Monte Carlo: strengths, weaknesses, and how to use it well

Behind every failure probability, every wealth fan, every "your plan works in 94% of cases", sits the same machine: Monte Carlo simulation. It generates thousands of possible futures, then counts what happens in them. It is the central tool of modern planning, the engine under every serious simulator. It is a beautiful instrument, as long as you know what it actually does. It does not predict the future. It works out the consequences of your assumptions, with a rigor no back-of-the-envelope calculation can match.

This page takes the machine apart. Where the method came from and how it runs, step by step. What it does better than the alternatives: historical replay, closed-form formulas, and gut feel. Its four structural weaknesses and how much each one really costs you. Then how to use it sensibly, which is what separates the reader who informs a decision from the one who lets a random number generator tell them a story. Two articles carry this one forward: the model families that feed the machine (Historical windows, bootstrap, parametric: the three families of models) and the corrections that make it relevant (Making a Monte Carlo relevant: blending, regimes, stress).

Where it comes from, and how it actually works§

The method was born in 1946 at Los Alamos. Stanislaw Ulam, recovering from an illness, was playing solitaire and wondered what fraction of games could be won. The exact combinatorics were hopeless. It struck him that he could simply play a large number of games and count. With von Neumann and Metropolis, the idea became a method. They named it after the casino where Ulam's uncle lost his money, and it cracked the neutron diffusion calculations of the bomb. The principle is general: when a system is too complex for a formula, run it thousands of times and look at the distribution of outcomes.

A retiree's problem is exactly that kind of system. A withdrawal plan has memory. The withdrawal in year 12 depends on the portfolio in year 12, which depends on the whole sequence before it, on the spending rules, on the buffer, on taxes, and on the pension that starts in year 15. No closed-form formula captures that once the plan has any realism to it. Simulation does. Simulators carry different amounts of detail, but the most complete ones (TPAW Planner or Portfolio Visualizer, say) run every path like this:

  1. Draw a sequence of real returns for the whole horizon, from the market model you picked: independent draws from a parametric law, ideally a Student-t (a bell-shaped cousin of the normal with fat tails, where extreme years stay far more frequent), blocks of history for a bootstrap, a real window for a historical cohort, Markov regimes for a sequence stress (Historical windows, bootstrap, parametric: the three families of models).
  2. Run year 1: apply the return to the portfolio, compute the withdrawal under the active rule (fixed indexed, flexible, guardrails, VPW, ABW and the rest, grossing every sale up for tax), draw on the buffer or refill it under its own rules, collect pensions and side income once they have started.
  3. Repeat year after year to the end of the horizon, or until the money runs out (ruin, with its date).
  4. Record everything: ruin or not, the date of ruin, ending wealth, the spending actually delivered each year, the time spent underwater.

Then start over a few thousand times and count. The share of ruined paths gives the failure probability. The year-by-year wealth gives the fan (Reading a fan chart without fooling yourself). The spending delivered gives the distribution of the standard of living actually lived, and so on down the list. There is nothing else in the box: exact accounting applied to futures drawn out of a hat. (How that loop is really implemented is in Under the hood: how the simulator computes this book.)

What Monte Carlo does better than anything else§

To size the tool up, set it against its three rivals.

Against pure historical replay (Bengen's method, Bengen, the Trinity study, and the birth of the safe withdrawal rate). American history holds barely a hundred years, and the 30-year windows cut out of it overlap almost completely: that comes to three or four genuinely independent retirements. Replay answers "how would my plan have done in the past?". By construction it can say nothing about futures that resemble no past. And it treats the worst historical vintage as a bound, when nothing says the worst has already happened. Monte Carlo generates tens of thousands of synthetic years instead. It explores the space of possibilities around your assumptions, not just the one path that happened. The two complement each other, which is why good simulators show them side by side.

Against closed-form formulas (analytic ruin expectations, rules of thumb). Formulas need unrealistic assumptions to stay solvable: normal returns, proportional withdrawals, no rules at all. Add a guardrail floor, a deferred pension, a buffer with a refill rule and tax on sales, and only simulation keeps up. That is its distinctive strength: plan complexity is free. A realistic rule costs a few lines of code, never an approximation.

Against intuition, last and most important. The human mind is catastrophically bad at compounding randomness over 40 years. Nobody has a feel for what 11% volatility does to a 3.8% withdrawal over 45 years, with a pension landing in year 16. The simulation does, exactly. Its psychological work cuts both ways. It sobers up the optimists, because the fan holds ruined paths even under "good" assumptions. And it settles the anxious, because the median of a sensible plan is surprisingly plush (The psychology of spending: why spending it is the hard part).

The four structural weaknesses, and how much each one matters§

MONTE CARLOWhat moves failure: the assumptions, not the number of draws0246810failure probability (%)μ moves plus or minus 0.5 point (undetectable in the data), at N = 10,000μ 4.5%8.7%μ 5.0%5.9%μ 5.5%4.1%×2.1 on failureN goes up tenfold, assumptions frozen (μ = 5.0%)N 1,000± 1.5 ptsN 4,000± 0.7 ptsN 10,000± 0.5 pts$1M, $32k a year real (3.2%), 35 years, Student-t σ 11%, df 5. Bars: 95% sampling error.
The same plan, one setting changed at a time. On top, μ moves half a point up, then half a point down. Failure goes from 4.1% to 8.7%, a factor of 2.1, and no data series can settle which of the three values is right. Below, the number of paths goes up tenfold with the assumptions frozen. The result does not move; only the uncertainty bar shrinks, from plus or minus 1.5 points at 1,000 paths to plus or minus 0.5 at 10,000. All the precision that matters is on the input side.

Weakness 1: garbage in, garbage out, with leverage. The output is hypersensitive to the inputs. Half a point of difference on μ, statistically undetectable (Forward-looking expected returns (Morningstar, Vanguard, the investment banks)), can double the failure probability. The simulator even amplifies the apparent precision: three decimals of failure, computed off a μ known to within a point. Severity: the worst of the four, and entirely manageable. Input discipline (forward-looking calibration, blending, anchors) and reading in ranges are the answer (Failure probability: reading it, choosing it, and not letting it run you). This is the whole reason for running several models.

Weakness 2: independent draws. Naive Monte Carlo draws each year independently of the last, coin flip by coin flip, with no memory. Real markets have memory. Bad years clump together into recessions and multi-year bear markets. Valuations create decade-long trends (Valuations, the CAPE, and what they say about your withdrawal rate). Volatility comes in clusters. The consequence is precise. At the same mean and variance, the i.i.d. model understates the odds of long mediocre stretches, the ones that kill retirees (Sequence of returns risk: the retiree's real enemy). It also overstates dispersion a little at very long horizons, because real markets mean-revert, which pulls the 30-year fan back in. Severity: real but quantifiable, worth a few points of failure probability. The fixes exist, and a good simulator carries three: blocks, Markov regimes and cohorts (Making a Monte Carlo relevant: blending, regimes, stress).

Weakness 3: the shape of the distribution. Plenty of commercial simulators draw their returns from a normal distribution. It assigns catastrophic years (−35% real) astronomically small probabilities, so 2008 becomes an "impossible" event. A Gaussian Monte Carlo is structurally optimistic about the tails. Severity: serious, but it corrects well. Draw from a Student-t whose df is fitted to the kurtosis of observed returns (kurtosis measures how fat a distribution's tails are: the higher it runs, the more frequent the extreme years). Disasters then live inside the model instead of sitting in its blind spot (Fat tails, crises, and the Student-t). Check this before you trust a tool: many do not even document the law they draw from (Simulator traps: independence, US bias, survivorship).

Weakness 4: what is not in the model does not exist. No draw contains confiscation, hyperinflation wiping out the currency you count in (Hyperinflation and extreme scenarios), a broker's fraud, a divorce, or long-term care at $300 a day. The model simulates the market risk of a plan whose parameters never change. Life's risks stay off screen. Severity: structural and irreducible. Which is why a simulator's output replaces neither your margins (The ten mistakes that wreck a FIRE plan), nor insurance, nor piloting along the way (When to worry, and when to let it ride).

Using it well: eight rules§

Here is the practical summary, forged by the literature (Kitces has written a great deal on the right use of Monte Carlo from the advisor's side).

1. Spend ten times more care on the inputs than on reading the outputs. An hour on the real budget and the calibration of μ (How much you need, Forward-looking expected returns (Morningstar, Vanguard, the investment banks)) beats ten hours contemplating fans.

2. Never read a single model. The range between models (central, sequence stress, broad sample, historical) is the information. One model on its own is just an opinion (Failure probability: reading it, choosing it, and not letting it run you).

3. Read it as a ranking. Comparing (plan A against plan B, lever X against lever Y) is what the tool is good at. Measuring ("my risk is 4.7%") is what it is bad at. The gaps are signal, the decimals noise.

4. Use it as a what-if machine, not an oracle. That is its real calling. What if I leave two years earlier? What if inflation adds half a point to spending? What if the pension is cut by 20%? The best tools industrialize this with an equivalent-moves solver, which converts every margin into dollars, into years, or into flexibility.

5. Look at the paths, not just the aggregates. A probability summarizes. The sample paths in the fan, the red ones included, show how failure arrives: fast, through an early crash, or slowly, through erosion (Reading a fan chart without fooling yourself). The failure mode dictates the defense, not the failure rate.

6. Simulate your real rules, not the rigid caricature. Then look at the standard of living delivered, not just survival. A flexible rule always "succeeds" in the ruin sense. The real question becomes what it makes you live on (Flexibility: myth and reality (what it can really absorb)).

7. Re-simulate rarely. The plan gets re-simulated at the annual review (The annual review: the retiree's checklist) or on an event, never every week. The output moves with the markets; your plan should not.

8. Decide on the pessimistic scenarios, live on the central one. The plan has to be acceptable under the harsh models, the broad sample and the sequence stress. But once the decision is made, the central case is the one describing your likely life. Mixing the two registers produces either reckless retirees or permanently anxious ones.

The essentials§

Going further§