Fisher equation

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 Failed to parse (Missing texvc executable; please see math/README to configure.): r_r

denote the real interest rate, Failed to parse (Missing texvc executable; please see math/README to configure.): r_n
denote the nominal interest rate, and let Failed to parse (Missing texvc executable; please see math/README to configure.): i
denote the inflation rate.

The Fisher equation is the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): r_n = r_r + i

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 1 + r_n = (1 + r_r)(1 + i).

Derivation

From

Failed to parse (Missing texvc executable; please see math/README to configure.): 1 + r_n = (1 + r_r)(1 + i)

follows

Failed to parse (Missing texvc executable; please see math/README to configure.): 1 + r_n = 1 + r_r + i + r_r i

and hence

Failed to parse (Missing texvc executable; please see math/README to configure.): r_n = r_r + i + r_r i

Drop Failed to parse (Missing texvc executable; please see math/README to configure.): r_r i

because Failed to parse (Missing texvc executable; please see math/README to configure.): r_r + i
is much larger than Failed to parse (Missing texvc executable; please see math/README to configure.): r_r i

Failed to parse (Missing texvc executable; please see math/README to configure.): r_n = r_r + i

is the result.

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.

Applications

The Fisher equation has important implications in trading inflation-indexed bonds, where changes in coupon payments are a result in changes in break even inflation and real interest rates.

See also

• Yield
• Yield curve
• Interest rate
• Inflation
• Fisher hypothesisde:Fisher-Effekt

it:Equazione di Fisher