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A Course on Statistics for Finance

Book Description

Taking a data-driven approach, A Course on Statistics for Finance presents statistical methods for financial investment analysis. The author introduces regression analysis, time series analysis, and multivariate analysis step by step using models and methods from finance.

The book begins with a review of basic statistics, including descriptive statistics, kinds of variables, and types of data sets. It then discusses regression analysis in general terms and in terms of financial investment models, such as the capital asset pricing model and the Fama/French model. It also describes mean-variance portfolio analysis and concludes with a focus on time series analysis.

Providing the connection between elementary statistics courses and quantitative finance courses, this text helps both existing and future quants improve their data analysis skills and better understand the modeling process.

Table of Contents

  1. Preliminaries
  2. Preface
  3. About the Author
  4. Part I Introductory Concepts and Definitions
    1. Chapter 1 Review of Basic Statistics
      1. 1.1 What Is Statistics?
        1. 1.1.1 Data Are Observations
        2. 1.1.2 Statistics Descriptions; Statistics Methods
        3. 1.1.3 Origins of Data
        4. 1.1.4 Philosophy of Data and Information
          1. 1.1.4.1 Data versus Information
          2. 1.1.4.2 Decisions
      2. 1.2 Characterizing Data
        1. 1.2.1 Types of Data
          1. 1.2.1.1 Modes and Ways
          2. 1.2.1.2 Types of Variables
          3. 1.2.1.3 Cross-Sectional Data versus Time Series Data
        2. 1.2.2 Raw Data versus Derived Data
          1. 1.2.2.1 Ratios
          2. 1.2.2.2 Indices
      3. 1.3 Measures of Central Tendency
        1. 1.3.1 Mode
        2. 1.3.2 Measuring the Center of a Set of Numbers
          1. 1.3.2.1 Median
          2. 1.3.2.2 Quartiles
          3. 1.3.2.3 Percentiles
          4. 1.3.2.4 Section Exercises
          5. 1.3.2.5 Mean
          6. 1.3.2.6 Other Properties of the Ordinary Arithmetic Average
          7. 1.3.2.7 Mean of a Distribution
        3. 1.3.3 Other Kinds of Averages
          1. 1.3.3.1 Root Mean Square
          2. 1.3.3.2 Other Averages
        4. 1.3.4 Section Exercises
      4. 1.4 Measures of Variability
        1. 1.4.1 Measuring Spread
          1. 1.4.1.1 Positional Measures of Spread
          2. 1.4.1.2 Range
          3. 1.4.1.3 IQR
        2. 1.4.2 Distance-Based Measures of Spread
          1. 1.4.2.1 Deviations from the Mean
          2. 1.4.2.2 Mean Absolute Deviation
          3. 1.4.2.3 Root Mean Square Deviation
          4. 1.4.2.4 Standard Deviation
          5. 1.4.2.5 Variance of a Distribution
      5. 1.5 Higher Moments
      6. 1.6 Summarizing Distributions
        1. 1.6.1 Partitioning Distributions
        2. 1.6.2 Moment-Preservation Method
      7. 1.7 Bivariate Data
        1. 1.7.1 Covariance and Correlation
          1. 1.7.1.1 Computational Formulas
          2. 1.7.1.2 Covariance, Regression Cooefficient, and Correlation Coefficient
        2. 1.7.2 Covariance of a Bivariate Distribution
      8. 1.8 Three Variables
        1. 1.8.1 Pairwise Correlations
        2. 1.8.2 Partial Correlation
      9. 1.9 Two-Way Tables
        1. 1.9.1 Two-Way Tables of Counts
        2. 1.9.2 Turnover Tables
        3. 1.9.3 Seasonal Data
          1. 1.9.3.1 Data Aggregation
          2. 1.9.3.2 Stable Seasonal Pattern
      10. 1.10 Summary
      11. 1.11 Chapter Exercises
        1. 1.11.1 Applied Exercises
        2. 1.11.2 Mathematical Exercises
      12. 1.12 Bibliography
      1. Table 1.1
      2. Table 1.2
      3. Table 1.3
      4. Table 1.4
      5. Table 1.5
    2. Chapter 2 Stock Price Series and Rates of Return
      1. 2.1 Introduction
        1. 2.1.1 Price Series
        2. 2.1.2 Rates of Return
          1. 2.1.2.1 Continuous ROR and Ordinary ROR
          2. 2.1.2.2 Advantages of Continuous ROR
          3. 2.1.2.3 Modeling Price Series
        3. 2.1.3 Review of Mean, Variance, and Standard Deviation
          1. 2.1.3.1 Mean
          2. 2.1.3.2 Variance
          3. 2.1.3.3 Standard Deviation
      2. 2.2 Ratios of Mean and Standard Deviation
        1. 2.2.1 Coefficient of Variation
        2. 2.2.2 Sharpe Ratio
      3. 2.3 Value-at-Risk
        1. 2.3.1 VaR for Normal Distributions
        2. 2.3.2 Conditional VaR
      4. 2.4 Distributions for RORs
        1. 2.4.1 t Distribution as a Scale-Mixture of Normals
        2. 2.4.2 Another Example of Averaging over a Population
        3. 2.4.3 Section Exercises
      5. 2.5 Summary
      6. 2.6 Chapter Exercises
      7. 2.7 Bibliography
      8. 2.8 Further Reading
      1. Table 2.1
      2. Table 2.2
    3. Chapter 3 Several Stocks and Their Rates of Return
      1. 3.1 Introduction
      2. 3.2 Review of Covariance and Correlation
      3. 3.3 Two Stocks
        1. 3.3.1 RORs of Two Stocks
        2. 3.3.2 Section Exercises
      4. 3.4 Three Stocks
        1. 3.4.1 RORs of Three Stocks
        2. 3.4.2 Section Exercises
      5. 3.5 m Stocks
        1. 3.5.1 RORs for m Stocks
        2. 3.5.2 Parameters and Statistics for m Stocks
      6. 3.6 Summary
      7. 3.7 Chapter Exercises
      8. 3.8 Bibliography
      9. 3.9 Further Reading
      1. Table 3.1
      2. Table 3.2
      3. Table 3.3
      4. Table 3.4
  5. Part II Regression
    1. Chapter 4 Simple Linear Regression; CAPM and Beta
      1. 4.1 Introduction
      2. 4.2 Simple Linear Regression
        1. 4.2.1 Data
        2. 4.2.2 An Introductory Example
      3. 4.3 Estimation
        1. 4.3.1 Method of Least Squares
          1. 4.3.1.1 Least Squares Criterion
          2. 4.3.1.2 Least Squares Estimator
        2. 4.3.2 Maximum Likelihood Estimator under the Assumption of Normality*
        3. 4.3.3 A Heuristic Approach
          1. 4.3.3.1 Observational Equations
          2. 4.3.3.2 Method of Reduction of Observations
        4. 4.3.4 Means and Variances of Estimators
          1. 4.3.4.1 Means of Estimators
          2. 4.3.4.2 Unbiasedness
          3. 4.3.4.3 Variance of the Least Squares Estimator
          4. 4.3.4.4 Nonlinear and Biased Estimators
        5. 4.3.5 Estimating the Error Variance
          1. 4.3.5.1 Computational Formulas
          2. 4.3.5.2 Decomposition of Sum of Squares
      4. 4.4 Inference Concerning the Slope
        1. 4.4.1 Testing a Hypothesis Concerning the Slope
        2. 4.4.2 Confidence Interval
      5. 4.5 Testing Equality of Slopes of Two Lines through the Origin
      6. 4.6 Linear Parametric Functions
      7. 4.7 Variances Dependent upon X *
      8. 4.8 A Financial Application: CAPM and “Beta”
        1. 4.8.1 CAPM
        2. 4.8.2 “Beta”
      9. 4.9 Slope and Intercept
        1. 4.9.1 Model with Slope and Intercept
        2. 4.9.2 CAPM with Differential Return
      10. 4.10 Appendix 4A: Optimality of the Least Squares Estimator
      11. 4.11 Summary
      12. 4.12 Chapter Exercises
        1. 4.12.1 Applied Exercises
        2. 4.12.2 Mathematical Exercises
      13. 4.13 Bibliography
      14. 4.14 Further Reading
      1. Figure 4.1
      1. Table 4.1
      2. Table 4.2
      3. Table 4.3
      4. Table 4.4
      5. Table 4.5
    2. Chapter 5 Multiple Regression and Market Models
      1. 5.1 Multiple Regression Models
        1. 5.1.1 Regression Function
        2. 5.1.2 Method of Least Squares
        3. 5.1.3 Types of Explanatory Variables
      2. 5.2 Market Models
        1. 5.2.1 Fama/French Three-Factor Model
        2. 5.2.2 Four-Factor Model
      3. 5.3 Models with Numerical and Dummy Explanatory Variables
        1. 5.3.1 Two-Group Models
        2. 5.3.2 Other Market Models
          1. 5.3.2.1 Two Betas
          2. 5.3.2.2 More Advanced Models
      4. 5.4 Model Building
        1. 5.4.1 Principle of Parsimony
        2. 5.4.2 Model-Selection Criteria
          1. 5.4.2.1 Residual Mean Square
          2. 5.4.2.2 Adjusted R-Square
        3. 5.4.3 Testing a Reduced Model against a Full Model
        4. 5.4.4 Comparing Several Models
        5. 5.4.5 Combining Results from Several Models
      5. 5.5 Chapter Summary
      6. 5.6 Chapter Exercises
        1. 5.6.1 Exercises for Two Explanatory Variables
        2. 5.6.2 Mathematical Exercises: Two Explanatory Variables
        3. 5.6.3 Mathematical Exercises: Three Explanatory Variables
        4. 5.6.4 Exercises on Subset Regression
        5. 5.6.5 Mathematical Exercises: Subset Regression
      7. 5.7 Bibliography
      1. Table 5.1
      2. Table 5.2
      3. Table 5.3
  6. Part III Portfolio Analysis
    1. Chapter 6 Mean-Variance Portfolio Analysis
      1. 6.1 Introduction
        1. 6.1.1 Mean-Variance Portfolio Analysis
        2. 6.1.2 Single-Criterion Analysis
      2. 6.2 Two Stocks
        1. 6.2.1 Mean
        2. 6.2.2 Variance
        3. 6.2.3 Covariance and Correlation
        4. 6.2.4 Portfolio Variance
          1. 6.2.4.1 Variance of a Sum; Variance of a Difference.
          2. 6.2.4.2 Portfolio Variance
        5. 6.2.5 Minimum Variance Portfolio
      3. 6.3 Three Stocks
      4. 6.4 m Stocks
      5. 6.5 m Stocks and a Risk-Free Asset
        1. 6.5.1 Admissible Points
        2. 6.5.2 Capital Allocation Lines
      6. 6.6 Value-at-Risk
        1. 6.6.1 VaR for Normal Distributions
        2. 6.6.2 Conditional VaR
      7. 6.7 Selling Short
      8. 6.8 Market Models and Beta
        1. 6.8.1 CAPM
        2. 6.8.2 Computation of Covariances under the CAPM
        3. 6.8.3 Section Exercises
      9. 6.9 Summary
        1. 6.9.1 Rate of Return
        2. 6.9.2 Bi-Criterion Analysis
        3. 6.9.3 Market Models
      10. 6.10 Chapter Exercises
        1. 6.10.1 Exercises on Covariance and Correlation
        2. 6.10.2 Exercises on Portfolio ROR
        3. 6.10.3 Exercises on Three Stocks
        4. 6.10.4 Exercises on Correlation and Regression
      11. 6.11 Appendix 6A: Some Results in Terms of Vectors and Matrices (Optional)*
        1. 6.11.1 Variates
        2. 6.11.2 Vector Differentiation
          1. 6.11.2.1 Some Rules for Vector Differentiation
          2. 6.11.2.2 Minimum-Variance Portfolio
          3. 6.11.2.3 Maximum Sharpe Ratio
        3. 6.11.3 Section Exercises
      12. 6.12 Appendix 6B: Some Results for the Family of Normal Distributions
        1. 6.12.1 Moment Generating Function; Moments
        2. 6.12.2 Section Exercises
      13. 6.13 Bibliography
      14. 6.14 Further Reading
      1. Figure 6.1
      1. Table 6.1
      2. Table 6.2
      3. Table 6.3
      4. Table 6.4
      5. Table 6.5
      6. Table 6.6
      7. Table 6.7
    2. Chapter 7 Utility-Based Portfolio Analysis
      1. 7.1 Introduction
        1. 7.1.1 Background
        2. 7.1.2 Types of Portfolio Analysis
      2. 7.2 Single-Criterion Analysis
        1. 7.2.1 Mean versus Variance Plot
        2. 7.2.2 Weights on the Risk-Free and Risky Parts of the Portfolio
        3. 7.2.3 Separation
      3. 7.3 Summary
      4. 7.4 Chapter Exercises
      5. 7.5 Bibliography
      1. Figure 7.1
      1. Table 7.1
  7. Part IV Time Series Analysis
    1. Chapter 8 Introduction to Time Series Analysis
      1. 8.1 Introduction
      2. 8.2 Control Charts
      3. 8.3 Moving Averages
        1. 8.3.1 Running Median
        2. 8.3.2 Various Moving Averages
        3. 8.3.3 Exponentially Weighted Moving Averages
        4. 8.3.4 Using a Moving Average for Prediction
          1. 8.3.4.1 Smoothed Value as a Predictor of the Next Value
          2. 8.3.4.2 A Predictor-Corrector Formula
          3. 8.3.4.3 MACD
      4. 8.4 Need for Modeling
      5. 8.5 Trend, Seasonality, and Randomness
      6. 8.6 Models with Lagged Variables
        1. 8.6.1 Lagged Variables
        2. 8.6.2 Autoregressive Models
      7. 8.7 Moving-Average Models
        1. 8.7.1 Integrated Moving-Average Model
        2. 8.7.2 Preliminary Estimate of θ
        3. 8.7.3 Estimate of θ
        4. 8.7.4 Integrated Moving-Average with a Constant
      8. 8.8 Identification of ARIMA Models
        1. 8.8.1 Pre-Processing
          1. 8.8.1.1 Transformation
          2. 8.8.1.2 Differencing
        2. 8.8.2 ARIMA Parameters p, d, q
        3. 8.8.3 Autocorrelation Function; Partial Autocorrelation Function
      9. 8.9 Seasonal Data
        1. 8.9.1 Seasonal ARIMA Models
        2. 8.9.2 Stable Seasonal Pattern
      10. 8.10 Dynamic Regression Models
      11. 8.11 Simultaneous Equations Models
      12. 8.12 Appendix 8A: Growth Rates and Rates of Return
        1. 8.12.1 Compound Interest
        2. 8.12.2 Geometric Brownian Motion
        3. 8.12.3 Average Rates of Return
        4. 8.12.4 Section Exercises: Exponential and Log Functions
      13. 8.13 Appendix 8B: Prediction after Data Transformation
        1. 8.13.1 Prediction
        2. 8.13.2 Prediction after Transformation
        3. 8.13.3 Unbiasing
        4. 8.13.4 Application to the Log Transform
        5. 8.13.5 Generalized Linear Models
      14. 8.14 Appendix 8C: Representation of Time Series
        1. 8.14.1 Operators
        2. 8.14.2 White Noise
        3. 8.14.3 Stationarity
        4. 8.14.4 AR
          1. 8.14.4.1 Variance
          2. 8.14.4.2 Covariances and Correlations
          3. 8.14.4.3 Higher-Order AR
        5. 8.14.5 MA
          1. 8.14.5.1 Variance
          2. 8.14.5.2 Correlation
          3. 8.14.5.3 Representing the Error Variables in Terms of the Observations
        6. 8.14.6 ARMA
      15. 8.15 Summary
      16. 8.16 Chapter Exercises
        1. 8.16.1 Applied Exercises
        2. 8.16.2 Mathematical Exercises
      17. 8.17 Bibliography
      18. 8.18 Further Reading
      1. Figure 8.1
      1. Table 8.1
      2. Table 8.2
      3. Table 8.3
      4. Table 8.4
      5. Table 8.5
    2. Chapter 9 Regime Switching Models
      1. 9.1 Introduction
      2. 9.2 Bull and Bear Markets
        1. 9.2.1 Definitions of Bull and Bear Markets
        2. 9.2.2 Regressions on Bull3
          1. 9.2.2.1 Two Betas, No Alpha
          2. 9.2.2.2 Two Betas, One Alpha
          3. 9.2.2.3 Two Betas, Two Alphas
        3. 9.2.3 Other Models for Bull/Bear
          1. 9.2.3.1 Two Means and Two Variances
          2. 9.2.3.2 Mixture Model
          3. 9.2.3.3 Hidden Markov Model
        4. 9.2.4 Bull and Bear Portfolios
      3. 9.3 Summary
      4. 9.4 Chapter Exercises
        1. 9.4.1 Applied Exercises
        2. 9.4.2 Mathematical Exercises
      5. 9.5 Bibliography
      6. 9.6 Further Reading
      1. Table 9.1
      2. Table 9.2
  8. Appendix A Vectors and Matrices
    1. A.1 Introduction
    2. A.2 Vectors
      1. A.2.1 Inner Product of Two Vectors
      2. A.2.2 Orthogonal Vectors
      3. A.2.3 Variates
      4. A.2.4 Section Exercises
    3. A.3 Matrices
      1. A.3.1 Entries of a Matrix
      2. A.3.2 Transpose of a Matrix
      3. A.3.3 Matrix Multiplication
      4. A.3.4 Section Exercises
      5. A.3.5 Identity Matrix
      6. A.3.6 Inverse
        1. A.3.6.1 Inverse of a Matrix
        2. A.3.6.2 Inverse of a Product of Matrices
      7. A.3.7 Determinant
    4. A.4 Vector Differentiation
    5. A.5 Paths
    6. A.6 Quadratic Forms
    7. A.7 Eigensystem
    8. A.8 Transformation to Uncorrelated Variables
      1. A.8.1 Covariance Matrix of a Linear Transformation of a Random Vector
      2. A.8.2 Transformation to Uncorrelated Variables
      3. A.8.3 Transformation to Uncorrelated Variables with Variances Equal to One
    9. A.9 Statistical Distance
    10. A.10 Appendix Exercises
    11. A.11 Bibliography
    12. A.12 Further Reading
  9. Appendix B Normal Distributions
    1. B.1 Some Results for Univariate Normal Distributions
      1. B.1.1 Definitions
      2. B.1.2 Conditional Expectation
      3. B.1.3 Tail Probability Approximation
    2. B.2 Family of Multivariate Normal Distributions
    3. B.3 Role of D-Square
    4. B.4 Bivariate Normal Distributions
      1. B.4.1 Shape of the p.d.f
      2. B.4.2 Conditional Distribution of Y Given X
      3. B.4.3 Regression Function
    5. B.5 Other Multivariate Distributions
    6. B.6 Summary
      1. B.6.1 Concepts
      2. B.6.2 Mathematics
    7. B.7 Appendix B Exercises
      1. B.7.1 Applied Exercises
      2. B.7.2 Mathematical Exercises
    8. B.8 Bibliography
    9. B.9 Further Reading
  10. Appendix C Lagrange Multipliers
    1. C.1 Notation
    2. C.2 Optimization Problem
    3. C.3 Bibliography
    4. C.4 Further Reading
  11. Appendix D Abbreviations and Symbols
    1. D.1 Abbreviations
      1. D.1.1 Statistics
      2. D.1.2 General
      3. D.1.3 Finance
    2. D.2 Symbols
      1. D.2.1 Statistics
      2. D.2.2 Finance