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Here's the paper, this time via University College, Dublin, which I trust will be around for a while.
How Unlucky is 25-Sigma?
By
Kevin Dowd, John Cotter, Chris Humphrey and Margaret Woods§
March 24 2008
1. IntroductionBy
Kevin Dowd, John Cotter, Chris Humphrey and Margaret Woods§
March 24 2008
One of the more memorable moments of last summer’s credit crunch came when the CFO of Goldman Sachs, David Viniar, announced in August that Goldman’s flagship GEO hedge fund had lost 27% of its value since the start of the year. As Mr. Viniar explained, “We were seeing things that were 25-standard deviation moves, several days in a row.”1 One commentator wryly noted:
That Viniar. What a comic. According to Goldman’s mathematical models, August, Year of Our Lord 2007, was a very special month. Things were happening that were only supposed to happen once in every 100,000 years. Either that … or Goldman’s models were wrong (Bonner, 2007b).But sadly Goldman were not alone. In 2007 alone, massive losses were announced by Bear Stearns, UBS, Merrill Lynch and Citigroup, and then there were the earlier financial disasters – 1987, Daiwa, Barings, Long-Term Capital, the dotcoms, Russia, East Asia, and so on – and afterwards Société Générale and Bear Stearns again in early 2008, with rumours of more yet to come. Citi’s case was particularly interesting.
To quote from the same commentator:
Gary Crittenden, Citi’s chief financial officer, claimed … that the firm was simply a victim of unforeseen events. … No mention was made of the previous five years, when Citi was busily consolidating mortgage debt from people who weren’t going to repay … pronouncing it ‘investment grade’ … mongering it to its clients … and stuffing it into its own portfolio … while paying itself billions in fees and bonuses. No, according to the masters of the universe, downgrades by Moody’s and Fitch’s were completely unexpected … like the eruption of Vesuvius; even the gods were caught off guard.
Apparently, as of September 30th, Citigroup’s subprime portfolio was worth every penny of the $55 billion that Citi’s models said it was worth. Then, whoa, in came one of those 25-sigma events. Citi was whacked by a once-ina- blue-moon fat tail.
Who could have seen that coming? (Bonner, 2007c).
___________________________________________________________________________________
§ Dowd and Woods: Centre for Risk and Insurance Studies, Nottingham University Business School,
Jubilee Campus, Nottingham NG8 1BB, UK. Cotter: Centre for Financial Markets, School of Business,
University College Dublin, Carysfort Avenue, Blackrock, Co. Dublin, Ireland. Humphrey: School of
Accounting and Finance, University of Manchester, Crawford House, Oxford Road, Manchester M13
9PL, UK. Corresponding author: Kevin.Dowd@nottingham.ac.uk.
1 Reported in the Financial Times, August 13, 2007.
Be all this as it may, one thing is for sure: there are certainly a lot of very unlucky financial institutions around.
2. How Unlikely is a 25 sigma event?
The once-in-a-100,000 year figure was quoted in a number of places,2 and suggests that Goldman, Citi and so on must have been very unlucky indeed. But exactly how unlikely is a 25-sigma shock?3
To start with, lets assume that losses are normally distributed – assume that losses obey the classic bell curve – and ask the question: what is the probability of a loss that is, say, 2 standard deviations or more away from the mean, i.e., what is the probability of a 2-sigma loss event?
The answer is given in Figure 1: the probability associated with a 2 sigma event is equal to the mass of the right-hand tail of the distribution demarcated at the point where the number of standard deviations from the mean is equal to 2, and this is equal to 2.275%.4 We might therefore expect to see a 2-sigma loss event on one trading day out of 1/2.275%=43.956 days, i.e., on approximately 1 day out of 44 days.
A 2-sigma event is unlikely to occur on any given day, but we would expect to see a few of them in any given year.5
...MORE (8 page PDF)
