Compound Growth
An arithmetic (simple) average return overstates what you actually earn once volatility and compounding are involved. This is why the site's own Market Return input is already the median (geometric) figure, not the naive average.
All figures are in today's money: every rate on this page is real, i.e. above inflation. Why?
Assumed Distribution
The "naive" average of a return series: what most people assume they'll earn.
Your Own Return Sequence
Enter a year-by-year return sequence and see both means computed from it directly. The default is the classic +50% / -50% demonstration.
| Metric | Value |
|---|---|
| Geometric (Compounded) Return what you actually realize per year | 6.79% |
| Volatility Drag gap between the two rates | -1.21% |
| Arithmetic Path Final Value the naive expectation | €466,096 |
| Geometric Path Final Value what actually compounds | €372,185 |
Arithmetic vs Geometric Path
Same starting pot, compounded at the naive arithmetic rate versus the actual (volatility-adjusted) geometric rate, over the horizon.
A Random Return Sequence
One seeded random walk drawn from the site's global Market Return (median 5%/yr) and Volatility (15%/yr) settings, one draw per horizon year. Reroll in the top bar draws a new path. This is a DIFFERENT anchor from the arithmetic-mean input above: Market Return calibrates the median outcome (the site-wide convention every other calculator's market band uses), while the chart above demonstrates what a stated arithmetic mean implies once volatility drags it down to its geometric (compounded) equivalent.
Geometric: (€1,038,445 ÷ €100,000)^(1/20) − 1 = 12.4%
Gap: 1.1% realized vs 1.13% theoretical (σ² ÷ 2). Expect these to be close but rarely identical, since one draw is one sample.
| Year | Drawn Return | Resulting Pot |
|---|---|---|
| 1 | 12.2% | €112,176 |
| 2 | 32.9% | €149,047 |
| 3 | 38.4% | €206,210 |
| 4 | 19.8% | €246,975 |
| 5 | 16.2% | €286,892 |
| 6 | -5.1% | €272,167 |
| 7 | -0.8% | €269,884 |
| 8 | 25.0% | €337,287 |
| 9 | 27.9% | €431,347 |
| 10 | 20.9% | €521,557 |
| 11 | 26.7% | €660,635 |
| 12 | 25.3% | €827,910 |
| 13 | 29.3% | €1,070,253 |
| 14 | -6.7% | €998,907 |
| 15 | 13.1% | €1,129,662 |
| 16 | 0.2% | €1,131,460 |
| 17 | -7.6% | €1,045,930 |
| 18 | 23.8% | €1,294,631 |
| 19 | -8.4% | €1,186,161 |
| 20 | -12.5% | €1,038,445 |
- Volatility drag: an arithmetic mean of 8%/yr with 15% volatility actually compounds at 6.79%/yr. The higher the volatility, the bigger the gap, even when the arithmetic average never changes. For example, +50% then −50% averages 0% arithmetically, but a pot of 100 ends at 75.
- Why this matters for Market Return: elsewhere on this site "Market Return" is already treated as the median/geometric rate your pot actually compounds at, so it needs no further drag correction. This calculator shows what goes wrong if you instead type in a naive arithmetic average.
- Your own sequence: enter any year-by-year return series on the left and this calculator computes its simple (arithmetic) average and its true compound annual growth rate (geometric mean) directly from those numbers.
Geometric (compounded) return from a volatility-adjusted arithmetic mean
- g: geometric (actual, compounded) annual return: what you actually realise
- μ: arithmetic mean return, as a fraction (arithMeanPct ÷ 100)
- σ: log-volatility: std dev of yearly log returns ln(1 + r), as a fraction (volatilityPct ÷ 100)
Arithmetic (naive) path value at year y
A(y) = 100,000 × (1 + μ)^y
- A(y): pot value on the naive arithmetic path at year y
- P0: starting pot
- μ: see the return formula above
Geometric (actual) path value at year y
G(y) = 100,000 × (1 + g)^y
- G(y): pot value on the actual geometric path at year y
- P0: starting pot
- g: see the return formula above
Value lost to volatility drag at the horizon
gap = A(20) − G(20)
- gap: value lost to volatility drag by year H
- H: horizon, in years
- A(H): arithmetic path value at the horizon
- G(H): geometric path value at the horizon
Your own sequence: arithmetic vs geometric mean
- rᵢ: year i's entered return, %
- n: number of entered years
Assumptions behind every figure: how this site models the market →
Every model leaves things out. Here is what this one does not see:
- The arithmetic-to-geometric formula is a lognormal approximation driven by a single fixed volatility number; real return distributions have fat tails and skew this closed form does not capture.
- The "arithmetic path" line is a hypothetical no real investor experiences: nobody's pot grows at a smooth, unchanging rate every single year. It is shown only as the naive comparison point, not an achievable outcome.
- Each year in your own entered sequence is treated as a fixed, known number, not a random draw. This is a demonstration of the arithmetic-vs-geometric gap on data you supply, not a Monte-Carlo simulation of what other sequences might have produced.
- No fees, taxes, inflation, or contributions/withdrawals are modelled. This is pure return compounding, to isolate the volatility-drag effect from everything else.
- The random return sequence is one draw, not a distribution: a single sample of what could happen, not a range of outcomes. Rerolling changes the exact numbers every time but never the direction of the gap: the geometric mean is mathematically guaranteed to sit at or below the arithmetic mean of the same draw.