Introduction to statistical methods for financial models

Introduction to Statistical Methods for Financial Models T&F Cat #K31368 — K31368 C000— page i — 6/14/2017 — 22:05 CHAPMAN & HALL/CRC Texts in Statistical Science Series Series Editors Joseph K Blitzstein, Harvard University, USA Julian J Faraway, University of Bath, UK Martin Tanner, Northwestern University, USA Jim Zidek, University of British Columbia, Canada Statistical Theory: A Concise Introduction F Abramovich and Y Ritov Practical Multivariate Analysis, Fifth Edition A Afifi, S May, and V.A Clark Practical Statistics for Medical Research D.G Altman Interpreting Data: A First Course in Statistics A.J.B Anderson Introduction to Probability with R K Baclawski Problem Solving: A Statistician’s Guide, Second Edition C Chatfield Statistics for Technology: A Course in Applied Statistics, Third Edition C Chatfield Analysis of Variance, Design, and Regression : Linear Modeling for Unbalanced Data, Second Edition R Christensen Linear Algebra and Matrix Analysis for Statistics S Banerjee 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Additive Models: An Introduction with R, Second Edition S Wood Epidemiology: Study Design and Data Analysis, Third Edition M Woodward Practical Data Analysis for Designed Experiments B.S Yandell T&F Cat #K31368 — K31368 C000— page v — 6/14/2017 — 22:05 T&F Cat #K31368 — K31368 C000— page vi — 6/14/2017 — 22:05 Introduction to Statistical Methods for Financial Models Thomas A Severini Northwestern University Evanston, Illinois, USA T&F Cat #K31368 — K31368 C000— page vii — 6/14/2017 — 22:05 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-19837-1 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, 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Mathematical models Classification: LCC HG176.5 S49 2017 | DDC 332.072/7 dc23 LC record available at https://lccn.loc.gov/2017003073 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com T&F Cat #K31368 — K31368 C000— page viii — 6/14/2017 — 22:05 To Karla T&F Cat #K31368 — K31368 C000— page ix — 6/14/2017 — 22:05 356 References Caeiro, F and Mateus, A (2014) randtests: Testing randomness in R R package version 1.0 Campbell, J Y., Lo, A W., and MacKinlay, A C (1997) The Econometrics of Financial Markets Princeton University Press, Princeton, NJ Canty, A and Ripley, B (2015) boot: Bootstrap R (S-plus) Functions R package version 1.3-17 Carhart, M M (1997) On Persistence in Mutual Fund Performance Journal of Finance, 52:57–82 Carlin, B P and Louis, T A (2000) Bayes and Empirical Bayes Methods for Data Analysis CRC Press, Boca Raton, FL, second edition Chambers, J M., Cleveland, W S., Keiner, B., and Tukey, P A (1983) 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Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budget Review of Economics & Statistics, 47:13–37 Lo, A W (1991) Long-Term Memory in Stock Market Prices Econometrica, 59:1279–1313 Lo, A W (1997) A Nonrandom Walk Down Wall Street: Recent Advances in Financial Technology TIAA-CREF Research Dialogues, 52:1–7 Lo, A W and MacKinlay, A C (2002) A Non-Random Walk Down Wall Street Princeton University Press, Princeton, NJ Lu, T.-T and Shiou, S.-H (2002) Inverses of × Block Matrices Computers & Mathematics with Applications, 43:119–129 T&F Cat #K31368 — K31368 A001— page 358 — 6/14/2017 — 22:05 References 359 Malkiel, B G (1973) A Random Walk Down Wall Street W W Norton & Company, New York, NY Malkiel, B G (2003) A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing W W Norton, New York, NY Markowitz, H (1952) Portfolio Selection Journal of Finance, 7:77–91 Markowitz, H M (1987) Mean-Variance Analysis in Portfolio 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1:17–19 Sclove, S L (2013) A Course on Statistics for Finance CRC Press, Boca Raton, FL Severini, T A (2005) Elements of Distribution Theory Cambridge University Press, Cambridge, UK Severini, T A (2016) A Nonparametric Approach to Measuring the Sensitivity of an Asset’s Return to the Market Annals of Finance, 12:179–199 Shanken, J and Zhou, G (2007) Estimation and Testing Beta Pricing Models: Alternative Methods and their Performance in Simulations Journal of Financial Economics, 84:40–86 Sharpe, W F (1963) A Simplified Model for Portfolio Analysis Management Science, 9:277–293 Sharpe, W F (1964) Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk Journal of Finance, 19:425–442 Sharpe, W F (1991) The Arithmetic of Active Management Financial Analysts Journal, 47:7–9 Shumway, T (2000) Course Notes for Bus Admin 855 http://www-personal umich.edu/∼shumway/courses.dir/ba855.dir/Notes1.pdf Stewart, J (2015) Calculus Brooks Cole, Boston, MA, eighth edition Tamhane, A C and Dunlop, D D (2000) Statistics and Data Analysis: From Elementary to Intermediate Prentice-Hall, Upper Saddle River, NJ T&F Cat #K31368 — K31368 A001— page 360 — 6/14/2017 — 22:05 References 361 Touloumis, A (2015) Nonparametric Stein-Type Shrinkage Covariance Matrix Estimators in High-Dimensional Settings Computational Statistics and Data Analysis, 83:251–261 Trapletti, A and Hornik, K (2016) tseries: Time Series Analysis and Computational Finance R package version 0.10-35 Treynor, J L and Black, F (1973) How to Use Security Analysis to Improve Portfolio Selection Journal of Business, 46:66–86 Turlach, B A and Weingessel, A (2013) quadprog: Functions to Solve Quadratic Programming Problems R package version 1.5-5 Vasicek, O A (1973) A Note on Using Cross-Sectional Information in Bayesian Estimation of Betas Journal of Finance, 28:1233–1239 Venables, W N and Ripley, B D (2002) Modern Applied Statistics with S Springer, New York, fourth edition Warnes, G R., Bolker, B., and Lumley, T (2015) gtools: Various R Programming Tools R package version 3.5.0 Wei, W W S (2006) Time Series Analysis: Univariate and Multivariate Methods Pearson, Boston, MA, second edition Woolridge, J M (2013) Introductory Econometrics: A Modern Approach South-Western, Mason, OH, fifth edition T&F Cat #K31368 — K31368 A001— page 361 — 6/14/2017 — 22:05 T&F Cat #K31368 — K31368 C000— page vi — 6/14/2017 — 22:05 Index A Active portfolio management, Treynor–Black method and, 292–307 adding single asset to market portfolio, 293–298 benchmark portfolio, 292 estimator bias 304–306 numerical computation of portfolio weights, 306–307 portfolio of N assets together with market portfolio, 298–302 properties of Treynor–Black portfolio, 302–304 Adjusted beta, 243–244 Adjusted prices, 9–14 Adjusted R-squared, 320 Appraisal ratio, 258–259, 295 Arbitrage pricing theory (APT), 328–333 arbitrage portfolio, 329 asymptotic arbitrage, 332–333 factor premiums and, 334 no-arbitrage assumption, 330 no-asymptotic arbitrage assumption, 332 Assets appraisal ratio of, 295 correctly priced, 232–237 excess return of, 83 investable weight factor of, 223 market capitalization of, 222 mispriced, 208–211 prices, random walk models for, 48–54 returns, negatively correlated, 72 risk-free, 81–84 risky, 84, 85, 90 volatility of, 20 Assets, N (portfolios of), 95–102 correlation matrix, 100 diversification, 101–102 eigenvalues, 100 inner product, 97 matrix notation, 96–101 nonnegative definite matrix, 99 random vector, 96 Autocorrelation function, 16 Autocovariance function, 16 B Benchmark portfolio, 292 Best linear predictor, 51 Beta, adjusted, 243–244 Bias corrected estimate, 264 omitted-variable, 335 “Bloomberg adjusted beta,” 243 Bonferroni inequality, 234 Bonferroni method, 235 Bootstrap method, 261, 304 Box–Ljung test, 55 C Capital asset pricing model (CAPM), 2, 197–220 applying the CAPM to a portfolio, 206–208 capital market line, 199 CAPM without risk-free asset, 211–214 363 T&F Cat #K31368 — K31368 A001— page 363 — 6/14/2017 — 22:05 364 describing the expected returns on a set of assets, 215–217 efficiency of market portfolio, role of, 210–211 implications of, 202–206 linear regression analysis, relationship to, 201–202 market capitalization, 209 market portfolio, 197 mispriced assets, 208–211 relationship between risk and reward, 205–206 security market line, 198–202 tangency portfolio, 198 zero-beta portfolio, 212, 214 Cap-weighted indices, 221–224 Cauchy–Schwarz inequality, 105 Central limit theorem (CLT), 149, 178 Conditional expectation, 41–45 Correlation matrix, 100 Covariance function, 15 D Data matrix, 152–155 Decay parameter, 157 Diversification, 101–102 Dividends, 5, 8–9 E Effective sampling size, 159 Efficient frontier, 105, 77, 118 Efficient market hypothesis, Efficient portfolio theory, 76, 95–144 affine combinations, 109 Cauchy–Schwarz inequality, 105 correlation matrix, 100 diversification, 101–102 efficient frontier, 105, 118–120 eigenvalues, 100 holding constraints, 136–137 inner product, 97 matrix notation, 96–101 minimum-risk frontier, 103–113 minimum-variance portfolio, 113–117 Index nonnegative definite matrix, 99 opportunity set, 103 portfolio constraints, 133–139 portfolios of N assets, 95–102 quadratic programming problem, 110 random vector, 96 risk-aversion criterion, 121–128 Sharpe ratio, 137–139 tangency portfolio, 129–132 two-fund theorem, 110 variance matrix, 101 zero-investment portfolios, 106 Eigenvalues, 100 Empirical Bayes estimation, 189 Estimation, 145–195 basic sample statistics, 145–151 central limit theorem, 149 data matrix, 152–155 decay parameter, 157 effective sampling size, 159 exponentially weighted moving average estimator, 157 mean vector and covariance matrix, 151–157 observation period, 145 parametric bootstrap, 190 plug-in estimator, 171 portfolio weights, estimation of, 171–174 return means and standard deviations, estimation of, 146–148 sample covariance matrix, properties of, 156–157 sample covariances and correlations, 148–149 sampling horizon, 145 shrinkage estimators, 163–171 standard error, 150 statistical properties of estimators, 149–151 target matrix, 168 trace of a matrix, 156 T&F Cat #K31368 — K31368 A001— page 364 — 6/14/2017 — 22:05 Index using Monte Carlo simulation to study the properties of estimators, 174–189 weighted estimators, 157–163 Excess return of assets, 83 Exponentially weighted moving average (EWMA) estimator, 157 Extractor functions, 231 F Factor models, 3, 311–353 adjusted R-squared, 320 applications of, 343–349 arbitrage portfolio, 329 arbitrage pricing theory, 328–333 asymptotic arbitrage, 332–333 common factors, 316 economic factors, 321 estimation, 318–321 factor premiums, 333–343 factors, 321–328 factor sensitivities, 316 Fama–French three-factor model, 326 fundamental factors, 321, 325–328 “high minus low,” 326 imposing factor-sensitivity constraints, 346–349 limitations of single-index model, 311–315 model and its estimation, 315–321 no-arbitrage assumption, 330 no-asymptotic arbitrage assumption, 332 obtaining standard errors of premium estimates, 337–339 omitted-variable bias, 335 portfolios, 317–318 principal components analysis, 350 365 risk premium, 334 role of arbitrage pricing theory, 334–335 rolling regressions, 339–343 “small minus big,” 325 two-stage least-squares estimation, 335–337 using factor sensitivities to describe a portfolio, 345–346 value stock, 326 False discovery rate (FDR), 235–237 Fama–French three-factor model, 326 Federal Reserve Economic Data (FRED), Financial engineering, Float-adjusted index, 223 Freedman–Diaconis rule, 32 Fundamental analysis, G Geometric random walk, 52 Gross return, H “High minus low” (HML), 326 Holding constraints, 136–137 I Inner product, 97 Inter-quartile range (IQR) of data, 32 Iterated conditional expectations, 46 J Jensen’s alpha, 257–258 K K -period return, L Linear regression analysis, 201–202 Log-returns, 7–8 T&F Cat #K31368 — K31368 A001— page 365 — 6/14/2017 — 22:05 366 M Market capitalization, 209 Market model, 2, 221–272 adjusted beta, 243–244 appraisal ratio, 258–259 bias-corrected estimate, 264 “Bloomberg adjusted beta,” 243 Bonferroni inequality, 234 Bonferroni method, 235 bootstrap procedure, 261 cap-weighted indices, 221–224 comparison of portfolios, 266–268 correctly priced asset (hypothesis testing), 232–237 decomposition of risk, 237–239 diversification and, 247–254 estimation, 228–232 extractor functions, 231 false discovery rate, 235–237 float-adjusted index, 223 interpretation of βi , 228 investable weight factor of the asset, 223 Jensen’s alpha, 257–258 market capitalization, 222 market indices, 221–226 model and its estimation, 226–232 model formula, 230 portfolio performance, measurement of, 254–259 portfolio risk, 252–254 portfolios, application to, 244–247 portfolios of several assets, 249–252 price-weighted indices, 225–226 relationship to CAPM, 227–228 residual returns, 226 shrinkage estimation, 239–244 standard errors of estimated performance measures, 259–268 Index stock screening and multiple testing, 233–235 time-dependent portfolio weights, 246–247 Treynor ratio, 254–257 Market portfolio, 3, 197 Markowitz portfolio theory, 2, 91 Martingale model, 46–48 Mean function, 14 Mean squared error (MSE), 188 Mean-variance analysis, Minimum-risk frontier, 103–113 affine combinations, 109 calculating the weight vector of a portfolio on, 110–113 Cauchy–Schwarz inequality, 105 characterization, 106–109 efficient frontier, 105 opportunity set, 103 portfolios constructed from portfolios on, 109–110 quadratic programming problem, 110 zero-investment portfolios, 106 Minimum-variance portfolio, 110, 78, 113–117 Mispriced assets, 208–211 Model formula, 230 Modern portfolio theory, Monte Carlo simulation, properties of estimators studied using, 174–189 comparison of estimators, 186–189 description of sampling distribution of a statistic, 182–186 simulating a return vector, 178–182 MSE, see Mean squared error N N assets, portfolios of, 95–102 correlation matrix, 100 diversification, 101–102 T&F Cat #K31368 — K31368 A001— page 366 — 6/14/2017 — 22:05 Index 367 portfolios of two risky assets and a risk-free asset, 84–91 portfolio theory, 69 portfolio weights, 69 risk-aversion criterion, 79–81 risk-free assets, 81–84 Sharpe ratio, 87–88 tangency, 88–91 weights, estimation of, 171–174 zero-beta, 212, 214 O Portfolios of N assets, 95–102 Observation period, 145 correlation matrix, 100 Omitted-variable bias, 335 diversification, 101–102 Opportunity set, 75, 103 eigenvalues, 100 P inner product, 97 Parametric bootstrap, 190 matrix notation, 96–101 Passive investing, 292 nonnegative definite matrix, 99 Plug-in estimator, 171 random vector, 96 Portfolio constraints, 133–139 variance matrix, 101 holding constraints, 136–137 Prices, adjusted, 9–14 Sharpe ratio, 137–139 Price-weighted indices, 225–226 Portfolios, 1, 69–94; see also Efficient Principal components analysis, 350 portfolio theory eigenvalues, 100 inner product, 97 matrix notation, 96–101 nonnegative definite matrix, 99 random vector, 96 Net return, 5; see also Returns No-arbitrage assumption, 330 Nonnegative definite matrix, 99 Normal probability plot, 33 arbitrage, 329 basic concepts, 69–72 benchmark, 292 comparison of, 266–268 diversification, 71–72 efficient frontier, 77 efficient portfolios, 76–78 excess return of the asset, 83 “forced buy-in,” 74 “margin call,” 74 market, 197 Markowitz portfolio theory, 91 minimum-variance portfolio, 78–79 negative portfolio weights (short sales), 73–74 opportunity set, 75 optimal portfolios of two assets, 74–81 performance, measurement of, 254–259 portfolio selection problem, 69 Q Quadratic programming, 110, 127–128 Quantile–quantile plot (Q–Q plot), 33 Quantitative finance, R Random matrix theory, 189 Random vector, 96 Random walk hypothesis, 2, 41–67 application of random walk models to asset prices, 52–54 asset prices, random walk models for, 48–54 best linear predictor, 51 Box–Ljung test, 55 conditional expectation, 41–45 definitions of random walk, 49–52 T&F Cat #K31368 — K31368 A001— page 367 — 6/14/2017 — 22:05 368 drift, 50 efficient markets, 45–48 geometric random walk, 52 increments of the process, 49 iterated conditional expectations, 46 martingale model, 46–48 rescaled range test, 59–61 runs test, 58–59 sample autocorrelation function, test based on, 55 stock returns, 61–63 tests of, 54–61 variance-ratio test, 56–58 volatility, 50 Rescaled range test, 59–61 Residual returns, 226 Returns, 5–40 adjusted prices, 9–14 analyzing return data, 20–37 application to asset returns, 20 autocorrelation function, 16 autocovariance function, 16 basic concepts, 5–9 covariance function, 15 dividends, 5, 8–9 Freedman–Diaconis rule, 32 gross return, k -period return, log-returns, 7–8 mean function, 14 monthly returns, 24–26 net return, normal probability plot, 33 quantile–quantile plot, 33 return interval, 21 revenue, running means and standard deviations, 26–29 sample autocorrelation function, 29–32 sampling frequency of data, 21 second-order properties, 16 shape of return distribution, 32–37 Index stationarity, 15 statistical properties of, 14–20 stochastic process, 14 Sturges’ rule, 32 time series, 14 variance function, 14 volatility of the asset, 20 weak stationarity, 15–19 weak white noise, 19–20 Revenue, Risk-aversion criterion, 79–81, 121–128 finding wλ using quadratic programming, 127–128 properties of risk-averse portfolios, 124–127 Risk-free assets, 81–84 Risk premium, 334 Rolling regressions, 340 Root mean squared error (RMSE), 188 R-squared, adjusted, 320 Running means, 27 Runs test, 58–59 S Sample autocorrelation function, test based on, 55 covariance matrix, properties of, 156–157 covariances and correlations, 148–149 statistics, 145–151 Sampling distribution of a statistic, 182–186 frequency of data, 21 horizon, 145 size, effective, 159 Second-order properties (returns), 16 Sector funds, 309 Security market line (SML), 198–202 capital market line, 199 T&F Cat #K31368 — K31368 A001— page 368 — 6/14/2017 — 22:05 Index linear regression analysis, relationship to, 201–202 Sharpe ratio, 87–88, 129, 137 Shrinkage estimators, 163–171 Single-index model, 273–310 adding single asset to market portfolio, 293–298 applications to portfolio analysis, 286–291 benchmark portfolio, 292 correlation of asset returns under, 276–278 covariance structure of returns under, 275–281 estimation, 281–286 estimator bias, 304–306 limitations of, 311–315 matrix inverses, preliminary results on, 288–289 model, 273–275 numerical computation of portfolio weights, 306–307 partial correlation, 278–281 passive investing, 292 portfolio of N assets together with market portfolio, 298–302 sector funds, 309 Treynor–Black method, active portfolio management and, 292–307 weight vector of tangency portfolio under, 290–291 “Small minus big” (SMB), 325 SML, see Security market line Stationarity, 15 Statistical methods for financial models, introduction to, 1–4 capital asset pricing model, data analysis and computing, 3–4 efficient market hypothesis, factor model, 369 financial engineering, fundamental analysis, market model, market portfolio, Markowitz portfolio theory, mean-variance analysis, modern portfolio theory, portfolio, quantitative finance, random walk hypothesis, Stochastic process, 14 Stock returns, random walk model and, 61–63 Sturges’ rule, 32 T Tangency portfolio, 88, 129–132 Target matrix, 168 Time series, 14 Treynor–Black method, active portfolio management and, 292–307 adding single asset to market portfolio, 293–298 benchmark portfolio, 292 estimator bias, 304–306 numerical computation of portfolio weights, 306–307 portfolio of N assets together with market portfolio, 298–302 properties of Treynor–Black portfolio, 302–304 Treynor ratio, 254–257 Two-fund theorem, 110 U U.S Treasury Bill, 82 V Value stock, 326 Variance function, 14 Variance matrix, 101 Variance-ratio test, 56–58 Volatility of assets, 20 T&F Cat #K31368 — K31368 A001— page 369 — 6/14/2017 — 22:05 370 W Weak white noise, 19–20 Weighted estimators, 157–163 decay parameter, 157 effective sampling size, 159 exponentially weighted moving average estimator, 157 of mean vector and covariance matrix, 160–163 Index Z Zero-beta portfolio, 212, 214 Zero-investment portfolios, 106 T&F Cat #K31368 — K31368 A001— page 370 — 6/14/2017 — 22:05 ... 22:05 10 Introduction to Statistical Methods for Financial Models To see why this is true, consider one share of a particular stock and suppose that a dividend Dt is paid at time t Investors selling... and J.-B Denis Large Sample Methods in Statistics P.K Sen and J da Motta Singer Introduction to Statistical Methods for Financial Models T A Severini Spatio-Temporal Methods in Environmental Epidemiology... Nonparametric Statistical Methods, Fourth Edition P Sprent and N.C Smeeton Probability: Methods and Measurement A O’Hagan Data Driven Statistical Methods P Sprent Introduction to Statistical Limit