π Average Return Calculator
Arithmetic average versus the geometric return you actually earned.
Enter an amount to see the dollar growth year by year.
True Annualized Return
Estimates for planning only, not financial advice.
How to Use This Calculator
Enter each year's return as a percentage, one per line, in chronological order. The calculator computes both the arithmetic mean and the geometric mean, shows the volatility drag between them, and, if you enter a starting amount, grows it year by year so you can see the dollar path.
Type your Yearly Returns, one percentage per line. Use negative numbers for losing years, for example -5 for a 5 percent loss. Order does not affect the averages but does affect the year-by-year table.
Optionally enter a Starting Amount to translate the percentages into dollars. The table then shows the balance after each year.
Read the geometric mean first: that is the constant annual rate that reproduces your actual ending wealth. The arithmetic mean is shown for comparison.
Check the volatility drag, the gap between the two means. A large gap means the headline average flatters a bumpy ride.
The Two Formulas
Each return r is written as a decimal (10 percent = 0.10). The arithmetic mean treats every year independently and adds them. The geometric mean multiplies the yearly growth factors, which is exactly how wealth compounds, then takes the nth root to spread the total growth evenly across the years. A useful approximation: the drag is about half the variance of the returns, so wilder swings cost roughly half their variance in annualized return.
Worked Example
Yearly returns of 10, -5, 20, and 15 percent. Arithmetic mean = (10 - 5 + 20 + 15) / 4 = 10.00 percent. Geometric mean = (1.10 x 0.95 x 1.20 x 1.15)^(1/4) - 1 = (1.4421)^(0.25) - 1 = 9.58 percent. The volatility drag is 0.41 percentage points. On a $10,000 starting amount, the balance path is $11,000 after year 1, $10,450 after year 2, $12,540 after year 3, and $14,421 after year 4. Growing $10,000 to $14,421 in 4 years is a 44.21 percent total return, and 1.4421^(1/4) - 1 confirms the 9.58 percent annualized figure. Anyone quoting the 10 percent average is overstating what investors actually earned.
Where the Distinction Matters
Mutual fund fact sheets, backtests, and crypto influencers usually lead with whichever average looks better, and the arithmetic one always wins that contest. Pension plans and endowments forecast with the arithmetic mean but judge managers on the geometric one, which is the professionally honest split. For your own planning, the geometric mean answers "what did I get," while the arithmetic mean, adjusted for volatility, helps answer "what might I get." Whenever a single number has to describe multi-year performance, make it the geometric mean.
Estimates for planning only, not financial advice.
Frequently Asked Questions
What is the difference between arithmetic and geometric average return?
The arithmetic mean is the simple average of the yearly returns: add them up and divide by the number of years. The geometric mean is the constant yearly rate that would compound to the same final wealth. The arithmetic mean is always equal to or higher than the geometric mean, and the gap grows with volatility. Only the geometric mean tells you what you actually earned.
Why is the geometric mean lower than the arithmetic mean?
Because losses hurt more than equal gains help. A 50 percent gain followed by a 50 percent loss averages to 0 percent arithmetically, but your money went $100 to $150 to $75, a real loss. The geometric mean captures this asymmetry: (1.5 x 0.5)^(1/2) - 1 = -13.4 percent, which is the true annualized result. Volatility is a direct tax on compounded wealth.
Which average should I use to evaluate a fund?
Use the geometric mean (CAGR) to measure what investors actually earned over the period. Use the arithmetic mean only as an input for forecasting single-year outcomes or for statistical work like estimating expected return. Fund marketing sometimes quotes whichever looks better, so compute both from the yearly returns and trust the geometric one for history.
Can I just average the yearly percentages?
You can, but the result overstates your actual experience whenever returns vary. With yearly returns of 10, -5, 20, and 15 percent, the simple average is 10 percent, but $10,000 really grew to $14,421, which is a 9.58 percent annualized return. The 0.41-point gap is the volatility drag, and it gets bigger as the yearly swings get wilder.
What if one year has a -100% return?
A -100 percent return means the investment went to zero, and no later gain can recover it: the geometric mean becomes -100 percent regardless of other years. The calculator rejects any return at or below -100 percent as invalid input, since wealth cannot go negative in this model. If you are entering real data, check for typos first.
How does this relate to CAGR?
The geometric mean return over n years is exactly the CAGR (compound annual growth rate) for that period. CAGR = (ending value / beginning value)^(1/n) - 1, which is algebraically identical to taking the geometric mean of (1 + each yearly return) and subtracting 1. They are two names for the same number.