Time to Kick VaR to the Curb

Or, how two moment mean-variance killed modern finance theory.

July 5, 2010

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1021864_canThe next presentation at the Niagara Institutional Dialogue that I wish to comment on was a paper by Dr. William Shadwick, a Canadian financial researcher living in London, England, entitled The Right Answers to the Wrong Questions.  Bill is best known for the co-development of the widely used Omega statistic but together with his partner Dr. Ana Cascon has developed a portfolio of very useful statistics. His presentation was an ambitious one, consisting of 61 slides delivered in 50 minutes meaning the audience was deprived from seeing the most interesting tables at the end. This is one of the reasons I wish to report on his presentation because I think it makes a truly useful contribution to mathematical finance.

Shadwick’s point is that mathematical finance has developed by trying to emulate the theoretical precision of the hard sciences although it is impossible to do so with events that capture the actions of humans. His view is that mathematical finance has gotten it horribly wrong by answering all the wrong questions. In a lovely quote from Keynes he makes his point:

“We have involved ourselves in a colossal muddle, having blundered in the control of a delicate machine the working of which we do not understand.”

He makes his point early. VaR (Value at Risk) at 99% answers the question “what is the worst loss we should expect in 99 days of 100?” — or, stated in reverse, “what is the least we should expect to lose one day in 100?” He contends this is the wrong question; rather we should ask, “what should we expect to lose in that one day in 100?”

He contends that statisticians know the answer. They call it Expected Shortfall (ES). Further, the calculation of VaR relies on the normal distribution, an assumption that is hopelessly out of sync with financial data. He cited the work of innumerable well-known mathematicians, which show that financial data is far too prone to fat tails to be normal.

ES would suffer from the same problem, except that the technological developments to make it robust occurred before the crash of 1929! It is called Extreme Value Theory (EVT) and -get this – it has been in widespread use in the insurance industry for decades. It has almost been totally ignored by finance.

Shadwick took us through a brief history of mathematics in finance briefly citing the contributions from Bachelier, Fisher and Tippett, Cowles and Gnedenko all prominent mathematicians. When he got to the founder of Modern Portfolio Theory, Harry Markowitz, he made an essential point: Markowitz explicitly brought risk, measured by standard deviation into the discussion for the first time, which inadvertently sent modern finance theory headlong into the two moment world of mean-variance – and it has been paying dearly ever since. Shadwick very appropriately noted that the normality assumption was not made by Markowitz, and indeed, I have heard Markowitz say this on two occasions.

It is this assumption that is the killer in finance theory and it is an underlying factor in VaR, which was globally promulgated by shortsighted regulators at the BIS in Basel I as the metric banks must use to calculate risk.  We now know what folly that was!

Now comes the part missing in Shadwick’s presentation. In his unseen tables (they are available) he shows by employing EVT, the coming collapse was clearly showing up well ahead of its occurrence. His first set of tables analyzes the Dow Jones Index in the months leading up to the 1929 crash. Relying on the probabilities derived from a normal distribution was hopelessly wrong, but EVT was giving the signals well before the main event.

He then shows the ES with Citigroup’s stock. As early as the 27 of February 2007 the first breach showed the ES increasing from 3.93% to 5.92% (daily) and then steadily increasing from then until it went over 60%. The first really big losses in Citigroup’s stock occurred over mid to late September 2008 (!) at over 10% each day, growing to 18%, 23%, 26% and ending with a 39% one day loss.

Shadwick laments’ “we have the technology, why aren’t we using it?”  Indeed.

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Tris makes the point that the normality assumption (applying the statistical normal distribution to the behavior of financial markets returns) is "an underlying factor in VaR". I fully agree with him in his critique of the normality assumption - it is indeed unrealistic and empirically wrong, and is not made any better by the fact it still prevails in many risk models. It is not essential though for VaR calculations, and some institutions have developed more realistic models (using, for instance, long-term actual market data instead of making specific statistical assumptions) that produce figures that are much more in line with actual market behavior.

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