February 24, 2009

Wall Street's infatuation with Gauss

Felix Salmon has a readable article in Wired called "Recipe for Disaster: The Formula that Killed Wall Street" on David X. Li's wildly popular 2000 financial economics innovation, the Gaussian copula function, which was used to price mortgage-backed securities by estimating the correlation in Time to Default among different mortgages.

Li has an actuarial degree (among others), and that appears to have been his downfall: he assumed mortgage defaults were like Time to Death to a life insurance actuary: largely random events that could be modeled.

Steve Hsu's website Information Processing has a 2005 WSJ article on Li's Gaussian Cupola, for looking at events that are mostly independent but have a modest degree of correlation:

In 1997, nobody knew how to calculate default correlations with any precision. Mr. Li's solution drew inspiration from a concept in actuarial science known as the "broken heart": People tend to die faster after the death of a beloved spouse. Some of his colleagues from academia were working on a way to predict this death correlation, something quite useful to companies that sell life insurance and joint annuities.

"Suddenly I thought that the problem I was trying to solve was exactly like the problem these guys were trying to solve," says Mr. Li. "Default is like the death of a company, so we should model this the same way we model human life."

Uh, maybe, maybe not. There just isn't much in the field of life insurance where selling more life insurance increases the risk of death. The life insurance companies figured out the basics of moral hazard a long time ago: don't let people take out insurance policies on their business rivals or their ex-wives to whom they owe alimony. No tontines. Don't pay out on new policies who die by suicide.

In contrast, giving somebody a bigger mortgage directly raises the chance of default because they need more money to pay it back. Giving them a bigger mortgage because you are requiring a smaller down payment, in particular, raises the risk of default.

His colleagues' work gave him the idea of using copulas: mathematical functions the colleagues had begun applying to actuarial science. Copulas help predict the likelihood of various events occurring when those events depend to some extent on one another. Among the best copulas for bond pools turned out to be one named after Carl Friedrich Gauss, a 19th-century German statistician [among much else].

The Gaussian distribution (a.k.a., normal distribution or bell curve) works like this: Flip a coin ten times. How many heads did you get? Four. Write it down and do it again. Seven. Do it again. Five. As you keep repeating this flip-a-coin-ten-times experiment, the plot of the number of heads you get each time will slowly turn into a bell curve with a mean/median of five.

Now, that's really useful and widely applicable. Processes where you randomly select a sample will tend toward a bell curve distribution.

But the Housing Bubble didn't consist of fairly random events that everybody was trying pretty hard to avoid, like with life insurance. Instead, human beings were responding to incentives. The closest actuarial analogy might be the big insurance payouts that fire insurance companies got stuck with in the South Bronx in the 1970s when decayed businesses that were now worth less than their fire insurance payouts developed a statistically implausible tendency to burst into flames in the middle of the night.

As I said last fall:

Human life really isn't all that random. That's because human beings respond to incentives. If you treat human beings as if they are just mindless probabilistic events, whose risks you can diversify away by dealing with large numbers of them at a time, they will outsmart you. They will put down inflated incomes on their mortgage applications. They will claim to be owner-occupiers when they are just speculators who will rent out the property to Section 8 tenants when they get into a cash flow bind. They will bribe appraisers to report a higher than actual value.

Life insurance companies are in the selection business, not the influence business. Watching other people get rich buying and selling houses, however, influences behavior.

The life insurance actuarial model fails as an analogy for mortgages on other dimensions as well. For example, people die from a very large number of causes, making the distribution of deaths over time more Gaussian. Mortgages, in contrast, are more like being in the earthquake insurance business in California.

Further, Jim Morrison pointed out, the thing about life is that nobody gets out alive. In contrast, lots of people can imagine themselves selling the three houses in Temecula right at the top of the market and retiring to Dallas in comfort.

And there's a tournament aspect to competitive fields, such as homebuying. If you're in the Olympic boxing tournament and you get away with a few defensive lapses in your opening round match against a pudgy guy from Bhutan doesn't mean you can likely get away with them in the gold medal round against the Cuban. Similarly, when the median home price in California gets to $500k, it's not the same as when it was $200k. You can't use default data from when homes cost 40% as much. The margin for error has vanished.

Finally, the idea that just because there hadn't been a giant housing crash since WWII means there can't possibly be a giant housing crash is about 180 degrees backward. It's where there hasn't been a crash lately that you have to worry. What, did everybody expect the government to discourage home buying?

Statisticians need to be good with analogies, as well.


Anonymous said...


you are confusing independence (lack of correlation between observations) and randomness.

The mistake was to assume defaults had relatively low correlations, which is a fairly good assumption for life insurance because people tend to die independently, not in groups. (Wars are an exception and that is why insurance policies often won't pay claims for losses due to acts of war.)

But, it isn't a good assumption for defaults when defaults are driven by a systematic factor such as a nationwide housing bubble. Then, defaults can be highly correlated.

This is not randomness versus non randomness. Either way, defaults are still random. It's just that in one case the correlation is low, while in the other case it is high.

This also is not a failure of statistics or the Gaussian distribution. It is a failure to make the right assumptions and to build into the model the correct assumptions about the possibilty of correlated defaults. If they had recognized the possibility that housing prices might fall 30 percent across the country, then they would have gotten the correct result that defaults would be correlated. They assumed incorrectly that housing prices would not fall nationwide. They didn't even acknowledge that there was a non-zero chance it would happen. (Even the assumption that it was a 10 percent possibilty would have alerted them that there was a chance of an enormous meltdown and they might have at least made some contingency plans.)

Don't blame the modeling or the math or Gauss. Blame the assumptions that went into the models.

Anonymous said...

All of Wall Street actually wasn't infatuated with Gauss; the idiots in structured mortgage finance were. Everyone on Wall Street has known for years that securities returns are not normally distributed. That's why kurtosis and skewness have long been studied by entry level securities analysts. Why some people decided that more complex mortgage backed securities would exhibit Gaussian tendencies is an open question.

Frank said...

"the Housing Bubble didn't consist of random events that everybody was trying their best to avoid. Instead, human beings were responding to incentives."

Interesting observation, I'm guessing the distribution is still gaussian though. It's offset shifting along the time axis, having a shorter than expected lifetime to default. Look on the bright side, we now have an idea of what the incentives do to shorten that lifetime. But, the data probably isn't worth the trillions that just evaporated.

Anonymous said...

No, the Gaussian distribution does not assume randomness.

Anonymous said...

The frothings from Taleb are idiotic as usual. Had the world used his idea instead (*don't* infer correlation from historical data) things would obviously have been much better. Yes, sir!

The WSJ article from 12 Sept 2005 says that "the Federal Reserve Bank of New York has asked 14 big banks to meet with it this week about practices in the surging [credit derivatives] market". Is there any documentation of what happened at the meeting?

Anonymous said...

Flipping a coin ten times will give you a neat binomial distribution, not realy looking like a bell.
The more coins you flip the more this distribution will approach a real Gaussian distribution (Central Limit Theorem)

Anonymous said...

Monte Carlo distribution requires independent (non-coupled) occurrences. I think Gauss just take the data at face value.
Your point about the incentives is valid. Maybe Gauss applies for a set of houses with the same boundary conditions (incentives, value-appreciation/depreciation). I guess the criminal energy can be seen as normally distributed within a category of incentives. Basically it measures the distribution of criminal talent within an incentive class. (Obfuscating single-family houses in a sand state vs. peddling CDO’s on WS. The first is your greasy-haired, chrome-toothed second hand cars salesman type of guy. The other a slick, well-dressed crook with non-greased short hair. WTF)

Anonymous said...

Curve-fitting doesn't assume randomness.

The rather general comments suggest neither the blogger nor any commenter has actual familiarity with the model itself. (Nor do I.) Can anyone summarize the model and only afterward explain its defect(s)? Otherwise it's an argument by GCI (Greatest Common Idiot), i.e. "Someone misused it, ergo model bad."

My working guess is that, like most generally acccepted models, the model works fine.

Anonymous said...

The Economist special report on The Banking Crisis of several weeks ago also had a plausible description of the back end of the crisis, the investment bank purchases of MBSs. Basically, it is the same story: 1) the big banks used historical statistics that did not reflect the way the current market was working; and 2) the big banks knew little about housing markets.

The Economist report did not address the front end of the crisis, CRA and other corruptions of home lending.

But here is my problem with The-quants-did-it theory of even the back end of the crisis. There are always numbers nerds in financial firms, looking at statistics and coming up with predictions. There are supposed to be, and almost always are, bosses above the nerds who are talking to their friends in the relevant market, thinking through the basic problem at hand and qualifying the nerds' guidance on what it all means.

I worked as a low-level numbers nerd in a very small energy consulting firm at the turn of the 1990s. I would sit in front of a PC all day, slicing, dicing and tracking natural gas prices as the new market for spot gas evolved. I would spot all kinds of interesting nascent trends and relationships, and then rush to tell my bosses about them. They would review my conclusions, express interest and respect for the work, then say, in effect, "Yes, but..."

These superiors were in touch with other people, much more experienced in the market I was studying, who would inform them of special factors that were affecting current prices and how these factors were likely to change in the future. So my bosses would never bet their clients' businesses, much less their own company, on my hunches, patterns or relationships.

I am having an extremely hard time believing there were not the equivalents of these senior figures at very major and sophisticated institutions in the early 2000s, who at least suspected that there was more than relentless Gaussian patterns to default rates and costs.

Remember in interpreting The-quants-did-it stories, when a business creates a disaster, businessmen have an interest in looking naive or mistaken or even stupid, rather than willfully dishonest or incredibly careless.

Anonymous said...

Then take the underlying mathematical charlatanism and leverage it up 50 times.

The SEC Killed Wall Street On April 28, 2004


Anonymous said...

It's not infatuation with Gauss so much as with mathematical methods. In this wise, investors, both professionals and public, the banking industry, and regulatory agencies of government have long been influenced by economists' graphs, charts, and equations; these are nothing more (with no more efficacy) than incantations and ritual dances.

The Austrian School of economics has tried, with almost no success,
to explain such matters for nearly a hundred years. Very few care to listen; they're intent on magic.

My prediction for the future: more of the same, with occasional
"interesting" differences.

Anonymous said...

The phrase in the insurace / medical field is "Non Organic Failure to Thrive" or NOFT.

Jim Bowery said...

It seems to me the whole fiasco could have been avoided by applying The Capital Asset Pricing Model of Modern Portfolio Theory which incorporates market correlations in the Beta coefficient.

Am I missing something?

Anonymous said...

It's "cOpUla" not "cupola" (a cupola is a little dome on top of a roof).

It doesn't change the import of your words but it will hurt you with Google.

Anonymous said...

More from Hsu's blog:

Central limit theorem, securitization and CDOs

More on David Li

Copula, CDOs, structured finance

slides from talk on the credit crisis

PRCalDude said...

"Interesting observation, I'm guessing the distribution is still gaussian though."

Housing prices tend not to be gaussian distributed, which is why the median home price in an area is usually reported instead of the mean, which would be a much more meaningful statistic if the prices actually were gaussian.

Home prices follow a weibull or log-normal distribution.

Anonymous said...

Now I have to wonder, have the insurance companies factored events such as bird flu into their models?

Anonymous said...



Somehow that word, or one very like it, seems extremely appropriate here.

Anonymous said...

For anyone interested in the problems of predicting human behavior using Gaussian distribution assumptions, the best source is Nassim Taleb's "Fooled by Randomness". As Steve (and poster Terry) point out, human behavior has the potential to run off the Gaussian tracks because the participants can react strategically to one another's decisions, creating a self-perpetuating feedback loop. The problem can be especially pernicious if the existence of a systematic risk is not widely recognized, as in the case of the housing bubble.

Anonymous said...

Okay, in defense of the bell curve: We know that blacks and hispanics have IQ bell curve distributions which are at least [and probably more than] a standard deviation lower than Caucasian & Pacific Rim Asian IQ bell curves.

Furthermore, because of where the black & hispanic IQ bell curves live [centered no higher than 85, maybe down at 83.5, possibly as low as 79 (cf Guatemala), and apparently MOVING TO THE LEFT over time], and because we know the threshold IQ for any hope of a bare minimum of competency at the most basic of the Three R's [namely, an IQ of about 90], we therefore know that any government program designed to move large numbers of blacks & hispanics into the kinds of mortgages which only literate people could hope to afford [and highly literate people at that, like Master's or JD/MD/PhD levels of literacy] was doomed to failure from the outset.

Our problem now is that because of four decades of dysgenic fertility, hopelessly-low-IQ people are soon to be the majority of what America has to offer.

*PS: Of course, we also know that "global warming" is utter and complete nonsense, but that doesn't seem to prevent untold hundreds of millions of people from believing that it exists.

I'm honestly not sure how you fight this sort of institutionalized, delusional paganism - other than to walk away from it altogether, and make preparations for the battle that will ensue when it notices that you're gone and it decides to come after you.

Anonymous said...

Off topic, but I would sure appreciate it if Steve and the Sailersphere take a look at and comment on Eric deCarbonnel's website, http://www.marketskeptics.com. He's very prolific and well written and is tracking signs of a possible 'dollar event' around the corner.

-- Brendan

Anonymous said...

Looking at math instead of looking at the reality in front of your eyes can have bad effects.

For example, the poorly predicting mortgage formula not only fails to take sufficiently into account the phenomenon of incentives but also fails to pay any attention to HBD. Human differences.

On top of which, that particular systemic blind spot was strongly incentivized by legislative "carrots and sticks" such as the CRA.

Terry said

Defaults are still random.

You and I might differ about how much in life is random. Was this random?

Anonymous said...

Stop Math Now!...or not

"...Is financial math evil? I’m inclined to think we need more of it, not less. A simple model was applied too enthusiastically by bankers who didn’t understand the quants — maybe we’d have been spared the current credit crisis if asset managers had more mathematical sophistication."

Anonymous said...

Lucius Vorenus said...

*PS: Of course, we also know that "global warming" is utter and complete nonsense, but that doesn't seem to prevent untold hundreds of millions of people from believing that it exists.

I'm honestly not sure how you fight this sort of institutionalized, delusional paganism - other than to walk away from it altogether, and make preparations for the battle that will ensue when it notices that you're gone and it decides to come after you.

The Puritans had an answer. When the Church of England would not purify itself of its' catholic (small c) practices they set up a new colony away from England. Maybe Palin lives in Alaska for that very reason. Of course the faith of the Puritans descendants lasted until the arrival of the transcendentalists in the 1820's, making the entire experiment last about 200 years.

If we do set up a new country, what will prevent history from repeating? We can't run from ourselves you know.

Anonymous said...

As a Christian you should know that the Christian faith does not depend on your locale. We do not need an Israel to live as Christians. We are supposed to be the salt of the earth, i.e. live amongst the unbelievers to change the general culture for the better. The new dispensation (New Testament) is not territorially boud. It encompasses the whole earth. No need to run away and start another country. Unless things become like Zimbabwe or South Africa where life becomes unpractical. Then moving along becomes the reasonable thing to do.

Anonymous said...

Ronduck: The Puritans had an answer. When the Church of England would not purify itself of its' catholic (small c) practices they set up a new colony away from England. Maybe Palin lives in Alaska for that very reason. Of course the faith of the Puritans descendants lasted until the arrival of the transcendentalists in the 1820's, making the entire experiment last about 200 years. If we do set up a new country, what will prevent history from repeating? We can't run from ourselves you know.

Oh, but eventually our new nation will fail, just as our forefathers' nation has now failed before our very eyes: Every kingdom of Mammon is anchored to the nature of Mammon, which is Death itself.

But as the extinction so coveted by the nihilists is not an acceptable outcome in this most important of matters, the time has now come for us to part ways with a dying nation [and the nihilists who populate it] and to forge a new nation [which, in turn, will die and need rebirth, and so on until the end of time].

Anonymous said...

Lucius Vorenus:

I believe you're exactly right. Part of the reason is that, of those who have the "right stuf," too few will have whatever it is that enables understanding as to the origin of failure in the first place. The "right stuff" and the understanding are not mutually exclusive but the latter may be so rare and non-influential as to never attain society-wide acceptance and dominance. In that case, the very best that may be expected are repeated iterations, as you say, until the end of time.

Anonymous said...

No need to run away and start another country. Unless things become like Zimbabwe or South Africa where life becomes unpractical. Then moving along becomes the reasonable thing to do.

But there lies the problem. Where are we allowed to go? It seems we are not morally entitled to control North America, Australia, New Zealand or South Africa.

But the logic defying rules of po-mo PC mean we are not allowed to remain a majority in Europe either.

Where do we go?

Space would be the long term answer I suppose.

Anonymous said...

Wall St needed more math, not less.

Taleb may have been fine as a quant and an options trader, but as a "public intellectual" is a spectacular idiot. His ramblings are promoted because the press are even dumber than he is.
His book on options trading (orthodox Wall St quant stuff) is much more enlightening than the asinine Fooled By Randomness / Black Swan.

Anonymous said...

Ivy League:

Is it your opinion that the demise of Long Term Capital Management was due to some bit of mathematical inadequacy on the part of the Nobel-winning quant theorists according to which it was managed?

When forecasts of financial magnitudes miss their mark by a percent or two (say, 3% actual vs.
4% predicted), do we see this described as a mistake of 25%, as it is or of 1%, which it is numerically)? Quant economists of one or another type have dominated public discourse both in academia and in government and financial circles for almost 90 years now. I don't think astrologers and necromancers would have done worse.

One of the core problems (though having nothing to do with whether the methods have been suggested by one or another specifically quantitative theory) is that the very idea of "economic policy" in toto is concerned, not with bringing about general prosperity but with differing plans to interfere with those conditions which would have obtained on the unhampered market. It is, at all times, the intention of such policymakers (and enforcers) to favor certain (though not always identifiable) market participants at the expense of not-individually-identified others. Though the net is always a loss (in what could be thought of as "total gross welfare"), the discomfiture of very many is invisible and inaudible.

The current difficulties have undoubtedly been aggravated by greed, fraudulent practices, and unwise fiscal policies but the basic malaise is systemic, not episodic. And, at the root of the problem, which will one day lead to collapse on a heretofore unimaginable scale, is the very idea that mathematics has any legitimate connection with economics.

Anonymous said...

LTCM was not mathematically high-tech and neither were the Merton and Scholes Nobel prizewinning works. They had a simple economic intuition that they didn't properly test for robustness against relatively obvious, if a priori rare, scenarios. Oops.

Renaissance (Jim Simons), Princeton Newport (Ed Thorpe), DE Shaw, etc are better examples of quantitative trading. Those guys actually made money.

There are some shops that made a fortune on the mortgage markets due to competent and risk-aware modelling of a very mathematical kind. That's in addition to the more traditional traders who, less quantitatively, merely presumed that a bubble was about to explode.

Taleb claims to have been in the latter category. Good for his bank account, but getting lucky on the markets in no way vindicates his idiotic and overconfident pronouncements.