ReckonBoxHow the average return calculator works
The average return is the arithmetic mean of a run of yearly returns: add them up and divide by how many there are. Enter one figure per year and this returns that average alongside the annualised rate, which is the number describing what an investor actually ended up with.
Those two figures are almost never the same, and the distance between them is the whole reason this page exists.
The two means
Corporate Finance Institute's worked series: 15%, 17.5%, 3%, 10%, 5% and 8% over six years. They sum to 58.5, so the average is 9.75%.
Average return = sum of returns / n
The annualised figure multiplies the growth factors together and takes the nth root:
Annualised return = (product of (1 + r) )^(1/n) - 1
On the same six years that gives 9.63%. CFI puts the distinction in one line, that annualised return compounds while average return ignores compounding. calculator.net, which ranks first for this term, publishes no formula at all and no example separating the two.
Why the geometric mean is always lower
Losses and gains aren't symmetric, and that asymmetry is the entire mechanism.
A stock falling 20% from $100 to $80 needs a 25% gain to get back to $100. Not 20%. So a year of -20% followed by a year of +20% averages to zero while the balance sits 4% down.
The two means are equal only when every year is identical. Any variation at all pushes the annualised figure below the average, and the more variation, the wider the gap.
| Yearly returns | Average | Annualised | Gap |
|---|---|---|---|
| 9, 11, 9, 11 | 10% | 10.00% | 0.00 pts |
| 15, 17.5, 3, 10, 5, 8 | 9.75% | 9.63% | 0.12 pts |
| -30, 50, -30, 50 | 10% | 2.47% | 7.53 pts |
The top and bottom rows have the same average return. An investor in the second one compounds at a quarter of the rate. That is what a quoted average can hide.
Estimating the gap from volatility
The gap has a name, volatility drag, and it tracks the variance of the returns rather than their average. Kitces gives the relationship:
Annualised return is approximately the average return minus half the variance
Checked against two real series over 2007 to 2016: the S&P 500 averaged 8.75% with a standard deviation of 18.86%, so the estimate is 8.75% minus 3.56%/2, or 6.97%, against an actual annualised 6.94%. The Barclays Aggregate Bond index averaged 4.39% with a 3.12% standard deviation, estimating 4.34% against an actual 4.35%. Both land within a few basis points.
Kitces is careful that this is an approximation, since a limited sample doesn't perfectly follow the underlying distribution. So this tool prints the estimate beside the exact geometric figure and shows how far apart they landed, which is the only honest way to display a shortcut.
One choice worth stating, because every other tool leaves it silent: the standard deviation here is the sample one, dividing by n minus 1. Using the population version instead would shrink the variance and shift the drag estimate.
The extreme case
A single year of -100% takes the balance to zero, and nothing compounds after that.
Feed in 10%, -100%, 50% and the average return still reads as a positive number while the annualised return is -100%. No sequence of good years recovers from a wipeout, because compounding is multiplicative and zero absorbs everything.
That case is worth sitting with. It's the clearest demonstration that the two figures answer genuinely different questions, and that only one of them tracks money.
Which one gets reported
Fund factsheets and index reports quote the annualised figure, because it's the one matching what a continuously invested holder actually got. An average return quoted over a volatile period will always flatter the result.
The annualised return here is the same calculation as CAGR, approached from different inputs. CAGR starts from a beginning and an ending value; this starts from the yearly returns. Give either one consistent data and they agree.
What this calculator does not do
It doesn't handle money going in or out along the way. These are pure period returns on a holding left alone. Once there are contributions and withdrawals on real dates, the question becomes an XIRR one.
It doesn't forecast. A past average and a past standard deviation describe what happened, and nothing in the arithmetic makes them predictive.
It doesn't adjust for inflation, fees or tax. Net those out of the yearly returns before entering them if they matter.
Results are arithmetic on the numbers you entered, not financial advice. For decisions about your own money, speak to a licensed financial adviser. To grow a balance forward at a fixed rate instead, use the compound interest calculator.
Frequently asked questions
What is the average return? The average return is the arithmetic mean: add the yearly returns and divide by how many there are. Returns of 15, 17.5, 3, 10, 5 and 8 percent sum to 58.5 across six years, so the average is 9.75 percent.
What is the difference between average return and annualised return? Average return ignores compounding and annualised return accounts for it. The annualised figure multiplies the growth factors together and takes the nth root, which is the rate that genuinely turns the starting amount into the ending one. On that same six year run the average is 9.75 percent and the annualised figure is 9.63 percent, so the average overstates what an investor actually earned.
Why is the geometric mean always lower than the arithmetic mean? Because losses and gains are not symmetric. A stock falling 20 percent from 100 to 80 needs a 25 percent gain to get back to 100, not 20 percent. So a year of -20 percent followed by a year of +20 percent averages to zero while the balance is down 4 percent. The two means are equal only when every year is identical, and any variation at all pushes the geometric figure below the arithmetic one.
What is volatility drag? Volatility drag is the amount compounding loses to variation, which is exactly the gap between the two means. It grows with volatility rather than with the average: two portfolios both averaging 10 percent a year can end up far apart if one is much bumpier. Returns of 9, 11, 9 and 11 percent compound to 10.00 percent a year, while -30, 50, -30 and 50 percent share the same 10 percent average and compound to just 2.47 percent.
How do you estimate the annualised return from the average? Subtract half the variance from the arithmetic mean. Kitces checks this against the S&P 500 over 2007 to 2016, where an 8.75 percent average and an 18.86 percent standard deviation give an estimated 6.97 percent against an actual 6.94 percent, and against the Barclays Aggregate Bond index, where 4.39 percent and 3.12 percent give 4.34 percent against an actual 4.35 percent. The relationship is an approximation, so this tool prints the estimate beside the exact figure and shows how far apart they landed.
Which figure do funds report? The annualised one. Fund factsheets and index reports quote annualised returns because that is the figure matching what an investor holding throughout actually ended up with. An average return quoted over a volatile period will always look better than the money did.
What is the difference between annualised return and CAGR? They are the same calculation approached from different inputs. CAGR starts from a beginning value and an ending value; the annualised return here starts from the series of yearly returns. Feed either one consistent data and they agree.
Why does one terrible year matter so much? Because compounding is multiplicative. A year of -100 percent takes the balance to zero and nothing recovers from it, however good the remaining years are, so the annualised return is -100 percent while the arithmetic average still reads as a positive number. That extreme case is the clearest demonstration that the two figures answer different questions.