Statistics and Finance

Transcription

1 David Ruppert Statistics and Finance An Introduction Springer

2 Notation... xxi 1 Introduction References Probability and Statistical Models Introduction Axioms of Probability Independence Bayes law Probability Distributions Random variables Independence Cumulative distribution functions Quantiles and percentiles Expectations and variances Does the expected value exist? Functions of Random Variables Random Samples The Binomial Distribution Location, Scale, and Shape Parameters Some Common Continuous Distributions Uniform distributions Normal distributions The lognormal distribution Exponential and double exponential distributions Sampling a Normal Distribution Chi-squared distributions t-distributions F-distributions Order Statistics and the Sample CDF Y

3 xii Normal probability plots Skewness and Kurtosis Heavy-Tailed Distributions Double exponential distributions t-distributions have heavy tails Mixture models Pareto distributions Distributions with Pareto tails Law of Large Numbers and Central Limit Theorem Multivariate Distributions Correlation and covariance Independence and covariance The multivariate normal distribution Prediction Best linear prediction Prediction error in linear prediction Multivariate linear prediction Conditional Distributions Best prediction Normal distributions: Conditional expectations and variance Linear Functions of Random Variables Two linear combinations of raiidom variables Independence and variances of Sums Application to normal distributions Estimation Maximum likelihood estimation Standard errors Fisher information Bayes estimation* Robust estimation* Confidence Intervals Confidence interval for the mean Confidence intervals for the variance and Standard deviation Confidence intervals based on Standard e 2.20 Hypothesis Testing Hypotheses, types of errors, and rejection regions P-values Two-sample t-tests Statistical versus practical significance Tests of normality Likelihood ratio tests Summary Bibliographic Notes $

4 xiii References Problems Returns Introduction Net returns Gross returns Log returns Adjustment for div 3.2 Behavior Of Returns The Random Walk Model i.d. normal returns The lognormal model Random walks Geometric random The effect of the dri Are log returns normally distributed? Do the GE daily returns look like a geometric random walk? Origins of the Random Walk Hypothesis Fundamental analysis Technical analysis Efficient Markets Hypothesis (EMH) Three types of efficiency Testing market efficiency Discrete and Continuous Compounding Summary Bibliographic Notes References Problems Time Series Models......lOl 4.1 Time Series Data Stationary Processes Weak white noise Predicting white nois Est imat ing Parameters 4.3 AR( 1) Processes Properties of a stationary Convergence to the st Nonstationary AR(1) p 4.4 Estimation of AR(1) Processes Residuals and model checking AR(1) model for GE daily log returns AR(p) Models

5 xiv AR(6) model for GE daily log returns Moving Average (MA) Processes MA(1) processes General MA processes MA(2) model for GE dai ns ARIMA Processes The backwards Operator ARMA processes Fitting ARMA processes: returns The differencing Operator From ARMA processes to A processes ARIMA(2,1, 0) model for rices Model Selection AIC and SBC GE daily log returns: Choosing the AR Order Three-Month Treasury Bill Rates Forecast ing Forecasting GE daily log returns and log prices Summary... Bibliographie Notes... References Problems Port ;folio Theory Trading Off Expected Return and Risk One Risky Asset and One Risk-Free Asset Estimating E(R) and 0~ Two Risky Assets Risk versus expected return Estimating means, Standard deviations, and covariances Combining Two Risky Assets with a Risk-Free Asset Tangency portfolio with two risky assets Combining the tangency portfolio with the risk-free asset Effect of p Risk-Efficient Portfolios with N Risky Assets* Efficient-portfolio mathematics The minimum variance portfolio Selling short Back to the math - Finding the tangency portfolio Examples Quadratic Programming* s the Theory Useful? Utility Theory*... Summary

6 Y... xv 5.10 Bibliographic Notes References Problems Regression Introduction Straight line regression Least Squares Estimation Estimation in straight line regression Variance of ßi Estimation in multiple linear regression Standard Errors, T-Values, and P-Values Analysis Of Variance, R2, and F-Tests AOV table Sums of Squares (SS) and R Degrees of freedom (DF) Mean Sums of Squares (MS) and testing Adjusted R Sequential and partial Sums of Squares Regression Hedging* Regression and Best Linear Prediction Model Selection Collinearity and Variance Inflation Centering the Predictors Nonlinear Regression The General Regression Model Troubleshooting Influence diagnostics and residuals Residual analysis Transform-Both-Sides Regression* How TBS works Power transformations The Geometry of Transformations* Robust Regression* Summary Bibliographic Notes References Problems The Capital Asset Pricing Model Introduction to CAPM The Capital Market Line (CML) Betas and the Security Market Line Examples of betas Comparison of the CML with the SML

7 xvi The Security Characteristic Line Reducing unique risk by diversification Can beta be negative? Are the assumptions sensible? Some More Portfolio Theory Contributions to the market portfolio s risk Derivation of the SML... Estimatiori of Beta and Testing the CAPM Regression using returns instead of excess returns Interpretation of alpha Factor Models... o Analysis Estimati tations and covariances of asset three-factor model Cross-sectional factor models... An Interesting Question* s Beta Constant?*... Summary Bibliographic Notes... References Problems Options Pricing Introduction Ca11 Options Thc Law of One Price Arbitrage Time Value of Money and Present Value Pricing Calls - A Simple Binomial Example Two-Step Binomial Option Pricing Arbitrage Pricing by Expectation A General Binomial Tree Model Martingales Martingale or risk-neutral measur The risk-neutral world From Trees to Random Walks and Brownian Motion Getting more realistic A three-step binomial tree More time steps Properties of Brownian motion Geometric Brownian Motion Using the Black-Scholes Formula How does the Option price depend on the inputs? Early exercise of calls is never optimal

8 xvii Are there returns on nontrading days? Implied Volatility Volatility smiles and polynomial regre Puts Pricing puts by binomial trees Why are puts different from calls? Put-call parity The Evolution of Opti Leverage of Options a The Greeks Problems Fixed Income Securities Introduction Zero-Coupon Bonds Price and returns fluctuate with the interest 9.3 Coupon Bonds A general formula Yield to Maturity General method for yield to maturi MATLAB functions Spot rates Term Structure Introduction: Interest rates depend maturity Describing the term structure Continuous Compounding Continuous Forward Rates Sensitivity of Price to Yield Duration of a Coupon bond Estimation of a Continuous Forward Rate* Summary Bibliographic Notes References Problems Resampling Introduction Confidence Intervals for the Mean Resampling and Efficient Portfolios The global asset allocation Problem

9 xviii Uncertainty about mean-variance efficient portfolios What if we knew the expected returns? What if we knew the covariance matrix? What if we had more data? Bagging* Summary Bibliographic Notes References Problems Value-At-Risk The Need for Risk Management VaR with One Asset Nonparametric estimation of VaR Parametric estimation of VaR Estimation of VaR assuming Pareto tails* Estimating the tail index* Confidence intervals for VaR using resampling VaR for a derivative VaR for a Portfolio of Assets Portfolios of stocks only Portfolios of one stock and an Option on that stock Portfolios of one stock and an Option on another stock Choosing the Holding Period and Confidence VaR and Risk Management Summary Bibliographic Notes References Problems GARCH Models Introduction Modeling Conditional Means and Variances ARCH( 1) Processes The AR( l)/arch( I) Model ARCH(q) Models GARCH(p, q) Models GARCH Processes Have Heavy Tails Comparison of ARMA and GARCH Processes Fitting GARCH Models I-GARCH Models What does it mean to have an infinite variance? GARCH-M Processes E-GARCH The GARCH Zoo* P

10 xix Applications of GARCH in Finance Pricing Options Under Generalized GARCH ProCesSes* Summary Bibliographic Notes References Problems Nonparametric Regression and Splines Introduction Choosing a Regression Method Nonparametric regression Linear Nonlinear parametric regression Comparison of linear and nonparametric regression Linear Splines Linear splines with one knot Linear splines with many knots Other Degree Splines Quadratic splines pth degree splines Least Squares Estimation Selecting the Spline Parameters Estimating the volatility function Additive Models* Penalized Splines* Penalizing the jumps at the knots Cross-Validation The effective number of Parameters Generalized cross-validation AIC Penalized splines in MATLAB Summary Bibliographic Notes References Problems Behavioral Finance Introduction Defense of the EMH Challenges to the EMH Can Arbitrageurs Save the Day? What Do the Data Say? Excess price volatility The overreaction hypothesis Reactions to earnings announcements J A

11 xx Counter-arguments to pricing anomalies Reaction to non-news Market Volatility and Irrational Exuberance Best prediction The Current Status of Classical Finance Bibliographic Notes References Problems Glossary Index