EXTREME VALUE THEORY, The freakish event foretold
By Robert Matthews
Published: September 4 2003
Call it Shakespeare's Rule: when bad things happen, "they come not single spies, but in battalions". After economic recession and the Sars epidemic, Hong Kong looked set to learn the truth of that line from Hamlet this week, in the form of Typhoon Dujuan.
In the event, there was relatively little damage - largely because everyone was ready for it. For those in the insurance and risk management business, the real nightmare is the freak event that strikes out of the blue - and against which there seems no protection.
Now there is growing excitement about a mathematical technique that, though not quite a crystal ball, can help lessen the chances of being wrong-footed by fate.
Extreme value theory is starting to have an impact on areas as diverse as financial risk planning and marine safety and to judge from the range of applications described in a new book*, this is just the beginning.
At first glance, EVT seems to do the impossible: to predict events so extreme that they have never been seen before, from freak floods to financial disasters.
It took the genius of the Cambridge mathematician Ronald Fisher, the father of modern statistics, even to think it might be possible. In 1928 he showed that extreme events follow a distribution: a curve that captures their relative frequency.
Take the example of human height: the heights of people follow the familiar bell-shaped Gaussian curve, which reflects the fact that while most people have near-average heights, some are much taller or shorter. Just what percentage of people fall into these extreme categories can be estimated from the shape of the curve.
What Fisher showed was that extreme values themselves follow a distribution - a curve that allows the chances of even more extreme values to be estimated from past data.
The potential uses of such a technique were clearly legion but for many years EVT was regarded as a mathematical curiosity, requiring colossal amounts of data to be put to use.
There were also concerns about the technical assumptions underlying EVT. But during the past decade or so, mathematicians have worked to allay those qualms, allowing the power of EVT to be used in many applications.
Among the first to benefit have been insurance companies faced with gauging the chances of extreme events such as floods, storms and hurricanes. Overestimate the risk and the unrealistically high premiums could scare off business; under- estimate it and the insurance company will suffer the consequences.
For many years the industry has used rules of thumb such as the "20-80" formula, which asserts that 20 per cent of events are responsible for more than 80 per cent of the total cost of claims. Using EVT to analyse historical data, Paul Embrechts, a financial mathematician, and his colleagues at the Swiss Federal Institute of Technology (ETH) in Zurich, have shown that while this rule works fairly well for many insurance sectors, it fails disastrously for others.
For example, EVT shows that hurricanes follow a "0.1-95" rule - showing that the real threat is the 1-in-1,000-type hurricane, which can swallow up 95 per cent of the total cover in one go. Knowing this allows insurers to set more appropriate premiums, to the benefit of both themselves and their clients.
EVT is also making inroads into the world of financial risk management, still smarting from the Long Term Capital Management debacle of 1998. Wrong-footed by the meltdown of the Russian economy, the huge hedge fund had to be bailed out with $3.5bn from a consortium of 15 banks.
To guard against a repeat performance, risk managers keep watch on their so-called value at risk (Var), the biggest loss likely to strike over a fixed time period (typically 10 trading days) with a probability of, say, one in 100. But Vars are only as good as the probability curves on which they are based and EVT suggests that using the familiar bell curve is not good enough, leading to underestimates of the true risks of taking a big hit.
EVT is still in its infancy. Mathematicians are still exploring its potential, especially for predicting several extreme events all striking at the same time. Even so, this technique is already helping to save lives as well as fortunes (see left).
It may not be a crystal ball - but its formulas can help us all sleep more easily at night.
*Extreme Values in Finance, Telecommunications and the Environment, edited by Barbel Finkenstadt and Holger Rootzen, published by Chapman & Hall/CRC. The author is visiting reader in science at Aston University, Birmingham
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Practical applications of a mathematical theory on extreme occurrences
The lives of 16m inhabitants of the Netherlands are protected by a formula. With more than half its land below sea level, the country is surrounded by a series of sea dikes, designed to cope with the worst nature can throw at them using the mathematics of extreme value theory (EVT).
Following the disastrous storm surge of February 1953, which killed 1,800 people and destroyed 47,000 homes, the Dutch government demanded the construction of new sea defences that would protect the nation for centuries to come.
Analysing historical records of extreme events, scientists came up with a 5-metre height standard for the improved dikes. EVT was then used to confirm that the chances of another disaster in the near future were vanishingly small.
EVT is also at the heart of new marine safety recommendations to prevent tragedies such as the loss of the cargo ship MV Derbyshire, which sank with all its 44 crew in a typhoon off the south coast of Japan in 1980. In 2000, an official inquiry found that the vessel's forward hatch cover had given way in heavy seas, allowing water to flood in. The conclusion exonerated the drowned captain and crew of the Derbyshire, who had been blamed for the tragedy.
It was a conclusion based partly on research by Prof Jonathan Tawn and Dr Janet Heffernan of Lancaster University, who used EVT to examine scenarios in which the ship was exposed to waves violent enough to break the hatch covers.
In research carried out with Lloyd's Register, the two academics also used EVT to show that hatch strengths for Derbyshire-sized ships should be increased by 35 per cent, over and above the doubling recommended following the tragedy. A few months later, in December 2001, the giant bulk-carrier Christopher sank off the Azores with 27 crew. A final radio message reported that the forward hatch cover had collapsed.
It was an eerie echo of the fate of the Derbyshire - and perhaps demonstrated that even the most sophisticated theory offers no protection if it is not acted upon. |
