People type this into search every day: CAGR calculator.
The usual shortcut divides the total percent change by the number of years. That is a straight split. Compound annual growth is the single yearly rate that lands on the ending value if each year builds on the last one. The two answers diverge as soon as the span is longer than a year.
This page uses the CAGR calculator. It looks backward at two values. It is not a forecast, and it is not a recommendation to buy or sell anything.
The rate that connects the two ends
Investor.gov’s glossary defines compound annual growth rate as the constant annual rate that would take a starting value to an ending value. The companion compound interest entry is the forward idea: earnings on principal and on earlier earnings. CAGR solves that relationship for the rate.
CAGR = (ending value ÷ starting value) ^ (1 ÷ years) − 1
Growth multiple = ending ÷ starting
Total change = (multiple − 1) × 100
The calculator prints the rate to two decimal places at most, the total change to one decimal place at most, and the multiple to two decimal places at most.
The starting value has to be above zero, because the formula divides by it. Years have to be above zero, because the exponent divides by them. The ending value can be zero. It cannot be negative. A negative ending amount has no real annual compound rate in this tool.
Deposits and withdrawals are invisible to the formula. It treats the ending value as if the starting value grew on its own. The compound savings post is the forward version, with a rate you already have. This page starts from the two balances and finds the rate.
Fig 1
Two balances, one yearly rate
The exponent is 1 divided by the years
Divide the ending value by the starting value, raise that multiple to 1 ÷ years, then subtract 1. That rate is the CAGR.
Example 1: $10,000 to $16,105.10 in five years
These are the calculator defaults.
- starting value: $10,000
- ending value: $16,105.10
- years: 5
Multiple: 16105.10 ÷ 10000 = 1.61051
That multiple is 1.10 raised to the fifth power:
1.10² = 1.21
1.10³ = 1.331
1.10⁴ = 1.4641
1.10⁵ = 1.61051
CAGR: 1.61051 ^ (1/5) − 1 = 10%
Total change: (1.61051 − 1) × 100 = 61.051%, which prints as 61.1%
Multiple on the card: 1.61×, because the display keeps two decimal places. 1.61051 rounds to 1.61.
Calculator result
Compound annual growth rate
- Starting value
- $10,000.00
- Ending value
- $16,105.10
- Years
- 5 years
10%
compound annual growth rate
- Total change
- 61.1%
- Growth multiple
- 1.61×
- Years in the exponent
- 5
The path that compounds at 10% once a year, with no deposits, is:
| Year | Balance |
|---|---|
| 0 | $10,000.00 |
| 1 | $11,000.00 |
| 2 | $12,100.00 |
| 3 | $13,310.00 |
| 4 | $14,641.00 |
| 5 | $16,105.10 |
Each line is the previous line × 1.10. Year 5 matches the ending value in the default.
Fig 2 · 1/6
A 10% compound path
$10,000 compounded once a year. Axis labels round.
Balance
- Balance
| Point | Balance |
|---|---|
| 0y | $10k |
| 1y | $11k |
| 2y | $12k |
| 3y | $13k |
| 4y | $15k |
| 5y | $16k |
Year 0 is the $10,000 start.
Year 5 is $16,105.10.
The gain is 16105.10 − 10000 = $6,105.10. The first year contributes $1,000. The later years contribute 6105.10 − 1000 = $5,105.10. That later piece is interest on interest, which is why a straight split of the 61.051% does not equal 10%.
Fig 3 · 1/2
Where the $6,105.10 gain comes from
The 10% path from $10,000 over five years
Year 1
$1,000.00
Years 2–5
$5,105.10
Year 1 adds $1,000.00.
Years 2 through 5 add the other $5,105.10.
1000 ÷ 6105.10 = 16.4%. 5105.10 ÷ 6105.10 = 83.6%.
Example 2: dividing the total percent by the years
The common mistake takes example 1’s total change and splits it.
Exact total change: 61.051%
Straight split: 61.051 ÷ 5 = 12.2102%, or 12.21% if you keep two decimals.
The CAGR is 10%.
The straight split is higher because it counts each year’s gain against the original $10,000. On the compound path, year 5’s gain is $1,464.10 on a $14,641 balance. Calling the whole five years “12.21% a year” overstates the constant rate that reproduces $16,105.10. The rate that hits $16,105.10 is 10%, which example 1 multiplied out year by year.
Over one year the two methods match. Example 6 is that case.
Example 3: a loss, $10,000 down to $7,000 in four years
- starting value: $10,000
- ending value: $7,000
- years: 4
Multiple: 7000 ÷ 10000 = 0.70
Total change: (0.70 − 1) × 100 = −30%, which prints as -30%
CAGR: 0.70 ^ (1/4) − 1 = −8.53%
Calculator result
Compound annual growth rate
- Ending value
- $7,000.00
- Years
- 4 years
-8.53%
compound annual growth rate
- Total change
- -30%
- Growth multiple
- 0.7×
- Years in the exponent
- 4
A straight split of the loss is −30 ÷ 4 = −7.5% a year. The compound rate is −8.53%, a deeper annual drop. The same pattern as example 2, pointed the other way: splitting the total change understates how steep a compound decline has to be to arrive at $7,000.
The balances that compound at that −8.53% rate are about $10,000, $9,146.91, $8,366.60, $7,652.86, and $7,000. Year 4 is the ending value you typed. The middle years are the smooth path. The real account may have fallen in a lump in year 3. CAGR does not show that lump.
Example 4: doubling in seven years, against a 10% path
- starting value: $10,000
- ending value: $20,000
- years: 7
Multiple: 2×
Total change: 100%
CAGR: 2 ^ (1/7) − 1 = 10.41%
Calculator result
Compound annual growth rate
- Ending value
- $20,000.00
- Years
- 7 years
10.41%
compound annual growth rate
- Total change
- 100%
- Growth multiple
- 2×
- Years in the exponent
- 7
A 10% path does not quite double in seven years. 1.10⁷ = 1.9487171, so $10,000 becomes $19,487.17. That is 20000 − 19487.17 = $512.83 short of a double.
“About 10%” and “doubled in seven years” are different inputs. The calculator’s rate for an exact double in seven years is 10.41%. The default five-year example is 10%, and its ending value is $16,105.10.
Example 5: a flat balance, and a known 8% you can reverse
No growth. Start $25,000, end $25,000, 10 years.
Multiple: 1×
Total change: 0%
CAGR: 1 ^ (1/10) − 1 = 0%
Ten years of a flat balance is a 0% compound rate. It is not a missing input.
A rate you already know. Start $20,000. Compound at 8% once a year for 6 years.
1.08⁶ × 20000 = $31,737.49, to the nearest cent.
Feed $20,000, $31,737.49, and 6 years back into the formula.
The calculator prints 8%, total change 58.7%, multiple 1.59×.
31737.49 ÷ 20000 = 1.5868745, which rounds to 1.59 on the card. (1.5868745 − 1) × 100 = 58.687%, which prints as 58.7%. The rate comes back as 8%. The display rounding shows up on the total change and the multiple. The clean 8% rate prints as 8%.
Example 6: one year, where CAGR equals the total change
- starting value: $5,000
- ending value: $5,450
- years: 1
Multiple: 5450 ÷ 5000 = 1.09
Total change: 9%
CAGR: 1.09 ^ (1/1) − 1 = 9%
The exponent is 1. Raising a number to the first power leaves it alone, so the compound rate and the total percent change are the same figure. Example 2’s gap appears only when the years are not 1.
Example 7: four times the money, and three times the money
Twenty years, four times. Start $8,000, end $32,000.
Multiple: 32000 ÷ 8000 = 4×
Total change: 300%
Years: 20
CAGR: 4 ^ (1/20) − 1 = 7.18%
A straight split would say 300 ÷ 20 = 15% a year. The compound rate is 7.18%. The gap is wider than example 2 because the span is longer and the multiple is larger.
Ten years, three times. Start $15,000, end $45,000.
Multiple: 3×
Total change: 200%
CAGR: 3 ^ (1/10) − 1 = 11.61%
Straight split: 200 ÷ 10 = 20%. The CAGR is 11.61%.
Fig 4 · 1/4
Ending value is not the rate
Different spans. The axis rounds to thousands.
Ending value
- Ending value
| Category | Ending value |
|---|---|
| 10% × 5y | $16k |
| Double, 7y | $20k |
| Triple, 10y | $45k |
| ×4, 20y | $32k |
Five years at 10% ends at $16,105.10.
Seven years to an exact double ends at $20,000.
Ten years to triple the $15,000 start ends at $45,000.
Twenty years to quadruple the $8,000 start ends at $32,000.
The $45,000 bar is the largest ending value and its CAGR, 11.61%, is the highest of these four. The $32,000 bar is a smaller balance from a smaller start, over more years, at 7.18%. Comparing the ending balances without the starts and the years is the wrong ranking.
Fig 5 · 1/5
The compound rate on each path
Printed CAGR for each path
CAGR
$10k to $16,105, 5y
10%
Double in 7y
10.41%
Triple in 10y
11.61%
×4 in 20y
7.18%
$40k to $52k, 12y
2.21%
The default five-year path is 10%.
Doubling in seven years is 10.41%. Tripling in ten is 11.61%.
Quadrupling in twenty years is 7.18%. A 30% gain over twelve years is 2.21%.
The 2.21% bar is example 8.
Example 8: a long, smaller gain
- starting value: $40,000
- ending value: $52,000
- years: 12
Multiple: 52000 ÷ 40000 = 1.30
Total change: 30%
CAGR: 1.30 ^ (1/12) − 1 = 2.21%
Straight split: 30 ÷ 12 = 2.5%.
The gap is 2.5 − 2.21 = 0.29 percentage points, narrower than the five-year gap in example 2, because 30% of growth has less compounding inside it than 61% does. It is still not zero. Over 12 years, 2.21% is the rate that lands on $52,000. 2.5% would overshoot.
Example 9: money added along the way
Start with $10,000. Add $5,000 of new cash during the five years. End at $18,000.
If you type starting $10,000, ending $18,000, and 5 years, the calculator does what you asked:
Multiple: 18000 ÷ 10000 = 1.8×
Total change: 80%
CAGR: 1.8 ^ (1/5) − 1 = 12.47%
That 12.47% treats the extra $5,000 as growth. The ending balance is only $18,000 − $5,000 = $3,000 above the $15,000 of cash that went in. The tool cannot see the deposit. Investor.gov’s definition is a rate between a starting value and an ending value.
Use CAGR on a balance that had no external cash in or out, or on a series that already removes those flows. The savings goal calculator is the tool for a monthly deposit. This one is not.
Example 10: six months typed as half a year, or as six years
The balance goes from $10,000 to $11,000. The total change is 10% either way. The years box changes the rate.
Half a year. Years = 0.5.
CAGR: 1.10 ^ (1/0.5) − 1 = 1.10² − 1 = 21%
Multiple 1.1×. Total change 10%. A 10% gain in six months is a 21% compound annual rate, because a full year of that pace would be two of those steps.
Six in the years box. Someone with a six-month gain types 6.
CAGR: 1.10 ^ (1/6) − 1 = 1.6%
Same multiple, same 10% total change, a much smaller rate. The exponent is the whole mistake. Six months is 0.5 years in this formula, or 6 ÷ 12. It is not 6 years.
Example 11: zero, and a negative ending value
Ending value $0, start $10,000, 5 years.
Multiple: 0. CAGR: 0 ^ (1/5) − 1 = −100%. Total change −100%. Multiple 0×.
An account that went to zero has a −100% compound rate over that span. That is a real result.
Ending value below zero. Start $10,000, end −$100, 3 years. The calculator returns Need a non-negative ending value. There is no real number you can raise to a power here that represents a negative balance as a compound annual rate of a positive start.
Starting value $0. End $10,000, 5 years. The calculator returns Need a starting value. The formula divides by the start. A business that began at zero needs a different measure than CAGR.
The scenarios side by side
| Example | Start | End | Years | CAGR | Total change | Multiple |
|---|---|---|---|---|---|---|
| 1. Defaults | $10,000 | $16,105.10 | 5 | 10% | 61.1% | 1.61× |
| 2. Straight split of example 1 | $10,000 | $16,105.10 | 5 | 12.21% is not the CAGR | 61.051% exact | 1.61051 |
| 3. Loss | $10,000 | $7,000 | 4 | −8.53% | −30% | 0.7× |
| 4. Exact double | $10,000 | $20,000 | 7 | 10.41% | 100% | 2× |
| 5. Flat | $25,000 | $25,000 | 10 | 0% | 0% | 1× |
| 5. Known 8% reversed | $20,000 | $31,737.49 | 6 | 8% | 58.7% | 1.59× |
| 6. One year | $5,000 | $5,450 | 1 | 9% | 9% | 1.09× |
| 7. Quadruple | $8,000 | $32,000 | 20 | 7.18% | 300% | 4× |
| 7. Triple | $15,000 | $45,000 | 10 | 11.61% | 200% | 3× |
| 8. Long small gain | $40,000 | $52,000 | 12 | 2.21% | 30% | 1.3× |
| 9. Deposit hidden in the end | $10,000 | $18,000 | 5 | 12.47% | 80% | 1.8× |
| 10. Half a year | $10,000 | $11,000 | 0.5 | 21% | 10% | 1.1× |
| 10. Six typed as years | $10,000 | $11,000 | 6 | 1.6% | 10% | 1.1× |
| 11. Ended at zero | $10,000 | $0 | 5 | −100% | −100% | 0× |
The inflation post moves a dollar the other direction, from a price level you type. The savings goal post asks what to set aside each month. CAGR asks what constant rate connects two amounts you already have.
Open the CAGR calculator with a starting value that does not include money you added later. If the two ends are a price index rather than an account, the same formula is the annualized change, and it is still not a prediction of next year.