How big is ‘bad luck’?
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When Tim Harford published a second edition last year of his splendid book, The undercover economist, he added a chapter on the GFC. This week he’s made the chapter, Rotten eggs and rotten investments, available for free via download.
I’m always interested in the mis-use of statistics. I was struck by these paras (page 9) where he explains how investments that were thought to be ridiculously safe turned out to be incredibly dangerous.
To give you a sense of just how badly wrong the sums of the financial mathematicians became in reality, ponder the words of David Viniar, the Chief Financial Officer of Goldman Sachs. At the beginning of the credit crunch, he explained the sudden loss of 30 per cent of the value of a Goldman Sachs spin-off fund by saying, ‘We were seeing things that were 25-standard deviation moves, several days in a row.’
Viniar meant that Goldman Sachs had been unlucky. But just how unlucky, exactly? The financial economist Kevin Dowd has calculated that (given some reasonable assumptions) we’d expect to see three 25-standard deviation days in a row roughly once every
28,900,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000 000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000 years.
For reference, the universe is about 13,000,000,000 years old. Bad luck is not really an explanation. Somewhere, somehow, Goldman Sachs had got its sums wrong and then the inexorable complexity of the financial instruments it was dealing with magnified the error almost beyond comprehension.
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That calculation isn’t strictly true – it relies on the events not being correlated. In reality, though, a 25-S.D. move on day one would indicate something catastrophic – which in turn would substantially increase the likelihood of a similar move on day two.
On a Chebyshev basis (rather than 25 sigma), a 25-S.D. move would happen once in 625 results – let’s say once in 1,250 because we’re only looking at the downside.
So on day 1, something happens that happens every five years, give or take (assuming 250 trading days a year, which I know is high). The odds of another similar event happening THE NEXT DAY are going to be substantially less than 1/1,250 – they’re probably closer to 1/2 or 1/3. Given two days of runs like that, the probability of it extending into a third day might be better-than-even. Let’s say that the odds of that three-day run are 1/(1250*3*2), or one in 7,500. So it’s actually about a once-in-thirty-year bad run. The universe is about 13,000,000,000 years old, so it’s not unreasonable to think that sort of thing might happen once or twice.
Now, as long as the likelihood of that fall remains the same over the long term, it still may be a 25-S.D. event, but the fact that the events aren’t _randomly_ distributed means that the calculations aren’t appropriate in this case.
That’s not to say that there shouldn’t be risk management in place to handle a once-in-thirty-year bad run, or that judgement shouldn’t have stepped in to say that it was time to hedge, but Harford’s not doing himself any favours by deliberately misinterpreting the statement.
But 25 SD moves? Think. Suppose an airline found that an aircraft crashed, and then the subsequent investigation found that the crash was due to a major structural deficiency. And its whole fleet had the same structural deficiency, and it was so serious that it could not be rectified. What would happen to its share price? A 30% drop? More like a 90% drop, probably.
I would suggest that that would be either well more than a 25 SD drop, or else the terminology for a SD may not be applicable. Much more likely the latter.
You think my example is not reasonable? Consider BOAC which, after three crashes, found that the design of the Comet 1 airliner was flawed and the aircraft were totally unsafe for the job they were supposed to do. Luckily it had plenty of other airliners, so was not put out of business. IIRC the aircraft were then transferred to the defence department (whatever name) and used for non-pressurized low level flying.
But I doubt that Goldman Sachs ever did a proper financial analysis of the bundled mortgages that it was buying. More like “A mortgage is safe as houses – with all those mortgages bundled together, we cannot lose on this!”
Must read that chapter and consider if the rest of the book may be worthwhile!
The chapter does point out that the banks assumed that the events were independent and that the 25 SD moves showed that they weren’t. It also makes the point that the bankers failed to see the significance and hence referred to it as bad luck.
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