The exact equation and the shortcut
Money lent at a nominal rate i grows by a factor of (1 + i). What it can buy grows by (1 + r), the real rate, while prices rise by (1 + π). So (1 + i) = (1 + r)(1 + π). Multiply it out and the nominal rate is i = r + π + r × π; drop the last, small term and you have the familiar shortcut i ≈ r + π.
At a 7% nominal rate and 3% inflation the exact real rate is 1.07 ÷ 1.03 − 1 = 3.883%, while the shortcut says 4%. The shortcut always overstates the real rate when inflation is positive, by roughly the cross term.
Worked example
i = 7%, π = 3%
r = 1.07 ÷ 1.03 − 1
r = 3.883% (shortcut: 4%)
The nominal rate needed for a target real return
Turn the question around: what nominal rate delivers a 4% real return if inflation will be 3%? The exact answer is 1.04 × 1.03 − 1 = 7.12%. The extra 0.12 percentage points above 4 + 3 is the cross term — the real return earned on the inflation compensation itself.
Lenders who price by adding expected inflation to a target real rate leave that term out. At 3% inflation it is negligible; at 30% it is several percentage points.
Worked example
r = 4%, π = 3%
i = 1.04 × 1.03 − 1
i = 7.12%
Where the shortcut fails
At 50% inflation, a 60% nominal rate gives a real return of 1.60 ÷ 1.50 − 1 = 6.67%, not the 10% the subtraction suggests — an overstatement of a third. At 100% inflation, a 110% nominal rate is a 5% real return, not 10%.
The shortcut is fine for comparing returns across high-inflation countries. In economies with high inflation, such as Argentina and Turkey in recent years, the exact form is the only honest one, and small differences in nominal rates can hide large differences in real returns. The Fisher equation calculator prints both forms and the gap between them.
Worked example
i = 60%, π = 50%
r = 1.60 ÷ 1.50 − 1
r = 6.67% (shortcut: 10%)
Reading expected inflation from bond yields
US Treasury Inflation-Protected Securities pay a real yield, so the difference between an ordinary Treasury yield and a TIPS yield of the same maturity is the inflation the market is pricing in — the breakeven rate. On 17 September 2026 the ten-year Treasury yielded 4.94% and the ten-year TIPS 2.61%, and the published ten-year breakeven was their difference, 2.33%.
The exact Fisher inversion gives a little less, 1.0494 ÷ 1.0261 − 1 = 2.27%. Breakevens also carry liquidity and inflation-risk premiums, so they estimate expected inflation rather than measure it.
Worked example
i = 4.94%, r = 2.61%
π = 1.0494 ÷ 1.0261 − 1
π = 2.27% (published difference: 2.33%)
The Fisher effect: one-for-one, eventually
The Fisher effect is the claim that a rise in expected inflation raises nominal interest rates by the same amount, leaving real rates to be set by saving and investment. The long sweep of history fits it: nominal rates climbed with the inflation of the 1970s and fell with it afterwards.
Month to month the relationship is loose. Frederic Mishkin's 1992 study found US data consistent with a long-run Fisher effect but not a short-run one — nominal rates and inflation share a trend without moving together in the short run, because central banks, risk and expectations intervene.
Taxes make inflation costlier than the equation shows
Interest is taxed in nominal terms, including the part that only compensates for inflation. Taxed at 30%, a 7% nominal return is 4.9% after tax, and with 3% inflation that is a real return of 1.049 ÷ 1.03 − 1 = 1.84% — less than half the pre-tax real return of 3.88%.
To keep an after-tax real return steady when inflation rises, nominal rates have to rise by more than inflation — by the change in inflation divided by (1 − the tax rate). Michael Darby described this in 1975, and it is why even moderate inflation weighs heavily on savers in taxable accounts.
Worked example
i = 7%, τ = 30%, π = 3%
(1 + 0.07 × 0.7) ÷ 1.03 − 1
= 1.84%
Common mistakes
Subtracting inflation from the nominal rate at high inflation.The error grows with inflation; above about 10% use the exact form.Mixing horizons.A one-year nominal rate needs one-year expected inflation; a ten-year yield needs ten-year expectations.Treating breakeven inflation as a forecast.It includes liquidity and risk premiums as well as expected inflation.Ignoring tax on the inflation part of interest.After tax, the real return can be a fraction of the pre-tax one.