What "average return" should mean
Say a fund returned 12% one year, lost 8% the next, gained 23%, gained 5%, then lost 15%, and someone asks what it averaged over those five years. Add the five percentages and divide by five and you get 3.4% a year, the arithmetic mean, the number most people reach for first because it is the same "add them up, divide by the count" trick taught for any list of numbers. It is also, for a series of compounding returns, the wrong answer to "what did my money actually average", and this calculator exists to show why, and to give you the number that gets it right instead.
Two means, worked through on the same numbers
The arithmetic mean treats each year's return as a plain number to be summed, ignoring that a return does not add to the year before it, it multiplies. The geometric mean fixes that by working in growth factors instead: turn each return into "1 plus the return", multiply every year's factor together, take the nth root for n years, then subtract 1 to turn it back into a percentage.
Run the five returns above, 12%, −8%, 23%, 5% and −15%, through the geometric formula and the factors 1.12, 0.92, 1.23, 1.05 and 0.85 multiply out to about 1.1311, whose fifth root, minus 1, lands at roughly 2.50% a year, noticeably below the 3.4% arithmetic figure. Ten thousand pounds compounding through those five actual returns, in that actual order, grows to about $11,311.47, and 2.50% is precisely the single steady annual rate that would have produced that same $11,311.47 from $10,000 over five years. The arithmetic mean of 3.4% does not; compounded for five years it would overshoot the true result, which is exactly the gap this calculator draws out on the chart.
Why the arithmetic mean lies
The clearest way to see the gap is the smallest possible example: a fund that gains 50% one year and loses 50% the next. Add those and divide by two and the arithmetic mean is a flat 0%, which reads as "you broke even". Multiply the growth factors instead, 1.5 times 0.5 is 0.75, whose square root minus 1 is about −13.4%. Ten thousand pounds put through that exact sequence becomes $7,500, a real loss of $2,500, not the break-even the arithmetic mean implied. The geometric mean's −13.4% describes that outcome correctly; the arithmetic mean's 0% describes an outcome that never happened.
The gap between the two means has a name, volatility drag, and a well-used approximation: geometric mean ≈ arithmetic mean − variance ÷ 2, where variance is worked out from the same returns expressed as decimals. For the 50%/−50% pair, the variance of 0.5 and −0.5 around a mean of 0 is 0.25, so the approximation predicts a drag of about 12.5 percentage points, close to, though not exactly, the true 13.4-point gap; the shortfall is the approximation cutting a corner that only matters once swings get this large, not a different formula. The pattern holds generally: the more a series of returns swings up and down, the further its arithmetic mean strays from what actually happened, even when the true average, the geometric mean, is unchanged.
Negative returns, and the −100% wipeout
Ordinary negative years work through the same multiplication as positive ones, a −15% year is simply a growth factor of 0.85, no different in kind from a 5% year's 1.05. A year of exactly −100% is the one case that behaves differently, because it turns the growth factor to zero, and zero multiplied by anything, any later year's factor included, stays zero. This calculator treats that honestly: once a wipeout appears anywhere in the series, the running value collapses to zero from that year on, the geometric mean of the whole series comes out as a genuine −100%, and no later gain, however large, can undo it. That is arithmetic, not pessimism; a total loss really does end the compounding story for whatever was lost. A return cannot go below −100% without borrowed money in play, which is why the input here is capped at that floor.
From start and end value: the CAGR shortcut
Sometimes the year-by-year returns are not to hand, only a start value, an end value, and how many years separated them, a statement, savings account or portfolio valuation two dates apart. The compound annual growth rate answers exactly the same question as the geometric mean above, just from different raw material.
Ten thousand growing to twenty-five thousand over seven years works out as (25,000÷10,000)1/7 − 1, which is 2.5 raised to the power one-seventh, minus 1, about 13.98% a year. That is the single steady annual rate which, compounded for seven years, turns ten thousand into twenty-five thousand exactly. This shortcut assumes nothing else moved in or out along the way; it is the same arithmetic the year-by-year tab performs, just collapsed into one jump instead of several, so it needs a start value above zero and at least a fraction of a year to divide by.
Total return and annualised return are different questions
Ten thousand becoming twenty-five thousand is a total return of 150%, the whole journey's result in one number, however long it took. CAGR spreads that same 150% evenly across every year held, compounding rather than simply dividing seven into it, which is why 150% over seven years becomes roughly 13.98% a year rather than a plain 21.4%. Quoting the 150% figure next to another investment's one-year return, or next to a return earned over a different number of years, compares two different things dressed up as one; annualising both first is what makes the comparison fair, and it is the reason fund factsheets almost always quote an annualised, geometric figure rather than a raw multi-year total.
Does the order of returns matter?
For the arithmetic mean, the geometric mean and the final value alone, no. Multiplication does not care what order its factors arrive in, so 12%, then −8%, then 23% ends at exactly the same value as −8%, then 23%, then 12%; only the path drawn on the chart changes shape, not its destination. That stops being true the instant money is added or withdrawn partway through. Once a deposit lands right before a bad year, or a withdrawal happens right before a good one, the order in which returns and cash flows land changes the actual pound-and-pence outcome, not just the picture, a problem known as sequence-of-returns risk. This calculator's two tabs assume a single lump sum with no cash flows in between; the moment that stops being true, reach for the IRR Calculator, which is built to weigh cash flows by when they actually happened.
Choosing your inputs and reading the results
For the yearly-returns tab, list each year's percentage change in order; the calculator reads off the arithmetic mean, the geometric mean, the volatility drag between them, and what ten thousand of your chosen currency would have become. For the start-and-end tab, enter the two values and the years between them as a statement would show them; the CAGR returned is the fair, annualised figure to compare against another investment's own annualised rate or against inflation. Either way, the geometric mean or CAGR matches what actually happened to the money; treat the arithmetic mean as a caution sign, not the answer itself.
Questions people ask
Why is the geometric mean always lower than the arithmetic mean?
Whenever returns vary at all, the geometric mean sits at or below the arithmetic mean of the same numbers, a mathematical fact for any set of non-identical positive figures. The two are only equal when every return in the series is exactly the same, in which case there is nothing for volatility to drag down.
Which one should I actually use?
The geometric mean, whenever the question is what your money actually did over more than one period. The arithmetic mean has its own honest uses, such as estimating a single future period's expected return in some statistical models, but as a description of realised, compounding performance across several years it overstates the outcome, sometimes badly.
What does "volatility drag" mean?
It is the gap the arithmetic mean hides: the more a series of returns swings up and down, the further its arithmetic mean pulls away from the geometric mean that matches the real outcome, even when the true average return has not changed. The rough approximation is geometric mean ≈ arithmetic mean minus variance divided by two, worked out on this page from your own entered returns.
Can the geometric mean be calculated if a return is exactly −100%?
Yes, and honestly: a −100% year turns that year's growth factor to zero, and the running value stays at zero for every year after it, no matter how large a later gain is entered. The geometric mean of the whole series comes out as a genuine −100% in that case, which correctly reflects that nothing is left to compound from.
Does the order I enter my yearly returns in change the answer?
Not for the arithmetic mean, the geometric mean, or the final value, since multiplying and adding numbers gives the same result whatever order they arrive in. It only changes the shape of the growth chart. Order starts to matter for the actual outcome only once money is added or withdrawn between the returns, which this calculator does not model; that is sequence-of-returns risk, and the IRR Calculator is the tool built for it.
How is this different from the CAGR shown on the ROI Calculator?
The same formula, reached from a different starting point. The ROI Calculator annualises a single invested-and-returned figure over a holding period; the start-and-end tab here does the identical CAGR arithmetic. The yearly-returns tab goes further, working from an actual list of annual returns to show the arithmetic and geometric means side by side and where the gap between them comes from.
Are these figures adjusted for inflation or fees?
No. Every figure here is a nominal return in whatever currency you selected, before fees and before inflation, unless you have already built those into the numbers you entered. Subtracting inflation the exact way needs the Fisher equation, (1 plus nominal) equals (1 plus real) times (1 plus inflation), or roughly nominal minus inflation as a shortcut for small rates; the Inflation Calculator runs that working in full.
This calculator is an arithmetic and educational tool. It is informational only, not financial or investment advice, and none of its figures are a forecast, a recommendation, or a guarantee of any future return.