Lecture 5 Index Model. ÃŸ i = index of a securities â€™ 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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ÃŸi = index of a securitiesâ€™ particular return to the factor

m = Unanticipated movement related to security returns

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

Single Factor Model

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

Portfolioâ€™s variance:

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

When n gets large, becomes negligible

The Variance of an Equally Weighted Portfolio with Risk Coefficient Î²p in the Single-Factor Economy

Estimating the Index Model

Excess Returns (i)

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Excess returns

on market index

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Ri = ai + ÃŸiRm + 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 priceâ€™s change to the change of market price

Gilletteâ€™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, Ïƒ2 ( 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

Helps determine whether security is a good or bad buy

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):

Efficient Frontiers with the Index Model and Full-Covariance Matrix

Comparison of Portfolios from the Single-Index and Full-Covariance Models