Conditional Drawdown at Risk (CDaR)
Conditional Drawdown at Risk extends DaR by measuring the expected drawdown in the worst-case tail of the drawdown distribution. It is the drawdown analogue of CVaR and inherits its desirable properties, including coherence and convexity, making it suitable for portfolio optimization.
Overview
Just as CVaR improves upon VaR by capturing the expected severity of tail losses, CDaR improves upon DaR by measuring the average drawdown in the worst of drawdown scenarios. 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. The key insight is that CDaR can be formulated as a linear program, analogous to the Rockafellar-Uryasev formulation for CVaR, enabling efficient computation even for large portfolios.
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 can be expressed as the conditional expectation of the maximum drawdown given that it exceeds the DaR threshold:
This shows that CDaR answers the question: "Given that the drawdown exceeds the DaR threshold, what is the expected magnitude of the drawdown?" This is directly analogous to how CVaR measures the expected loss given that the loss exceeds VaR.
Discrete (Practical) Estimation
For empirical estimation from a sample of maximum drawdown observations, CDaR is computed as:
The first term is the DaR threshold, and the second term adds the average excess drawdown beyond that threshold, weighted by the tail probability. This formula is the drawdown equivalent of the standard CVaR estimation formula.
Linear Programming Formulation
Following the Rockafellar-Uryasev approach, CDaR minimization can be cast as a linear program. For a portfolio with weights and time periods with scenarios:
where are auxiliary variables capturing the excess drawdown beyond the threshold . This convex formulation enables efficient optimization for large-scale portfolio problems.
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 MDD distribution
- CDaR: Expected MDD beyond DaR
- Coherent and convex
The ordering always holds, ensuring CDaR provides a more conservative risk estimate than DaR while remaining less extreme than the absolute worst-case drawdown.
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: Reliable estimation requires extensive historical data or a large number of simulated scenarios.
- Computational cost: The LP formulation grows with the number of scenarios and time periods, potentially becoming 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
- Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). "Drawdown Measure in Portfolio Optimization." International Journal of Theoretical and Applied Finance, 8(1), 13-58.
- Goldberg, L. R., & Mahmoud, O. (2017). "Drawdown: From Practice to Theory and Back Again." Mathematics and Financial Economics, 11(3), 275-297.
- Rockafellar, R. T., & Uryasev, S. (2002). "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking & Finance, 26(7), 1443-1471.
- Uryasev, S. (2000). "Conditional Value-at-Risk: Optimization Algorithms and Applications." Financial Engineering News, 14, 1-5.