Entropic Value at Risk

Value at Risk tells you a loss threshold, and Expected Shortfall averages the losses beyond it, but both can be awkward to optimize inside a portfolio model because they involve sorting or tail-cutting the data. Entropic Value at Risk (EVaR) sidesteps that by working with the moment-generating function of the loss distribution instead — a smooth object with none of those kinks — and it comes out as a guaranteed upper bound on Expected Shortfall at the same confidence level, so if you control EVaR you automatically control the more standard measures too.

Formally, for a loss $L$ and confidence level $1-\alpha$:

$$\mathrm{EVaR}_{1-\alpha}(L) = \inf_{z>0} \left\{ \frac{1}{z} \ln\!\left( \frac{M_L(z)}{\alpha} \right) \right\},$$

where $M_L(z) = \mathbb{E}[e^{zL}]$ is the moment-generating function of $L$. In plain English: it searches over a tilting parameter $z$ for the tightest possible exponential bound on how bad the tail can get, using the same Chernoff-bound machinery that shows up in large-deviations theory.

The payoff is computational: EVaR is convex in the portfolio weights whenever the underlying loss distribution is well-behaved, so a portfolio optimizer can add an EVaR constraint and still solve a clean convex program — something that's harder to guarantee directly for Expected Shortfall on some distributions. The cost is conservatism: EVaR is always at least as large as Expected Shortfall at the same level, so a book managed to an EVaR limit will look "safer" on paper than one managed to the equivalent CVaR limit, even though the underlying risk hasn't changed.