Folio Lab/Documentation

Conditional Drawdown at Risk (CDaR)

Conditional Drawdown at Risk extends DaR by taking the mean of the worst tail of the drawdown function, on one portfolio path. It is the CVaR of that function, and it inherits coherence and convexity, which makes it suitable for portfolio optimization. FolioLab reports it at the 95% level as a positive magnitude.

Overview

Just as CVaR improves upon VaR by capturing the expected severity of tail losses, CDaR improves upon DaR by taking the average of the worst of the drawdown observations. This provides a more complete picture of tail drawdown risk than the threshold-based DaR.

CDaR was developed by Chekhlov, Uryasev, and Zabarankin (2005) as part of their comprehensive framework for drawdown-based portfolio optimization. Definition 3.4 of that paper sets the conditional drawdown as the CVaR of the drawdown function of Definition 3.1, which is a function of time on one sample path. Page 23 states the reading directly: at the measure is the average of the worst 5% of drawdowns. Proposition 3.1 gives a nonnegative magnitude.

The tail is therefore taken across the dates of one path, not across the maximum drawdowns of many paths. A single backtest supplies exactly one path, so the cross-path object does not exist for it while this one does.

The key computational insight is that CDaR can be formulated as a linear program, analogous to the Rockafellar-Uryasev formulation for CVaR, which enables efficient computation even for large portfolios. FolioLab uses that formulation in the MinCDaR optimizer. The reported metric described below is a direct sample calculation, not an optimization.

In practice, CDaR is particularly valuable for investors who need to control worst-case drawdowns while maintaining a coherent and mathematically tractable risk framework. It is used in hedge fund risk management, pension fund liability-driven investing, and any context where sustained capital erosion must be carefully managed.

Mathematical Formulation

Integral Definition

CDaR at confidence level is defined as the average of all DaR values from level to 1:

This integral representation mirrors the relationship between CVaR and VaR. It averages over the entire tail of the drawdown distribution, providing a more robust measure than the single quantile used by DaR.

Conditional Expectation Form

Equivalently, CDaR is the conditional expectation of the drawdown at a date, given that the drawdown at that date exceeds the DaR threshold:

The expectation runs over the dates of one path. CDaR answers the question: "On the dates where the portfolio was deeper than the DaR threshold, how deep was it on average?" This is directly analogous to how CVaR measures the expected loss given that the loss exceeds VaR.

How FolioLab computes it

FolioLab forms the signed drawdown series against the running peak, selects the dates at or below the negated DaR threshold, and reports the absolute value of their mean:

The result is a positive magnitude at the same scale as the drawdown itself. The ordering holds by construction, because every selected date is at or beyond the threshold.

Two details separate this calculation from the paper. First, FolioLab takes a simple mean over the selected tail. Equation 3.9 of Chekhlov, Uryasev and Zabarankin uses a weighted form that splits a probability atom sitting exactly at the threshold. For a long daily series the two agree closely, and the page does not claim Equation 3.9.

Second, the series is built as on a compound equity curve. Definition 3.1 builds it on the uncompounded cumulative rate of return, and the paper notes that a relative drawdown does not carry the same properties. FolioLab uses the common library form, and it is not literally Definition 3.1.

Linear Programming Formulation

Following the Rockafellar-Uryasev approach, CDaR minimization can be cast as a linear program. For a portfolio with weights over dates, with one drawdown observation per date and a free threshold variable :

where are auxiliary variables capturing the excess drawdown beyond the threshold , and the running peak that defines enters through its own linear constraints. This convex formulation enables efficient optimization for large-scale portfolio problems, and it is the form the MinCDaR optimizer solves.

CDaR in the Risk Measure Hierarchy

CDaR sits within a natural hierarchy of drawdown-based risk measures, mirroring the return-based hierarchy:

Return-Based Measures

  • VaR: Quantile of loss distribution
  • CVaR: Expected loss beyond VaR
  • Coherent and convex

Drawdown-Based Measures

  • DaR: Quantile of the drawdown function
  • CDaR: Mean drawdown beyond DaR
  • CDaR is coherent and convex

The ordering always holds on a given path, ensuring CDaR provides a more conservative risk estimate than DaR while remaining less extreme than the deepest single drawdown on that path.

Advantages & Limitations

Advantages

  • Coherent drawdown measure: Satisfies subadditivity, ensuring diversification benefits are properly reflected in drawdown risk.
  • Tail sensitivity: Captures the full severity of worst-case drawdowns, not just the threshold.
  • Convex optimization: Can be minimized via linear programming, enabling efficient portfolio optimization.
  • Comprehensive risk view: Combines the path-dependent nature of drawdown measures with the probabilistic rigor of tail risk measures.

Limitations

  • Data requirements: The tail holds only 5% of the dates, and those dates usually sit inside one or two drawdown episodes, so a short series gives an unstable estimate.
  • Computational cost: The LP formulation adds one auxiliary variable and one constraint per date, so it grows with the length of the series and can become expensive.
  • Serial dependence sensitivity: Results depend heavily on the assumed return dynamics and autocorrelation structure.
  • Limited analytical results: Unlike parametric CVaR, there are few closed-form expressions for CDaR under standard distributions.

References

  1. Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). "Drawdown Measure in Portfolio Optimization." International Journal of Theoretical and Applied Finance, 8(1), 13-58.
  2. Goldberg, L. R., & Mahmoud, O. (2017). "Drawdown: From Practice to Theory and Back Again." Mathematics and Financial Economics, 11(3), 275-297.
  3. Rockafellar, R. T., & Uryasev, S. (2002). "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking & Finance, 26(7), 1443-1471.
  4. Uryasev, S. (2000). "Conditional Value-at-Risk: Optimization Algorithms and Applications." Financial Engineering News, 14, 1-5.

Not investment advice. Past performance is not indicative of future results.