Report

CHAPTER 8 Index Models Investments, 8th edition Bodie, Kane and Marcus Slides by Susan Hine McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. Advantages of the Single Index Model • Reduces the number of inputs for diversification • Easier for security analysts to specialize 8-2 Single Factor Model ri E (ri ) i m ei ßi = index of a securities’ particular return to the factor m = Unanticipated movement related to security returns ei = Assumption: a broad market index like the S&P 500 is the common factor. 8-3 Single-Index Model • Regression Equation: Rt (t ) i t RM (t ) ei (t ) • Expected return-beta relationship: E ( Ri ) i i E ( RM ) 8-4 Single-Index Model Continued • Risk and covariance: – Total risk = Systematic risk + Firm-specific risk: i2 i2 M2 2 (ei ) – Covariance = product of betas x market index risk: 2 Cov(ri , rj ) i j M – Correlation = product of correlations with the market index Corr (ri , rj ) i j M2 i j i M2 j M2 i M j M Corr (ri , rM ) xCorr (rj , rM ) 8-5 Index Model and Diversification • Portfolio’s variance: 2 P 2 P (eP ) 2 M 2 • Variance of the equally weighted portfolio of firm-specific components: 2 1 2 1 2 (eP ) (ei ) (e) n i 1 n n 2 • When n gets large, negligible (eP ) 2 becomes 8-6 Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp in the Single-Factor Economy 8-7 Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006 8-8 Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP 8-9 Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard 8-10 Figure 8.4 Excess Returns on Portfolio Assets 8-11 Alpha and Security Analysis • Macroeconomic analysis is used to estimate the risk premium and risk of the market index • Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2 ( e i ) • Developed from security analysis 8-12 Alpha and Security Analysis Continued • The market-driven expected return is conditional on information common to all securities • Security-specific expected return forecasts are derived from various security-valuation models – The alpha value distills the incremental risk premium attributable to private information • Helps determine whether security is a good or bad buy 8-13 Single-Index Model Input List • Risk premium on the S&P 500 portfolio • Estimate of the SD of the S&P 500 portfolio • n sets of estimates of – Beta coefficient – Stock residual variances – Alpha values 8-14 Optimal Risky Portfolio of the SingleIndex Model • Maximize the Sharpe ratio – Expected return, SD, and Sharpe ratio: n 1 n 1 i 1 i 1 E ( RP ) P E ( RM ) P wi i E ( RM ) wi i 1 2 n 1 n 1 2 2 2 2 2 (eP ) 2 M wi i wi (ei ) i 1 i 1 1 P P2 M2 SP E ( RP ) P 8-15 Optimal Risky Portfolio of the SingleIndex Model Continued • Combination of: – Active portfolio denoted by A – Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M – Modification of active portfolio position: 0 w * A – When wA 1 (1 A ) w 0 A A 1, w w * A 0 A 8-16 The Information Ratio • The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy): A s P s M (e ) A 2 2 2 8-17 Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix 8-18 Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models 8-19 Table 8.3 Merrill Lynch, Pierce, Fenner & Smith, Inc.: Market Sensitivity Statistics 8-20 Table 8.4 Industry Betas and Adjustment Factors 8-21