Why is the average of +50% and −50% not zero?

Because the second percentage applies to a different amount. $10,000 up 50% is $15,000; down 50% from there is $7,500. The arithmetic average says zero and the investor is down a quarter. Percentages of different bases cannot be averaged, which is the single most common error in return reporting.

The compound annual growth rate — the constant yearly rate that would take the starting value to the ending value over the period. It is the geometric mean of the yearly returns rather than the arithmetic one, and it is the only "average" that reproduces the actual result when compounded.

When is the arithmetic average the right one?

When you are estimating a single future year rather than describing a past run. The arithmetic mean is the best estimate of next year’s return; the geometric mean is what a portfolio actually earned across many years. Both are correct answers to different questions, and quoting the first as the second is where funds flatter themselves.

Does CAGR account for deposits?

No, and using it where money moved in or out gives a meaningless figure. For a portfolio with contributions, the internal rate of return — a money-weighted return — is the right measure, because it accounts for how much was invested when.

Average Return Calculator

With starting value 10000 usd, ending value 18000 usd, years held 7, a year that gained 50 percent and 1 more field, average return comes to 8.760 % — annualised return. It is reached in 8 steps, the last of which is (pow(18000 / 10000, 1 / 7) - 1) * 100, and each one is printed on the page with its numbers filled in. The formula is the one published by US SEC Investor.gov, not an approximation fitted to it.

The compound annual growth rate of an investment, next to the arithmetic average that overstates it.

Formula and sources checked 27 August 2026 · How we check

Starting value 10000, Ending value 18000, Years held 7, A year that gained 50

8.760 %

Annualised return for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.

It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.

Starting valueusdEnding valueusdYears heldA year that gainedpercentFor the illustration below: two consecutive years of these returns.A year that lostpercent

Annualised return

8.760 %

Total return: (18000 / 10000 - 1) * 10080 %
Compound annual growth rate: (pow(18000 / 10000, 1 / 7) - 1) * 1008.76 %
Total return divided by the years: 80 / 711.429 %
How much the simple average overstates it: 11.428571 - 8.75957472.669 %
Years to double at this rate: log(2) / log(1 + 8.7595747 / 100)8.255
Arithmetic average of the two years: (50 + -50) / 20 %
What those two years actually did: (pow((1 + 50 / 100) * (1 + -50 / 100), 0.5) - 1) * 100-13.397 %
What $10,000 becomes over those two years: 10000 * (1 + 50 / 100) * (1 + -50 / 100)$7,500

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Worked example

$10,000 becoming $18,000 over seven years is a compound annual growth rate of 8.76%, not the 11.43% a total-divided-by-years figure suggests. The two-year illustration is starker: up 50% then down 50% averages zero and leaves $7,500.

How to work it out yourself

1.Use the starting and ending values with no deposits or withdrawals in between. Where money went in or out, the compound growth rate is not the right measure — a money-weighted return is.
2.Read the CAGR rather than the simple average. It is the constant rate that would have produced the same result, which is what "average return" is trying to mean.
3.Treat any advertised average return with the same suspicion. A fund quoting an arithmetic average is quoting a number no investor received.

The formula

Total return(18000 / 10000 - 1) * 100
Compound annual growth rate(pow(18000 / 10000, 1 / 7) - 1) * 100
Total return divided by the years80 / 7
How much the simple average overstates it11.428571 - 8.7595747
Years to double at this ratelog(2) / log(1 + 8.7595747 / 100)
Arithmetic average of the two years(50 + -50) / 2
What those two years actually did(pow((1 + 50 / 100) * (1 + -50 / 100), 0.5) - 1) * 100
What $10,000 becomes over those two years10000 * (1 + 50 / 100) * (1 + -50 / 100)

Source: US SEC Investor.gov — compound interest and investment returns

Questions people actually ask

Why is the average of +50% and −50% not zero?: Because the second percentage applies to a different amount. $10,000 up 50% is $15,000; down 50% from there is $7,500. The arithmetic average says zero and the investor is down a quarter. Percentages of different bases cannot be averaged, which is the single most common error in return reporting.
What is CAGR?: The compound annual growth rate — the constant yearly rate that would take the starting value to the ending value over the period. It is the geometric mean of the yearly returns rather than the arithmetic one, and it is the only "average" that reproduces the actual result when compounded.
When is the arithmetic average the right one?: When you are estimating a single future year rather than describing a past run. The arithmetic mean is the best estimate of next year’s return; the geometric mean is what a portfolio actually earned across many years. Both are correct answers to different questions, and quoting the first as the second is where funds flatter themselves.
Does CAGR account for deposits?: No, and using it where money moved in or out gives a meaningless figure. For a portfolio with contributions, the internal rate of return — a money-weighted return — is the right measure, because it accounts for how much was invested when.

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