How real returns work

A nominal rate counts currency units. A real rate counts what those units buy. They are linked by multiplication rather than addition:

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

Rearranged for the number you want:

$$r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + i} - 1$$

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.

At 7 percent with 3.2 percent inflation: $1.07 / 1.032 = 1.0368217$, so the real return is 3.68 percent a year. Subtraction would have said 3.80 percent. The shortcut is 0.12 points too high.

Put it on money. Ten years of 7 percent on $10,000 gives $19,671.51 on the statement. Grow the same $10,000 at the unrounded real rate 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 a few dollars off that over the ten years, so the calculator compounds the unrounded figure.

The real return calculator on this page does the division and puts the answer on a balance. The real return explorer is the same pair of balances as two curves: drag inflation and watch them separate.

A savings account paying 4.5 percent while inflation runs at 3.2 percent still gains ground: $1.045 / 1.032$ leaves 1.26 percent a year real. Five years turns $10,000 into $12,461.82 on the statement, worth $10,645.91 in today's money. The saver is ahead, but by far less than the headline rate suggests. Subtraction says 1.30 percent, only 0.04 points out, because the gap between the two rates has shrunk and the error shrinks in step with it.

Cash sitting in an account paying 2 percent while inflation runs at 3.2 percent is the case the statement never mentions. $1.02 / 1.032 = 0.9883721$, so the real return is -1.16 percent a year. Ten years at 2 percent grows the balance to $12,189.94. Ten years at the real rate leaves buying power of $8,896.20. The number on the statement went up. The quantity of goods that number covers went down.

Below inflation the shortcut changes sign and makes the loss look 0.04 points worse than it is. Divide in both directions and the sign takes care of itself.

In the United States and most systems like it, tax falls on the nominal return, not on the real one. Take the tax off first and enter the after-tax rate. A 7 percent return taxed at 20 percent leaves 5.6 percent nominal. This page then divides that by inflation. Doing inflation first and tax second answers a question the tax code does not ask.

Compounding frequency is a separate question from inflation. 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. Convert it with how APR and APY work and bring the APY here, not the APR.

One rule keeps a plan honest: pick a side and stay on it. Either state the goal in today's money and grow the savings at the real rate, or inflate the goal to what it will cost then and grow the savings at the quoted rate. Doing both counts inflation twice, and doing neither counts it once too few.

How inflation factors work is the other identity: what a basket costs later, and what cash still buys. Nominal against real return is when to use which figure. Inflation and purchasing power is the long concept page this formula sits under.

Subtraction says 7 minus 3.2 is 3.80 percent. Division says 3.68 percent. The gap is 0.12 points, and it is not a rounding error. It is the cross term $r_{\text{real}} \times i$. Inflation applies to the gain as well as to the starting money, so the gain has to be discounted too. Subtraction quietly assumes it does not.

The error is $3.68 \times 3.2$ percent of a percent, which is 0.12 points. It grows with both rates. At ordinary savings rates it is a few hundredths. At 7 percent against 3.2 percent it is already enough to move a ten-year balance. On $10,000 the statement reaches $19,671.51. The real buying power is $14,356.24. Growing at the 3.80 percent shortcut instead of 3.68 percent would overstate what that statement buys.

Below inflation the shortcut changes sign. A 2 percent account against 3.2 percent inflation is minus 1.16 percent real, and subtraction would have said minus 1.20 percent, making the loss look 0.04 points worse than it is. Divide in both directions and the sign takes care of itself.

A plan that inflates the goal to what it will cost later, and then grows the savings at a real rate, counts inflation twice. A plan that leaves the goal in today's money and grows the savings at the quoted nominal rate counts inflation once too few. Either state the goal in today's money and grow at the real rate, or inflate the goal and grow at the quoted rate. Mixing the two is the usual way a spreadsheet looks funded and then is not.

Tax sits in front of this page, not inside it. In the United States and most systems like it, tax falls on the nominal return. 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 this page then divides that by inflation. Doing inflation first and tax second answers a question the tax code does not ask.

How inflation factors work is the other identity: what a basket costs later, and what cash still buys. Nominal against real return is when to use which figure. Inflation and purchasing power is the long concept page this formula sits under.

It is not a forecast of inflation, not a tax engine, and not a ranking of accounts. The inflation rate you enter is the one that covers the same period as the return. A monthly print annualised in your head is a different object.

The three sheets are 7 percent against 3.2 percent (3.68 percent real; $19,671.51 nominal and $14,356.24 real on $10,000 over 10 years), 4.5 percent against 3.2 percent over 5 years (1.26 percent real; $12,461.82 and $10,645.91), and 2 percent against 3.2 percent over 10 years (-1.16 percent real; $12,189.94 and $8,896.20). This is educational material, not financial advice.