Nominal vs Real Interest Rate: The Fisher Equation

The real interest rate is the nominal interest rate minus inflation, so a loan at 7% when prices rise 3% earns the lender a real return of about 4%. That relationship is the Fisher equation: real rate ≈ nominal rate − inflation, or, in its exact form, (1 + nominal) = (1 + real) × (1 + inflation). When actual inflation turns out higher than the inflation people expected when they signed a loan, borrowers gain and lenders lose; when it turns out lower, the transfer runs the other way.

This guide focuses on the Fisher equation itself: where the shortcut comes from, when it breaks down, and how expected and actual inflation decide who wins on a loan. For the broader picture of how rates are set, see interest rates explained, and for real versus nominal values across GDP and wages, see real vs nominal explained.

The two rates

The nominal interest rate is the rate written on the loan, the one the bank advertises. It measures how many extra dollars you get back.

The real interest rate measures how much extra purchasing power you get back, after accounting for the fact that each dollar buys less when prices rise. See the real vs nominal interest rate comparison for a side by side table.

The Fisher equation: approximate and exact

The Fisher equation is named for the economist Irving Fisher. Write i for the nominal rate, r for the real rate, and π for inflation.

Exact form: (1 + i) = (1 + r) × (1 + π)

The logic: one dollar lent today grows to (1 + i) dollars next year. Prices grow by the factor (1 + π), so in today's purchasing power the lender gets back (1 + i) / (1 + π), which equals (1 + r).

Where the shortcut comes from. Multiply out the right side:

1 + i = 1 + r + π + (r × π)

so i = r + π + (r × π). When both rates are small, the last term is tiny. With a 4% real rate and 3% inflation, r × π = 0.04 × 0.03 = 0.0012, or 0.12 percentage point. Drop it and you get the version AP Macro uses:

Approximate form: r ≈ i − π, or equivalently i ≈ r + π

When the shortcut is close and when it is not

Low inflation. Nominal rate 5%, inflation 2%.

• Approximate: 5% − 2% = 3%
• Exact: 1.05 / 1.02 = 1.0294, so the real rate is 2.94%

The two answers differ by 0.06 point, which is why the approximation is standard for normal rates.

High inflation. Nominal rate 30%, inflation 25%.

• Approximate: 30% − 25% = 5%
• Exact: 1.30 / 1.25 = 1.04, so the real rate is 4%

Now the shortcut is off by a full percentage point. In countries with very high inflation, the exact form matters. On the AP exam, use the subtraction unless a question tells you otherwise.

Solving for the nominal rate. A lender wants a 3% real return and expects 4% inflation.

• Approximate: 3% + 4% = 7%
• Exact: 1.03 × 1.04 = 1.0712, so the nominal rate is 7.12%

The Fisher equation calculator checks either direction.

Expected vs actual inflation

The Fisher equation can be filled in with two different inflation numbers, and the difference is where the economics happens.

Expected inflation is what borrowers and lenders believe inflation will be when they agree on a loan. Lenders add it to the real return they want, so the nominal rate on a new loan reflects expected inflation. Economists call the real rate built from expected inflation the ex ante real rate, meaning "before the fact."

Actual inflation is what prices really do over the life of the loan. Plug it in and you get the ex post real rate, "after the fact," which is the real return the lender actually earned.

If inflation turns out exactly as expected, the two real rates match and nobody is surprised. If inflation comes in higher or lower than expected, the realized real rate moves away from what both sides planned on, and purchasing power shifts from one side of the loan to the other.

Who gains and who loses from unexpected inflation

The rule fits in one table.

Why borrowers gain from surprise inflation. A fixed nominal loan is repaid in dollars. If prices rise faster than expected, each repayment dollar buys less than the lender planned on, so the borrower hands back less purchasing power than both sides agreed to in real terms.

The same logic applies to anyone owed a fixed number of dollars. Savers holding fixed-rate bonds and workers or retirees whose incomes are fixed in dollars for a period lose purchasing power when inflation surprises upward. This redistribution is one of the costs of inflation covered in AP Macro topic 2.5, and the guide to what causes inflation covers the other costs.

Ways lenders protect themselves. A lender can charge a higher nominal rate when it expects more inflation, offer an adjustable rate that resets as conditions change, or index the loan to prices. U.S. Treasury Inflation-Protected Securities (TIPS), for example, adjust their principal with the consumer price index.

Worked example: a $10,000 loan with surprise inflation

A bank lends $10,000 for one year at a fixed 7% nominal rate. Both the bank and the borrower expect 3% inflation.

Step 1: the expected real rate.

Approximate: 7% − 3% = 4%. The bank expects to gain about 4% in purchasing power.

Step 2: inflation surprises. Actual inflation over the year is 6%.

Realized real rate, approximate: 7% − 6% = 1%

Realized real rate, exact: 1.07 / 1.06 = 1.0094, or 0.94%

Step 3: the repayment in purchasing power. The borrower repays $10,000 × 1.07 = $10,700, the same number of dollars either way.

• What the bank expected that $10,700 to be worth, in start-of-year dollars: $10,700 / 1.03 = $10,388.35
• What it is actually worth: $10,700 / 1.06 = $10,094.34

Step 4: the transfer. $10,388.35 − $10,094.34 = $294.01 of purchasing power, measured in start-of-year dollars, shifted from the bank to the borrower. Nobody broke the contract. The surprise in inflation alone moved the value.

Run it the other way. If actual inflation had been only 1%, the realized real rate would be about 7% − 1% = 6%, and the bank would gain at the borrower's expense.

Where each rate shows up on AP graphs

A frequent source of lost points is putting the wrong rate on an axis. The rule in AP Macro topic 4.2:

• The money market graph uses the nominal interest rate.
• The loanable funds market graph uses the real interest rate. The loanable funds explainer walks through that graph.

Capital flows between countries also respond to the real rate, which is covered in real interest rates and capital flows.

How it shows up on the AP exam

Questions usually ask you to:

• Calculate a rate. Given two of nominal rate, real rate, and inflation, find the third using real ≈ nominal − inflation.
• Identify the winner. Given expected and actual inflation, say whether borrowers or lenders gain, and explain why using purchasing power.
• Explain a lender's choice. Say why a lender who expects higher inflation charges a higher nominal rate.
• Match the rate to the graph. Label the money market with the nominal rate and loanable funds with the real rate.

Common mistakes:

• Adding when you should subtract. The real rate is nominal minus inflation.
• Using actual inflation when the question asks about the rate on a new loan. Lenders set nominal rates using expected inflation.
• Saying all inflation hurts lenders. Only unexpected inflation redistributes. Inflation that was fully expected is already built into the nominal rate.
• Forgetting that the real rate can be negative. A 2% nominal rate with 5% inflation is a real rate of about −3%.
• Swapping the axis labels on the money market and loanable funds graphs.