1. Chapter Eight Index Models
Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

2. Chapter Overview Advantages of a single-factor model
Risk decomposition
Systematic vs. firm-specific
Single-index model and its estimation
Optimal risky portfolio in the index model
Index model vs. Markowitz procedure

3. Single-Index Model (Market Model)
Suppose ri = ai + βi rM, (1) ai = a component of security i’s return that is not related to the market return; rM = the market return; βi = the sensitivity of security i’s return to the market return. Let ai = αi + ei , where αi = E(ai) (2) Substituting (2) into (1), we have ri = αi + βi rM + ei ,

4. Single-Index Model (Market Model)
ri = αi + βi rM + ei ,
ri = stock i’s return
 rM = market return
 βi = sensitivity of stock i’s return to the market
return
ei = return component due to stock specific events

5. Market Model vs Portfolio Analysis
From ri = αi + βi rM + ei and rj = αj + βj rM + ej
COV(ri rj) = COV(αi + βi rM + ei , αj + βj rM + ej)
= COV(βi rM, βj rM)
= βi βj COV(rM, rM)
= βi βj VAR(rM)
Hence, бij2 = βi βj бM2
VAR(ri) = COV(αi + βi rM + ei , αi + βi rM + ei)
= COV(βi rM + ei , βi rM + ei)
Hence, бi2 = βi2бM б2(ei)

6. Portfolio Optimization Problem
Max θ = [E(rM) – rf]/ бM
 where E(rM) = Σwi E(ri)
бM2 = ΣΣwiwj бi,j
Σwi = 1
Input data: E(ri), rf, бi,j
Number of input data = n (n2 – n)/2 + n
= (n +1)(1 + n/2)

7. Number of input data under the assumption of the market model
Input data: E(ri), rf, βi, бM2, б2(ei)
= n n n = 3n + 2

8. Single-Index Model Regression Equation: Rit = αi + βi RMt + eit
where Rit = rit – rft and RMt = rMt - rft

9. Single-Index Model Regression equation:
Expected return-beta relationship:

10. Figure 8.2 Excess Returns on HP and S&P 500

11. Figure 8.3 Scatter Diagram of HP, the S&P 500, and HP’s SCL

12. Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

13. Table 8.1 Interpreting the Output
Correlation of HP with the S&P 500 is
The model explains about 52% of the variation in HP
HP’s alpha is 0.86% per month (10.32% annually) but it is not statistically significant
HP’s beta is , but the 95% confidence interval is 1.43 to 2.53

14. Figure 8.4 Excess Returns on Portfolio Assets