earn 10 percent real return in either security. Therefore, you are indifferent between the dollar- and euro-denominated securities.

• Example #2: The second example has the same real returns on the dollar- and euro-denominated securities, which are 10 percent and 6 percent, respectively. However, in this case, the expected exchange rate is the same as the current exchange rate, so the expected change in the exchange rate is zero.

Now the real return on the dollar-denominated security exceeds that on the euro-denominated security by 4 percent. In this case, you want to invest in the dollar-denominated security.

• Example #3: This example indicates the real returns on the dollar- and euro-denominated securities as 10 percent and 12 percent, respectively. However, the expected change in the exchange rate is –4 percent, which indicates a 4 percent appreciation in the dollar.

This reduces your real return on the euro-denominated security in dollars to 8 percent, which is lower than the real return on the dollar-denominated security. In this case, you want to invest in the dollar-denominated security.

• Example #4: The last example indicates that the real return on the dollar- and euro-denominated securities is 10 percent and 6 percent, respectively. The expected change in the exchange rate is 8 percent, which indicates a depreciation of the dollar by 8 percent.

This means that your expected real return on the euro-denominated security is 4 percent greater than that on the dollar-denominated security. In this case, you want to invest in the euro-denominated security.