Monte Carlo simulation — the range of outcomes, not a single line

Two thousand independent paths are generated month by month, each month drawing a random return from a log-normal distribution. Your annual figures are converted to monthly ones geometrically rather than by division: the monthly drift is the natural log of one plus the annual return, divided by twelve, which for 7% a year is 0.564% a month and not the 0.583% that dividing would give. Monthly volatility is the annual figure divided by the square root of twelve, because variance grows with time and standard deviation with its square root. The paths are then sorted at each year to produce the percentile bands.

The expected return you enter is treated as the MEDIAN outcome, not the average. Because compounding is right-skewed the mean path sits higher than the middle line, and median is what a reader actually means by "expected". Returns are drawn independently each month, so the model contains no momentum, no mean reversion and no crash that clusters — real markets have all three, which makes the fan too narrow at the extremes rather than too wide. Volatility and return are fixed for the whole horizon, and nothing is deducted for cost or tax. The random draw is seeded from your inputs, so the same assumptions always redraw the same fan.

Why does the chart not change when I re-render the page?

Because the randomness is seeded from the inputs rather than taken from the clock. An unseeded simulation reshuffles every time anything on the page changes, so the fan flickers and two readings of identical assumptions disagree with each other — which makes the tool useless for comparing two scenarios. Change an input and the fan changes; leave the inputs alone and it is stable.

Is the median line the outcome I should plan for?

Plan for the lower bands. Half of all outcomes fall below the median by definition, and a plan that only works at the median is a plan that fails half the time. The tenth percentile is the more useful line for anything with a deadline attached, because it is roughly the bad-but-not-catastrophic case, and a plan that survives there survives most of what markets have historically done.

Why is it too optimistic about really bad outcomes?

Because independent monthly draws cannot produce a crash that lasts. Real market falls cluster — bad months arrive next to other bad months — and returns have fatter tails than a normal distribution allows. The consequence is that the worst paths here are milder and shorter than the worst paths history actually delivered, so treat the bottom of the fan as a floor on optimism rather than a worst case.

• Sequence of returns risk — the same returns in a different order
• Portfolio backtest — what a mix actually returned
• FIRE calculator with real broker costs