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How to Calculate CAGR by Hand, Step by Step

By Reeliy Editorial Team · · 3 min read

The compound annual growth rate (CAGR) answers one question: what constant yearly growth rate would take a value from where it started to where it ended? It smooths out the ups and downs in between, which makes two investments, sales figures or populations easy to compare.

You need only three numbers and a calculator with a power or root key. This guide gives the formula, a worked example done two ways, a case that shows what CAGR hides, and the mistakes people make with the number of periods.

The formula and the steps

CAGR = (end value ÷ start value)^(1 ÷ n) − 1n is the number of years (periods) between the start and end values.

  1. Divide the end value by the start value.
  2. Raise that ratio to the power 1 ÷ n, which is the same as taking the n-th root.
  3. Subtract 1.
  4. Multiply by 100 to express it as a percentage.

Worked example: a value grows from 10,000 to 16,000 over 6 years. The ratio is 1.6. The sixth root of 1.6 is 1.081484, so CAGR = 0.081484, or about 8.15% a year.

If your calculator has no root key, use logarithms: ln(1.6) = 0.470004; divided by 6 that is 0.078334; and e raised to 0.078334 is 1.081484. Same answer. As a check, 10,000 × 1.0814846 = 16,000.

Count periods, not data points

The most common error is using the wrong n. If you have a value at the end of 2019 and another at the end of 2024, there are five years between them, so n = 5 even though six year-end figures exist.

Worked example: a value of 2,500 grows to 4,100 across those five years. The ratio is 1.64 and CAGR = 1.641/5 − 1 = 10.40%. Using n = 6 by mistake would give 8.59%, which understates the growth.

For periods that are not whole years, convert to years first: 18 months is n = 1.5, and 90 days is roughly n = 0.25.

What CAGR hides

CAGR ignores the path taken between the two end points. Suppose a value is 100, then 150, then 90, then 120 at the end of years 0 to 3. The yearly changes were +50%, −40% and +33.33%.

The CAGR of the whole run is (120 ÷ 100)1/3 − 1 = 6.27%. The simple average of the three yearly changes is 14.44%, which overstates the outcome badly. A steady 6.27% a year would reach the same end point as that very bumpy ride.

CAGR is most useful for comparing results over different lengths of time. Worked example: option A grows from 10,000 to 15,000 in 5 years, and option B grows from 10,000 to 18,000 in 8 years. B has the larger total gain (80% against 50%), but its CAGR is 1.81/8 − 1 = 7.62%, while A's is 1.51/5 − 1 = 8.45%. A grew faster per year, and the longer period is what made B's total look better.

This is why CAGR is a growth summary, not a measure of risk. Two series with the same CAGR can have very different swings along the way.

When not to use it, and how to use it forward

  • Deposits or withdrawals in between. CAGR assumes nothing is added or taken out. For an investment with cash flows, use a money-weighted measure such as the IRR calculator.
  • Zero or negative values. The ratio must be positive for the root to work.
  • Very short periods. Annualising a few months of growth gives a number that is unlikely to repeat.

You can also run CAGR forward as a scenario, not a prediction. Growing 20,000 at 7% for 10 years gives 20,000 × 1.0710 = 39,343.03. To find how long a target takes, rearrange to n = ln(end ÷ start) ÷ ln(1 + CAGR): going from 10,000 to 16,000 at 8% takes ln(1.6) ÷ ln(1.08) = 6.11 years.

Use the CAGR calculator to check your hand result, and the any root calculator for the n-th root step. For the same idea from the borrower or saver's perspective, see compound interest, step by step.

This article is general education, not financial advice. Past growth, however it is measured, does not predict future results.

Try the tools
  • CAGR Calculator: Enter the value an investment started at, the value it ended at and the number of periods in between. The calculator returns the steady yearly growth rate that links the two.
  • Root Calculator: Type a number and the degree of the root you want. Square roots, cube roots, fourth roots and fractional roots all work the same way.
  • IRR Calculator: Enter the amount you invest, choose how many years the project runs, and fill in the cash flow for each year. The calculator finds the yearly rate at which the investment exactly breaks even.

This article is general information, not professional advice. See our Disclaimer and Editorial Policy.

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