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:real interest rate ≈ nominal interest rate − inflation rate.
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:contracted nominal interest rate ≈ real interest rate + expected inflation rate.
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.
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 πt+1 is defined—as usual—as the percentage change in the price level from period t to period t + 1.πt+1 = (Pt+1 − Pt)/Pt.
If a period is one year, then the price level next year is equal to the price this year multiplied by (1 + π):Pt+1 = (1 + πt) × Pt.
The Fisher equation says that these two contracts should be equivalent:(1 + i) = (1 + r) × (1 + π).
As an approximation, this equation impliesi ≈ r + π.
To see this, multiply out the right-hand side and subtract 1 from each side to obtaini = r + π + rπ.
If r and π are small numbers, then rπ is a very small number and can safely be ignored. For example, if r = 0.02 and π = 0.03, then rπ = 0.0006, and our approximation is about 99 percent accurate.