Folio Lab/Documentation

Minimum Entropic DaR

Minimise Entropic Drawdown-at-Risk (EDaR), the entropic counterpart of Conditional Drawdown-at-Risk (CDaR). EDaR is a smooth, coherent measure of drawdown tail risk and is preferred for capital-preservation mandates where maximum-drawdown control matters more than volatility control.

Overview

CDaR (Chekhlov, Uryasev and Zabarankin, 2005) reframes tail risk in drawdown space rather than return space: instead of measuring the worst-quantile of one-period losses, it measures the worst -quantile of running drawdowns from the cumulative-wealth high-water mark. CDaR is a coherent path-dependent risk measure and is the natural target for capital-preservation mandates.

EDaR is the entropic version of CDaR (Ahmadi-Javid, 2012; Cajas, 2021): apply the EVaR construction to the drawdown process instead of the return process. The result is a smooth, coherent tail-of-drawdown measure that upper-bounds CDaR pointwise. As with EVaR vs CVaR, EDaR is more conservative on heavy-tailed loss processes and is numerically friendlier than CDaR.

FolioLab implements minimum-EDaR via skfolio's drawdown-risk optimiser with RiskMeasure.EDAR. The confidence level is configurable; standard choices are 0.95 and 0.99 for institutional reporting.

Mathematical Formulation

Drawdown process

Let be the cumulative return of the portfolio. The running drawdown at time is

and equals zero whenever the portfolio is at a new high-water mark.

EDaR via the EVaR construction

EDaR is EVaR applied to the empirical drawdown sequence. As in the EVaR case the expression is jointly convex in , and skfolio handles it via a disciplined-convex-programming reformulation (Cajas, 2021).

Optimisation problem

The drawdown sequence is a function of the entire path of , so the problem is path-dependent. skfolio reformulates it as a convex programme with auxiliary variables tracking the running maximum, and the result is solved by CVXPY with MOSEK or CLARABEL.

EDaR vs CDaR

EDaR upper-bounds CDaR by construction: a min-EDaR portfolio is at least as conservative on the right tail of the drawdown distribution as a min-CDaR portfolio at the same confidence level. The bound is tight for sub-exponential drawdown distributions and looser for heavier tails.

On Indian equity universes EDaR portfolios tend to put more weight on low-beta, dividend-bearing names and on assets that historically did not participate in cyclical drawdowns. The trade-off versus CDaR is that EDaR is more numerically stable (smooth gradients, well-defined sensitivities) at the cost of a structurally more conservative allocation.

Advantages & Limitations

Advantages

  • Path-aware: Captures sustained-loss exposure that one-period CVaR misses.
  • Coherent and smooth: Easier to attribute and differentiate than CDaR.
  • Tighter than CDaR: Upper bounds CDaR pointwise.
  • Capital-preservation friendly: Targets the metric clients actually feel.

Limitations

  • Path dependence: Optimisation is over the full return path, not point-in-time moments.
  • Computational cost: More auxiliary variables than CVaR; larger conic problem.
  • History-bound: EDaR uses the empirical drawdown path; sample noise is high in short histories.
  • Conservative: Heavier-tailed than CDaR can cost long-run return.

References

  • Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). "Drawdown Measure in Portfolio Optimization." International Journal of Theoretical and Applied Finance, 8(1), 13-58.
  • Ahmadi-Javid, A. (2012). "Entropic Value-at-Risk: A New Coherent Risk Measure." Journal of Optimization Theory and Applications, 155(3), 1105-1123.
  • Cajas, D. (2021). "Entropic Portfolio Optimization: A Disciplined Convex Programming Framework." SSRN Working Paper.
  • Zabarankin, M., Pavlikov, K., & Uryasev, S. (2014). "Capital Asset Pricing Model (CAPM) with Drawdown Measure." European Journal of Operational Research, 234(2), 508-517.
  • Goldberg, L. R., & Mahmoud, O. (2017). "Drawdown: From Practice to Theory and Back Again." Mathematics and Financial Economics, 11(3), 275-297.
  • Palomar, D. P. (2025). Portfolio Optimization: Theory and Application. Cambridge University Press, Chapter 10 (Portfolios with Alternative Risk Measures), Section 10.5 (Drawdown Portfolios).
  • skfolio documentation — skfolio.RiskMeasure.EDAR.