Chebyshev and chickens

Felix Salmon asks what the hell David Viniar, Goldman’s chief financial officer, meant when he said:

We were seeing things that were 25-standard deviation moves, several days in a row [last week].

Clearly, if Mr Viniar is thinking in terms of bell-curves, this is just cretinous. Even a 9 standard deviation event is unlikely to have happened even once a second since the universe began.
Mr Viniar can't be that stupid. He knows financial returns are not normally distributed, but have fat tails; extreme events are more likely than a bell curve suggests.
More likely, he's thinking of Chebyshev's inequality. It says that in any data sample,  no more than 1/k² of the values are more than k standard deviations away from the mean. On this view, a 25 standard deviation event is a one in 625 probability.
Does this make Mr Viniar look less stupid?
Not really. If we define "several"  as  "three", then the chance of getting three successive 1 in 625 probabilities is roughly one in 244 million.
No, however we cut it, Mr Viniar is talking wibble. He just under-estimated risk.
There's one very stupid way of doing this. Imagine you're a chicken. Every day, the farmer feeds you. After a while, you figure: "My returns from the farmer are pretty stable, as I seem to get roughly the same amount of corn every day. Being a chicken  is a low-risk business."
The following day, the farmer breaks your neck.
Any hedge fund who took default risk (say by holding CDOs) and boasted about its Sharpe ratio based on post-2003 returns would have done much the same as this; you can't measure default risk by looking at past returns.
I had assumed that no-one was stupid enough to fall for this trick; one reason why I was underwhelmed by Taleb's The Black Swan - which laboured similar points - was that I thought he was just stating the bleedin' obvious.
But perhaps I was wrong.

Comments

[dearieme]

Does this Chebyshev result hold for a Cauchy distribution?

[dearieme]

Scrap that question - bad case of mathematical poncing about.

[James Killus]

I had the same reaction to The Black Swan, although I did limit my opinion to the notion that scientists and statisticians didn't make these kinds of stupid mistakes (they argue about what the underlying distribution _really_ is to annoying and exhausting detail).

Still, the idea that someone as stupid as Viniar was playing with vast sums of money does give one pause.

[dearieme]

"scientists and statisticians didn't make these kinds of stupid mistakes (they argue about what the underlying distribution _really_ is to annoying and exhausting detail)": except, James, when they are 'climate scientists' busily 'correcting' temperature measurements.

[Mike Woodhouse]

The hedge fund manager who has just reduced his investors' wealth by some unspecified amount does not feel their pain, especially if he's extracted large performance fees for a few years previously - he's under no obligation to repay his bonuses. So he takes exposure to large low-probability downside risk because it has a positive expectation for him due to his interests being misaligned with those of the clients he is ostensibly serving.

Misalignment of objectives, my favourite grumpy old man topic. There's a book in there, I tell you.

[Tom Womack]

So, how do you realign the objectives? There are the tales of the early days of (Goldman?) where the Grand Old Man insisted that every trader keep all his bonuses in the funds he managed, with the vague possibility of withdrawing only to buy property and only once and only after explaining yourself to the Grand Old Man, but restrictions on trader behaviour are incredibly vulnerable to a race to the bottom in a market with more than one employer of traders.

[Barry]

Mike, good point. Also, in terms of distributions, I think that what's happening is that people are assuming that the current situation continues, and that a market won't suddenly become rational.

We've seen it with dot-coms, we saw it with LTCM. I'm sure that somebody casually acquainted with the financial markets can throw up 10 more examples off of the top of their hear.

[MARK T]

While they may not have to pay back (though some do) the concept of a hurdle rate means the have to make back up the losses before they can start charging again, and if you close the fund down you will struggle to open another one. Anticipate a lot of hedge funds traders going back to investment banks next year.

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