(recently hatched chicks of *Cygnus atratus*, the Australian black swan)*The last entry went into the flood/drought cycles of the Nile and how attempts to tame this behavior have created long-standing political tensions in the region. During the time of the British protectorate, hydrologists were brought in to aid in reservoir design. One of them, H.E. Hurst, made a very interesting discovery. Note: I assume in this blog entry that the reader has a nodding familiarity with the classical bell-shaped curve of the "normal", or "Gaussian", distribution. If that isn't the case, it is no big deal---just keep in mind that many of our assumptions about the world, when we have to quantify them, are based on the assumption that the Gaussian holds true. ***Hurst in Egypt**

As a result of the critical importance of the Nile to agriculture, the Egyptians had documented many, many years of data points on the river's annual level. Hurst had grown up near Leicester, England and studied physics at Oxford. He went to Egypt to work on dam project proposals and he received the nickname "Father Nile" from the Egyptians out of their respect for his work on reservoirs and dam design (work that was used by Nasser's engineers in the construction of High Aswan).

Hurst knew that the Nile's *average* annual discharge was 92.4 billion cubic meters of water. However, the variation was large---in a wet year, the river could discharge 151 billion cubic meters, while during a drought the number could be 42 billion. As we discussed when we turned to Aswan, the engineering solution to these cycles is well-known---build a dam that can hold back a certain period length of "wet year" water, and then release this water during dry periods. What Hurst needed to do, first and foremost, was to determine the *maximum range* of the river, the distance from the highest flood marks to the lowest drought marks.**T^.5 and the Random Walk**

Engineers trained in traditional (Gaussian) statistics would normally model a problem like this by assuming that rainfall patterns were independent from one year to the next and follow a so-called "random walk". A powerful mathematical shorthand would then be licensed---something called the "t to the one-half" rule. Written more explicitly, the rule states that the dispersion (range) of a random walking variable will be its standard deviation, or sigma, multiplied by the square root of time, which of course is the same as saying time to the 1/2 power, or t^.5. For brevity, I will just refer to it as the t^.5 rule. T^.5 is a convenient scaling feature of the Gaussian distribution, and its use stems back to the French gambling statistician Bachelier and his coin-tossing experiments, as well as the formulation of geometric Brownian motion by Albert Einstein in the early 1900s.

If you imagine a drunk following a statistically correct random walk, lurching back and forth as he attempts to make it to his car, the t^.5 rule tells you how far the drunk can stumble to the left before he has to stumble back to the right, towards his original path of travel.

Here is a practical example of the t^.5 rule in action: over the last six months, a stock has generated returns (actually, for technical reasons the log of returns is probably what will be used in this kind of methodology, but that's not important right now) with a monthly standard deviation, or "volatility" in finance-speak, of 2%. We want to assess how "risky" the stock is, using this calculation of volatility, but we need to somehow use the monthly data we have to determine the annual volatility (we care about annual performance, not monthly oscillations).

The t^.5 rule comes to our rescue here: we could take the monthly standard deviation, the 2%, and multiply it by the square root of t---in this case, t=12, because there are twelve months in a year. If we make a number of heroic, dangerously naive assumptions about something called stationarity, we can determine how volatile the behavior of the stock *should* be over the next year. To establish the asset's range, we would add this annual volatility (from the sigma*t^.5 calculation we just performed) to the asset's expected return---in itself quite hard to predict ---for the higher bound, and then subtract the sigma*t^.5 calculation from the expected return to get the lower bound (which is probably what we are truly concerned about if we are long-biased investors).

This example was chosen because it reveals the t^.5 rule being misapplied. Financial market prices are not stationary and do not follow a random walk. However, much of modern finance is built on this kind of assumption. As we will soon see, the (mis)use of this rule is, in fact, endemic within modern financial theory. Why? Perhaps because it works in the geometric Brownian motion of equilibrium physics, and economics occasionally suffers from "physics-envy". Perhaps because it is a tidy and convenient mathematical tool. Whatever the reason, it comes with a steep price tag attached: the underlying distribution must be the normal/Gaussian, or at the very least a close approximation to it. If the rule is invoked and the distribution is not normal, great surprises lay in wait...**Rescaled Range Analysis **

Hurst found that the range of the Nile increased much faster than would be predicted by the t^.5 rule of the Gaussian random walk. To avoid obfuscating the basic point and becoming tedious, I will just say, somewhat simplistically, that Hurst took the Nile data he had and walked forward through time on the charts, looking at the highest and lowest values that were achieved and then determining what exponent power of t would---as an estimation---be necessary in order to generate that type of behavior.

The basic formulation that Hurst used was R/S=(a*N)^H, where:

R/S=Rescaled range

a=a constant

N=number of observations

H=Hurst exponent (given this name by a French mathematician who came a bit later)

The important takeaway idea is this: H is going to range between 0 and 1. If H=.5, then you do in fact have your t^.5 rule and a random walk. If H is *greater* than .5, then the time-series you are working with displays **persistent**, or trend-reinforcing properties---the range will be greater than it would be in a Gaussian distribution, and the tails of the underlying distribution will be "fatter". In other words, there will be a higher frequency of so-called extreme events than we would normally predict. If H is *less* than .5, then the time-series is **anti-persistent**, or mean-reverting, and we will see a tighter dispersion, a smaller overall range, than a random walk would predict. Extreme events will be very rare, even rarer than they are with a Gaussian distribution.

(I am making a few overgeneralizations here in order to keep things clean. Technically, it is possible---unlikely and weird, but possible---to have H=.5, a random walk that obeys the t^.5 rule, but a non-Gaussian distribution. However, we know that if H>.5 or H<.5, the underlying distribution cannot be Gaussian. So H=.5 is necessary for us to say that we are dealing with a Gaussian, but insufficient to absolutely prove that we are. Even if the distribution were to somehow be non-Gaussian, however, we would know that its behavior was moderate in terms of extremes to the point that a Gaussian assumption was quite safe)

According to portfolio manager and Chaos specialist Edgar Peters, Hurst found that the Nile's discharge produced a time-scaling exponent of .9, indicating a very strong tendency for floods to follow floods and for droughts to follow droughts. The discharge is directly correlated with the amount of rainfall that the Nile's sourcewaters receive, so we can easily understand why the drought periods have caused terrible humanitarian tragedies, such as the great famine in Ethiopia during the 1980s. The Nile does not just hit you once---like vintage Tyson, it throws hard blows in non-random combinations.

Hurst went on to apply rescaled range analysis to a variety of other natural phenomena, including sun spot activity, tree rings, lakebed sediment deposits, water levels of other lakes and rivers, rainfall measurements, and temperature readings. In all of the cases that he studied, he found that the ranges widened more aggressively than they would if they were following random walks with statistically independent data points. Nature heavily features trend-reinforcing, or persistent behavior. In fact, Hurst found that a particular number---H=.73---was showing up again and again. Hurst explained his findings by saying that natural phenomena were frequently* long-memory processes*---positive feedback loops cause events of today to influence events of tomorrow, in cascades of dependence that drive directional drifts to go on longer than would be realistically possible under the independent steps of a random walk.

A practical application of Hurst's work was the finding that the chances of freakishly high or low points on the range of one of these phenomena were much higher than they would be if the phenomena followed the random walk of a staggering drunk. This meant that engineers needed to take true extreme values directly into account when specifying their project designs: for instance, Hurst's applied methods would suggest that, to prepare for a 100 year extreme drought, a hydrologist would want a reservoir that could store about **20 times** the annual standard deviation of rainfall, rather than the 3 times that you would need if the rainfall's statistical distribution was normal or approximately normal. Hurst's plan was to build a series of dam/reservoir systems upriver of Egypt, probably in Ethiopia, in order to create those reserves without causing extreme flooding to the reservoir basin area, or losing an excessive amount of water to evaporation. We know that Nasser preferred an alternative proposal---a high dam that was completely within Egyptian control.**Mandelbrot, Taleb, DeVany**

(the Mandelbrot Set, an iconic pattern of fractal geometry)

In the 1960s, a French mathematician named Benoit Mandelbrot came across the Hurst papers while he was teaching economics at Harvard. Mandelbrot had found that commodity prices---specifically cotton prices---were not obeying t^.5, and he started applying rescaled range to the financial markets. He is actually the one who named the exponent that is estimated in rescaled range the "H" exponent, in honor of both Hurst and a mathematician named Ludwig Holder who had been working on some similar problems (today we often just call it the "Hurst exponent").

Mandelbrot found that financial market prices also tended to show Hurst exponents greater than .5, once again indicating persistence in the time-series. The levels of persistence varied from asset to asset: interest rate futures contracts tended to have quite high H exponents (over .7), for example, while utility stocks tended to be in the .55-.65 range.

Mandelbrot later systematized his observations about financial market behavior into a conceptual architecture, the Fractal Market Hypothesis, and coined two descriptive terms: the "Noah Effect", named for the great destructive flood of the Book of Genesis and describing the tendency for a time-series to show sudden, violent change; and the "Joseph Effect", describing the trending feature and named for the Biblical prophecy of seven years of famine striking Egypt after seven years of prosperity. Mandelbot developed a metric called "alpha" for measuring the Noah Effect, while the Joseph Effect is still estimated from the Hurst exponent.

The departure from the Gaussian distribution becomes more and more pronounced as you go further and further out into the "tails" and encounter the extreme events: in reviewing prices of the Dow Jones Industrial Average from 1916 to 2003, for example, Mandelbrot found that the Gaussian assumption would predict that there were 58 days when the Dow should move more than 3.4%; in reality, there were over 1,000 of these days. The normal distribution would have given 6 days of swings greater than 4.5%; in reality, there were 366. Index swings of greater than 7% should happen once every *300,000 years*; during that historical time there were actually 48 days like this. When we get to the 29.2% drop that occurred on 19 October, 1987, we reach a 22 standard deviation move and a probability of occurrence of less than 1 in 10^50---1 followed by 50 zeroes, a number far, far greater than the number of trading days that have been available in the entire 13.7 billion year history of the Universe.

Edgar Peters has since used rescaled range analysis on a number of different financial asset prices and other economic time-series. He has found that markets tend to feature more noise at the shorter time scales: the Hurst exponent for the S&P500 varies from about .59 when you use 1-day intervals to .78 when you use 30-day intervals, so the "trend" component of the S&P seems to become stronger as the time horizon lengthens (up to a point). 30-year Treasury Bonds have H of about .7. G7 currencies come in around .65. As Peters notes in *Chaos and Order in the Capital Markets*, "These results will come as no surprise to currency traders. Currency markets are characterized by abrupt changes traceable to central bank intervention---attempts by governments to control the value of each respective currency, contrary to natural market forces. Currencies have a reputation as 'momentum trading' vehicles in which technical analysis has more validity than usual. R/S analysis bears out the market lore that currencies have trends..."

There are a number of explanations for why markets feature the Noah and Joseph Effects. One approach sees them as self-organizing, complex systems that feature a number of interdependencies and feedback loops, and which can reach a state of criticality: when they are critical, or "loaded", a small disturbance can have a magnified, large-scale effect---this "sensitive dependence on initial conditions" is a signature of Chaos. Another approach seeks less to explain the reasons why as to describe a more accurate model for market behavior: this is the "power law" perspective, which says that the magnitude of market price changes reflects some kind of scaling mathematical relationship (for example, in a system that scales according to a power law of 10 a move of 2% may be 10 times more likely than a 3% move, which in turn may be 10 times more likely than a 4% move, and so on).

Power laws have been found in distributions created by wars, earthquakes (obviously a very timely subject which warrants further exploration), bandwidth demand, traffic jams, and a baffling variety of other places. Their major feature for our discussion here is that they have far more extreme events than you would find in a normal distribution, and this can have profound effects on the "average" that the system reports.

For example of how a power law can change the average in remarkable ways, consider the case of income distribution: imagine having 99 "average" people in a room and then adding Bill Gates. The value of that one, extraordinarily wealthy individual would cause everyone in the group to be worth over $500 million, *on average*. In contrast, putting the world's tallest man with 99 individuals of normal height would have very little effect on the average height for the group, because the tallest man is constrained by physical limits and cannot be 1,000 feet, or 100 feet, or even 10 feet tall. The overall distribution that results from averaging height in this way is Gaussian.

Mandelbrot has a number of great descriptions of power law relationships in his books and papers. He is considered to have been an instrumental figure in the development of Chaos Theory and a closely associated field, fractal geometry, and an attempt to discuss his many contributions would go well beyond the scope of this blog entry. Several key figures in finance can be linked directly to him, but for our purposes today we will turn next to a particular individual he mentored, a man named Nassim Taleb.

Taleb is a highly experienced options trader and decision theorist who initially made his reputation by cashing in on Eurodollar options during the '87 Crash. Taleb had purchased the options when they were far out-of-the-money (i.e., would only become valuable if an extreme market shock event occurred), and he had been observing for some time that multi-sigma events---incredibly unlikely according to the Gaussian assumption---occurred regularly in the major financial markets, often on quiet financial news days.

After witnessing the legendary portfolio manager Victor Niederhoffer blowing up his previously very successful hedge fund by selling disaster insurance (naked puts on the S&P500) to other investors and, possibly, engaging in the Martingale betting that we have previously discussed, Taleb developed a trading strategy that was explicitly designed to be "blow-up proof". The strategy uses a rolling portfolio of out-of-the-money option positions to profit from Mandelbrot's "Noah Effect"---sudden, discontinuous, demonic jumps and crashes in the markets (there are other approaches for attempting to profit from trend-reinforcing persistence, or the "Jacob Effect", and I will discuss these in some detail in the future. For now, I will say that, in my opinion, consistent profits from the Noah Effect require insurance-type option strategies like Taleb espoused, and that consistent profits from the Jacob Effect require systematic trend-capture strategies). Taleb has gone from trading to a sort of modern Bohemian lifestyle in which he is a man-about-town pursuing his interest in decision-making, particularly decision-making under conditions of risk, pressure, and uncertainty. It seems to be the core intellectual meditation on which his many other interests revolve.

In *The Black Swan*, Taleb provides a number of interesting anecdotes that relate to the impact of huge moves on the average performance of a system. For instance, most of the returns of the S&P500 index over the last fifty years can be traced back to just 10 big up days in the market, yet we tend to view the stock market in terms of its "average" drift.

Since the publication of his first book for general audiences, *Fooled by Randomness,* and certainly since the release of The Black Swan and his prediction that Fannie Mae and Freddie Mac were ticking time bombs, Taleb has become quite hostile to mainstream finance and risk management. He has variously accused members of the financial economics profession of being charlatans, frauds, imbeciles, criminals, and worse. Taleb has been particularly vicious towards the financial media's penchant for insipid ex post explanatory narrative fallacies, and the very poor accuracy rates attained by "professional" economic and political forecasters. The former should be mostly viewed as entertainment, but the latter is a very important subject for investment and trading system design, and we will deal with the evidence regarding the effectiveness of various political and economic forecasting methods at some length in the future.

I personally rather like Taleb, am a fan of his work, and feel that there are many good lessons to be learned from him and from his cerebral lifestyle, though I feel that he has increasing turned to certain sensationalist rhetorical gimmicks as he has evolved into more of a public intellectual/philosopher/social critic and less of a trader.

Another interesting individual from academia who has done a lot of work with Chaos in its financial market applications is Arthur DeVany. DeVany taught economics in California and conducted an analysis of the movie industry, wherein he found that the frequencies of extreme "tail-events"---wildly profitable films---did not follow the normal distribution curve, but instead followed a power law. DeVany also found that the forecast accuracies in the movie industry are as low as they are in economics and politics. He wrote a very nice book, *Hollywood Economics*, about his findings, and I think he has the same general disdain towards Gaussian-based economics and finance that Nassim Taleb has---in fact, the two are friends. He also shares Taleb's tremendous self-confidence, although DeVany seems to be a bit more charitable to his colleagues in mainstream economics.

DeVany's work in *Hollywood Economics *quantitatively validates an observation that was made by legendary movie industry maven William Goldman: "Nobody knows anything. Why did Paramount say yes to *Raiders of the Lost Ark*? Because nobody knows anything. And why did all the other studios say no? Because nobody knows anything. And why did Universal, the mightiest studio of all, pass on *Star Wars*?...Because nobody---not now, not ever---knows the least goddamn thing about what is or isn't going to work at the box office."

In private correspondence, DeVany has mentioned his interest in "living on kurtosis"---i.e., building a lifestyle around preparation for the extreme events. Following these principles, DeVany has become his own test subject as he undergoes the discipline of a physical conditioning, diet, and overall health approach based on evolutionary principles and a study of what the human body was "selected" by Darwinian processes to thrive on and achieve.

Some of this is heavily based on applications of Chaos theory to human physiology, as we would expect from a man with DeVany's intellectual interests, with the thesis being that Chaos can actually be good for us and that we evolved under conditions which featured lots of walking and outdoors play, punctuated by random, intense bursts of very serious (i.e., life-threatening) physical demands.

To name just two recent examples of Chaos being applied to medicine:

-bioengineers and neurologists studying seizures have found that EEG patterns in epilepsy sufferers in the throes of seizures tend to be, counter-intuitively, *less* chaotic than they are under normal conditions.

-a cardiologist named Ary Goldberger found that the most regular, Gaussian heart rate diagrams were found in people suffering from congestive heart failure, and that our heart rate patterns tend to become less chaotic---less healthy, in these terms---as we age. DeVany has used studies like this to form an opinion that long-duration, steady-state aerobic training may be detrimental to health if it trains the Chaos out of the heart rate.

A final figure in our colorful cast of Chaos finance academics who I'd like to mention is James Orlin Grabbe. Grabbe did his PhD in economics at Harvard and became an authority on pricing foreign exchange derivatives. He taught at Wharton, where he specialized in trying to find and develop currency traders, and wrote a brilliant text on macroeconomics, *International Financial Markets*. My third-edition (1996) copy of that book has a piece of the Mandelbrot Set on the cover and is filled with tantalizing, occasionally even delicious insights that any mathematically-inclined trader or strategic investor could appreciate. Here's one: "Consider the yen/dollar exchange rate for 1971-1980 and 1980-1992. Embrechts (1994) reports finding a value of H=.64 for the first period, and H=.62 for the second. This suggests persistence in the exchange rate returns, and nonperiodic cycles. Periods of gains and losses tend to bunch up, much like Nile discharge levels. It also means that the scale of the distribution of 100-day returns is about 18 or 19 times as large as the distribution over 1 day (since 100^.64=19.05 and 100^.62=17.38). In the (random walk) Brownian motion case, the scale would only be ten times as large (100^.5=10). So if you were doing risk analysis or 'what if' scenarios for your bank, the Brownian motion assumption (assumption of independent normal increments) would give too small a probability for changes over that time horizon."

Always a libertarian/anarcho-capitalist, Grabbe appears to have become radicalized sometime in the late 1990s. He moved to Costa Rica with his beloved cat and began a "digital money trust" project, essentially a free banking/offshore finance experiment heavily leveraging encryption technology, Chaos, and e-gold. By 2001 or so, Grabbe's website pages began to regularly feature soft porn pictorials and an increasingly bitter view of the effects that government interventions were having on social and economic activity. I was greatly saddened to learn that this eccentric, fascinating individual died in 2008 at his home in San Jose, Costa Rica. Grabbe had begun work on a second textbook that would feature more in-depth research into game theory applications to trading, as well as continuing the recurrent theme of Chaos.

Because of recent, terrible events in Haiti, my next blog entry will be about power law distributions in earthquakes and the chances of increasing our level of prediction accuracy where these disasters are concerned. There are many similarities between financial market crashes and earthquakes, and models from the two study areas are frequently shared. After that, I plan on getting back to some practical investment implications of the Hurst/Mandelbrot findings, hedge fund strategies that can make money from these market properties, and a school of economics---the Austrian---that can shed light on why markets behave this way and why some Chaos there, as in human heart rates, is a good thing (and why attempts to tame the Chaos will ultimately fail unless the innovation capacity of the economy is itself virtually destroyed in the process). At some point in the near future, I am going to turn to the battlefield and some models that are appropriate for the study of war.

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