Where can I find bilateral real exchange rates? For example per USD?
I can only find real effective exchange rates, but it is an index, not a bilateral rate.
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Check out this link from the United States Department of Agriculture.
This data set contains annual and monthly data for exchange rates. It includes both bilateral nominal and real exchange rates for 79 countries, plus the European Union, as well as real trade-weighted exchange rate indexes for many commodities and aggregations. All series are updated quarterly. Data series start at the beginning of 1970 and run to the last available data point.
As pointed out by @luchonacho, real exchange rates here are derived by multiplying the nominal exchange rate by the ratio of the U.S. to local currency Consumer Price Index
The real exchange rate is an index.
For example, the nominal exchange rate between British pounds and US dollars is 0.75 pounds per dollar (as of today; e.g. £20 are roughly $27). To calculate the real exchange rate, you have to multiply this by the relative prices of these two countries, after which you end up with a dimensionless index. This index tells you the relative value of the same basket of goods in the two countries. In a very idealised world were the law of one price holds, such index is one.
Both the bilateral exchange rate and the effective or multilateral exchange rate are defined in terms of currencies. Not their real counterparts.