# Lecture 5 Index Model - PowerPoint PPT Presentation

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Lecture 5 Index Model. ß i = index of a securities ’ particular return to the factor m = Unanticipated movement related to security returns e i = Assumption: a broad market index like the S&P 500 is the common factor. Single Factor Model. Single-Index Model. Regression Equation:

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Lecture 5 Index Model

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## Lecture 5 Index Model

m = Unanticipated movement related to security returns

ei = Assumption: a broad market index like the S&P 500 is the common factor.

### Single-Index Model

Regression Equation:

Expected return-beta relationship:

### Single-Index Model Continued

• Risk and covariance:

• Total risk = Systematic risk + Firm-specific risk:

• Covariance = product of betas x market index risk:

• Correlation = product of correlations with the market index

### Index Model and Diversification

Portfolio’s variance:

Variance of the equally weighted portfolio of firm-specific components:

When n gets large, becomes negligible

### Estimating the Index Model

Excess Returns (i)

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Excess returns

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Ri = ai + ßiRm + ei

### Estimating Beta

• The standard procedure for estimating betas is to use single index model

Rj = a + b Rm

• where a is the intercept and b is the slope of the regression.

• The slope of the regression corresponds to the beta of the stock, and measures the the sensitivity of the stock price’s change to the change of market price

### Gillette’s Beta

• Period used: September 1998 to August 2003

• Return Interval = Monthly

• Market Index: S&P 500 Index

• ReturnsGillette = 0.02% + 0. 40 ReturnsS&P500

(0.011)

• Intercept =0.02%

• Slope = 0.40= Beta

• R squared = 5.5%

• Problem: low confidence

### Alpha and Security Analysis

Macroeconomic analysis is used to estimate the risk premium and risk of the market index

Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2 ( e i )

Developed from security analysis

### Alpha and Security Analysis Continued

• The market-driven expected return is conditional on information common to all securities

• Security-specific expected return forecasts are derived from various security-valuation models

• The alpha value distills the incremental risk premium attributable to private information

### Single-Index Model Input List

• Risk premium on the S&P 500 portfolio

• Estimate of the SD of the S&P 500 portfolio

• n sets of estimates of

• Beta coefficient

• Stock residual variances

• Alpha values

### Optimal Risky Portfolio of the Single-Index Model

• Maximize the Sharpe ratio

• Expected return, SD, and Sharpe ratio:

### Optimal Risky Portfolio of the Single-Index Model Continued

• Combination of:

• Active portfolio denoted by A

• Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M

• Modification of active portfolio position:

• When

### The Information Ratio

The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):