Thursday, April 12, 2012

Modern Portfolio Theory & CAPM.

Conventional wisdom amongst investors recommends maintaining a diversified portfolio. The reasoning behind this, however, is not so common knowledge. One theory put forth by Harry Markowitz, Modern Portfolio Theory, asserts that asset returns represent normally-distributed random variables, each with their own variances, and quantifiable risk being represented by standard deviation (Markowitz, 1999). While variants of Modern Portfolio Theory are still in use in the financial industry, and its main theorists had won the Nobel Prize for its creation, it has recently been the subject of increased criticism, citing its dependence on rational investor behaviour and market efficiency (The Economist, 2009). Building on Markowitz’s earlier work on Portfolio Theory,William Sharpe (1964) proposed the Capital Asset Pricing Model (CAPM) for pricing risky securities. Sharpe says an investor might think of the CAPM this way. "First, I would say to you, ‘what do you think you should pay to get a dollar in bad times? Should you pay more than to get a dollar in good times?' And you would probably say, ‘Yes, I think so.' If you said, ‘No,' I'd say, ‘Well there aren't as many dollars available in bad times so don't you think the price of scarcer goods should be higher than more plentiful goods?' And hopefully you'd see that this makes sense. So I'd try to convince you that, all things being equal, the price of a dollar in bad times should be higher so, therefore, the return on what you pay for it should be worse. You pay more to get the dollar so the return on your investment is going to be worse."Sharpe is also known for creating the Sharpe Ratio, which is a risk-adjusted measure of investment performance. This continues to be one of the most widely-used performance measures for investment managers. Sharpe originally called the Sharpe Ratio the Reward to Variability Ratio but other scholars insisted on naming the idea for its founder.
William Sharpe


Before Markowitz formalized his portfolio theory in 1952 (Markowitz, 1999), investors had already held the belief that diversification of portfolios could reduce risk while preserving an adequate level of returns. Portfolio theory assumes that investors are risk-averse; that is, given two securities with identical returns, investors will prefer the security with the lower risk (Markowitz, 1999). What Markowitz (1952) contributed, however, was a formalization to diversification. Prior to Portfolio Theory, informed investors understood that it was in their best interests to maintain a diverse set of securities in a given portfolio. Markowitz (1952) demonstrated that correlation between securities can be quantified and as long as securities are not perfectly positively correlated, their combination in a portfolio will result in reduced overall portfolio risk. His work put risk at the center of investing, and attempted to measure the appropriate amount of risk to undertake, as higher returns are dependent on greater risk, and the greater the risk, the greater the possibility of loss. Further, he explained that the variance of the overall portfolio is a function of the variances and covariances of the individual securities comprising the portfolio (Markowitz, 1999). His landmark 1952 paper showed how a subset of the possible portfolio compositions, the efficient frontier, represented the lowest level of risk for a given level of return.Markowitz showed how an investor’s portfolio choice can be reduced to balancing just two dimensions: the expected return on the portfolio, and its variance or standard deviation, depending on the circumstances.At the RAND Corporation, he researched optimization techniques, developing the critical line algorithm for the identifications of the optimal mean-variance portfolios. This was found to be lying on what was later named the Markowitz Frontier.A Markowitz Efficient Portfolio is one where no added diversification can lower the portfolio’s risk for a given return expectation, while the Markowitz Efficient Frontier is the set of all portfolios that will give the highest expected return for each given level of risk. These concepts of efficiency were essential to the development of the capital asset pricing model .Building on the quantified risk concept described by Markowitz (1952), Sharpe described a pricing theory for risky securities comprising two components: a risk-free rate of return equal to the return of a security with no default risk (such as a US Treasury bill) and systematic risk (“beta”) coefficient of risk responsiveness relative to market risk premium (Sharpe, 1964). The result of this relationship is encapsulated graphically in the Security Market Line.
Harry Markowitz


 While Markowitz (1999) pointed out that even Shakespeare included wisdom about portfolio diversification in The Merchant of Venice, it was not until Markowitz (1952) and Sharpe (1964) that widely accepted quantitative models of the relationships between risk, return, and asset pricing gained acceptance. Although Portfolio Theory and the Capital Asset Pricing Model are key theories in finance, contemporaries of Markowitz (1952) and Sharpe (1964) have identified a number of opportunities for these theories to be developed further. It is possible, for example, that returns may not be best represented by a normally-distributed random variable in all cases. There are also special groups in the population that behave contrariwise to the typical “rational investor” either for psychological or, in some cases, even religious reasons: Islam is the fastest-growing religion in the US (Dar & Presley, 1999). Islamic law forbids earning interest on debt but allows—even encourages—investors to earn a return for taking on risk in their investments (Chiu, Newberger, & Paulson, 2005). Consequently, there are legitimate explanations for rational investors to make decisions that under previous assumptions would be classified as nonrational. Very interesting Game theory is another revolutionary development of the 20 th Century that at one time was hoped to elucidate investor behaviour. Due to the complexities of large systems, particularly involving human behaviour, it too has proved to be somewhat underwhelming in terms of predictive ability. I have long believed that such studies of economics are excellent tools to identify themes and understand behaviours, but that human behaviour is too complex for these models to propose appropriate interventions. While blood levels of a certain nutrient may show a deficiency is causing a certain symptomology, supplementation of this nutrient may not have the desired effect because of complex interactions that were not readily apparent from observations. I believe this is so with economics and financial markets as well. Even if individual causal interactions could be identified from existing models, they might still not be applicable in situ
Thus, I believe that the concepts Markowitz (1952) and Sharpe (1964) put forth in their theories, such as quantification of risk and interactions of risky assets, are important for future research; yet, research must still be done to find parsimonious models with predictive power, as well as interventions that can be practically applied to real-world investment decisions.

Sharpe's View of the Future of Financial Economics
In his recent book, Investors and Markets, Sharpe called for a new approach to the teaching and practice of Finance. "It's time to move beyond the simplifying assumption that probability distributions of economic events are normal or log-normal and that people only need to consider the mean and the standard deviation of a distribution," he says.
"I think, increasingly, we need to go beyond that - and there's no reason that we can't. You can have distributions that are as complex as you like and you can have characterizations of people's preferences that are reasonably complex. So I think practitioners should go beyond simple paradigms and I have been calling for academics teaching in business schools to do so."

Wednesday, April 4, 2012

Nassim Nicholas Taleb against Gaussian Curve

Note:This article is made from theories and books of Nassim Nicholas Taleb.

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.

Nassim Nicholas Taleb, known for his aggressive attitude towards Financial Industry.Taleb is a modern day Nietzsche. This is a man who suffers fools impatiently, and his intellect makes his hauteur largely justified.
Nassim Taleb is a guy who made a very groundbreaking theory of "Black Swans" , an original idea which few economists and financial advisers understand (as all theories are based on Gaussian Distribution) and his idea is based on Probability of Wild Events.


Henri Poincare was suspicious of Gaussian curve,Gaussian was initially established for cosmology and atomic uncertainty. But apart from physicists 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 is trying since 2005-2006 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 (2^3=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.
Nassim Taleb and Beniot Mandelbrot suggest that Gaussian curve is not so useful for calculating randomness of man-made variables and especially the financial market.

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. In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk. Dr Mandelbrot 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.
Beniot Mandelbrot


Henri Poincare



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 advisor 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 formulae 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.