How two endpoint values become one annualized growth rate
CAGR answers what constant annual compound rate would turn one positive starting value into an ending value over the stated time. It compresses a start-to-end change into an annual comparison number.
CAGR is the single annual compound rate that connects two endpoints. A 21% total gain over two years is 10% per year compounded, since 1.10 × 1.10 = 1.21. The smooth line is an equivalent path only; the real investment could have risen and fallen between observations.
The annualized rate, total return and mathematically equivalent path are shown together. The page labels the path as hypothetical, helping readers avoid mistaking smooth interpolation for actual yearly performance.
How starting value, ending value and elapsed years set CAGR
CAGR uses only the ratio between the two values and the time between them. The size of the values does not matter, but the years entered matter a great deal.
- Starting value
- The value at the beginning of the period, which must be positive. Only its ratio to the ending value counts, so 1,000 to 1,210 and 10 to 12.10 both give 10% a year over 2 years. A negative starting figure, such as a loss-making year, cannot be annualized with this formula.
- Ending value
- The value at the end, on the same basis as the start, for example both including reinvested dividends or both excluding them. A lower ending value gives a negative rate: 1,000 falling to 800 over 2 years is −10.5573% a year, not −10%. Zero is accepted and is reported as −100%.
- Elapsed years
- Decimals are accepted, and this is where most errors start. The same 21% total gain is 21% a year over 1 year, 10% over 2 and 6.5602% over 3, but 46.41% over half a year, because a short period is scaled up to a full year. Measure the actual time between valuation dates.
- Total return beside CAGR
- The unannualized change, 21% in the example, is shown next to the annual rate. Reading both keeps a multi-year total from being quoted as a yearly figure, and keeps a short-period CAGR from looking like a result that was actually earned.
Steps
- Enter the positive starting value and nonnegative ending value on the same basis.
- Enter the elapsed time in years; decimals are allowed.
- Read the annualized percentage and compare it with the separate total return.
Reading tip: Use starting and ending figures on the same valuation basis. If you added money between the dates, the endpoint formula attributes those deposits to growth and becomes misleading.
Comparisons that need growth expressed per year
Compare endpoint growth
Put multi-year start-to-end changes on a comparable annualized scale.
Review a portfolio period
Distinguish annualized endpoint growth from the total change over the whole holding period.
Model a steady path
See the constant-rate trajectory implied by two observed endpoints without claiming it occurred.
Spot a decline
Express a lower ending value as a negative annualized growth rate.
How the calculation works
- G
- Overall ending-to-starting value multiple across the entire interval.
- t
- Elapsed number of years, including decimal fractions when entered.
- start
- Positive beginning value on the same valuation basis as the end.
- end
- Nonnegative ending value before any unmodeled cash-flow adjustments.
For a positive end value the curve is a hypothetical constant-rate path. A zero end value is shown as two endpoints only; CAGR never reveals interim volatility or deposits.
What every part means
- G
- Overall ending-to-starting value multiple across the entire interval.
- t
- Elapsed number of years, including decimal fractions when entered.
- start
- Positive beginning value on the same valuation basis as the end.
- end
- Nonnegative ending value before any unmodeled cash-flow adjustments.
Calculate it step by step
The starting value must be positive, so the growth multiple is defined.
Find the constant factor that would compound t times to the total multiple.
Subtract the original one unit and convert the decimal rate to a percentage.
Years can be fractional. The zero-ending-value case is reported as −100% by endpoint convention; only the two endpoints are plotted because an intermediate path cannot be inferred.
See how Calculation Guide tests examples and handles calculation limits.
See it in practice
- Starting value
- 1000 currency units
- Ending value
- 1210 currency units
- Elapsed time
- 2 years
Start 1,000; end 1,210; t = 2: G = 1.21; CAGR = (1.21^(1/2) − 1) × 100 = 10%, while total return is 21%.
- Starting value1,000
- Ending value1,210
Interpretation & limitations
How to read your result
Compare CAGR only when valuation bases and periods are sensible. A negative percentage means the end value is below the start; the total-return fact shows the nonannualized change separately.
Where this estimate stops
CAGR ignores interim cash flows, volatility, sequence risk, taxes, fees and inflation. It cannot describe year-by-year performance from only two endpoint values.
Frequently asked questions
01Does CAGR tell me what happened each year?
No. It draws an equivalent smooth path between two endpoints and hides the actual swings between them.
02Can CAGR be negative?
Yes. A lower ending value gives a negative annualized rate; a zero ending value gives −100%.
03Why is 21% total growth over two years only 10% CAGR?
Compounding 10% twice gives 1.1² = 1.21, a 21% total gain.
04Can I enter 2.5 years?
Yes. The exponent uses exactly the decimal number of years entered.
05Does the line show historical observations?
No. It is an interpolated constant-growth illustration between the two values.
06What if I deposited more money during the period?
This formula cannot separate added capital from return. Use a cash-flow-aware return measure for that case.
07Can I calculate CAGR for a figure that started negative, like a loss turning into a profit?
No. The starting value must be positive, because the formula takes a root of ending value divided by starting value, and a negative ratio has no real annual rate. For a loss that became a profit, report the absolute change, or start from the first positive year and state that choice.