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# Fisher equation

The Fisher equation estimates the relationship between nominal and real interest rates under inflation. This equation is primarily used in YTM calculations of bonds or IRR calculations of investments.

Let rr denote the real interest rate, rn denote the nominal interest rate, and let π denote the rate of inflation.

The Fisher equation is the following:

rn = rr + π

The equation can be used in either ex-ante (before) or ex-post (after) analysis. Ex-ante analysis is done in the beginning of investment (either before or immediate after investing). Because the actual rate of inflation is unknown, the expected value of inflation is used for π. Ex-post analysis is done at the end of the investment to check if the investment was worthwhile. The actual rate of inflation is known and therefore used for π.

This equation is named after Irving Fisher who was famous for his works on the theory of interest. This equation existed before Fisher, but Fisher proposed a better approximation which is given below. The estimated equation can be derived from the proposed equation

1 + rn = (1 + rr)(1 + π).

## Derivation

From

1 + rn = (1 + rr)(1 + π)

follows

1 + rn = 1 + rr + π + rr π

and hence

i = r + π + r π.

Drop rπ because r + π is much larger than rπ:

i = r + π

is the result.