“Gaussian or Bell curve is named because Friedrich Gauss proposed this curve . Bell curve, or Gaussian model, has since pervaded our business and scientific culture terms like sigma, variance, standard deviation, correlation, R-square and Sharpe.
Bell curve is meaningless in money and stocks but you may see it on German notes , financial markets run on the theories which are based on Gaussian Distribution . Bell curve is used in risk management though in many banks by Black Suit wearing officer. Assume that average height of men has a mean of 1.5 m and standard deviation 0.22 m and they follow a Normal /Bell distribution. Note that 220 cm standard deviation(.220 m) is randomness here and a very high one for a computer programmer.Gaussian yields its properties rather rapidly (a way to get a solution rather accuracy ), standard deviation in Bell curve faces a head wing where probabilities move rapidly as you move far away from mean. But my way of calculation does not make probability change ,it stays same over a range(unlike Bell curve). If I tell you that combined height of 2 men is 14 feet then you will think of 7 feet for each not 11 and 3 feet for them. People like to think in an easiest way and avoid randomness as 7 feet as for frequent and mind see it easier to conceive. Bell curves used in extreme events may cause a lot of disaster. Measures of uncertainty that are based on Bell curve disregard the impact the sharp jumps and inequalities and using them is like getting grass (grass disaster) and missing out the trees (Big Black Swans). This is why economics is based on Equilibrium , it allows you to treat economics as Gaussian . Assume you have a sample of 1000 people(giants and dwarfs) , your average will not be changed if you add another giant as your average will not be altered but if you add a mega giant it may be. So a single event will not change anything.
Randomness if Gaussian is tameable and is not altered by a single addition or removal. Casino people make such calculation and sleep well in night, no single gambler with a big hit will not change it and you will never see one gambler getting 1 Billion. The Gaussian family also contains Poisson and the distribution are the ones where mean and standard deviation describe everything and you don’t need any other thing.In fact, while the occasional and unpredictable large deviations are rare, they cannot be dismissed as “outliers” because, cumulatively,their impact in the long term is so dramatic.Mediocre events get fine or acceptable with Gaussian Distribution because big trees are not present in such events. I say that one should not use Gaussian in extreme events. But once you get Bell curve in head it’s hard to avoid.
A group of thinkers consisting Karl Marx and others , they all worked on Socialism and were looking for “Golden mean” of everything ,like height ,wealth and economy etc. A golden saying my Dad said once :Virtue lies in moderation ,all should embrace mediocrity. People have a golden mean and so people deviate from these like a normal mind and steady health is best but many deviate .Some men are sick and some are intellectual and some are bulky and intelligent. Being an average man means that one should be mediocre in thinking but God has made every man equal and have given gifts to few and given flaws so to tell people as in Equilibrium. Though divergent society does change people like wars disable people and prosper some.”
“Henri Poincare was suspicious of Gaussian curve,Gaussian was initially established for cosmology and atomic uncertainty. But apart from pysicists mathematicians began to use it because maths people trusted physics people. Doing science for sake of knowledge does not mean you will be successful. Gentlemen scientists like Lord Cavendish ,Lord Kelvin, Ludwig Wittgenstein, and Uber philosopher Bertrand Russell are those who will think twice in using Gaussian curve. Bell curves are used in medicine in yes -no events because they are mediocre events.Even Sir Karl Popper also considered how new observations affected knowledge – such as spotting a black swan when it was thought all swans were white.”
Gaussian Fallacies are Everywhere
Gaussian yields its properties rather rapidly (a way to get a solution rather accuracy ), standard deviation in Bell curve face a head wing where probabilities move rapidly as you move far away from mean. But my way of calculation thus not make probability change ,it stays remain same over a range(unlike Bell curve). If I tell you that combined height of 2 men is 14 feet then you will think of 7 feet for each not 11 and 3 feet for them. People like to think in an easiest way and avoid randomness as 7 feet as for frequent and mind see it easier to conceive. Bell curves used in extreme events may cause a lot of disaster. Measures of uncertainty that are based on Bell curve disregard the impact the sharp jumps and inequalities and using them is like getting grass (grass disaster) and missing out the trees (Black Swans,black swans are rare events that carry a massive impact).
Nassim Taleb stipulates that financial markets do not follow Gaussian curves and that academics around the world , all the financial theories and all the monetary policy makers do not understand this concept and so cannot access its validity and that only guy to promote such thinking was Beniot Mandelbrot.
So, while weight, height and calorie consumption are Gaussian, wealth is not. Nor are income, market returns, size of hedge funds, returns in the financial markets, number of deaths in wars or casualties in terrorist attacks. Almost all man-made variables are wild or carry massive randomness(Black Swans).The unknown process and factors influence the financial markets and that according to the Central Limit Theorem ,these unknown influences become or accumulate to normal distribution. The reason that systematic risk is based on Normal distribution, so systematic risk is what rules financial risk and Bell curve does not follow it as it is evident from the 2007-2008 Financial crises.
The problem is that measures of uncertainty using the bell curve simply disregard the possibility of sharp jumps or discontinuities and, therefore, have no meaning or consequence.Using them is like focusing on the grass and missing out on the (gigantic) trees .This is why economics is based on Equilibrium , it allows you to treat economics as Gaussian . Assume you have a sample of 1000 people(giants and dwarfs) , your average will not be changed if you add another giant as your average will not be altered but if you add a mega giant it may be. So a single event will not change anything.Mediocre events get fine or acceptable with Gaussian Distribution because big trees are not present in such events.
Financial Crisis of 2008:A key factor that led to the collapse of the banking industry in 2008 was the increasing use of mathematical models, spurred by the desire to exert total control over risk. These models, in all its elegance and beauty, badly underestimated the occurrence of extreme events.
Of particular note is a modeling technique called the Gaussian copula, which puts a price on the risk of multiple assets (or in this case, mortgages) defaulting at the same time. Upon its introduction by a quant named David Li, the popularity of this model sky-rocketed and the banking industry embraced it gleefully as the final piece to the risk management jigsaw that the industry had been piecing together. Ratings agencies such as Moody’s and the S&P readily adopted it in formulating company credit ratings, and the model finally found its way into the Basel II regulatory framework, as the guideline to calculate capital requirements for banks based on structured credit that they hold.
If you use Bell curve in top stocks and genetic measures, extreme events if calculated Bell curve may cause disaster. Head -tail on a coin is a random walk (left or right or win or loose) is a mediocre event so we use Gaussian curve. Tree diagrams are based on multiple tosses or multiple balls chosen etc or two or more dices tossed etc. In tree diagram if net is one Win it can have many cases (23=8) .Use of this tree diagram makes us closer to Normal/Bell curve as we know that condition of Gaussian is that N (no of objects) should be greater than 10 etc and also Poisson is followed by these tosses of coins but it will get to Normal after some time.
We have moved from observation to mathematics ,something abstract is like thermometer where 25 degree Celsius is pleasant and 40 degrees is hot and you do not need to know what temperature is. Also remember that standard deviation is not average standard deviation of a curve. Standard deviation is between +1 to -1 ,it is a scale ,Standard deviation and Variance(standard deviation 2) or sigma variates dramatically when you get away from average .Scaling to a sigma is used as well.In real life people don’t take account of past probability ,though past winning has effect on future probability but Bell curve doesn’t take into account of it. Models are made to scale standard deviation.
A major theme of Nassim Taleb is that models of uncertainty are too precise, and this thread has a long history. Taleb’s sometime co-author Benoit Mandelbrot has been trying to sell the world on the big idea of fractals in finance for several decades. James Gleick’s Chaos outlined the essence of Benoit Mandelbrot’s fractals, which takes a simple few lines of inputs to create graphics of insane complexity yet also beautiful recursive symmetry, in many cases eerily similar to nature (eg, ferns, snowflakes). In dynamic systems, you have chaotic systems that are purely deterministic though sufficiently complex that they appear random. These systems have large jumps, or phase shifts, reminiscent of market crashes or sudden bankruptcies; they have butterfly effects where small changes produce big differences in outcomes. Mandelbrot and others have been trying to apply these ideas to financial markets for many decades now (since 1962!), and the effort has not gained any traction, in spite of many papers applying this concept (search skew or kurtosis in any financial journal and you will see many papers). Mandelbrot’s big idea in finance is that finance relies on a profoundly flawed assumption, mainly that market prices are normally distributed.
The markets are non-linear, dynamic systems, subject to the rules of Chaos Theory. Market prices are highly random, with a short to intermediate term trend component. They are highly dependent on initial conditions. Markets also show qualities of fractals — self-similar in the sense that the individual parts are related to the whole.Due to the non-Gaussian behavior of the markets the methods from Chas Theory, Fractals and Quantum Physics(probability calculations from quantum mechanics) are being used in Finance.
There were only 2 Mathematicians who were actually able to understand randomness in a practical way and understand the flaws in them: 1.Beniot Mandelbrot 2.Henri Poincare
The poet of randomness : Beniot Mandelbrot : French philosopher, other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness. Black Swans were dealt by him in a philosophical and aesthetic way. Nassim Taleb and Mandelbrot have common things like Black Swans and Wild Randomness.Dr Mandelbrot also claimed that financial-market movements, too, have fractal forms, rather than the familiar bell shapes of “normal” distribution that Gauss described.Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called “Mandelbrot set” and later worked on shapes and fractals of maths and also worked on Chaos Theory.
Alternative to Gaussian/Bell curve would be using power laws and fractals instead of the Gaussian distribution. The idea of power laws and fractals in the financial markets is first pioneered by Benoit Mandelbrot, and subsequently popularized by Nassim Nicholas Taleb. This theory states that the markets are not just random—they are turbulent.Randomness associated with Gaussian distributions is too polite, too courteous, and is too unrealistic. Turbulent markets, on the other hand, incorporate a “wild” kind of randomness into consideration, which is characterized by sudden large jumps in volatility.EMH(Efficient Market Hypothesis ),the core concept of Finance also assumes Gaussian curve for its validity,another flaw in Finance.
If stock markets were Gaussian then stock market crashes would have happened once in a Billion years. Mandelbrot’s randomness methods make the statistics methods look useless. After the stock market crash William Sharpe and Markowitz model was given a Nobel Prize and this portfolio model was based on Gaussian Distribution. If in this world such method can get Noble then anything in this world is possible , anyone can become President etc.
Fractals distributions do better than Bell curve in avoiding the Big Black Swans .Sometimes a Fractal can make you believe it is Gaussian. Normally extreme events fit into Fractal category , fractals thus have very high standard deviation . Statistical Physics is what is good for use in Fractals Methods and Econometric and Gaussian methods are not to be used in Fractal Distribution. Businessmen have big egos, the Fractal Ego. So Fractal is any event described mathematically and is an extreme event and has high standard deviation, just like Black Swanevent.(Nassim Taleb also worked with Mandelbrot on randomness of Black Swan events ).
The Gaussian bell curve variations face a headwind that makes probabilities drop at a faster and faster rate, as you move away from the mean, while “scalables” or Mandelbrotian variations do not have such restriction.
True total intellectual people are what I look for, erudition is what I look for in people.Mandelbrot linked randomness to geometry and made randomness a more natural science.Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness.”
Henri Poincare is said to be underrated, he was the best mathematical thinker of all time,a true polymath and the man who published in every branch of math and science. Every time I see picture of Einstein I think of Poincare because I think Poincare was better than Einstein(another form of narrative fallacy).It took almost a century to understand his theories ,Poincare was the first thinker to go against Gaussian or Normal Bell curve.Poincare was suspicious of of Gaussian as he knew that extreme events don’t follow Bell curve.Poincare was the master of theory of relativity and atomic structure and even Einstein had to read him before he published as he was foremost authority on relativity.Many claim that Poincare was the first one to give idea of relativity but he never made it big to get prominence.Poincare also started the study of fractals and their use in physics,math and randomness etc,Poincare worked on Theory of Probability and also on Geometry,Chaos Theory and Astronomy/Mathematical Physics.Mandlbrolt later continued his work 100 years after his death. Poincre was first big gun to understand mathematical techniques and limits involved in randomness and hence forecasting limits. Poincare’s research on solar system got a Prize which was the highest academic prize at that time. Poincare was the adviser of Louis Bachelier, the pioneer of Financial Maths. Poincare suggested that as you project in the future you may need an increasing amount of precision ,near precision is not possible . Think of forecasting as in terms of tree branches, this grows in multiple ways and doubling every time so such increasing amount requires a lot of precision.
Merton made his famous formula based on Gaussian and so a flattering thing. Steve Ross an economist, famed to be more intellectual than Merton gave Nassim applause on his Black Swan theory work in a seminar in U.S. Portfolio theory users can’t tell me how can they accept the use of Gaussian curve with large deviations(high standard deviations) in stocks.Gaussian and high sigma cannot go together but all economists have been using it since a long time.
Robert Merton and Scholes made their company LTCM (Long Term Capital Management), they employed top quants and used complex methods based on portfolio theory. Later in Russia when there was market crash and it made big impact on U.S market and thus making extreme event in U.S market and everything got busted along with LTCM. Someone using Gaussian in our U.S market or Wall Street (market which can experience extreme events) is a madman in my world.
Source: The Black Swan.