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## PowerPoint Slideshow about ' Lecture 5 Index Model ' - tiffany-leda

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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 ModelSingle-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

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

Estimating the Index Model Coefficient

Excess Returns (i)

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Excess returns

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Ri = ai + ßiRm + ei

Estimating Beta Coefficient

- 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 Coefficient ’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 Coefficient

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 Coefficient

- 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 Coefficient

- 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 Coefficient

- Maximize the Sharpe ratio
- Expected return, SD, and Sharpe ratio:

Optimal Risky Portfolio of the Single-Index Model Continued Coefficient

- 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 Coefficient

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

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

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

Industry Betas and Adjustment Factors Sensitivity Statistics

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