Monte Carlo Simulation

Introduction: Monte Carlo Simulation

A Monte Carlo Simulation is a statistical technique used to model the probability of different outcomes in processes that involve uncertainty and randomness. By running a large number of simulations with random variables, it produces a range of possible results along with their likelihood, making it a valuable tool for risk assessment and decision-making.

Background

The method was developed in the 1940s during the Manhattan Project and was named after the Monte Carlo casino in Monaco, reflecting its reliance on random sampling. Today, it is widely applied in fields such as finance, engineering, science, and project management to evaluate uncertainty in complex systems.

Key Elements / Features

The process of Monte Carlo Simulation typically involves:

1. Defining the problem – Identify the model or process with uncertain inputs.
2. Assigning probability distributions – Specify possible values and likelihoods for each uncertain variable.
3. Running simulations – Generate thousands or more random samples from the distributions.
4. Calculating outcomes – Apply the model to each set of inputs.
5. Analysing results – Summarise in terms of probabilities, ranges, and confidence intervals.

Formula:

\(
\hat{\theta} = \dfrac{1}{N} \sum_{i=1}^{N} f(x_i)
\)

Where:
– \(N\) = number of simulations
– \(x_i\) = random samples from input distributions
– \(f(x_i)\) = function or model evaluated at each sample
– \(\hat{\theta}\) = Monte Carlo estimate of the expected value

This formula shows that Monte Carlo Simulation approximates an expectation by averaging results from repeated random trials.

Applications / Examples

Monte Carlo Simulation is applied across many industries:

• Finance – Modelling stock prices, portfolio returns, or investment risk.
• Project management – Estimating completion times and cost overruns.
• Manufacturing & quality – Predicting defect rates and production variability.
• Supply chain – Simulating demand fluctuations and delivery performance.
• Healthcare & science – Analysing uncertainty in treatment outcomes or experimental data.

Example:
A project manager with uncertain task durations runs 10,000 simulations.

• In 75% of simulations, the project finishes within 12 weeks.
• In 95% of simulations, it finishes within 14 weeks.
  This provides a clearer picture of schedule risk compared to a single estimate.

Relevance / Impact

Monte Carlo Simulation provides organisations with a data-driven basis for planning, forecasting, and optimisation. By showing the probability of different outcomes, it reduces reliance on assumptions and improves confidence in strategic decisions.

See also

• Data Distributions
• Risk Analysis
• Forecasting Models
• ARIMA