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Step-by-step mathematical derivation with typeset equations and expandable detail.
Sections
Nominal versus real returns
The Fisher equation links the nominal interest rate , the real interest rate , and the expected inflation rate . The distinction matters because lenders and borrowers care about purchasing power, not dollar amounts. A nominal return of 6% is worth much less if prices rise 5% over the same period. The real interest rate strips out inflation to measure the true return on saving or the true cost of borrowing in terms of goods and services.
Irving Fisher formalized this in the early 20th century. The key insight is that a lender who demands a real return of must charge a nominal rate that compensates for both the real return and the expected erosion of purchasing power due to inflation. If the lender expects prices to rise by over the loan period, the nominal rate must be set high enough so that, after inflation, the lender still earns in real terms.
The rate quoted in financial contracts, unadjusted for inflation.
The inflation-adjusted return: the increase in purchasing power from lending or saving.
The rate at which the price level is expected to rise over the relevant horizon.
Exact and approximate Fisher equations
Consider investing one dollar today. At the nominal rate , the dollar grows to nominal dollars after one period. In real terms, each of those dollars is worth in today's purchasing power, so the real gross return is . Setting this equal to the required real gross return gives the exact Fisher equation. This is a no-arbitrage condition: in equilibrium, the nominal return must exactly compensate for both the real return and expected inflation.
Expanding the exact equation yields . The cross-term is the product of two small numbers (typically both under 0.10), so it is negligible in practice. Dropping it gives the familiar approximation . This linear form is widely used in macroeconomics because it is analytically convenient and accurate to within a few basis points when inflation and real rates are moderate.
Exact Fisher equation: the nominal gross return equals the product of the real gross return and the gross inflation rate.
Approximate Fisher equation: the nominal rate is roughly the sum of the real rate and expected inflation, valid when both are small.