Chapter Eight Index Models

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Presentation transcript:

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.

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

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 ,

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

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 = βi2бM2 + б2(ei)

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 + 1 + (n2 – n)/2 + n = (n +1)(1 + n/2)

Number of input data under the assumption of the market model Input data: E(ri), rf, βi, бM2, б2(ei) = n + 1 + n + 1 + n = 3n + 2

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

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

Figure 8.2 Excess Returns on HP and S&P 500

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

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

Table 8.1 Interpreting the Output Correlation of HP with the S&P 500 is 0.7238 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 2.0348, but the 95% confidence interval is 1.43 to 2.53

Figure 8.4 Excess Returns on Portfolio Assets