Fisher equation

In financial mathematics and economics, the Fisher equation expresses the relationship between nominal interest rates, real interest rates, and inflation. Named after Irving Fisher, an American economist, it can be expressed as real interest rate ≈ nominal interest rate − inflation rate.

In more formal terms, where ${\displaystyle r}$ equals the real interest rate, ${\displaystyle i}$ equals the nominal interest rate, ${\displaystyle \pi }$ equals the inflation rate, and ${\displaystyle c}$ equals the compounding period, then ${\displaystyle (1+ci)=(1+cr)(1+c\pi )}$.

Therefore,

${\displaystyle r={\frac {i-\pi }{1+c\pi }}}$

and, more importantly,

${\displaystyle \operatorname {ln} (1+ci)=\operatorname {ln} (1+cr)+\operatorname {ln} (1+c\pi )}$.

This asserts that the nominal force of interest is the sum of the real force of interest and the force of inflation. Since ln(1+x)~x, for low interest and inflation rates, the force of interest/inflation can be approximated by the interest/inflation rate.

The approximation ${\displaystyle r=i-\pi }$ is often used instead since the nominal interest rate, real interest rate, and inflation rate are usually close to zero.