Monte Carlo simulation: how safe is your plan?

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A Monte Carlo simulation answers questions that a single backtest cannot: how likely is it that my money will last 30 years? What range of outcomes should I expect? Instead of one path, thousands of possible return sequences are generated at random and evaluated. The result is not a forecast, but a map of the possibilities under clearly stated assumptions.

How the simulation works

1. Define the returns: from a distribution (mean and volatility) or by drawing historical months (bootstrap).
2. For each path, apply the return, contributions and withdrawals month by month or year by year.
3. Calculate many paths (thousands) and evaluate them: success rate, median, percentiles, point at which the money runs out.

The name comes from the casino in Monaco: as in roulette, chance decides each individual path, but across many paths a reliable picture of the distribution emerges. In finance, Boyle (1977) was the first to use the method, to value options; today it is the standard tool for retirement planning and endowments. A comprehensive textbook on random numbers, variance reduction and error estimation is Glasserman (2003).

Example: 30 years of withdrawals

For simplicity, the example uses annual steps. Median final wealth is positive, yet more than a third of the paths fail. This range is exactly what you learn from the simulation: a plan that “works on average” can go wrong in many futures, especially if the first years are bad (sequence risk).

Reading the results

• Success rate: the share of paths in which the money lasts until the end. 80–90% is often considered solid; 100% is usually too cautious.
• Percentiles: the 10th percentile shows a bad but not extreme path, the 90th percentile a good one. The median (50%) is the typical path, not the average. With skewed distributions, the average is well above it.
• Fan: the distribution over time shows when things get tight.
• Nominal and real: final wealth of $200,000 in 30 years is only just over $110,000 in today's purchasing power at 2% inflation. For plans, the real view is what counts.
• When the plan fails: whether the money runs out at 84 or at 94 makes a big difference. The distribution of years until the money is used up says more than a single rate.

Which returns? Four methods

The bootstrap goes back to Efron (1979): you act as if the observed data were the whole distribution and draw from it again and again. For time series, ordinary drawing of individual months is problematic, because it chops up phases such as long bear markets. Kunsch (1989) therefore proposed drawing blocks of consecutive observations; Politis and Romano (1994) made the block length random (the stationary bootstrap). More on this in the article bootstrap.

That extreme returns are more frequent than the normal distribution allows (fat tails) was already shown by Mandelbrot (1963) using commodity prices. A simulation with a normal distribution therefore underestimates the probability of large losses. See skewness and kurtosis.

For advanced readers: precision and alternatives

How many paths? The success rate is itself an estimate. Its standard error is √(p × (1 − p) / N). With a success rate of 80%, the 95% range is ±1.8 percentage points with 2,000 paths and ±0.8 percentage points with 10,000 paths. More paths only make the result more precise, though, not more correct. The much larger error lies in the assumptions about return and volatility.

Parameter uncertainty: an expected return of 5% is itself only an estimate. If you vary it at random (for example by drawing a different return from a range for each path), you get wider fans and more honest success rates.

Analytical shortcut: Milevsky and Robinson (2000) calculate the probability that wealth with fixed withdrawals is used up with an approximation based on the lognormal distribution, without any simulation. This is suitable for quick rough estimates and for checking whether a simulation produces plausible values.

Common paths: if you compare several portfolios, you should simulate them with the same random months. Then differences in the success rate are genuine differences between the portfolios and not an accident of the draw.

Choosing sensible inputs

1. Time horizon: do not plan up to average life expectancy, but beyond it. If you retire at 65, you should plan for 30 to 35 years. Half of all people live longer than the average.
2. Withdrawals in today's purchasing power: enter the spending you need today and let it rise with inflation. Fixed nominal amounts lose almost half their purchasing power over 30 years.
3. Deduct pensions and other income: only what has to come out of the portfolio is simulated. Social Security or a pension reduces the required withdrawal, and with it the risk, considerably.
4. Do not forget costs: fund costs, custody fees and taxes on withdrawals reduce the return. A return that is half a percentage point lower changes the success rate over 30 years noticeably.
5. Several scenarios: run the simulation with cautious and with historical assumptions. If both success rates are above your target, the plan is robust; if they are far apart, it depends heavily on the assumptions.

Limitations

The simulation is only as good as its assumptions. Normally distributed returns underestimate crises; a bootstrap from historical months captures them better, but repeats the past with its interest rates, valuations and inflation phases. Return and volatility should be chosen on the cautious side, for example with an eye on today's valuations. And people adapt: if you withdraw less in bad years, you have a much higher success rate than a rigid rule shows (see withdrawal rate).

A success rate of 85% does not mean that you have a 15% probability of going broke. It means: if the future matches the assumptions and you never react, the money does not last until the final year in 15 out of 100 paths.

How Portfolio Lab calculates it

Monte Carlo simulates in monthly steps using the historical monthly returns of your portfolio. Four methods are available: bootstrap of individual months, block bootstrap with 12-month blocks, normal distribution and Student-t with 5 degrees of freedom (scaled to the same volatility); for the parametric methods, you can enter your own values for return and volatility p.a. You choose 500 to 10,000 paths (default 2,000) and 1 to 60 years. The monthly contribution or withdrawal is in today's purchasing power and, if you wish, rises with inflation; a path counts as successful if there is still wealth left at the end. The random numbers use a fixed seed, so the same input gives the same result. The page shows the 10th, 25th, 50th, 75th and 90th percentiles, nominal and real. Financial goals and ALM use the same simulation. Details are in the Monte Carlo methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.