Chapter 8: Index Models

Advantages of single factor market

Reduces number of inputs for diversification, easier for security analysts to specialize

Beta

Response of an individual security's return to the common factor, m; measure of systematic risk

m

common macroeconomic factor

ei

firm-specific surprises

Variance of single index model

Systematic risk + Firm specific risk

Covariance of single index model

Product of betas * market index risk

Correlation of single index model

Product of correlations with the market index

When n gets large, variance:

Becomes really small and firm specific risk is diversified away

Information ratio

The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio/passive strategy. The contribution of the active portfolio depends on the ratio of its alpha to its residual standard deviation. Measures the extra return we can obtain from security analysis

Why the single index model is better than the full covariance/Markowitz model

Using the full covariance matrix invokes estimation risk of thousands of terms. Also practical and decentralizes macro and security analysis

Average beta for all securities

1

Residual

ei

Part of the stock return not explained by the explanatory variable (market index return). Measures the impact of firm-specific event

The higher the information rate, the

Better.1 is outstanding; .75 is very good, .5 is good

Market neutral investment strategy

Seeks to avoid risk by hedging

Inputs for single index model

Estimate of standard deviation of S&P 500, risk premium on S&P 500 portfolio, and n sets of estimates of beta, stock residual variances, and alpha

Drawbacks of Markowitz procedure

Requires a huge number of estimates to fill the covariance matrix, and doesn't provide any guideline for forecasting security risk premium that are essential for constructing the efficient frontier of risky assets, high potential for error

Input list for 50 stocks

n=50 estimates of expected returns

n=50 estimates of variances

(n^2-n)/2=1,225 estimates of covariances

Joint normally distributed

When security returns can be well approximated by normal distributions that are correlated across securities. At any time, security returns are driven by 1 or more common variables

Correlation of securities for m and ei

M=Correlation between all securities because all are affected my macroeconomic events

ei=Uncorrellated across firms because it is specific to certain firms

Single factor model

Model of security returns that acknowledges only one common factor

Single index model

A model of stock returns that decomposes influences on returns into a systematic factor, as measured by the return on a broad market index, and firm-specific factors

A single factor model of the economy classifies sources of uncertainty as:

Systematic (macroeconomic) factors or firm-specific (microeconomic) factors

The index model is estimated by applying regression analysis to

Excess rates of return

Regression line

Security characteristic line

Full risky portfolio is a mixture of the

Active portfolio and the passive market-index portfolio

According to the index model, covariances among security pairs are:

Usually positive and due to the influence of a single common factor represented by the market index return

In a single factor model, the return on a stock in a particular period will be related to:

Macroeconomic and firm-specific events

Security returns:

A) are usually positively correlated with each other.

B) are based on both macro events and firm-specific events.

C) are based on firm-specific events only.

D) A and B

E) A and C

D

Slope of security characteristic/regression line

Beta

Intercept of security characteristic/regression line

Alpha

Positive alpha

Security is underpriced; good

Negative alpha

Security is overpriced; short sell the security

All securities are assumed to have an alpha value of

0

What does r squared value tell us

How much of a return is because of the market

Information ratio calculation

Alpha/Standard deviation of diversifiable risk

What does residual standard deviation give us

Firm specific risk