Quantile threshold
VaR_alpha(L) = inf{l : P(L <= l) >= alpha}VaR is the loss quantile at the selected confidence level.
Risk
Value at Risk (VaR)
A loss threshold: “on a normal day, losses should not exceed this amount with X% confidence.”
Mental model
VaR tells you where the selected loss tail begins; it does not tell you how deep that tail runs.
VaR is a percentile (quantile) of the loss distribution over a fixed horizon, for example the 99th percentile of one-day losses.
It answers: “How bad is a bad day?” but not “How bad can it get beyond that bad day?” — tail severity is left out.
Explore the mechanism
Where does the selected loss tail begin, and what information does that threshold leave out?
Simulated daily returns under a normal model. The shaded red tail shows the 5% worst days. The solid red line marks VaR as a loss threshold; the dashed red line marks the average loss on those bad days (Expected Shortfall).
Numbers
VaR (loss, 95% level) ≈ 2.58%
Expected Shortfall (ES) ≈ 3.32%
Portfolio VaR at this level (for 10m) ≈ 0.26m; ES ≈ 0.33m
Mean return ≈ 0.04%, volatility σ ≈ 1.60%
Tail probability (empirical) ≈ 5% ( ≈ 12.6 days per year)
Interpretation
The grey bars show a normal return distribution centred on μ with spread σ. The solid red line is VaR: only about 5% of simulated days fall to the left. The dashed red line (ES) is the average of those bad days.
As you increase σ, the histogram widens, and both VaR and ES move further left (bigger losses). The portfolio slider translates those percentages into money: a risk manager will often read “95% 1-day VaR of 2.58% on a 10m book” directly as an amount at risk on a normal day.
Control analogy: VaR is like a design limit for typical disturbances on a system state. ES tells you how severe the excursions are once you actually cross that limit — which is why regulators now often prefer ES to VaR for capital.
UNDERSTAND
What to notice
In practice, VaR is estimated from models (normal, t, GARCH), historical data, or Monte Carlo, and results can change a lot when tails are fat.
Engineering analogy: designing a bridge to withstand loads up to a chosen stress percentile, but not explicitly specifying what happens in truly extreme events.
The quantitative structure
Notation
L- loss over the chosen horizon
alpha- confidence level of the VaR quantile
This model assumes
- The loss distribution and horizon are defined consistently with the risk question.
- The quantile estimator has enough representative data or a defensible fitted model.
A small example
A simple historical 95% VaR
Twenty equally weighted historical losses are sorted from smallest to largest. The 19th loss is 8 and the worst loss is 20.
- At 95%, one observation out of twenty lies beyond the selected threshold.
- The empirical 95% VaR is therefore 8 under this order-statistic convention.
- The loss of 20 remains possible and is not described by the VaR number.
ResultVaR reports the threshold 8, not the worst observed loss 20.
Where it appears
- setting portfolio risk limits
- reporting model-based loss thresholds
- comparing risk across desks on a common horizon
Where the intuition breaks
What this view leaves out
- VaR does not measure the severity of losses beyond the selected quantile.
Common mistakes
“VaR is the maximum possible loss.”
Not quite.VaR is a quantile threshold; losses beyond it can be substantially larger.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Value at Risk (VaR)
Value at Risk (VaR)YOU ARE HERE
Contrasts withScenario and Stress EngineVaR summarises probabilistic loss under a model, whereas stress testing examines explicit adverse scenarios; both are core views of portfolio risk.
Contrasts withVaR vs CVaR (Expected Shortfall)VaR identifies a loss threshold, while Expected Shortfall summarises losses beyond that threshold.
UsesCovariance Estimation and CorrelationParametric portfolio VaR commonly uses covariance and correlation to aggregate risk across positions.
UsesHigh-Dimensional Covariance ShrinkageStabilised covariance estimates can improve parametric VaR when the portfolio contains many correlated risk factors.
Validated byValidation Theory: Backtest Power and Multiple TestingRisk models such as VaR require backtesting, and test power matters when interpreting apparent model adequacy.
Contrasts with · Scenario and Stress Engine
VaR summarises probabilistic loss under a model, whereas stress testing examines explicit adverse scenarios; both are core views of portfolio risk.
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