Gene Expression

The function that f**ked the world?

Recipe for Disaster: The Formula That Killed Wall Street:

In hindsight, ignoring those warnings looks foolhardy. But at the time, it was easy. Banks dismissed them, partly because the managers empowered to apply the brakes didn’t understand the arguments between various arms of the quant universe. Besides, they were making too much money to stop.

They didn’t know, or didn’t ask. One reason was that the outputs came from “black box” computer models and were hard to subject to a commonsense smell test. Another was that the quants, who should have been more aware of the copula’s weaknesses, weren’t the ones making the big asset-allocation decisions. Their managers, who made the actual calls, lacked the math skills to understand what the models were doing or how they worked. They could, however, understand something as simple as a single correlation number. That was the problem.

Pr[TA < 1, TB < 1] = ?2(?-1(FA(1)),?-1(FB(1),?)), the Gaussian copula. I’m waiting for the book to come out titled “The World Is Not Normal.” Here’s a PDF of the paper referenced in the article, a quick skim doesn’t suggest that it’s really that opaque a piece of financial mathematics. The problem seems to be the same as with Value at Risk which was profiled in The New York Times Magazine. Long Term Capital Management was claiming regular occurrences of “10 sigma events” when it was going through its meltdown in 1998. Risk has a tail as long and vicious as a Diplodocus.

Update: See comments at Marginal Revolution.


  1. #1 John Emerson
    February 23, 2009

    Speaking as a Luddite, I’d like to mention that our intellectual elite fucked up. Everyone was sleazing through. The non-quants were winging it, the quants were happy to get paid, the media and political class were cheerleading, and as long as the bubble kept going everyone was smiling. And now some of them are burned, and most of us will be bruned worse.

    I don’t understand any of the math, so I could just stand and watch. But doesn’t this show that IQ by itself is worthless? The quants were presumably brilliant, and I doubt that the finance people were at all dumb, but between venality and incomprehension of concret reality, they took the whole world down with them.

  2. #2 Joshua Zelinsky
    February 24, 2009

    The vast majority of the time the world is normal to a close approximation. The problem is that occasionally when things aren’t normal they are really not normal.

  3. #3 statsquatch
    February 24, 2009

    There has to me more to it than that. Everyone knows that survival distributions are long tailed. Why would anyone think taking the inverse normal CDF of two empirical survival CDFs would give you two normal variables? And how would you estimate the correlation? Nobody models multivariate survival data that way.

  4. #4 Thorfinn
    February 24, 2009

    It seems like a bigger problem was the “Besides, they were making too much money to stop.” Incentive plans that take the long-term into account may have been more helpful than extra math background.

  5. #5 JM
    February 24, 2009

    John the problem is really a clash between financial knowledge and the maths.

    Bad quants don’t know finance and assume that the model is the world. Good quants know that the model doesn’t capture all risk and good traders know it too, and are very well aware of fat tail risks.

    The problem is that when you hit the market you have to price your deals competively, so if there is a “standard” formula for pricing risk in the market (Black Scholes is the classic example) all of your competitors and your customers are going to use that as a rule of thumb.

    Now anyone with any intelligence knows that formula underprices risk at the extremes but because everyone tries to trade close to at the money they don’t worry too much. The Black Scholes model is pretty accurate at the middle. So long as they can keep their portfolio away from extremes they know they’re close enough.

    They then (if they’re at least sentient) do ad hoc stuff to compensate for the inadequacy of the log-normal assumption. The classic example again is in Black Scholes where “volatilty smiles” are introduced. In basic Black Scholes volatility is assumed to be constant, but it is known that deep out of the money or in the money deals exhibit far greater risk (ie. fat tails where the probability of loss is greater than a normal distribution would predict), so an arbitrary increase in volatility is introduced at the extremes (unless you are a certain head trader at a major UK bank in the mid-90’s, do that and you’ll go on to lose a few hundred million pounds for your employer)

    So now there is a tension between the model and the amount of arbitrary tweaks you introduce to protect yourself from unquantified risks lurking out there in reality. If you tweak too much (ie. are too conservative) you won’t be competative and won’t do any business, tweak too little and you’ll do alright for a while but eventually blow up.

    (Actually you’ll probably make out like a bandit until the inevitable catastrophe – google “capital destruction partners”)

    To date the reaction to this problem has been to resort to the rational market hypothesis and look at the prices being shown by everyone else. Since the market, by definition, knows more than you do its price should be accurate.

    So to do business all you have to do is shade your price a bit and customers will come to your door.

    But if everyone shades their price the market can develop a bias to low pricing of risk.

    Quants in interest rate markets have been dealing with this sort of problem for years and have elaborated and extended the basic model to quantify all of the nuances and most of those more elaborate models price risk much more finely.

    Now we get to the next problem. What I’ve described above captures market price risk but not credit risk. Market price risk is semi-observable, you just look in the market and derive the price volatility from price movements.

    But credit risk is not so easy. There is nothing observable. For that you have to see what people are willing to pay. The discount the market applies represents the perceived risk (so long as you believe in rational markets).

    So if a company’s bond is selling at a 10% discount you get a picture of how risky they are. But you don’t really know, so you have to rely on rating agencies.

    Then you start creating derivatives on that risk (Credit Default Swaps which are like insurance, and the less common Total Return Swap which is like common stock)

    There is nothing in principal wrong with either of those two instruments (although they are hard to manage and require a lot of close attention to legal fine print).

    But. An insurance company selling straight up insurance policies has to show to a regulator that it has the financial backing to meet claims.

    A hedge fund selling a CDS does not. It is unregulated. It can sell these things till the cows come home without ever having to demonstrate any backing at all. That’s what regulation means – not some big brother looking over their shoulder, but rather that they have to put up money to show financial capacity.

    The market tried to compensate for this a bit when purchasers of CDS’s asked the sellers to post collatoral. That should have worked but it actually made things worse as when sellers were put under pressure and couldn’t increase collatorel their own credit rating fell which caused more customers to demand even more collatoral.

    That’s a death spiral.

    The last piece of the puzzle is that because the credit risk was tradable (but not in a transparent market) you never knew if your insurer (the seller of a CDS) was sufficiently hedged to withstand a default, and more importantly you didn’t know who they were exposed to. It’s possible that they would lose their hedge if the very event you were insuring against occurred.

    That’s where correlation comes in and the Gaussian model fails.

    My belief is that it was pretty widely known that the Gaussian Copula didn’t fully capture these correlations (as represented by the systemic links), but since that model created the “rule-of-thumb” price it was impossible to trade in the market by deviating from it.

    Regulation – by forcing participants to back up their bets – would have compensated, but it didn’t exist.

    Our host’s comment re LTCM reporting regular 10 sigma events is also pertinent:- a good risk management framework would have red-flagged those events. Management that doesn’t ask itself “why were there 10 once-in-a-billion-years events last month” would have been forced to explain and address the problem.

  6. #6 razib
    February 24, 2009

    My belief is that it was pretty widely known that the Gaussian Copula didn’t fully capture these correlations (as represented by the systemic links),

    you obviously know a lot more about this than me, but my cursory reading over the past 6 months suggests exactly this. one of the reasons people were irritated with the black swan is that it said what a lot of people already knew, except with a lot of prose and publicity. all that being said, look where we are today. the problem isn’t the quantitative models, the shortcomings are pretty obvious, but the social ecology which allows one to act rationally on those short comings.

  7. #7 dearieme
    February 24, 2009

    At bottom, however entertaining the Central Limit Theorem is, Gaussian distributions are used because sums of squared discrepancies are continuous and differentiable, and their derivatives give you nice linear maths. That’s all very well, but if God has decided that the world isn’t like that, you’re stuffed.

  8. #8 John Emerson
    February 24, 2009

    At the URL: “A funny math joke just destroyed half of everything”.

    So now there is a tension between the model and the amount of arbitrary tweaks you introduce to protect yourself from unquantified risks lurking out there in reality.

    Much of economics needs tweaks in order to be applicable. Assume we have a can opener. The jokes are suddenly less funny.

    I interpret it as 1.) management failure to integrate the quants and the finance 2.) motivated by wishfulness, laziness, and greed, within 3.) a perverse, unregulated bubble market which was 4.) in part the result of ideologically-motivated deregulation, in 5.) an atmosphere of irrational optimism which was a matter of social psychology.

    The most rational response for someone in finance would have been to get out, and enough people had done that everyone would now be better off. But someone who got out too soon would be worse off than someone who got out at exactly the right time, so people tended to wait too long.

  9. #9 John Emerson
    February 24, 2009

    It may be that Taleb was also letting out everyone’s dirty little secret. A lot of people knew, but they weren’t necessarily saying. If they had been talking, they would have become as famous as Talib.

    People have been talking about this kind of stuff for decades, but economics has gotten away with ignoring it, because a.) the old-timers controlled hiring, firing, and promotion, and b.) you can’t argue with success.

    Maybe now that we’ve got failure, maybe the profession’s invulnerability will be gone.

  10. #10 Eric Dennis
    February 24, 2009

    That article is atrocious. I won’t get into all the technical errors contained in it.

    The pure Gaussian copula ceased to be the market standard around 2005. Ever since markets on index tranches started to be quoted, people realized that the normal distribution was just not rich enough to capture the prices the *market* was puting on these tranches.

    Someone a while ago came up with this idea that some mad scientist must be behind the financial cataclysm. There must always be a single evil individual, and he should definitely have been the kind of guy stuffed into lockers in high school.

    Ironically, this template is not far from the truth, at least the proximate, economic truth. That guy, however, is not David Li. It’s the guy who thought his random guess ought to replace an entire market of savers and borrowers: Alan Greenspan (with an assist from Bernanke).

  11. #11 John Emerson
    February 24, 2009

    Plenty of blame to go around, Mr. Quant.

  12. #12 razib
    February 24, 2009

    Someone a while ago came up with this idea that some mad scientist must be behind the financial cataclysm. There must always be a single evil individual, and he should definitely have been the kind of guy stuffed into lockers in high school.

    i think the attempt shoehorn the cause into a large effect goes beyond individuals, but into sectors and what not. as it is, it seems like there were systemic issues, as they say….

  13. #13 gcochran
    February 24, 2009

    I thought at the time that it was just another real estate bubble, like others before but larger. Sure to pop. I don’t remember doing any fancy modeling: am I inferior?

  14. #14 John Emerson
    February 24, 2009

    Like others before but larger

    Quantitative becomes qualitative sometimes.


  15. #15 Eric Dennis
    February 24, 2009

    Plenty of blame to go around, Mr. Quant.

    That is the mental smokescreen people throw up when the unravelling process seems too daunting. Actually, there isn’t plenty of blame.

    Economics is currently in the state that biology was in when Lamarckism was still a going concern. People with a sound understanding of macroeconomics understood years ago that the present crisis was coming, and precisely who was to blame (the Fed), even if the details of how it would play out were hard to presage.

    Unfortunately for laymen, such people do not include the vast majority of economics professors or CNBC talking heads or Fed chairmen.

  16. #16 John Emerson
    February 24, 2009

    Let me guess, you’re an Austrian.

  17. #17 toto
    February 25, 2009

    I’m waiting for the book to come out titled “The World Is Not Normal.”

    Benoit Mandelbrot, “Fractals and Scaling In Finance”.

    Back in the 60s, Mandelbrot did the simplest of things: just plot stock price variations over hours, weeks, years or decades. He found two things:

    1) They’re similar at all scales and

    2) They’re emphatically not normal (power laws, fat tails, yadda yadda).

    Unfortunately these results were published in French. Perhaps now more people will wise up to the consequences. (See also this recent non-technical essay, co-written with with Taleb).

  18. #18 John Emerson
    February 25, 2009

    Mandelbrot published a lot in English 1961-1971. See Mirowski, “The Effortless Economy of Science”. All of Mirowski’s stuff is good. In this book, he points out that econ has been stealing ideas from other sciences forever, but rarely the other way around. Ernest Gellner made the same point: We’ll know that a social science is a real science when its methods start to be borrowed by the physical sciences.

    Mandelbrot actually published some of his first work in economics, but wasn’t really taken seriously there. After that, he went to physics and found acceptance. That’s Mirowski’s story anyway.

    (For the record, evolutionary genetics may also use models originating in econ.)

    This has all been frustrating to me, because I don’t have the math and always have economists pulling rank on my that way. (I’m interested in economics because I’ve found that the world is ruled by economists). But there have been a lot of people for a long time saying that econ’s models are inappropriate or irrelevant and the math inappropriate, unnecessary, or even flawed, and by and large they’ve been marginalized. Professional economists’ scientific common sense hasn’t had to change, until now, because you can patch and kludge a theoretical problems for a long time before shit hits the fan.

  19. #19 Eric Dennis
    February 25, 2009


    Yes, I’m an Austrian. And, despite that, I do have the math. Unfortunately, the mathematicization of modern macroeconomics has been a useless diversion. It’s all Ptolemaic epicycles — as if Kepler and Galileo (much less Newton) had never been born. That the epicycles are expressed in terms of some kind of control-theoretic stochastic differential equations doesn’t impress me.

    Actually I’m less pessimistic about the utility of mathematics in this context than, say, Mises. But there is no question that you have to settle some basic matters of principle before any mathematical theory can be taken seriously.

  20. #20 John Emerson
    February 25, 2009

    Possibly, now that the orthodoxy has been shaken and discredited, there can now be a serious all-comers debate about the fundamentals of economics. While things were going well the combination of institutional dominance and “can’t argue with success!” kept economics invulnerable. Hopefully that has changed.

    I don’t think that anyone will, or should, come out of the debate intact, including the Austrians, Marxists, and other dissidents. but the driticisms of mainstream neo-classicism have been piling up for decades, and maybe now the dam will burst.

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