# Fisher equation

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The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates inflation. It is named after Irving Fisher, who was famous for his works on the theory of interest. In finance, the Fisher equation is primarily used in YTM calculations of bonds or IRR calculations of investments. In economics, this equation is used to predict nominal and real interest rate behavior.

Letting Template:Mvar denote the real interest rate, Template:Mvar denote the nominal interest rate, and let Template:Mvar denote the inflation rate, the Fisher equation is:

${\displaystyle i\approx r+\pi }$

This is a linear approximation, but as here, it is often written as an equality:

${\displaystyle i=r+\pi }$

The Fisher equation can be used in either ex-ante (before) or ex-post (after) analysis. Ex-post, it can be used to describe the real purchasing power of a loan:

${\displaystyle r=i-\pi }$

Rearranged into an expectations augmented Fisher equation and given a desired real rate of return and an expected rate of inflation πe (with superscript Template:Mvar meaning "expected") over the period of a loan, it can be used as an ex-ante version to decide upon the nominal rate that should be charged for the loan:

${\displaystyle i=r+\pi ^{e}}$

This equation existed before Fisher,[1][2][3] but Fisher proposed a better approximation which is given below. The approximation can be derived from the exact equation:

${\displaystyle 1+i=(1+r)(1+\pi ).}$

## Derivation

### Monetary policy

The Fisher equation plays a key role in the Fisher hypothesis, which asserts that the real interest rate is unaffected by monetary policy and hence unaffected by the expected inflation rate. With a fixed real interest rate, a given percent change in the expected inflation rate will, according to the equation, necessarily be met with an equal percent change in the nominal interest rate in the same direction. Contrary models assert that, for example, a rise in expected inflation would result in only a smaller rise in the nominal interest rate Template:Mvar and thus a decline in the real interest rate Template:Mvar.