Value-at-Risk

Definition

Value-at-Risk (VaR) at level $\beta$ is the $\beta$-quantile of a cost random variable $X$: the smallest threshold $z$ such that the cost stays at or below $z$ with probability at least $1-\beta$. It is the most basic tail risk measure — a single point on the loss distribution — and, in the convention of akella2024risk, $\beta \to 1$ is risk-neutral while $\beta \to 0$ is maximally risk-averse. Crucially VaR is not coherent: it fails subadditivity, so jointly assessed risks may be understated.

Key Equations

${\mathrm{VaR}}_{\beta }(X):=\inf \{z\in \mathbb{R}\mid F_X(z)\ge 1-\beta \}=F_X^{-1}(\beta ),$
the inverse-CDF (quantile) of $X$ — gloss: the loss level exceeded with probability $\beta$. It is the loosest member of the ordered chain ${\mathrm{VaR}}_{\beta }(X)\le {\mathrm{CVaR}}_{\beta }(X)\le {\mathrm{EVaR}}_{\beta }(X)$, so ${\mathrm{VaR}}_{\beta }(X)\le 0$ already certifies the chance constraint $\mathbb{P}(X\le 0)\ge 1-\beta$.

Source Support

• akella2024risk — tutorial-survey definition of VaR (Def. after Def. 1), its quantile form, the non-coherence (subadditivity-failure) caveat, and its place in the VaR ≤ CVaR ≤ EVaR ordering.

Related Topics

• conditional_value_at_risk — the coherent tail-average that upper-bounds VaR and fixes its non-coherence.
• coherent_risk_measures — the axiom set (subadditivity et al.) that VaR violates.
• chance_constraints — VaR${}_{\beta }(X)\le 0$ is exactly a chance constraint $\mathbb{P}(X\le 0)\ge 1-\beta$.
• tail_risk_measures — VaR is the base member of this taxonomy.

Open Questions

• Where, in a free-flying inspection collision-risk constraint, does VaR’s non-subadditivity actually bite versus where the cheaper quantile suffices?