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Overview of Circumstances & Strategy

By Hector Harris,2014-05-17 09:52
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Overview of Circumstances & Strategy

    A Non-Technical Introduction To Benoit Mandelbrot’s

    Multi-Fractal View of Financial Markets Verses Modern Financial Models

    By Clifford J. Short III, AAMS AWMA CPM

    Scope

Accurately measuring and predicting risk and investment return has been the elusive „holy grail‟ of

    financial market analysts and technicians since the beginning of capital markets. From diverse backgrounds

    such as astronomy, physics, and applied mathematics, to name a few, „students‟ of the financial markets

    have developed and modified models to comprise the analytical tools used today. Based deeply in normal

    distribution, rational markets and rational participants, continuous probability, and smooth price changes,

    Modern Portfolio Theory is the foundation of today‟s asset management and market rationale.

The current standard financial models for investing and market analysis suggest utilizing variance

    measurements and betas of different securities and classifying investment portfolios by their probability of

    risk--- we can build an “efficient” portfolio to target a specific return with a desired level of risk. Further,

    price changes are statistically independent and normally distributed.

To the contrary, Benoit Mandelbrot, the eighty-year-old Polish-born mathematician awarded the Japan

    Prize in Science and Technology and the Wolf Prize in Physics, sees the markets and their movements

    much differently. The „inventor‟ of fractal geometry---a mathematic discipline that attempts to measure and explain “the roughness” of observations---submits market prices do not follow a bell curve (normal distribution) and prices are very much dependent and have „memories‟.

    The Basis of Modern Financial Analysis

To start, the Efficient Market Hypothesis ----in an efficient market, all relevant information is already (or

    will adjust to new information quickly) priced into a security today---is the argument for Modern Portfolio

    Theory. Simply, equilibrium is current or corrects itself almost immediately to information, preventing one

    from consistently outperforming the overall market--- nullifying the idea of Alpha (value-added long term

    active security selection). Today‟s modern risk measurements (Beta, Sharpe, Standard Deviation, CAPM,

    Arbitrage Pricing Theory) derive their formulae from this assumption.

Developed by Harry Markowitz in the 1950‟s, Modern Portfolio Theory is derived from Brownian Motion

    Theory. This physics theory was „discovered‟ by Robert Brown, a Scottish botanist, by observing under a

    microscope the “erratic way that tiny pollen grains jiggled in a sample of water.” After comprehensive

    study, he discovered this erratic motion over time developed into a defined pattern---later defined as

    normal distribution. From this came the universally applicable linear measurement the „bell curve‟ and the observation that distribution will revert to the mean over a period of time. Subsequently, in 1900 a French

    mathematician Louis Bachelier began his efforts in the new field of probability theory and applying

    Brownian Motion Theory in finance---French government bond prices. His approach foundation--

    “Random Walk”---postulated prices go up and down with equal probability. Standard deviation was used to

    explain and quantify the scatter of data outside the mean. Hence, with the „bell curve‟ the “abundant small changes cluster in the center of the bell while the infrequent big changes (outliers) dot the edges”.

    Bachelier‟s key model hinged on price changes being statistically independent and normally distributed.

    What is Risk?

Risk, in the financial markets, is viewed as price volatility and the possible loss of capital. The two types of

    risk pertaining to the financial markets are systematic and unsystematic. Systematic refers overall market

risk, interest rate risk, exchange rate risk---macro event risks that effect the entire market with no way to

    diversify out the risk. Unsystematic risk (diversifiable risk) points to security-specific risk---risks the

    specific company or entity accommodate--e.g. liquidity risk, default risk, poor management.

    A Sampling of Risk Measurements of Modern Financial Analysis

    ? Probably the most widely recognized and used measurement for a security is Beta. Beta measures

    a security‟s risk relative to the overall market based on past volatility (linear regression). For

    instance, a Beta of 1 is „perfectly‟ correlated to the overall market--will do no better or worse

    relative to the market---while a Beta of less than 1 displays less volatility to the market and a Beta

    of more than 1 will be much more volatile as the number increases above 1.

    ? Standard Deviation is the mathematical apparatus used to measure the variability of a security‟s

    return around its average‟s (mean) total risk return--- “A simple mathematical yardstick for

    measuring the scatter of data”. Utilizing the standard deviation model and graphing the

    observations, a bell shape will occur. Again, “small clusters in the center and infrequent big

    changes at the edges”. As the German mathematician Carl Friedrich Gauss imposed , the bell

    shape is „normal‟ and any other shape is an anomaly. Subsequently, standard deviation considers

    volatility above the mean just as harmful as volatility below the mean.

    ? The Sharpe Ratio, developed by William Sharpe, to become known as the Capital Asset Pricing

    Model (CAPM) takes Markowitz‟s “cumbersome portfolio calculations” and reduces them to just

    a handful. CAPM basically sets „a hurdle rate‟ for a security‟s expected return. Simply, the value

    of an asset is based on how it functions within a market portfolio and how its risk relates to the

    market‟s risk. The returns are based on the relativity to returns of a risk-less asset (usually

    identified as U.S. government T-bills). The return above the risk-less security is labeled excess

    return.

    ? A broader, more comprehensive model has been developed, again, rooted in the original thesis of

    normal distribution. The Arbitrage Pricing Theory (APT) uses not one factor, like Beta, but

    utilizes multiple factors. In fact, APT uses as many factors as the analyst desires: Beta

    market/book effect, Beta price/earnings effect, Beta state of the economy, and Beta “for any

    element that could affect a security‟s price”. Essentially, APT dictates that the return of a risky

    asset is a linear function of a number of macro economic and financial factors. The value of the

    return responds to the changing values of the factors.

    ? And probably the most celebrated formula in the last thirty years: The groundbreaking Black-

    Scholes model for pricing options. Devised by the Nobel Prize winning duo Fischer Black and

    Merton Scholes in the late 1960‟s and in use by 1973, this model is based on the thesis “you do

    not need to know what the stock price will finally be when the option expires, but what the traders

    themselves know, the terms of the option and how volatile the stock is.” They followed Bachelier,

    Markowitz, and Sharpe on assuming a stock‟s risk and volatility can be gauged by the „bell curve‟

    standard. It asserts volatility will remain the same through the life of the option, prices do not

    jump‟. Still widely used by the „street‟ (with acknowledgement of its limitations---especially in

    pricing exotic options and company-offered options which are both privately arranged

    transactions), but with much less blind faith.

In review, our standard financial models which risk measurements are predicated on the assumption returns

    are normally distributed, price changes are statistically independent, and the markets and its participants are

    rational. Additionally, utilizing measurements of variance and betas of different securities and classifying

    investment portfolios by their probability of risk, one can build an „efficient‟ portfolio to target a specific

    return with a desired level of risk.

    A Sampling of Outcomes Based on Modern Financial Analysis

    Mental note: “small clusters in the center and infrequent big changes on the edges” (?)

    ? From 1916 to 2003 there should have been 58 days when the Dow Jones Index Average moved

    more than 3.4%. There were 1,001

    ? There should have been 6 days the Dow swung beyond 4.5%. There were 366. th? Dow swings of more than 7% should have come once every 300,000 years. In the 20 century

    there were 48 days of swings exceeding 7%.

    ? In August 1998 there were 3 trading days where the Dow dropped 3.5%, 4.4%, and 6.8%.

    According to our current standard models the 6.8% drop should only happen 1 in 20 million. “If

    you traded every day for 100,000 years, you would not have expected that to happen once. The

    odds of having those 3 declines in one month: 1 in 500 billion.

    ? In 1997 the Dow cratered 7.7% in one day. The odds: 1 in 50 billion.

    ? In July of 2002 the Dow declined sharply three time within 7 trading days. The odds: 1 in 4

    trillion.

    ? On October 19, 1987 (“the worst day of trading in over 100 years”) the Dow fell 29%. The odds; 1 thin 10 to the 50 power--- “a number outside the scale of nature”.

    Benoit Mandelbrot and His Multi-Fractal View

Multi-fractal geometry is the study and measurement of the irregular, convoluted or „roughness‟ in our

    environment--- think the coast line of any water lined continent viewed from a thousand feet, or a Jackson

    Pollack painting, or the chart of a market index over the past five years. Utilizing dimension as a tool of

    measurement, rather than an „inherent property‟, fractal mathematics „is about spotting repeating patterns,

    quantifying them, and manipulating them--- „a tool of both analysis and synthesis‟. Subsequently, fractals

    are inherently visual starting with an initiator--- usually a traditional geometric shape. Following the

    initiator is the generator--- „a template from which the fractal will be made‟. And the rule of recursion: the actual process of building the fractal--- basically, one small simple shape giving birth to the same shape

    hundreds, thousands, hundreds of thousands, and so on forming a complex reverberation: „an echo of the

    whole‟ (think DNA). To the point, fractal geometry is a „proper theory of the irregular‟ and measurement of

    the „uncontrollable element of life‟. Life is comprehensively „jagged‟. And, too, the financial markets.

Beginning in 1961 with the study of Cotton prices, wind turbulence, and power-law distributions,

    Mandelbrot discovered distributions have „fat tails‟ and „prices can and do vary by leaps and bounds‟

    unlike the nice neat idea of normal distributions. He discovered applying his fractal theory to the financial

    markets was a supreme fit. Mandelbrot comments: „It is an extraordinary feature of science that the most

    diverse, seemingly unrelated phenomena can be described with the same mathematical tools. The same

    quadratic equations with which the ancients drew right angles to build their temples can be used today by a

    banker to calculate the yield to maturity of a new, two-year bond‟.

Mandelbrot, and the failure of our modern financial analysis, shows prices in fact do not follow the bell

    curve and rather than move in a „random walk‟ seamlessly, frequently „jump‟ (discontinuity) and a far more

    volatile and risky than we preach. Additionally, different kinds of price series „exhibit different degrees of memory‟. To that, the power of fractal geometry comes from its ability to express a „great deal of

    complicated, irregular data?‟.

    The Snapshot of Mandelbrot’s Rules of Market Behavior

Rule No. 1: Markets are risky. Riskier than currently given credit or effectively measured. As the examples

    of the previous Dow debacles indicates, extreme price swings are not the anomalies we believed them to be,

    but „are the norm in financial markets, not aberrations?‟ Price movements do not follow the well-mannered, smooth bell curve assumed by modern financial models. „They follow a more violent curve.‟

    Not unlike measuring wind turbulence, there are „eddies‟ and frequent discontinuous movements to prices

    in the markets that have been and are mis-categorized as anomalies and infrequent outliers. Fractal

    geometry indeed expresses clearly risk has and is being mis-measured. Our current understanding of cause

    and effect is murky at best and catastrophically wrong at worst.

Rule No. 2: Trouble Runs in Streaks. „Market turbulence tends to cluster.‟ As professional traders

    experience, a wild volatile trading day on Day 1 may well be followed by a wilder trading day on Day 2.

    As Mandelbrot observes, „Their (traders) intuition--- not included in the standard model of efficient

    markets--- is entirely validated by the multi-fractal model‟. Multi-fractals make turbulence a „fundamentally new way of analyzing finance‟. Markets stop appearing rational, „well-behaved‟ patterns. They are in fact, dynamic, unpredictable and sometimes „dangerous systems for transferring wealth‟. Price

    charts are commonly measured by their volatility. However, volatility itself is volatile (an „echo of the

    whole‟). Mandelbrot‟s fractal approach is leveraged on spotting the regularity inside the irregular--- „the

    pattern in the formless‟.

    Rule No. 3: Markets Have Personality. „Prices are not driven solely by real-world events, news, and people‟. A new kind of dynamic surfaces when the „gathering‟ of all the traders and investors of the universe converge on the markets daily. Literally thousands of different motives, ideas, strategies, tactics, ambitions,

    needs and wants rush the markets that affect them endogenously. The fundamental process by which prices

    react to news does not change (mathematically stationary)--- completely contradictory to the random-walk

    theory.

    Rule No. 4: Markets Mislead. „Patterns are the fool‟s gold of financial markets‟. They appear predictable and bankable under modern financial analysis. Hence, „bubbles and crashes are inherent to the markets‟.

    They are the culmination and by-product of the „human need to find patterns in the patternless‟. People

    often think and behave in hierarchies „with repetition and scaling.‟ As Mandelbrot submits, „the brain

    highlights what it imagines as patterns and disregards contradicting information‟. Human nature strives to

    see order and will create it where it cannot find it. Fractals do not discount patterns, but, in fact patterns are

    the essence of fractal geometry. Fractal geometry states a theory of clearly measuring patterns, not

    subjectively and subsequently Gerry rigging observations to form patterns.

    Rule No. 5: Market Time is Relative. There is a relativity of time in the financial markets--- „distinct from linear clock time.‟ Simply, and as traders well understand, in periods of high volatility, time seems to speed

    up multi-fold while the inverse is true in periods of low volatility. The foundation of the common trading

    terms „fast‟ and „slow‟ markets is the compression and expansion of time (scaling) within the confines of

    the financial markets. It has an effect on how accurate modern financial analysis measures the observations,

    but, unfortunately, ignores the fact, disregarding relative time as a factor in analyzing the data observed---

    not even as an outlier, but as non-existent. Fractal mathematics has the opportunity to not only measure

    this component, but make it part of the overall analysis for comprehensive modeling.

    Conclusion

The modern financial models‟ theories and assumptions that prices move independently, smoothly, and

    markets are rational as well as its participants have been tested, if not throttled by Mandelbrot‟s multi-

    fractal approach and observations. Clearly there are limitations and stressors that frequently surface to

    condemn the „conventional wisdom‟ of modern portfolio theory and its modified incarnations. In

    Mandelbrot‟s words, “We know more about how our car engines work than about how our global financial

    system functions?we lurch from crisis to crisis”. Mandelbrot‟s intriguing assumptions strongly suggest a

    need for invention and change, but admits „clearly, fractal investment analysis has more questions than

    answers today‟. One thing that is clear to Mandelbrot without exception is „the very heart of finance is

    fractal‟.

    References

Dynamic Portfolio Theory and Management (Oberuc E., Richard) 2004

The Fractal Geometry of Nature (Mandelbrot, Benoit) 1982

The (Mis)Behavior of Markets (Mandelbrot, Benoit and Hudson L., Richard) 2004

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