Simulating Portfolio Outcomes

Here is a question no formula on earth can answer cleanly: I have $500,000, I'll add$10,000 a year until I retire, then draw $40,000 a year for 30 years — will the money last? Markets return “about 7%” with “about 15% volatility,” which sounds like enough information to settle it. It isn’t. Punch 7% into a calculator and you get a single, smooth, exponential line gliding upward to a comfortable number. That line is a fiction. Real markets don’t return 7% every year; they return +24%, then −12%, then +6%, in an order nobody can predict — and the order turns out to matter enormously when you’re adding or pulling money out along the way.

Monte Carlo refuses to pretend. Instead of one fake smooth path, it simulates thousands of jagged, plausible 30-year futures — each one drawing a fresh random return every year, compounding the balance, adding your contributions and subtracting your withdrawals. Then it counts. In how many of these thousands of futures did you run out of money? The answer isn’t a number on a chart. It’s a probability — “you have a 78% chance of making it” — and that is the only honest way to answer a question this haunted by luck.

The question with no closed form

Analogy. A weather forecaster doesn’t tell you “it will be exactly 71 degrees on June 5th, 2056.” She can’t — the system is too chaotic to pin a single number three decades out. What she can say is “given these conditions, there’s a 30% chance of rain.” Long-horizon wealth is the same kind of chaotic system, and it deserves the same kind of answer: not a point, but a probability.

Definition. A wealth-path simulation projects a portfolio’s balance forward year by year, where each year’s return is a random draw from an assumed distribution rather than a fixed number. Starting from balance $W_0$, each year applies a random growth factor and a cash flow:

$W_{t+1} = W_t \cdot (1 + r_t) + C_t$

where $r_t$ is the random return drawn for year $t$ (say from a normal distribution with mean $\mu$ and volatility $\sigma$, or a fatter-tailed one) and $C_t$ is the cash flow — a positive contribution while you’re saving, a negative withdrawal once you’re spending it down. Run this recursion for 30 years and you get one possible future. Run it thousands of times with fresh random draws each time and you get the full fan of what might happen.

Worked example. Start with $500,000, draw three years of returns +20$10,000 at the end of each year. Year 1: $500{,}000 \times 1.20 + 10{,}000 = 610{,}000$. Year 2: $610{,}000 \times 0.90 + 10{,}000 = 559{,}000$. Year 3: $559{,}000 \times 1.08 + 10{,}000 = 613{,}720$. That’s one path. The next simulated path draws a totally different trio of returns and lands somewhere else entirely. The whole point is that no single path is “the” answer — the cloud of them is.

The cone of outcomes

Analogy. Picture a hurricane forecast map: a single dot today, then a cone of possible tracks that flares wider the further out you look. Nobody believes the storm will follow the exact center line — but the cone tells you the realistic spread, and the edge tells you the worst-case landfall you’d better prepare for. A Monte Carlo wealth fan is that hurricane cone, drawn for your money.

Definition. At each future year, take the balances of all the simulated paths and compute their percentiles: the 10th, 25th, 50th, 75th, and 90th. The median (p50) path is the honest “typical” outcome — half the futures end up above it, half below. The p90 band is the lucky outcome; the p10 band is the “markets were unkind” outcome. Plot all of them across time and you get a fan that starts as a single point and widens with the horizon — because uncertainty compounds: a small return difference early on multiplies into a huge balance difference 30 years later.

The island below is exactly this object. Drag the average-return and volatility sliders and watch the fan breathe: more volatility flares the cone wider (more spread between the lucky and unlucky bands), a higher mean lifts the whole fan. The median, p10, and p90 readouts are your planning trio — typical, bad, and good.

The crucial habit. Don’t plan around the median alone — plan around the p10. The median says “if luck is average, here’s your balance.” The p10 says “if the dice come up cold, here’s how thin things get.” A retirement plan that only survives the median is a plan that fails roughly half the time. The whole reason you simulate is to see that unlucky band and build a margin for it.

Pitfall. Reading the fan as if the median path is guaranteed. It isn’t a guarantee; it’s a coin-flip boundary. Half of all futures land below it. Treating p50 as “the plan” quietly ignores the entire bottom half of the cone — which is precisely the half that can ruin you.

Average is a liar — use the median

Analogy. Walk into a bar where everyone earns $40,000 a year, then Jeff Bezos walks in. The *average* income in the room is now in the hundreds of millions; the *typical* person in the room still earns$40,000. The average got hijacked by one outlier. Compounded wealth has the same disease — a handful of spectacularly lucky paths drag the average upward, while the person living a typical outcome sees something far lower.

Definition. Because returns multiply rather than add, terminal wealth is right-skewed (roughly lognormal): bounded by zero on the downside but with a long tail of enormous lucky outcomes on the upside. That skew makes the mean (arithmetic average) terminal wealth sit above the median. The mean answers “average across all futures including the jackpots”; the median answers “the middle future, the one a typical investor actually lives.” For planning, the median is the honest number — you are far more likely to be the typical investor than the jackpot one.

Worked example. Three equally likely multiplicative outcomes for an investment: it could $\times 2$, $\times 1.1$, or $\times 0.5$. Start with $100{,}000$. The three terminal balances are $200{,}000$, $110{,}000$, and $50{,}000$.

The mean terminal balance is $(200{,}000 + 110{,}000 + 50{,}000)/3 = 120{,}000$. The median is the middle value, $110{,}000$. The mean ($120k$) sits above the median ($110k$) — pulled up by the single lucky path. Now stretch this over 30 compounding years instead of one step and the gap explodes: the mean can overstate the typical experience by a wide margin, because a few paths that compound a string of great years run away to absurd balances and yank the average with them.

Sequence-of-returns risk — the killer insight

Analogy. Two hikers descend the same mountain with the same total elevation loss — but one hits the steepest, iciest stretch while their pack is heaviest and their legs are freshest of energy to spend, and the other hits it at the very end. Same elevation, same trail, wildly different injuries. When you’re withdrawing from a portfolio, the order of returns is that icy stretch: the same set of yearly returns can leave you rich or broke depending purely on when the bad years land.

Definition. Sequence-of-returns risk is the danger that a poor order of returns — specifically a crash early, while the balance is large and withdrawals are draining it — permanently damages a portfolio, even if the long-run average return is identical to a benign scenario. It exists only because of cash flows. With no contributions or withdrawals, order doesn’t matter at all — multiplication commutes, so $1.2 \times 0.8 = 0.8 \times 1.2$. The moment you pull money out, order becomes everything, because a crash early means you sell more shares at the bottom to fund the same withdrawal, and those shares are gone — they can’t recover when the market does.

Worked example. A retiree starts with $1,000,000 and withdraws$50,000 at the end of each year. Two scenarios use the exact same three returns — −30%, +10%, +20% — in opposite orders. The arithmetic average return is identical in both. Watch the endings diverge.

Same three returns, same average — yet Scenario A (crash first) ends at $748,000 and Scenario B (crash last) ends at$800,500, a gap of more than $52,000 after only three years. Stretch it over a 30-year retirement and an early crash can be the difference between a portfolio that lasts and one that’s exhausted a decade early. An average-return calculator cannot see any of this — it reports the same number for both scenarios, because to it they are the same. Only simulation, which respects the actual sequence, exposes the danger.

Probability of success, not a single number

Analogy. A surgeon doesn’t promise “you will survive.” She says “this procedure has a 92% success rate.” That framing is more honest, not less — it admits the irreducible role of luck and lets you weigh the risk with eyes open. A Monte Carlo plan delivers exactly this: not “you’ll have $1.4M,” but “your plan succeeds in 82% of simulated futures.”

Definition. The probability of success is the fraction of simulated paths in which the portfolio meets its objective — staying solvent for the full horizon, or reaching a target balance, without hitting zero. Its complement is the probability of ruin: the fraction of paths that run out of money before the horizon ends. If 8,200 of 10,000 simulated retirements survive all 30 years, your plan has an 82% success rate and an 18% ruin rate.

Worked example. Run 10,000 simulated 30-year retirements. Suppose 7,500 of them still have money at year 30 and 2,500 hit zero early. Success rate $= 7{,}500 / 10{,}000 = 75\%$. That single figure carries more planning value than any point estimate: it tells you that a quarter of plausible futures fail, which might push you to spend a little less, work a year longer, or hold more bonds — levers you can pull now, while it’s cheap. A plan reported as “$1.4M expected” hides all of this behind a comforting average; “75% success” forces the uncomfortable, useful truth into the open.

Pitfall. Chasing 100%. No realistic plan is bulletproof — pushing the success rate toward 100% usually means withdrawing so little or holding so much cash that you starve your own lifestyle to insure against a future that probably won’t happen. The art is choosing an acceptable success rate (often 80–90%) and the trade-offs that buy it, not demanding certainty markets can never give.

Garbage in, garbage out — at retirement scale

A Monte Carlo wealth plan is only as honest as the assumptions feeding it, and the failure mode is the familiar one: the output looks gorgeously precise — “82.4% success” — no matter how rotten the inputs. The danger is that a thousand jagged paths feel more trustworthy than a single line, so a bad assumption laundered through simulation looks even more authoritative than it would on a calculator.

The usual sins:

• Too-high a mean return. Plugging in 10% because that’s what the last bull market delivered. A single optimistic point of average return, compounded over 30 years, can flip a plan from failing to passing — on paper only.
• Too-low a volatility. Understating $\sigma$ shrinks the cone and hides the unlucky band where ruin actually lives. The plan looks safe because you erased the scary outcomes from the simulation.
• Thin tails. Assuming clean normal returns when real markets have fat tails — crashes bigger and more frequent than a normal curve predicts. This systematically understates the probability of the early-crash sequences that do the most damage.
• Ignoring inflation. A balance that “lasts 30 years” in nominal dollars may buy half as much at the end. Withdrawals and goals must be modeled in real terms or grown with inflation.
• Assuming i.i.d. returns. Real returns have regimes, volatility clustering, and mean-reversion. Treating each year as an independent identical draw is a convenient lie that misses how bad years bunch together.

Each of these makes the success rate look better than reality. Fix the inputs — realistic mean, honest volatility, fat tails, real (inflation-adjusted) cash flows — before you trust a single decimal of the output.

Putting it together

The question “will my money last?” has no closed-form answer because real returns are random and their order interacts with your contributions and withdrawals. Monte Carlo answers it by simulating thousands of jagged 30-year futures and summarizing the cross-section: the cone of outcomes (percentile fan that flares with the horizon — plan around the p10, not just the median), the gap between mean and median terminal wealth (right-skew means the average flatters; the median is the typical truth), sequence-of-returns risk (for a withdrawer, an early crash is permanently lethal in a way no average-return tool can see), and the probability of success (a far more honest output than any single number). And through it all, the iron law: garbage assumptions — too-high a mean, too-low a volatility, thin tails, ignored inflation — produce a confidently precise success rate that is simply wrong.

Next up — Pricing Path-Dependent Options — we turn the same simulated paths to a new job: valuing options whose payoff depends on the entire journey, not just the endpoint. Asian options that average the whole route, barrier options that trigger if the price ever touches a level — instruments with no clean formula, priced by simulating thousands of paths and averaging their discounted payoffs.