This calculator finds the single yearly rate that carries a figure from its starting value to its ending value. Because each year builds on the last, it works for sales, customers, populations and prices, not just money.
The US population makes a good test. The 2020 census counted 331,449,281 people, up 7.4% from 308,745,538 in 2010. Spread over the decade, that's about 0.71% a year.
Count the steps between years, not the years
Counting n is the easy place to slip. From 2010 to 2020 there are ten yearly steps, even though the span touches eleven calendar years. Enter 11 and the rate comes out too low:
| Series | Years entered | Total change | Yearly rate |
|---|---|---|---|
| US population, 2010 to 2020 census | 10 yr | 7.4% | 0.71% |
| Same counts, years miscounted | 11 yr | 7.4% | 0.65% |
| West Virginia population, 2010 to 2020 | 10 yr | −3.2% | −0.32% |
The last row is a shrinking series. West Virginia's population fell 3.2% between the two censuses, so its yearly rate is negative. The formula handles a decline without any change; the rate simply carries a minus sign.
The starting year can flatter the rate
A growth rate is only as honest as its endpoints. Picture a business with sales of $800,000 in 2019 that fell to $600,000 in 2020, then reached $1,100,000 in 2024. Here's the rate to 2024 from each possible start:
| Start year | Sales that year | Years to 2024 | Yearly rate to $1,100,000 |
|---|---|---|---|
| 2019 | $800,000 | 5 yr | 6.58% |
| 2020 (the dip) | $600,000 | 4 yr | 16.36% |
| 2021 | $750,000 | 3 yr | 13.62% |
| 2022 | $900,000 | 2 yr | 10.55% |
Start in the dip year and the business seems to grow 16.36% a year. Start one year earlier and it's 6.58%. When someone quotes a growth rate, ask which year it starts from, and why that one.
The rate also ignores the path between the endpoints. Two companies with the same start and end get the same rate. One may have grown steadily, the other in a single jump.
How long until it doubles?
The rule of 72 gives a quick answer: divide 72 by the yearly rate. Nebraska's Department of Banking and Finance describes the result as the number of years your money takes to double.
It's an approximation, though. Here's the rule next to the exact doubling time, log(2) ÷ log(1 + rate). Both come from the rule of 72 calculator:
| Yearly growth | Rule of 72 (years) | Exact doubling time (years) |
|---|---|---|
| 2% | 36.0 | 35.00 |
| 4% | 18.0 | 17.67 |
| 8% | 9.0 | 9.01 |
| 12% | 6.0 | 6.12 |
| 18% | 4.0 | 4.19 |
| 36% | 2.0 | 2.25 |
The rule is almost exact near 8%. It runs a year long at 2% and three months short at 36%. That's close enough for mental math, but use the exact figure in a written plan.
A few points of growth, compounded for decades
Rate gaps look small in a single year and huge after twenty. Here's $100,000 of yearly sales growing at 12%, 9% and 6%:
$100,000 of yearly sales at three growth rates
Compounded once a year for 24 years
- 12% a year $1,517,863
- 9% a year $791,108
- 6% a year $404,893
Show the yearly figures
| Year | 12% a year | 9% a year | 6% a year |
|---|---|---|---|
| 6 | $197,382 | $167,710 | $141,852 |
| 8 | $247,596 | $199,256 | $159,385 |
| 12 | $389,598 | $281,266 | $201,220 |
| 18 | $768,997 | $471,712 | $285,434 |
| 24 | $1,517,863 | $791,108 | $404,893 |
The rule of 72 predicts doubling every 6, 8 and 12 years at those rates. The yearly figures under the chart show each line near $200,000 at exactly those points.
By year 24, the 12% line is $1,517,863, almost four times the 6% line. Nothing in the first few years hints at a gap that wide.
When there's no growth rate to find
The formula needs a start and an end above zero. A business that swings from a $50,000 loss to a $20,000 profit has no compound growth rate. The calculator returns an error rather than a misleading number.
In that case, report the change in dollars. For a single step between two positive figures, the year-over-year growth calculator gives the plain percentage change.