Folio Lab/Documentation

Minimum Entropic VaR

Minimise Entropic Value-at-Risk (EVaR), a coherent tail-risk measure introduced by Ahmadi-Javid (2012). EVaR is the tightest upper bound on CVaR obtainable from the Chernoff bound and is more sensitive to extreme losses than CVaR while remaining a convex programme.

Overview

Conditional Value-at-Risk (CVaR; Rockafellar and Uryasev, 2000, 2002) is the standard coherent tail-risk measure. It is the average loss in the worst -fraction of scenarios. CVaR has many attractive properties (Artzner, Delbaen, Eber and Heath, 1999) but is non-smooth at the VaR boundary, which can lead to numerical issues in some optimisers.

Ahmadi-Javid (2012) introduced Entropic Value-at-Risk (EVaR) as a coherent, strictly stronger upper bound on CVaR derived from the Chernoff inequality: for every loss and confidence level . EVaR is smooth (it is the optimal value of a convex programme involving the moment-generating function of the loss), and it admits a clean disciplined-convex-programming formulation that solvers handle efficiently. Ahmadi-Javid and Fallah-Tafti (2019) develop the portfolio-optimisation theory; Cajas (2021) gives the disciplined-convex implementation that powers skfolio.

FolioLab implements minimum-EVaR via skfolio's tail-risk optimiser withRiskMeasure.EVAR. The confidence level is configurable; standard choices are 0.95 and 0.99 to match institutional VaR / CVaR reporting.

Mathematical Formulation

Chernoff bound on tail probability

For any loss random variable with finite moment-generating function , the Chernoff bound gives

Setting the right-hand side equal to and solving for the smallest that achieves the bound gives the EVaR.

EVaR definition (Ahmadi-Javid, 2012)

EVaR is a one-dimensional infimum of a convex function in ; the inner expression is jointly convex in , which makes the portfolio version of the problem convex too.

Empirical EVaR (sample formulation)

Using the empirical distribution over historical scenarios, EVaR of the loss reduces to

This is jointly convex in , and skfolio handles it via a disciplined-convex-programming reformulation in CVXPY (Cajas, 2021). The result is solved by a conic solver such as MOSEK or CLARABEL.

Optimisation problem

EVaR vs CVaR

Because EVaR upper-bounds CVaR, a min-EVaR portfolio is at least as conservative on tail loss as a min-CVaR portfolio at the same confidence level. The bound is tight when the loss is approximately exponential or sub-Gaussian; on heavy-tailed equity returns EVaR is strictly larger than CVaR and the resulting portfolio is more defensive.

EVaR is also smooth (the inner inf is differentiable in interior points) while CVaR is non-smooth at the VaR threshold. This makes EVaR more numerically friendly to first-order solvers and yields cleaner gradients in risk-attribution work.

Advantages & Limitations

Advantages

  • Coherent tail measure: Satisfies Artzner et al.'s axioms.
  • Tighter than CVaR: Always bounds CVaR from above.
  • Smooth: Differentiable in , friendlier to gradient-based methods.
  • Disciplined convex: Handled cleanly by CVXPY-based solvers.

Limitations

  • Conservative: Bound looseness can cost return on benign distributions.
  • Confidence-level sensitivity: EVaR and EVaR can give meaningfully different portfolios.
  • Computational cost: Conic-programming solver required.
  • Exponential moments: Sample MGF estimates are noisy in short histories.

References

  • Ahmadi-Javid, A. (2012). "Entropic Value-at-Risk: A New Coherent Risk Measure." Journal of Optimization Theory and Applications, 155(3), 1105-1123.
  • Ahmadi-Javid, A., & Fallah-Tafti, M. (2019). "Portfolio Optimization with Entropic Value-at-Risk." European Journal of Operational Research, 279(1), 225-241.
  • Cajas, D. (2021). "Entropic Portfolio Optimization: A Disciplined Convex Programming Framework." SSRN Working Paper.
  • Rockafellar, R. T., & Uryasev, S. (2002). "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking & Finance, 26(7), 1443-1471.
  • Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). "Coherent Measures of Risk." Mathematical Finance, 9(3), 203-228.
  • Palomar, D. P. (2025). Portfolio Optimization: Theory and Application. Cambridge University Press, Chapter 10 (Portfolios with Alternative Risk Measures), Section 10.4 (Tail-Based Portfolios).
  • skfolio documentation — skfolio.RiskMeasure.EVAR.