Estimate potential outcomes by simulating a process many times with random variables.
Understanding Monte Carlo Simulations
A Monte Carlo simulation is a computational technique used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. It is named after the famous casino in Monaco, renowned for its games of chance.
How It Works
The core idea is to repeatedly sample random inputs from probability distributions and use these inputs to calculate a result. By performing this process thousands or millions of times, we can generate a distribution of possible outcomes, allowing us to understand the range of possibilities and their likelihoods. This is particularly useful in fields like finance, project management, engineering, and science where uncertainty is inherent.
The Math Behind the Calculator
This calculator simplifies a common application of Monte Carlo simulations: forecasting potential future values of an asset or metric. The core of the simulation for each run involves a geometric Brownian motion model, commonly used in finance:
Initial Value (S₀): The starting point of our simulation (e.g., current stock price, project cost).
Mean (μ): The expected average rate of change over a time period. In finance, this is often the expected return.
Standard Deviation (σ): This measures the volatility or dispersion of the returns around the mean. Higher standard deviation implies greater uncertainty and wider potential outcomes.
Time Periods (T): The number of steps or intervals over which the simulation progresses.
Number of Simulations (N): The total number of independent random paths to simulate.
The formula for a single step in a discrete geometric Brownian motion model can be approximated as:
Δt is the length of a single time step (1 / timePeriods if time periods are in years).
exp() is the exponential function.
Z is a random variable drawn from a standard normal distribution (mean 0, standard deviation 1).
Our calculator simplifies this by performing a series of random steps to reach the final time period for each simulation. The results are then aggregated to provide statistical insights.
Interpreting the Results
After running the simulations, the calculator will display:
Average Outcome: The mean of all the final values from the simulations. This gives an expected future value based on the inputs.
Minimum Outcome: The lowest value achieved across all simulations.
Maximum Outcome: The highest value achieved across all simulations.
Likely Range (e.g., 95% Confidence): This shows a range within which the outcome is likely to fall with a certain probability (calculated using percentiles of the simulated outcomes).
Risk Management: Quantifying potential losses or upsides in investments or projects.
Project Management: Assessing the probability of completing a project by a certain deadline or within a budget, considering task duration uncertainties.
Scientific Modeling: Simulating complex systems with inherent randomness.
function runMonteCarlo() {
var initialValue = parseFloat(document.getElementById("initialValue").value);
var mean = parseFloat(document.getElementById("mean").value);
var standardDeviation = parseFloat(document.getElementById("standardDeviation").value);
var timePeriods = parseInt(document.getElementById("timePeriods").value);
var numSimulations = parseInt(document.getElementById("numSimulations").value);
var results = [];
var deltaT = 1 / timePeriods; // Assuming time periods are in years, simplifying step size
if (isNaN(initialValue) || isNaN(mean) || isNaN(standardDeviation) || isNaN(timePeriods) || isNaN(numSimulations) ||
initialValue <= 0 || standardDeviation < 0 || timePeriods <= 0 || numSimulations <= 0) {
document.getElementById("result").innerHTML = "Please enter valid positive numbers for all fields.";
document.getElementById("result").style.display = "block";
document.getElementById("result").style.backgroundColor = "#dc3545"; // Error red
return;
}
for (var i = 0; i < numSimulations; i++) {
var currentValue = initialValue;
for (var t = 0; t < timePeriods; t++) {
var z = randomStandardNormal(); // Get a random number from standard normal distribution
var drift = (mean – 0.5 * Math.pow(standardDeviation, 2)) * deltaT;
var randomShock = standardDeviation * Math.sqrt(deltaT) * z;
currentValue *= Math.exp(drift + randomShock);
}
results.push(currentValue);
}
results.sort(function(a, b) { return a – b; });
var averageOutcome = results.reduce(function(sum, val) { return sum + val; }, 0) / numSimulations;
var minOutcome = results[0];
var maxOutcome = results[results.length – 1];
// Calculate percentiles for likely range (e.g., 5th and 95th percentile)
var p5 = results[Math.floor(numSimulations * 0.05)];
var p95 = results[Math.ceil(numSimulations * 0.95) – 1]; // Adjust index for 0-based array
var resultHtml = "Simulation Results:" +
"Average Outcome: " + averageOutcome.toFixed(2) + "" +
"Minimum Outcome: " + minOutcome.toFixed(2) + "" +
"Maximum Outcome: " + maxOutcome.toFixed(2) + "" +
"Likely Range (5%-95%): " + p5.toFixed(2) + " – " + p95.toFixed(2) +
"(Based on " + numSimulations.toLocaleString() + " simulations over " + timePeriods + " periods)";
document.getElementById("result").innerHTML = resultHtml;
document.getElementById("result").style.display = "block";
document.getElementById("result").style.backgroundColor = "var(–success-green)"; // Reset to success green
}
// Function to generate a random number from a standard normal distribution (Box-Muller transform)
function randomStandardNormal() {
var u1 = 0, u2 = 0;
while(u1 === 0) u1 = Math.random(); // Avoid log(0)
while(u2 === 0) u2 = Math.random();
return Math.sqrt( -2.0 * Math.log( u1 ) ) * Math.cos( 2.0 * Math.PI * u2 );
}