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Probability and Statistics for Finance

A comprehensive look at how probability and statistics is applied to the investment process Finance has become increasingly more quantitative, drawing on techniques in probability and statistics that many finance practitioners have not had exposure to before. In order to keep up, you need a firm understanding of this discipline. Probability and Statistics for Finance addresses this issue by showing you how to apply quantitative methods to portfolios, and in all matter of your practices, in a clear, concise manner. Informative and accessible, this guide starts off with the basics and builds to an intermediate level of mastery. ? Outlines an array of topics in probability and statistics and how to apply them in the world of finance ? Includes detailed discussions of descriptive statistics, basic probability theory, inductive statistics, and multivariate analysis ? Offers real-world illustrations of the issues addressed throughout the text The authors cover a wide range of topics in this book, which can be used by all finance professionals as well as students aspiring to enter the field of finance.
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Table of Contents

Preface xv About the Authors xviiCHAPTER 1 Introduction 1 Probability vs. Statistics 4 Overview of the Book 5 Part One Descriptive Statistics 15 Chapter 2 Basic Data Analysis 17 Data Types 17 Frequency Distributions 22 Empirical Cumulative Frequency Distribution 27 Data Classes 32 Cumulative Frequency Distributions 41 Concepts Explained in this Chapter 43 Chapter 3 Measures of Location and Spread 45 Parameters vs. Statistics 45 Center and Location 46 Variation 59 Measures of the Linear Transformation 69 Summary of Measures 71 Concepts Explained in this Chapter 73 Chapter 4 Graphical Representation of Data 75 Pie Charts 75 Bar Chart 78 Stem and Leaf Diagram 81 Frequency Histogram 82 Ogive Diagrams 89 Box Plot 91 QQ Plot 96 Concepts Explained in this Chapter 99 CHAPTER 5 Multivariate Variables and Distributions 101 Data Tables and Frequencies 101 Class Data and Histograms 106 Marginal Distributions 107 Graphical Representation 110 Conditional Distribution 113 Conditional Parameters and Statistics 114 Independence 117 Covariance 120 Correlation 123 Contingency Coefficient 124 Concepts Explained in this Chapter 126 CHAPTER 6 Introduction to Regression Analysis 129 The Role of Correlation 129 Regression Model: Linear Functional Relationship Between Two Variables 131 Distributional Assumptions of the Regression Model 133 Estimating the Regression Model 134 Goodness of Fit of the Model 138 Linear Regression of Some Nonlinear Relationship 140 Two Applications in Finance 142 Concepts Explained in this Chapter 149 CHAPTER 7 Introduction to Time Series Analysis 153 What Is Time Series? 153 Decomposition of Time Series 154 Representation of Time Series with Difference Equations 159 Application: The Price Process 159 Concepts Explained in this Chapter 163 Part Two Basic Probability Theory 165 CHAPTER 8 Concepts of Probability Theory 167 Historical Development of Alternative Approaches to Probability 167 Set Operations and Preliminaries 170 Probability Measure 177 Random Variable 179 Concepts Explained in this Chapter 185 Chapter 9 Discrete Probability Distributions 187 Discrete Law 187 Bernoulli Distribution 192 Binomial Distribution 195 Hypergeometric Distribution 204 Multinomial Distribution 211 Poisson Distribution 216 Discrete Uniform Distribution 219 Concepts Explained in this Chapter 221 CHAPTER 10 Continuous Probability Distributions 229 Continuous Probability Distribution Described 229 Distribution Function 230 Density Function 232 Continuous Random Variable 237 Computing Probabilities from the Density Function 238 Location Parameters 239 Dispersion Parameters 239 Concepts Explained in this Chapter 245 CHAPTER 11 Continuous Probability Distributions with Appealing Statistical Properties 247 Normal Distribution 247 Chi-Square Distribution 254 Student's t-Distribution 256 F-Distribution 260 Exponential Distribution 262 Rectangular Distribution 266 Gamma Distribution 268 Beta Distribution 269 Log-Normal Distribution 271 Concepts Explained in this Chapter 275 CHAPTER 12 Continuous Probability Distributions Dealing with Extreme Events 277 Generalized Extreme Value Distribution 277 Generalized Pareto Distribution 281 Normal Inverse Gaussian Distribution 283 α-Stable Distribution 285 Concepts Explained in this Chapter 292 CHAPTER 13 Parameters of Location and Scale of Random Variables 295 Parameters of Location 296 Parameters of Scale 306 Concepts Explained in this Chapter 321 Appendix: Parameters for Various Distribution Functions 322 Chapter 14 Joint Probability Distributions 325 Higher Dimensional Random Variables 326 Joint Probability Distribution 328 Marginal Distributions 333 Dependence 338 Covariance and Correlation 341 Selection of Multivariate Distributions 347 Concepts Explained in this Chapter 358 Chapter 15 Conditional Probability and Bayes' Rule 361 Conditional Probability 362 Independent Events 365 Multiplicative Rule of Probability 367 Bayes' Rule 372 Conditional Parameters 374 Concepts Explained in this Chapter 377 CHAPTER 16 Copula and Dependence Measures 379 Copula 380 Alternative Dependence Measures 406 Concepts Explained in this Chapter 412 Part Three Inductive Statistics 413 Chapter 17 Point Estimators 415 Sample, Statistic, and Estimator 415 Quality Criteria of Estimators 428 Large Sample Criteria 435 Maximum Likehood Estimator 446 Exponential Family and Sufficiency 457 Concepts Explained in this Chapter 461 Chapter 18 Confidence Intervals 463 Confidence Level and Confidence Interval 463 Confidence Interval for the Mean of a Normal Random Variable 466 Confidence Interval for the Mean of a Normal Random Variable with Unknown Variance 469 Confidence Interval for the Variance of a Normal Random Variable 471 Confidence Interval for the Variance of a Normal Random Variable with Unknown Mean 474 Confidence Interval for the Parameter p of a Binomial Distribution 475 Confidence Interval for the Parameter λ of an Exponential Distribution 477 Concepts Explained in this Chapter 479 Chapter 19 Hypothesis Testing 481 Hypotheses 482 Error Types 485 Quality Criteria of a Test 490 Examples 496 Concepts Explained in this Chapter 518 Part Four Multivariate Linear Regression Analysis 519 CHAPTER 20 Estimates and Diagnostics for Multivariate Linear Regression Analysis 521 The Multivariate Linear Regression Model 522 Assumptions of the Multivariate Linear Regression Model 523 Estimation of the Model Parameters 523 Designing the Model 526 Diagnostic Check and Model Significance 526 Applications to Finance 531 Concepts Explained in this Chapter 543 CHAPTER 21 Designing and Building a Multivariate Linear Regression Model 545 The Problem of Multicollinearity 545 Incorporating Dummy Variables as Independent Variables 548 Model Building Techniques 561 Concepts Explained in this Chapter 565 CHAPTER 22 Testing the Assumptions of the Multivariate Linear Regression Model 567 Tests for Linearity 568 Assumed Statistical Properties about the Error Term 570 Tests for the Residuals Being Normally Distributed 570 Tests for Constant Variance of the Error Term (Homoskedasticity) 573 Absence of Autocorrelation of the Residuals 576Concepts Explained in this Chapter 581 Appendix A Important Functions and Their Features 583 Continuous Function 583 Indicator Function 586 Derivatives 587 Monotonic Function 591 Integral 592 Some Functions 596 Appendix B Fundamentals of Matrix Operations and Concepts 601 The Notion of Vector and Matrix 601 Matrix Multiplication 602 Particular Matrices 603 Positive Semidefinite Matrices 614 APPENDIX C Binomial and Multinomial Coefficients 615 Binomial Coefficient 615 Multinomial Coefficient 622 APPENDIX D Application of the Log-Normal Distribution to the Pricing of Call Options 625 Call Options 625 Deriving the Price of a European Call Option 626 Illustration 631 References 633 Index 635

About the Author

SVETLOZAR T. RACHEV, PhD, DSC, is Chair Professor at theUniversity of Karlsruhe in the School of Economics and BusinessEngineering, and Professor Emeritus at the University ofCalifornia, Santa Barbara, in the Department of Statistics andApplied Probability. He was cofounder of Bravo Risk ManagementGroup, acquired by FinAnalytica, where he currently serves as ChiefScientist. MARKUS HOCHSTOTTER, PhD, is an Assistant Professor inthe Department of Econometrics and Statistics, University ofKarlsruhe. FRANK J. FABOZZI, PhD, CFA, CPA, is Professor in the Practice ofFinance and Becton Fellow at the Yale School of Management andEditor of the Journal of Portfolio Management. He is an AffiliatedProfessor at the University of Karlsruhe's Institute of Statistics,Econometrics and Mathematical Finance, and is on the AdvisoryCouncil for the Department of Operations Research and FinancialEngineering at Princeton University. SERGIO M. FOCARDI, PhD, is a Professor of Finance at EDHECBusiness School and founding partner of the Paris-based consultingfirm Intertek Group plc.

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