Explainer, in Risk & Portfolio Construction
Value at Risk (VaR)
Value at Risk at confidence 1−α and horizon h is the threshold L such that P(loss > L) = α over horizon h. Common parameterizations: 1-day 95% VaR, 10-day 99% VaR (Basel). Three estimation methods: parametric (assume Gaussian), historical simulation (empirical quantile), and Monte Carlo. Each makes different assumptions about the return distribution.
1 question. Published May 10, 2026. By AI Fin Hub Research.
Definition
Value at Risk (VaR)
Value at Risk at confidence 1−α and horizon h is the threshold L such that P(loss > L) = α over horizon h. Common parameterizations: 1-day 95% VaR, 10-day 99% VaR (Basel). Three estimation methods: parametric (assume Gaussian), historical simulation (empirical quantile), and Monte Carlo. Each makes different assumptions about the return distribution.
Why it matters
VaR is the regulator's risk number. Basel capital, broker-dealer margin, and most institutional risk limits are written against it. The catch: VaR is a single quantile, not the loss distribution. Two portfolios with identical VaR can have wildly different expected losses beyond VaR — which is what expected shortfall fixes.
How it works
Pick horizon, confidence level, and method. Historical: take the empirical α-quantile of the past N-period returns. Parametric: assume returns are Gaussian, VaR = μ − z_α · σ. Monte Carlo: simulate paths from a chosen model, take the empirical quantile of simulated PnL. Backtest: count exceedances; significantly more or fewer than the expected α·N is a model rejection.
Example
Equity portfolio, 1-day 95% VaR, parametric
Daily mean return μ
0.04%
Daily volatility σ
1.1%
z_0.95
1.645
VaR (95%, 1-day)
1.645 × 1.1% − 0.04% = 1.77%
On a $10M portfolio, VaR is roughly $177k. Roughly five days a quarter you should expect to lose more than that. If you see ten, the model is wrong.
Key takeaways
VaR is a quantile, not an expectation; the average loss given a breach can be much larger.
Historical VaR fails when the future doesn't look like the recent past.
VaR backtesting (Kupiec, Christoffersen) catches mis-specified models — use it.
Related terms
Run the numbers next
- Playgrounds VaR Backtest — Kupiec & Christoffersen
Paste P&L + VaR series and run Kupiec POF, Christoffersen independence, and joint conditional-coverage tests. Likelihood-ratio χ² p-values.
- Calculators Returns Distribution Analyzer
Paste a returns CSV. Histogram, normal QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.
Questions people ask next
Why isn't VaR coherent?
Because it can violate sub-additivity — the VaR of a portfolio can exceed the sum of the component VaRs, which contradicts the intuition that diversification reduces risk. Expected shortfall fixes this by being a coherent risk measure.
Sources
- An Overview of Value at Risk , Federal Reserve Board FEDS Working Paper (1996)
- Basel III: International regulatory framework for banks , Bank for International Settlements
Keep the topic connected
- Learn Expected Shortfall (CVaR)
Expected shortfall: the average loss given a VaR breach. Why regulators are migrating from VaR and what ES catches that VaR misses.
- Learn Volatility
Volatility as the standard deviation of returns: realized vs implied, the annualization gotcha, and why volatility-of-volatility matters.