correlation coefficient

59 results back to index


Analysis of Financial Time Series by Ruey S. Tsay

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Asian financial crisis, asset allocation, Black-Scholes formula, Brownian motion, capital asset pricing model, compound rate of return, correlation coefficient, data acquisition, discrete time, frictionless, frictionless market, implied volatility, index arbitrage, Long Term Capital Management, market microstructure, martingale, p-value, pattern recognition, random walk, risk tolerance, short selling, statistical model, stochastic process, stochastic volatility, telemarketer, transaction costs, value at risk, volatility smile, Wiener process, yield curve

A Lagrange multiplier statistic 371 ••••••••••••••••••••••• •• • • • • • • • • ••••••••••• • ••••••••••••••••••••• •• •• • •••••••• •••• •••••• • ••• •• ••••••••••• •••••• • • ••• • • • • • ••• • •• • • •••••••• • • • ••• ••• • • •••• ••••• •••• ••••••• •••••••••• • •••••• •• • •• • •• •• •• •• • •• • •• •• ••• • • ••• • •• • ••••••• ••• • ••• ••••• • •• •••• ••• • ••••• •••••••••• • • • • ••• • • • • • • • • • • • • •• • ••••• •• ••• •• ••••• • •••••• •••••••••••••• • ••• •• • • • •••••• • •• ••• ••••• • •••• •• •• ••••• ••••••• •• • • • •••••• •• • 0.4 0.5 rho(t) 0.6 0.7 0.8 GARCH MODELS FOR BIVARIATE RETURNS 1940 1950 1960 1970 year 1980 1990 2000 Figure 9.5. The sample correlation coefficient between monthly log returns of IBM stock and the S&P 500 index. The correlation is computed by a moving window of 120 observations. The sample period is from January 1926 to December 1999. was proposed recently by Tse (2000) to test constant correlation coefficients in a multivariate GARCH model. A simple way to relax the constant-correlation constraint within the GARCH framework is to specify an exact equation for the conditional correlation coefficient. This can be done by two methods using the two reparameterizations of Σt discussed in Section 9.1. First, we use the correlation coefficient directly. Because the correlation coefficient between the returns of IBM stock and S&P 500 index is positive and must be in the interval [0, 1], we employ the equation ρ21,t = exp(qt ) , 1 + exp(qt ) (9.23) where a1,t−1 a2,t−1 qt = 0 + 1 ρ21,t−1 + 2 √ , σ11,t−1 σ22,t−1 where σii,t−1 is the conditional variance of the shock ai,t−1 .

. , k. In other words, D = diag{ 11 (0), . . . , kk (0)}. The concurrent, or lag-zero, cross-correlation matrix of rt is defined as ρ0 ≡ [ρi j (0)] = D−1 Γ0 D−1 . More specifically, the (i, j)th element of ρ0 is Cov(rit , r jt ) i j (0) ρi j (0) = = , std(r ii (0) j j (0) it )std(r jt ) which is the correlation coefficient between rit and r jt . In a time series analysis, such a correlation coefficient is referred to as a concurrent, or contemporaneous, CROSS - CORRELATION 301 correlation coefficient because it is the correlation of the two series at time t. It is easy to see that ρi j (0) = ρ ji (0), −1 ≤ ρi j (0) ≤ 1, and ρii (0) = 1 for 1 ≤ i, j ≤ k. Thus, ρ(0) is a symmetric matrix with unit diagonal elements. An important topic in multivariate time series analysis is the lead-lag relationships between component series.

From the definition, Cov(rit , r j,t− ) i j () ρi j () = = , std(rit )std(r jt ) ii (0) j j (0) (8.4) which is the correlation coefficient between rit and r j,t− . When > 0, this correlation coefficient measures the linear dependence of rit on r j,t− , which occurred prior to time t. Consequently, if ρi j () = 0 and > 0, we say that the series r jt leads the series rit at lag . Similarly, ρ ji () measures the linear dependence of r jt and ri,t− , and we say that the series rit leads the series r jt at lag if ρ ji () = 0 and > 0. Equation (8.4) also shows that the diagonal element ρii () is simply the lag- autocorrelation coefficient of rit . Based on this discussion, we obtain some important properties of the crosscorrelations when > 0. First, in general, ρi j () = ρ ji () for i = j because the two correlation coefficients measure different linear relationships between {rit } and {r jt }.

 

The Intelligent Asset Allocator: How to Build Your Portfolio to Maximize Returns and Minimize Risk by William J. Bernstein

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, backtesting, capital asset pricing model, computer age, correlation coefficient, diversification, diversified portfolio, Eugene Fama: efficient market hypothesis, fixed income, index arbitrage, index fund, Long Term Capital Management, p-value, passive investing, prediction markets, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, South Sea Bubble, the scientific method, time value of money, transaction costs, Vanguard fund, Yogi Berra, zero-coupon bond

Most of the points lie on nearly a straight line; a poor return for one was invariably associated with a poor return for the other. The correlation coefficient of .777 for these two assets is quite high. This graph demonstrates that adding U.S. small stocks to a portfolio of U.S. large stocks does not diminish risk very much, as a poor return for one will be very likely associated with a poor return for the other. Figure 3-4 plots two loosely correlated assets—U.S. large stocks (S&P 500) and foreign large stocks (EAFE Index). Although there does appear to be a loose relation between the two, it is far from perfect. The correlation coefficient of this pair is .483. Lastly, Figure 3-5 plots two very poorly correlated assets (correlation coefficient of .068): Japanese small stocks and REITs. This plot is a “scattergram” with no discernable pattern.

The Behavior of Multiple-Asset Portfolios 37 (This is the same reason why big offices have messier politics than small ones. A three-person office has three interpersonal relationships; a 10-person office has 45 relationships.) Real assets are almost always imperfectly correlated. In other words, an above-average return in one is somewhat more likely to be associated with an above-average return in the other. The degree of correlation is expressed by a correlation coefficient. This value ranges from ⫺1 to ⫹1. Perfectly correlated assets have a correlation coefficient of ⫹1, and uncorrelated assets have a coefficient of 0. Perfectly inversely (or negatively) correlated assets have a coefficient of ⫺1. The easiest way to understand this is to plot the returns of two assets against each other for many periods, as is done in Figures 3-3, 3-4, and 3-5. Each figure plots the 288 monthly returns for each asset pair for the 24-year period from January 1975 to December 1998.

This plot is a “scattergram” with no discernable pattern. A good or bad result for one of these assets tells us nothing about the result for the other. Why is this so important? As already discussed the most diversification benefit is obtained from uncorrelated assets. The above Math Details: How to Calculate a Correlation Coefficient In this book’s previous versions, I included a section on the manual calculation of the correlation coefficient. In the personal computer age, this is an exercise in masochism.The easiest way to do this is with a spreadsheet. Let’s assume that you have 36 monthly returns for two assets, A and B. Enter the returns in columns A and B, next to each other, spanning rows 1 to 36 for each pair of values. In Excel,enter in a separate cell the formula ⫽ CORREL(A1:A36, B1:B36) In Quattro Pro, the formula would be @CORREL(A1..A36, B1..B36) Both of these packages also contain a tool that will calculate a “correlation grid” of all of the correlations of an array of data for more than two assets.Those of you who would like an explanation of the steps involved in calculating a correlation coefficient are referred to a standard statistics text. 40 The Intelligent Asset Allocator analysis suggests that there is not much benefit from mixing domestic small and large stocks and that there is great benefit from mixing REITs and Japanese small stocks.

 

Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Asian financial crisis, asset allocation, backtesting, capital asset pricing model, collateralized debt obligation, commodity trading advisor, compound rate of return, constrained optimization, corporate governance, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, discrete time, distributed generation, diversification, diversified portfolio, dividend-yielding stocks, fixed income, high net worth, implied volatility, index arbitrage, index fund, interest rate swap, iterative process, linear programming, London Interbank Offered Rate, Long Term Capital Management, market fundamentalism, merger arbitrage, Mexican peso crisis / tequila crisis, p-value, Ponzi scheme, quantitative trading / quantitative finance, random walk, risk-adjusted returns, risk/return, Sharpe ratio, short selling, stochastic process, systematic trading, technology bubble, transaction costs, value at risk

These two indices display the highest skewness and kurtosis; the former is the only index to exhibit negative returns over the entire sample. Table 6.3 examines the correlation coefficients between the different CTA indices as well as between the CTA indices and the first two return moments of the Russell 3000 (Russell squared). The results for the entire sample as well as the subsamples confirm our earlier findings. The correlation coefficient between the CTA index, the Financial and Metal Traders Index, the Systematic Traders Index, and the Diversified Traders Index are positive and close to 1 for all the different periods. The Currency Trader Index and the Discretionary Index have the lowest correlation coefficient with the other CTA indices. The coefficients are still positive between all the indices and for all the subperiods, but the correlation coefficient is much smaller. Over the entire period, all of the CTA indices have a small and negative correlation coefficient with the Russell 3000 index and a positive relation with the square of the Russell 3000 returns.

To match EGR’s assumption of homoskedasticity, data sets were generated with the standard deviation set at 2. Heteroskedasticity was created by letting the values of σ be 5, 10, 15, and 20, with one-fourth of the observations using each value. This allowed us to compare the Spearman correlation coefficient calculated for data sets with and without homoskedasticity. The funds were ranked in ascending order of returns for period one (first 12 months) and period two (last 12 months). From each 24-month period of generated returns, Spearman correlation coefficients were calculated for a fund’s rank in both periods. For the distribution of Spearman correlation coefficients to be suitably approximated by a normal, at least 10 observations are needed. Because 120 pairs are used here, the normal approximation is used. Mean returns also were calculated for each fund in period one and period two, and then ranked.

Returns-protection diversifiers have relatively high correlations in both the up and down markets with a generic asset class (such as the S&P 500 Index). 2. Returns-enhancing diversifiers possess correlations with the same generic asset class in an up market but are relatively less correlated in a down market. 3. “Ineffective” diversifiers are assets that do not add value, even though they may possess significant correlation coefficients with the generic asset class. CTA Strategies for Returns-Enhancing Diversification 339 To illustrate, a hedge fund strategy that has a negative correlation coefficient in an up-market regime and positive correlation coefficient in a down-market regime provides diversification with no incremental returns. We classify this in the third category, that is, as an ineffective diversifier. Indeed, a strategy with such a characteristic will have the opposite effect of a good diversifier as it weakens the returns on an uptrend and exaggerates the negative returns of the portfolio.

 

pages: 755 words: 121,290

Statistics hacks by Bruce Frey

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Berlin Wall, correlation coefficient, Daniel Kahneman / Amos Tversky, distributed generation, en.wikipedia.org, feminist movement, game design, Hacker Ethic, index card, Milgram experiment, p-value, place-making, RFID, Search for Extraterrestrial Intelligence, SETI@home, Silicon Valley, statistical model

The groups of people didn't score exactly the same on both scales, of course, and the rank order isn't even the same, but, relatively speaking, the position of each person to each of the other people when it comes to cheese attitude is about the same as when it comes to cheesecake attitude. The Association's marketer has support for her hypothesis. Computing a Correlation Coefficient Just eyeballing two columns of numbers from a sample, though, is usually not enough to really know whether there is a relationship between two things. The marketing specialist in our example wants to use a single number to more precisely describe whatever relationship is seen. The correlation coefficient takes into account all the information we used when we looked at our two columns of numbers in Table 2-1 and decided whether there was a relationship there. The correlation coefficient is produced through a formula that does the following things: Looks at each score in a column Sees how distant that score is from the mean of that column Identifies the distance from the mean of its matching score in the other column Multiplies the paired distances together Averages the results of those multiplications If this were a statistics textbook, I'd have to present a somewhat complicated formula for calculating the correlation coefficient.

This is very close to 1.0, which is the strongest a positive correlation can be, so the cheese-to-cheesecake correlation represents a very strong relationship. Interpreting a Correlation Coefficient Somewhat magically, the correlation formula process produces a number, ranging in value from -1.00 to +1.00, that measures the strength of relationship between two variables. Positive signs indicate the relationship is in the same direction. As one value increases, the other value increases. Negative signs indicate the relationship is in the opposite direction. As one value increases, the other value decreases. An important point to make is that the correlation coefficient provides a standardized measure of the strength of linear relationship between two variables [Hack #12]. The direction of a correlation (whether it is negative or positive) is the artificial result of the direction of the scale one chooses to use to measure the variables.

Let's imagine that a small college decides to use scores on the American College Test (ACT) as a predictor of college grade point average (GPA) at the end of students' first years. The admissions office goes back through a few years of records and gathers the ACT scores and freshman GPAs for a couple hundred students. They discover, to their delight, that there is a moderate relationship between these two variables: a correlation coefficient of .55. Correlation coefficients are a measure of the strength of linear relationships between two variables [Hack #11], and .55 indicates a fairly large relationship. This is good news because the existence of a relationship between the two makes ACT scores a good candidate as a predictor to guess GPA. Simple linear regression is the procedure that produces all the values we need to cook up the magic formula that will predict the future.

 

pages: 519 words: 102,669

Programming Collective Intelligence by Toby Segaran

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

correlation coefficient, Debian, en.wikipedia.org, Firefox, full text search, information retrieval, PageRank, prediction markets, recommendation engine, slashdot, web application

Euclidean distance A clear implementation of this formula is shown here: def euclidean(p,q): sumSq=0.0 # add up the squared differences for i in range(len(p)): sumSq+=(p[i]-q[i])**2 # take the square root return (sumSq**0.5) Euclidean distance is used in several places in this book to determine how similar two items are. Pearson Correlation Coefficient The Pearson correlation coefficient is a measure of how highly correlated two variables are. It is a value between 1 and −1, where 1 indicates that the variables are perfectly correlated, 0 indicates no correlation, and −1 means they are perfectly inversely correlated. Figure B-2 shows the Pearson correlation coefficient. Figure B-2. Pearson correlation coefficient This can be implemented with the following code: def pearson(x,y): n=len(x) vals=range(n) # Simple sums sumx=sum([float(x[i]) for i in vals]) sumy=sum([float(y[i]) for i in vals]) # Sum up the squares sumxSq=sum([x[i]**2.0 for i in vals]) sumySq=sum([y[i]**2.0 for i in vals]) # Sum up the products pSum=sum([x[i]*y[i] for i in vals]) # Calculate Pearson score num=pSum-(sumx*sumy/n) den=((sumxSq-pow(sumx,2)/n)*(sumySq-pow(sumy,2)/n))**.5 if den==0: return 1 r=num/den return r We used the Pearson correlation in Chapter 2 to calculate the level of similarity between people's preferences.

, Learning from Clicks, Learning from Clicks, Setting Up the Database, Feeding Forward, Training with Backpropagation, Training Test, Training Test artificial, Learning from Clicks, Learning from Clicks, Setting Up the Database, Feeding Forward, Training with Backpropagation, Training Test, Training Test backpropagation, Training with Backpropagation connecting to search engine, Training Test designing click-training network, Learning from Clicks feeding forward, Feeding Forward setting up database, Setting Up the Database training test, Training Test neural network classifier, Exercises neural networks, Neural Networks, Neural Networks, Neural Networks, Neural Networks, Training a Neural Network, Training a Neural Network, Training a Neural Network, Strengths and Weaknesses, Strengths and Weaknesses backpropagation, and, Training a Neural Network black box method, Strengths and Weaknesses combinations of words, and, Neural Networks multilayer perceptron network, Neural Networks strengths and weaknesses, Strengths and Weaknesses synapses, and, Neural Networks training, Training a Neural Network using code, Training a Neural Network news sources, A Corpus of News newsfeatures.py, Selecting Sources, Downloading Sources, Downloading Sources, Downloading Sources, Converting to a Matrix, Using NumPy, The Algorithm, Displaying the Results, Displaying the Results, Displaying by Article, Displaying by Article getarticlewords function, Downloading Sources makematrix function, Converting to a Matrix separatewords function, Downloading Sources shape function, The Algorithm showarticles function, Displaying the Results, Displaying by Article showfeatures function, Displaying the Results, Displaying by Article stripHTML function, Downloading Sources transpose function, Using NumPy nn.py, Setting Up the Database, Setting Up the Database, Setting Up the Database, Setting Up the Database searchnet class, Setting Up the Database, Setting Up the Database, Setting Up the Database, Setting Up the Database generatehiddennode function, Setting Up the Database getstrength method, Setting Up the Database setstrength method, Setting Up the Database nnmf.py, The Algorithm difcost function, The Algorithm non-negative matrix factorization (NMF), Supervised versus Unsupervised Learning, Clustering, Non-Negative Matrix Factorization, Non-Negative Matrix Factorization, Non-Negative Matrix Factorization, Using Your NMF Code factorization, Supervised versus Unsupervised Learning goal of, Non-Negative Matrix Factorization update rules, Non-Negative Matrix Factorization using code, Using Your NMF Code normalization, Normalization Function numerical predictions, Building Price Models numpredict.py, Building a Sample Dataset, Building a Sample Dataset, Defining Similarity, Defining Similarity, Defining Similarity, Defining Similarity, Subtraction Function, Subtraction Function, Weighted kNN, Weighted kNN, Cross-Validation, Cross-Validation, Cross-Validation, Heterogeneous Variables, Scaling Dimensions, Optimizing the Scale, Optimizing the Scale, Uneven Distributions, Estimating the Probability Density, Graphing the Probabilities, Graphing the Probabilities, Graphing the Probabilities createcostfunction function, Optimizing the Scale createhiddendataset function, Uneven Distributions crossvalidate function, Cross-Validation, Optimizing the Scale cumulativegraph function, Graphing the Probabilities distance function, Defining Similarity dividedata function, Cross-Validation euclidian function, Defining Similarity gaussian function, Weighted kNN getdistances function, Defining Similarity inverseweight function, Subtraction Function knnestimate function, Defining Similarity probabilitygraph function, Graphing the Probabilities probguess function, Estimating the Probability Density, Graphing the Probabilities rescale function, Scaling Dimensions subtractweight function, Subtraction Function testalgorithm function, Cross-Validation weightedknn function, Weighted kNN wineprice function, Building a Sample Dataset wineset1 function, Building a Sample Dataset wineset2 function, Heterogeneous Variables NumPy, Using NumPy, Using NumPy, Simple Usage Example, NumPy, Installation on Other Platforms, Installation on Other Platforms installation on other platforms, Installation on Other Platforms installation on Windows, Simple Usage Example usage example, Installation on Other Platforms using, Using NumPy O online technique, Strengths and Weaknesses Open Web APIs, Open APIs optimization, Optimization, Group Travel, Representing Solutions, Representing Solutions, Representing Solutions, Representing Solutions, The Cost Function, The Cost Function, The Cost Function, Random Searching, Hill Climbing, Simulated Annealing, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Optimizing for Preferences, Optimizing for Preferences, The Cost Function, The Cost Function, Network Visualization, Network Visualization, Counting Crossed Lines, Drawing the Network, Exercises, Exercises, Exercises, Exercises, Exercises, Exercises, Exercises, Optimizing the Scale, Exercises, Optimization, Optimization annealing starting points, Exercises cost function, The Cost Function, Optimization exercises, Exercises genetic algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms crossover or breeding, Genetic Algorithms generation, Genetic Algorithms mutation, Genetic Algorithms population, Genetic Algorithms genetic optimization stopping criteria, Exercises group travel cost function, Exercises group travel planning, Group Travel, Representing Solutions, Representing Solutions, Representing Solutions, The Cost Function, The Cost Function car rental period, The Cost Function departure time, Representing Solutions price, Representing Solutions time, Representing Solutions waiting time, The Cost Function hill climbing, Hill Climbing line angle penalization, Exercises network visualization, Network Visualization, Counting Crossed Lines, Drawing the Network counting crossed lines, Counting Crossed Lines drawing networks, Drawing the Network layout problem, Network Visualization network vizualization, Network Visualization pairing students, Exercises preferences, Optimizing for Preferences, Optimizing for Preferences, The Cost Function, The Cost Function cost function, The Cost Function running, The Cost Function student dorm, Optimizing for Preferences random searching, Random Searching representing solutions, Representing Solutions round-trip pricing, Exercises simulated annealing, Simulated Annealing where it may not work, Genetic Algorithms optimization.py, Group Travel, Representing Solutions, Representing Solutions, The Cost Function, Random Searching, Hill Climbing, Simulated Annealing, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Optimizing the Scale annealingoptimize function, Simulated Annealing geneticoptimize function, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms, Genetic Algorithms elite, Genetic Algorithms maxiter, Genetic Algorithms mutprob, Genetic Algorithms popsize, Genetic Algorithms getminutes function, Representing Solutions hillclimb function, Hill Climbing printschedule function, Representing Solutions randomoptimize function, Random Searching schedulecost function, The Cost Function P PageRank algorithm, Real-Life Examples, The PageRank Algorithm pairing students, Exercises Pandora, Real-Life Examples parse tree, Programs As Trees Pearson correlation, Hierarchical Clustering, Viewing Data in Two Dimensions hierarchical clustering, Hierarchical Clustering multidimensional scaling, Viewing Data in Two Dimensions Pearson correlation coefficient, Pearson Correlation Score, Pearson Correlation Coefficient, Pearson Correlation Coefficient code, Pearson Correlation Coefficient Pilgrim, Mark, Universal Feed Parser polynomial transformation, The Kernel Trick poplib, Exercises population, Genetic Algorithms, What Is Genetic Programming?, Creating the Initial Population, Genetic Algorithms diversity and, Creating the Initial Population Porter Stemmer, Finding the Words on a Page Pr(Document), Exercises prediction markets, Real-Life Examples price models, Building a Sample Dataset, Building a Sample Dataset, k-Nearest Neighbors, Exercises, Exercises, Exercises, Exercises, Exercises, Exercises, Exercises, Exercises building sample dataset, Building a Sample Dataset eliminating variables, Exercises exercises, Exercises item types, Exercises k-nearest neighbors (kNN), k-Nearest Neighbors laptop dataset, Exercises leave-one-out cross-validation, Exercises optimizing number of neighbors, Exercises search attributes, Exercises varying ss for graphing probability, Exercises probabilities, Calculating Probabilities, Starting with a Reasonable Guess, Probability of a Whole Document, A Quick Introduction to Bayes' Theorem, Combining the Probabilities, Graphing the Probabilities, Conditional Probability assumed probability, Starting with a Reasonable Guess Bayes' Theorem, A Quick Introduction to Bayes' Theorem combining, Combining the Probabilities conditional probability, Calculating Probabilities graphing, Graphing the Probabilities of entire document given classification, Probability of a Whole Document product marketing, Other Uses for Learning Algorithms public message boards, Filtering Spam pydelicious, Simple Usage Example, Simple Usage Example, pydelicious installation, Simple Usage Example usage example, Simple Usage Example pysqlite, Building the Index, Persisting the Trained Classifiers, Installation on All Platforms, Installation on All Platforms, pysqlite, Simple Usage Example importing, Persisting the Trained Classifiers installation on other platforms, Installation on All Platforms installation on Windows, Installation on All Platforms usage example, Simple Usage Example Python, Style of Examples, Python Tips advantages of, Style of Examples tips, Python Tips Python Imaging Library (PIL), Drawing the Dendrogram, Python Imaging Library, Installation on Windows, Installation on Windows, Installation on Windows installation on other platforms, Installation on Windows usage example, Installation on Windows Windows installation, Installation on Windows Python, genetic programming and, Programs As Trees, Programs As Trees, Representing Trees in Python, Building and Evaluating Trees, Displaying the Program building and evaluating trees, Building and Evaluating Trees displaying program, Displaying the Program representing trees, Representing Trees in Python traversing complete tree, Programs As Trees Q query layer, Design of a Click-Tracking Network querying, Querying, Querying query function, Querying R radial-basis function, The Kernel Trick random searching, Random Searching random-restart hill climbing, Hill Climbing ranking, What's in a Search Engine?

, Exercises marketing, Other Uses for Learning Algorithms mass-and-spring algorithm, The Layout Problem matchmaker dataset, Matchmaker Dataset, Difficulties with the Data, Decision Tree Classifier, Categorical Features, Creating the New Dataset, Creating the New Dataset, Applying SVM to the Matchmaker Dataset categorical features, Categorical Features creating new, Creating the New Dataset decision tree algorithm, Decision Tree Classifier difficulties with data, Difficulties with the Data LIBSVM, applying to, Applying SVM to the Matchmaker Dataset scaling data, Creating the New Dataset matchmaker.csv file, Matchmaker Dataset mathematical formulas, Euclidean Distance, Euclidean Distance, Pearson Correlation Coefficient, Weighted Mean, Tanimoto Coefficient, Conditional Probability, Gini Impurity, Entropy, Variance, Gaussian Function, Dot-Products conditional probability, Conditional Probability dot-product, Dot-Products entropy, Entropy Euclidean distance, Euclidean Distance Gaussian function, Gaussian Function Gini impurity, Gini Impurity Pearson correlation coefficient, Pearson Correlation Coefficient Tanimoto coefficient, Tanimoto Coefficient variance, Variance weighted mean, Weighted Mean matplotlib, Graphing the Probabilities, matplotlib, Installation, Simple Usage Example installation, Installation usage example, Simple Usage Example matrix math, Clustering, A Quick Introduction to Matrix Math, A Quick Introduction to Matrix Math, What Does This Have to Do with the Articles Matrix?

 

pages: 312 words: 35,664

The Mathematics of Banking and Finance by Dennis W. Cox, Michael A. A. Cox

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

barriers to entry, Brownian motion, call centre, correlation coefficient, inventory management, iterative process, linear programming, meta analysis, meta-analysis, P = NP, pattern recognition, random walk, traveling salesman, value at risk

So, from above we know that the gradient of the line is estimated by n n n 1 xi yi − xi yi n i=1 i=1 i=1 â = n n 2 2 1 xi − xi n i=1 i=1 This may be written in a more compact form since the variance (section 5.6) is n n 2 2 1 xi − xi n i=1 i=1 var(x) = n−1 The covariance may be similarly written as: n cov(x, y) = (xi − x̄) (yi − ȳ) i=1 = n−1 n xi yi i=1 n n 1 − xi yi n i=1 i=1 n−1 (*) We can then write the equation for the estimated gradient in a more compact form using this new notation: â = cov(x, y) var(x) 13.3 CORRELATION COEFFICIENT A closely related term to the covariance is the correlation coefficient (r (x, y)), which is simply r (x, y) = cov(x, y) std(x) std(y) where the standard deviation of x (section 5.6) is given by n 2 n 2 1 i=1 xi − n i=1 xi std(x) = n−1 There would be a similar expression for y. The correlation coefficient is a measure of the interdependence of two variables. The coefficient ranges in value from −1 to +1, indicating perfect negative correlation at −1, absence Linear Regression 105 of correlation at zero, and perfect positive correlation at +1. Two variables are positively correlated if the correlation coefficient is greater than zero and the line that is drawn to show a relationship between the items sampled has a positive gradient.

Two variables are positively correlated if the correlation coefficient is greater than zero and the line that is drawn to show a relationship between the items sampled has a positive gradient. If the correlation coefficient is negative, the variables are negatively correlated and the line drawn will have a negative gradient. The final option is that the correlation coefficient vanishes, and the gradient vanishes since â = 0, in which case the variables are completely uncorrelated. This means that there is no relationship between the two variables. An example of this might be the time taken to process a batch of transactions and the movements in interest rates. As a rough guide, 100r 2 is the percentage of the total variation of the y population that is accounted for by their relationship with x. More precisely there is generally considered to be a significant correlation if the correlation coefficient, or r-value, exceeds the critical value (rcrit ) at n − 2 degrees of freedom.

Index a notation 103–4, 107–20, 135–47 linear regression 103–4, 107–20 slope significance test 112–20 variance 112 abscissa see horizontal axis absolute value, notation 282–4 accuracy and reliability, data 17, 47 adaptive resonance theory 275 addition, mathematical notation 279 addition of normal variables, normal distribution 70 addition rule, probability theory 24–5 additional variables, linear programming 167–70 adjusted cash flows, concepts 228–9 adjusted discount rates, concepts 228–9 Advanced Measurement Approach (AMA) 271 advertising allocation, linear programming 154–7 air-conditioning units 182–5 algorithms, neural networks 275–6 alternatives, decisions 191–4 AMA see Advanced Measurement Approach analysis data 47–52, 129–47, 271–4 Latin squares 131–2, 143–7 linear regression 110–20 projects 190–2, 219–25, 228–34 randomised block design 129–35 sampling 47–52, 129–47 scenario analysis 40, 193–4, 271–4 trends 235–47 two-way classification 135–47 variance 110–20, 121–7 anonimised databases, scenario analysis 273–4 ANOVA (analysis of variance) concepts 110–20, 121–7, 134–47 examples 110–11, 123–7, 134–40 formal background 121–2 linear regression 110–20 randomised block design 134–5, 141–3 tables 110–11, 121–3, 134–47 two-way classification 136–7 appendix 279–84 arithmetic mean, concepts 37–45, 59–60, 65–6, 67–74, 75–81 assets classes 149–57 reliability 17, 47, 215–18, 249–60 replacement of assets 215–18, 249–60 asymptotic distributions 262 ATMs 60 averages see also mean; median; mode concepts 37–9 b notation 103–4, 107–20, 132–5 linear regression 103–4, 107–20 variance 112 back propagation, neural networks 275–7 backwards recursion 179–87 balance sheets, stock 195 bank cashier problem, Monte Carlo simulation 209–12 Bank for International Settlements (BIS) 267–9, 271 banks Basel Accord 262, 267–9, 271 failures 58 loss data 267–9, 271–4 modelling 75–81, 85, 97, 267–9, 271–4 profitable loans 159–66 bar charts comparative data 10–12 concepts 7–12, 54, 56, 59, 205–6, 232–3 discrete data 7–12 examples 9–12, 205–6, 232–3 286 Index bar charts (cont.) narrative explanations 10 relative frequencies 8–12 rules 8–9 uses 7–12, 205–6, 232–3 base rates, trends 240 Basel Accord 262, 267–9, 271 bathtub curves, reliability concepts 249–51 Bayes’theorem, probability theory 27–30, 31 bell-shaped normal distribution see normal distribution bi-directional associative memory 275 bias 1, 17, 47–50, 51–2, 97, 129–35 randomised block design 129–35 sampling 17, 47–50, 51–2, 97, 129–35 skewness 41–5 binomial distribution concepts 55–8, 61–5, 71–2, 98–9, 231–2 examples 56–8, 61–5, 71–2, 98–9 net present value (NPV) 231–2 normal distribution 71–2 Pascal’s triangle 56–7 uses 55, 57, 61–5, 71–2, 98–9, 231–2 BIS see Bank for International Settlements boards of directors 240–1 break-even analysis, concepts 229–30 Brownian motion 22 see also random walks budgets 149–57 calculators, log functions 20, 61 capital Basel Accord 262, 267–9, 271 cost of capital 219–25, 229–30 cash flows adjusted cash flows 228–9 future cash flows 219–25, 227–34, 240–1 net present value (NPV) 219–22, 228–9, 231–2 standard deviation 232–4 central limit theorem concepts 70, 75 examples 70 chi-squared test concepts 83–4, 85, 89, 91–5 contingency tables 92–5 examples 83–4, 85, 89, 91–2 goodness of fit test 91–5 multi-way tables 94–5 tables 84, 91 Chu Shi-Chieh’s Ssu Yuan Y Chien 56 circles, tree diagrams 30–5 class intervals concepts 13–20, 44–5, 63–4, 241–7 histograms 13–20, 44–5 mean calculations 44–5 mid-points 44–5, 241–7 notation 13–14, 20 Sturges’s formula 20 variance calculations 44–5 classical approach, probability theory 22, 27 cluster sampling 50 coin-tossing examples, probability theory 21–3, 53–4 collection techniques, data 17, 47–52, 129–47 colours, graphical presentational approaches 9 combination, probability distribution (density) functions 54–8 common logarithm (base 10) 20 communications, decisions 189–90 comparative data, bar charts 10–12 comparative histograms see also histograms examples 14–19 completed goods 195 see also stock . . . conditional probability, concepts 25–7, 35 confidence intervals, concepts 71, 75–81, 105, 109, 116–20, 190, 262–5 constraining equations, linear programming 159–70 contingency tables, concepts 92–5 continuous approximation, stock control 200–1 continuous case, failures 251 continuous data concepts 7, 13–14, 44–5, 65–6, 251 histograms 7, 13–14 continuous uniform distribution, concepts 64–6 correlation coefficient concepts 104–20, 261–5, 268–9 critical value 105–6, 113–20 equations 104–5 examples 105–8, 115–20 costs capital 219–25, 229–30 dynamic programming 180–82 ghost costs 172–7 holding costs 182–5, 197–201, 204–8 linear programming 167–70, 171–7 sampling 47 stock control 182–5, 195–201 transport problems 171–7 trend analysis 236–47 types 167–8, 182 counting techniques, probability distribution (density) functions 54 covariance see also correlation coefficient concepts 104–20, 263–5 credit cards 159–66, 267–9 credit derivatives 97 see also derivatives Index credit risk, modelling 75, 149, 261–5 critical value, correlation coefficient 105–6, 113–20 cumulative frequency polygons concepts 13–20, 39–40, 203 examples 14–20 uses 13–14 current costs, linear programming 167–70 cyclical variations, trends 238–47 data analysis methods 47–52, 129–47, 271–4 collection techniques 17, 47–52, 129–47 continuous/discrete types 7–12, 13–14, 44–5, 53–5, 65–6, 72, 251 design/approach to analysis 129–47 errors 129–47 graphical presentational approaches 1–20, 149–57 identification 2–5, 261–5 Latin squares 131–2, 143–7 loss data 267–9, 271–4 neural networks 275–7 qualities 17, 47 randomised block design 129–35 reliability and accuracy 17, 47 sampling 17, 47–52 time series 235–47 trends 5, 10, 235–47 two-way classification analysis 135–47 data points, scatter plots 2–5 databases, loss databases 272–4 debentures 149–57 decisions alternatives 191–4 Bayes’theorem 27–30, 31 communications 189–90 concepts 21–35, 189–94, 215–25, 228–34, 249–60 courses of action 191–2 definition 21 delegation 189–90 empowerment 189–90 guesswork 191 lethargy pitfalls 189 minimax regret rule 192–4 modelling problems 189–91 Monty Hall problem 34–5, 212–13 pitfalls 189–94 probability theory 21–35, 53–66, 189–94, 215–18 problem definition 129, 190–2 project analysis guidelines 190–2, 219–25, 228–34 replacement of assets 215–18, 249–60 staff 189–90 287 steps 21 stock control 195–201, 203–8 theory 189–94 degrees of freedom 70–1, 75–89, 91–5, 110–20, 136–7 ANOVA (analysis of variance) 110–20, 121–7, 136–7 concepts 70–1, 75–89, 91–5, 110–20, 136–7 delegation, decisions 189–90 density functions see also probability distribution (density) functions concepts 65–6, 67, 83–4 dependent variables, concepts 2–5, 103–20, 235 derivatives 58, 97–8, 272 see also credit . . . ; options design/approach to analysis, data 129–47 dice-rolling examples, probability theory 21–3, 53–5 differentiation 251 discount factors adjusted discount rates 228–9 net present value (NPV) 220–1, 228–9, 231–2 discrete data bar charts 7–12, 13 concepts 7–12, 13, 44–5, 53–5, 72 discrete uniform distribution, concepts 53–5 displays see also presentational approaches data 1–5 Disraeli, Benjamin 1 division notation 280, 282 dynamic programming complex examples 184–7 concepts 179–87 costs 180–82 examples 180–87 principle of optimality 179–87 returns 179–80 schematic 179–80 ‘travelling salesman’ problem 185–7 e-mail surveys 50–1 economic order quantity see also stock control concepts 195–201 examples 196–9 empowerment, staff 189–90 error sum of the squares (SSE), concepts 122–5, 133–47 errors, data analysis 129–47 estimates mean 76–81 probability theory 22, 25–6, 31–5, 75–81 Euler, L. 131 288 Index events independent events 22–4, 35, 58, 60, 92–5 mutually exclusive events 22–4, 58 probability theory 21–35, 58–66, 92–5 scenario analysis 40, 193–4, 271–4 tree diagrams 30–5 Excel 68, 206–7 exclusive events see mutually exclusive events expected errors, sensitivity analysis 268–9 expected value, net present value (NPV) 231–2 expert systems 275 exponent notation 282–4 exponential distribution, concepts 65–6, 209–10, 252–5 external fraud 272–4 extrapolation 119 extreme value distributions, VaR 262–4 F distribution ANOVA (analysis of variance) 110–20, 127, 134–7 concepts 85–9, 110–20, 127, 134–7 examples 85–9, 110–20, 127, 137 tables 85–8 f notation 8–9, 13–20, 26, 38–9, 44–5, 65–6, 85 factorial notation 53–5, 283–4 failure probabilities see also reliability replacement of assets 215–18, 249–60 feasibility polygons 152–7, 163–4 finance selection, linear programming 164–6 fire extinguishers, ANOVA (analysis of variance) 123–7 focus groups 51 forward recursion 179–87 four by four tables 94–5 fraud 272–4, 276 Fréchet distribution 262 frequency concepts 8–9, 13–20, 37–45 cumulative frequency polygons 13–20, 39–40, 203 graphical presentational approaches 8–9, 13–20 frequentist approach, probability theory 22, 25–6 future cash flows 219–25, 227–34, 240–1 fuzzy logic 276 Garbage In, Garbage Out (GIGO) 261–2 general rules, linear programming 167–70 genetic algorithms 276 ghost costs, transport problems 172–7 goodness of fit test, chi-squared test 91–5 gradient (a notation), linear regression 103–4, 107–20 graphical method, linear programming 149–57, 163–4 graphical presentational approaches concepts 1–20, 149–57, 235–47 rules 8–9 greater-than notation 280–4 Greek alphabet 283 guesswork, modelling 191 histograms 2, 7, 13–20, 41, 73 class intervals 13–20, 44–5 comparative histograms 14–19 concepts 7, 13–20, 41, 73 continuous data 7, 13–14 examples 13–20, 73 skewness 41 uses 7, 13–20 holding costs 182–5, 197–201, 204–8 home insurance 10–12 Hopfield 275 horizontal axis bar charts 8–9 histograms 14–20 linear regression 103–4, 107–20 scatter plots 2–5, 103 hypothesis testing concepts 77–81, 85–95, 110–27 examples 78–80, 85 type I and type II errors 80–1 i notation 8–9, 13–20, 28–30, 37–8, 103–20 identification data 2–5, 261–5 trends 241–7 identity rule 282 impact assessments 21, 271–4 independent events, probability theory 22–4, 35, 58, 60, 92–5 independent variables, concepts 2–5, 70, 103–20, 235 infinity, normal distribution 67–72 information, quality needs 190–4 initial solution, linear programming 167–70 insurance industry 10–12, 29–30 integers 280–4 integration 65–6, 251 intercept (b notation), linear regression 103–4, 107–20 interest rates base rates 240 daily movements 40, 261 project evaluation 219–25, 228–9 internal rate of return (IRR) concepts 220–2, 223–5 examples 220–2 interpolation, IRR 221–2 interviews, uses 48, 51–2 inventory control see stock control Index investment strategies 149–57, 164–6, 262–5 IRR see internal rate of return iterative processes, linear programming 170 j notation 28–30, 37, 104–20, 121–2 JP Morgan 263 k notation 20, 121–7 ‘know your customer’ 272 Kohonen self-organising maps 275 Latin squares concepts 131–2, 143–7 examples 143–7 lead times, stock control 195–201 learning strategies, neural networks 275–6 less-than notation 281–4 lethargy pitfalls, decisions 189 likelihood considerations, scenario analysis 272–3 linear programming additional variables 167–70 concepts 149–70 concerns 170 constraining equations 159–70 costs 167–70, 171–7 critique 170 examples 149–57, 159–70 finance selection 164–6 general rules 167–70 graphical method 149–57, 163–4 initial solution 167–70 iterative processes 170 manual preparation 170 most profitable loans 159–66 optimal advertising allocation 154–7 optimal investment strategies 149–57, 164–6 returns 149–57, 164–6 simplex method 159–70, 171–2 standardisation 167–70 time constraints 167–70 transport problems 171–7 linear regression analysis 110–20 ANOVA (analysis of variance) 110–20 concepts 3, 103–20 equation 103–4 examples 107–20 gradient (a notation) 103–4, 107–20 intercept (b notation) 103–4, 107–20 interpretation 110–20 notation 103–4 residual sum of the squares 109–20 slope significance test 112–20 uncertainties 108–20 literature searches, surveys 48 289 loans finance selection 164–6 linear programming 159–66 risk assessments 159–60 log-normal distribution, concepts 257–8 logarithms (logs), types 20, 61 losses, banks 267–9, 271–4 lotteries 22 lower/upper quartiles, concepts 39–41 m notation 55–8 mail surveys 48, 50–1 management information, graphical presentational approaches 1–20 Mann–Whitney test see U test manual preparation, linear programming 170 margin of error, project evaluation 229–30 market prices, VaR 264–5 marketing brochures 184–7 mathematics 1, 7–8, 196–9, 219–20, 222–5, 234, 240–1, 251, 279–84 matrix plots, concepts 2, 4–5 matrix-based approach, transport problems 171–7 maximum and minimum, concepts 37–9, 40, 254–5 mean comparison of two sample means 79–81 comparisons 75–81 concepts 37–45, 59–60, 65–6, 67–74, 75–81, 97–8, 100–2, 104–27, 134–5 confidence intervals 71, 75–81, 105, 109, 116–20, 190, 262–5 continuous data 44–5, 65–6 estimates 76–81 hypothesis testing 77–81 linear regression 104–20 normal distribution 67–74, 75–81, 97–8 sampling 75–81 mean square causes (MSC), concepts 122–7, 134–47 mean square errors (MSE), ANOVA (analysis of variance) 110–20, 121–7, 134–7 median, concepts 37, 38–42, 83, 98–9 mid-points class intervals 44–5, 241–7 moving averages 241–7 minimax regret rule, concepts 192–4 minimum and maximum, concepts 37–9, 40 mode, concepts 37, 39, 41 modelling banks 75–81, 85, 97, 267–9, 271–4 concepts 75–81, 83, 91–2, 189–90, 195–201, 215–18, 261–5 decision-making pitfalls 189–91 economic order quantity 195–201 290 Index modelling (cont.) guesswork 191 neural networks 275–7 operational risk 75, 262–5, 267–9, 271–4 output reviews 191–2 replacement of assets 215–18, 249–60 VaR 261–5 moments, density functions 65–6, 83–4 money laundering 272–4 Monte Carlo simulation bank cashier problem 209–12 concepts 203–14, 234 examples 203–8 Monty Hall problem 212–13 queuing problems 208–10 random numbers 207–8 stock control 203–8 uses 203, 234 Monty Hall problem 34–5, 212–13 moving averages concepts 241–7 even numbers/observations 244–5 moving totals 245–7 MQMQM plot, concepts 40 MSC see mean square causes MSE see mean square errors multi-way tables, concepts 94–5 multiplication notation 279–80, 282 multiplication rule, probability theory 26–7 multistage sampling 50 mutually exclusive events, probability theory 22–4, 58 n notation 7, 20, 28–30, 37–45, 54–8, 103–20, 121–7, 132–47, 232–4 n!

 

Stocks for the Long Run, 4th Edition: The Definitive Guide to Financial Market Returns & Long Term Investment Strategies by Jeremy J. Siegel

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, backtesting, Black-Scholes formula, Bretton Woods, buy low sell high, California gold rush, capital asset pricing model, cognitive dissonance, compound rate of return, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, dividend-yielding stocks, equity premium, Eugene Fama: efficient market hypothesis, fixed income, German hyperinflation, implied volatility, index arbitrage, index fund, Isaac Newton, joint-stock company, Long Term Capital Management, loss aversion, market bubble, mental accounting, new economy, oil shock, passive investing, prediction markets, price anchoring, price stability, purchasing power parity, random walk, Richard Thaler, risk tolerance, risk/return, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, short selling, South Sea Bubble, technology bubble, The Great Moderation, The Wisdom of Crowds, transaction costs, tulip mania, Vanguard fund

This will be true if bond and stock returns are negatively correlated, which means that bond yields and stock prices move in opposite directions. The diversifying strength of an asset is measured by the correlation coefficient. The correlation coefficient, which theoretically ranges between –1 and +1, measures the correlation between an asset’s return and the return of the rest of the portfolio. The lower the correlation coefficient, the better the asset serves as a portfolio diversifier. Assets with negative correlations are particularly good diversifiers. As the correlation coefficient between the asset and portfolio returns increases, the diversifying quality of the asset declines. The correlation coefficient between annual stock and bond returns for six subperiods between 1926 and 2006 is shown in Figure 2-4. From 1926 through 1965 the correlation was only slightly positive, indicating that bonds were fairly good diversifiers for stocks.

An asset with a low correlation with the rest of the market provides better diversification than an asset with a high correlation. The correlation of returns between stocks or portfolios of stocks is measured by the correlation coefficient. A good case for investors is if there is no correlation between the stock returns of two countries, and the correlation coefficient is equal to zero. In this case, an investor who allocates his or her portfolio equally between each country can reduce his or her risk by almost one-third, compared to investing in a single country. As the correlation coefficient increases, the gains from diversification dwindle, and if there is perfect synchronization of returns, the correlation coefficient equals 1 and there is no gain (but no loss) from diversification. “Efficient” Portfolios: Formal Analysis How do you determine how much should be invested at home and abroad?

From 1926 through 1965 the correlation was only slightly positive, indicating that bonds were fairly good diversifiers for stocks. From 1966 through 1989 the correlation coefficient jumped to +0.34, and from 1990 through 1997 the correlation increased further to +0.55. This means that the diversifying quality of bonds diminished markedly from 1926 to 1997. There are good reasons why the correlation became more positive during this period. Under the gold-based monetary standard of the 1920s and early 1930s, bad economic times were associated with falling commodity prices; when the real economy was sinking, stocks declined and the real value of government bonds rose. Under a paper money standard, bad economic times are more likely to be associated with inflation, not deflation, as the government at- CHAPTER 2 Risk, Return, and Portfolio Allocation FIGURE 31 2–4 Correlation Coefficient between Monthly Stock and Bond Returns tempts to offset economic downturns with expansionary monetary policy.

 

pages: 447 words: 104,258

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

algorithmic trading, asset allocation, asset-backed security, backtesting, banking crisis, Black Swan, Black-Scholes formula, Brownian motion, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discounted cash flows, discrete time, diversification, fixed income, implied volatility, interest rate derivative, interest rate swap, margin call, market microstructure, martingale, p-value, passive investing, quantitative trading / quantitative finance, random walk, risk/return, Sharpe ratio, short selling, statistical model, stochastic process, stochastic volatility, time value of money, transaction costs, value at risk, volatility smile, Wiener process, yield curve, zero-coupon bond

A famous example is if (x, y) can only value (0, 1), (0, −1), (1, 0) or (−1, 0), with equal probability. The linear correlation of x and y is 0, although they are clearly dependent: if x = 0, y can only value 1 or −1, and if x ≠ 0, y = 0. Coming back to the general case of φx(x) and φy(y) being not Gaussian, this inference cannot be made. Typically, the classic “rank correlation” coefficient of Spearman shows the way to get round the problem: this rank correlation consists in a linear correlation coefficient of the variates, 5 now transformed in a non-linear way, by a probability transformation, that is, their respective cumulative marginal distributions: with The Spearman correlation is a correlation measure that can be computed from these relationships and from the general formula for ρx, y above, but, as a step further, we can link above Φx(x), Φy(y) and Φ(x, y) relationships in a more general way that defines C – named a copula of two variables x and y – as a cumulative probability function of the marginal cumulative probabilities Φx(x), Φy(y) of x and y.6 A copula is thus a general measure of co-dependence between two variates, which is independent of their individual marginal distribution – see Figure 13.5.

It is indeed based on several restrictive hypotheses: Hypotheses related to financial assets: Asset returns r are modeled by a random variable, distributed as a Gaussian probabilities distribution, fully determined by its first two moments, namely its expected value E and its variance V, although instead of V, the theory makes use of the corresponding standard deviation measure STD (STD = ). Returns of different financial assets i and j are correlated by the linear correlation coefficient ρij. Markets are efficient1 – practically speaking, we observe that the more liquid a market, the more efficient it is. The theory is built on mid prices (average of the market quoted bid and offer (or ask) prices): the market bid–offer spread is thus not considered here. Various costs such as brokerage fees, taxes, and so on are not taken into account (they are too much affected by local circumstances, market features, and the investor's situation).

For example, in 2006, based on successive daily close prices, the return and risk of L'Oreal were 20% and 19% respectively. Figure 4.3 Example of a stock showed in a (r, σ) graph 4.3.3 The Markowitz model Markowitz's goal was to optimize the budget allocation to a portfolio P of n stocks Si(ri, σi), weighted by wi, with 0 ≤ wi ≤ 1 and ∑wi = 1, so that for P: (4.1) that is, where the ρij correlation coefficients are computed by In a (r, σ) chart, it is possible, for a given past period of data to locate by a point any Si(ri, σi), but also any possible weighted combination of up to n stocks, defining points that represent portfolios, among which the optimal ones have to be identified. Performing this graph representation shows that there is a (non-linear) “frontier” of possible portfolios presenting the highest return, for different risks.

 

pages: 119 words: 10,356

Topics in Market Microstructure by Ilija I. Zovko

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Brownian motion, continuous double auction, correlation coefficient, financial intermediation, Gini coefficient, market design, market friction, market microstructure, Murray Gell-Mann, p-value, quantitative trading / quantitative finance, random walk, stochastic process, stochastic volatility, transaction costs

CORRELATION AND CLUSTERING IN THE TRADING OF THE MEMBERS OF THE LSE it assumes normally distributed disturbances, whereas we have discrete ternary values. Later in the text we use a bootstrap approach to test the significance. Now, however, we test the significance of the correlation coefficients using a standard algorithm as in ref. (Best and Roberts, 1975). The algorithm calculates the approximate tail probabilities for Spearman’s correlation coefficient ρ. Its precision unfortunately degrades when there are ties in the data, which is the case here. With this caveat in mind, as a preliminary test, we find that, for example, for on-book trading in Vodafone for the month of May 2000, 10.3% of all correlation coefficients are significant at the 5% level. Averaging over all stocks and months, the average percentage of significant coefficients for on-book trading is 10.5% ± 0.4%, while for off-book trading it is 20.7% ± 1.7%.

Averaging over all stocks and months, the average percentage of significant coefficients for on-book trading is 10.5% ± 0.4%, while for off-book trading it is 20.7% ± 1.7%. Both of these averages are substantially higher than the 5% we would expect randomly with a 5% acceptance level of the test. 4.2 Significance and structure in the correlation matrices The preliminary result of the previous section that some correlation coefficients are non-random is further corroborated by testing for non-random structure in the correlation matrices. The hypothesis that there is structure in the correlation matrices contains the weaker hypothesis that some coefficients are statistically significant. The test for structure in the matrices would involve multiple joint tests for the significance of the coefficients. An alternative method, however, is to examine the eigenvalue spectrum of the correlation matrices. Intuitively, one can understand the relation between the two tests by remembering that eigenvalues λ are roots of the characteristic equation det(A − λ1) = 0, and that the determinant is a sum !

However, being stronger, they are perhaps of a more simple nature: The second largest eigenvalue is almost never significant for off-book trading, while on the on-book market it is quite often significant. 4.2.3 Clustering of trading behaviour The existence of significant eigenvalues allows us to use the correlation matrix as a distance measure in the attempt to classify institutions into groups of similar or dissimilar trading patterns. We apply clustering techniques using a metric chosen so that two strongly correlated institutions are ’close’ and anti-correlated institutions are ’far away’. A functional form fulfilling this requirement and satisfying the properties of being a metric is (Bonanno et al., 2000) # (4.2) di,j = 2 · (1 − ρi,j ), where ρi,j is the correlation coefficient between strategies i and j. We have tried several reasonable modifications to this form but without obvious differences in the results. Ultimately the choice of this metric is influenced by the fact that it has been successfully used in other studies (Bonanno et al., 2000). We use complete linkage clustering, in which the distance between two clusters is calculated as the maximum distance between its members.

 

pages: 321 words: 97,661

How to Read a Paper: The Basics of Evidence-Based Medicine by Trisha Greenhalgh

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

call centre, complexity theory, conceptual framework, correlation coefficient, correlation does not imply causation, deskilling, knowledge worker, meta analysis, meta-analysis, microbiome, New Journalism, p-value, personalized medicine, placebo effect, randomized controlled trial, the scientific method

If you do, you might be stuck with non-parametric tests, which aren't as much fun (see section ‘What sort of data have they got, and have they used appropriate statistical tests?’). 4. Ignore all withdrawals (‘drop outs’) and non-responders, so the analysis only concerns subjects who fully complied with treatment (see section ‘Were preliminary statistical questions addressed?’). 5. Always assume that you can plot one set of data against another and calculate an ‘r-value’ (Pearson correlation coefficient) (see section ‘Has correlation been distinguished from regression, and has the correlation coefficient (‘r-value’) been calculated and interpreted correctly?’), and that a ‘significant’ r-value proves causation (see section ‘Have assumptions been made about the nature and direction of causality?’). 6. If outliers (points that lie a long way from the others on your graph) are messing up your calculations, just rub them out. But if outliers are helping your case, even if they appear to be spurious results, leave them in (see section ‘Were ‘outliers’ analysed with both common sense and appropriate statistical adjustments?’).

Correlation, regression and causation Has correlation been distinguished from regression, and has the correlation coefficient (‘r-value’) been calculated and interpreted correctly? For many non-statisticians, the terms correlation and regression are synonymous, and refer vaguely to a mental image of a scatter graph with dots sprinkled messily along a diagonal line sprouting from the intercept of the axes. You would be right in assuming that if two things are not correlated, it will be meaningless to attempt a regression. But regression and correlation are both precise statistical terms that serve different functions [2]. The r-value (or to give it its official name, ‘Pearson’s product–moment correlation coefficient') is among the most overused statistical instruments in the book. Strictly speaking, the r-value is not valid unless certain criteria, as given here, are fulfilled. 1.

Every r-value should be accompanied by a p-value, which expresses how likely an association of this strength would be to have arisen by chance (see section ‘Have ‘p-values’ been calculated and interpreted appropriately?’), or a confidence interval, which expresses the range within which the ‘true’ R-value is likely to lie (see section ‘Have confidence intervals been calculated, and do the authors' conclusions reflect them?’). (Note that lower case ‘r’ represents the correlation coefficient of the sample, whereas upper case ‘R’ represents the correlation coefficient of the entire population.) Remember, too, that even if the r-value is an appropriate value to calculate from a set of data, it does not tell you whether the relationship, however strong, is causal (see subsequent text). The term regression refers to a mathematical equation that allows one variable (the target variable) to be predicted from another (the independent variable).

 

pages: 297 words: 91,141

Market Sense and Nonsense by Jack D. Schwager

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, Bernie Madoff, Brownian motion, collateralized debt obligation, commodity trading advisor, conceptual framework, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, fixed income, high net worth, implied volatility, index arbitrage, index fund, London Interbank Offered Rate, Long Term Capital Management, margin call, market bubble, market fundamentalism, merger arbitrage, pattern recognition, performance metric, pets.com, Ponzi scheme, quantitative trading / quantitative finance, random walk, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, Sharpe ratio, short selling, statistical arbitrage, statistical model, transaction costs, two-sided market, value at risk, yield curve

In this chapter, we take a closer look at correlation and some of the ways it is often misinterpreted. Correlation Defined The correlation coefficient, typically denoted by the letter r, measures the degree of linear relationship between two variables. The correlation coefficient ranges from −1.0 to +1.0. The closer the correlation coefficient is to +1.0, the closer the relationship is between the two variables. A perfect correlation of 1.0 would occur only in artificial situations. For example, the heights of a group of people measured in inches and the heights of the same group of people measured in feet would be perfectly correlated. The closer the correlation coefficient is to −1.0, the stronger the inverse correlation is between the two variables. For example, average winter temperatures in the U.S.

For example, average winter temperatures in the U.S. Northeast and heating oil usage in that region would be inversely related variables (variables with a negative correlation coefficient). If two variables have a correlation coefficient near zero, it indicates that there is no significant (linear) relationship between the variables. It is important to understand that the correlation coefficient only indicates the degree of correlation between two variables and does not imply anything about cause and effect. Correlation Shows Linear Relationships Correlation reflects only linear relationships. For example, Figure 9.1 illustrates the returns of a hypothetical stock index option selling strategy (selling out-of-the-money calls and puts) versus Standard & Poor’s (S&P) returns. Calls that expire below the strike price and puts that expire above the strike price would generate profits equal to the premium collected.

Although Figure 9.1 clearly reflects a strong relationship between the strategy and S&P returns, the correlation between the two is actually zero! Why? Because correlation reflects only linear relationships, and there is no linear relationship between the two variables. Figure 9.1 Strategy Returns versus S&P Returns The Coefficient of Determination (r2) The square of the correlation coefficient, which is called the coefficient of determination and is denoted as r2, has a very specific interpretation: It represents the percentage of the variability of one variable explained by the other. For example, if the correlation coefficient (r) of a fund versus the S&P is 0.7, it implies that nearly half the variability of the fund’s returns is explained by the S&P returns (r2 = 0.49). For a mutual fund that is a so-called closet benchmarker—a fund that maintains a portfolio very similar to the S&P index with only minor differences—the r2 would tend to be very high (e.g., above 0.9).

 

pages: 436 words: 140,256

The Rise and Fall of the Third Chimpanzee by Jared Diamond

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

agricultural Revolution, Atahualpa, Columbian Exchange, correlation coefficient, double helix, Drosophila, European colonialism, invention of gunpowder, invention of the wheel, invention of writing, out of africa, phenotype, Scientific racism, Search for Extraterrestrial Intelligence, the scientific method, trade route, V2 rocket

Psychologists have tackled this question by examining many married couples, measuring everything conceivable about their physical appearance and other characteristics, and then trying to make sense out of who married whom. A simple numerical way of describing the result is by means of a statistical index called the correlation coefficient. If you line up 100 husbands in order of their ranking for some characteristic (say, their height), and if you also line up their 100 wives with respect to the same characteristic, the correlation coefficient describes whether a man tends to be at the same position in the husbands' line-up as his wife is in the line-up of wives. A correlation coefficient of plus one would mean perfect correspondence: the tallest man marries the tallest woman, the thirty-seventh tallest man marries the thirty-seventh tallest woman, and so on. A correlation coefficient of minus one would mean perfect matching by opposites: the tallest man marries the shortest woman, the thirty-seventh tallest man marries the thirty-seventh shortest woman, and so on.

A correlation coefficient of minus one would mean perfect matching by opposites: the tallest man marries the shortest woman, the thirty-seventh tallest man marries the thirty-seventh shortest woman, and so on. Finally, a correlation coefficient of zero would mean that husbands and wives assort completely randomly by height: a tall man is as likely to marry a short woman as a tall woman. These examples are for height, but correlation coefficients can also be calculated for anything else, such as income and IQ. If you measure enough things about enough couples, here is what you will find. Not surprisingly, the highest correlation coefficients—typically around +0.9—are for religion, ethnic background, race, socioeconomic status, age, and political views. That is, most husbands and wives prove to be of the same religion, ethnic background, and so on. Perhaps you also will not be surprised that the next highest correlation coefficients, usually around +0.4, are for measures of personality and intelligence, such as extroversion, neatness, and IQ.

Those other traits include ones as diverse as breadth of nose, length of ear lobe or middle finger, circumference of wrist, distance between eyes, and lung volume! Experimenters have made this finding for people as diverse as Poles in Poland, Americans in Michigan, and Africans in Chad. If you do not believe it, try noting eye colours (or measuring ear lobes) the next time you are at a dinner party with many couples, and then get your pocket calculator to give you the correlation coefficient. Coefficients for physical traits are on the average +0.2- not so high as for personality traits (+0.4) or religion (+0.9), but still significantly higher than zero. For a few physical traits the correlation is even higher than 0.2-for instance, an astonishing 0.61 for length of middle finger. At least unconsciously people care more about their spouse's middle finger length than about his or her hair colour and intelligence!

 

pages: 589 words: 69,193

Mastering Pandas by Femi Anthony

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Amazon Web Services, correlation coefficient, correlation does not imply causation, Debian, en.wikipedia.org, Internet of things, natural language processing, p-value, random walk, side project, statistical model

For more information on correlation and dependency, refer to http://en.wikipedia.org/wiki/Correlation_and_dependence. The correlation measure, known as correlation coefficient, is a number that captures the size and direction of the relationship between the two variables. It can vary from -1 to +1 in direction and 0 to 1 in magnitude. The direction of the relationship is expressed via the sign, with a + sign expressing positive correlation and a - sign negative correlation. The higher the magnitude, the greater the correlation with a one being termed as the perfect correlation. The most popular and widely used correlation coefficient is the Pearson product-moment correlation coefficient, known as r. It measures the linear correlation or dependence between two x and y variables and takes values between -1 and +1. The sample correlation coefficient r is defined as follows: This can also be written as follows: Here, we have omitted the summation limits.

However, note that the intercept value is not really meaningful as it is outside the bounds of the data. We can also only make predictions for values within the bounds of the data. For example, we cannot predict what the chirpFrequency is at 32 degrees Fahrenheit as it is outside the bounds of the data; moreover, at 32 degrees Fahrenheit, the crickets would have frozen to death. The value of R, the correlation coefficient, is given as follows: In [38]: R=np.sqrt(result.rsquared) R Out[38]: 0.83514378678237422 Thus, our correlation coefficient is R = 0.835. This would indicate that about 84 percent of the chirp frequency can be explained by the changes in temperature. Reference of this information: The Song of Insects http://www.hup.harvard.edu/catalog.php?isbn=9780674420663 The data is sourced from http://bit.ly/1MrlJqR. For a more in-depth treatment of single and multi-variable regression, refer to the following websites: Regression (Part I): http://bit.ly/1Eq5kSx Regression (Part II): http://bit.ly/1OmuFTV Summary In this chapter, we took a brief tour of the classical or frequentist approach to statistics and showed you how to combine pandas along with the stats packages—scipy.stats and statsmodels—to calculate, interpret, and make inferences from statistical data.

 

pages: 408 words: 85,118

Python for Finance by Yuxing Yan

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset-backed security, business intelligence, capital asset pricing model, constrained optimization, correlation coefficient, distributed generation, diversified portfolio, implied volatility, market microstructure, P = NP, p-value, quantitative trading / quantitative finance, Sharpe ratio, time value of money, value at risk, volatility smile

Find out the meaning of zscore() included in the stats submodule (SciPy), and offer a simple example of using this function. 18. What is the market risk (beta) for IBM in 2010? (Hint: the source of data could be from Yahoo! Finance.) 19. What is wrong with the following lines of code? >>>c=20 >>>npv=np.npv(0.1,c) [ 121 ] Introduction to NumPy and SciPy 20. The correlation coefficient function from NumPy is np.corrcoef(). Find more about this function. Estimate the correlation coefficient between IBM, DELL, and W-Mart. 21. Why is it claimed that the sn.npv() function from SciPY() is really a Present Value (PV) function? 22. Design a true NPV function using all cash flows, including today's cash flow. 23. The Sharpe ratio is used to measure the trade-off between risk and return: Sharpe = R − Rf σ Here, R is the expected returns for an individual security, and R f is the expected risk-free rate. σ is the volatility, that is, standard deviation of the return on the underlying security.

The color of the arrow is black. For more detail about the function, just type help(plt.annotate) after issuing import matplotlib.pyplot as plt. From the preceding graph, we see that the fluctuation, uncertainty, or risk of our equal-weighted portfolio is much smaller than those of individual stocks in its portfolio. We can also estimate their means, standard deviation, and correlation coefficient. The correlation coefficient between those two stocks is -0.75, and this is the reason why we could diversify away firm-specific risk by forming an even equal-weighted portfolio as shown in the following code: >>>import scipy as sp >>>sp.corrcoef(A,B) array([[ 1. , -0.74583429], [-0.74583429, 1. ]]) In the preceding example, we use hypothetical numbers (returns) for two stocks. How about IBM and W-Mart?

First, let us look at a hypothetical case by assuming that we have 5 years' annual returns of two stocks as follows: Year Stock A Stock B 2009 0.102 0.1062 2010 -0.02 0.23 2011 0.213 0.045 2012 0.12 0.234 2013 0.13 0.113 We form an equal-weighted portfolio using those two stocks. Using the mean() and std() functions contained in NumPy, we can estimate their means, standard deviations, and correlation coefficients as follows: >>>import numpy as np >>>A=[0.102,-0.02, 0.213,0.12,0.13] >>>B=[0.1062,0.23, 0.045,0.234,0.113] >>>port_EW=(np.array(ret_A)+np.array(ret_B))/2. >>>round(np.mean(A),3),round(np.mean(B),3),round(np.mean(port_EW),3) (0.109, 0.146, 0.127) >>>round(np.std(A),3),round(np.std(B),3),round(np.std(port_EW),3) (0.075, 0.074, 0.027) In the preceding code, we estimate mean returns, their standard deviations for individual stocks, and an equal-weighted portfolio.

 

pages: 416 words: 118,592

A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing by Burton G. Malkiel

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

accounting loophole / creative accounting, Albert Einstein, asset allocation, asset-backed security, backtesting, Bernie Madoff, BRICs, capital asset pricing model, compound rate of return, correlation coefficient, Credit Default Swap, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, Elliott wave, Eugene Fama: efficient market hypothesis, experimental subject, feminist movement, financial innovation, fixed income, framing effect, hindsight bias, Home mortgage interest deduction, index fund, invisible hand, Isaac Newton, Long Term Capital Management, loss aversion, margin call, market bubble, mortgage tax deduction, new economy, Own Your Own Home, passive investing, pets.com, Ponzi scheme, price stability, profit maximization, publish or perish, purchasing power parity, RAND corporation, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, short selling, Silicon Valley, South Sea Bubble, The Myth of the Rational Market, The Wisdom of Crowds, transaction costs, Vanguard fund, zero-coupon bond

Still, at least at certain times, some stocks and some classes of assets do move against the market; that is, they have negative covariance or (and this is the same thing) they are negatively correlated with each other. THE CORRELATION COEFFICIENT AND THE ABILITY OF DIVERSIFICATION TO REDUCE RISK Correlation Coefficient Effect of Diversification on Risk +1.0 No risk reduction is possible. +0.5 Moderate risk reduction is possible. 0 Considerable risk reduction is possible. –0.5 Most risk can be eliminated. –1.0 All risk can be eliminated. Now comes the real kicker; negative correlation is not necessary to achieve the risk reduction benefits from diversification. Markowitz’s great contribution to investors’ wallets was his demonstration that anything less than perfect positive correlation can potentially reduce risk. His research led to the results presented in the preceding table. As shown, it demonstrates the crucial role of the correlation coefficient in determining whether adding a security or an asset class can reduce risk.

When higher returns can be achieved with lower risk by adding international stocks, no investor should fail to take notice. Some portfolio managers have argued that diversification has not continued to give the same degree of benefit as was previously the case. Globalization led to an increase in the correlation coefficients between the U.S. and foreign markets as well as between stocks and commodities. The following three charts indicate how correlation coefficients have risen over the first decade of the 2000s. The charts show the correlation coefficients calculated over every twenty-four-month period between U.S. stocks (as measured by the S&P 500-Stock Index) and the EAFE index of developed foreign stocks, between U.S. stocks and the broad (MSCI) index of emerging-market stocks, and between U.S. stocks and the Goldman Sachs (GSCI) index of a basket of commodities such as oil, metals, and the like.

Broad international diversification would have been of enormous benefit to U.S. investors, even during “the lost decade.” DIVERSIFICATION INTO EMERGING MARKETS HELPED DURING “THE LOST DECADE”: CUMULATIVE RETURNS FROM ALTERNATIVE MARKETS Source: Vanguard, Datastream, Morningstar. Moreover, safe bonds proved their worth as a risk reducer. The graph Time Varying Stock–Bond Correlation shows how correlation coefficients between U.S. Treasury bonds and large capitalization U.S. equities fell during the 2008–09 financial crisis. Even during the horrible stock market of 2008, a broadly diversified portfolio of bonds invested in the Barclay’s Capital broad bond index returned 5.2 percent. There was a place to hide during the financial crisis. Bonds have proved their worth as an effective diversifier. TIME VARYING STOCK–BOND CORRELATION Data: 10Y Treasury return is calculated from 10Y Treasury yields.

 

pages: 369 words: 128,349

Beyond the Random Walk: A Guide to Stock Market Anomalies and Low Risk Investing by Vijay Singal

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Andrei Shleifer, asset allocation, capital asset pricing model, correlation coefficient, cross-subsidies, Daniel Kahneman / Amos Tversky, diversified portfolio, endowment effect, index arbitrage, index fund, locking in a profit, Long Term Capital Management, loss aversion, margin call, market friction, market microstructure, mental accounting, merger arbitrage, new economy, prediction markets, price stability, profit motive, random walk, Richard Thaler, risk-adjusted returns, risk/return, Sharpe ratio, short selling, transaction costs, Vanguard fund

Once you choose to invest in both stocks, the return becomes riskless because one of the two stocks does well and the other does poorly regardless of the kind of weather. The key in diversification of risk is correlation. Notice that the returns from beachwear and video rental always go in the opposite direction. If one of them does well, the other does not. Therefore, adding stocks that do not behave like other stocks in your portfolio is good and can reduce risk. The correlation is measured by what is called a correlation coefficient. The correlation coefficient varies between –1 and +1. The two stocks in the above example have a correlation of –1. Unfortunately, most stocks have a positive correlation, and many of them have a correlation with the market portfolio that is close to +1. The challenge in diversifying risk is to find stocks that have a correlation of less than +1. However, if you own only one stock, such as the stock of the company you work for, it is easy to find other stocks that are not well correlated with that stock.

This important point underscores the trade-off between risk and return: investors are happy to give up some return if the reduction in risk is sufficient. Therefore, it is not always necessary to ensure that the return is preserved. A general rule to evaluate whether a new asset should be included in an existing portfolio is based on the risk-return trade-off relationship: E(Rn ) = R f + σ n ρn, p σp × E(Rp ) − R f  where E(R) is the return from an asset, s is the standard deviation, r is the correlation coefficient, and the subscripts n and p refer to the new stock and existing portfolio. Rf is the return on the risk-free asset. If the new asset’s return is greater than the right-hand side in the above equation, then the asset should be included in the existing portfolio, otherwise not. That condition can be rewritten as below: E( Rn ) − R f σn > E(Rp ) − R f σp × ρn, p Evidence Before looking at the evidence, consider the potential benefits from international investing and the source of those benefits.

That condition can be rewritten as below: E( Rn ) − R f σn > E(Rp ) − R f σp × ρn, p Evidence Before looking at the evidence, consider the potential benefits from international investing and the source of those benefits. Assume that the dollar return on U.S. stocks is 12 percent with a standard deviation of 18 percent, and the dollar return on non-U.S. stocks is also 12 percent with a standard deviation of 18 percent. Since the U.S. markets and foreign markets are not well correlated, let the correlation coefficient be 0.60. Putting the U.S. stocks and the non-U.S. stocks in a 50-50 combination would generate a new world portfolio with the following characteristics: Rw = w1RUS + w2 Rnon −US = 0.50 × 12% + 0.50 × 12% = 12% σ w = w12σ 12 + w22σ 22 + 2w1w2 ρσ 1σ 2 = 0.50 2 × 0.18 2 + 0.50 2 × 0.18 2 + 2 × 0.50 × 0.50 × 0.60 × 0.18 × 0.18 = 0.16 236 Beyond the Random Walk The new world portfolio has a return of 12 percent and a risk of 16 percent.

 

The Art of Computer Programming by Donald Ervin Knuth

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Brownian motion, complexity theory, correlation coefficient, Eratosthenes, Georg Cantor, information retrieval, Isaac Newton, iterative process, John von Neumann, Louis Pasteur, mandelbrot fractal, Menlo Park, NP-complete, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, sorting algorithm, Turing machine, Y2K

Similar remarks apply to the subtract-with-borrow and add- with-carry generators of exercise 3.2.1.1-14. K. Serial correlation test. We may also compute the following statistic: This is the "serial correlation coefficient," a measure of the extent to which Uj+i depends on Uj. Correlation coefficients appear frequently in statistical work. If we have n quantities Uo, Ui, ..., t/n-i and n others Vo, Vi, ..., Vn_i, the correlation coefficient between them is defined to be c = All summations in this formula are to be taken over the range 0 < j < n; Eq. B3) is the special case Vj = C/(j+i) mod n- The denominator of B4) is zero when JJo = U\ = ¦ ¦ ¦ = Un-\ or V$ = V\ = ¦ ¦ ¦ = Vn-\\ we exclude that case from discussion. 3.3.2 EMPIRICAL TESTS 73 A correlation coefficient always lies between —1 and +1. When it is zero or very small, it indicates that the quantities Uj and Vj are (relatively speaking) independent of each other, whereas a value of ±1 indicates total linear depen- dependence.

., Vn_i, let their mean values be «=- V v = - V vk. n ^ n ^ ' n 0<k<n 0<k<n a) Let U'k = Uk — u, V"fe' = Vk — v. Show that the correlation coefficient C given in Eq. B4) is equal to ? u'kvL 0<k<n b) Let C = N/D, where N and D denote the numerator and denominator of the expression in part (a). Show that N2 < D2, hence — 1 < C < 1; and obtain a formula for the difference D2 - N2. [Hint: See exercise 1.2.3-30.] c) If C = ±1, show that aUk + CVk = t, 0 < k < n, for some constants a, C, and r, not all zero. 18. [M20] (a) Show that if n = 2, the serial correlation coefficient B3) is always equal to —1 (unless the denominator is zero), (b) Similarly, show that when n = 3, the serial correlation coefficient always equals — \. (c) Show that the denominator in B3) is zero if and only if Uq = U\ = • • • = Un-i- 19. [M30] (J.

Therefore it is desirable to have C in Eq. B3) close to zero. In actual fact, since U0U1 is not completely independent of U1U2, the serial correlation coefficient is not expected to be exactly zero. (See exercise 18.) A "good" value of C will be between \xn — 2an and \xn + 2an, where l" ">2' B5) "-=;rrr °l=(n-1) (n - 2)' ">2' We expect C to be between these limits about 95 percent of the time. The formula for a\ in B5) is an upper bound, valid for serial correlations between independent random variables from an arbitrary distribution. When the C/'s are uniformly distributed, the true variance is obtained by subtracting %r-n~2 + O(n~7/3 logn). (See exercise 20.) Instead of simply computing the correlation coefficient between the obser- observations (Uo, U\, ..., Un-\) and their immediate successors (U\,..., Un-i,Uo), we can also compute it between (Uo, U\,..., Un-\) and any cyclically shifted sequence (Uq,...

 

pages: 130 words: 11,880

Optimization Methods in Finance by Gerard Cornuejols, Reha Tutuncu

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, call centre, constrained optimization, correlation coefficient, diversification, finite state, fixed income, frictionless, frictionless market, index fund, linear programming, Long Term Capital Management, passive investing, Sharpe ratio, transaction costs, value at risk, Y2K

We will discuss his model in more detail later. Here we give a brief description of the model and relate it to QPs. Consider an investor who has a certain amount of money to be invested in a number of different securities (stocks, bonds, etc.) with random returns. For each security i, i = 1, . . . , n, estimates of its expected return, µi , and variance, σi2 , are given. Furthermore, for any two securities i and j, their correlation coefficient ρij is also assumed to be known. If we represent the proportion of the total funds invested in security i by xi , one can compute the expected return and the variance of the resulting portfolio x = (x1 , . . . , xn ) as follows: E[x] = x1 µ1 + . . . + xn µn = µT x, and V ar[x] = X ρij σi σj xi xj = xT Qx i,j where ρii ≡ 1, Qij = ρij σi σj for i 6= j, Qii = σi2 , and µ = (µ1 , . . . , µn ).

Chapter 5 QP Models and Tools in Finance 5.1 Mean-Variance Optimization In the introductory chapter, we have discussed Markowitz’ theory of mean-variance optimization (MVO) for the selection of portfolios of securities (or asset classes) in a manner that trades off the expected returns and the perceived risk of potential portfolios. Consider assets S1 , S2 , . . . , Sn (n ≥ 2) with random returns. Let µi and σi denote the expected return and the standard deviation of the return of asset Si . For i 6= j, ρij denotes the correlation coefficient of the returns of assets Si and Sj . Let µ = [µ1 , . . . , µn ]T , and Q be the n × n symmetric covariance matrix with Qii = σi2 and Qij = ρij σi σj for i 6= j. Denoting the proportion of the total funds invested in security i by xi , one can represent the expected return and the variance of the resulting portfolio x = (x1 , . . . , xn ) as follows: E[x] = x1 µ1 + . . . + xn µn = µT x, and V ar[x] = X ρij σi σj xi xj = xT Qx, i,j where ρii ≡ 1.

Ω The variance of a random variable X is defined by h V ar[X] = E (X − E[X])2 i = E[X 2 ] − (E[X])2 . The standard deviation of a random variable is the square-root of its variance. 108 APPENDIX C. A PROBABILITY PRIMER For two jointly distributed random variables X1 and X2 , their covariance is defined to be Cov(X1 , X2 ) = E [(X1 − E[X1 ])(X2 − E[X2 ])] = E[X1 X2 ] − E[X1 ]E[X2 ] The correlation coefficient of two random variables is the ratio of their covariance to the product of their standard deviations. For a collection of random variables X1 , . . . , Xn , the expected value of the sum of these random variables is equal to the sum of their expected values: " E n X i=1 # Xi = n X E[Xi ]. i=1 The formula for the variance of the sum of the random variables X1 , . . . , Xn is a bit more complicated: " V ar n X i=1 # Xi = n X i=1 V ar[Xi ] + 2 X 1≤i<j≤n Cov(Xi , Xj ).

 

pages: 315 words: 93,628

Is God a Mathematician? by Mario Livio

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Antoine Gombaud: Chevalier de Méré, Brownian motion, cellular automata, correlation coefficient, correlation does not imply causation, cosmological constant, Dava Sobel, double helix, Edmond Halley, Eratosthenes, Georg Cantor, Gerolamo Cardano, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, John von Neumann, music of the spheres, probability theory / Blaise Pascal / Pierre de Fermat, The Design of Experiments, the scientific method, traveling salesman

For a given value of the temperature, one cannot predict precisely the number of forest fires that will break out, since the latter depends on other variables such as the humidity and the number of fires started by people. In other words, for any value of the temperature, there could be many corresponding numbers of forest fires and vice versa. Still, the mathematical concept known as the correlation coefficient allows us to measure quantitatively the strength of the relationship between two such variables. The person who first introduced the tool of the correlation coefficient was the Victorian geographer, meteorologist, anthropologist, and statistician Sir Francis Galton (1822–1911). Galton—who was, by the way, the half-cousin of Charles Darwin—was not a professional mathematician. Being an extraordinarily practical man, he usually left the mathematical refinements of his innovative concepts to other mathematicians, in particular to the statistician Karl Pearson (1857–1936).

If the correlation between them is very close, a very long cubit would usually imply a very tall stature, but if it were not very close, a very long cubit would be on the average associated with only a tall stature, and not a very tall one; while, if it were nil, a very long cubit would be associated with no especial stature, and therefore, on the average, with mediocrity. Pearson eventually gave a precise mathematical definition of the correlation coefficient. The coefficient is defined in such a way that when the correlation is very high—that is, when one variable closely follows the up-and-down trends of the other—the coefficient takes the value of 1. When two quantities are anticorrelated, meaning that when one increases the other decreases and vice versa, the coefficient is equal to–1. Two variables that each behave as if the other didn’t even exist have a correlation coefficient of 0. (For instance, the behavior of some governments unfortunately shows almost zero correlation with the wishes of the people whom they supposedly represent.) Modern medical research and economic forecasting depend crucially on identifying and calculating correlations.

 

pages: 586 words: 159,901

Wall Street: How It Works And for Whom by Doug Henwood

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

accounting loophole / creative accounting, affirmative action, Andrei Shleifer, asset allocation, asset-backed security, bank run, banking crisis, barriers to entry, borderless world, Bretton Woods, British Empire, capital asset pricing model, capital controls, central bank independence, corporate governance, correlation coefficient, correlation does not imply causation, credit crunch, currency manipulation / currency intervention, David Ricardo: comparative advantage, debt deflation, declining real wages, deindustrialization, dematerialisation, diversification, diversified portfolio, Donald Trump, equity premium, Eugene Fama: efficient market hypothesis, experimental subject, facts on the ground, financial deregulation, financial innovation, Financial Instability Hypothesis, floating exchange rates, full employment, George Akerlof, George Gilder, hiring and firing, Hyman Minsky, implied volatility, index arbitrage, index fund, interest rate swap, Internet Archive, invisible hand, Isaac Newton, joint-stock company, Joseph Schumpeter, kremlinology, labor-force participation, late capitalism, law of one price, liquidationism / Banker’s doctrine / the Treasury view, London Interbank Offered Rate, Louis Bachelier, market bubble, Mexican peso crisis / tequila crisis, microcredit, minimum wage unemployment, moral hazard, mortgage debt, mortgage tax deduction, oil shock, payday loans, pension reform, Plutocrats, plutocrats, price mechanism, price stability, prisoner's dilemma, profit maximization, Ralph Nader, random walk, reserve currency, Richard Thaler, risk tolerance, Robert Gordon, Robert Shiller, Robert Shiller, shareholder value, short selling, Slavoj Žižek, South Sea Bubble, The Market for Lemons, The Nature of the Firm, The Predators' Ball, The Wealth of Nations by Adam Smith, transaction costs, transcontinental railway, women in the workforce, yield curve, zero-coupon bond

Sometimes it doesn't matter whether the bad news is true; if the short can take a position and undertake a successful disinformation campaign, he or she can profitably cover the short. 35. For the "real" sector, however, borders still matter, and the "global assembly line" is a bit of an exaggeration. 36. The correlation coefficient is a measure of how tightly two sets of numbers are related to each other, ranging from -1 (a perfect mirror image) through 0 (no relation at all) to PLAYERS +1 (perfect lockstep). A correlation coefficient under 0.2 marks a fairly cacophanous relation, but figures over 0.9 signify great intimacy. 37. In fact, many foreign investments made in the U.S. during the 1980s have had apparently dismal rates of return. The dollar's decline has savaged financial invesments, and real investments haven't done much better.

In dollar terms, 70% of all foreign debt issued in the U.S. between 1926 and 1929 (excluding Canada) went bad — compared with a default rate of "only" 30% on corporate debt issued in the late 1920s. Most of the sovereign defaulters, by the way, had good ratings from Moody's (Cantor and Packer 1995). But now those defaults are a distant memory, and today's capital markets look seamless. Statistics confirm the decreasing importance of borders for the financial markets."*^ In the 1970s, the correlation coefficient between interest rates on 10-year U.S. government bonds and German bonds of similar maturity was 0.191, but from 1990 to 1994, it was 0.934; Japan and the U.S., 0.182 and 0.965, respectively; and the U.S. and the U.K., 0.590 and 0.949 (Bank of England data, reported in Goldstein et al. 1994, p. 5).-^'' While it would be an exaggeration to say that there's now a single global credit market, we're definitely moving in that direction.

One doesn't want to get too carried away naturalizing temperament and values, but the model seems particularly to drive away women and nonwhites, at least in America, because of its chilly irreality. It may just be that sex and race are simply convenient markers for hierarchy —that economics is an ideology of privilege, and the already privileged, or those who wish to become apologists for the privileged, are drawn to its study. WALL STREET 8. Correlation coefficients for the various versions of q suggested in the text are all well over .92. The correlation for the simple equity q (market value of stock divided by tangible assets, as shown in the charts and used in the text) and the values for 1960-74 reported in Tobin and Brainard C1977) is .97. 9- It's interesting that investment rose during what are usually considered the bad years of the 1970s.

 

pages: 467 words: 154,960

Trend Following: How Great Traders Make Millions in Up or Down Markets by Michael W. Covel

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, asset allocation, Atul Gawande, backtesting, Bernie Madoff, Black Swan, buy low sell high, capital asset pricing model, Clayton Christensen, commodity trading advisor, correlation coefficient, Daniel Kahneman / Amos Tversky, delayed gratification, deliberate practice, diversification, diversified portfolio, Elliott wave, Emanuel Derman, Eugene Fama: efficient market hypothesis, fiat currency, fixed income, game design, hindsight bias, housing crisis, index fund, Isaac Newton, John Nash: game theory, linear programming, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, market fundamentalism, market microstructure, mental accounting, Nash equilibrium, new economy, Nick Leeson, Ponzi scheme, prediction markets, random walk, Renaissance Technologies, Richard Feynman, Richard Feynman, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, short selling, South Sea Bubble, Stephen Hawking, systematic trading, the scientific method, Thomas L Friedman, too big to fail, transaction costs, upwardly mobile, value at risk, Vanguard fund, volatility arbitrage, William of Occam

It is the historical tendency of one thing to move in tandem with another.” The correlation coefficient is a number from –1 to +1, with –1 being the perfectly opposite behavior of two investments (for example, up 5 percent every time the other is down 5 percent). The +1 reflects identical investment results (up or down the same amount each period). The further away from +1 one gets (and thus closer to –1), the better a diversifier one investment is for the other. But because his firm is keenly aware of keeping things simple, it also provides another description of correlation: the tendency for one investment to “zig” while another “zags.”27 I took the monthly performance numbers of trend followers and computed their correlation coefficients. Comparing correlations provided evidence that trend followers trade typically the same markets in the same way at the same time.

He explained the reason—most investors pulled out at the wrong time. They traded with their gut and treated drawdowns as a cancer, rather than the natural ebb and flow of trading.” Interestingly, there is another perspective on drawdowns that few people consider. When you look at trend following performance data—for example, Dunn’s track record—you can’t help but notice that certain times are better than others to invest with Dunn. Correlation coefficient: A statistical measure of the interdependence of two or more random variables. Fundamentally, the value indicates how much of a change in one variable is explained by a change in another.25 Smart clients of Dunn look at his performance chart and buy in when his fund is experiencing a drawdown. Why? Because if he is down 30 percent, and you know from analysis of past performance data that his recovery from drawdowns is typically quick, why not “buy” Dunn while he is on sale?

We’ve all evolved and developed systems that are very different from those we were taught, and that independent evolution suggests that the dissimilarities to trading between turtles are always increasing.”29 A Turtle correlation chart paints a clear picture. The relationship is solid. The data (Chart 3.6) is the judge: CHART 3.6: Correlation Among Turtle Traders Chesapeake Chesapeake Eckhardt Hawksbill JPD Rabar 1 0.53 0.62 0.75 0.75 Eckhardt 0.53 1 0.7 0.7 0.71 Hawksbill 0.62 0.7 1 0.73 0.76 JPD 0.75 0.7 0.73 1 0.87 Rabar 0.75 0.71 0.76 0.87 1 Correlation coefficients gauge how closely an advisor’s performance resembles another advisor. Values exceeding 0.66 might be viewed as having significant positive performance correlation. Consequently, values exceeding –0.66 might be viewed as having significant negative performance correlation. Chesapeake Capital Corporation Eckhardt Trading Co. Hawksbill Capital Management JPD Enterprises Inc. Rabar Market Research Of course, there is more to the story than just correlation.

 

The Handbook of Personal Wealth Management by Reuvid, Jonathan.

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, banking crisis, BRICs, collapse of Lehman Brothers, correlation coefficient, credit crunch, cross-subsidies, diversification, diversified portfolio, estate planning, financial deregulation, fixed income, high net worth, income per capita, index fund, interest rate swap, laissez-faire capitalism, land tenure, market bubble, merger arbitrage, new economy, Northern Rock, pattern recognition, Ponzi scheme, prediction markets, risk tolerance, risk-adjusted returns, risk/return, short selling, side project, sovereign wealth fund, statistical arbitrage, systematic trading, transaction costs, yield curve

As a result, forestry is considered to have strong diversification potential and the capability of reducing an investment portfolio’s overall risk. Forestry in the United States has been repeatedly shown to have a negative correlation coefficient with, among other financial assets, common stocks, corporate and government bonds, and the S&P 500 (see Table 2.3.1), and in certain studies reduced real portfolio risk by an average of 5 per cent.1 We observe that forestry generally forms a minor element of an overall investment portfolio, perhaps no more than 5–10 per cent as a maximum. Table 2.3.1 Timberland correlation coefficients, 1959–78 Investment correlation coefficient Timberland Residential housing Farm real estate S&P 500 index OTC stocks Preferred stock average No-load mutual fund average Municipal bonds Treasury Bills Long-term corporate bonds Commodity futures average 1.0000 –0.0905 0.5612 –0.4889 –0.4917 –0.3533 –0.6351 –0.0900 0.3118 –0.2704 0.8988 Source: Zinkhan, FC, Sizemore, WR, Mason, GH and Ebner, TJ (1992) Timberland Investments, Portland, Oregon, Timber Press. _______________________________________ CURRENT OPPORTUNITIES IN FORESTRY 81 ឣ Liquidity Certain forestry investments such as COEIC funds and exchange-traded funds (see below) are traded daily on markets such as the London Stock Exchange and Alternative Investment Market (AIM).

 

pages: 206 words: 70,924

The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, David Ricardo: comparative advantage, discovery of penicillin, discrete time, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, floating exchange rates, full employment, Henri Poincaré, implied volatility, index fund, Isaac Newton, John von Neumann, Joseph Schumpeter, Long Term Capital Management, Louis Bachelier, margin call, market clearing, martingale, means of production, moral hazard, naked short selling, price stability, principal–agent problem, quantitative trading / quantitative finance, RAND corporation, random walk, risk tolerance, risk/return, Ronald Reagan, shareholder value, Sharpe ratio, short selling, stochastic process, The Chicago School, the scientific method, too big to fail, transaction costs, tulip mania, Works Progress Administration, yield curve

These assumptions allow us to calculate the expected return of the portfolio Rp by summing across all securities: E ( Rp ) = ∑ wi E (Ri ) i and the portfolio variance: s 2p = ∑ wi2 s i2 + ∑ ∑ wi w j si sj r ij i i j ≠i Notice that all of the coefficients on the right-hand side of the portfolio variance expression are necessarily positive, except for the correlation coefficient. We can readily see that portfolio variance is minimized if the correlation coefficient ij 1. We can generalize this risk minimization procedure through the matrix algebra for which Markowitz developed efficient solution algorithms that were more easily computable. This matrix algebra approach that minimizes variance for a given return R and wealth w becomes: min s 2 = min XVX T ∋ r = (W − X.1)rf + XR X X where the wealth constraint r = ( W − X.1) rf + XR affirms that wealth is invested in a risky portfolio R that returns R and a risk-free asset that returns rf.

Marschak had framed the The Theory 23 problem and indicated the direction for its solution. Most significantly for financial pricing theory, he went on: [W]e reinterpret [the decision variables] to mean not future yields but parameters [e.g., moments and joint moments] of the jointfrequency distribution of future yields. Thus, x may be interpreted as the mathematical expectation of first year’s meat consumption, y may be its standard deviation, z may be the correlation coefficient between meat and salt consumption … etc. … It is sufficiently realistic, however, to confine ourselves, for each [return] to two parameters only: the mathematical expectation … and the coefficient of variation [“risk”].8 Marschak proposed a simple approach to the consideration of the interplay between return and risk by confining its description to first moments, known as means, and second moments of returns, labeled variances and covariances.

 

Data Mining: Concepts and Techniques: Concepts and Techniques by Jiawei Han, Micheline Kamber, Jian Pei

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

bioinformatics, business intelligence, business process, Claude Shannon: information theory, cloud computing, computer vision, correlation coefficient, cyber-physical system, database schema, discrete time, distributed generation, finite state, information retrieval, iterative process, knowledge worker, linked data, natural language processing, Netflix Prize, Occam's razor, pattern recognition, performance metric, phenotype, random walk, recommendation engine, RFID, semantic web, sentiment analysis, speech recognition, statistical model, stochastic process, supply-chain management, text mining, thinkpad, web application

For 1 degree of freedom, the χ2 value needed to reject the hypothesis at the 0.001 significance level is 10.828 (taken from the table of upper percentage points of the χ2 distribution, typically available from any textbook on statistics). Since our computed value is above this, we can reject the hypothesis that gender and preferred_reading are independent and conclude that the two attributes are (strongly) correlated for the given group of people. Correlation Coefficient for Numeric Data For numeric attributes, we can evaluate the correlation between two attributes, A and B, by computing the correlation coefficient (also known as Pearson's product moment coefficient, named after its inventer, Karl Pearson). This is(3.3) where n is the number of tuples, ai and bi are the respective values of A and B in tuple i, Ā and are the respective mean values of A and B, σA and σB are the respective standard deviations of A and B (as defined in Section 2.2.2), and Σ(aibi) is the sum of the AB cross-product (i.e., for each tuple, the value for A is multiplied by the value for B in that tuple).

An attribute (such as annual revenue, for instance) may be redundant if it can be “derived” from another attribute or set of attributes. Inconsistencies in attribute or dimension naming can also cause redundancies in the resulting data set. Some redundancies can be detected by correlation analysis. Given two attributes, such analysis can measure how strongly one attribute implies the other, based on the available data. For nominal data, we use the χ2 (chi-square) test. For numeric attributes, we can use the correlation coefficient and covariance, both of which access how one attribute's values vary from those of another. χ2 Correlation Test for Nominal Data For nominal data, a correlation relationship between two attributes, A and B, can be discovered by a χ2 (chi-square) test. Suppose A has c distinct values, namely a1, a2, … ac. B has r distinct values, namely b1, b2, … br. The data tuples described by A and B can be shown as a contingency table, with the c values of A making up the columns and the r values of B making up the rows.

Covariance of Numeric Data In probability theory and statistics, correlation and covariance are two similar measures for assessing how much two attributes change together. Consider two numeric attributes A and B, and a set of n observations {(a1, b1), …, (an, bn)}. The mean values of A and B, respectively, are also known as the expected values on A and B, that is, and The covariance between A and B is defined as(3.4) If we compare Eq. (3.3) for rA, B (correlation coefficient) with Eq. (3.4) for covariance, we see that(3.5) where σA and σB are the standard deviations of A and B, respectively. It can also be shown that(3.6) This equation may simplify calculations. For two attributes A and B that tend to change together, if A is larger than Ā (the expected value of A), then B is likely to be larger than (the expected value of B). Therefore, the covariance between A and B is positive.

 

The Concepts and Practice of Mathematical Finance by Mark S. Joshi

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Black-Scholes formula, Brownian motion, correlation coefficient, Credit Default Swap, delta neutral, discrete time, Emanuel Derman, implied volatility, incomplete markets, interest rate derivative, interest rate swap, London Interbank Offered Rate, martingale, millennium bug, quantitative trading / quantitative finance, short selling, stochastic process, stochastic volatility, the market place, time value of money, transaction costs, value at risk, volatility smile, yield curve, zero-coupon bond

If we now compare Yt with X (l), the fact that Xil) was used to construct Yt means that their movements are correlated. In particular, we have that E((Yt - YS)(Xil) - X(1))) = PE((Xtl) - Xs 1))2) + 1 - p2]E((Xi2)- X(Z))(Xt`1 Xs1))). (11.4) Since V) and X(2) are independent, the second expectation is zero and so 1E((Yt - (Xrl) YS) - Xs 1))) = p(t - s). (11.5) As Yt - YS and Xtl) - XSl) both have variance t - s, this means that the correlation coefficient is p. Thus we have constructed a Brownian motion whose increments are correlated to those of X(1) with correlation p. More generally, we could construct a Brownian motion from any vector a = (al,ak) with a? = 1, by taking Ek=l ajX(J). 11.3 The higher-dimensional Ito calculus 263 The existence of such correlated Brownian motions will be crucial in pricing multi-asset options. In general, we may want a whole vector of Brownian motions with a specified correlation matrix.

This means that (Wr+At - (yVr+At - Wt(k)) = AtpjkZk + Ot 1 - pjkejkZk. (11.12) The second term has mean zero and variance of order At2 so we can discard it as small, whereas the first term has mean pjk At and variance of order At2 and therefore contributes. This gives us a new rule for the multi-dimensional Ito calculus: dWrj)dW(k) = pjkdt. To summarize, we have Theorem 11.1 (Multi-dimensional Ito lemma) Let Wtj) be correlated Brownian motions with correlation coefficient pjk between the Brownian motions WU) and Wtk). Let Xj be an Ito process with respect to Wt W. Let f be a smooth function; we then have that af of 11 at j=1 11 1 T a2f + 2 j,k=1 ax axi` (t, X1, ax j (t,X1,...,X,1)dXj ..., X7,)dX jdXk, (11.14) with dWtj)dWtk) = pjkdt. When collecting terms, the final double sum will be absorbed into the dt term. We still need to think a little about what a process of the form 12 dYt = ltdt + ajdWtj), (11.16) j=1 means.

Perfect correlation means the vectors point the same way, perfect negative correlation means they point the opposite way, and zero correlation means they are orthogonal. In (11.20), the first vector has length O'1, and the second length oa2. When we add two vectors, v1, V2, the square of the length of the resultant vector is IIv1112 + 2 cos(9)IIv111. i1v211 + IIv2112, where 9 is the angle between the vectors. If we interpret the correlation coefficient as being the cosine of the angle between the two Brownian motions, then this means that the new volatility is just the length of the vector obtained by summing the vectors for each Brownian motion. 266 Multiple sources of risk More generally, we could construct a Brownian motion from any vector a=(a1,...,ak) Ek=1 ajX( ). with a? = 1, by taking When we have a process driven by k > 2 Brownian motions, we obtain a similar expression to (11.20).

 

pages: 363 words: 28,546

Portfolio Design: A Modern Approach to Asset Allocation by R. Marston

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, Bretton Woods, capital asset pricing model, capital controls, carried interest, commodity trading advisor, correlation coefficient, diversification, diversified portfolio, equity premium, Eugene Fama: efficient market hypothesis, family office, financial innovation, fixed income, German hyperinflation, high net worth, hiring and firing, housing crisis, income per capita, index fund, inventory management, Long Term Capital Management, mortgage debt, passive investing, purchasing power parity, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, Sharpe ratio, Silicon Valley, superstar cities, transaction costs, Vanguard fund

c05 P2: c/d QC: e/f JWBT412-Marston T1: g December 8, 2010 17:36 Printer: Courier Westford 88 PORTFOLIO DESIGN 1.0 Correlation Coefficients P1: a/b 0.8 10 year 0.6 5 year 0.4 0.2 Dec-74 Dec-79 Dec-84 Dec-89 Dec-94 Dec-99 Dec-04 Dec-09 FIGURE 5.9 Correlations between S&P 500 and EAFE Measured over Five and Ten Year Periods, 1970–2009 Data Sources: MSCI, © Morningstar, and S&P. and MSCI Pacific has an even lower correlation of 0.43. But for the last 10 years alone ending in 2009, the correlation between EAFE and the S&P rises to 0.87. There are correspondingly large increases in correlations between the S&P and the regional MSCI indexes. When did this increase in correlations occur? Consider Figure 5.9 which shows five- and 10-year correlation coefficients between the EAFE and S&P 500 indexes. Since the EAFE index starts only in 1970, the graph begins in 1975 for the five-year correlation and in 1980 for the 10-year correlation. The figure is noteworthy in several respects. First, the correlations vary widely over time whether they are measured over five- or ten-year intervals. The five-year correlation begins above 60 percent and at times falls below 30 percent.

Data Sources: Barclays Capital and Russell 9 10 P1: a/b c08 P2: c/d QC: e/f JWBT412-Marston T1: g December 8, 2010 Strategic Asset Allocation 17:51 Printer: Courier Westford 153 with the lowest allocation of 10 percent in stocks to the portfolio that is invested wholly in stocks.7 Large-cap growth stocks, as represented by the Russell 1000 Growth Index, do not appear in any of the 10 portfolios. The Russell 1000 Growth Index is dominated by the Russell 1000 Value Index. This result should not be surprising given the analysis in Chapter 4. Russell 1000 Value has a higher return and a lower standard deviation than Russell 1000 Growth. What’s more, the two indexes are highly correlated with a correlation coefficient of 0.82. The optimizer finds that one series is totally dominated by the other. So the optimizer rejects one whole asset class. The optimizer is also not fond of small-cap stocks. The Russell 2000 Index has a small weighting in the lowest risk portfolios, and its role disappears in portfolios with larger allocations to stocks. The point of this experiment is not to show the inferiority of large-cap growth stocks or small-cap stocks.

The four databases together have 3,924 live funds at the end of December 2002. The overlap among the three largest databases was analyzed after eliminating the funds that only appeared in MSCI. The percentages were rounded to the nearest decimal. 12. As explained in Chapter 5, the underperformance of EAFE relative to U.S. stocks is almost entirely due to Japan. 13. Recall that beta is equal to the correlation coefficient times the ratio of the standard deviation of the asset relative to the standard deviation of the benchmark. The beta is 0.36 = 0.77 ∗ (0.071/0.152). 14. Since the average return on the risk-free Treasury bill is 3.8 percent and the average return on the Russell 3000 is 9.2 percent, the alpha = 11.8 percent – [3.8% + 0.36∗(9.2% – 3.8%)] = 6.1%. 15. Over the same period, the correlation between the S&P 500 index and the Russell 1000 large-cap index is 1.00 and the correlation between the S&P 500 and the Russell 3000 all-cap index is 0.99.

 

pages: 545 words: 137,789

How Markets Fail: The Logic of Economic Calamities by John Cassidy

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Andrei Shleifer, anti-communist, asset allocation, asset-backed security, availability heuristic, bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Black-Scholes formula, Bretton Woods, British Empire, capital asset pricing model, centralized clearinghouse, collateralized debt obligation, Columbine, conceptual framework, Corn Laws, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, Daniel Kahneman / Amos Tversky, debt deflation, diversification, Elliott wave, Eugene Fama: efficient market hypothesis, financial deregulation, financial innovation, Financial Instability Hypothesis, financial intermediation, full employment, George Akerlof, global supply chain, Haight Ashbury, hiring and firing, Hyman Minsky, income per capita, incomplete markets, index fund, invisible hand, John Nash: game theory, John von Neumann, Joseph Schumpeter, laissez-faire capitalism, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, market clearing, mental accounting, Mikhail Gorbachev, Mont Pelerin Society, moral hazard, mortgage debt, Naomi Klein, Network effects, Nick Leeson, Northern Rock, paradox of thrift, Ponzi scheme, price discrimination, price stability, principal–agent problem, profit maximization, quantitative trading / quantitative finance, race to the bottom, Ralph Nader, RAND corporation, random walk, Renaissance Technologies, rent control, Richard Thaler, risk tolerance, risk-adjusted returns, road to serfdom, Robert Shiller, Robert Shiller, Ronald Coase, Ronald Reagan, shareholder value, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, statistical model, technology bubble, The Chicago School, The Great Moderation, The Market for Lemons, The Wealth of Nations by Adam Smith, too big to fail, transaction costs, unorthodox policies, value at risk, Vanguard fund

Unfortunately, during periods of great stress, the relationships between different asset classes tend to change dramatically. As the big hedge fund Long-Term Capital Management discovered to its cost during the international financial crisis of 1998, many assets that seem to have little or nothing in common suddenly move in the same direction. Prior to the blowup, for example, the correlation coefficient between certain bonds issued by the governments of the Philippines and Bulgaria was just 0.04: as the crisis unfolded, their correlation coefficient rose to 0.84. (A correlation coefficient of zero means two assets have no relationship; a coefficient of one means they move in perfect unison.) During a period of market upheaval, as a Wall Street saying has it, “all correlations go to one.” Investors panic and sell many different types of assets at the same time. When this happens, even a bank or financial institution that appears to be well diversified can suffer losses much bigger than a VAR model would have predicted, especially if it is highly leveraged (as Long-Term Capital was).

.”: Quoted in Jenny Anderson, “Merrill Painfully Learns the Risks of Managing Risk,” New York Times, October 12, 2007. 274 In its 1994 . . . : Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed. (New York: McGraw-Hill, 2000), 107. 274 “In contrast with traditional . . .”: Ibid., xxii. 275 “It helps you understand . . .”: Quoted in Joe Nocera, “Risk Mismanagement,” New York Times Magazine, January 2, 2009. 277 the correlation coefficient . . . : Linda Allen, Jacob Boudoukh, and Anthony Saunders, Understanding Market, Credit, and Operational Risk: The Value at Risk Approach (Hoboken, N.J.: Wiley-Blackwell, 2004), 103. 278 “We remind our readers . . .”: “CreditMetrics Technical Document,” RiskMetrics, April 1997, available at www.riskmetrics.com/publications/techdocs/cmtdovv.html. 278 “The relative prevalence of . . .”: Allen et al., Understanding Market, 35. 278 “I believe that . . .”: “Against Value at Risk: Nassim Taleb Replies to Philippe Jorion,” 1997, available at www.fooledbyrandomness.com/jorion.html. 279 “business planning relied on . . .”: UBS, “Shareholder Report on UBS’s Write-Downs,” 34. 279 “even though delinquency . . .”: Ibid., 38–39. 280 “would overturn . . .”: Gillian Tett, Fool’s Gold: How the Bold Dream of a Small Tribe at J.P.

 

pages: 335 words: 94,657

The Bogleheads' Guide to Investing by Taylor Larimore, Michael Leboeuf, Mel Lindauer

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, buy low sell high, corporate governance, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, Donald Trump, endowment effect, estate planning, financial independence, financial innovation, high net worth, index fund, late fees, Long Term Capital Management, loss aversion, Louis Bachelier, margin call, market bubble, mental accounting, passive investing, random walk, risk tolerance, risk/return, Sharpe ratio, statistical model, transaction costs, Vanguard fund, yield curve

Some bond funds invest in government bonds, some in corporate bonds, and others in municipal bonds. While some bond funds invest in highly rated investment-grade bonds, still others invest in lower-rated junk bonds. For more information on the various types of bonds, see Chapter 3. When investments (like stocks and bonds) don't always move together, they're said to have a low correlation coefficient. Understanding the correlation coefficient principal isn't really that difficult. The correlation numbers for any two investments can range from +1.0 (perfect correlation) to -1.0 (negative correlation) Basically, if two stocks (or funds) normally move together at the same rate, they're said to be highly correlated, and when two investments move in the opposite directions, they're said to be negatively correlated.

When two investments each randomly go their separate ways, independent of the movement of the other one, there is said to be no correlation between them, and their correlation figure would be shown as 0. Finally, when two investments always move in the opposite direction, they would have a negative correlation, which would be represented by a rating of -1.0. In actual practice, you'll find that most investment choices available to you will have a correlation coefficient somewhere between 1.0 (perfect correlation) and 0 (noncorrelated). It's very difficult to find negatively correlated asset classes that have similar expected returns. The closer the number is to 1.0, the higher the correlation between the two assets, and the lower the number, the less correlation there is between the two investments. So, a correlation figure of 0.71 would mean the two assets are not perfectly correlated, but a fund with a correlation figure of 0.52 would offer still more diversification, since it has an even lower number.

 

pages: 266 words: 86,324

The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Alfred Russel Wallace, Antoine Gombaud: Chevalier de Méré, Atul Gawande, Brownian motion, butterfly effect, correlation coefficient, Daniel Kahneman / Amos Tversky, Donald Trump, feminist movement, forensic accounting, Gerolamo Cardano, Henri Poincaré, index fund, Isaac Newton, law of one price, pattern recognition, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, Richard Feynman, Richard Feynman, Ronald Reagan, Stephen Hawking, Steve Jobs, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, V2 rocket, Watson beat the top human players on Jeopardy!

The coefficient of correlation is a number between -1 and 1; if it is near ± 1, it indicates that two variables are linearly related; a coefficient of 0 means there is no relation. For example, if data revealed that by eating the latest McDonald’s 1,000-calorie meal once a week, people gained 10 pounds a year and by eating it twice a week they gained 20 pounds, and so on, the correlation coefficient would be 1. If for some reason everyone were to instead lose those amounts of weight, the correlation coefficient would be -1. And if the weight gain and loss were all over the map and didn’t depend on meal consumption, the coefficient would be 0. Today correlation coefficients are among the most widely employed concepts in statistics. They are used to assess such relationships as those between the number of cigarettes smoked and the incidence of cancer, the distance of stars from Earth and the speed with which they are moving away from our planet, and the scores students achieve on standardized tests and the income of the students’ families.

 

All About Asset Allocation, Second Edition by Richard Ferri

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, asset-backed security, barriers to entry, Bernie Madoff, capital controls, commodity trading advisor, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, equity premium, estate planning, financial independence, fixed income, full employment, high net worth, Home mortgage interest deduction, implied volatility, index fund, Long Term Capital Management, Mason jar, mortgage tax deduction, passive income, pattern recognition, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, Sharpe ratio, too big to fail, transaction costs, Vanguard fund, yield curve

That said, I will also warn you that there are no two asset classes that relate the same way to each other all the time. These relationships are dynamic, and they can and do change without warning. Selecting investments that do not go up and down at the same time (or most of the time) can be made easier with correlation analysis. This is a mathematical measure of the tendency of one investment to move in relation to another. The correlation coefficient is a mathematically derived number that measures this tendency toward comovement relative to the investments’ average return. If two investments each move in the same direction at the same time above their average returns, they have a positive correlation. If they each move in opposite directions below their average returns, they have a negative correlation. If the movement of one investment relative to its average return is independent of the other, the two investments are noncorrelated.

Negative correlation is theoretically ideal when selecting investments for a portfolio, but you are not going to find it in the real world. These pairs of investments just do not exist. Correlation is measured using a range between ⫹1 and ⫺1. Two investments that have a correlation of ⫹0.3 or greater are considered positively correlated. When two investments have a correlation of ⫺0.3 or less, this is considered negative correlation. A correlation coefficient between ⫺0.3 and ⫹0.3 is considered noncorrelated. When two investments are noncorrelated, either the movement of one does not track the movement of the other or the tracking is inconsistent and shifts between positive and negative. Figure 3-4 represents two investments that are noncorrelated; sometimes they move together, and sometimes they do not. There is a diversification benefit from investing in noncorrelated assets.

Chartered Financial Analyst (CFA) An investment professional who has met competency standards in economics, securities, portfolio management, and financial accounting as determined by the Institute of Chartered Financial Analysts. Closed-End Fund A mutual fund that has a fixed number of shares, usually listed on a major stock exchange. Commodities Unprocessed goods, such as grains, metals, and minerals, traded in large amounts on a commodities exchange. Consumer Price Index (CPI) A measure of the price change in consumer goods and services. The CPI is used to track the pace of inflation. Correlation Coefficient A number between ⫺1 and 1 that measures the degree to which two variables are linearly related. Cost Basis The original cost of an investment. For tax purposes, the cost basis is subtracted from the sale price to determine any capital gain or loss. Glossary 323 Country Risk The possibility that political events (e.g., a war, national elections), financial problems (e.g., rising inflation, government default), or natural disasters (e.g., an earthquake, a poor harvest) will weaken a country’s economy and cause investments in that country to decline.

 

pages: 193 words: 47,808

The Flat White Economy by Douglas McWilliams

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

access to a mobile phone, banking crisis, Big bang: deregulation of the City of London, bonus culture, cleantech, cloud computing, computer age, correlation coefficient, Edward Glaeser, en.wikipedia.org, Erik Brynjolfsson, eurozone crisis, George Gilder, hiring and firing, income inequality, informal economy, knowledge economy, low skilled workers, Network effects, new economy, offshore financial centre, Peter Thiel, Productivity paradox, Silicon Valley, smart cities, special economic zone, Steve Jobs, working-age population

b) Growing Together: London and the UK Economy 2005 This report looks at a range of links between the London economy and those of the rest of the UK.18 It concludes that London’s growth is not at the expense of the rest of the UK, but that London and other UK regions and countries are interdependent. Table 6.1: Correlation between economic growth in London and the rest of the UK, 1983–2004 Regions and countries of Great Britain Correlation coefficient South East 0.80 East England 0.81 South West 0.64 East Midlands 0.45 West Midlands 0.73 North West 0.73 Yorkshire and Humberside 0.56 North East 0.22 Wales 0.55 Scotland 0.27 Northern Ireland 0.36 Table 6.2: Percentage change in employment, 1989–2001 Regions and countries of Great Britain % change South East 23.7 South West 21.2 East of England 18.8 Scotland 17.4 London 15.3 East Midlands 12.5 Wales 11.7 West Midlands 10.8 Yorkshire and Humberside 10.2 North West 9.9 North East 7.2 A unique feature of this analysis is research into the correlations between GVA growth in London and other regions and countries.

It is also interesting that the three emerging economies for which we have data in this sample – China, Mexico and South Korea – have much lower labour shares of income than in the advanced economies (which might be a bit of evidence to support Marx’s contention that capitalism in the long term would ultimately bid profits down to a level that is too low to permit economic growth). However, these emerging economies currently have much faster rates of economic growth. What is the evidence that higher profits boost economic growth? A crude statistical analysis for the OECD economies for which data is available shows a negative correlation coefficient of -0.31 between the labour income share average from 2000 to 2006 and the rate of economic growth from 2001 to 2008. What this says is that there is a statistically significant negative correlation between the labour income share and GDP growth. In other words, the higher the share of profits, the faster the rate of growth. Although the capital theorists differ, two undoubted heavyweights in Karl Marx and John Maynard Keynes definitely believed that the higher profits and hence investment, the faster the rate of economic growth.

 

pages: 654 words: 191,864

Thinking, Fast and Slow by Daniel Kahneman

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Atul Gawande, availability heuristic, Black Swan, Cass Sunstein, Checklist Manifesto, choice architecture, cognitive bias, complexity theory, correlation coefficient, correlation does not imply causation, Daniel Kahneman / Amos Tversky, delayed gratification, demand response, endowment effect, experimental economics, experimental subject, Exxon Valdez, feminist movement, framing effect, hindsight bias, index card, job satisfaction, John von Neumann, libertarian paternalism, loss aversion, medical residency, mental accounting, meta analysis, meta-analysis, nudge unit, pattern recognition, pre–internet, price anchoring, quantitative trading / quantitative finance, random walk, Richard Thaler, risk tolerance, Ronald Reagan, The Chicago School, The Wisdom of Crowds, transaction costs, union organizing, Walter Mischel, Yom Kippur War

Each adviser’s scoof ဆre for each year was his (most of them were men) main determinant of his year-end bonus. It was a simple matter to rank the advisers by their performance in each year and to determine whether there were persistent differences in skill among them and whether the same advisers consistently achieved better returns for their clients year after year. To answer the question, I computed correlation coefficients between the rankings in each pair of years: year 1 with year 2, year 1 with year 3, and so on up through year 7 with year 8. That yielded 28 correlation coefficients, one for each pair of years. I knew the theory and was prepared to find weak evidence of persistence of skill. Still, I was surprised to find that the average of the 28 correlations was .01. In other words, zero. The consistent correlations that would indicate differences in skill were not to be found. The results resembled what you would expect from a dice-rolling contest, not a game of skill.

If all you know about Tom is that he ranks twelfth in weight (well above average), you can infer (statistically) that he is probably older than average and also that he probably consumes more ice cream than other children. If all you know about Barbara is that she is eighty-fifth in piano (far below the average of the group), you can infer that she is likely to be young and that she is likely to practice less than most other children. The correlation coefficient between two measures, which varies between 0 and 1, is a measure of the relative weight of the factors they share. For example, we all share half our genes with each of our parents, and for traits in which environmental factors have relatively little influence, such as height, the correlation between parent and child is not far from .50. To appreciate the meaning of the correlation measure, the following are some examples of coefficients: The correlation between the size of objects measured with precision in English or in metric units is 1.

Of course they do, and the effects have been confirmed by systematic research that objectively assessed the characteristics of CEOs and their decisions, and related them to subsequent outcomes of the firm. In one study, the CEOs were characterized by the strategy of the companies they had led before their current appointment, as well as by management rules and procedures adopted after their appointment. CEOs do influence performance, but the effects are much smaller than a reading of the business press suggests. Researchers measure the strength of relationships by a correlation coefficient, which varies between 0 and 1. The coefficient was defined earlier (in relation to regression to the mean) by the extent to which two measures are determined by shared factors. A very generous estimate of the correlation between the success of the firm and the quality of its CEO might be as high as .30, indicating 30% overlap. To appreciate the significance of this number, consider the following question: Suppose you consider many pairs of firms.

 

pages: 220 words: 73,451

Democratizing innovation by Eric von Hippel

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

additive manufacturing, correlation coefficient, Debian, hacker house, informal economy, inventory management, iterative process, James Watt: steam engine, knowledge economy, meta analysis, meta-analysis, Network effects, placebo effect, principal–agent problem, Richard Stallman, software patent, transaction costs, Vickrey auction

Ogawa determined how much of the design for each was done by the user firm and how much by the manufacturer firm. Controlling for profit expectations, he found that increases in the stickiness of user information were associated with a significant increase in the amount of need-related design undertaken by the user (Kendall correlation coefficient = 0.5784, P < 0.01). Conversely he found that increased stickiness of technology-related information was associated in a significant reduction in the amount of technology design done by the user (Kendall correlation coefficients = 0.4789, P < 0.05). In other words, need-intensive tasks within product-development projects will tend to be done by users, while solutionintensive ones will tend to be done by manufacturers. Low-Cost Innovation Niches Just as there are information asymmetries between users and manufacturers as classes, there are also information asymmetries among individual user firms and individuals, and among individual manufacturers as well.

 

pages: 295 words: 66,824

A Mathematician Plays the Stock Market by John Allen Paulos

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Benoit Mandelbrot, Black-Scholes formula, Brownian motion, business climate, butterfly effect, capital asset pricing model, correlation coefficient, correlation does not imply causation, Daniel Kahneman / Amos Tversky, diversified portfolio, Donald Trump, double entry bookkeeping, Elliott wave, endowment effect, Erdős number, Eugene Fama: efficient market hypothesis, four colour theorem, George Gilder, global village, greed is good, index fund, invisible hand, Isaac Newton, John Nash: game theory, Long Term Capital Management, loss aversion, Louis Bachelier, mandelbrot fractal, margin call, mental accounting, Nash equilibrium, Network effects, passive investing, Paul Erdős, Ponzi scheme, price anchoring, Ralph Nelson Elliott, random walk, Richard Thaler, Robert Shiller, Robert Shiller, short selling, six sigma, Stephen Hawking, transaction costs, ultimatum game, Vanguard fund, Yogi Berra

Even a portfolio of stocks from the same sector will be less volatile than the individual stocks in it, while a portfolio consisting of Wal-Mart, Pfizer, General Electric, Exxon, and Citigroup, the biggest stocks in their respective sectors, will provide considerably more protection against volatility. To find the volatility of a portfolio in general, we need what is called the “covariance” (closely related to the correlation coefficient) between any pair of stocks X and Y in the portfolio. The covariance between two stocks is roughly the degree to which they vary together—the degree, that is, to which a change in one is proportional to a change in the other. Note that unlike many other contexts in which the distinction between covariance (or, more familiarly, correlation) and causation is underlined, the market generally doesn’t care much about it.

Brian auditors Aumann, Robert availability error average values compared with distribution of incomes risk as variance from averages average return compared with median return average value compared with distribution of incomes buy-sell rules and outguessing average guess risk as variance from average value averaging down Bachelier, Louis Bak, Per Barabasi, Albert-Lazló Bartiromo, Maria bear markets investor self-descriptions and shorting and distorting strategy in Benford, Frank Benford’s Law applying to corporate fraud background of frequent occurrence of numbers governed by Bernoulli, Daniel Beta (B) values causes of variations in comparing market against individual stocks or funds strengths and weaknesses of technique for finding volatility and Big Bang billiards, as example of nonlinear system binary system biorhythm theory Black, Fischer Black-Scholes option formula blackjack strategies Blackledge, Todd “blow up,” investor blue chip companies, P/E ratio of Bogle, John bonds Greenspan’s impact on bond market history of stocks outperforming will not necessarily continue to be outperformed by stocks Bonds, Barry bookkeeping. see accounting practices bottom-line investing Brock, William brokers. see stock brokers Buffett, Warren bull markets investor self-descriptions and pump and dump strategy in Butterfly Economics (Ormerod) “butterfly effect,” of nonlinear systems buy-sell rules buying on the margin. see also margin investments calendar effects call options. see also stock options covering how they work selling strategies valuation tools campaign contributions Capital Asset Pricing Model capital gains vs. dividends Central Limit Theorem CEOs arrogance of benefits in manipulating stock prices remuneration compared with that of average employee volatility due to malfeasance of chain letters Chaitin, Gregory chance. see also whim trading strategies and as undeniable factor in market chaos theory. see also nonlinear systems charity Clayman, Michelle cognitive illusions availability error confirmation bias heuristics rules of thumb for saving time mental accounts status quo bias Cohen, Abby Joseph coin flipping common knowledge accounting scandals and definition and importance to investors dynamic with private knowledge insider trading and parable illustrating private information becoming companies/corporations adjusting results to meet expectations applying Benford’s Law to corporate fraud comparing corporate and personal accounting financial health and P/E ratio of blue chips competition vs. cooperation, prisoner’s dilemma complexity changing over time horizon of sequences (mathematics) of trading strategies compound interest as basis of wealth doubling time and formulas for future value and present value and confirmation bias definition of investments reflecting stock-picking and connectedness. see also networks European market causing reaction on Wall Street interactions based on whim interactions between technical traders and value traders irrational interactions between traders Wolfram model of interactions between traders Consumer Confidence Index (CCI) contrarian investing dogs of the Dow measures of excellence and rate of return and cooperation vs. competition, prisoner’s dilemma correlation coefficient. see also statistical correlations counter-intuitive investment counterproductive behavior, psychology of covariance calculation of portfolio diversification based on portfolio volatility and stock selection and Cramer, James crowd following or not herd-like nature of price movements dart throwing, stock-picking contest in the Wall Street Journal data mining illustrated by online chatrooms moving averages and survivorship bias and trading strategies and DeBondt, Werner Deciding What’s News (Gans) decimalization reforms decision making minimizing regret selling WCOM depression of derivatives trading, Enron despair and guilt over market losses deviation from the mean. see also mean value covariance standard deviation (d) variance dice, probability and Digex discounting process, present value of future money distribution of incomes distribution of wealth dynamic of concentration UN report on diversified portfolios. see stock portfolios, diversifying dividends earnings and proposals benefitting returns from Dodd, David dogs of the Dow strategy “dominance” principle, game theory dot com IPOs, as a pyramid scheme double-bottom trend reversal “double-dip” recession double entry bookkeeping doubling time, compound interest and Dow dogs of the Dow strategy percentages of gains and losses e (exponential growth) compound interest and higher mathematics and earnings anchoring effect and complications with determination of inflating (WCOM) P/E ratio and stock valuation and East, Steven H.

 

pages: 263 words: 75,455

Quantitative Value: A Practitioner's Guide to Automating Intelligent Investment and Eliminating Behavioral Errors by Wesley R. Gray, Tobias E. Carlisle

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Andrei Shleifer, asset allocation, Atul Gawande, backtesting, Black Swan, capital asset pricing model, Checklist Manifesto, cognitive bias, compound rate of return, corporate governance, correlation coefficient, credit crunch, Daniel Kahneman / Amos Tversky, discounted cash flows, Eugene Fama: efficient market hypothesis, forensic accounting, hindsight bias, Louis Bachelier, p-value, passive investing, performance metric, quantitative hedge fund, random walk, Richard Thaler, risk-adjusted returns, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, short selling, statistical model, systematic trading, The Myth of the Rational Market, time value of money, transaction costs

Profitable Months Proportion of monthly performances that have a positive return. Rolling 5-Year Wins Proportion of rolling 5-year periods that a designated strategy beats the identified benchmarks. Rolling 5-Year Wins Proportion of rolling 10-year periods that a designated strategy beats identified benchmarks. Cumulative Drawdown Sum of the rolling 5-year period worst drawdowns for the designated strategy. Correlation Correlation coefficient for a designated strategy and the identified benchmarks, which demonstrates the extent to which a designated strategy and the identified benchmarks move together. RISK AND RETURN Table 12.2 sets out the standard statistical analyses of the Quantitative Value strategy's performance and risk profile, comparing it to the Magic Formula, the Standard & Poor's (S&P) 500 and the MW Index, the market capitalization–weighted index of the universe from which we draw the stocks in the model portfolios.

Profitable Months Proportion of monthly performances that have a positive return. Rolling 5-Year Wins Proportion of rolling 5-year periods that a designated strategy beats the identified benchmarks. Rolling 10-Year Wins Proportion of rolling 10-year periods that a designated strategy beats identified benchmarks. Cumulative Drawdown Sum of the rolling 5-year period worst drawdowns for the designated strategy. Correlation Correlation coefficient for a designated strategy and the identified benchmarks, which demonstrates the extent to which a designated strategy and the identified benchmarks move together. About the Authors Wesley R. Gray, PhD, is the founder and executive managing member of Empiritrage, LLC, an SEC-Registered Investment Advisor, and Turnkey Analyst, LLC, a firm dedicated to educating and sharing quantitative investment techniques to the general public.

 

Beginning R: The Statistical Programming Language by Mark Gardener

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

correlation coefficient, distributed generation, natural language processing, New Urbanism, p-value, statistical model

subset = group %in% “sample” If the data includes a grouping variable, the subset instruction can be used to select one or more samples from this grouping. The commands summarized in Table 6-3 enable you to carry out a range of correlation tasks. In the following sections you see a few of these options illustrated, and you can then try some correlations yourself in the activity that follows. Simple Correlation Simple correlations are between two continuous variables and you can use the cor() command to obtain a correlation coefficient like so: > count = c(9, 25, 15, 2, 14, 25, 24, 47) > speed = c(2, 3, 5, 9, 14, 24, 29, 34) > cor(count, speed) [1] 0.7237206 The default for R is to carry out the Pearson product moment, but you can specify other correlations using the method = instruction, like so: > cor(count, speed, method = 'spearman') [1] 0.5269556 This example used the Spearman rho correlation but you can also apply Kendall’s tau by specifying method = “kendall”.

. $ weight: num 115 117 120 123 126 129 132 135 139 142 ... You need to use attach() or with() commands to allow R to “read inside” the data frame and access the variables within. You could also use the $ syntax so that the command can access the variables as the following example shows: > cor(women$height, women$weight) [1] 0.9954948 In this example the cor() command has calculated the Pearson correlation coefficient between the height and weight variables contained in the women data frame. You can also use the cor() command directly on a data frame (or matrix). If you use the data frame women that you just looked at, for example, you get the following: > cor(women) height weight height 1.0000000 0.9954948 weight 0.9954948 1.0000000 Now you have a correlation matrix that shows you all combinations of the variables in the data frame.

In the Pearson correlation you are assuming that the data are normally distributed and are looking to see how close the relationship is between the variables. In regression you are taking the analysis further and assuming a mathematical, and therefore predictable, relationship between the variables. The results of regression analysis show the slope and intercept values that describe this relationship. The R squared value that you obtain from the regression is the square of the correlation coefficient from the Pearson correlation, which demonstrates the similarities between the methods. The result shows you the coefficients for the regression, that is, the intercept and the slope. To see more details you should save your regression as a named object; then you can use the summary() command like so: > fw.lm = lm(count ~ speed, data = fw) > summary(fw.lm) Call: lm(formula = count ~ speed, data = fw) Residuals: Min 1Q Median 3Q Max -13.377 -5.801 -1.542 5.051 14.371 Coefficients: Estimate Std.

 

pages: 431 words: 132,416

No One Would Listen: A True Financial Thriller by Harry Markopolos

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

backtesting, barriers to entry, Bernie Madoff, call centre, centralized clearinghouse, correlation coefficient, diversified portfolio, Emanuel Derman, Eugene Fama: efficient market hypothesis, family office, fixed income, forensic accounting, high net worth, index card, Long Term Capital Management, Louis Bachelier, offshore financial centre, Ponzi scheme, price mechanism, quantitative trading / quantitative finance, regulatory arbitrage, Renaissance Technologies, risk-adjusted returns, risk/return, rolodex, Sharpe ratio, statistical arbitrage, too big to fail, transaction costs

That meant that he knew from his order flow what stocks were going to go up, which obviously would have been extremely beneficial when he was picking stocks for his basket. We found out later that several hedge funds believed he was doing this. I created hypothetical baskets using the best-performing stocks and followed his split-strike strategy, selling the call option to generate income and buying the put option for protection. The following week I’d pick another basket. I expected the correlation coefficient—the relationship between Bernie’s returns and the movement of the entire S&P 100—legitimately to be around 50 percent, but it could have been anywhere between 30 percent and 80 percent and I would have accepted it naively. Instead Madoff was coming in at about 6 percent. Six percent! That was impossible. That number was much too low. It meant there was almost no relationship between those stocks and the entire index.

After going through my work, Dan told us that whatever Madoof, as he referred to him, was doing, he was not getting his results from the market. Pointing to the 6 percent correlation and the 45-degree return line, he said, “That doesn’t look like it came from a finance distribution. We don’t have those kinds of charts in finance.” I was right, he agreed. Madoof’s strategy description claimed his returns were market-driven, yet his correlation coefficient was only 6 percent to the market and his performance line certainly wasn’t coming from the stock market. Volatility is a natural part of the market. It moves up and down—and does it every day. Any graphic representation of the market has to reflect that. Yet Madoff’s 45-degree rise represented a market without that volatility. It wasn’t possible. Bernie Madoff was a fraud. And whatever he was actually doing, it was enough to put him in prison.

Broyhill meets Scott Franzblau and mobsters on Ponzi scheme on Ponzi scheme vs. front-running post Bernie Madoff arrest public acknowledgment of role and Rene-Thierry de la Villehuchet on reporting to SEC on reverse engineering role of sailing disaster on SEC failure Wall Street Journal warns individual investors Casey, Judy Cattle trading scam Charles, Prince Chelo, Neil: business education of careers at Rampart continued activities of early career of impact of Bernie Madoff case on information gathering leaves Rampart OPRA tapes on payment for order flow on Ponzi scheme vs. front-running post Bernie Madoff arrest public acknowledgment of role on quants on reporting to SEC reviews strategy analysis role of talks to Amit Vijayvergiya Wall Street Journal warns individual investors Cheung, Meaghan Chicago Art Museum Chicago Board of Options Exchange (CBOE) Chinese vitamin suppliers scandal Citigroup Client redemptions Clinton, Hillary Cohen, Steve Collars Commodities straddle Commodity Futures Trading Commission Congress: Chuck Schumer call to SEC Harry Markopolos testimony investigation by SEC established by SEC in hearings by Congressional Record Contacts and relationships Cook, Boyd Correlation coefficient Corruption: as business as usual drug cartels incompetence vs. municipal bonds organized crime regulatory reporting on Russian mafia vs. stupidity Wall Street crimes See also Taxpayers Against Fraud; whistleblowers Court, Andy Covered call writing program Cox, Christopher Criminal investigation CSPAN3 Cuomo, Andrew Danger, concerns about Darien Capital Management Data analysis Data collection DeBello, Nicole de la Villehuchet, Bertrand de la Villehuchet, Claudine de la Villehuchet, Rene-Thierry: on Bernie Madoff and Frank Casey and Harry Markopolos meets Frank Casey suicide Department of Justice Derivative experts Devoe, George diBartolomeo, Dan Dickens, Charles Direct accounts Discrepancies Documentation Documentation and literature Dominelli, David Donnelly, Joe Drosos, Elaine Drug cartels Due diligence Dumb equity Ebbers, Bernie Efficient markets hypothesis Electronic security Electronic trading European banks European investors Excuses for investing with Bernie Madoff Fairfield Emerald Fairfield Greenwich Group Fairfield Greenwich Sentry Fund False Claims Act cases Fax machines Federal Bureau of Investigation (FBI) Feeder funds Fielder, David Financial frauds.

 

pages: 268 words: 89,761

Unhealthy societies: the afflictions of inequality by Richard G. Wilkinson

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

attribution theory, clean water, correlation coefficient, experimental subject, full employment, fundamental attribution error, Gini coefficient, income inequality, income per capita, Indoor air pollution, invisible hand, land reform, means of production, purchasing power parity, rising living standards, upwardly mobile

If increases in GNPpc over time were simply understated, it might be thought that this would not mask a statistical relationship between health and GNPpc: the extent to which societies benefited from qualitative changes in output would be a constant function of their growth rates. If this were so, then understated growth would change the units rather than weaken the correlation between the two. It would tend to make any given increase in income appear more health effective. In technical terms: rather than weakening the correlation coefficient it would increase the size of the regression coefficient. However, it could be argued that the spread of better products does not depend simply on the expenditure which results from the few per cent of income growth. Much nearer to the truth is that, as earlier forms of goods are made obsolete and replaced in the shops by new models and lines, the whole flow of expenditure is applied to the current range of goods, including new goods and ones in which the quality has changed.

Looking at data from nine industrialised countries, Kunst and Mackenbach (1994) found that, ‘The rank order of countries in terms of income inequalities strongly corresponds to their rank order in terms of inequalities in mortality.’ Since then van Doorslaer et al. have reported that differences in self-reported illness were greatest in countries whose income differences were greatest (van Doorslaer Income distribution and health 89 et al. 1996). The relationship between measures of inequality in income and in illness was very close: across the USA and the eight European countries for which they had data, the correlation coefficient was 0.87. The methods used in each study were quite different. Kunst and Mackenbach classified people according to occupation in one of their studies and by education in another, and they concluded that occupational and educational differences in mortality were greater in countries where income differences were greater. In contrast, van Doorslaer et al. used data from surveys giving details of incomes and self-reported health for the same individuals.

 

pages: 287 words: 44,739

Guide to business modelling by John Tennent, Graham Friend, Economist Group

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

correlation coefficient, discounted cash flows, double entry bookkeeping, iterative process, purchasing power parity, RAND corporation, shareholder value, the market place, time value of money

The coefficient of determination R2 (which is calculated automatically by most spreadsheet packages) indicates how much of the variation in Y is explained by the explanatory variables. The greater the value of R2, the more the variation in the dependent variable is explained by the selected independent variables. The square root of the coefficient of determination is the product moment correlation coefficient in the case of linear regression of a straight line. The product moment correlation is a number between 1 and ⫺1. If r ⫽ 1 then there is a perfect, positive relationship between the dependent and explanatory variable. A perfect relation implies that every data point lies on a straight line. If r ⫽ ⫺1 then a perfect negative relationship exists, and if r ⫽ 0 there is no relationship. Estimating the coefficients To demonstrate a number of linear regression estimation techniques, it is necessary to develop a forecast for gross connections based on the historical data set out in Chart 10.12.

The resulting graph, the regression line and the regression equation and R2 value are shown in Chart 10.14. Chart 10.14 Regression equation for monthly gross connections against time The R2 value is very low at 0.315. This implies that only 31.5% of the variation in gross connections is explained by time, so any forecast based on this regression equation will be liable to considerable error. The correlation coefficient is the square root of R2⫽SQR(0.315)⫽0.561. Although this value is low it is possible to show, using significance testing, that time is still a significant determinant of gross connections. Regression techniques 99 This procedure is quick and simple to use. However, to develop a forecast it is necessary to use the equation of the straight line. The use of the graphical procedure does not allow this unless the formula is reproduced manually by extrapolating the coefficients by hand and entering them in a spreadsheet.

 

pages: 339 words: 112,979

Unweaving the Rainbow by Richard Dawkins

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Any sufficiently advanced technology is indistinguishable from magic, Arthur Eddington, complexity theory, correlation coefficient, David Attenborough, discovery of DNA, double helix, Douglas Engelbart, I think there is a world market for maybe five computers, Isaac Newton, Jaron Lanier, Mahatma Gandhi, music of the spheres, Necker cube, p-value, phenotype, Ralph Waldo Emerson, Richard Feynman, Richard Feynman, Ronald Reagan, Solar eclipse in 1919, Steven Pinker, Zipf's Law

Different astrologers, after all, presumably have access to the same books. Even if their verdicts are wrong, you'd think their methods would be systematic enough at least to agree in producing the same wrong verdicts! Alas, as shown in a study by G. Dean and colleagues, they don't even achieve this minimal and easy benchmark. For comparison, when different assessors judged people on their performance in structured interviews, the correlation coefficient was greater than 0.8 (a correlation coefficient of 1.0 would represent perfect agreement, –1.0 would represent perfect disagreement, 0.0 would represent complete randomness or lack of association; 0.8 is pretty good). Against this, in the same study, the reliability coefficient for astrology was a pitiable 0.1, comparable to the figure for palmistry (0.11), and indicating near total randomness. However wrong astrologers may be, you'd think that they would have got their act together to the extent of at least being consistent Apparently not.

 

pages: 394 words: 85,734

The Global Minotaur by Yanis Varoufakis, Paul Mason

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

banking crisis, Berlin Wall, Big bang: deregulation of the City of London, Bretton Woods, business climate, capital controls, Carmen Reinhart, central bank independence, collapse of Lehman Brothers, collateralized debt obligation, colonial rule, corporate governance, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, debt deflation, declining real wages, deindustrialization, eurozone crisis, financial innovation, first-past-the-post, full employment, Hyman Minsky, industrial robot, Joseph Schumpeter, Kenneth Rogoff, labour market flexibility, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, market fundamentalism, Mexican peso crisis / tequila crisis, mortgage debt, new economy, Northern Rock, paper trading, planetary scale, post-oil, price stability, quantitative easing, reserve currency, rising living standards, Ronald Reagan, special economic zone, Steve Jobs, structural adjustment programs, systematic trading, too big to fail, trickle-down economics, urban renewal, War on Poverty, Yom Kippur War

This book is not the place to enter into the proof in any detail. If interested, please consult Y. Varoufakis, J. Halevi and N. Theocarakis (2011) Modern Political Economics: Making sense of the post-2008 world, London and New York: Routledge. 11. In more technical language, the formulae used to assemble the CDOs assumed that the correlation coefficient between the probability of default across a CDO’s different tranches or slices was constant, small and knowable. 12. Doubt about the constancy of the correlation coefficient (see previous footnote) would have cost them their jobs, particularly as their supervisors did not really understand the formula but were receiving huge bonuses while it was being used. 13. See George Soros (2009) The Crash of 2008 and What It Means: The new paradigm for financial markets, New York: Public Affairs.

 

pages: 353 words: 88,376

The Investopedia Guide to Wall Speak: The Terms You Need to Know to Talk Like Cramer, Think Like Soros, and Buy Like Buffett by Jack (edited By) Guinan

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, asset allocation, asset-backed security, Brownian motion, business process, capital asset pricing model, clean water, collateralized debt obligation, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, discounted cash flows, diversification, diversified portfolio, dividend-yielding stocks, equity premium, fixed income, implied volatility, index fund, interest rate swap, inventory management, London Interbank Offered Rate, margin call, market fundamentalism, mortgage debt, passive investing, performance metric, risk tolerance, risk-adjusted returns, risk/return, shareholder value, Sharpe ratio, short selling, statistical model, time value of money, transaction costs, yield curve, zero-coupon bond

Related Terms: • Bond • Municipal Bond • Yield to maturity—YTM • Debt Financing • Yield Correlation What Does Correlation Mean? In the investment world, correlation is a statistical measure of how two securities move in relation to each other. Correlations are used in advanced portfolio management. Investopedia explains Correlation Correlation is expressed as the correlation coefficient, which ranges between –1 and +1. Perfect positive correlation (a correlation coefficient of +1) means that as one security moves up or down, the other security will move lockstep in the same direction. Perfect negative correlation means that when one security moves in one direction, the other security will move by an equal amount in the opposite direction. If the correlation is 0, the movements of the securities are said to have no correlation; they are completely random.

 

pages: 397 words: 109,631

Mindware: Tools for Smart Thinking by Richard E. Nisbett

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

affirmative action, Albert Einstein, availability heuristic, big-box store, Cass Sunstein, choice architecture, cognitive dissonance, correlation coefficient, correlation does not imply causation, cosmological constant, Daniel Kahneman / Amos Tversky, dark matter, endowment effect, experimental subject, feminist movement, fundamental attribution error, glass ceiling, Henri Poincaré, Isaac Newton, job satisfaction, lake wobegon effect, libertarian paternalism, loss aversion, low skilled workers, Menlo Park, meta analysis, meta-analysis, quantitative easing, Richard Thaler, Ronald Reagan, Socratic dialogue, Steve Jobs, Steven Levy, the scientific method, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions, William of Occam, Zipcar

A correlation of .7 corresponds to the association between height and weight—substantial but still not perfect. A correlation of .8 corresponds to the degree of association you find between scores on the math portion of the Scholastic Aptitude Test (SAT) at one testing and scores on that test a year later—quite high but still plenty of room for difference between the two scores on average. Correlation Does Not Establish Causality Correlation coefficients are one step in assessing causal relations. If there is no correlation between variable A and variable B, there (probably) is no causal relation between A and B. (An exception would be when there is a third variable C that masks the correlation between A and B when there is in fact a causal relation between A and B.) If there is a correlation between variable A and variable B, this doesn’t establish that variation in A causes variation in B.

Coding Is the Key to Thinking Statistically I’m going to ask you some questions concerning your beliefs about what you think the correlation is between a number of pairs of variables. The way I’ll do that is to ask you how likely it is that A would be greater than B on one occasion given that A was greater than B on another occasion. Your answers in probability terms can be converted to correlation coefficients by a mathematical formula. Note that if you say “50 percent” for a question below, you’re saying that you think there’s no relationship between behavior on one occasion and behavior on another. If you say “90 percent,” you’re saying that there is an extremely strong relationship between behavior on one occasion and behavior on another. For the first question below about spelling ability, if you think that there is no consistency between spelling performance on one occasion and spelling performance on another occasion, you would say “50 percent.”

 

pages: 743 words: 189,512

The Big Fat Surprise: Why Butter, Meat and Cheese Belong in a Healthy Diet by Nina Teicholz

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, correlation coefficient, correlation does not imply causation, Gary Taubes, Indoor air pollution, meta analysis, meta-analysis, phenotype, placebo effect, randomized controlled trial, Robert Gordon, the scientific method, Upton Sinclair

In 1999, when the Seven Countries study’s lead Italian researcher, Alessandro Menotti, went back twenty-five years later and looked at data from the study’s 12,770 subjects, he noticed an interesting fact: the category of foods that best correlated with coronary mortality was sweets. By “sweets,” he meant sugar products and pastries, which had a correlation coefficient with coronary mortality of 0.821 (a perfect correlation is 1.0). Possibly this number would have been higher had Menotti included chocolate, ice cream, and soft drinks in his “sweets” category, but those fell under a different category and, he explained, would have been “too troublesome” to recode. By contrast, “animal food” (butter, meat, eggs, margarine, lard, milk, and cheese) had a correlation coefficient of 0.798, and this number likely would have been lower had Menotti excluded margarine. (Margarine is usually made from vegetable fats, but researchers at the time tended to lump it in with animal foods because it looked so much like butter.)

“a remarkable and troublesome omission”: Katerina Sarri and Anthony Kafatos, letter to the editor, “The Seven Countries Study in Crete: Olive Oil, Mediterranean Diet or Fasting?” Public Health Nutrition 8, no. 6 (2005): 666. “we should not” . . . “the ideal thing all the time”: Daan Kromhout, interview with author, October 4, 2007. he knew it would go unnoticed: Keys, Aravanis, and Sdrin, “Diets of Middle-Aged Men in Two Rural Areas of Greece,” 577. category of foods . . . which had a correlation coefficient: Alessandro Menotti et al., “Food Intake Patterns and 25-Year Mortality from Coronary Heart Disease: Cross-Cultural Correlations in the Seven Countries Study,” European Journal of Epidemiology 15, no. 6 (1999): 507–515. “too troublesome” to recode: Alessandro Menotti, interview with author, July 24, 2008. “Keys was very opposed to the sugar idea”: Kromhout, interview. “He was so convinced that fatty acids” . . .

 

The Age of Turbulence: Adventures in a New World (Hardback) - Common by Alan Greenspan

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

air freight, airline deregulation, Albert Einstein, asset-backed security, bank run, Berlin Wall, Bretton Woods, business process, call centre, capital controls, central bank independence, collateralized debt obligation, collective bargaining, conceptual framework, Corn Laws, corporate governance, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, cuban missile crisis, currency peg, Deng Xiaoping, Dissolution of the Soviet Union, Doha Development Round, double entry bookkeeping, equity premium, everywhere but in the productivity statistics, Fall of the Berlin Wall, fiat currency, financial innovation, financial intermediation, full employment, Gini coefficient, Hernando de Soto, income inequality, income per capita, invisible hand, Joseph Schumpeter, labor-force participation, labour market flexibility, laissez-faire capitalism, land reform, Long Term Capital Management, Mahatma Gandhi, manufacturing employment, market bubble, means of production, Mikhail Gorbachev, moral hazard, mortgage debt, new economy, North Sea oil, oil shock, open economy, pets.com, Potemkin village, price mechanism, price stability, Productivity paradox, profit maximization, purchasing power parity, random walk, reserve currency, risk tolerance, Ronald Reagan, shareholder value, short selling, Silicon Valley, special economic zone, the payments system, The Wealth of Nations by Adam Smith, Thorstein Veblen, too big to fail, total factor productivity, trade liberalization, trade route, transaction costs, transcontinental railway, urban renewal, working-age population, Y2K

THE M O D E S OF C A P I T A L I S M at the same time, Germany ranks among the highest in terms of the freedom of its people to open and close businesses, property-rights protection, and the overall rule of law. France (number forty-five) and Italy (number sixty) have profiles that are similarly mixed. The ultimate test of the usefulness of such a scoring process is whether it correlates with economic performance. And it does. The correlation coefficient of 157 countries between their "Economic Freedom Score" and the log of their per capita incomes is 0.65, impressive for such a motley body of data.* Thus, we are left with a critical question: Granted that open competitive markets foster economic growth, is there an optimum trade-off between economic performance and the competitive stress it imposes on the one hand, and the civility that, for example, the continental Europeans and many others espouse?

Accordingly the weighted correlation between national saving rates and domestic investment rates for countries or regions representing virtually all of the world's gross domestic product, a measure of the degree of home bias, declined from a coefficient of around 0.95 in 1992, where it had hovered since 1970, to an estimated 0.74 in 2005. (If in every country saving equaled investment—that is, if there were 100 percent home bias—the correlation coefficient would be 1.0. On the other hand, if there were no home bias, and the amount of domestic saving bore no relationship to the amount and location of investments, the coefficient would be 0.)* Only in the past decade has expanding trade been associated with the emergence of ever-larger U.S. trade and current account deficits, matched by a corresponding widening of the aggregate external surpluses of many of our trading partners, most recently including China.

The piling up of dollar claims against U.S. residents is already leading to concerns about "concentration risk"—the too-many-eggs-in-one-basket worry that could prompt foreign holders to exchange dollars for other currencies, even when the dollar investments yield more. Although foreign investors *The persistent divergence subsequent to t h e creation of t h e euro of m a n y prices of identical goods a m o n g m e m b e r countries of t h e euro area is analyzed in John H. Rogers (2002). For t h e case of U.S. and Canadian prices, see Charles Engel and John H. Rogers ( 1 9 9 6 ) . t T h e correlation coefficient measures of h o m e bias have flattened o u t since 2 0 0 0 . So have t h e measures of dispersion. This is consistent w i t h t h e United States' accounting for a rising share of deficits. 361 More ebooks visit: http://www.ccebook.cn ccebook-orginal english ebooks This file was collected by ccebook.cn form the internet, the author keeps the copyright. THE AGE OF T U R B U L E N C E have not yet significantly slowed their financing of U.S. capital investments, since early 2002 the value of the dollar relative to other currencies has declined, as has the share of dollar assets in some measures of global cross-border portfolios.* If the current disturbing drift toward protectionism is contained and markets remain sufficiently flexible, changing terms of trade, interest rates, asset prices, and exchange rates should cause U.S. saving to rise relative to domestic investment.

 

pages: 823 words: 220,581

Debunking Economics - Revised, Expanded and Integrated Edition: The Naked Emperor Dethroned? by Steve Keen

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

accounting loophole / creative accounting, banking crisis, banks create money, barriers to entry, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black Swan, Bonfire of the Vanities, butterfly effect, capital asset pricing model, cellular automata, central bank independence, citizen journalism, clockwork universe, collective bargaining, complexity theory, correlation coefficient, credit crunch, David Ricardo: comparative advantage, debt deflation, diversification, double entry bookkeeping, en.wikipedia.org, Eugene Fama: efficient market hypothesis, experimental subject, Financial Instability Hypothesis, Fractional reserve banking, full employment, Henri Poincaré, housing crisis, Hyman Minsky, income inequality, invisible hand, iterative process, John von Neumann, laissez-faire capitalism, liquidity trap, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, market clearing, market microstructure, means of production, minimum wage unemployment, open economy, place-making, Ponzi scheme, profit maximization, quantitative easing, RAND corporation, random walk, risk tolerance, risk/return, Robert Shiller, Robert Shiller, Ronald Coase, Schrödinger's Cat, scientific mainstream, seigniorage, six sigma, South Sea Bubble, stochastic process, The Great Moderation, The Wealth of Nations by Adam Smith, Thorstein Veblen, time value of money, total factor productivity, tulip mania, wage slave

Just as significantly, the unemployment rate stabilized when the decline in debt-financed demand turned around. Though the huge fiscal and monetary stimulus packages also played a role, changes in debt-financed demand dominate economic performance. One statistical indicator of the importance of debt dynamics in causing both the Great Depression and the Great Recession and the booms that preceded them is the correlation coefficient between changes in debt and the level of unemployment. Over the whole period from 1921 till 1940, the correlation coefficient was minus 0.83, while over the period from 1990 till 2011, it was minus 0.91 (versus the maximum value it could have taken of minus one). A correlation of that scale, over time periods of that length, when economic circumstances varied from bust to boom and back again, is staggering. 13.31 Debt-financed demand and unemployment, 1990–2011 The Credit Impulse confirms the dominant role of private debt.

In Sharpe’s words: In order to derive conditions for equilibrium in the capital market we invoke two assumptions. First, we assume a common pure rate of interest, with all investors able to borrow or lend funds on equal terms. Second, we assume homogeneity of investor expectations: investors are assumed to agree on the prospects of various investments – the expected values, standard deviations and correlation coefficients described in Part II. Needless to say, these are highly restrictive and undoubtedly unrealistic assumptions. However, since the proper test of a theory is not the realism of its assumptions but the acceptability of its implications, and since these assumptions imply equilibrium conditions which form a major part of classical financial doctrine, it is far from clear that this formulation should be rejected – especially in view of the dearth of alternative models leading to similar results.

 

Trend Commandments: Trading for Exceptional Returns by Michael W. Covel

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, Bernie Madoff, Black Swan, commodity trading advisor, correlation coefficient, delayed gratification, diversified portfolio, en.wikipedia.org, Eugene Fama: efficient market hypothesis, family office, full employment, Lao Tzu, Long Term Capital Management, market bubble, market microstructure, Mikhail Gorbachev, moral hazard, Nick Leeson, oil shock, Ponzi scheme, prediction markets, quantitative trading / quantitative finance, random walk, Sharpe ratio, systematic trading, the scientific method, transaction costs, tulip mania, upwardly mobile, Y2K

Consider this nearly 40-year track record of trend trading wealth building: Chart 1: Bill Dunn Unit Value Log Scale DUNN Composite Performance: $500,000 $570,490 10 Drawdowns Greater Than -25% October 1974 through January 2011 Average Major Drawdown: 38% $100,000 -40% Compound Annual Rate of Return -27% -60% -34% $62,375 DUNN Composite: 19.09% S&P 500 (Total return) : 11.85% -29% -34% -45% -35% -51% -43% $10,000 -30% Correlation Coefficient -0.05 Past Performance is Not Necessarily Indicative of Future Results -28% -45% Includes Notional and Proprietary Funds All Net of Pro Forma Fees and Expenses -52% ' 70 ' 71 ' 72 ' 73 ' 74 ' 75 ' 76 ' 77 ' 78 ' 79 ' 80 ' 81 ' 82 ' 83 ' 84 ' 85 ' 86 ' 87 ' 88 ' 89 ' 90 ' 91 ' 92 ' 93 ' 94 ' 95 ' 96 ' 97 ' 98 ' 99 ' 00 ' 01 ' 02 ' 03 ' 04 ' 05 ' 06 ' 07 ' 08 ' 09 ' 10 ' 11 $1,000 That picture is worth a thousand words.

 

pages: 204 words: 58,565

Keeping Up With the Quants: Your Guide to Understanding and Using Analytics by Thomas H. Davenport, Jinho Kim

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Black-Scholes formula, business intelligence, business process, call centre, computer age, correlation coefficient, correlation does not imply causation, Credit Default Swap, en.wikipedia.org, feminist movement, Florence Nightingale: pie chart, forensic accounting, global supply chain, Hans Rosling, hypertext link, invention of the telescope, inventory management, Jeff Bezos, margin call, Moneyball by Michael Lewis explains big data, Netflix Prize, p-value, performance metric, publish or perish, quantitative hedge fund, random walk, Renaissance Technologies, Robert Shiller, Robert Shiller, self-driving car, sentiment analysis, six sigma, Skype, statistical model, supply-chain management, text mining, the scientific method

., records in a database) into groups (called clusters) so that objects within clusters are similar in some manner while objects across clusters are dissimilar to each other. Clustering is a main task of exploratory data mining, and a common technique for statistical data analysis used in many fields. Correlation: The extent to which two or more variables are related to one another. The degree of relatedness is expressed as a correlation coefficient, which ranges from −1.0 to +1.0. Correlation = +1 (Perfect positive correlation, meaning that both variables always move in the same direction together) Correlation = 0 (No relationship between the variables) Correlation = −1 (Perfect negative correlation, meaning that as one variable goes up, the other always trends downward) Correlation does not imply causation. Correlation is a necessary but insufficient condition for casual conclusions.

 

pages: 306 words: 78,893

After the New Economy: The Binge . . . And the Hangover That Won't Go Away by Doug Henwood

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

accounting loophole / creative accounting, affirmative action, Asian financial crisis, barriers to entry, borderless world, Branko Milanovic, Bretton Woods, capital controls, corporate governance, correlation coefficient, credit crunch, deindustrialization, dematerialisation, deskilling, ending welfare as we know it, feminist movement, full employment, gender pay gap, George Gilder, glass ceiling, Gordon Gekko, greed is good, half of the world's population has never made a phone call, income inequality, indoor plumbing, Internet Archive, job satisfaction, joint-stock company, Kevin Kelly, labor-force participation, liquidationism / Banker’s doctrine / the Treasury view, manufacturing employment, means of production, minimum wage unemployment, Naomi Klein, new economy, occupational segregation, pets.com, profit maximization, purchasing power parity, race to the bottom, Ralph Nader, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, Silicon Valley, Simon Kuznets, statistical model, structural adjustment programs, Telecommunications Act of 1996, telemarketer, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, total factor productivity, union organizing, War on Poverty, women in the workforce, working poor, Y2K

That estimate was arrived at by dividing the Fed's X4HTK2 output index, part of the industrial production series, by an estimate of hours worked. Hours worked was estimated by multiplying the BL5 figure for total employment by average weekly hours in each of the component industries, adding them together, and subtracting the result from a similar estimate of total manufacturing hours worked. While not exact, the approximation is pretty good; an estimate of total manufacturing productivity using this technique had a correlation coefficient of .86 with the official index. See text for discussion. the way we live and work, sometimes to the good, sometimes not. Do they make a 28% annual contribution to the growth of human happiness? Closely related to the productivity argument is a claim about innovation: that we Hve in a time of new product development without any his- torical precedent. This is a remarkably amnesiac claim.

 

pages: 209 words: 13,138

Empirical Market Microstructure: The Institutions, Economics and Econometrics of Securities Trading by Joel Hasbrouck

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

barriers to entry, conceptual framework, correlation coefficient, discrete time, disintermediation, distributed generation, experimental economics, financial intermediation, index arbitrage, interest rate swap, inventory management, market clearing, market design, market friction, market microstructure, martingale, price discovery process, price discrimination, quantitative trading / quantitative finance, random walk, Richard Thaler, second-price auction, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, two-sided market, ultimatum game

Equating the certainty equivalents of being hit and not being hit gives: B1 = µ1 − ασ1 [(1 + 2n1 ) σ1 + 2ρn2 σ2 ] , 2 (11.3) where ρ = Corr(X1 , X2 ). Thus, the dealer will bid less aggressively if the securities are positively correlated. This conforms to the usual intuition that positive correlation aggravates total portfolio risk. On the other hand, if we assume (as before) that the dealer is starting at his optimum, then B1 = P1 − ασ12 /2. Surprisingly, this is the same result as in the one-security case. In particular, the correlation coefficient drops out. This is a consequence of offsetting effects. The optimal n1 and n2 in equation (11.3) depends negatively on ρ, leaving the bracketed term invariant to changes in ρ. (Although this offset is a general feature of the problem, the complete disappearance of ρ in the final expression for the bid is a consequence of CARA utility.) 11.3 Empirical Analysis of Dealer Inventories 11.3.1 A First Look at the Data Changes in the dealer’s position reveal the dealer’s trades, which may disclose strategy and profitability.

 

pages: 935 words: 267,358

Capital in the Twenty-First Century by Thomas Piketty

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

accounting loophole / creative accounting, Asian financial crisis, banking crisis, banks create money, Berlin Wall, Branko Milanovic, British Empire, capital controls, Capital in the Twenty-First Century by Thomas Piketty, carbon footprint, central bank independence, collapse of Lehman Brothers, conceptual framework, corporate governance, correlation coefficient, David Ricardo: comparative advantage, demographic transition, distributed generation, diversification, diversified portfolio, European colonialism, eurozone crisis, Fall of the Berlin Wall, financial intermediation, full employment, German hyperinflation, Gini coefficient, high net worth, Honoré de Balzac, immigration reform, income inequality, income per capita, index card, inflation targeting, informal economy, invention of the steam engine, invisible hand, joint-stock company, Joseph Schumpeter, market bubble, means of production, mortgage debt, mortgage tax deduction, new economy, New Urbanism, offshore financial centre, open economy, pension reform, purchasing power parity, race to the bottom, randomized controlled trial, refrigerator car, regulatory arbitrage, rent control, rent-seeking, Robert Gordon, Ronald Reagan, Simon Kuznets, sovereign wealth fund, Steve Jobs, The Nature of the Firm, the payments system, The Wealth of Nations by Adam Smith, Thomas Malthus, Thorstein Veblen, trade liberalization, very high income, We are the 99%

But this is a different issue from skill and earned income mobility, which is what is of interest here and is the focal point of these measurements of intergenerational mobility. The data used in these works do not allow us to isolate mobility of capital income. 28. The correlation coefficient ranges from 0.2–0.3 in Sweden and Finland to 0.5–0.6 in the United States. Britain (0.4–0.5) is closer to the United States but not so far from Germany or France (0.4). Concerning international comparisons of intergenerational correlation coefficients of earned income (which are also confirmed by twin studies), see the work of Markus Jantti. See the online technical appendix. 29. The cost of an undergraduate year at Harvard in 2012–2013 was $54,000, including room and board and various other fees (tuition in the strict sense was $38,000).

According to the available data, the answer seems to be no: the intergenerational correlation of education and earned incomes, which measures the reproduction of the skill hierarchy over time, shows no trend toward greater mobility over the long run, and in recent years mobility may even have decreased.26 Note, however, that it is much more difficult to measure mobility across generations than it is to measure inequality at a given point in time, and the sources available for estimating the historical evolution of mobility are highly imperfect.27 The most firmly established result in this area of research is that intergenerational reproduction is lowest in the Nordic countries and highest in the United States (with a correlation coefficient two-thirds higher than in Sweden). France, Germany, and Britain occupy a middle ground, less mobile than northern Europe but more mobile than the United States.28 These findings stand in sharp contrast to the belief in “American exceptionalism” that once dominated US sociology, according to which social mobility in the United States was exceptionally high compared with the class-bound societies of Europe.

 

pages: 701 words: 199,010

The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal by Ludwig B. Chincarini

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

affirmative action, asset-backed security, automated trading system, bank run, banking crisis, Basel III, Bernie Madoff, Black-Scholes formula, buttonwood tree, Carmen Reinhart, central bank independence, collapse of Lehman Brothers, collateralized debt obligation, collective bargaining, corporate governance, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discounted cash flows, diversification, diversified portfolio, family office, financial innovation, financial intermediation, fixed income, Flash crash, full employment, Gini coefficient, high net worth, hindsight bias, housing crisis, implied volatility, income inequality, interest rate derivative, interest rate swap, labour mobility, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, low skilled workers, margin call, market design, market fundamentalism, merger arbitrage, Mexican peso crisis / tequila crisis, moral hazard, mortgage debt, Northern Rock, Occupy movement, oil shock, price stability, quantitative easing, quantitative hedge fund, quantitative trading / quantitative finance, Ralph Waldo Emerson, regulatory arbitrage, Renaissance Technologies, risk tolerance, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, Sharpe ratio, short selling, sovereign wealth fund, speech recognition, statistical arbitrage, statistical model, systematic trading, The Great Moderation, too big to fail, transaction costs, value at risk, yield curve, zero-coupon bond

., ) and you’ll have a riskless portfolio with a positive return. Unfortunately, it is impossible to have an average correlation of −1 with more than two strategies in a portfolio. As you add more strategies, however, the conditions necessary for a very low-risk portfolio grow less stringent. Consider a portfolio of n trading strategies. Assuming a similar variance for each strategy, , and that any pair of strategies has the same correlation coefficient, . If all strategies are equally weighted (that is, ) and individual strategy returns are positive, the portfolio variance is: (A.4) Construct a portfolio with a large number of positions that have an average correlation of zero, and the portfolio risk decreases toward zero. For the purposes of this analysis, assume that this was LTCM’s driving concept. It may seem that this is too simple an explanation of how LTCM operated.

A simple value-at-risk (VaR) formula for the above structure is: (A.9) where represents the expected return of the levered portfolio, represents the standard deviation of the levered portfolio, Vt represents the initial portfolio value, and k represents the confidence level critical value, assuming a normal distribution (i.e., k = 1.96 for a 97.5% confidence interval).9 Table A.1 presents the potential VaR calculations at a 99% confidence level for a normal distribution (k = 2.33) and a capital base of $4.8B (the amount that LTCM had at the beginning of 1998). The VaR numbers are presented as monthly numbers. Given the correlation coefficient, this represents what might have been expected to occur in any given month at LTCM. TABLE A.1 Sensitivity of VaR to Strategy Correlations Table A.1 shows that an unlevered fund’s standard deviation was 0.0951% per month and 0.6723% per month with a correlation of 0 and 1 respectively. The equivalent annualized volatility was 0.3294% and 2.3290% respectively. Generally this illustrates the portfolio that LTCM sought: a high Sharpe ratio and very low unlevered risk.

 

pages: 317 words: 106,130

The New Science of Asset Allocation: Risk Management in a Multi-Asset World by Thomas Schneeweis, Garry B. Crowder, Hossein Kazemi

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, backtesting, Bernie Madoff, Black Swan, capital asset pricing model, collateralized debt obligation, commodity trading advisor, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, diversified portfolio, fixed income, high net worth, implied volatility, index fund, interest rate swap, invisible hand, market microstructure, merger arbitrage, moral hazard, passive investing, Richard Feynman, Richard Feynman, Richard Feynman: Challenger O-ring, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, short selling, statistical model, systematic trading, technology bubble, the market place, Thomas Kuhn: the structure of scientific revolutions, transaction costs, value at risk, yield curve

The break-even (Rc) and excess break-even rate of return (EBK) is often computed as follows: ⎛ E (Rp ) − Rf ⎞ E ( Rc ) = ⎜ ⎟⎠ ( ρcp ) σ c + Rf ⎝ σp ⎤ ⎡⎛ E (Rp ) − Rf ⎞ EBK = Rc − ⎢⎜ ⎟⎠ ( ρcp ) σ c + Rf ⎥ ⎝ σ p ⎦ ⎣ where E(Rc) = Break-even rate of return required for the asset to improve the Sharpe Ratio of alternative index p Rc = Rate of return on asset c Rf = Riskless rate of return E(Rp) = Rate of return on alternative index p ρcp = Correlation coefficient between asset c and alternative benchmark p σc = Standard deviation of asset c σp = Standard deviation of alternative index p First, it is important to realize that the above expression is based on the assumption that only mean and variance matter in evaluating the risk-return profile of a portfolio. Second, one must be familiar with the potential prob- 44 THE NEW SCIENCE OF ASSET ALLOCATION lems that can arise in using this expression.

 

pages: 345 words: 87,745

The Power of Passive Investing: More Wealth With Less Work by Richard A. Ferri

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

asset allocation, backtesting, Bernie Madoff, capital asset pricing model, cognitive dissonance, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, endowment effect, estate planning, Eugene Fama: efficient market hypothesis, fixed income, implied volatility, index fund, Long Term Capital Management, passive investing, Ponzi scheme, prediction markets, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, too big to fail, transaction costs, Vanguard fund, yield curve

chartered financial analyst (CFA) An investment professional who has met competency standards in economics, securities, portfolio management, and financial accounting as determined by the Institute of Chartered Financial Analysts. closed-end fund A mutual fund that has a fixed number of shares, usually listed on a major stock exchange. commodities Unprocessed goods, such as grains, metals, and minerals, traded in large amounts on a commodities exchange. consumer price index (CPI) A measure of the price change in consumer goods and services. The CPI is used to track the pace of inflation. correlation coefficient A number between −1 and 1 that measures the degree to which two variables are linearly related. cost basis The original cost of an investment. For tax purposes, the cost basis is subtracted from the sale price to determine any capital gain or loss. country risk The possibility that political events (e.g., a war, national elections); financial problems (e.g., rising inflation, government default); or natural disasters (e.g., an earthquake, a poor harvest) will weaken a country’s economy and cause investments in that country to decline.

 

pages: 389 words: 109,207

Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, anti-communist, asset allocation, Benoit Mandelbrot, Black-Scholes formula, Brownian motion, buy low sell high, capital asset pricing model, Claude Shannon: information theory, computer age, correlation coefficient, diversified portfolio, en.wikipedia.org, Eugene Fama: efficient market hypothesis, high net worth, index fund, interest rate swap, Isaac Newton, Johann Wolfgang von Goethe, John von Neumann, Long Term Capital Management, Louis Bachelier, margin call, market bubble, market fundamentalism, Marshall McLuhan, New Journalism, Norbert Wiener, offshore financial centre, publish or perish, quantitative trading / quantitative finance, random walk, risk tolerance, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, short selling, speech recognition, statistical arbitrage, The Predators' Ball, The Wealth of Nations by Adam Smith, transaction costs, traveling salesman, value at risk, zero-coupon bond

For reasons mathematical, psychological, and sociological, it is a good idea to use a money management system that is relatively forgiving of estimation errors. Fat Tails and Leverage Suppose you’re betting on a simultaneous toss of coins believed to have a 55 percent chance of coming up heads, as depicted on the previous page. But on this toss, only 45 percent of the coins are heads. Call it a “fat tail” event, or a failure of correlation coefficients, or a big dumb mistake in somebody’s computer model. What then? The Kelly bettor cannot be ruined in a single toss. (He is prepared to survive the worst-case scenario, of zero heads.) In this situation, with many coins, the Kelly bettor will stake just short of his full bankroll. He wins only 45 percent of the wagers, doubling the amount bet on each coin that comes up heads. The Kelly bettor therefore preserves at least 90 percent of his bankroll.

 

pages: 303 words: 67,891

Advances in Artificial General Intelligence: Concepts, Architectures and Algorithms: Proceedings of the Agi Workshop 2006 by Ben Goertzel, Pei Wang

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

AI winter, artificial general intelligence, bioinformatics, brain emulation, combinatorial explosion, complexity theory, computer vision, conceptual framework, correlation coefficient, epigenetics, friendly AI, information retrieval, Isaac Newton, John Conway, Loebner Prize, Menlo Park, natural language processing, Occam's razor, p-value, pattern recognition, performance metric, Ray Kurzweil, Rodney Brooks, semantic web, statistical model, strong AI, theory of mind, traveling salesman, Turing machine, Turing test, Von Neumann architecture, Y2K

Analysis by Comparison with Psychometric Data In order to further validate our findings, we compared our principal dimensions found in the English core against the dimensions of the ANEW dataset: pleasure5, arousal and dominance. The ANEW list contains 1,034 words, 479 of which were found in the English core. The scatter plot of our PC #1 versus the first dimension of ANEW, which is the mean value of pleasure, is represented in Figure 7. The plot shows strong correlation, with similar bimodal distributions in both PC #1 and the ANEW-pleasure dimensions. Pearson correlation coefficient r = 0.70. Figure 7. Scatter plot demonstrating strong correlation of PC #1 with the first dimension of ANEW: pleasure. The dashed line is a linear fit. The two clusters (“positive” and “negative”) are separated in each dimension. How can we match PCs with ANEW dimensions? Our correlation analysis shows that PC #1 is the best match (i.e., most highly correlated among all PCs) for ANEWpleasure, and vice versa (r = 0.70, p = 10-70).

 

pages: 385 words: 128,358

Inside the House of Money: Top Hedge Fund Traders on Profiting in a Global Market by Steven Drobny

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, asset allocation, Berlin Wall, Bonfire of the Vanities, Bretton Woods, buy low sell high, capital controls, central bank independence, Chance favours the prepared mind, commodity trading advisor, corporate governance, correlation coefficient, Credit Default Swap, diversification, diversified portfolio, family office, fixed income, glass ceiling, high batting average, implied volatility, index fund, inflation targeting, interest rate derivative, inventory management, Long Term Capital Management, margin call, market bubble, Maui Hawaii, Mexican peso crisis / tequila crisis, moral hazard, new economy, Nick Leeson, oil shale / tar sands, oil shock, out of africa, paper trading, Peter Thiel, price anchoring, purchasing power parity, reserve currency, risk tolerance, risk-adjusted returns, risk/return, rolodex, Sharpe ratio, short selling, Silicon Valley, The Wisdom of Crowds, too big to fail, transaction costs, value at risk, yield curve, zero-coupon bond

LTCM was at the forefront of investing at the time and offers insight into some of the failings of risk management systems. Risk management systems based on historical prices are one way to look at risk but are in no way faultless. Financial market history is filled with theoretically low probability or fat tail events. In LTCM’s case, its risk systems calculated roughly a 1-in-6-billion chance of a major blowup. Ironically, however, one correlation the brilliant minds of LTCM neglected to consider was the correlation coefficient of positions that were linked for no other reason than the fact 2.50 AAA Spread BAA Spread Yield (%) 2.00 Spreads Blow Out 1.50 1.00 0.50 4 Ju l-9 4 Oc t-9 4 Ja n95 Ap r-9 5 Ju l-9 5 Oc t-9 5 Ja n96 Ap r-9 6 Ju l-9 6 Oc t-9 6 Ja n97 Ap r-9 7 Ju l-9 7 Oc t-9 7 Ja n98 Ap r-9 8 Ju l-9 8 Oc t-9 8 Ja n99 Ap r-9 9 Ju l-9 9 Oc t-9 9 -9 Ap r Ja n- 94 0.00 FIGURE 2.13 Corporate Spreads to Treasuries, 1994–1999 Source: Bloomberg. 26 INSIDE THE HOUSE OF MONEY GREENSPAN ON LTCM How much dependence should be placed on financial modeling, which, for all its sophistication, can get too far ahead of human judgment?

 

How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Albert Einstein, algorithmic trading, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, asset allocation, asset-backed security, backtesting, bank run, banking crisis, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business process, buy low sell high, capital asset pricing model, centre right, collateralized debt obligation, corporate governance, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, discounted cash flows, disintermediation, diversification, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, full employment, George Akerlof, Gordon Gekko, hiring and firing, implied volatility, index fund, interest rate derivative, interest rate swap, John von Neumann, linear programming, Loma Prieta earthquake, Long Term Capital Management, margin call, market friction, market microstructure, martingale, merger arbitrage, Nick Leeson, P = NP, pattern recognition, pensions crisis, performance metric, prediction markets, profit maximization, purchasing power parity, quantitative trading / quantitative finance, QWERTY keyboard, RAND corporation, random walk, Ray Kurzweil, Richard Feynman, Richard Feynman, Richard Stallman, risk-adjusted returns, risk/return, shareholder value, Sharpe ratio, short selling, Silicon Valley, six sigma, sorting algorithm, statistical arbitrage, statistical model, stem cell, Steven Levy, stochastic process, systematic trading, technology bubble, The Great Moderation, the scientific method, too big to fail, trade route, transaction costs, transfer pricing, value at risk, volatility smile, Wiener process, yield curve, young professional

They termed this effect money illusion. Our paper examined a corollary of their result: In the presence of money illusion, the correlation between stock and bond returns will be abnormally high during periods of high inflation. For the United States, it was shown that inflation had exactly this effect on stock/bond correlations during the postwar era. As a result, asset allocation strategies that are based on the high correlation coefficients calculated using data from the 1970s and early 1980s can be expected to generate inefficient portfolios in regimes of low inflation. JWPR007-Lindsey 82 May 7, 2007 16:44 h ow i b e cam e quant Ray LeClair and I wrote a paper, “Revenue Recognition Certificates: A New Security,” in which we explored the concept and potential benefits of a new type of security.14 This security provides returns as a specified function of a firm’s sales or gross revenues over a defined period of time, say 10 years, and then expires worthless.

 

pages: 1,351 words: 385,579

The Better Angels of Our Nature: Why Violence Has Declined by Steven Pinker

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

1960s counterculture, affirmative action, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, availability heuristic, Berlin Wall, Bonfire of the Vanities, British Empire, Broken windows theory, California gold rush, Cass Sunstein, citation needed, clean water, cognitive dissonance, colonial rule, Columbine, computer age, conceptual framework, correlation coefficient, correlation does not imply causation, crack epidemic, cuban missile crisis, Daniel Kahneman / Amos Tversky, David Brooks, delayed gratification, demographic transition, desegregation, Doomsday Clock, Douglas Hofstadter, Edward Glaeser, en.wikipedia.org, European colonialism, experimental subject, facts on the ground, failed state, first-past-the-post, Flynn Effect, food miles, Francis Fukuyama: the end of history, fudge factor, full employment, ghettoisation, Gini coefficient, global village, Henri Poincaré, impulse control, income inequality, informal economy, invention of the printing press, Isaac Newton, lake wobegon effect, libertarian paternalism, loss aversion, Marshall McLuhan, McMansion, means of production, mental accounting, meta analysis, meta-analysis, Mikhail Gorbachev, mutually assured destruction, open economy, Peace of Westphalia, Peter Singer: altruism, QWERTY keyboard, race to the bottom, Ralph Waldo Emerson, random walk, Republic of Letters, Richard Thaler, Ronald Reagan, Rosa Parks, Saturday Night Live, security theater, Skype, Slavoj Žižek, South China Sea, statistical model, stem cell, Steven Levy, Steven Pinker, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, theory of mind, transatlantic slave trade, transatlantic slave trade, Turing machine, ultimatum game, uranium enrichment, V2 rocket, Walter Mischel, WikiLeaks, women in the workforce

The graph also shows the trend for Canada since 1961. Canadians kill at less than a third of the rate of Americans, partly because in the 19th century the Mounties got to the western frontier before the settlers and spared them from having to cultivate a violent code of honor. Despite this difference, the ups and downs of the Canadian homicide rate parallel those of their neighbor to the south (with a correlation coefficient between 1961 and 2009 of 0.85), and it sank almost as much in the 1990s: 35 percent, compared to the American decline of 42 percent.132 The parallel trajectory of Canada and the United States is one of many surprises in the great crime decline of the 1990s. The two countries differed in their economic trends and in their policies of criminal justice, yet they enjoyed similar drops in violence.