Geeks With Blogs
Josh Reuben

I recently read an informative and succint book - A mathematician Plays the Market, by John Allen Paulos.

Heres my summary of key points:
Behavioral finance
·        Anticipating other's anticipations
·        A trading strategy can yield the illusion of effectiveness, when only chance is at work.
·        Keynes - short-term investors anticipate what average opinion expects the average opinion to be.
·        Distinction between being smart & rich, and distinction between being right & being right about the market. Involves logic and psychology
·        E.g. a group trying to guess 80% the average value of a number between 0 & 100 - some might guess 40, then others might count on this and guess 32 etc. – Meta-reasoning about other's reasoning until the optimal response of 0 is reached (the only number equal to 80% of itself) - the Nash equilibrium results when individuals modify their actions until they can no longer benefit from changing them given what other's actions are. It is also time independent - if someone guesses 0 right away it will be wrong as not enough meta-reasoning has yet taken place. Equilibrium naturally develops - "nobody goes there anymore, it's too crowded".
·        CCI - consumer confidence index - measures consumer's propensity to consume & their confidence in their own economic future - surveys people's beliefs about other people's beliefs
·        Common knowledge - when everyone knows what everyone else knows
·        Confirmation bias - search for good news
·        Adages - don't try to catch a falling knife (don't be self-delusional), don't put all eggs in 1 basket
·        Irrational exuberance - overoptimistic self-delusion in speculation
·        Behavioral finance - do not assume all players are fully rational homo-economicus. People's reactions to ultimatum games are influenced by various emotions & cognitive biases & may be counterproductive & irrational. Anchoring effect - we become attached to any number we hear (e.g. a price target, or a 52 week high). Books: Tversky & Kahneman - judgment under uncertainty; Thaler - the winner's curse ; Gilovich - how we know what isn't so.
Cognitive bias
·        Availability error - the inclination to view any story through the lens of a similar story
·        Confirmation bias - we check a hypothesis by observing instances that confirm it & ignore those that don't. Such selective thinking reinforces the anchoring effect - we look for reasons. We gravitate towards people who have similar perspectives (e.g. chat rooms)
·        Status quo bias - keep the money where it is
·        Endowment effect - inclination to endow ones holdings with more value than its worth.
·        Loss aversion - The flipside is minimizing possible regret - most people tend to assume less risk to obtain gains than they do to avoid losses - people are likely to take $5 rather than taking a Markov coin flip between gaining either $0 or $10; contrast with people are likely to take a coin flip between losing either $0 or $10 rather than definitely losing $5 
·        Mental accounts - we categorize money in irrational ways - people who lose a $100 ticket are less likely to buy a new one than people who lose a $100 on the way to buy a ticket.
·        Survivorship bias - investments that go out of business are dropped from the average - skews past returns upwards and induces irrational exuberance
·        Cognitive biases persist because they result in heuristic rules of thumb that can save time & energy. These can become evolutionarily hardwired.
·        Pump & dump - tout an asset hyperbolically, then sell it for a profit. Works best in bull markets where people are susceptible to greed & for thinly traded stocks where a few buyers can have a pronounced effect. The bear market analog is short & distort where people are susceptible to fear & anxiety
Technical analysis
·        Trends, crowds & waves
·        Discerning patterns of short-term market directions & then devising rules
·        Momentum investing - following the crowd – herd-ish behavior. Disdaining the crowd is hubris. 
·        Murkily justified, quasi-mathematical mysticism
·        Similar to the Fibonacci numbers (as are many finite state automata patterns occurring in nature) and the golden ratio (symbolized by the Greek letter phi - represents the irrational number 1.618... - Its decimal representation never repeats. Note: 1.618 = 1 + 1/1.618) - whereby a point bisects a straight line such that the ratio of the short segment to the long segment is equivalent to the ratio of the long segment to the whole line. Relationship: the ratio of any Fibonacci number to its predecessor is close to the golden ratio & gets closer with larger Fibonacci numbers
·        Elliot wave theory - the market rises on 5 distinct consecutive waves and declines in 3. The problem arises when investors try to identify where on the wave they are. The theory becomes too complicated to be falsified.
·        Value of the moving average - less volatility than the stock price itself. Variations may account for different weighting for different days or periods of volatility. Smooth out the daily fluctuations to expose broader trends & generate buy-sell rules - e.g. buy when it exceeds X-day moving average. The rule can work well when the stock fluctuates about a long-term upward or downward sloping course.
·        Problem - cost allot in commissions for a hovering stock price - must modify the rule so that the trade yields significant profit. Another problem is deciding when in the day to trade.
·        Can fine tune via voluminous time-series data-mining, by comparing moving averages over different intervals & trading when they cross, or calculate the integral using X-minute moving averages
·        If the efficient market hypothesis holds then information about a stock is almost instantaneously incorporated into its price, such that its future moves will be determined by random external events. Its past behavior is irrelevant & its future movement is unpredictable.
·        Resistance & support - obstacles to further upward/downward movement. Resistance - People who lose money remember what they paid or could have paid for a falling stock, so when it reaches that level they are likely to sell to recoup their loss, thus pushing the price back down. Support - people who miss out on making money remember what they paid or could have paid for a rising stock, so when it falls back to that level they are likely to buy to get the gains, thus driving the price back up.
·        Technical analysts recommend buying on the bump & bounce, or better yet if the price breaks through a resistance or support level
·        Variants based on price pattern volatility have peculiar names - e.g. head & shoulders. Lack foundational justification in psychological or financial principles.
·        There is some evidence that momentum strategies & short-term trend following outperforms a blind index fund - see Jegadeesh, Titman - could be caused by behavioral finance: investor overreactions or short-term persistence of the impact of corporate earnings reports. Rules based on moving averages, resistance & support are effective. In the long term, the prognosis changes: individual stock prices display a slight negative correlation.
·        See books: random walk down Wall Street, non-random walk down Wall Street.
·        If the random-walk hypothesis is wrong, then there are patterns of cross-correlations over time between stocks (not necessarily causal) which is an exploitable opportunity.
·        Other technical anomalies caused by behavioral finance: calendar effects (e.g. unusual good/bad returns at the turn of a financial period).
·        Blackjack is the only casino game of chance whose outcomes depend on past outcomes - can count cards. Roulette wheels & dice are Markov. Blackjack is much simpler than the market which is dependent on many more factors + actions & beliefs of other investors
·        Parrondo's paradox: winning through losing - with 2 games resulting in steady losses over time, when the games are played in succession in random order result in a steady gain. Variations of these games might yield counterintuitive investment strategies.
Chance & efficient markets
·        If the movement of stock prices is random , or near random, then technical analysis is bullshit. A correlation may have statistical significance (unlikelihood of occurring by chance) it is not practically significant because of the presence of so many confounding variables
·        There is a cognitive bias to place meaning in random events.
·        Efficient market hypothesis - Eugene Fama - at any given time, stock prices reflect all relevant information about a stock. Forced by competition. The weak form maintains that all information about past market prices are already reflected in the stock price --> technical analysis is useless. Strong form maintains that all publically available information about a company is already reflected in its stock price --> fundamental analysis is useless. Strongest form maintains that all information is already reflected in its stock price --> insider trading is useless, which is clearly bunkum! Like stating that if the light bulb needed changing then market would have already done it.
·        Efficient market theorists believe in passive & diversified investments such as -broad-gauged index funds -- see John Bogle.
·        The investing horde's actions (on rapidly disseminated information) causes the overall market to rapidly respond to any new info - is there a window of exploitable activity?. As markets become more efficient they become less predictable because if present stock prices already reflect all available information then future stock prices are unpredictable --> random walk hypothesis whereby each movement is independent of the previous.
·        There is over time an upward trend, as if the flipped coin were slightly biased. What moves stock is new technological developments.
·        Sources of such randomness are penny stocks.
·        An odd fact about a series of coin flips is that the proportion of time the number of heads exceeds the number of tails is seldom close to 50%. A stock thats fallen on a truly random trajectory is as likely to fall further as it is to rise, & vice versa. The rarity in which the lead overtakes & switches sides is independent of the 50% probability of heads, and of the phenomenon of regression to the mean.
·        Some stellar investors are lucky and stay ahead, , some have long streaks of luck, others have a reputation of prior (but random) success and become market trendsetters that influence others.
·        Stock trades are called bids & asks
·        See level 2 screens
·        Benford's law - numbers have 1 as a 1st non-zero digit 30% of time, 2 as 1st non-zero digit 18% of time, 3 about 12.5%, with larger numbers progressively less often. Used to catch accounting fraud.
·        Krauthammer dubbed A beautiful mind as 'disturbing nerd chic'
Fundamental analysis
·        A sober investment strategy , but at odds with the irrational herd behavior of the market
·        Euler's number e is the root of all money - geometric / exponential growth. The formula 1+e^(pi.i) =0 uses the 5 most important constants in a single equation.
·         compound interest. let t be a time period in years. The amount after annually compounded interest A=P(1+r)^t , and after quarterly compounded interest A=P(1+r/4)^4t. Can be rewritten as A=Pe^rt for continuous compounding. Doubling time - the time for a sum of money to double in value. Given by the rule of 72: the number of years = 72/(r*100) - e.g. for r=0.8 , it will take 9 years. For continuous compounding, use 70 instead of 72.
·        Discounting - the process of determining the present value of future money. Allows comparison of different amounts of money received at different times based upon r - use to evaluate the present or future value of an income stream by multiplying or dividing by the appropriate power of (1+r). Use to figure out how much to save.
·        a stocks fundamental value - price should be equivalent to the discounted stream of dividends you can expect to receive from holding onto it indefinitely. If the stock doesn't pay dividends or if you plan on selling it & realizing capital gains, its price should be roughly equivalent to the discounted value of the expected sell price + the discounted value of any dividends. Most stock prices are higher than this. During booms investors are more concerned with capital-gains than dividends. Bottom-line investment - the max price you should pay for a stock is the present value of all future gains from it. The discounting of future dividends & future stock price is dependent on your estimations of future interest rates, dividend policies, and other fundamentals. It is also distorted by behavioral cognitive biases.
·        Speculators often get conned by Ponzi schemes - myopically discounting the future at an absurdly steep rate, looking only at near term outcomes.
·        The arithmetic mean of N different rates of return is what we normally think of as the average ; the geometric mean of N different rates of return is calculated like compound interest: is equivalent to the rate that if received N times in succession, would be equivalent to receiving N different rates of return in succession --> astronomical returns or losses.
·        Lucky investors skew the average up. An investment's value can never sink below 0. E.g. the average worth of a $10K investment that can either grow 80% or shrink -60% a week (probability of 50% either direction) after 1 year (with disproportionately more up weeks than down weeks ) is $1.4 million , but its most likely worth (with 26 up weeks & 26 down weeks) is $1.95. N=2, so the (Nth root of the product [(1+80%)x(1-60%)] )-1 is SQRT(1.8 x 0.4)-1 = -0.15 (a loss of 15% each week).
·        Majority of investors receive worse than average returns & mutual funds misleadingly stress their average returns.
·        Stock evaluation - worth what a stock returns to its holder in dividends & price increases by based on the company's earnings (which will be paid out in dividends, or increase its price by growing the company or retiring debt). Estimate the amount of cash the stock will generate over its lifetime (the stock evaluation), then discount this stream of payments to the present. If the company earnings are growing, and further growth is projected, and general economic growth is good, and interest rates are low, then buy.
·        Use P/E ratio (also called the multiple) to determine a reasonable stock price - divide the stock price P (taken from the news) by the company's earnings per outstanding share E in the past year (not so clear-cut - Enron!) . It is both a prediction - measure of expectation of future earnings, & an appraisal - the price you must pay to receive the company's earnings (via dividends & price appreciation). A company with a high P/E ratio must perform to keep it. shrinkage occurs as startups evolve into blue-chips in the shape of a logarithmic S curve. Note: P/E ratio in isolation does not indicate whether a stock is under or over-valued - need to compare to its previous value, & to the industry as a whole. The average P/E for the entire market is approx. 20. Companies losing money have a negative P/E. Strong economies support high P/E ratios.
·        PEG ratio - P/E/G , where G is the expected annual growth rate of earnings * 100. A low PEG indicates an undervalued stock (a high G relative to P/E). The lower the better. Motley Fool & Peter Lynch recommend buying stocks when PEG<=0.5 and selling when PEG>=1.5 - with several exceptions. Finding stocks with a low PEG is not easy.
·        Dow-Jones industrial average, S&P 500 average
·        Growth investing - chasing of fast growing companies with high P/E ratios, such as high-tech, Telco, big pharms. Prone to investor overreactions. Value investment gets better returns.
·        Value investing - does yield moderately better rates of return than a broad-gauged index fund. Stocks with low P/E ratios are undervalued and yield better returns than those with high P/E ratios, dependent on other factors such as risk, company size & type. Focus on Solid humdrum companies of the type Warren Buffet likes: oil, finance, utilities, manufacturing. Dogs of the Dow / foolish four - buy the ten/ four stocks with the lowest P/E ratios - established firms unlikely to belly-up. These strategies did beat the market, but as with all such strategies , increased returns diminished as more people adopted it.
·        Transaction costs tend to eat up allot of this return , & competing investors tend to shrink it over time.
·        P/B - price to book ratio - where B is the book value per share: its total assets minus total liabilities & intangible assets, divided by the #shares. Less volatile than P/E. Like E, B is somewhat malleable. Eugene Fama & ken French endorse it. The decile with the lowest P/B had an average return of 21.4
·        P/S - price to sales ratio - a low P/S is an even stronger predictor - James O'Shaughnessy.
·        Foreign markets have high returns.
·        Regression to the mean - tendency for an extreme value of a partially chance-dependent quantity to be followed by a value closer to the average. The sequel to a low ratio is not as likely to be low. 
·        Fraudulent Accounting practices - can distort the E, B, S values! If an accounting firm both audits and consults to a company, there is a clear conflict of interest - e.g. WorldCom reported expenses as capital investments --> reported losses as profits
Options, risk & volatility
·        Requires estimation of small probabilities.
·        Derivatives - financial instruments whose values are derived from an underlying asset.
·        Stock options counter-intuitively reduce risk, like an insurance policy - hedge against the small chance of a decline in the strike price of the underlying. The strike price determines whether the option is in-the-money & has value.
·        Losses are limited to what you have paid for them, but the potential gains are unlimited for calls & very substantial for puts. Selling an option is a bet that the underlying will only rise/fall slightly.
·        Covered calls - investment hedge strategy of buying shares and simultaneously selling calls on it - e.g. buy at stock price $25 & sell 6mth calls with strike price $30. If the stock doesn't rise to $30, you keep the proceeds from the sale of the calls; else you can sell your stock to the buyer of the calls, limiting the risk in selling the calls.
·        Louis Bachelier - 1900 - dissertation that the stock market has normal-distribution random fluctuations according to Brownian motion. Prescient, but incorrect as it did not account for the effect of compounding stock returns, which results in a lognormal-distribution.
·        Black-Scholes - 1973 formula for option valuation. Depends (directly?) upon 5 parameters: the present stock price, the length of time until expiry, the interest rate, the strike price of the option, & the volatility of the underlying.
·        Short-selling - selling of stocks not yet owned, betting that the stock will decline & one can buy the shares back at a lower price in the future. Risky - the price may rise. Can be corrective to the overly optimistic bias of the market.
·        Buying on margin - buy stocks with borrowed money from your broker. Federal regulations require max 50% of the total market value is allowed to be on margin.
·        Hedge funds - small, private lightly regulated investment portfolios that do allot of short-selling, buying on margin & other complicated arbitrages (near simultaneous buying & selling of the same underlying in order to profit from tiny price discrepancies).
·        Long-Term Capital Management Inc. - in 1998, this firm with founders Merton & Scholes, failed to hedge, & collapsed! The lack of liquidity in world markets should have been anticipated.
·        Expected value - (mu) average value weighted according to the probabilities. E.g. a stock has 6% ROI 1/2 of the time, -2% ROI 1/3 of the time, 28% ROI 1/6 of the time: weighted average = 0.06 0.5 + -0.02x0.33 +0.28x0.17 =0.07 - i.e. 7%. A stock may likely do well, but still have a negative average value - e.g. rise by 1% with 95% chance, and fall by 60% with 5% chance -while the value expected is 1%, the expected value is -2.1%. 2 complimentary trading strategies: 1) usually small gains, but sometimes big losses; 2) usually small losses, but sometimes big gains.
·        Quantifying risk (deviations from the mean) and volatility – standard dev. (sigma) is the most common measure of risk as the average deviation from the mean, and is the square root of the variance. in a normal distribution, 2/3 (68%) of values lie within 1 s.d. of the expected value, & 95% of values lie within 2 s.d. of the expected value. E.g. a conservative stock with expected value of 5.4% and a s.d. of 3.2% --> 68% of time ROI will be between 2.2% & 8.6%, and 95% of time ROI will be between -1.0% & 11.8%
·        Six sigma - 6 s.d. from the expected average, product defects are miniscule. So unlikely that it is not included in most statistics tables
·        Central limit theorem - states that the averages & sums of a sufficient number of chance dependent quantities are always normally distributed. However, stock ROIs are not necessarily normally distributed!
Portfolio diversification
·        Bonds are safer & less volatile than stocks, but have a lower rate of return (this is not guaranteed for the future) . The average rate of return before inflation between 1802 & 1997 for stocks was 8.4%, and for bonds was 4.5%. Stomach churning volatility - s.d. of 17.5% --> 2/3 of the time the rate of return is between -9.1% & 25.9%. Over time the returns even out and the s.d. shrinks.
·        The equity-risk premium is the amount by which stock ROI must exceed bond ROI to attract investors. Stock ROI has been higher than for bonds because their prices are relatively lower because they are riskier. This could change if stocks are viewed as less risky.
·        St Petersburg paradox - repeat a coin flip until a heads comes up on the Nth flip, at which point you win 2^N dollars. The probability of a sequence of independent events such as TTTTTTTH is the product of all their respective events p^N. Expected value Sum(2^N . p^N) fails to capture our intuition
·        Bernoulli – people's emotion of any increase/decrease in wealth is disproportionate to the quantity of wealth previously possessed - the less you have the more you feel it. Utility drops off. Utility functions of money vary across people & time, wealth & age. Log utility function reflects the slowly diminishing satisfaction.
·        Volatility - cannot just weigh the volatilities of stocks in the portfolio - because their performances often have interdependencies and inverse relationships. Look for stocks whose values balance each other out, so that the volatility of the complete portfolio is 0. As long as stocks aren't positively correlated, volatility will decrease. Check the Covariance, not necessarily causation - the aim is to use the association, not understand it. For a diversified portfolio, look for negative covariance.
·        Covariance - expected value of the deviation from the mean of one stock multiplied by deviation from the mean of another stock i.e. [(X-µx).(Y-µy)], where µx is the mean of X. If the stocks vary together then their product will be positive, and if they vary inversely then covariance will be negative - this is what we want, as it reduces the variance of a portfolio.
·        variance of a portfolio - based upon the covariance & stock weights in the portfolio. eg of a 2 equity portfolio where p% is in stock X and q% is in stock Y - involves squaring the sum of 2 terms ((a+b)^2=a^2+2ab+b^2). The variance of the portfolio (pX+qY) is the expected value of [(pX-pµx)+(qY-qµy)]^2 (i.e. of the squares of its deviations from the mean) = [p^2.Variance(X) + q^2.Variance(Y) +2pq.Covariance(X,Y)]. Aim to minimize risk without hurting the rate of return.
·        As a mutual fund is simply a set of stocks, there are more funds than stocks!
·        Betas - Harry Markowitz - efficient frontier of portfolios - for each level of risk (i.e. volatility - based on s.d) amongst all the portfolios having a given level of risk, select the portfolio with the highest expected rate of return.
·        Variation 1 of portfolio selection theory - The more risky a portfolio on the efficient frontier curve, the higher its expected rate of return - because most investors are risk adverse, making risky stocks cheaper. Investors decide upon a risk level that they are comfortable with, and then choose the portfolio at that level which has the highest rate of return. Portfolio performance is measured as the ratio of the excess return of a portfolio (the difference between its expected return & the return of a risk free T-bill) to the portfolio's volatility. Complications - can combine risk free investment in T-bills (which pay a fixed rate of return & have 0 volatility) with a risky portfolio.
·        Variation 2 of portfolio selection theory -claims that there is only one optimal portfolio on the efficient frontier, which when combined with a risk free investment (such as T-bills) will have the highest rate of return for any given risk level. All investors choose the same optimal portfolio and adjust (according to acceptable risk) the percentage p% that goes into the T-bills & the (100-p)% that goes into the portfolio.
·        Both variations are computationally intensive.
·        Variation 3 of portfolio selection theory - Sharpe's single index model - relates a portfolio's rate of return not to the covariance of all possible pairs of stocks in the portfolio, but simply measures its volatility to a market representative index
·        Capital Asset Pricing Model - expected excess return (the difference between expected portfolio ROR Rp and risk free T-bill ROR Rf) is equivalent to the expected excess return of the market Rm  minus Rf multiplied by the portfolio's relative volatility (beta): Rp-Rf = ß(Rm-Rf).
·        Eg a beta of 1.5 means the portfolio gains/losses 1.5% for every 1% market gain/loss. The more volatile the riskier. Beta is determined by the gradient of the linear relation between market changes and the portfolio changes. Betas can vary.
·        Portfolios are less risky than individual stocks, but still risky - risk has 2 components: 1) systematic risk related to general market movement the portfolio's beta and 2) portfolio-specific risk, which can be factored out with the appropriate negative covariance amongst underlying's.
Chaos theory
·        Statistical independence often fails - actions affect each other & agents learn from & respond to each other.
·        Behavioral finance - Insider training may cause price movements. Like bluffing in poker, affecting the belief states of other traders. When private info becomes common knowledge, it brings with it convoluted beliefs about other's beliefs. Subterranean info processing between partial insiders. Socioeconomic, geopolitical factors, media spin & analyses.
·        A more macro level interaction among investors occurs between technical traders & value traders. When value traders perceive stocks to be undervalued, they start buying, raising prices & starting a trend which the technical traders follow, increasing prices even further. Soon the market is seen as overvalued by the value traders, who begin to sell, oscillating the market in the other direction. Contrarian value traders have a stabilizing effect on the market, while lockstep technical & algorithmic traders increase volatility.
·        Chaos theory - nonlinear dynamics whose parts influence each other disproportionately. Small stimuli Butterfly economics & irrational trend following. Market complexity emerges from the simple rules of interaction - patterns emerge from volatile trading fluctuations & interdependent economic variables. Trajectories often follow an aperiodic & unpredictable course that is fractal - discovered by Mandelbrot to be curves, surfaces & high D objects which contain increasingly complex pattern repetitions the closer one looks.
·        Most stock movements are small fluctuations that follow a normal curve with a slight long term upward bias. This curve has fatter tails than a normal curve to account for bubbles & crashes. the flocking affect (Erdos) - how connectedness of network nodes skews distribution. Power law (Barabasi) - the probability that a node has k links is approximately proportionate to 1/k^3. The number of nodes with k links declines quickly as k increases --> curve has fatter tails than a normal curve to account for bubbles & crashes. 1/k^m laws are commonly found in complex systems in nature. Zipf's law - the frequency of a word to its rank order is 1/k^1. Connected nodes (e.g. a trader, an interest rate, a media outlet) exert influence on each other with different & disproportionate levels of influence which change over time. E.g. herd-like contagion, volatility clustering.
·        The market is a dynamic system that readjusts for equilibrium between efficient market hypothesis and the sluggish market hypothesis - the more people that act in accordance to belief that it holds, the less it holds, & v.v. efficient market hypothesis holds most of the time because most investors disbelieve it.
·        Prisoner's dilemma - keep your investment methods secret, otherwise the level of complexity increases for the market as a whole in an arms race, leading to zero excess returns & a reliance on chance. People search for special knowledge which eventually becomes common knowledge. The cooperative option is to remain silent, the most likely option is for both to confess, the best outcome as a group is to remain silent; the best outcome for an individual is to confess as others remain silent.
·        Complexity - Chaitin & Kolmogorov defined a binary sequence's complexity as the shortest program that will generate it. Random sequences manifest no regularity or order - never repeats or exhibits a pattern. No way to compress - program must dumbly list the sequence. Technical & fundamental analyses believe that pockets of order exist. Chaitan paraphrased Godel's incompleteness theorem of mathematical logic in saying that if the market were random we might not be able to prove it as it is beyond our complexity horizon. Outperforming the market requires being on the cusp of the complexity horizon. The more efficient the market, the greater the complexity of price movements, and the closer to randomness.
·        Game theory - Newcombe's paradox between 2 reasonable choices: dominance principle (2 is better than 1) vs. maximization of expected value.
·        Don't believe the hype, don't put too many eggs in one basket, don't buy on margin, insure against sudden drops using puts. Remember that shifting behavior underlies the math.
Posted on Wednesday, May 11, 2011 5:24 AM Quant Programming | Back to top

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The knowledge we can learn from Mathematics is very important. - Mark Zokle
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