Lecture 5 index model
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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

Lecture 5 Index Model


Single factor model

ßi = index of a securities’ 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

Single-Index Model

Regression Equation:

Expected return-beta relationship:


Single index model continued

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

Index Model and Diversification

Portfolio’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 p in the single factor economy

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


Estimating the index model

Estimating the Index Model

Excess Returns (i)

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Excess returns

on market index

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Ri = ai + ßiRm + ei


Estimating beta

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

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

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

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

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

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

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 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

Efficient Frontiers with the Index Model and Full-Covariance Matrix


Comparison of portfolios from the single index and full covariance models

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


Merrill lynch pierce fenner smith inc market sensitivity statistics

Merrill Lynch, Pierce, Fenner & Smith, Inc.: Market Sensitivity Statistics


Industry betas and adjustment factors

Industry Betas and Adjustment Factors


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