Efficient Conditional Drawdown At Risk Frontier¶

Conditional drawdown at risk is a measure often used in portfolio optimization for effective risk management introduced by Cheklov, Uryasev and Zabarankin in 2000 ¹,². The CDar (conditional drawdown-at-risk) is the average of all drawdowns for all the instances that drawdown exceeded a certain threshold. Drawdown is a measure of downside risk.

Conditional drawdown is the average of all drawdowns, or cumulative losses, in excess of a certain threshold. That threshold is referred to as drawdown-at-risk. It is very similar to how expected shortfall is defined.

CDar in the context of portfolio optimization was first proposed by Alexeei Chekhlov, Stanislav Uryasev and Michael Zabarankin in 2003. The main goal of their paper was to show that the optimization of CDar can be solved using linear programming, a kind of problems that cvxpy is very good at solving.

Following the specifics present at the PyPortfolioOpt documentation, the notation required to formulate the drawdown concept mathematically is the following:

\begin{equation} \textrm{CDaR}(\mathbf{w}, \beta) = \frac{1}{1-\beta} \int_{D(w,r,t) \geq \alpha(w)} D(w,r,t) p(r(t)) d r(t) \end{equation} In the above expression the quantities are

• \(\mathbf{w}\) is the vector of portfolio weights
• \(\mathbf{r}\) is the vector of cumulative asset returns (daily), with p.m.f \(p(\mathbf{r}(t))\).
• \(D(\mathbf{w},\mathbf{r},t) = \max_{r < t}\left( \mathbf{w}^T r(\tau) - \mathbf{w}^T\mathbf{r}(t)\right)\) is the drawdown of the portfolio.
• \(\alpha\) is the portfolio drawdown (DaR) with confidence \(\beta\).

The portfolio conditional drawdown at risk is based on using portoflio drawdowns rathar than returns. For the definition look here and the original PyPortfolioOpt version.

Here we offer three specific classes to handle the efficient CDar frontier, namely the portfolio with the Minimum CDar, and the portfolios of lowest CDar given target return or the highest return given CDar.

Minimum CDar portfolio¶

The minimum CDar portfolio identifies the point with the least Conditional Drawdown at risk along the CDar efficient frontier. The optimization problem is solved through the MinimumCDar class.

Efficient return on mean-cdar frontier¶

The efficient return CDar-frontier portfolio identifies the point with the least Conditional Drawdown at risk along the CDar efficient frontier, conditioned on having at least the specified target return. The optimization problem is solved through the CDarEfficientReturn class.

Efficient risk on mean-cdar frontier¶

The efficient risk CDar-frontier portfolio identifies the point with the highest return having at most the target Conditional Drawdown at risk. The optimization problem is solved through the CDarEfficientRisk class.

References¶

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