Part 4 (1/2)

The lower panels of Figure 10 are histograure 9 Panel D, for the normal distribution, reproduces the fa in any one day is about 45 per cent The real data, panel C, ht not look that different; however, closer inspection reveals that it has what are known as fat tails: the extreher and smaller values As seenexceeded 45 per cent on 54 different days over the 60-year period - so an event that according to theory should almost never happen, actually happened about once per year on average

Now, it has often been argued that the markets are normal most of the time, with only the occasional lapse into abnormal behaviour Perhaps Black Monday and the credit crunch are inherently unpredictable events that come out of nowhere If so, then there isn't much we can do about these ”Black Swans,” to use Taleb's term But if we look a little closer at the financial data, we see that it does have, not regularity, but a kind of character

Figure 10 Panel A shows daily percentage price changes in the S&P 500 index over a period of nearly 60 years Panel B is what the price changes would look like if they followed a normal distribution with the saes Panel D is a histogra normal distribution

For example, financial crashes are often compared to earthquakes This is no loose metaphor: in a very real sense, financial crashes feel like an earthquake happening in slow ure 11 is a zoomed view of the S&P 500 data for the period of the recent credit crunch The lower panel shows 50the earthquake of January 17, 1995, in Kobe, japan There is a strikingly similar appearance to the data So when one of the traders at Lehman Brothers told a BBC reporter in September 2008 that ”It is terriblelike aaccurate6 But the correspondence goes even deeper than that, for it turns out that the frequency of both phenomena is described by the same kind of mathematical law If you double the size of an earthquake, it becomes about four times rarer This is called a power law, because the probability depends on the size to the power 2 (A number raised to the power 2 is the number squared; a number raised to the power 3 is the number cubed; and so on) Financial crashes are similar Numerous studies have dees for major international indices follow a power-law distribution with a power of approximately 37 Power-law distributions are scale-free, in the sense that there is no ”typical” or ”norer an event is, the less likely it becomes In many respects, power-law distributions are therefore the opposite of the normal distribution The bell curve is concentrated about the mean, with a well-defined standard deviation The power law, in contrast, is scale-free, but biased towards seophysicist, they would say that earthquakes don't exist - there is just a constant low level of vibration in the earth) Given that we can't predict earthquakes any better than we can predict financial crashes, it is again te to see crashes as isolated events However, the scale-free nature of financial data implies that this is not the case There is no clear boundary between nore, the s To understand this more clearly, it le, which also has a power law hidden within it

Figure 11 Top panel is a zooure, showing the time period of the 2008 credit crunch Lower panel shows seisical data from the 1995 earthquake in Kobe, japan8

Fractal ure 12 shows a le in which the odd nu only the even nuer versions of the triangle (corresponding to es towards a peculiar geoasket, discovered by the Polish ure is shown in the lower panel

The usual way to construct this figure is to start with a triangle; divide into four sle This process is then repeated for the three reles, and so on, ad infinituasket is an early example of what eometric objects that have the property of self-similarity - no matter how far you zoom in, the object continues to reveal finer detail and structure, and different scales have a similar appearance There is no preferred or ”norles

As Mandelbrot noted, financial data is also self-similar - if you look at data taken over years, months, weeks, days, or even seconds, it is hard to know from the shape of the plot what the tiasket is close to being a visual history of stock-market crashes

To see this, suppose that each of the white triangles corresponds to a price change, with the size of the top edges equal to the size of the percentage change in a single day The figure is doe events, but as you zoom in further and further you can see that there are innules that correspond to est price changes tend to be crashes, which happen in tith 100, then the triangle at the centre has size 50 Let's say that corresponds to a fall of 50 per cent in one day - the worst iinable crash, I hope, and the one we haven't had yet in a major index Then there are three crashes of half the size (25 per cent); nine crashes of a quarter the size (125 per cent); and 27 crashes of an eighth the size (625 per cent) In general, every tie, there are three tiure 12 Top panel shows Pascal's triangle with the odd nuer versions of the figure, the result begins to reseasket (lower panel) Market price fluctuations follow a siles

This relationshi+p is equivalent to a power law, similar to that for earthquakes except that the power used is slightly different It also captures the behaviour of the financial es in the S&P 500, ranked in descending order (solid line) The dotted line is the result you would get by assues follow the saasket It is obviously not a perfect fit (this can be i the power) However, it isthe normal distribution, shown by the dashed line, which are clearly far too small We therefore see that the presence of extreme market crashes is no e triangles in the gasket: they all belong to the picture, and are created by the saest price changes in the S&P 500, ranked in descending order (solid line), with the 100 largest price changes from the noret by assues follow the saest crash of 50 per cent is assules in the gasket coroups of the saest size, three of the next largest size, and so on

It isn't just financial crashes and earthquakes that have these fractal patterns; the lengths of coastlines, the size of craters on the moon, the diaements of species in ecosystems, and many other natural and man-made systems show fractal statistics As ill see in Chapter 7, the sizes of businesses follow a fractal pattern - there aremultinationals at the top Pareto's 80-20 laas an observation that wealth scales fractally Indeed, fractals are a kind of signature of co far from equilibriuanised criticality10

Going critical

The classic exaanised criticality is the huine a conical sandpile with sides of a certain slope If the slope is too steep, then the pile will be unstable and adding just a single grain could cause it to collapse This is the chaotic state, which shows sensitivity to initial conditions On the other hand, if the slope is very shallow, then adding a few extra grains to the top of the pile won't cause much of a disturbance In this state the systerains of sand, then the sandpile will eventually converge, or self-organise, to a critical state In a sense, this state is maximally efficient, because it has the steepest sides and reaches as high as possible without beco further grains of sand will create avalanches that range in size froe, and that follow a power-law, scale-free distribution The system is not chaotic or stable, but on the border between the two

One can therefore hypothesise that markets, like the earth's crust or species in ecosysteanise to a critical state, and so exhibit fractal statistics This is an interesting insight, and one that links financial markets to the natural world Unfortunately, it isn'tthe odds of a crash in the future11 One reason is that we can know the exact distribution only bydata If you know that the data is normally distributed, then it is possible to quickly esti standard techniques But if the data is scale-free, then the largest and most important events occur extreives us only a few major crashes This makes it much more difficult to come up with accurate statistical estimates for the probability of similar events12 Also, fractal statistics tell you only about the distribution of price changes, not the ti We know that earthquakes follow a power-law distribution, but we still have no idea when the next one will strike13 (One useful tip, though, is that volatility does show clustering - if the markets are stormy, it's not safe to expect them to calm down soon) Finally, a model that was valid in the 1960s or 1980s won't be valid in the 2010s, because the entire econoulators, and so on will have changed

Traders and investors like to have sireat appeal of the norle number, the standard deviation It is possible to come up with more elaborate versions of the bell curve that include fat tails (extre, but this introduces extra co proble estiuesses Soorithms, but they tend to be focused on specialised trades that exploit isolated pockets of predictability A dirty secret of quantitative finance is that the tools used are usually quite siely for show So in practice, the broader field of riskhasn't le

A good exa technique, Value at Risk or VaR, seen earlier As the naests, it is used to estimate the worst-case loss that an institution could face on a given financial position The ulators, and credit rating institutions, and is important for investors and analysts Risk is esti fro on the case, and applying standard statistical techniques to give a likelihood for a particular loss This is usually expressed in tere confidence level A 3-standard deviation event is one that has a 999 per cent probability of not happening, so this defines a kind of maximum limit on the expected loss A 165-standard deviation event has a 95 per cent probability of not happening

Despite its popularity, the ular basis As mentioned in the introduction, Gold 2007 On February 29, 2008, Bear Stearns reported a VaR of 62 million at 95 per cent confidence By mid-March, its share price had dropped fro a loss of 8 billion Bear Stearns were of course aware of the drawbacks of VaR, but as they stated in their final report to the Security and Exchange Commission, ”the coy for the quantification of risk in the financial services industry despite these limitations”14 Indeed, it often see behaviour and provide an excuse when things go wrong

Even before the crisis, there have beenextra variables (aka fudge factors) into the model to account for unseen risks However, because the risks are unseen and therefore unobservable, there is no reliable way to forecast their future values15 The calculated risk will also depend on the exact values of these variables,the techniques difficult to share on an industry-wide basis Which is why risk still tends to be expressed in terh the concept has little er, as hedge fund er David Einhorn put it, is that quantitative risk ive users ”a false sense of securitylike an air bag that works all the ti the possibility of extre disaster more likely The next way to is of co or nitude of the next crash, but to model the financial system and find ways to reduce the likelihood and impact of extre, but we canto normal

One useful observation froanic systems, there is a trade-off between efficiency and robustness17 In the chaotic regiies are small and follow the normal distribution If left to their own devices, the systems will often evolve towards the critical state on the boundary between chaos and order Here the fluctuations follow a power-law, scale-free distribution: the syste pushed to the maximum, but it is not robust because it is susceptible to extreme fluctuations

In the same way, our financial syste profits for banks and investors over the short ter to do with the efficient market hypothesis, which is about nores) However, its fluctuations follow a power-law distribution, and it is susceptible to crashes that have a severe iiant sandpile, investors are piling more and more money on to the top, in the hope that they candown In this state, extreme events aren't aberrations - they're part of the landscape

An interesting question, then, is whether it is possible to improve the balance between stability and efficiency: to create a financial system that is less profitable in the short term, but also less prone to harmful collapses After all, the financial syste that we have created ourselves, so we should be able to engineer it in such a way that it behaves in a more stable fashi+on Unlike a sandpile, we have at least some influence over our destiny

Four steps iulate the introduction of new financial products The finance industry has becoulated in the last few decades, based on the dogovernments propose new rules, the banks complain that this will stifle innovation in the field of financial engineering But in other kinds of engineering or technology, regulations are strictly applied because they savein the stable regi off into chaos This reduces efficiency by soer to get to h the costs

As discussed further in Chapter 6, afactor to the credit crunch was the proliferation of new financial products (ie schemes) that allowed risk to be sliced and diced and sold off to third parties The risk calculations were perforly weren't up to the job In fact these new products, such as credit default swaps and collateralised ations, turned out to have the sae factory has on a diseased aniuised it and helped it propagate Furtherulated by the financial authorities, which was a large part of their attraction Their adoption was rather like the phar that there was no need to perfor it to market

Actually, some healthcare companies (or quacks) try to do this all the tiulatory bodies have to be on their toes During the 2009 swine flu pande Adainst 120 products that claimed to be able to cure or prevent the disease These included a 2,995 ”photon genie” to stimulate the immune system, and a bottle of ”Silver Shampoo” that would kill the airborne swine flu virus if it settled on your hair18 The FDA said that these clainificant threat to public health because they may create a false sense of security and possibly prevent people fro proper ht seem that financial ulate However, this iely due to the carefully maintained myths that markets are efficient and optimal, so any attempt to interfere with their function will be counterproductive As Adair Turner, chair of the UK's Financial Services Authority, observes, ”the whole efficient ulation systeh the intellectual zeitgeist” The abandonulators ”in aspace, because you don't have an intellectual systeulation doesn't ers The first step is to change their position, fro new financial products unless and until they are proven faulty, to (the default in other critical industries likethem unless they can be shown to provide a erous side-effects It costs about a billion dollars to bring a new cancer drug to ulatory hurdles before it can be adopted as a therapy People don't autoulators aredevelopers, but they seee

The next ime is to reduce the incentives for bankers tooff in the short teruaranteed to eventually blow up The asyer: they have a great upside (the bonuses) and a , but there are no negative bonuses, and by that time the person will probably be on the beach anyway) After the credit crunch there was great public de a portion of bonuses for a few years to liulators haveput it, paraphrasing Winston Churchill: ”Never in the field of financial endeavour has so ht add, with so little real reforestion, mentioned also in Chapter 3, is that credit creation and leverage should be controlled The total aross domestic product, was approximately three times what it was in the 1980s22 That's fine when , but any unexpected events quickly becoulations shouldn't apply just to banks, but to any institution that creates credit, including derivative e is of course linked to perceived risk: if you can convince a lending institution that a proposed investment has low risk, they will lend you more money Finally, then, the risk models used by banks and financial institutions should be modernised to better reflect the fractal nature of the markets and the possibility of extreme events Techniques from areas of matheenerating realistic stress tests for financial products or institutions; however, it is equally ie the limitations of mathematical models of any type23 The er reserves - ie keep some money under the mattress, even when risk seems low - and develop scepticism about their ability to predict the future Just because a formula says it's safe, that doesn't ement techniques such as experience-honed intuition, coue

Perhaps theto realise with complex systems is that models can actually be counterproductive if they are taken too literally Risk assess illusion of control, but can turn out to be highly dangerous Overconfidence in ers that lurk below the surface Just as engineers and biological systein of safety, and boat-builders over-design their shi+ps rather than build them to withstand only the ”norainst unexpected shocks

To summarise, the trade-off between efficiency and robustness in coests that we can lower the risk level of the economy, if we are prepared to accept lower levels of short-teres of the sort discussed in Chapter 2, this will also require a high degree of regulation This shouldn't be surprising - all forulated Economists talk about the invisible hand, but if you look at your own hand, everything about it - the temperature, the blood pressure, the salinity of the cells, and so on - is subject to a fierce degree of regulation that would put any financial regulator to shao too far in the other direction - ant to steady the econo this balance is to change the ”intellectual zeitgeist” by absorbing new ideas from science

It often feels that our financial systehts from complexity, network theory, nonlinear dynamics, and fractal statistics may help us find our way back to a ime Of course, finance is inherently risky, and when it coer mathematical models are only one piece of the puzzle Risk is ultimately a product of hu neat mathematical equations