Value at Risk (VaR) is a statistical measure used in financial risk management to quantify the level of financial risk within a firm, portfolio, or position over a specific time frame. It estimates the maximum potential loss that could be incurred on an investment or portfolio over a given holding period, at a specified confidence level, under normal market conditions.
Essentially, VaR answers the question: "What is the maximum amount I could lose with X% probability over Y days?"
How VaR is Calculated (Parametric or Variance-Covariance Method)
The most common method for calculating VaR, especially for portfolios with normally distributed returns, is the parametric method (also known as the variance-covariance method). This method relies on historical data to estimate the expected return, volatility, and then uses these to forecast potential losses.
The formula for VaR using this method is:
VaR = Portfolio Value * (Expected Return - Z-score * Volatility)
However, a more direct interpretation of the *potential loss* (which is what most users are interested in) often focuses on the deviation from the expected return, using the standard deviation (volatility) scaled by the Z-score and the time horizon. A simplified approach for maximum loss estimation at a given confidence level, assuming normal distribution, is:
VaR (Loss) = Portfolio Value * Z-score * Volatility * sqrt(Time Horizon / Trading Days in Year)
Where:
Portfolio Value: The current market value of the investment or portfolio.
Expected Annual Return: The average historical or projected return of the portfolio on an annual basis. While used in more complex VaR models, for a simple maximum loss calculation, the focus is often on volatility.
Annual Volatility: The standard deviation of the portfolio's annual returns. This measures how much the returns typically fluctuate.
Confidence Level: The probability that the actual loss will not exceed the VaR estimate. Common levels are 95% or 99%.
Z-score: The number of standard deviations from the mean required to reach the specified confidence level in a standard normal distribution. For example:
90% confidence level => Z-score ≈ 1.282
95% confidence level => Z-score ≈ 1.645
99% confidence level => Z-score ≈ 2.326
Time Horizon: The period for which the VaR is calculated (e.g., 1 day, 1 week, 1 month).
Trading Days in Year: Typically assumed to be 252 for financial markets.
How to Use This Calculator:
Portfolio Value ($): Enter the total current market value of your investment portfolio.
Expected Annual Return (%): Input the expected average annual return for your portfolio.
Annual Volatility (%): Enter the standard deviation of your portfolio's annual returns.
Confidence Level (%): Select your desired confidence level (e.g., 95 for a 95% confidence).
Time Horizon (Days): Specify the number of days for which you want to estimate the potential loss.
The calculator will then estimate the maximum potential loss you might experience over the specified time horizon at your chosen confidence level. A VaR of $50,000 at a 95% confidence level for a 1-day period means there is a 5% chance that losses will exceed $50,000 on any given day.
Limitations:
It's crucial to understand that VaR is not a perfect measure. It assumes normal market conditions and, for parametric VaR, often assumes normal distribution of returns, which may not hold true during extreme market events. VaR does not tell you the *magnitude* of loss if the threshold is breached (that's the domain of Expected Shortfall). It's a tool to estimate potential downside risk, not a guarantee of maximum loss.
function getZScore(confidenceLevel) {
if (confidenceLevel = 100) {
return 1.645; // Default to 95% if input is invalid
}
var confidence = confidenceLevel / 100.0;
// Using approximations for common Z-scores
if (Math.abs(confidence – 0.90) < 0.01) return 1.282;
if (Math.abs(confidence – 0.95) < 0.01) return 1.645;
if (Math.abs(confidence – 0.99) < 0.01) return 2.326;
// For other values, a more precise inverse CDF calculation would be needed,
// but for a simple calculator, we'll provide common ones.
// If not a common value, we might return an average or default.
// For simplicity, let's return 1.645 for any other value or handle it with a warning.
// A more robust solution would use a statistical library or lookup table.
console.warn("Using default Z-score for less common confidence level. For precise values, consider using a statistical library.");
return 1.645; // Fallback
}
function calculateVaR() {
var portfolioValue = parseFloat(document.getElementById("portfolioValue").value);
var expectedReturn = parseFloat(document.getElementById("expectedReturn").value);
var volatility = parseFloat(document.getElementById("volatility").value);
var confidenceLevel = parseFloat(document.getElementById("confidenceLevel").value);
var timeHorizon = parseFloat(document.getElementById("timeHorizon").value);
var resultDiv = document.getElementById("result-value");
var errorMessage = "";
if (isNaN(portfolioValue) || portfolioValue <= 0) {
errorMessage += "Please enter a valid positive Portfolio Value.";
}
if (isNaN(volatility) || volatility < 0) {
errorMessage += "Please enter a valid non-negative Volatility.";
}
if (isNaN(confidenceLevel) || confidenceLevel = 100) {
errorMessage += "Please enter a Confidence Level between 0 and 100.";
}
if (isNaN(timeHorizon) || timeHorizon <= 0) {
errorMessage += "Please enter a valid positive Time Horizon in days.";
}
// Expected return is not strictly necessary for the simplified loss calculation, but we can validate it.
if (isNaN(expectedReturn)) {
errorMessage += "Please enter a valid Expected Annual Return.";
}
if (errorMessage !== "") {
resultDiv.innerHTML = "Error: " + errorMessage;
resultDiv.style.color = "#dc3545"; // Red for errors
return;
}
// Convert percentages to decimals
var annualVolatilityDecimal = volatility / 100.0;
// Expected return is often omitted in simple loss calculation, focusing on deviation
// var annualExpectedReturnDecimal = expectedReturn / 100.0;
var zScore = getZScore(confidenceLevel);
// Adjust volatility for the time horizon
var tradingDaysInYear = 252; // Standard assumption
var timeHorizonFactor = Math.sqrt(timeHorizon / tradingDaysInYear);
var dailyVolatility = annualVolatilityDecimal * timeHorizonFactor;
// Calculate VaR as potential loss from the current portfolio value
// VaR (Loss) = Portfolio Value * Z-score * Volatility_for_period
// We use daily volatility derived from annual for the given time horizon.
var varValue = portfolioValue * zScore * dailyVolatility;
// Ensure VaR is presented as a positive loss value
varValue = Math.abs(varValue);
resultDiv.innerHTML = "$" + varValue.toFixed(2);
resultDiv.style.color = "#28a745"; // Green for success
}