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Monte Carlo Simulation

Monte Carlo simulation is the next level beyond scenario analysis. Rather than test a few discrete cases, it assigns a probability distribution to each uncertain input, then draws thousands of random combinations and recomputes the project outcome for every draw. The output is a full distribution of NPV or free cash flow, complete with a mean, a spread and the probability of a negative result. Common input distributions include the uniform, where every value in a range is equally likely, and the triangular, which peaks at a most-likely value. Practitioners run it through spreadsheet add-ins such as Oracle Crystal Ball or @Risk.

Why it matters

Scenario analysis lights up three points on a map. Monte Carlo floods the whole territory. By letting every uncertain driver vary at once across its own distribution, and repeating that thousands of times, you trace out the entire range of outcomes and how often each occurs. The headline is no longer a single NPV but a curve, from which you can read the chance the project loses money. The hard intellectual work is upfront, in choosing each input’s distribution, which forces honest thinking about what could go wrong and how likely it is.

Formulas

Simulated NPV draw
NPV(k)=∑t=0nFCFt(x1(k),…,xm(k))(1+r)t\mathrm{NPV}^{(k)} = \sum_{t=0}^{n} \frac{\mathrm{FCF}_{t}\big(x_1^{(k)}, \dots, x_m^{(k)}\big)}{(1 + r)^{t}}
On iteration kk, each input xi(k)x_i^{(k)} is drawn from its distribution, the cash flows are recomputed, and one NPV results. Thousands of draws build the full distribution.
Probability of a positive NPV
P(NPV>0)≈1N∑k=1N1{NPV(k)>0}P(\mathrm{NPV} > 0) \approx \frac{1}{N} \sum_{k=1}^{N} \mathbf{1}\{\mathrm{NPV}^{(k)} > 0\}
The share of the NN simulated outcomes that exceed zero estimates the certainty of a value-creating result. Crystal Ball reports this as the certainty level.

Worked examples

Scenario

A 10,000-iteration simulation of Earthilizer’s 2016 free cash flow returns a mean of about A$87,700, a range from roughly A$29,000 to A$162,000, and a certainty level showing 99.47 percent of outcomes lie above zero. How should the analyst read this?

Solution

The distribution is wide, which confirms genuine dispersion in first-year cash flow, yet almost all of that mass sits above zero, so a negative 2016 FCF is very unlikely under the assumed input distributions. The mean near A$87,700 is the simulation’s best single estimate, and the 99.47 percent certainty is the model’s answer to how confident the firm can be. These figures are from a Crystal Ball simulation and are illustrative of how the output is interpreted.

NoteA normality check such as the Jarque-Bera test can be applied to the simulated distribution before its percentiles are trusted.

Common mistakes

  • ✗Monte Carlo gives a precise, objective answer. The output is only as good as the input distributions, which are themselves judgement calls, so garbage in still means garbage out.
  • ✗A narrow output distribution proves the project is safe. The spread reflects the assumed input ranges. Understating those ranges produces a falsely tight and overconfident NPV distribution.
  • ✗Simulation removes the need for sensitivity and scenario analysis. It extends them. Sensitivity still identifies which drivers to model carefully and a tornado diagram still ranks their influence.
  • ✗More iterations make the forecast more accurate. Extra iterations only smooth the estimated distribution. They cannot fix wrong distributional assumptions about the inputs.

Revision bullets

  • •Assigns a probability distribution to each uncertain input
  • •Draws thousands of random combinations and recomputes the outcome each time
  • •Produces a full NPV distribution with a mean, spread and downside probability
  • •Uniform spreads probability evenly, triangular peaks at a most-likely value
  • •Run via add-ins such as Oracle Crystal Ball or @Risk in a spreadsheet
  • •Output quality depends entirely on the chosen input distributions

Quick check

The main output of a Monte Carlo simulation of a project is

In a Monte Carlo model, a triangular distribution for an input means that

Connected topics

SensitivityScenario AnalysisBreak-EvenRisk vs Uncertainty

Sources

  1. Titman & Martin, Ch. 3
    Titman, S., & Martin, J. D. Valuation: The Art and Science of Corporate Investment Decisions. Pearson.
    Introduces simulation analysis, input distributions and the interpretation of simulated cash-flow and NPV distributions.
  2. Oracle Crystal Ball
    Oracle. Crystal Ball, Spreadsheet-Based Application for Predictive Modeling and Simulation. Oracle Corporation.
    Spreadsheet add-in used in the course to run Monte Carlo simulation and report the certainty level.
How to cite this page
Dr. Phil's Quant Lab. (2026). Monte Carlo Simulation. Derivatives Atlas. https://phucnguyenvan.com/concept/sabv-monte-carlo-simulation
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