1. 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

2. Advantages of the Single Index Model
Reduces the number of inputs for diversification.
Easier for security analysts to specialize.

3. 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.

4. 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

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

6. 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

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

8. 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

9. Index Model and Diversification

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

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

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

13. Table 8.1 Regression Statistics for the SCL of Hewlett-Packard

14. Figure 8.4 Excess Returns on Portfolio Assets

15. 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

16. Sharpe Ratio for the Combined Portfolio

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

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

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

20. Table 8.4 Industry Betas and Adjustment Factors