My last peroration concerned the way one major component of the economic fiasco of 2008 could have been made visible. A diagram, on a single (if large) piece of paper, of the way credit default swaps (CDS) connected major financial institutions could have made it obvious at a glance that the major financial institutions were joined at the hip when it came to risk of defaults. When it came to risk, half a dozen firms might be renamed as a single "Wall Street, Inc." This is only one piece of the puzzle, but an important one.

Not only was this diagram missing when needed to avert disaster, the information to draw it was practically unobtainable. Auditors might check that an individual firm had in fact made provisions to transfer risk from a major default, but they were not tasked with a simultaneous audit of all connected firms. In fact, details of which counterparties were used for which CDS, and the total exposure possible through a network of links, were treated as trade secrets. Competitors were not supposed to find out who you went to to cover risk, or how bad your situation was when their help averted disaster. Even so, competitors probably knew more than the vast majority of investors with a stake in your own firm. Fierce competition between rival financial empires made them, perhaps unwittingly, allies in giving investors the old 14-carat bamboozle.

The walls between competitors came down far enough to construct such a diagram when the imminent collapse of AIG, and those firms guaranteeing it against defaults, brought a series of urgent meetings of CEOs at the Federal Reserve Bank of New York. Regulators had previously been taken by surprise when it turned out to be impossible to save Lehman Brothers within known time constraints without federal guarantees against losses. Even the most highly-skilled examiners with full access to the books could not state "the value of the firm is somewhere between X and Y". (If they had, their own banks would have bid X for the firm.) They were unable to decide if the firm had any positive value with only a few days of work. (How was anyone less skilled, or with less access, to make a decision -- by consulting their horoscope?)

The potential mess if AIG went down was even worse. When Henry Paulson, Secretary of the Treasury, explained in detail to President Bush why it was necessary to go to Congress and seek the authority to intervene the President is said to have asked "Can an insurance company do all that?" Insurance is a way of managing financial risk, which leads us back to money.

The problem here was the intimate relationship between value and risk, assets and liabilities. Anyone familiar with double-entry bookkeeping could tell you that you don't gain money by taking out a loan, the money you have to spend is balanced by the liability to repay the loan. Making that liability disappear is equivalent to printing money. Moving major risks off balance sheets was more effective than tunneling into Fort Knox.

While a CDS can, in principle, be written for any amount, in practice they have been confined to the high end of finance where people worry about the failure of colossal institutions. Down closer to the level where most of us operate we find those collateralized debt obligations (CDO) which fueled the speculative bubble in sub-prime mortgages from inception.

As a 'black box' the small part of a financial institution which took in risky mortgages, and repackaged them into CDOs, was doing magic. The financial products which emerged might be described in a bewildering language of derivatives, tranches, mezzanines, etc. The simple fact was that assets with associated risks went in one end, and assets almost devoid of risk came out. If the firm was not assuming those risks, someone was printing money, or passing risks to unwitting others.

I'll try to describe the defective basis of the statistical models of risk in widespread use below. If you don't follow what I write, don't despair. At this point I'm going to remove much of the motivation for understanding this. For the purposes of separating people from money, the important characteristic of all these models was that very few understood their assumptions and implications.

One theoretical model for pricing options is called the Black-Scholes model. This even has an impressive partial differential equation, to awe those who understand differential equations, and earned a Nobel Prize. Another popular risk model was based on the so-called Gaussian Copula, which concerned correlation between random variables. There are far more extensive literatures on these than I can claim to understand. What I can understand is one common misconception of people using them concerning very unlikely events. (If you want an entertaining account of the statistics of unlikely events, try "The Black Swan" by Nassim Nicolas Taleb.)

The simplest statistical model for extremely rare events is the Poisson distribution. This kind of randomness is completely described by a single parameter. Thick books on queuing theory are replete with equations based on the idea of Poisson arrivals in queues. Unfortunately, this is a mathematical way of stating that there is no correlation at all between different arrivals in a queue, (something which must seem very doubtful to anyone who has tried to pick something up at a grocery store on the way home from work.)

The next step up on the scale of random models is a Gaussian distribution, which requires two parameters called mean and variance (or, equivalently, standard deviation) to explain everything about it. This is so apparently ubiquitous in arising where least expected that a deep theorem, called the Central Limit Theorem (CLT), was developed to explain this aspect. Very rare events in this view are part of a 'lean tail' where probabilities become very small. Alternative distributions modifying this distribution are sometimes said to have a 'fat tail'. Simply tweaking the numbers by fattening lean tails is not enough, the fundamental flaw goes to the very heart of the derivation.

The standard proof for the CLT assumes it is produced by many individual random variables which may or may not be Gaussian distributed. These must be independent, identically distributed, and possess individual means. (If you can't imagine how a distribution could fail to possess a mean, you should probably avoid transfinite set theory.) Identical is a pretty good description of elementary particles, but less useful for people. The real catch comes from the requirement of statistical independence. These models work best in a world without financial news to produce correlated behavior of investors, (which is conspicuous as I write.)

How such a basic flaw became the basis for dominant economic activity should be a fruitful topic for future historians. Still, as said above, the important characteristic of these arguments was that they were completely opaque to most people. How many members of Congress or Parliament understand the proof of the Central Limit Theorem, and its inherent limitations? Assuming, as I have suggested above, this is not truly a necessary requirement for managing public or private monies, how many business and political leaders can recognize the spiel of a conman?

Not only was this diagram missing when needed to avert disaster, the information to draw it was practically unobtainable. Auditors might check that an individual firm had in fact made provisions to transfer risk from a major default, but they were not tasked with a simultaneous audit of all connected firms. In fact, details of which counterparties were used for which CDS, and the total exposure possible through a network of links, were treated as trade secrets. Competitors were not supposed to find out who you went to to cover risk, or how bad your situation was when their help averted disaster. Even so, competitors probably knew more than the vast majority of investors with a stake in your own firm. Fierce competition between rival financial empires made them, perhaps unwittingly, allies in giving investors the old 14-carat bamboozle.

The walls between competitors came down far enough to construct such a diagram when the imminent collapse of AIG, and those firms guaranteeing it against defaults, brought a series of urgent meetings of CEOs at the Federal Reserve Bank of New York. Regulators had previously been taken by surprise when it turned out to be impossible to save Lehman Brothers within known time constraints without federal guarantees against losses. Even the most highly-skilled examiners with full access to the books could not state "the value of the firm is somewhere between X and Y". (If they had, their own banks would have bid X for the firm.) They were unable to decide if the firm had any positive value with only a few days of work. (How was anyone less skilled, or with less access, to make a decision -- by consulting their horoscope?)

The potential mess if AIG went down was even worse. When Henry Paulson, Secretary of the Treasury, explained in detail to President Bush why it was necessary to go to Congress and seek the authority to intervene the President is said to have asked "Can an insurance company do all that?" Insurance is a way of managing financial risk, which leads us back to money.

The problem here was the intimate relationship between value and risk, assets and liabilities. Anyone familiar with double-entry bookkeeping could tell you that you don't gain money by taking out a loan, the money you have to spend is balanced by the liability to repay the loan. Making that liability disappear is equivalent to printing money. Moving major risks off balance sheets was more effective than tunneling into Fort Knox.

While a CDS can, in principle, be written for any amount, in practice they have been confined to the high end of finance where people worry about the failure of colossal institutions. Down closer to the level where most of us operate we find those collateralized debt obligations (CDO) which fueled the speculative bubble in sub-prime mortgages from inception.

As a 'black box' the small part of a financial institution which took in risky mortgages, and repackaged them into CDOs, was doing magic. The financial products which emerged might be described in a bewildering language of derivatives, tranches, mezzanines, etc. The simple fact was that assets with associated risks went in one end, and assets almost devoid of risk came out. If the firm was not assuming those risks, someone was printing money, or passing risks to unwitting others.

I'll try to describe the defective basis of the statistical models of risk in widespread use below. If you don't follow what I write, don't despair. At this point I'm going to remove much of the motivation for understanding this. For the purposes of separating people from money, the important characteristic of all these models was that very few understood their assumptions and implications.

One theoretical model for pricing options is called the Black-Scholes model. This even has an impressive partial differential equation, to awe those who understand differential equations, and earned a Nobel Prize. Another popular risk model was based on the so-called Gaussian Copula, which concerned correlation between random variables. There are far more extensive literatures on these than I can claim to understand. What I can understand is one common misconception of people using them concerning very unlikely events. (If you want an entertaining account of the statistics of unlikely events, try "The Black Swan" by Nassim Nicolas Taleb.)

The simplest statistical model for extremely rare events is the Poisson distribution. This kind of randomness is completely described by a single parameter. Thick books on queuing theory are replete with equations based on the idea of Poisson arrivals in queues. Unfortunately, this is a mathematical way of stating that there is no correlation at all between different arrivals in a queue, (something which must seem very doubtful to anyone who has tried to pick something up at a grocery store on the way home from work.)

The next step up on the scale of random models is a Gaussian distribution, which requires two parameters called mean and variance (or, equivalently, standard deviation) to explain everything about it. This is so apparently ubiquitous in arising where least expected that a deep theorem, called the Central Limit Theorem (CLT), was developed to explain this aspect. Very rare events in this view are part of a 'lean tail' where probabilities become very small. Alternative distributions modifying this distribution are sometimes said to have a 'fat tail'. Simply tweaking the numbers by fattening lean tails is not enough, the fundamental flaw goes to the very heart of the derivation.

The standard proof for the CLT assumes it is produced by many individual random variables which may or may not be Gaussian distributed. These must be independent, identically distributed, and possess individual means. (If you can't imagine how a distribution could fail to possess a mean, you should probably avoid transfinite set theory.) Identical is a pretty good description of elementary particles, but less useful for people. The real catch comes from the requirement of statistical independence. These models work best in a world without financial news to produce correlated behavior of investors, (which is conspicuous as I write.)

How such a basic flaw became the basis for dominant economic activity should be a fruitful topic for future historians. Still, as said above, the important characteristic of these arguments was that they were completely opaque to most people. How many members of Congress or Parliament understand the proof of the Central Limit Theorem, and its inherent limitations? Assuming, as I have suggested above, this is not truly a necessary requirement for managing public or private monies, how many business and political leaders can recognize the spiel of a conman?