Fisher equation

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NOTE: this is not Fisher's equation in differential equations

The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under inflation. In finance, this 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.

Let $r r$ denote the real interest rate, $r n$ denote the nominal interest rate, and let $π$ denote the inflation rate.

The Fisher equation is the following:

$r n = r r + π$

The equation can be used in either ex-ante (before) or ex-post (after) analysis.

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

$1 + r n = (1 + r r)(1 + π).$

[edit] Derivation

From

$1 + r n = (1 + r r)(1 + π)$

follows

$1 + r n = 1 + r r + π + r r π$

and hence

$r n = r r + π + r r π$

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

$r n = r r + π$

is the result.

[edit] Example

The market rate of return on the 4.25% UK government bond maturing on 7 March 2036 is currently 3.81% per annum. Let's assume that this can be broken down into a real rate of exactly 2% and an inflation premium of 1.775% (no premium for risk, as government bond is considered to be "risk-free"):

1.02 x 1.01775 = 1.0381

This article implies that you can ignore the third term (0.02 x 0.01775 = 0.00035 or 0.035%) and just call the nominal rate of return 3.775%, on the grounds that that is almost the same as 3.81%.

At a nominal rate of return of 3.81% pa, the value of the bond is £107.84 per £100 nominal. At a rate of return of 3.775% pa, the value is £108.50 per £100 nominal, or 66p more.

The average size of actual transactions in this bond in the market in the final quarter of 2005 was £10 million. So a difference in price of 66p per £100 translates into a difference of £66,000 per deal.