Tag Archives: fat tails

Modelling leverage and the financial crisis

Science reports on agent based models of financial leverage -Leverage: The Root of All Financial Turmoil:

Given that in the buildup to the recent global economic meltdown hedge funds had been leveraging their deals by ratios of 30-to-1 (that is, borrowing $30 for every $1 of their own that they put in), it may seem obvious that massive leverage leads to trouble. But Stefan Thurner, an econophysicist and director of the complex systems research group at the Medical University of Vienna, Austria, and colleagues say their model shows that many of the distinctive statistical properties of financial markets emerge together as rates of leverage climb. “Leverage is the driver,” Thurner says. “That wasn’t obvious.”

Financial markets behave in ways that, econophysicists say, classical economic theory cannot explain. Classical economics assumes that the fluctuations in stock prices conform to a so-called Gaussian distribution—a bell curve that gives little probability to large swings. In reality, the distribution has “fat tails” that make big changes more likely, and the shapes of those tails conform to a mathematical formula known as a power law. Classical economics assumes that the fluctuations are uncorrelated from one moment to the next, whereas big swings in prices tend to come together in the so-called clustering of volatility.

To try to explain those characteristics, over the past 5 years Thurner and colleagues have developed an “agent-based model” of a market. In such a computer model, virtual agents of various types interact according to certain rules, like robots playing a game. The researchers included hedge funds that could borrow to make their investments; banks to loan the money; “noise investors” who, like day traders, simply react to the market and have no other insight into the value of assets; and general investors who played the role of, for example, state pension funds.

The model contains more than a dozen adjustable parameters. However, Thurner and colleagues found that the maximum level of leverage exerts a curious, unifying effect. If they forbade leverage, the market behaved largely as classical economics would predict. But as they increased the maximum leverage, the characteristics of real markets emerged together. “We can explain the fat tails, the right [power law], the clustering of volatility, all this,” Thurner says. And when the leverage limit climbed to levels of 5-to-1 and beyond, the market became unstable and hedge funds went bust much more often.

“I thought it was rather brilliant,” says Christoph Jan Hamer, an econophysicist with Solvency Fabrik in Köln, Germany. Hamer says he was impressed with a detail of the model: If leverage is high, then a tiny fluctuation created by the noise traders can trigger a much bigger swing. But Christian Hirtreiter of the University of Regensburg says, “I would think that leverage itself is not the problem. I would think it is a symptom of the problem.”

Thurner, who managed a hedge fund that tanked, says that limiting leverage should help prevent crashes. He admits, however, that he would not have embraced that idea when the market was still going strong.

Economist on fat tails and finance

A special report on the future of finance in The Economist Fallible mathematical models: In Plato’s cave:

… although the normal distribution closely matches the real world in the middle of the curve, where most of the gains or losses lie, it does not work well at the extreme edges, or “tails”. In markets extreme events are surprisingly common—their tails are “fat”. Benoît Mandelbrot, the mathematician who invented fractal theory, calculated that if the Dow Jones Industrial Average followed a normal distribution, it should have moved by more than 3.4% on 58 days between 1916 and 2003; in fact it did so 1,001 times. It should have moved by more than 4.5% on six days; it did so on 366. It should have moved by more than 7% only once in every 300,000 years; in the 20th century it did so 48 times.

In Mr Mandelbrot’s terms the market should have been “mildly” unstable. Instead it was “wildly” unstable. Financial markets are plagued not by “black swans”—seemingly inconceivable events that come up very occasionally—but by vicious snow-white swans that come along a lot more often than expected.

This puts VAR in a quandary. On the one hand, you cannot observe the tails of the VAR curve by studying extreme events, because extreme events are rare by definition. On the other you cannot deduce very much about the frequency of rare extreme events from the shape of the curve in the middle. Mathematically, the two are almost decoupled.

The drawback of failing to measure the tail beyond 99% is that it could leave out some reasonably common but devastating losses. VAR, in other words, is good at predicting small day-to-day losses in the heart of the distribution, but hopeless at predicting severe losses that are much rarer—arguably those that should worry you most.

When David Viniar, chief financial officer of Goldman Sachs, told the Financial Times in 2007 that the bank had seen “25-standard-deviation moves several days in a row”, he was saying that the markets were at the extreme tail of their distribution. The centre of their models did not begin to predict that the tails would move so violently. He meant to show how unstable the markets were. But he also showed how wrong the models were.

Modern finance may well be making the tails fatter, says Daron Acemoglu, an economist at MIT. When you trade away all sorts of specific risk, in foreign exchange, interest rates and so forth, you make your portfolio seem safer. But you are in fact swapping everyday risk for the exceptional risk that the worst will happen and your insurer will fail—as AIG did. Even as the predictable centre of the distribution appears less risky, the unobserved tail risk has grown. Your traders and managers will look as if they are earning good returns on lower risk when part of the true risk is hidden. They will want to be paid for their skill when in fact their risk-weighted returns may have fallen.

How to deal with uncertainty in climate change economics

From the economist Martin L. Weitzman‘s website, a new draft paper On Modeling and Interpreting the Economics of Catastrophic Climate Change(pdf).  He proposes a method for including unlikely but extreme events (fat tails) in cost-benefit analyses, such as the uncertainty surrounding climate sensitivity.  Considering the possibility of such events can completely change the results of an analysis, and favour action as a type of catastrophe insurance.

Abstract: Using climate change as a prototype example, this paper analyzes the implications of structural uncertainty for the economics of low-probability high-impact catastrophes. The paper is an application of the idea that having an uncertain multiplicative parameter, which scales or amplifes exogenous shocks and is updated by Bayesian learning, induces a critical tail fattening of posterior-predictive distributions. These fattened tails can have very strong implications for situations (like climate change) where a catastrophe is theoretically possible because prior knowledge cannot place sufficiently narrow bounds on overall damages. The essence of the problem is the difficulty of learning extreme-impact tail behavior from finite data alone. At least potentially, the ináuence on cost-benefit analysis of fat-tailed uncertainty about climate change, coupled with extreme unsureness about high-temperature damages, can outweigh the influence of discounting or anything else.

The paper concludes:

In principle, what might be called the catastrophe-insurance aspect of such a fat-tailed unlimited-exposure situation, which can never be fully learned away, can dominate the social-discounting aspect, the pure-risk aspect, or the consumption-smoothing aspect. Even if this principle in and of itself does not provide an easy answer to questions about how much catastrophe insurance to buy (or even an easy answer in practical terms to the question of what exactly is catastrophe insurance buying for climate change or other applications), I believe it still might provide a useful way of framing the economic analysis of catastrophes.