Stressed Value-at-Risk To Be Required by the Revised Basel II Capital Framework

It can no longer be disputed that the catastrophic losses suffered by financial institutions worldwide as a result of the credit crisis of 2007-8 were not anticipated by the industry standard methodology for predicting worst case losses, known as Value-at-Risk, or VaR. The failure of VaR to predict the actual worst case losses that were suffered mostly in 2008 meant that financial institutions that used the methodology had set aside inadequate capital reserves. The lack of adequate reserves led directly to the spectacular failure of two legendary Wall Street investment banks, Bear Sterns (which was acquired by J. P. Morgan at a fire sale price) and Lehman Brothers, which ceased to exist in September of 2008, triggering extreme volatility in the equity markets and the largest one-day loss to date on the Dow Jones industrial average. Other victims include the American Insurance Group (AIG), Freddie Mac and Fannie Mae, all of which were taken over by the federal government. The thundering herd of Merrill Lynch was lassoed by Bank America, which also suffered staggering losses in the credit markets. Citigroup, the house that Sandy Weill built, was reduced to 'zombie bank' status as its share value plunged from the mid-fifties to penny stock levels. Collateral damage spread around the globe as credit markets devolved into a classic liquidity squeeze, rendering the true market value of hundreds of billions of dollars of credit derivatives unknowable. Citigroup alone lost over 40 billion dollars as a result of the collapse in the value of the credit derivatives it was holding when the markets seized up.

 

None of these losses was predicted by the VaR methodologies that were specified by the original Basel II accords in 1996, and subsequently adopted by most banks as an industry standard risk measure, with the possible exception of the Goldman Sachs implementation.

 

In an attempt to address the clear tendency of VaR methodologies to seriously underestimate actual market risk, the expanded Basel Committee on Banking Supervision has approved a package of amendments to the Basel II accord to strengthen the 1996 rules governing  the three pillars of the Basel II framework. One of these amendments addresses the Pillar 1 VaR shortcomings directly. To quote from a BIS website article (http://www.bis.org/publ/bcbs158.htm):

 

"An additional response to the crisis is the introduction of a stressed value-at-risk requirement. Losses in most banks’ trading books during the financial crisis have been significantly higher than the minimum capital requirements under the former Pillar 1 market risk rules. The Committee therefore requires banks to calculate a stressed value-at-risk taking into account a one-year observation period relating to significant losses, which must be calculated in addition to the value-at-risk based on the most recent one-year observation period. The additional stressed value-at-risk requirement will also help reduce the procyclicality of the minimum capital requirements for market risk"

 

The pressing technical issue now facing financial institutions that intend to comply with the amended Basel II framework is to understand how to calculate a valid stressed VaR number. A simplistic approach might be to increase the assumed volatilities of the securities in a portfolio, This would have the effect of lengthening the tails of the Gaussian (normal) loss distributions that underlie all standard VaR calculations. A more sophisticated approach might include employing 'fat-tailed' distributions instead of the Gaussian to try to model the extreme loss tail events more accurately. These 'extreme value theory' distributions have names like Gumbel, Generalized Pareto, Weibull, Frechet, and the Tukey g&h.

 

However, simply fattening the tails of the estimated loss distributions is only a half step towards a true stressed VaR calculation. In order to calculate stressed VaR accurately it is also necessary to stress the correlation matrix used in all VaR methodologies. (The correlation matrix describes the tendency of the prices of the various securities in a portfolio to change together or in opposite directions.) It is a repeated observation that during times of extreme volatility, such as occurs during every market crash, correlations are dramatically perturbed relative to their 'normal' historical values. In general, most correlations tend to increase during market crises, asymptotically approaching 1.0 during periods of complete meltdown, such as occurred in 1987, 1998 and 2008. (This is due mainly to a phenomenon called "tail contagion" whose existence is known only to a relatively small group of Wall Street quantitative risk analysts.) Unfortunately, stressing the correlation matrix is not as straightforward as stretching the tails of the loss distributions. The VaR calculation engine requires a correlation matrix that satisfies the mathematical property of positive definiteness, which is a fancy way of saying that all of the correlations are internally consistent with each other. Noisy or erroneous historical price data can result in matrices that are not positive definite. Perturbing the correlation matrix, which is necessary for a true stressed VaR calculation, will invariably result in correlation matrices that also violate the internal consistency requirement. If the matrix is not positive definite the VaR math will  fail, so methods have to be devised to modify the stressed matrix until it becomes positive definite. There are a variety of techniques available to transform a non-positive definite matrix into one that is positive definite, all of which involve modifying one of more of the off-diagonal elements. However, there is only one known workable technique for finding an optimally modified correlation matrix, that is, the unique positive definite matrix that requires the least net modification to the original stressed, non-positive definite matrix.

 

The amended Basel II framework requiring stressed VaR takes effect as of Dec. 31 of 2010. In the final analysis, this requirement is still inadequate to capture all of the the extreme losses that can be inflicted on financial institutions during a market panic. For this the stressed VaR calculations should be augmented by a suite of  worst case scenario tests that can better capture the nonlinear dynamics that dominate market behavior during a major meltdown. But in the interim the new Basel II amendments are certainly a step in the right direction.