Monte Carlo Simulation, Explained
Updated ·2 min read·Reviewed by the StockTools.ai Research Team
- ▸A Monte Carlo simulation runs thousands of randomized versions of the future instead of one straight line.
- ▸It gives you a range — a 10th, 50th, and 90th percentile — plus the probability of hitting a goal, not one false-precision number.
- ▸Two portfolios with the same average return can end decades apart because of the order of good and bad years.
- ▸It is a study of ranges and sequence risk, not a forecast — a normal-returns model even understates rare crashes.
Why one number lies to you
A standard retirement or savings calculator assumes your money earns the exact same return every single year — say a flat 7%. Markets never behave that way. The straight line it draws is not the average outcome so much as a fantasy that will almost certainly not happen, and it hides the thing you most need to see: the spread of what could happen.
A Monte Carlo simulation replaces that single line with thousands of possible futures. Each one draws a different random sequence of yearly returns from your assumptions, compounds it, and records where you ended up. Instead of "you will have $X," you get a distribution: a pessimistic 10th-percentile outcome, a median, an optimistic 90th, and the odds of clearing a specific target.
Run your own simulation
Each path draws a fresh sequence of yearly returns from a normal distribution with your mean and volatility, compounds the balance, then applies your contribution: balanceₜ = balanceₜ₋₁ × (1 + r) + contribution, where r = mean + volatility × z. Reproducible seed: 12345. A normal model understates rare crashes and sequence risk — treat this as a study of ranges, not a forecast. Educational only, not financial advice.
How it works
You supply the inputs — a starting balance, contributions or withdrawals, an expected return, a volatility (how much returns swing year to year), and a time horizon. The simulation then plays out thousands of independent "lives" of your portfolio, each year drawing a return at random from a bell curve centered on your expected return with the width set by your volatility.
The result is usually shown as a percentile "cone" that widens over time — because uncertainty compounds. The middle line is the median path; the shaded band is the range between the unlucky and lucky outcomes. The wider that band, the more your plan depends on luck, which is exactly the risk a single-number projection conceals.
What it is good for — and its limits
Monte Carlo shines at questions of range and probability: "what is the chance I reach $1M?", "how much cushion do I have if returns disappoint?", "does my withdrawal plan survive a bad decade?" It is the engine behind stress-testing the 4% rule, and it makes sequence-of-returns risk visible in a way no straight-line tool can.
Its main limitation is honest to state: a normal (bell-curve) model understates the frequency of extreme crashes — real markets have fatter tails than the math assumes. So treat the probabilities as directional, not exact, and never as a guarantee. A 90% success rate means roughly one in ten simulated futures still failed.
FAQ
How many simulations are enough?
A few thousand paths is plenty for stable percentile estimates; going from 5,000 to 10,000 barely changes the answer. More paths reduce random noise in the output but do not make the underlying assumptions any more accurate.
What return and volatility should I use?
They are your assumptions. As rough reference points, a broad US stock index has historically returned around 10% a year (about 7% after inflation) with roughly 15–20% annual volatility; a 60/40 stock-bond mix is lower on both. Try a range of inputs and see how sensitive your plan is.
Is Monte Carlo better than the 4% rule?
They complement each other. The 4% rule is a simple heuristic from historical data; Monte Carlo lets you stress-test any withdrawal rate against your own assumptions and see the probability of success. Use the rule as a starting point and the simulation to pressure-test it.
Why does the range get wider over time?
Because uncertainty compounds. A good or bad year early on gets multiplied by all the years after it, so small differences fan out into large ones over decades. That widening cone is the visual signature of long-horizon risk.
Put it to work
Related guides
Sources & further reading
- ▸ Metropolis & Ulam (1949). The Monte Carlo Method. Journal of the American Statistical Association.
- ▸ Historical US equity return and volatility reference points: broad-market index data, ~10% nominal / ~7% real long-run average.
More to learn
Educational only — not financial advice. Concepts simplified for clarity; markets are messier than definitions.