CAGR calculator and formula
CAGR is the one steady yearly rate that turns a starting value into an ending value. Divide the end by the start, take the root for the number of years, subtract 1. $10,000 growing to $18,000 over 6 years is a CAGR of 10.29 percent a year.
Compound annual growth rate
10.29%
$10,000 reaches $18,000 in 6 years at that steady rate.
- Growth multiple
- 1.80x
- Total growth over the period
- 80.0%
- Gain in money
- $8,000.00
Balance at the end of each year
| Year | Balance | Multiple |
|---|---|---|
| 1 | $11,029 | 1.10x |
| 2 | $12,164 | 1.22x |
| 3 | $13,416 | 1.34x |
| 4 | $14,797 | 1.48x |
| 5 | $16,320 | 1.63x |
| 6 | $18,000 | 1.80x |
Count years of growth, not readings. Start of year one to end of year six is 6. A part year goes in as a fraction, so 18 months is 1.5.
The formula
is the value at the start, the value at the end, and the number of years between them. The answer comes out as a decimal, so 0.1029 means 10.29 percent a year.
What this calculator works out
Enter a starting value, an ending value and the number of years between them. It returns the one steady annual rate that connects the two, the total growth multiple, and the balance at the end of every year had growth run at exactly that rate.
CAGR is a smoothed rate, not a description of any single year. Nothing has to grow by that amount in any year for the number to be right. It answers one narrow question: what constant yearly rate would have produced this result over this many years?
The CAGR formula
Divide the ending value by the starting value, take the -th root of that, then subtract 1. A root is the same thing as a fractional power, which is how you type it into anything: the sixth root of 1.8 is .
Count years, not readings. A value at the start and a value at the end of the sixth year is 6 years of growth, not 7, and an exponent that is one year out changes the answer by well over a percentage point in the first example below.
Run the same arithmetic in the other direction and you have compound interest. That formula takes a rate and gives you an ending value. CAGR takes an ending value and gives you the rate.
Why the number of years changes the answer
The rate depends on the period as much as on the two values. The same $10,000 to $18,000 is 10.29 percent a year over 6 years and 21.64 percent a year over 3 years. Same start, same finish, and the shorter run has to grow faster every year to get there.
So a CAGR quoted without its period is not a number anyone can use. A fund reporting 21.64 percent over 3 years and 10.29 percent over 6 is not contradicting itself, it is reporting two windows on the same money. When you compare two investments, compare equal periods ending on the same date: a window that stops just before a bad quarter is a different claim from one that includes it.
What a CAGR hides
CAGR is a geometric average. It multiplies the yearly growth factors together rather than adding the returns up, which is what makes it reproduce the ending value, and also why it says nothing about the ride. Two holdings with the same CAGR can feel nothing alike: one grinding up a few percent a year, the other halving and then more than doubling to land in the same place. Halving and then merely doubling is a CAGR of zero, because 0.5 times 2 is 1.
It also assumes the money went in once at the start and came out once at the end. Deposits and withdrawals in between break that assumption, because a rate worked out from an opening and a closing balance counts your own deposits as investment growth. Use a stretch with no cash flows in it, or a measure built for cash flows arriving on different dates.
For interest rather than investment growth the same idea already has a name. An effective annual rate is the CAGR of an interest-bearing balance, which is what the APR against APY calculator works out.
Worked examples
\$10,000 to \$18,000 over 6 years
An investment was worth $10,000 at the start and $18,000 six years later, with nothing paid in and nothing taken out. What annual rate is that?
- Divide the ending value by the starting value: . The money grew to 1.8 times its starting size.
- Take the 6th root, which is the same as raising to the power : .
- Subtract 1: , which is 10.29 percent a year.
- Check it by growing the money back up: .
The CAGR is 10.29 percent a year. $10,000 compounding at that rate for 6 years lands on $18,000, which is total growth of 80 percent spread over the period.
The same money in half the time
Take the same $10,000 to $18,000, but suppose it happened over 3 years rather than 6. What is the CAGR now?
- The growth multiple does not change: .
- Only the exponent moves: where before it was .
- Subtract 1: , or 21.64 percent a year.
- Check it: comes back to 18000.
The CAGR is 21.64 percent a year, more than double the 10.29 percent that the same growth gives over 6 years. Halving the time more than doubles the rate, because each year has to carry more of the work.
The average return trap
A holding gains 60 percent in year one and loses 37.5 percent in year two. Those two returns average 11.25 percent a year. Starting from $10,000, what is the CAGR?
- Turn each year into a growth factor. Up 60 percent is a factor of 1.6. Down 37.5 percent is a factor of 0.625, because 100 minus 37.5 leaves 62.5 percent.
- Returns compound, so the factors multiply: .
- A total factor of 1 means the ending value equals the starting value: $10,000 in, $10,000 out, 2 years apart.
- Put that into the formula: .
The CAGR is 0 percent. The holding ends at $10,000, exactly where it started, while the average of the two yearly returns says 11.25 percent a year. The average is the mean of a list of numbers. The CAGR is the rate the money grew at, and here it grew by nothing.
Averaging the yearly returns instead of compounding them
Add up a run of yearly returns, divide by the number of years, and you have an arithmetic mean. It is always at least as large as the CAGR, and the more the yearly returns bounce around, the wider the gap gets.
The third worked example above is the clean case. Up 60 percent then down 37.5 percent averages 11.25 percent a year and leaves the money precisely where it started. Quoting the 11.25 percent is not a small overstatement, it is a rate that never happened.
The fix is one habit. Turn each year into a growth factor, multiply the factors together, then take the root. 1.6 times 0.625 is 1, and the square root of 1 is 1, so the rate is zero. A geometric average is the only average that reproduces the ending value, which is the entire reason for quoting a growth rate.
Common questions
Can a CAGR be negative?
Yes. When the ending value is below the starting value, the ratio is less than 1, so its root is less than 1 and subtracting 1 leaves a negative number. A negative CAGR is the steady annual rate at which the money shrank, and it is read exactly like a positive one.
What if the period is not a whole number of years?
Use the fraction. Eighteen months is 1.5 years, so the exponent is 1 over 1.5. Rounding a part year to a whole one shifts the answer more than people expect over short periods, because the time enters the formula through the exponent rather than through the values.
Does CAGR account for money paid in along the way?
No, and that is the assumption broken most often. It compares two values and treats everything between them as growth, so a rate taken from the opening and closing balance of an account you were paying into credits your own deposits to the investment. Pick a stretch with no deposits or withdrawals, or use a measure built for dated cash flows.
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.