Fisher Equation Calculator

What this calculator does

A bank pays you 6 percent. Prices rise 4 percent. How much better off are you? Not 6 percent, and as it happens not quite 2 percent either. Sorting that out is what the Fisher equation does, and this calculator applies it.

Three quantities are involved and they are locked together. The nominal rate is the number on the account or the loan, the one you can see. The inflation rate is how fast prices are climbing. The real rate is what is left once inflation has taken its share, which is the only one that tells you whether your buying power actually grew. Name any two and the third is fixed.

Solving for whichever rate you are missing

The dropdown at the top decides which of the three you are hunting for, and the inputs change to ask for the other two.

• Real interest rate. You know what you are being paid and what inflation is doing. This is the everyday question: is this account actually getting me anywhere?
• Nominal interest rate. You know the real return you want and the inflation you expect. This is the lender's question, or the saver's target: what headline rate do I need to demand to come out ahead by a given margin?
• Inflation rate. You know the nominal and real rates. This is the one that reads inflation expectations back out of market rates, which is roughly how the gap between ordinary and inflation-protected bonds is interpreted.

All three are the same relationship rearranged, so the calculator is not really three tools. It is one identity you can enter from any side.

Why it multiplies instead of subtracting

Nearly everyone carries the shortcut version in their head: real equals nominal minus inflation. It is close enough for conversation and it is not what this calculator uses, for a good reason.

The proper statement of the relationship is multiplicative, not additive:

(1 + nominal) = (1 + real) × (1 + inflation)

Multiply that out and you get nominal = real + inflation + (real × inflation). The familiar shortcut is just that expression with the last term thrown away. That final piece, the product of the two rates, is the bit the subtraction forgets, and it exists because inflation does not only erode your original money, it also erodes the interest you earned on it.

When both rates are small, the discarded term is tiny and nobody misses it. When they are large, it is not tiny at all. That is exactly when people are most likely to reach for the shortcut and most likely to be misled by it.

Where the shortcut starts to hurt

Watch the gap between the two methods open up as the numbers grow.

• Nominal 5 percent, inflation 2 percent: exact real rate 2.941 percent, shortcut says 3 percent. Off by 0.059 points, which nobody needs to care about.
• Nominal 12 percent, inflation 9 percent: exact 2.752 percent, shortcut says 3 percent. Off by 0.248 points, now visibly overstating your gain by about a twelfth of it.
• Nominal 60 percent, inflation 50 percent: exact 6.667 percent, shortcut says 10 percent. Off by 3.333 points, overstating the real return by half.

Notice the direction of the error. The shortcut always flatters, telling you that you did better than you did, and it flatters most in high-inflation conditions where the truth matters most. It works the same way in reverse: a 2 percent real return with 5 percent inflation needs a nominal rate of 7.1 percent, not 7, because the extra 0.1 covers the inflation on the interest itself.

The case where your savings grow and you still lose

The most useful thing this calculator can tell you is unwelcome. If inflation runs above your nominal rate, the real rate is negative, and a negative real rate means your money is buying less than it did even though the balance on the statement went up.

An account paying 3 percent while inflation runs at 6 percent produces a real rate of about −2.83 percent. After a year, the balance is comfortably larger and it buys noticeably less. Nothing on the statement says so, because statements report nominal figures and inflation never appears on them. This is not an exotic scenario either. Deposit rates have sat below inflation for long stretches in many countries.

That is the practical payoff of the whole exercise. Comparing your savings rate against inflation, rather than against zero or against last year's rate, is what tells you whether saving is preserving your position or quietly losing it. The nominal number tells you what you are being paid. The real number tells you what you are getting.

Questions people ask

Which rate matters more, nominal or real?

The real rate, for judging whether you are better off, since it is the one measured in buying power. The nominal rate matters for the cash amounts you will actually pay or receive.

Is nominal minus inflation ever good enough?

For a rough answer at low rates, yes, since the error is a fraction of a percentage point. At higher rates it overstates the real return significantly, which is why this calculator uses the exact form.

What inflation number should I use?

For a past period, the published consumer price inflation for that period. For a future one, your expectation, which is a genuine assumption, so it is worth trying a range rather than a single figure.

Can the real rate be negative?

Yes, whenever inflation exceeds the nominal rate. Your balance still rises, but it buys less than before, which is precisely what a negative real rate means.

References

The Fisher equation states that one plus the nominal rate equals one plus the real rate multiplied by one plus the inflation rate, and multiplying it out shows that the familiar shortcut of subtracting inflation from the nominal rate simply drops the product of the two rates, a term that is negligible only when both are small. Inflation itself is measured by consumer price indexes such as the one published by the U.S. Bureau of Labor Statistics.

1. Saylor Academy, Macroeconomics: Theory Through Applications, 16.14 The Fisher Equation: Nominal and Real Interest Rates. https://saylordotorg.github.io/text_macroeconomics-theory-through-applications/s20-14-the-fisher-equation-nominal-an.html
2. U.S. Bureau of Labor Statistics, Consumer Price Index. https://www.bls.gov/cpi/