Estimate future outcomes by simulating thousands of possible scenarios based on your input variables.
Simulation Results:
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Understanding Monte Carlo Simulations
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 widely used in finance, project management, engineering, and science to understand the impact of risk and uncertainty.
How It Works
The core idea behind a Monte Carlo simulation is to run a model many times, each time using a different set of random numbers drawn from probability distributions that represent the uncertainties in the input variables. By aggregating the results of these numerous simulations, we can build a distribution of possible outcomes.
The Mathematical Basis
For financial modeling, a common approach is to simulate the path of an asset price using a geometric Brownian motion model. The price at time t+1 can be approximated by the price at time t:
μ (mu) is the expected return or drift rate (average growth rate).
σ (sigma) is the volatility (standard deviation of returns).
Δt (delta t) is the time increment (1 / number of periods in a year, if simulating annual steps).
Z is a random variable drawn from a standard normal distribution (mean 0, standard deviation 1).
Our calculator simplifies this slightly by assuming Δt=1 for each period you define, and the input rates are annual, applied over the specified number of periods. The drift and volatility are used to generate random paths for the 'Initial Value' over the 'Number of Time Periods'. The 'Number of Simulations' determines how many distinct paths are generated.
Interpreting the Results
This calculator provides a simplified Monte Carlo output. It runs multiple simulations of the 'Initial Value' over a specified number of time periods, incorporating the 'Average Growth Rate' and 'Volatility'. The result displayed is the average final value across all simulations.
While this simplified calculator shows the average outcome, a more robust Monte Carlo analysis would also consider:
Probability Distributions: Calculating the mean, median, standard deviation, and percentiles (e.g., 5th and 95th percentile) of the final values.
Risk Metrics: Estimating Value at Risk (VaR) or Conditional Value at Risk (CVaR).
Scenario Analysis: Identifying best-case and worst-case outcomes.
Use Cases
Investment Planning: Estimating the potential future value of portfolios or retirement savings.
Financial Forecasting: Predicting future revenues, costs, or project profitability under uncertainty.
Risk Management: Quantifying potential losses or upside potential in financial instruments.
Option Pricing: Determining the fair value of complex financial derivatives.
Project Management: Assessing the probability of project completion on time and within budget.
function runMonteCarlo() {
var initialValue = parseFloat(document.getElementById("initialValue").value);
var driftRate = parseFloat(document.getElementById("driftRate").value) / 100; // Convert percentage to decimal
var volatility = parseFloat(document.getElementById("volatility").value) / 100; // Convert percentage to decimal
var timePeriods = parseInt(document.getElementById("timePeriods").value);
var numSimulations = parseInt(document.getElementById("numSimulations").value);
var results = [];
var totalFinalValue = 0;
// Basic input validation
if (isNaN(initialValue) || isNaN(driftRate) || isNaN(volatility) || isNaN(timePeriods) || isNaN(numSimulations) ||
initialValue <= 0 || timePeriods <= 0 || numSimulations <= 0) {
document.getElementById("simulationResult").textContent = "Invalid Input";
document.getElementById("resultDescription").textContent = "Please enter valid positive numbers for all fields.";
return;
}
// If timePeriods is 0, the final value is just the initial value.
if (timePeriods === 0) {
document.getElementById("simulationResult").textContent = initialValue.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 });
document.getElementById("resultDescription").textContent = "Final Value (Average of all simulations)";
return;
}
// Generate random numbers for standard normal distribution (Box-Muller transform)
function randomStandardNormal() {
var u1 = Math.random();
var u2 = Math.random();
return Math.sqrt(-2.0 * Math.log(u1)) * Math.cos(2.0 * Math.PI * u2);
}
for (var i = 0; i < numSimulations; i++) {
var currentValue = initialValue;
for (var t = 0; t < timePeriods; t++) {
var z = randomStandardNormal();
// Simplified geometric Brownian motion formula for discrete steps
// S(t+1) = S(t) * exp((mu – sigma^2/2)*dt + sigma*sqrt(dt)*Z)
// Assuming dt = 1 for each period input
var driftComponent = (driftRate – 0.5 * Math.pow(volatility, 2));
var randomComponent = volatility * z;
currentValue *= Math.exp(driftComponent + randomComponent);
}
results.push(currentValue);
totalFinalValue += currentValue;
}
var averageFinalValue = totalFinalValue / numSimulations;
document.getElementById("simulationResult").textContent = averageFinalValue.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 });
document.getElementById("resultDescription").textContent = "Average Final Value across " + numSimulations.toLocaleString() + " simulations";
}