﻿ The Formula for the Real Interest Rate
San José State University

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Thayer Watkins
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The Formula for the Real Interest Rate

The notion that the real interest rate is the nominal interest rate less the rate of inflation has been articulated so often that many have come to believe that it is the true formula. It is not. It is an approximation which under ordinary conditions of inflation is a good approximation. But under conditions of high inflation or deflation there are significant differences between this approximation and the true value of the real interest rate.

Let the nominal interest rate be denoted as i and the rate of inflation as f. A unit of money after one year will have grown to (1+i). The price level will have grown to be (1+f). The amount of goods that can be purchased after one year is (1+i)/(1+f). Thus the real interest rate r is given by:

#### (1+r) = (1+i)/(1+f)

The solution for r is therefore

#### r = (1+i)/(1+f) − 1 = (i − f)/(1+f)

Consider a case of hyperinflation in which f=100%. If the nominal interest rate is 110% the real rate interest is not 10% but instead 5% because

#### r = 0.1/(1+1) = 0.05

Now consider a situation like occurred in the early 1930's when the nominal interest was a very low half of one percent but there was deflation of 10%. The real interest rate was not 10.5% but instead 11.67% because

#### r = (0.005 − (−0.1)/(1−0.1) = 0.105/0.9 = 0.1167

That is a full 1.17% higher.

(To be continued.)