Fisher Equation Calculator
Convert between nominal and real interest rates using the Fisher equation — with both the exact and the r ≈ i − π approximation.
Exact r = +3.92%; the approximation r ≈ +4.00% differs by 0.08 pt. At these small rates the two are practically the same.
The Fisher equation links a nominal rate i, a real rate r, and inflation π by (1 + i) = (1 + r)(1 + π). Solving for the real rate gives r = (1 + i) ÷ (1 + π) − 1. So 6% nominal with 2% inflation yields (1.06 ÷ 1.02) − 1 = 3.92% real exactly, while the shortcut i − π gives ≈ 4%.
Why the real rate differs from the nominal rate
The nominal interest rate is the number a bank quotes — the growth of the money in your account. But if prices are rising, each of those future dollars buys less. The real interest rate strips inflation out to show the growth of your actual purchasing power. When inflation is positive the real rate sits below the nominal rate; if inflation outpaces the nominal rate, the real rate turns negative and you lose buying power even as your balance grows.
The exact Fisher relation multiplies the two growth factors rather than adding the rates, because compounding the return and the price change together is not the same as summing them. The familiar shortcut r ≈ i − π drops the cross term (r × π) and is only a first-order approximation.
Exact form (i = nominal, r = real, π = inflation, all as decimals). Approximation: r ≈ i − π, accurate only when the rates are small.
Worked example
A savings account pays 6% nominal while inflation runs at 2%.
- 1 Convert the rates to decimals. Nominal i = 6% = 0.06 and inflation π = 2% = 0.02.
- 2 Build the two growth factors. 1 + i = 1.06 and 1 + π = 1.02.
- 3 Divide to isolate the real factor. 1.06 ÷ 1.02 = 1.0392 — the real growth factor.
- 4 Subtract 1 for the exact real rate. 1.0392 − 1 = 0.0392, or 3.92% real.
- 5 Compare with the approximation. i − π = 6% − 2% = 4%. It overstates the real rate by 0.08 points here because it drops the r × π cross term.
Nominal and inflation → real rate: exact vs approximation
Exact uses r = (1 + i) ÷ (1 + π) − 1; approximation uses r ≈ i − π. The gap widens as the rates grow.
| Nominal i | Inflation π | Real r (exact) | Real r (approx i − π) |
|---|---|---|---|
| 3% | 2% | 0.98% | 1% |
| 6% | 2% | 3.92% | 4% |
| 10% | 4% | 5.77% | 6% |
| 25% | 20% | 4.17% | 5% |
| 100% | 50% | 33.33% | 50% |
Interpretation and pitfalls
The approximation only holds for small rates. Dropping the cross term r × π is harmless when both rates are a few percent, but in high-inflation regimes it is badly off — at 100% nominal and 50% inflation the exact real rate is 33.3% while i − π suggests 50%. Use the exact form whenever the rates are large.
A negative real rate is common. If inflation exceeds the nominal rate — say 2% nominal against 5% inflation — the real rate is negative, meaning cash and low-yield savings lose purchasing power over the year.
Consistency of expectations matters. The equation can be applied with expected inflation (giving the ex-ante real rate you plan around) or with realised inflation after the fact (the ex-post real rate you actually earned). Mixing the two produces a number that means neither.