Retirement Probability Calculator: Beyond Monte Carlo

How this compares to a Monte Carlo retirement calculator

A Monte Carlo retirement calculator rolls the dice thousands of times: it draws random return sequences from a model and counts how many survive. This calculator makes the same modelling assumption but skips the dice, computing the exact probability the simulation is trying to estimate. This is Monte Carlo at infinite trials.

That claim is checkable. For the classic 4% rule over 30 years, all stocks, the exact computation puts the risk of running out at 7.8% under a normal return model. A 200,000-path Monte Carlo run on the identical model lands at 7.7%, give or take 0.12 percentage points of sampling noise at 95% confidence. The exact answer sits inside the simulation’s error band, as it should, but it needs no error band of its own: there is no seed, no run-to-run wobble, and the answer never changes between visits.

The second difference is the distribution itself, and here it is an input rather than a fixed choice. Most Monte Carlo retirement tools draw from a normal (or lognormal) curve, which is where this page starts too. One toggle away is the historical distribution with its heavy left tail intact, which for the same scenario moves the measured risk from 7.8% to 8.5%. The cost of the bell-curve simplification is a number on the page rather than a buried assumption.

How the exact calculation works

The calculator tracks the entire probability distribution of your balance, not sample paths. Each simulated year has two steps. First the whole distribution is multiplied by the year’s uncertain return, which in logarithmic space is a convolution of the wealth distribution with the return distribution. Then the year’s spending is subtracted, and any probability mass that hits zero is moved into a permanent “ran out of money” bucket. Repeat for every year of the horizon and the bucket’s final weight is the exact probability of failure; what remains is the exact distribution of how much is left.

Working with probability mass rather than curve heights is what keeps the arithmetic honest: mass is conserved at every step (the engine verifies this to nine decimal places each year), so nothing is lost or double-counted in the transformations. The same machinery accepts any return distribution. It can keep the historical shape and shift its level (useful when a projection needs “history’s shape, but a lower expected return”), collapse to a single fixed return, which reproduces a deterministic projection exactly, or use any fitted curve. The two options exposed above, historical and normal fit, are the ones this page needs.

What the fat left tail changes, and what it does not

The fat left tail multiplies the chance of a single catastrophic year roughly fourfold against the bell curve, and it moves the 30-year chance of depletion by 0.7 percentage points. Historical annual real stock returns are not normally distributed: over 1871–2024, 4 of 154 years finished at or below −30% in real terms, a 2.6% frequency, where the fitted normal curve predicts 0.7%. The distribution’s skewness is -0.65 and its excess kurtosis +0.53, against zero of both for a normal distribution.

At the 4% rule over 30 years, the historical distribution puts the chance of depletion at 8.5% and the normal fit puts it at 7.8%. That is 0.7 percentage points, against roughly fourfold on a single year. Depletion depends on all thirty years together rather than on any one of them, and the normal fit is built from the same 154 years, matching their average yearly growth and their year-to-year spread. What separates the two is the shape of the extremes.

Both fits are approximations, and they fail in different places. Out at −30% the historical shape is carrying 4 data points: had one more of those 154 years landed there, the observed frequency would read 3.2% instead of 2.6%; had one fewer, 1.9%. The normal fit gets those same years wrong, and wrong in a known direction, but it draws its shape from all 154 rather than from the 4 in the tail. Since the whole choice is worth 0.7 percentage points on the 30-year answer, this page opens on the fit that a single year barely moves, and the return distribution toggle hands back the historical shape. If the odds hold either way, the tail’s shape is not what the plan depends on.

What this calculator assumes

A probability invites more trust than a point estimate, so it carries more obligation to say what it rests on. These are all of the model’s assumptions, each with its honest limitation attached.

• Each year is an independent draw. The model treats every year as a fresh draw from the return distribution, with no memory of the year before. This is the biggest assumption in the model, and the data neither rejects nor proves it; the independence section below shows the measurements.
• US markets, 1871–2024. The distribution is estimated from a single country over a single stretch of history, and the US was the best-performing major stock market of the 20th century largely in hindsight. A reader in any other market, or a future US reader, has no guarantee of drawing from the same distribution. This is the assumption most likely to be materially wrong, and no amount of exact computation repairs it.
• Bond-heavy mixes: the bands are likely slightly too narrow. Real bond returns inherit inflation’s streakiness (inflation is strongly autocorrelated, and deflating a smooth bond return by a persistent series injects that persistence). Stock-heavy and mixed portfolios measure clean, but at high bond weights true long-run variation is somewhat wider than independent draws imply, so the risk shown there errs on the low side.
• Everything is in today’s money. Returns are inflation-adjusted via the CPI series, and spending stays constant in real terms. There is no separate inflation guess to get wrong; actual historical inflation is already inside the real returns.
• One year, one step. A year is a single draw; there is no within-year sequencing. Someone who retires into a crash mid-year experiences a path the annual model cannot distinguish.
• Annual rebalancing to the target mix. Applying a weighted return to the whole balance quietly assumes the portfolio is rebalanced back to the chosen stock/bond split every year, the same convention the historical backtest uses. It is a real assumption, not a detail.
• Withdrawals are fixed in real terms. The model spends the same inflation-adjusted amount every year no matter what markets do. Real retirees usually cut spending after bad years, which is exactly why constant-spending ruin numbers overstate risk. The historical simulator on this site models a dozen dynamic withdrawal strategies for that question.
• No taxes, fees, or transaction costs. Index returns are used as-is. Fees reduce returns, so the model is optimistic by roughly the total cost drag on your portfolio.
• The horizon is an input, not an estimate of a lifetime. The number of years entered is a planning choice. The model runs to the end of it and stops, and nothing in it estimates or consumes a life expectancy: a horizon set too short simply ends the calculation early, and the odds shown are for surviving the years asked for and no others. This site ships period life tables by country and sex as a reference for choosing that number, described on the methodology page. This calculator does not read them.
• Failure means the balance reaches zero within the horizon. Depletion is an absorbing state: once the money is gone, it stays gone. The model does not simulate the person noticing trouble in year 20 and adjusting, which is one more reason the raw number reads pessimistic for anyone willing to adapt.
• The distribution is fitted, not the history replayed. Outcomes are computed from a distribution estimated over all 154 years, in a choice of two fits: the normal fit, the default here, which smooths the returns into the bell curve most Monte Carlo tools assume, or the historical fit, which keeps the fat left tail that actually occurred. The fat-tails section explains what the choice is worth in numbers, and why the smooth fit is the starting point.

For the fixed-withdrawal assumption in particular, the historical simulator’s dynamic withdrawal strategies show how much adapting your spending changes the picture.

Are market returns really independent from year to year?

As far as this dataset can tell, yes: the data does not reject independence, and the model’s biggest assumption survives its most direct test. The correlation between one year’s real stock return and the next, measured over 153 year-pairs, is 0.009, which is indistinguishable from zero.

A sharper test: shuffle the 154 observed years into 20,000 random orderings. Shuffling keeps the distribution exactly (same mean, same variance, same worst year) and destroys only the ordering, so it is precisely the independent-years hypothesis with the real returns. The actual historical ordering produced a 1.6% failure rate for the 4% rule in rolling 30-year windows; the shuffled orderings average 7% but range so widely that 21% of them did as well as or better than history (p = 0.207). History’s good result needs no special market memory to explain it; luck of the draw is sufficient.

One measurement leans the other way and is reported as an open question, not a conclusion: a 30-year variance ratio drifts below one, hinting long horizons might be tamer than independence implies. But any 30-year statistic in this dataset rests on about five non-overlapping windows, too few to conclude anything. The full table of measurements, and why the short-lag tests are the trustworthy ones, is on the methodology page.

Why this model and the historical backtest disagree

They answer different questions, and showing both is more honest than picking one. The historical backtest replays every actual retirement start year and reports how many survived: a true story about specific sequences, but statistically thin, because overlapping windows share almost all of their years, and a century and a half contains only about five non-overlapping 30-year retirements. The probability model instead weighs every sequence the return distribution could produce, including bad decades that never happened to land in the record. When you run a scenario above, the results panel shows both engines’ answers for it side by side. How much of the gap is method depends on which fit is selected: on the historical fit both are working from the same shape, so what is left between them is the difference between “what happened” and “what could have happened”. On the normal fit the model is also smoothing the tail the backtest replays, and that smoothing is part of the gap too.