CHAPTER 8 Index Models Investments Cover image Slides by Richard D. Johnson McGraw-Hill/Irwin Copyright © 2008 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.

Single Factor Model ri = E(Ri) + ßiF + e ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor.

a (ri - rf) = i + ßi(rm - rf) + ei Single Index Model Risk Prem Market Risk Prem or Index Risk Prem a = the stock’s expected return if the market’s excess return is zero i (rm - rf) = 0 ßi(rm - rf) = the component of return due to movements in the market index ei = firm specific component, not due to market movements

Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) Ri = i + ßi(Rm) + ei

Components of Risk Market or systematic risk: risk related to the macro economic factor or market index. Unsystematic or firm specific risk: risk not related to the macro factor or market index. Total risk = Systematic + Unsystematic

Measuring Components of Risk i2 = i2 m2 + 2(ei) where; i2 = total variance i2 m2 = systematic variance 2(ei) = unsystematic variance

Examining Percentage of Variance Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2 ßi2  m2 / 2 = 2 i2 m2 / i2 m2 + 2(ei) = 2

Index Model and Diversification

Figure 8.1 The Variance of a Portfolio with Risk Coefficient Beta in the Single-Factor Economy

Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006

Figure 8.3 Scatter Diagram of HP, S&P 500, and Security Characteristic Line (SCL) for HP

Table 8.1 Regression Statistics for the SCL of Hewlett-Packard

Figure 8.4 Excess Returns on Portfolio Assets

Using the Single-Index Model with Active Management The single-index model can be extended to optimize the portfolio with active management The portfolio consists of an active portfolio and a passive or index portfolio The weight of the active portfolio is determined by the information ratio

Sharpe Ratio for the Combined Portfolio

Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models

Table 8. 3 Merrill Lynch, Pierce, Fenner & Smith Inc Table 8.3 Merrill Lynch, Pierce, Fenner & Smith Inc.: Market Sensitivity Statistics

Table 8.4 Industry Betas and Adjustment Factors