Arithmetic mean per period
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Free Tool
Compound Returns Calculator
Compounding is geometric, and your period returns are a volatile sequence, not a flat rate. This tool projects the numbers you type, then isolates the one figure almost every other calculator hides: volatility drag, the gap between the average return you assume and the compounded return you actually keep. Enter a return per period and a per-period variation, and it shows the arithmetic mean beside the realised geometric return, side by side.
The account compounds geometrically while your profit and loss swings arithmetically, and the difference between the two is a tax called volatility. Compounding rewards the boring.
This is arithmetic, not a forecast. This projects the numbers you type. It is arithmetic, not a forecast, and no return is implied or achievable by using it. The return per period is your assumption, not a rate this or any tool can deliver. The teaching point is the mechanism of compounding and, above all, volatility drag, not how large a number you could reach.
Scenario
Teaching presets
Two of these presets carry the identical average return and differ only in volatility, so the endpoint gap you see is volatility drag and nothing else. None is a return you should expect to earn.
Your assumptions
Any figure. The tool works in multiples, so the starting number does not change the drag.
Your assumed average per period. This is an input, not a return the tool can produce.
Years, months, or trades, whatever your return per period is measured in.
Volatility and cash flow
Per-period variation around the average. The engine of drag. Set 0 for a smooth path; raise it to watch the geometric return fall below the average.
Optional. Once you add cash, the order of returns starts to matter.
Shows sequence-of-returns risk. Same set of returns, reordered.
Ending value, smooth at the average
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Ending value, with your volatility
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Arithmetic mean return
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Realised geometric return (CAGR)
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Volatility drag (the number most tools hide)
Geometric per period (kept)
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Drag per period
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Ending value lost to drag
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Two paths, one average return
Both curves carry the same arithmetic mean return you entered. The smooth one has no volatility; the jagged one has yours. They start together and the calmer path ends higher. The distance between the endpoints is volatility drag.
Milestones along the path
| Period | Smooth (₹) | With volatility (₹) |
|---|
Where the ending value comes from
| Component | ₹ |
|---|---|
| Ending value (with volatility) | − |
Illustrative projection of your inputs only. Contributions are the cash you add; the rest is compounding on your assumed return, reduced by the drag your volatility creates. Not a prediction.
Flags to read before trusting the ending number
The arithmetic here is trivial. The hard part is refusing to extrapolate a good year into a decade, and building a process that keeps the losing periods small so the geometric return stays close to the average. That discipline, sizing for survival and cutting the left tail, is what the method we teach is built around, because the geometric return is the only one that compounds.
The one principle
Compounding is geometric, and your period returns are a volatile sequence, not a constant rate. The number you compound at is not the average of your returns; it is lower, and the gap is volatility drag. Because a loss removes more than the same-sized gain adds back, larger swings mean a larger shortfall between the return you quote and the return you keep. That is why, for building an account, avoiding large losses matters more than chasing large gains: cutting the size of the drawdowns raises the geometric return even when the average is unchanged.
This is the highest-stakes arithmetic a trader ignores. The SEBI FY25 finding that over 91 percent of individual derivatives traders were net loss-making, with aggregate net losses near 1,05,603 crore rupees, is not only a story about being on the wrong side of trades. It is also a story about volatility: outsized position sizes turn ordinary swings into account-ending drawdowns, and a drawdown does not just cost its own percent, it resets the base that every future return has to rebuild from.
The math, derived
Straight compounding of a lump sum at a constant rate is the familiar future-value formula. Add an equal contribution each period and you add the future value of an annuity. Both assume a single fixed rate:
FV = P × (1 + r)n
with contributions C each period:
FV = P × (1 + r)n + C × [ (1 + r)n − 1 ] ÷ r
P principal, r return per period, n periods, C addition per period
with contributions C each period:
FV = P × (1 + r)n + C × [ (1 + r)n − 1 ] ÷ r
P principal, r return per period, n periods, C addition per period
The trouble is the phrase, a single fixed rate. Real returns are a sequence, and once they vary, no single closed form describes the path, so this tool steps through it period by period. The teaching computation is the relationship between the two ways of averaging that sequence:
arithmetic mean = (r₁ + r₂ + ... + rₙ) ÷ n
geometric mean = [ (1+r₁)(1+r₂)...(1+rₙ) ]1/n − 1
the geometric mean is the CAGR: the one constant rate giving the same endpoint
geometric ≈ arithmetic − (variance ÷ 2)
geometric mean = [ (1+r₁)(1+r₂)...(1+rₙ) ]1/n − 1
the geometric mean is the CAGR: the one constant rate giving the same endpoint
geometric ≈ arithmetic − (variance ÷ 2)
That last line is the whole lesson in one relationship. The compounded return you keep is approximately the average return minus half the variance of your returns. Drag scales with the square of volatility, so doubling the size of your swings roughly quadruples the tax. The approximation is a second-order expansion of the log return; the tool reports the exact geometric mean from the actual path, and the exact figure and the approximation agree closely at moderate volatility and diverge only when swings are extreme.
The proof in four numbers. Start at 1.00. Up 50 percent gives 1.50. Down 50 percent gives 0.75. The arithmetic mean of plus 50 and minus 50 is exactly zero, so a naive reading expects to break even, yet you have lost 25 percent. The geometric return per period is the square root of 0.75 minus 1, about minus 13.4 percent. Zero average, minus 25 percent kept. That is not a rounding effect or a fee; it is the shape of compounding, and it is why volatility is not free.
Volatility drag, drawn
Set two accounts to the identical average return and let one be smooth while the other swings. They do not end level. The volatile account spends part of every recovery just climbing back to where it was, and that repeated backtracking is money that never compounds. The wider the swings, the wider the final gap.
The gap widens with volatility
Drag is not linear in volatility, it is quadratic. A little dispersion costs almost nothing; a lot costs a great deal. Hold the arithmetic mean fixed and raise only the size of the swings, and the realised geometric return bends away from the average, slowly at first and then sharply.
Reference: what a fixed average keeps at rising volatility
The table holds the arithmetic mean per period at a constant 10 percent and raises only the volatility, then reports the realised geometric return and, over 20 periods, the fraction of the smooth ending value that survives. Read it as the price of dispersion at a fixed average, not as a return anyone should expect.
| Volatility per period | Arithmetic mean | Realised geometric return | Drag | Kept after 20 periods |
|---|---|---|---|---|
| ±0% | 10.0% | 10.00% | 0.00% | 100% |
| ±10% | 10.0% | 9.54% | 0.46% | 92% |
| ±20% | 10.0% | 8.17% | 1.83% | 71% |
| ±30% | 10.0% | 5.83% | 4.17% | 46% |
| ±40% | 10.0% | 2.47% | 7.53% | 24% |
| ±50% | 10.0% | −2.02% | 12.02% | 10% |
Notice the last row. At plus or minus 50 percent swings around a 10 percent average, the compounded return has gone negative: the account loses money over time despite a positive average return every period. The approximation predicts the shape well, half of a 0.50 variance is 0.125, close to the drag you see, and the exact figures above come from compounding the real path, not the approximation.
Reference: the loss-recovery asymmetry that drives the drag
Volatility drag is drawdown asymmetry applied repeatedly. A loss and its recovery are not mirror images: the gain needed to get back to even is always larger than the loss that put you there, and it grows without bound as losses deepen. This is why a single large drawdown is so much more expensive than its headline percent, and why keeping the left tail small is the highest-leverage thing a trader can do for the geometric return.
| Drawdown suffered | Gain needed to recover | What it means for compounding |
|---|---|---|
| −10% | +11.1% | A mild dent, quickly repaired. Drag stays small. |
| −20% | +25.0% | The recovery is already a quarter above the loss. |
| −33% | +49.3% | Half your capital's next job is just getting back to flat. |
| −50% | +100.0% | You must double the survivors to undo one halving. |
| −75% | +300.0% | A near-terminal hole; the geometric return may never recover. |
| −90% | +900.0% | Effectively unrecoverable within a normal horizon. |
For any drawdown figure of your own, the companion drawdown recovery calculator returns the exact gain required, and the risk of ruin calculator shows how a fixed bet size makes deep drawdowns more or less likely over a run of trades.
Failure modes: where the clean number still misleads
A compounding projection is a clean line on a chart, and its cleanliness is the danger. Five habits turn the ending figure into a story that will not survive contact with a real account.
1. Extrapolating a past return into the future. The most common error is taking a recent or long-run average and compounding it forward as if it were fixed and owed to you. It is neither. Past averages are drawn from a particular sequence of conditions that will not repeat, and dispersion around them is wide. The tool compounds whatever you type precisely because it must not pretend to know the future; the ending number is a consequence of your assumption, not evidence for it.
2. Confusing the average return with the compounded return. Quoting the arithmetic mean and then compounding it is double counting the good years. The number your capital actually experiences is the geometric mean, the CAGR, and it is always lower for any path that is not flat. When a return is advertised, the first question is whether it is the average, which flatters, or the compounded figure, which is what you keep.
3. Ignoring sequence-of-returns risk. For an untouched lump sum, order does not change the endpoint. The instant you add or withdraw money it changes it a great deal. Early weakness while you are still contributing is survivable and can even help; early weakness while you are drawing down can be fatal, because you liquidate more units at low prices. A projection that assumes a smooth path hides this entirely, which is why the tool lets you cluster weak returns early or late and watch the endpoint move.
4. Survivorship in the return assumption. The averages that make it into headlines and marketing are the ones that survived. Strategies and funds that blew up leave the sample, so the surviving average overstates what a fresh participant should expect. Feeding a survivor's return into a compounding tool bakes that upward bias into every year of the projection.
5. Treating volatility as free because the average looks fine. Two paths with the same average are not worth the same, and the higher-volatility one is worth strictly less to a compounder. Chasing a higher average by accepting much larger swings can lower the geometric return even as the arithmetic mean rises. The account is built by the geometric return, so the swings are not a cosmetic detail, they are a direct deduction from what you keep.
Common Questions
Frequently Asked Questions
What is volatility drag and why does it lower my compound return?
+Volatility drag is the gap between the arithmetic mean of your period returns and the geometric return you actually compound at. Compounding multiplies returns, and a percent loss removes more value than the same percent gain adds back: down 10 percent then up 10 percent leaves 0.99, not 1.00. Over a volatile sequence those asymmetries accumulate, so the realised geometric return sits below the simple average. A useful approximation is geometric is about arithmetic minus half the variance of returns, which means drag grows with the square of volatility. Two strategies with the identical average return keep different amounts of money if their volatility differs, and the calmer one keeps more.
How does the plus fifty percent then minus fifty percent example work?
+Start with 1.00. A plus 50 percent period takes it to 1.50. A minus 50 percent period takes 1.50 to 0.75. The arithmetic mean of plus 50 and minus 50 is zero, so a naive reading expects to break even, but the account is down 25 percent. That gap is volatility drag in its starkest form. The order does not change the endpoint here: minus 50 then plus 50 is 0.50 then 0.75, the same 0.75. The lesson is that the average return you quote is not the return you compound, and the larger the swings, the wider that difference becomes.
What is the difference between arithmetic mean return and CAGR?
+The arithmetic mean return is the simple average of each period's return, added up and divided by the number of periods. CAGR, the compound annual growth rate, is the single constant rate that turns the starting value into the ending value over the whole horizon: it is the geometric mean of the growth factors. For any sequence that is not perfectly flat, CAGR is lower than the arithmetic mean, and the gap is set by volatility. When someone advertises an average return, ask whether it is the arithmetic average, which flatters, or the compounded CAGR, which is what your capital actually experienced.
Does this calculator predict how much my money will grow?
+No. It projects the numbers you type and nothing else. It is arithmetic on your own assumptions, not a forecast, and no return is implied or achievable by using it. If you enter a return per period, the tool compounds exactly that assumption; it has no view on whether the assumption is realistic and cannot know what any market will do. The value of the tool is not the ending figure. It is the mechanism it exposes: how volatility quietly lowers the return you keep below the average you assumed, which is a fact of arithmetic that holds whatever numbers you choose.
Why does avoiding large losses matter more than chasing large gains?
+Because loss and recovery are not symmetric, and compounding punishes the downside. A 50 percent loss needs a 100 percent gain to recover; a 75 percent loss needs a 300 percent gain. A single large drawdown resets the base that every future return compounds on, so it costs far more than its headline percent. Volatility drag is the continuous version of the same effect: bigger swings mean a bigger gap between your average return and your compounded return. This is why a risk manager treats capital preservation as the first job. Cutting the size of losses raises the geometric return even if the average return is unchanged, and the geometric return is the one that builds the account.
What is sequence-of-returns risk?
+For a lump sum left untouched, the order of returns does not change the ending value: the same set of factors multiplies to the same result whatever the order. The moment you add or withdraw money, order starts to matter a great deal. Early losses on a portfolio you are still contributing to are less damaging, because later contributions buy in lower; early losses on a portfolio you are drawing down from can be terminal, because you sell more units at low prices and the account may never recover. This calculator lets you add a periodic contribution and toggle whether losses cluster early or late, so you can see the endpoint move even though the set of returns is identical.
Are the historical return figures on this page a promise of future returns?
+No. Any index or historical figure shown is context for how large the arithmetic-to-geometric gap has been in the past, not a projection. Past dispersion does not fix future dispersion, and extrapolating any past average forward is exactly the error the page warns against. The tool deliberately ships with no default that reads like an expected outcome; the preset scenarios exist to contrast a steady path with a volatile one at the same average, so the teaching point is the shape of the path, never a level of return you should expect to earn.
How is compound interest calculated with regular contributions?
+The starting capital grows by the future-value factor, principal times one plus the periodic rate raised to the number of periods. A stream of equal contributions is the future value of an annuity: each contribution compounds for the periods remaining after it is added, and the closed form is contribution times the quantity, one plus the rate to the number of periods minus one, divided by the rate. The two terms are added. When the rate varies period to period, as it does with any volatile return, there is no single closed form, so this tool steps through the sequence period by period, applying each period's return and adding the contribution, which is also how it can show volatility drag rather than assuming it away.
Where the facts come from
Sources
- The volatility-drag relationship. The compounded (geometric) return is approximately the arithmetic mean return minus half the variance of returns, a second-order expansion of the log return, equivalent to the drift correction in geometric Brownian motion. Standard treatment in the geometric-mean and variance-drain literature. kitces.com and bogleheads.org
- The FY25 loss base rate. SEBI study on the profit and loss of individual traders in the equity derivatives segment: over 91 percent net loss-making in FY25, with aggregate net losses of about 1,05,603 crore rupees, up roughly 41 percent from about 74,812 crore in FY24, across the top 13 brokers and around 96 lakh unique traders. sebi.gov.in
- Loss-recovery asymmetry. The gain required to recover a drawdown is one divided by one minus the loss fraction, minus one, an exact algebraic identity: a 50 percent loss needs a 100 percent gain, a 75 percent loss needs 300 percent. Reproduced on the companion drawdown-recovery tool.
Educational note. This tool computes figures from your own inputs; every output is an illustrative projection that depends entirely on the numbers you enter. It is arithmetic, not a forecast, and no return is implied or achievable by using it. The reference tables are mathematical models, not predictions of any strategy or account. Nothing here is a recommendation to trade or to buy or sell any security, and it is not investment advice. Bharath Shiksha is an educational publisher, not a SEBI-registered investment adviser or research analyst.