Entropic Value-at-Risk

Definition

Entropic Value-at-Risk (EVaR) at level $\beta$ is a coherent tail risk measure derived from the Chernoff bound on the tail probability of $X$. It is the tightest convex upper bound on VaR in the standard chain ${\mathrm{VaR}}_{\beta }\le {\mathrm{CVaR}}_{\beta }\le {\mathrm{EVaR}}_{\beta }$ (CVaR being the looser one), and admits a dual distributionally-robust reading as a worst-case expectation over a relative-entropy (KL-divergence) ambiguity ball. In the limit $\beta \to 0$ it recovers the essential supremum, ${\lim }_{\beta \to 0}{\mathrm{EVaR}}_{\beta }(X)=\mathrm{esssup}(X)$.

Key Equations

${\mathrm{EVaR}}_{\beta }(X):={\inf }_{z>0}\left[z^{-1}\ln \right(\mathbb{E}[e^{Xz}]/\beta \left)\right],$
the moment-generating-function form from the Chernoff bound — gloss: optimize the exponential tilt $z$ of the log-MGF. Because ${\mathrm{EVaR}}_{\beta }(X)\le 0\Rightarrow \mathbb{P}(X\le 0)\ge 1-\beta$, it gives the tightest distribution-free convex inner approximation of a chance constraint.

Source Support

• akella2024risk — EVaR definition (Entropic Value-at-Risk section), its Chernoff-bound derivation, the dual relative-entropy interpretation, the VaR ≤ CVaR ≤ EVaR ordering, and the $\mathrm{esssup}$ limit.

Related Topics

• conditional_value_at_risk — the looser coherent bound EVaR tightens.
• value_at_risk — the quantile EVaR upper-bounds (the bottom of the chain).
• coherent_risk_measures — EVaR satisfies all four coherence axioms.
• tail_risk_measures — EVaR is the tightest member of this taxonomy.

Open Questions

• The exponential cone makes EVaR the costliest to optimize; for free-flying receding-horizon collision constraints, when does its tighter envelope justify the solver cost over CVaR?