Monte Carlo simulation is a powerful computational technique used to model and analyze the behavior of complex systems and to estimate probabilities of various outcomes when there is inherent randomness or uncertainty. Named after the famous casino city, it uses repeated random sampling to obtain numerical results. Essentially, it's a way to understand the potential range of results and their likelihoods for a process with uncertain inputs.
The Math Behind the Simulation
At its core, a Monte Carlo simulation involves the following steps:
Define the Model: First, you define a mathematical model that represents the system or process you want to analyze. This model includes variables that are uncertain or random.
Identify Probability Distributions: For each uncertain variable, you assign a probability distribution that best describes its possible values and their likelihoods. Common distributions include the normal (Gaussian) distribution, uniform distribution, and log-normal distribution.
Generate Random Samples: The simulation then generates a large number of random samples for each uncertain variable, drawing from their defined probability distributions.
Run the Model: For each set of random samples (one sample for each uncertain variable), you run the model and calculate the outcome. This is one "trial" or "iteration" of the simulation.
Aggregate Results: You repeat step 4 many times (often thousands or millions of times). The collection of outcomes from all these trials forms a distribution of possible results.
Analyze Results: Finally, you analyze the aggregated results to understand the range of outcomes, calculate probabilities, find averages, and determine confidence intervals.
In the context of financial modeling, a common application is simulating future investment values. We often assume that asset returns follow a specific probability distribution (like a log-normal distribution, which is common for prices, or a normal distribution for returns themselves). The simulation then estimates the probability of achieving a certain return, the range of possible future values, and the likelihood of the investment losing value.
Calculator Breakdown
This calculator models a simple investment growth scenario over a specified number of periods. The key inputs are:
Initial Value: The starting amount of the investment.
Mean Return (per period): The average expected return for each period (e.g., daily, monthly, annually).
Standard Deviation of Return (per period): A measure of the volatility or dispersion of returns around the mean. Higher standard deviation means greater uncertainty.
Number of Periods: The total number of time steps over which the simulation runs.
Number of Simulations: The total number of random trials to perform. A higher number generally leads to more accurate and stable results.
The calculator uses a simplified approach where each period's return is sampled from a normal distribution. The final value for each simulation is calculated by compounding these random returns. The output provides insights into the potential future value of the investment, including:
Average Final Value: The mean of the final values across all simulations.
25th Percentile: The value below which 25% of the simulated final values fall.
75th Percentile: The value below which 75% of the simulated final values fall. (The range between the 25th and 75th percentiles is often called the Interquartile Range or IQR).
Probability of Gain: The percentage of simulations where the final value was greater than the initial value.
Use Cases
Financial Planning: Estimating future retirement account balances, college savings, or other long-term financial goals.
Investment Analysis: Understanding the risk and potential reward of different investment strategies.
Project Management: Estimating project completion times or costs when tasks have uncertain durations.
Risk Management: Quantifying potential losses or gains in various business scenarios.
Scientific Research: Modeling complex systems in physics, biology, engineering, and economics where random factors play a role.
By running multiple scenarios, Monte Carlo simulations help move beyond single-point estimates to a more realistic understanding of uncertainty and potential outcomes.
function runMonteCarloSimulation() {
var initialValue = parseFloat(document.getElementById("initialValue").value);
var meanReturn = parseFloat(document.getElementById("meanReturn").value);
var stdDevReturn = parseFloat(document.getElementById("stdDevReturn").value);
var numPeriods = parseInt(document.getElementById("numPeriods").value);
var numSimulations = parseInt(document.getElementById("numSimulations").value);
// Input validation
if (isNaN(initialValue) || initialValue <= 0) {
alert("Please enter a valid positive Initial Value.");
return;
}
if (isNaN(meanReturn)) {
alert("Please enter a valid Mean Return.");
return;
}
if (isNaN(stdDevReturn) || stdDevReturn < 0) {
alert("Please enter a valid non-negative Standard Deviation of Return.");
return;
}
if (isNaN(numPeriods) || numPeriods <= 0) {
alert("Please enter a valid positive Number of Periods.");
return;
}
if (isNaN(numSimulations) || numSimulations <= 0) {
alert("Please enter a valid positive Number of Simulations.");
return;
}
var finalValues = [];
var successfulSimulations = 0;
for (var i = 0; i < numSimulations; i++) {
var currentValue = initialValue;
for (var j = 0; j initialValue) {
successfulSimulations++;
}
}
// Sort final values for percentiles
finalValues.sort(function(a, b) { return a – b; });
// Calculate statistics
var sumFinalValues = 0;
for (var k = 0; k < finalValues.length; k++) {
sumFinalValues += finalValues[k];
}
var averageOutcome = sumFinalValues / finalValues.length;
var percentile25 = finalValues[Math.floor(finalValues.length * 0.25)];
var percentile75 = finalValues[Math.floor(finalValues.length * 0.75)];
var probabilityGain = (successfulSimulations / numSimulations) * 100;
// Display results
document.getElementById("averageOutcome").innerText = "Average Final Value: " + averageOutcome.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 });
document.getElementById("percentile25").innerText = "25th Percentile: " + percentile25.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 });
document.getElementById("percentile75").innerText = "75th Percentile: " + percentile75.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 });
document.getElementById("probabilityGain").innerText = "Probability of Gain: " + probabilityGain.toFixed(2) + "%";
}