Hong Life CalculatorsUnderstanding the Fisher equation calculator — concept and principle
This calculator applies the Fisher equation linking the nominal interest rate, the real interest rate and inflation. Solve for any one of the three, see the exact result and the common approximation, and how large the approximation error is. Use it in economics courses, to judge whether a savings rate beats inflation, or to set a nominal target that preserves buying power.
How it is calculated
- Exact: (1 + i) = (1 + r) × (1 + π).
- Real rate r = (1 + i) ÷ (1 + π) − 1.
- Nominal rate i = (1 + r)(1 + π) − 1.
- Inflation π = (1 + i) ÷ (1 + r) − 1.
- Approximation: i ≈ r + π (r ≈ i − π), dropping the small cross term r × π.
The principle
A nominal interest rate tells you how many more units of money you get; the real rate tells you how much more you can buy. If prices rise by π, each unit of money buys 1 ÷ (1 + π) as much, so the growth in buying power is (1 + i) ÷ (1 + π). Multiplying out gives i = r + π + rπ; when rates are small, rπ is tiny and the approximation is close. When inflation or rates are high, the cross term matters. Using expected inflation gives the ex-ante real rate; actual inflation gives the ex-post rate.
Worked example
Default inputs: solve for the real rate, nominal 5%, inflation 3%.
- Exact: r = 1.05 ÷ 1.03 − 1 ≈ 1.94%.
- Approximation: r ≈ 5% − 3% = 2%.
- Error of the approximation ≈ 0.058 percentage points.
| Nominal interest rate (%) | Exact result |
|---|---|
| 1.25 | -1.7% |
| 2.5 | -0.49% |
| 3.75 | 0.73% |
| 5 | 1.94% |
| 6.25 | 3.16% |
| 7.5 | 4.37% |
| 10 | 6.8% |
How to read the result
A positive real rate means savings at this nominal rate grow in buying power; a negative real rate means they lose it even though the balance rises. For low rates the approximation is fine for everyday use; for high inflation, use the exact result.
Common mistakes
- Using the approximation with high inflation, where it can be far off.
- Mixing an annual nominal rate with monthly inflation.
- Ignoring taxes: the after-tax real return can be much lower.
- Confusing expected inflation with past inflation.
Frequently asked questions
What is the Fisher equation?
(1 + nominal) = (1 + real) × (1 + inflation), named after the economist Irving Fisher.
Why does the approximation drift?
It ignores the product r × π, which grows when rates or inflation are high.
Can the real interest rate be negative?
Yes, whenever inflation is higher than the nominal rate.
Sources: R. Cooper & A. John, Macroeconomics: Theory Through Applications (Saylor Academy), §16.14 "The Fisher Equation: Nominal and Real Interest Rates"