Value-at-Risk (VaR)#
A threshold loss unlikely to be exceeded at a given confidence over a horizon.
Important
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What it is#
Value-at-Risk summarizes downside risk in a single number: the maximum loss over a holding period \(h\) that will not be exceeded with confidence \(\alpha\) (typically 95% or 99%) — anything worse occurs only with probability \(1 - \alpha\). Formally it is the \(\alpha\)-quantile of the loss distribution (the negative \(\alpha\)-quantile of returns):
with \(F_r\) the return CDF and \(L\) the loss.
Where it comes from#
VaR was introduced by J. P. Morgan’s RiskMetrics (1994) and enshrined by the Basel framework for bank regulatory capital. It is estimated by historical simulation (the empirical quantile over a rolling window), parametric methods (assume a normal / t distribution and scale by volatility, often via GARCH), or Monte Carlo.
Its blind spot#
VaR says nothing about how bad losses beyond the threshold are, and it is not coherent — it can violate subadditivity, so a diversified portfolio’s VaR may exceed the sum of its parts. Expected Shortfall (CVaR) — the average loss given VaR is breached — repairs both and is coherent.
Theme: Risk & Probabilistic Forecasting · All terminology
Hint
Mind map — connected ideas
Return Distribution · Risk Forecast · Quantile Level · Quantile Regression · Cumulative Distribution Function (CDF) · Probabilistic Scoring
Hint
More in Risk & Probabilistic Forecasting
Continuous Probabilistic Forecasts · Continuous Ranked Probability Score (CRPS) · Deterministic forecasts · Full Distribution · Pinball Loss (a.k.a. Quantile Loss) · Point Forecasts · Predicting Percentiles · Prediction Intervals (PI) · Probabilistic Forecasts · Probabilistic Scoring · Probability Forecasts · Quantile Forecasts · Quantile Level · Quantile Regression
See also
Source article Adapted (context, re-expressed) in our own words from: Value-at-Risk (VaR) (insightful-data-lab.com).