Monte Carlo Simulation

Definition¶

Monte Carlo simulation is a quantitative technique that builds a distribution of possible outcomes by running a large number of trials, each drawing its inputs from specified probability distributions, rather than relying on a single point estimate or a small set of discrete scenarios. Each trial (or iteration) randomly samples a value for every input from its assigned distribution, calculates the resulting output, and repeats the process — typically several thousand times or more — to build up a full picture of how likely different outcomes are, not just what a single "base case" would produce.

Why It Matters¶

Most financial models are built to produce a single point estimate: one NPV, one IRR, one valuation. That single figure necessarily obscures the underlying uncertainty in the inputs that produced it. Sensitivity analysis and scenario analysis each address this partially — one variable at a time, or a handful of coherent alternative cases — but neither directly answers a genuinely probabilistic question, such as "what is the probability this outcome falls below a specific threshold." Monte Carlo simulation exists specifically to answer that class of question, by treating each uncertain input as a probability distribution rather than a single assumed value, and propagating that uncertainty all the way through to a full distribution of possible results.

How Monte Carlo Simulation Works¶

1. Identify the key uncertain inputs to the model — the variables whose uncertainty is material enough to warrant explicit probabilistic treatment, rather than being held at a fixed base case assumption.
2. Assign a probability distribution to each input — for example, a normal distribution around a central estimate, a triangular distribution defined by a minimum, most-likely, and maximum value, or a uniform distribution across a defined range.
3. Run a large number of iterations. In each iteration, a value is randomly drawn from each input's assigned distribution, the model is recalculated using those drawn values, and the resulting output is recorded.
4. Aggregate the results across all iterations into a distribution of possible outcomes, from which summary statistics — mean, median, percentiles, and threshold probabilities — are derived.

For i = 1 to N (thousands of iterations):
    Draw x1 from Distribution(Input 1)
    Draw x2 from Distribution(Input 2)
    ...
    Calculate Output_i = f(x1, x2, ...)
Record the full set of Output_1 ... Output_N as the output distribution

Monte Carlo vs. Sensitivity and Scenario Analysis¶

Monte Carlo simulation differs from the other two in kind, not just degree: it does not choose a small number of specific points or cases in advance. Instead, it defines the shape of uncertainty for each input and lets the simulation itself generate the resulting range and likelihood of outcomes.

Interpreting the Output¶

The raw output of a Monte Carlo simulation is a large set of individual results, one per iteration, typically summarized as:

• Mean or median — the central tendency of the output distribution
• Percentiles (e.g., P10, P50, P90) — the value below which a given percentage of iterations fell
• Probability of exceeding or falling below a threshold — the proportion of iterations producing a result on one side of a specified reference point, such as zero, a required hurdle rate, or a target purchase price

This last statistic is what most clearly distinguishes Monte Carlo from sensitivity or scenario analysis: it can attach an actual probability to an outcome across the full range of plausible input combinations, which neither of the other two techniques can do directly on their own.

Applications in Financial Modelling¶

Monte Carlo simulation is a general-purpose technique, applicable wherever a model depends on several uncertain inputs whose combined effect is of interest. In financial modelling and investment analysis specifically, it is most commonly applied to:

• Capital budgeting and investment appraisal, where it can express the probability that a project's NPV falls below zero, or that its IRR falls below the hurdle rate, described in Risk Analysis in Investment Appraisal.
• DCF valuation, where growth, margin, discount rate, and terminal value assumptions are each modelled as distributions rather than single point estimates. This is a large enough and sufficiently distinct application — with its own considerations around correlated inputs, terminal-value distribution bounds, and practical run-time approaches — that it is addressed as its own dedicated guide: Monte Carlo Simulation for DCF Valuation.
• Project finance and infrastructure cash flow modelling, where demand, cost, and financing assumptions can each carry material, independently modellable uncertainty.

Common Errors¶

Best Practices¶

Reserve Monte Carlo simulation for questions that are genuinely probabilistic in nature, rather than applying it by default to every model. Document the distribution type and parameters chosen for each simulated input, and disclose whether and how correlation between related inputs has been addressed. Report the full output distribution — not just its mean — including the specific percentiles and threshold probabilities most relevant to the decision being supported.

Continue Reading¶

Related Pillars¶

• Investment Analysis and Capital Budgeting

Related Glossary¶

• Sensitivity Analysis
• Scenario Analysis

Related Technical Guides¶

• Risk Analysis in Investment Appraisal
• Monte Carlo Simulation for DCF Valuation — the DCF-specific, applied treatment of this general technique