Efficient Conditional Value At Risk Frontier

Conditional value at risk is a measure often used in portfolio optimization for effective risk management. The CVar (conditional value-at-risk) also called expected shortfall is a popular measure of tail risk. CVaR is derived by taking a weighted average of the "extreme" losses in the tail of the distribution of possible returns, beyond the value at risk (VaR) cutoff point \(\beta\).

For example, if we calculate the CVaR to be 10% for \(\beta=0.95\), we can be \(95\%\) confident that the worst-case average daily loss will be 3 %.

In other words, one can see CVar in terms of percentiles, so that it boils down to the average of all losses so severe that they only occur \((1−\beta)\%\) of the times.

We will adopt the following notation:

  • w for the vector of portfolio weights
  • r for a vector of asset returns (daily), with probability distribution \(p(r)\).
  • \(L(w, r) = - w^T r\) for the loss of the portfolio
  • \(\alpha\) for the portfolio value-at-risk (VaR) with confidence \(\beta\).

The CVaR can then be written as:

\[\begin{equation} CVaR(w, \beta) = \frac{1}{1-\beta} \int_{L(w, r) \geq \alpha (w)} L(w, r) p(r)dr. \end{equation}\]

This is a nasty expression to optimize because we are essentially integrating over VaR values. The key insight of Rockafellar and Uryasev (2001) 1 is that we can can equivalently optimize the following convex function:

\[\begin{equation} F_\beta (w, \alpha) = \alpha + \frac{1}{1-\beta} \int [-w^T r - \alpha]^+ p(r) dr, \end{equation}\]

where \([x]^+ = \max(x, 0)\). The authors prove that minimising \(F_\beta(w, \alpha)\) over all \(w, \alpha\) minimises the CVaR. Suppose we have a sample of T daily returns (these can either be historical or simulated). The integral in the expression becomes a sum, so the CVaR optimization problem reduces to a linear program:

\[\begin{equation*} \begin{aligned} & \underset{w, \alpha}{\text{minimise}} & & \alpha + \frac{1}{1-\beta} \frac 1 T \sum_{i=1}^T u_i \\ & \text{subject to} & & u_i \geq 0 \\ &&& u_i \geq -w^T r_i - \alpha. \\ \end{aligned} \end{equation*}\]

This formulation introduces a new variable for each datapoint (similar to Efficient Semivariance), so you may run into performance issues for long returns dataframes. At the same time, you should aim to provide a sample of data that is large enough to include tail events.

Minimum CVar portfolio 📖

Efficient return on mean-cvar frontier 📖

Efficient risk on mean-cvar frontier 📖

References

  1. Chekhlov, A.; Rockafellar, R.; Uryasev, D. (2005). Drawdown measure in portfolio optimization ↩