This is “The Fisher Equation: Nominal and Real Interest Rates”, section 16.14 from the book Theory and Applications of Macroeconomics (v. 1.0).

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When you borrow or lend, you normally do so in dollar terms. If you take out a loan, the loan is denominated in dollars, and your promised payments are denominated in dollars. These dollar flows must be corrected for inflation to calculate the repayment in real terms. A similar point holds if you are a lender: you need to calculate the interest you earn on saving by correcting for inflation.

The **Fisher equation** provides the link between nominal and real interest rates. To convert from nominal interest rates to real interest rates, we use the following formula:

To find the real interest rate, we take the nominal interest rate and subtract the inflation rate. For example, if a loan has a 12 percent interest rate and the inflation rate is 8 percent, then the real return on that loan is 4 percent.

In calculating the real interest rate, we used the actual inflation rate. This is appropriate when you wish to understand the real interest rate actually paid under a loan contract. But at the time a loan agreement is made, the inflation rate that will occur in the future is not known with certainty. Instead, the borrower and lender use their *expectations* of future inflation to determine the interest rate on a loan. From that perspective, we use the following formula:

We use the term *contracted nominal interest rate* to make clear that this is the rate set at the time of a loan agreement, not the realized real interest rate.

- To correct a nominal interest rate for inflation, subtract the inflation rate from the nominal interest rate.

Imagine two individuals write a loan contract to borrow *P* dollars at a nominal interest rate of *i*. This means that next year the amount to be repaid will be *P* × (1 + *i*). This is a standard loan contract with a nominal interest rate of *i*.

Now imagine that the individuals decided to write a loan contract to guarantee a constant real return (in terms of goods not dollars) denoted *r*. So the contract provides *P* this year in return for being repaid (enough dollars to buy) (1 + *r*) units of real gross domestic product (real GDP) next year. To repay this loan, the borrower gives the lender enough money to buy (1 + *r*) units of real GDP for each unit of real GDP that is lent. So if the inflation rate is π, then the price level has risen to *P* × (1 + π), so the repayment in dollars for a loan of *P* dollars would be *P*(1 + *r*) × (1 + π).

Here (1 + π) is one plus the inflation rate. The inflation rate π_{t}_{+1} is defined—as usual—as the percentage change in the price level from period *t* to period *t* + 1.

If a period is one year, then the price level next year is equal to the price this year multiplied by (1 + π):

The Fisher equation says that these two contracts should be equivalent:

(1 +As an approximation, this equation implies

To see this, multiply out the right-hand side and subtract 1 from each side to obtain

If *r* and π are small numbers, then *r*π is a very small number and can safely be ignored. For example, if *r* = 0.02 and π = 0.03, then *r*π = 0.0006, and our approximation is about 99 percent accurate.