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Minimum CDaR Optimization

Portfolio optimization that minimizes Conditional Drawdown-at-Risk, a risk measure based on the severity of portfolio drawdowns rather than return volatility.

Overview

Conditional Drawdown-at-Risk (CDaR) is a drawdown-based risk measure analogous to Conditional Value-at-Risk (CVaR), but applied to the drawdown process rather than the return distribution. While CVaR measures the expected loss in the tail of the return distribution, CDaR measures the expected drawdown conditional on the drawdown exceeding a certain threshold.

Drawdown-based risk measures are particularly appealing to practitioners because drawdowns directly correspond to the experience of investors: large peak-to-trough declines in portfolio value are the primary source of investor discomfort and often trigger forced liquidations or redemptions. By directly minimizing drawdown risk, CDaR optimization produces portfolios that are explicitly designed to limit severe underwater periods.

Like CVaR, CDaR optimization can be formulated as a linear program, making it computationally tractable for realistic portfolio sizes.

Drawdown Definitions

Portfolio Drawdown

The drawdown at time measures the percentage decline from the running maximum of the portfolio value :

The drawdown is always non-negative: . When the portfolio is at its all-time high, . When the portfolio has declined 20% from its peak, . For a portfolio with cumulative returns , the drawdown can equivalently be expressed using the running maximum of cumulative returns.

Maximum Drawdown

The maximum drawdown over the entire investment horizon is:

While widely used, maximum drawdown depends on a single worst-case path and can be unstable across different sample periods. CDaR provides a more robust characterization of drawdown risk.

Drawdown-at-Risk (DaR)

Analogous to VaR, the Drawdown-at-Risk at confidence level is the -quantile of the drawdown distribution:

Conditional Drawdown-at-Risk (CDaR)

CDaR is the expected drawdown conditional on the drawdown exceeding the DaR threshold:

CDaR captures the average severity of the worst fraction of drawdowns. At the extremes, equals the average drawdown and equals the maximum drawdown, so CDaR interpolates between these two measures as varies.

Linear Programming Formulation

Following the approach of Chekhlov, Uryasev, and Zabarankin (2005), CDaR minimization can be cast as a linear program. Given time periods with portfolio returns determined by weights and asset return vectors , define the cumulative portfolio return up to time as:

LP with Auxiliary Variables

Introducing auxiliary variables for the running maximum, for the DaR threshold, and for excess drawdowns:

The variables track the running maximum of cumulative returns, and the drawdown at time is . The variables capture the excess drawdown beyond , and the objective averages these excesses over the worst fraction.

Special Cases

Average Drawdown Minimization

Setting minimizes the average drawdown over the entire sample period. This produces portfolios that avoid sustained underwater periods.

Maximum Drawdown Minimization

As , CDaR converges to the maximum drawdown. This is the most conservative drawdown-based objective, focusing entirely on the single worst drawdown episode.

Advantages

  • Investor-relevant risk: Drawdowns directly correspond to the investment experience and pain felt by investors.
  • Path-dependent: Captures the serial correlation and persistence of losses that variance-based measures miss.
  • LP tractable: Can be solved as a linear program, enabling efficient computation.
  • Flexible confidence level: The parameter allows smooth interpolation between average drawdown and maximum drawdown optimization.
  • Coherent: CDaR is a coherent risk measure applied to the drawdown process, inheriting the desirable axiomatic properties.

Limitations

  • Path dependence: The drawdown process depends on the order of returns, so the optimization is sensitive to the specific historical path observed.
  • Sample requirements: Requires long time series to capture meaningful drawdown events; short histories may not contain sufficient stress episodes.
  • Computational complexity: The LP formulation has more constraints than the CVaR LP due to the running maximum variables.
  • Lookback bias: Optimizing over historical drawdowns may overfit to past crisis patterns that may not repeat.

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.
  • Zabarankin, M., Pavlikov, K., & Uryasev, S. (2014). "Capital Asset Pricing Model (CAPM) with Drawdown Measure." European Journal of Operational Research, 234(2), 508-517.