Monte Carlo Simulation for Stock Returns, Explained Simply

Every valuation model ends with a single confident number. This stock is worth $84. That number is built on a stack of assumptions: revenue grows 12% a year, margins hold, the market keeps paying the same multiple. Change any one of them slightly and $84 becomes $61 or $109.

The number was never the answer. It was one answer, out of thousands the same model could have produced.

Monte Carlo simulation is the technique that shows you the rest of them.

What a Monte Carlo Simulation Actually Is

Imagine you want to know how long your commute takes. You could time it once and call it 34 minutes. Or you could time it every day for a year and end up with something more useful: usually 31 to 40 minutes, occasionally 55 when the highway backs up, almost never under 28.

The second answer is worse for bragging and far better for deciding when to leave.

A Monte Carlo simulation does the second thing to a financial model. Instead of feeding it one growth rate, you feed it a range of plausible growth rates. Same for margins, same for the discount rate, same for every assumption that is really a guess. Then you run the model thousands of times, each run drawing a different combination from those ranges.

Ten thousand runs produce ten thousand fair values. Together they form a distribution: a shape that tells you not just what the business might be worth, but how confident that estimate deserves to be.

The name comes from the casino in Monaco. The technique was developed by physicists working on nuclear weapons in the 1940s, who needed to model processes too tangled to solve with an equation. Finance borrowed it later.

Why One Fair Value Number Is Not Enough

A single fair value estimate has a specific failure mode: it looks equally precise whether the business is predictable or wildly uncertain.

Consider two companies, both modeled at a fair value of $50 while trading at $40.

The first is a utility. Regulated returns, contracted revenue, demand that barely moves. Run its model ten thousand times and the answers cluster tightly, maybe $46 to $55. The $50 means something.

The second is a pre profit biotech whose value depends on one drug trial. Run its model ten thousand times and you get $8 in some runs and $180 in others. The average might still be $50. That $50 describes almost none of the actual possibilities.

Same fair value, same 25% gap to price, completely different decisions. A single number cannot tell those two situations apart. A distribution tells them apart instantly.

The Inputs That Get Randomized

You do not randomize everything. Cash in the bank is cash in the bank. What gets a range is anything that depends on the future:

• Revenue growth. Not "12%" but something like "somewhere between 4% and 20%, most likely around 11%," shaped by the company's own history and its industry.
• Operating or gross margin. Margins mean revert more than people expect. A business earning 40% margins in a competitive market rarely holds them for a decade.
• The discount rate. This encodes risk and the cost of capital. Small changes here move long dated cash flows a lot. See WACC explained for how that rate gets built.
• The exit multiple or terminal growth rate. Usually the single biggest driver of a DCF result, and usually the assumption people think about least.

The ranges are not invented. They come from the company's own track record, the spread across its peers, and how volatile the industry has been. A business with fifteen years of steady 8% growth gets a narrow range. One that grew 90% last year and shrank the year before gets a wide one.

What Ten Thousand Runs Give You

Running the model once gives you a point. Running it ten thousand times gives you four things a point cannot:

A central estimate that survived stress. The median of the distribution is a fair value that held up across thousands of assumption combinations, not one lucky path.

A range. The 10th and 90th percentiles bracket the outcomes. If they sit at $44 and $58, the model is reasonably confident. If they sit at $12 and $190, the honest conclusion is that nobody knows what this business is worth, including the model.

The share of outcomes below today's price. If the stock trades at $40 and 8% of simulated values land below $40, that is a measurable statement about downside. It is not a prediction that you will lose money 8% of the time. It is a statement about how much of the model's own uncertainty leaves you overpaying.

Which assumption is doing the damage. Because every run records its inputs, you can see which variable drives the spread. Often one assumption accounts for most of the uncertainty, and it is rarely the one you argued about.

How to Read the Output

The shape matters more than the average.

A narrow, symmetric distribution means the model is confident and the business is predictable. The central estimate is usable.

A wide distribution means the model is telling you it does not know. Widen your margin of safety or move on. A wide range is not a reason to trust the midpoint harder.

A right skewed distribution, with a long tail of high outcomes, is the shape of asymmetry: limited downside, meaningful upside. Most outcomes are modest, a few are large. This is the shape worth hunting for, and it is what risk reward analysis is trying to capture.

A left skewed distribution is the opposite and the most dangerous. Most runs look fine, a few are disasters. Averages hide this. Distributions do not.

A Worked Example in Plain Numbers

Take a software company trading at $60.

The single point model says: 15% revenue growth, 25% operating margin, 9% discount rate, and a fair value of $78. Thirty percent upside. Easy.

Now give each input a range. Growth between 8% and 22%. Margin between 18% and 30%. Discount rate between 8% and 11%. Run it ten thousand times.

The output might come back like this: median $74, 10th percentile $51, 90th percentile $103. Eighteen percent of runs land below the $60 price.

Nothing here says do not buy. But the picture changed. The upside is real and the tail is fat on the right, which is attractive. And roughly one run in five ends with the business worth less than you paid, which is the part the $78 headline number quietly deleted.

That is the entire value of the technique. Same model, same inputs, one honest extra dimension.

What Monte Carlo Cannot Do

It does not predict the future. It maps the consequences of your assumptions. If those assumptions are wrong, ten thousand runs produce ten thousand wrong answers with impressive precision.

It usually assumes inputs move independently, and in real businesses they do not. Revenue collapses and margins collapse together. Naive simulations understate how bad the bad cases get.

It underestimates extreme events. Standard distributions assign near zero probability to things that happen every decade or so. Fraud, a regulatory reversal, a technology that makes the product irrelevant. No simulation catches those.

And it will not save a bad model. Randomizing the inputs of a model that misunderstands the business just spreads the error around.

How Convex Uses It

Monte Carlo is step 5 of the eight step framework behind every analysis on Convex, and it sits deliberately after the qualitative work, not before it. Classify the business, grade the quality, read the signals, estimate fair value, and only then simulate.

The simulation output feeds the two things you actually see: how wide the estimate's range is, and where today's price sits inside it. That is also what defines a buy zone, the price range where the gap between price and value is wide enough to absorb being somewhat wrong.

Ten thousand runs is not a precision claim. It is the opposite. It is the part of the method that admits how much any single estimate can miss by, and puts a number on it.

Frequently Asked Questions

How many simulations are enough?

Results stabilize surprisingly fast. A few thousand runs usually converge, and ten thousand is a common standard because it is cheap and leaves the tails better populated. Running a million does not make the assumptions any better.

Is Monte Carlo better than a DCF?

It is not an alternative to a DCF. It is a DCF run thousands of times with varying inputs. The underlying valuation logic is identical. What changes is that the output is a range instead of a point.

Can I do this without programming?

A basic version works in a spreadsheet using a random function and a data table, which is enough to build the intuition. Practical use on many companies means automating it, which is why it usually shows up inside analysis tools rather than in personal spreadsheets.

Does a low probability of loss mean a stock is safe?

No. It means that within the model's own assumptions, few simulated outcomes fall below the current price. Everything the model failed to consider sits outside that number. Treat it as a measure of the model's confidence, never as a measure of real world risk.