Conditional VaR

"Finance is an area that's dominated by rare events," says Nassim Taleb*. "The tools we have in quantitative finance do not work in what I call the Black Swan domain."

To illustrate what the Black Swan stands for, let’s look back at the past couple of months. Many funds or portfolios have performed below the 95% confidence interval (below their VaR) several days in a row. These extreme events do not seem as extreme anymore… at least not as rare as ordinary quantitative finance techniques would tell us.

The Value at Risk is a simple and convenient measure used by banks to calculate the maximum loss they can suffer each day. It is defined as the percentile of loss distribution, i.e. the smallest value such that the probability that losses exceed this value is greater or equal to a given threshold.

The Value at Risk, failed to detect the gravity of the U.S. subprime crisis. "VaR, stress tests and other risk measures significantly underestimated the magnitude of actual loss from the unprecedented credit market environment," according to Merrill third-quarter filing with the U.S. Securities and Exchange Commission. "In the past, these AAA ABS CDO securities had never experienced a significant loss in value."

Conditional VaR: an alternative measure of extreme losses

An alternative to the VaR has emerged as an important tool in risk and asset management: the Conditional Value-at-Risk. The CVaR at a given confidence level is the expected loss given that the loss is greater than or equal to the VaR at that level. In other words, it is the average of all the losses below the considered quantile (5% for the CVaR 95% for instance). As opposed to the CVaR, the VaR does not tell you anything about how bad things get in the worst 5% cases.


Furthermore, the CVaR is a simple convenient representation of risks that features nice properties:

• - It measures the downside risk (the "bad" risk)
• - It is applicable to non-symmetric loss distributions
• - It accounts for risks beyond VaR (more conservative than VaR)
• - It is continuous with respect to the confidence level
• - It is convex with respect to the portfolio positions : this property is very important on the portfolio optimization framework, as it ensures the stability of the solution and the convergence of the numerical implementation
• - It is a coherent measure of risk: translation invariant, sub-additive, positively homogeneous, monotonic (see reference [1]). The subadditivity allows to measure the diversification impact of adding new portfolios or business lines. With a coherent risk measure, one can allocate capital in a fair way to the various businesses.
  
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Optimizing the CVaR

Another interesting property of the CVaR is that it is well adapted to the active risk management (or risk budgeting) approaches in Asset Management when dealing with non-normally distributed assets. Uryasev & Rockafellar (cf. [2]) approximate the CVaR optimization of a portfolio as a Linear Program (LP). This LP approach is based on MC simulations (stochastic optimization) and can be used to solve very large optimization problems. Yet, linear programming can be the source of instabilities.

At RaisePartner, we chose to implement a different approach for CVaR optimization. Our experience with portfolio optimization issues led us to privilege the strongly convex formulation of the optimization problems. The convex property of an optimization problem ensures the stability of its solutions and the convergence and efficiency of the numerical implementation.

We came up with a convex formulation of the CVaR optimization problem, with directly interpretable parameters. As a consequence, we get an implementation with nice geometric properties and avoid the costly MC simulation implied by the stochastic optimization approaches.


* Nassim Nicholas Taleb is an essayist, researcher, and former trader.

References

[1] Artzner, P., Delbaen, F., Eber, J.-M. Heath D. Coherent Measures of Risk, Mathematical Finance, 9 (1999), 203--228.

[2] Rockafellar R.T. and S. Uryasev (2000): Optimization of Conditional Value-at-Risk. The Journal of Risk. Vol. 2, No. 3, 2000, 21-41