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Real return after inflation calculator

A real return is what a return buys after inflation. Divide, do not subtract: real = (1 + nominal) / (1 + inflation) - 1. At 7 percent with 3.2 percent inflation the real return is 3.68 percent a year, not the 3.80 percent subtraction gives. Over 10 years $10,000 grows to $19,671.51 and buys $14,356.24.

Real return a year

3.68%

7.00% growth with 3.20% inflation. Subtracting one from the other would say 3.80%.

Statement balance after 10 years
$19,671.51
What it buys in today's money
$14,356.24
Subtracting instead of dividing
3.80%, out by 0.12 points too high
%

The rate you are quoted, before inflation. If it compounds more often than once a year, convert it to an effective annual rate first.

%
$
yr

The formula

rreal=1+rnominal1+i−1r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + i} - 1

rnominalr_{\text{nominal}} is the return you are quoted, ii is inflation over the same period, and rrealr_{\text{real}} is what the money actually buys. Both rates go in as decimals, so 7 percent is 0.07.

What this calculator works out

Enter the return you were quoted, inflation over the same period, a balance and a number of years. It returns the real return a year, the balance the statement will show, and what that balance buys in today's money.

The two balances are the point of the page. One is the number you will see and the other is what the number is worth, and only the second says whether you are better off. On $10,000 at 7 percent for 10 years with inflation at 3.2 percent, the statement reaches $19,671.51 and the buying power reaches $14,356.24.

The Fisher relation

A nominal return is two things at once: the part that covers rising prices, and the part that is genuine gain. They multiply rather than add, so the relation is:

1+rnominal=(1+rreal)(1+i)1 + r_{\text{nominal}} = (1 + r_{\text{real}})(1 + i)

Rearranged for the number you want:

rreal=1+rnominal1+i−1r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + i} - 1

Division, not subtraction. Inflation applies to the gain as well as to the money you started with, so the gain has to be discounted too. Subtracting inflation from the rate quietly assumes it does not, which is why the shortcut answer is off by the real return times the inflation rate whenever inflation is above zero.

What the real rate does to a balance

Rates are easy to shrug at, so put them on money. Ten years of 7 percent on $10,000 gives $19,671.51 on the statement. Grow the same $10,000 at the real rate, which is 3.6822 percent before it is rounded to 3.68, and you get $14,356.24, which is what that statement balance will buy at the end. Rounding the rate to two places first would shave about three dollars off that over the ten years, so the calculator compounds the unrounded figure.

Nothing was charged. The gap is inflation working on the whole balance, year after year, at the same time as the return works on it. Longer horizons widen the gap, because the real rate compounds as well.

The growth part is ordinary compounding. The compound interest calculator produces the nominal figure, and this page says what it is worth.

When inflation is higher than the return

A real return can be negative while the balance still rises. At 2 percent with inflation at 3.2 percent the real return is -1.16 percent a year: $10,000 becomes $12,189.94 after 10 years and buys $8,896.20 of today's goods.

That is the case worth checking on cash. An account paying below inflation loses buying power every year it is held, and the statement never mentions it, because the number on the statement goes up.

Compounding frequency is a separate question from inflation, and mixing the two produces a rate that is wrong twice. This page compounds once a year, so the figure it wants is an effective annual rate. A bank quoting a yearly rate that adds interest monthly is quoting an APR, which is lower than what the account pays over a year. Convert it with the APR against APY calculator and bring the APY here, not the APR. Both are quoted as yearly rates, which is what makes that swap so easy to make by accident.

The adjustment this calculator performs has a name in economics. Subtracting inflation from a nominal rate gives the real rate, and the exact form divides rather than subtracts: the real interest rate.

Worked examples

A 7 percent return with inflation at 3.2 percent

Your investments return 7 percent over a year while prices rise 3.2 percent. What did you gain in buying power, and what does that do to $10,000 held for 10 years?

  1. Write both rates as decimals: nominal 0.07, inflation 0.032.
  2. Divide rather than subtract: 1.071.032=1.0368217\frac{1.07}{1.032} = 1.0368217.
  3. Take away the 1: the real return is 0.0368217, which is 3.68 percent a year.
  4. Subtraction would have said 7−3.2=3.807 - 3.2 = 3.80 percent, so the shortcut is 0.12 points too high.
  5. Grow the balance at the nominal rate: 10000×1.071010000 \times 1.07^{10}, which is $19,671.51.
  6. Grow it at the real rate instead: 10000×1.03682171010000 \times 1.0368217^{10}, which is $14,356.24.

The real return is 3.68 percent a year. After 10 years the statement says $19,671.51, and it buys what $14,356.24 buys today.

A savings account at 4.5 percent

A savings account pays 4.5 percent while inflation runs at 3.2 percent. Is $10,000 in it gaining ground over 5 years?

  1. Decimals again: nominal 0.045, inflation 0.032.
  2. 1.0451.032=1.0125969\frac{1.045}{1.032} = 1.0125969, so the real return is 1.26 percent a year.
  3. Subtraction says 4.5−3.2=1.304.5 - 3.2 = 1.30 percent, only 0.04 points out. Inflation is the same 3.2 percent as in the first example, so what shrank is the gap between the two rates, from 3.8 points to 1.3 points, and the error shrinks in step with it.
  4. Five years at the quoted rate: 10000×1.045510000 \times 1.045^{5}, which is $12,461.82.
  5. Five years at the real rate: 10000×1.0125969510000 \times 1.0125969^{5}, which is $10,645.91.

The account gains 1.26 percent a year in buying power. Five years turns $10,000 into $12,461.82 on the statement, worth $10,645.91 in today's money, so the saver is ahead, but by far less than the headline rate suggests.

An account paying 2 percent, which is below inflation

Cash sits in an account paying 2 percent while inflation runs at 3.2 percent. The balance rises every year. Is the money growing?

  1. Decimals: nominal 0.02, inflation 0.032.
  2. 1.021.032=0.9883721\frac{1.02}{1.032} = 0.9883721, and taking away the 1 leaves -0.0116279.
  3. The real return is -1.16 percent a year, so the money buys less each year even though the balance is bigger.
  4. Subtraction says 2−3.2=−1.202 - 3.2 = -1.20 percent. Below inflation the shortcut tips the other way and makes the loss look 0.04 points worse than it is.
  5. Ten years at 2 percent: 10000×1.021010000 \times 1.02^{10}, which is $12,189.94.
  6. Ten years at the real rate: 10000×0.98837211010000 \times 0.9883721^{10}, which is $8,896.20.

The real return is -1.16 percent a year. The balance grows to $12,189.94 while its buying power falls to $8,896.20 in today's money. A number rising on a statement is not the same thing as getting richer.

The mistake that costs the most

Subtracting inflation from the nominal rate. It is close, and being close is exactly what lets it survive a check.

The error has a formula of its own. Subtraction minus the true answer is (r−i)×i1+i(r - i) \times \frac{i}{1 + i}, so it grows with the gap between the rates and with inflation itself. At 7 percent against 3.2 percent the shortcut says 3.80 percent when the answer is 3.68 percent, 0.12 points too high. At 4.5 percent against 3.2 percent it is 0.04 points out. Neither looks like much on its own, and both compound along with everything else in a long projection, so a plan built on the shortcut drifts high the whole way.

When the return is below inflation the same error changes sign and makes the loss look bigger than it is. Divide in both directions and the sign takes care of itself.

Common questions

Which inflation rate should I enter?

The headline consumer price index over the same period as the return, which is what published real return figures use. Your own rate can differ from the basket: rent, tuition and medical costs move on their own schedule. If most of your spending sits in one of those, use that rate instead and the answer becomes personal to you.

Does tax come off before or after inflation?

Before, in the United States and the United Kingdom and most systems like them: tax falls on the nominal return, not on the real one, so take the tax off first and enter the after-tax rate. A 7 percent return taxed at 20 percent leaves 5.6 percent nominal, and dividing that by 3.2 percent inflation leaves about 2.33 percent real, roughly a third of the 7 percent headline. A few countries index the taxable gain to inflation instead, so check the rule where you file.

Is a negative real return the same as losing money?

Not on the statement, only in what the money buys. The balance still rises, and every figure the bank prints is correct. What falls is the quantity of goods that balance covers, which is the thing you hold savings for. It is a real loss with no line item to point at.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.