Fisher equation

This article is about an equation from financial mathematics. For the unrelated partial differential equation, see Fisher's equation.

The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under 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 $r$ denote the real interest rate, $i$ denote the nominal interest rate, and let $π$ denote the inflation rate, the Fisher equation is:

i\approx r+\pi

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

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:

r=i-\pi

Rearranged into an expectations augmented Fisher equation and given a desired real rate of return and an expected rate of inflation $π e$ (with superscript $e$ 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:

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:

1+i=(1+r)(1+\pi ).

Derivation

Although time subscripts are sometimes omitted, the intuition behind the Fisher equation is the relationship between nominal and real interest rates, through inflation, and the percentage change in the price level between two time periods. So assume someone buys a $1 bond in period $t$ while the interest rate is $i t$ . If redeemed in period $t + 1$ , the buyer will receive $(1 + i t)$ dollars. But if the price level has changed between period $t$ and $t + 1$ , then the real value of the proceeds from the bond is therefore

(1+r_{{t+1}})={\frac {1+i_{t}}{1+\pi _{{t+1}}}}

From here the nominal interest rate can be solved for.

{\begin{aligned}1+i_{t}&=\left(1+r_{{t+1}}\right)\left(1+\pi _{{t+1}}\right)\\&=1+r_{{t+1}}+\pi _{{t+1}}+r_{{t+1}}\pi _{{t+1}}\end{aligned}}

Therefore,

{\begin{aligned}i_{t}&=r_{{t+1}}+\pi _{{t+1}}+r_{{t+1}}\pi _{{t+1}}\\&\approx r_{{t+1}}+\pi _{{t+1}}\end{aligned}}

The last line follows from the assumption that both real interest rates and the inflation rate are fairly small, (perhaps on the order of several percent, although this depends on the application) therefore $r t +1 + π t +1$ is much larger than $r t +1 π t +1$ and so $r t +1 π t +1$ can be dropped.

More formally, this linear approximation is given by using two 1st order Taylor expansions, namely:

{\begin{aligned}{\frac {1}{1+x}}&\approx 1-x,\\(1+x)(1+y)&\approx 1+x+y.\end{aligned}}

Combining these yields the approximation:

1+r={\frac {1+i}{1+\pi }}\approx (1+i)(1-\pi )\approx 1+i-\pi ,

and hence

r\approx i-\pi .

These approximations, valid only for small changes, can be replaced by equalities, valid for any size changes, if logarithmic units are used, notably centinepers, which are infinitesimally equal to percentages (approximately equal for small values); other logarithmic units differ by scale factors.

Example

The market rate of return on the 4.25% UK government bond maturing on 8 March 2050 is 3.81% per year. 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 \times 1.01775 = (1 + 0.02) \times (1 + 0.01775) = 1.0381.

This article implies that you can ignore the least significant term in the expansion ( $0.02 \times 0.01775 = 0.00035$ 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% per year, the value of the bond is £107.84 per £100 nominal. At a rate of return of 3.775% per year, 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

Cost–benefit analysis

As detailed by Steve Hanke, Philip Carver, and Paul Bugg (1975),^[4] cost benefit analysis can be greatly distorted if the exact Fischer equation is not applied. Prices and interest rates must both be projected in either real or nominal terms.

For the purpose of cost–benefit analysis inflation can be consistently handled in either of two ways. First, when calculating the present value of expected net benefits, prices and interest rates can be calculated in real terms. That is, no inflationary components are included in either the prices or the interest rates. The second approach includes inflation in both the price and the interest rate calculations; calculations are made in nominal terms. As detailed below, both approaches are equivalent as long as both prices and interest rates are projected in real terms, or both projected in nominal terms.

For example, assume that $Z i$ represents the undiscounted expected net benefits at the end of year $t$ , evaluated at constant prices, and $R t, I t$ , and $r t$ are the real rate of interest, the expected rate of inflation, and the nominal rate of interest for year $t, t = 1, ..., n$ , respectively. The present value of the expected net benefits $PVNB$ is given by

{\text{PVNB}}={\frac {Z_{1}}{1+R_{1}}}+{\frac {Z_{2}}{(1+R_{1})(1+R_{2})}}+\cdots +{\frac {Z_{n}}{(1+R_{1})\cdots (1+R_{n})}}

where no inflation components are included in either prices or the interest rate. Alternatively, the present value of expected net benefits is given by

{\text{PVNB}}={\frac {Z_{1}(1+I_{1})}{1+r_{1}}}+{\frac {Z_{2}(1+I_{1})(1+I_{2})}{(1+r_{1})(1+r_{2})}}+\cdots +{\frac {Z_{n}(1+I_{1})\cdots (1+I_{n})}{(1+r_{1})\cdots (1+r_{n})}}

or through the relationship dictated by the exact Fischer equation

{\begin{aligned}{\text{PVNB}}&={\frac {Z_{1}(1+I_{1})}{(1+R_{1})(1+I_{1})}}+{\frac {Z_{2}(1+I_{1})(1+I_{2})}{(1+R_{1})(1+R_{2})(1+I_{1})(1+I_{2})}}+\cdots \\[8pt]&{}\qquad \cdots +{\frac {Z_{n}(1+I_{1})\cdots (1+I_{n})}{(1+R_{1})\cdots (1+R_{n})(1+I_{1})\cdots (1+I_{n})}}\end{aligned}}

By observing the above equations, it is clear that the present value of net benefits derived by either equation will be identical. This alleviates any question concerning whether to conduct cost–benefit analysis in terms of constant or nominal prices.

Inflation-indexed bonds

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

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 decline in the nominal interest rate.