Long-Run Value-at-Risk Documentation
Understanding Value-at-Risk
Value-at-Risk (VaR) is a statistical measure that quantifies the potential loss in value of an asset or portfolio over a defined period for a given confidence interval. It answers a fundamental question in risk management: "How large could losses be in a typical bad period, and how often might losses be even worse than that?"
Key Insight
A $1 million, 1% Value-at-Risk means that, over the specified period, losses are expected to exceed $1 million only about 1 time out of 100 (though when they do, the loss could be much larger).
VaR has become the industry standard for measuring market risk, used by financial institutions worldwide for risk reporting, capital allocation, and regulatory compliance. However, it is important to understand both its power and its limitations.
Explore VaR Parameters
Try adjusting the sliders to see how confidence level and time horizon affect VaR. Notice how longer horizons increase VaR due to the "square root of time" rule.
Confidence Level
95%
Time Horizon
1 trading day
VaR Threshold:
2.07%
Expected Shortfall:
2.60%
Loss Distribution Visualization
What This Means: At 95% confidence, there is only a 5.0% chance of losing more than 2.07% over this time horizon. If losses do exceed VaR, the average loss would be 2.60% (the Expected Shortfall).
Note: This chart is for demonstration purposes only and assumes a normal distribution. Real return distributions may exhibit skewness, excess kurtosis, and other higher-order moments not reflected here.Key Parameters of VaR
Confidence Level
The probability that the actual loss will not exceed the VaR estimate. Higher confidence levels produce larger VaR values.
95% VaR: 5% chance of exceeding this loss
99% VaR: 1% chance of exceeding this loss (more conservative)
Time Horizon
The period over which the potential loss is measured. Longer horizons typically result in larger VaR values.
Short-run (1-10 days): Used for trading desks, daily risk limits
Long-run (30-365 days): Used for portfolio management, strategic planning
Try It Yourself
Use the calculator below to compute VaR for your own portfolio. Enter your portfolio value and estimated annual volatility, then see how the VaR changes with different confidence levels and time horizons.
Simple VaR Calculator
Enter your portfolio details below to calculate Value-at-Risk. The calculator uses the parametric (variance-covariance) method assuming normally distributed returns.
Portfolio Parameters
Enter your total portfolio value in USD
Typical stocks: 15-25%, bonds: 5-10%
Risk Parameters
Confidence Level
Time Horizon
VaR Results
95% VaR (1 day):
$20,723
(2.07% of portfolio)Expected Shortfall (ES):
$25,988
(2.60% of portfolio)In Plain English: With 95% confidence, your portfolio should not lose more than $20,723 over the next 1 day. There is a 5.0% chance of exceeding this loss.
Formula Used: VaR is calculated as: Portfolio Value × Daily Volatility × √Time Horizon × Z-score. Daily volatility is derived from annual volatility assuming 252 trading days.
VaR = Portfolio × σdaily × √T × zα
Historical Context
VaR emerged in the 1990s and became formalized through the Basel Accords, which established international banking regulations (Jorion, 2006). Basel I (1988) introduced capital requirements, while Basel II (2004) and Basel III (2010) refined how banks must use VaR for risk assessment and capital adequacy calculations.
The 2007-2008 financial crisis revealed limitations in short-horizon VaR measures, leading to increased interest in long-run risk metrics that account for volatility dynamics and mean reversion. V-Lab's Long-Run VaR models address these concerns by incorporating sophisticated volatility forecasting.
VaR vs. Expected Shortfall
Value-at-Risk (VaR)
The maximum loss that will not be exceeded with a specified confidence level. For example, 99% VaR is the loss that will only be exceeded 1% of the time.
Limitation: VaR says nothing about the magnitude of losses beyond the threshold.
Expected Shortfall (ES)
The expected (average) loss given that the loss exceeds the VaR threshold. It answers: "If things go badly, how bad can they get?"
Advantage: ES is a "coherent" risk measure (Artzner et al., 1999) and captures tail risk more completely.
Long-Run VaR Models
Long-Run VaR Models
GJR-GARCH-based models for long-horizon Value-at-Risk estimation
Use Cases: Long-horizon risk measurement, portfolio stress testing, regulatory capital requirements, and multi-period VaR forecasting