How the result is calculated
A value that grows by a percentage each year does not grow by the same amount each year: every year's growth is measured on a larger base than the last. So when something went from 10,000 to 20,000 over ten years, the useful question is not "how much did it grow on average" but "what steady yearly rate, compounding, gets from the first figure to the second". That rate is the compound annual growth rate, or CAGR.
CAGR(E ÷ S)1 ÷ t − 1
- S
- the starting value
- E
- the ending value
- t
- the number of years between them, fractions allowed
Because the formula links four quantities, any one of them can be found from the other three, and the calculator's three modes do exactly that. Growth rate is the formula above. End value rearranges it to E = S × (1 + CAGR)^t, which is ordinary compound growth from a known rate. Years needed solves for t with logarithms, t = ln(E ÷ S) ÷ ln(1 + CAGR), which is how long a rate takes to reach a target.
Worked example
| Item | Amount | How it is obtained |
|---|---|---|
| Starting value, S | $10,000.00 | Starting value |
| Ending value, E | $20,000.00 | Ending value |
| Growth multiple, E ÷ S | 2.0 | 20,000 ÷ 10,000 |
| Yearly multiple | 1.071773 | 2.01 ÷ 10 |
| CAGR | 7.18% | 1.071773 − 1 |
| Simple average | 10.00% | a 100% rise ÷ 10 years, which overstates the rate |
Check it by compounding: 10,000 × 1.071773^10 is 20,000. At 10% a year, the simple average, the value would have reached 25,937.
| Item | Amount | How it is obtained |
|---|---|---|
| Growth multiple needed | 3.0 | 30,000 ÷ 10,000 |
| ln(3.0) | 1.098612 | natural logarithm |
| ln(1.07) | 0.067659 | natural logarithm of one year's growth |
| Years needed | 16.24 | 1.098612 ÷ 0.067659 |
The table under the result shows sixteen whole years and then the part of the seventeenth in which the value crosses 30,000.
Reading the result
The headline is whichever quantity you asked for: a rate, a value or a number of years. Beside it, the growth multiple is the end value divided by the start, the total change is the same thing as a percentage, and the simple average divides that total by the years. The simple average is shown precisely because it is the figure people reach for by mistake, and it is always higher than the CAGR when growth is positive.
The chart and table draw the steady path the rate implies. That path joins your two points and nothing else. A share that rose 40% in one year and fell 20% in the next has the same CAGR as one that rose 5.8% in each, and the table cannot tell them apart.
What changes the result
The length of the period matters more than people expect. Doubling over 5 years is 14.87% a year; doubling over 10 is 7.18%; over 20 it is 3.53%. Halving the time does more than double the rate needed, because the compounding has fewer years to work.
The choice of start and end dates matters too. A CAGR measured from a market low to a high looks spectacular, and the same investment measured from a high to a low looks dreadful. Any CAGR quoted to you comes with two dates, and it is worth asking why those two were chosen.
Volatility lowers the CAGR you actually get for a given average of yearly returns. Up 50% and then down 50% averages 0% a year, but 100 becomes 75: a CAGR of minus 13.4% a year. The bigger the swings, the bigger the gap between the arithmetic average of the years and the compound rate.
What is not included
CAGR describes a single sum from a start date to an end date. Deposits and withdrawals in between break it, because money paid in raises the end value without being growth. For an account you added to, a money-weighted return over each dated deposit is the right measure.
It also says nothing about risk or about the route taken, and it does not allow for tax, fees or inflation unless the two values you enter already do. A nominal CAGR of 3% while prices rose 3% a year is no growth in what the money buys.
Using the result
Use CAGR to compare growth over different periods on one footing: a fund's ten-year record against another's seven, a company's revenue against its rivals', the rise in your home's value against inflation. Use the reverse modes to set a realistic plan: if a target needs 15% a year for a decade, it needs a return that very few investments have sustained, and the honest response is usually more time or a smaller target rather than a riskier bet.
When you are shown a CAGR, check the dates, check whether money was added along the way, and remember that the smooth line is a summary. The years themselves were not smooth.