1 / 19

# Lecture 5 Index Model - PowerPoint PPT Presentation

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:

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

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

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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 Factor Model

Regression Equation:

Expected return-beta relationship:

• 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

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)

.

.

.

.

.

.

Security

Characteristic

Line

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Excess returns

on market index

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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

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

• Maximize the Sharpe ratio

• Expected return, SD, and Sharpe ratio:

• 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):