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Real vs Nominal Rates: the Fisher Equation

The nominal interest rate is what is quoted. The real interest rate is what is left after inflation. The Fisher equation links them, with the real rate roughly the nominal rate minus expected inflation. Real rates, not nominal ones, drive borrowing and lending decisions.

Try it yourself

The Fisher equation

The nominal rate links the real rate and expected inflation, set when the loan is priced: i ≈ r + πᵉ, exactly i = (1+r)(1+πᵉ) − 1 = r + πᵉ + r·πᵉ. The cross-term r·πᵉ is the gap the approximation drops, and it grows with inflation. Later, realized inflation decides the actual real return r = (1+i)/(1+π) − 1.

Nominal rate iapprox 5.00%exact 5.06%
Nominal rate i, decomposedr 2.00%πᵉ 3.00%approx ends+ r·πᵉ = +0.06% gapEx-post real return as realized inflation moves-2%0%2%4%6%8%ex-ante r2.00%real rexpected πᵉcross-term r·πᵉ
Exact − approx gap (cross-term) +0.06%Inflation surprise π − πᵉ +0.00%
Real rate r2.00%
Expected inflation πᵉ (sets the nominal contract)3.00%
Realized inflation π (decides the real return)3.00%
The nominal rate is locked at 5.06% (exact), versus 5.00% from the approximation. The +0.06% gap is the dropped cross-term r·πᵉ, which widens as inflation rises. With no inflation surprise, the ex-post real return equals the ex-ante real rate 2.00%.
Try this.Hold r = 2% and πᵉ = 3% (nominal locked at 5.06%), then push realized inflation above 3%. The lender's real return drops below the 2% they expected: an inflation surprise transfers purchasing power from lender to borrower. Now raise expected inflation toward 12% and watch the exact-vs-approx gap widen well past the 0.06-point gap seen at 3%.

Why it matters

If you earn 7% but prices rise 3%, your purchasing power only grew about 4%. Borrowers and lenders care about that 4%, the real return, because that is what actually buys more goods.

Formulas

Fisher equation (approximation)
ir≈i−πei_r \approx i - \pi^e
ii is the nominal rate, πe\pi^e expected inflation, and iri_r the real rate. The exact form is (1+ir)=(1+i)/(1+πe)(1 + i_r) = (1+i)/(1+\pi^e).

Worked examples

Scenario

A loan carries a nominal rate of 7% and expected inflation is 3%. What is the real rate?

Solution

Approximately 7% minus 3% = 4%. The exact Fisher calculation gives (1.07/1.03) - 1 = 3.88%, so the approximation is close at low inflation.

Common mistakes

  • ✗A high nominal rate always means expensive borrowing. If inflation is also high, the real cost can be low or even negative.
  • ✗Real and nominal rates are interchangeable. Decisions depend on the real rate. The nominal rate can mislead when inflation is high.

Revision bullets

  • •Nominal rate is quoted, real rate is after inflation
  • •Real rate is roughly nominal minus expected inflation
  • •Real rates drive economic decisions

Quick check

Nominal rate 5%, expected inflation 6%. The real rate is approximately

Connected topics

Present ValueBond Market

Sources

  1. Mishkin (2018), Ch. 4
    Mishkin, F. S. The Economics of Money, Banking, and Financial Markets. 12th ed. Pearson, 2018. ISBN 978-1-292-26885-9.
    Defines real vs nominal interest rates and the Fisher equation.
How to cite this page
Dr. Phil's Quant Lab. (2026). Real vs Nominal Rates: the Fisher Equation. Derivatives Atlas. https://phucnguyenvan.com/concept/mb-fisher
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