1. Single Index and Multifactor Models
Chapter 10
Single Index and Multifactor Models

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. Security Characteristic Line
Excess Returns (i)
SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excess returns
on market index . . . . . . . . . . . . . . . . .
Ri =  i + ßiRm + ei

7. Using the Text Example from Table 10-1
Excess
GM Ret.
Excess
Mkt. Ret.
Jan.
Feb. .
Dec
Mean
Std Dev
5.41
-3.44 .
2.43
-.60
4.97
7.24
.93 .
3.90
1.75
3.32

8.   rGM - rf = + ß(rm - rf) ß Estimated coefficient
Regression Results 
rGM - rf = ß(rm - rf)  ß
Estimated coefficient
Std error of estimate
Variance of residuals =
Std dev of residuals = 3.550
R-SQR = 0.575
-2.590
(1.547)
1.1357
(0.309)

9. 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

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

11. 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

12. Index Model and Diversification

13. Risk Reduction with Diversification
St. Deviation
Unique Risk
s2(eP)=s2(e) / n
bP2sM2
Market Risk
Number of Securities

14. Industry Prediction of Beta
Merrill Lynch Example
Use returns not risk premiums
a has a different interpretation
a = a + rf (1-b)
Forecasting beta as a function of past beta
Forecasting beta as a function of firm size, growth, leverage etc.

15. Use factors in addition to market return
Multifactor Models
Use factors in addition to market return
Examples include industrial production, expected inflation etc.
Estimate a beta for each factor using multiple regression.
Fama and French
Returns a function of size and book-to-market value as well as market returns.