# The Fisher Equation The Fisher Equation is a mathematical formula that describes the theoretical relationship between the interest rates we observe and the expected rate of inflation.

## What is the Fisher Equation?

The Fisher Equation is named after Irving Fisher, a tremendously insightful classical economist whose work in capital theory influenced many of today’s leading economists.  In 1895, Fisher began an in-depth examination of the effect of inflation on interest rates at the request of the American Economic Association (AEA).  He later published his studies in a book titled The Theory of Interest.  Despite the fact that the book was published in 1930, it remains one of the best discussions of the inherent determinants of interest rates.  In it, he provides a clear and simple explanation regarding why interest rates change over time.

Fisher began with the assumption that people who lend money recognize that it is not simply cash that is being lent, but command of real goods.  Thus, the rate of return they require when lending cash is not driven simply by a return on the money lent, but by an increase in their control of real goods.  If lenders expect prices to increase (i.e., inflation), they demand both a “real rate of return” and compensation for the expected increase in prices.

You can think of the “real rate of return” as the true price of credit.  In the absence of expected inflation (i.e., if the inflation rate is expected to be zero), it is the rate that lenders would require for forgoing current consumption.  It is the percentage increase in purchasing power (their control of real goods) that they demand simply for lending the money.  From the aspect of the borrowers, it is the interest rate they must pay for the opportunity to purchase goods today rather than having to wait until they can accumulate the cash to do so.  If lenders expect an increase in price levels, they will demand compensation for their loss of purchasing power in addition to real rate of return.

Thus, the Fisher effect states that inflationary expectations can lead to higher interest rates.  A simplified version of the Fisher effect equation is provided below:

Nominal interest rate on a riskless asset = real interest rate + expected inflation

## How the Fisher Equation Is Derived

The derivation of the Fisher equation is best explained by means of an example.  Let’s assume that both the lenders and the borrowers have looked into a crystal ball and know with certainty that the rate of inflation in the coming year will be 5%.  Then for every \$1 lent, the lenders will require \$1.05 at the end of the year as compensation for inflation; otherwise they would experience a loss in purchasing power since the good that costs \$1 today will cost them \$1.05 at the end of the year.  Let’s also assume that lenders require a 2% increase in purchasing power just for lending the money.  Then the total payment they will demand for each dollar lent will be \$1.05 x 1.02 = \$1.071, or a total return of 7.1%.  We can express this in general terms as follows:

Nominal interest rate = (1 + real interest rate) times (1 + expected inflation) – 1

There are a couple of important points to note about this equation.  The nominal interest rate is the observed rate on a riskless asset, such as a U.S. Treasury bill.  You can find these rates quoted on the internet and in some business publications, such as The Wall Street Journal.  The real interest rate cannot be observed, but is calculated ex post facto.  As mentioned earlier, it is the return lenders demand for forgoing current consumption.  The formula for the Fisher effect is usually expressed as a mathematical equation:

irf = (1 + ir)(1 + π) – 1,

where irf = the nominal risk-free rate, ir = the real interest rate, and π = expected inflation.  In this equation the real interest rate and the inflation rate are compounded, following the mathematical logic.  However, not much accuracy is lost by omitting the cross term in the multiplication since it is usually a small number.  Doing so, we arrive at a simplified version of the Fisher equation:

irf  ≈ ir + π

Using the numbers in our original example, irf  2% + 5% = 7%, which is slightly less than the 7.1% we originally calculated.  In practice, this simplified version is the one commonly used.  This version states that the observed interest rate on a riskless asset is approximately equal to the sum of the real interest rate and the expected rate of inflation.  Since the real interest rate is relatively stable compared to the rate of inflation, the Fisher effect leads to the conclusion that the greatest determinant of interest rate changes is expected inflation.

## How To Use The Fisher Equation

There are a number of ways that the Fisher equation can be applied in practice.  The interest rates we observe on loans that are not risk free, such as car loans, mortgages, business loans, credit cards, and so forth are all based on this simple equation.  All must offer the risk-free rate of interest, irf, and then an additional premium, based on the riskiness of the loan.  Therefore, the rate you must pay to borrow money to finance your business, for example, is equal to the real interest rate that the lender demands as compensation for forgoing current consumption plus expected inflation plus a risk premium based on the level of risk the lender assigns the loan.  Financial managers of corporations must understand how changes in inflationary expectations can affect their financing costs.

Investors have to be mindful of expected inflation when making their investment decisions.  For example, assume an investor purchases a 12-month U.S. Treasury bill that offers a 3% rate of return and that the real rate of interest is 1%.  Applying the Fisher effect, we know that this implies the expected rate of inflation is 2%:

irf ≈ ir + π

3% ≈ 1% + π

π ≈ 2%

If inflation during the year is 5%, the investor will have actually lost money since he will have less purchasing power.  He was compensated 1% for forgoing current consumption and 2% for loss of purchasing power, but he lost 5% in purchasing power, so his real return on this investment was a negative 2%.

Investment analysts can use prevailing nominal returns on risk-free assets to determine the market’s consensus on future inflation.  For example, if an investment analyst observes that the real interest rate has been stable at about 0.8% for the past several years and doesn’t expect it to change, he can look at the prevailing yields on Treasury securities with different maturities to calculate an estimate of the what the market is expecting inflation to be:

 Fisher Effect Worksheet Maturity Year (#) Yield on Treasury Bond Maturing in Year # Real Interest Rate Implied Average Yearly Inflation 1 1.31% 0.80% 0.51% 2 1.49% 0.80% 0.69% 3 1.63% 0.80% 0.83% 4 1.75% 0.80% 0.95% 5 1.94% 0.80% 1.14%

He can compare these consensus figures to what he expects inflation will be, based on other analytics he has completed, and make trades accordingly.