VaR Calculator (Value at Risk)
Calculate the maximum potential loss of your investment portfolio at a given confidence level. Value at Risk (VaR) is a critical risk management metric used by investors, traders, and financial institutions worldwide.
Interpretation
With 95% confidence, you should not lose more than $2,318 (2.32%) over 1 trading day. There is a 5% chance of exceeding this loss.
VaR at 90% Confidence
VaR at 95% Confidence
VaR at 99% Confidence
Expected Shortfall (CVaR)
Portfolio Return Distribution
VaR by Time Horizon
VaR Sensitivity Analysis
How VaR changes with different volatility levels:
| Volatility | VaR (90%) | VaR (95%) | VaR (99%) | Risk Level |
|---|
VaR at Different Confidence Levels
Table of Contents
What is Value at Risk (VaR)?
Value at Risk (VaR) is a statistical measure that quantifies the potential loss in value of a portfolio over a defined period for a given confidence interval. It answers the question: "What is the maximum amount I could lose with X% certainty over Y time period?"
For example, if a portfolio has a 1-day 95% VaR of $10,000, it means there is a 95% probability that the portfolio will not lose more than $10,000 in a single trading day. Conversely, there is a 5% chance that losses could exceed $10,000.
The VaR Formula
The parametric (variance-covariance) VaR formula assumes returns are normally distributed:
Or for loss (taking absolute value):
VaR = Portfolio Value × |Z × σ × √t - μ|
Where:
• μ = Expected return (daily)
• Z = Z-score for the confidence level
• σ = Standard deviation (volatility) of daily returns
• t = Time horizon in days
• √t = Square root of time (for scaling)
Converting Annual to Daily:
Daily Return = Annual Return / 252
Daily Volatility = Annual Volatility / √252
VaR Calculation Methods
There are three primary methods for calculating VaR:
| Method | Description | Pros | Cons |
|---|---|---|---|
| Parametric (Variance-Covariance) | Assumes returns follow a normal distribution; uses mean and standard deviation | Fast computation, easy to understand | Assumes normality, underestimates tail risk |
| Historical Simulation | Uses actual historical returns to estimate potential losses | No distribution assumption, captures fat tails | Requires extensive data, past may not predict future |
| Monte Carlo Simulation | Generates thousands of random scenarios based on specified distributions | Most flexible, handles complex portfolios | Computationally intensive, model dependent |
Our calculator uses the parametric method, which is most common for quick calculations and educational purposes.
Understanding Confidence Levels
The confidence level determines how conservative the VaR estimate is:
- 90% Confidence: VaR will be exceeded about 1 in 10 days (about 25 times per year)
- 95% Confidence: VaR will be exceeded about 1 in 20 days (about 12-13 times per year)
- 99% Confidence: VaR will be exceeded about 1 in 100 days (about 2-3 times per year)
Higher confidence levels produce larger VaR figures, as they account for more extreme scenarios.
Z-Scores in VaR Calculation
The Z-score converts the confidence level to a standard deviation multiplier:
| Confidence Level | Z-Score | Interpretation |
|---|---|---|
| 90% | 1.282 | Loss exceeds 1.28 std deviations 10% of the time |
| 95% | 1.645 | Loss exceeds 1.65 std deviations 5% of the time |
| 99% | 2.326 | Loss exceeds 2.33 std deviations 1% of the time |
| 99.9% | 3.090 | Loss exceeds 3.09 std deviations 0.1% of the time |
Expected Shortfall (CVaR)
Expected Shortfall, also called Conditional VaR (CVaR) or Expected Tail Loss, measures the average loss when losses exceed the VaR threshold. It provides information about the severity of losses in the tail of the distribution.
Where:
• φ(Z) = Standard normal density function at Z
• c = Confidence level (e.g., 0.95 for 95%)
Simplified approximation for normal distribution:
CVaR ≈ VaR × 1.253 (at 95% confidence)
CVaR is considered superior to VaR because it captures tail risk—it tells you not just the threshold, but how bad things can get when you exceed it.
How VaR is Used
- Risk Management: Setting limits on trading positions and portfolio risk exposure
- Capital Requirements: Banks use VaR to determine regulatory capital reserves (Basel Accords)
- Performance Measurement: Risk-adjusted returns can be evaluated using VaR
- Portfolio Optimization: Constructing portfolios with target VaR levels
- Reporting: Communicating risk to stakeholders and regulators
Limitations of VaR
- Not a worst-case measure: VaR doesn't tell you how much you could lose in extreme scenarios beyond the confidence level
- Assumes normality: Real returns often have fat tails (extreme events occur more frequently than normal distribution predicts)
- Backward looking: Based on historical data that may not reflect future conditions
- Not subadditive: Portfolio VaR can exceed the sum of individual VaRs (diversification might not reduce risk as expected)
- Time horizon sensitivity: Scaling VaR assumes independence of returns, which may not hold
The 2008 financial crisis exposed major flaws in VaR models when actual losses far exceeded VaR predictions. This led to increased adoption of CVaR and stress testing.
Calculation Example
• Portfolio Value: $100,000
• Annual Expected Return: 10%
• Annual Volatility: 15%
• Time Horizon: 1 day
• Confidence Level: 95% (Z = 1.645)
Step 1: Convert to Daily Values
Daily Return = 10% / 252 = 0.0397%
Daily Volatility = 15% / √252 = 0.945%
Step 2: Calculate VaR
VaR = $100,000 × (1.645 × 0.945% - 0.0397%)
VaR = $100,000 × (1.555% - 0.0397%)
VaR = $100,000 × 1.515%
VaR = $1,515
Interpretation: With 95% confidence, the portfolio should not lose more than $1,515 in a single trading day.
Frequently Asked Questions
What is a typical VaR for a diversified portfolio?
A diversified portfolio typically has a 1-day 95% VaR of 1-3% of portfolio value. More aggressive portfolios might have VaR of 3-5% or higher. Conservative portfolios often target VaR below 1%.
How do I calculate VaR for multiple days?
For multi-day VaR, multiply the 1-day VaR by the square root of the number of days. For example, a 10-day VaR equals the 1-day VaR multiplied by √10 ≈ 3.16. This assumes returns are independent.
Is higher or lower VaR better?
Lower VaR indicates less potential loss, which is generally preferred for risk management. However, lower VaR often means lower expected returns. The optimal VaR depends on your risk tolerance and investment objectives.
Why is 95% confidence level commonly used?
The 95% confidence level represents a practical balance between being conservative enough to be useful while not being so extreme that VaR becomes uninformative. It means expected exceedance about once per month on average (roughly 12 trading days per year).
Can VaR be negative?
VaR is typically expressed as a positive number representing potential loss. If your calculation shows a negative VaR, it means even at the specified confidence level, you expect a gain rather than a loss (your expected return exceeds the risk threshold).
How do banks use VaR for regulatory capital?
Under Basel regulations, banks must hold capital equal to at least 3 times their 10-day 99% VaR. This capital buffer helps ensure banks can absorb losses during market stress without becoming insolvent.