Monte Carlo Simulation Calculator
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Monte Carlo Simulation Calculator
Simulation Results
—
Average Outcome after years
—
95th Percentile Outcome after years
—
5th Percentile Outcome after years
Based on simulations.
Understanding Monte Carlo Simulations
The Monte Carlo simulation is a powerful 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's widely used in finance, physics, engineering, and project management to understand risk and potential future scenarios.
How it Works
At its core, a Monte Carlo simulation involves:
- Defining a Model: Creating a mathematical model that describes the system or process you want to analyze. This model includes inputs that are subject to uncertainty.
- Defining Probability Distributions: For each uncertain input, specifying its possible range and the probability of each value occurring. This is often done using statistical distributions (e.g., normal distribution for returns, uniform distribution for costs).
- Running Trials: Performing a large number of random trials (simulations). In each trial, random values are drawn from the probability distributions of the uncertain inputs, and the model is run with these values.
- Aggregating Results: Collecting the outcomes from all the trials. By analyzing the distribution of these outcomes, we can determine the probability of various results, calculate expected values, identify potential risks, and understand the range of possibilities.
Applications in Finance
In finance, Monte Carlo simulations are invaluable for:
- Portfolio Management: Estimating the future value of investment portfolios under various market conditions.
- Option Pricing: Determining the fair value of complex financial derivatives.
- Risk Management: Assessing the likelihood of significant losses or the probability of achieving financial goals.
- Retirement Planning: Simulating thousands of potential investment paths to see if retirement savings are likely to last.
The Math Behind This Calculator
This calculator simulates the future value of an initial amount based on a defined expected annual return (drift) and annual volatility. It uses a simplified geometric Brownian motion model, where the change in value over a small time interval dt is given by:
dS = μS dt + σS dW
Where:
S is the current valueμ (mu) is the drift (expected rate of return)σ (sigma) is the volatilitydt is a small time stepdW is a random variable drawn from a standard normal distribution (mean 0, variance 1)
For discrete time steps (e.g., annually), the formula can be approximated. The calculator generates random numbers from a normal distribution to simulate the daily or yearly fluctuations around the expected drift, influenced by volatility. By running many simulations, it provides an average outcome, as well as percentile outcomes (like the 5th and 95th) to give a range of plausible future values.
Average Outcome represents the mean of all simulated final values.
95th Percentile Outcome means that in 95% of the simulations, the final value was at or below this amount. This indicates a more optimistic potential outcome.
5th Percentile Outcome means that in 5% of the simulations, the final value was at or below this amount. This highlights a potential downside risk.
function runMonteCarlo() {
var initialValue = parseFloat(document.getElementById("initialValue").value);
var drift = parseFloat(document.getElementById("drift").value);
var volatility = parseFloat(document.getElementById("volatility").value);
var timeHorizon = parseInt(document.getElementById("timeHorizon").value);
var numSimulations = parseInt(document.getElementById("numSimulations").value);
// Basic input validation
if (isNaN(initialValue) || isNaN(drift) || isNaN(volatility) || isNaN(timeHorizon) || isNaN(numSimulations) ||
initialValue <= 0 || timeHorizon <= 0 || numSimulations <= 0 || volatility < 0) {
alert("Please enter valid positive numbers for all parameters. Volatility cannot be negative.");
return;
}
var simulationResults = [];
var dt = 1 / 252; // Assuming daily steps for finer granularity, 252 trading days in a year
for (var i = 0; i < numSimulations; i++) {
var currentValue = initialValue;
for (var year = 0; year < timeHorizon; year++) {
var dailyDrift = drift / 252;
var dailyVolatility = volatility / Math.sqrt(252);
for (var day = 0; day < 252; day++) {
var randomNormal = randomStandardNormal();
var dailyChange = currentValue * (dailyDrift + dailyVolatility * randomNormal);
currentValue += dailyChange;
// Prevent value from going below a reasonable minimum (e.g., zero or a small fraction)
if (currentValue < 0.01) {
currentValue = 0.01;
}
}
}
simulationResults.push(currentValue);
}
// Sort results to find percentiles
simulationResults.sort(function(a, b) { return a – b; });
// Calculate average
var sum = 0;
for (var j = 0; j < simulationResults.length; j++) {
sum += simulationResults[j];
}
var averageOutcome = sum / simulationResults.length;
// Calculate percentiles
var percentile95Index = Math.floor(0.95 * simulationResults.length);
var percentile5Index = Math.floor(0.05 * simulationResults.length);
var percentile95Outcome = simulationResults[percentile95Index];
var percentile5Outcome = simulationResults[percentile5Index];
// Display results
document.getElementById("averageOutcome").innerText = formatCurrency(averageOutcome);
document.getElementById("percentile95").innerText = formatCurrency(percentile95Outcome);
document.getElementById("percentile5").innerText = formatCurrency(percentile5Outcome);
document.getElementById("avgTimeHorizon").innerText = timeHorizon;
document.getElementById("p95TimeHorizon").innerText = timeHorizon;
document.getElementById("p5TimeHorizon").innerText = timeHorizon;
document.getElementById("simulationsRun").innerText = numSimulations;
document.getElementById("result-container").style.display = "block";
}
// Helper function to generate a random number from a standard normal distribution (mean 0, variance 1)
// Using the Box-Muller transform
function randomStandardNormal() {
var u = 0, v = 0;
while (u === 0) u = Math.random(); // Converting [0,1) to (0,1)
while (v === 0) v = Math.random();
return Math.sqrt(-2.0 * Math.log(u)) * Math.cos(2.0 * Math.PI * v);
}
// Helper function to format numbers as currency
function formatCurrency(amount) {
return amount.toLocaleString('en-US', { style: 'currency', currency: 'USD' });
}