CHAPTER 8 Index Models
Advantages of the Single Index Model
Reduces the number of inputs for diversification
Easier for security analysts to specialize
Cost:
accuracy
Inputs for Markowitz Model
To perform the necessary calculations you need the following data pieces for a portfolio
with N assets.
N estimates of returns
N estimates of variances
(N2 – N) / 2 estimates of covariances
A 30 asset portfolio requires
30 estimates of returns and variances
435 estimates of covariance
Single Factor Model
ri  E(ri )  i * m  ei
βi = response of an individual security’s return to the common factor, m. Beta
measures systematic risk.
m = a common macroeconomic factor that affects all security returns. The S&P
500 is often used as a proxy for m.
ei = firm-specific surprises
Single-Index Model
Regression Equation:
R(t )  i  i * RM  ei
Expected return-beta relationship:
E(R)  i  i * E(RM )
Risk and covariance:
Variance = Systematic risk and Firm-specific risk:
i2  i2 * M2  2 (ei )
Covariance = product of betas x market index risk:
COV(ri ,rj )  i *  j * M2
Correlation between 2 assets = product of correlations with the market index
Corr(ri , rj )  Corr(ri , rM ) * Corr(ri jrM )
Index Model and Diversification
Variance of the equally weighted portfolio of firm-specific components:
When n gets large, σ2(ep) becomes negligible and firm specific risk is diversified away.
 2 ( eP ) 
1 2
*  ( e)
n
Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp
We have seen this graph many times before. It shows the reduction in risk as portfolios
increase in size.
Figure 8.3 Scatter Diagram of HP, the S&P 500, and HP’s Security Characteristic Line (SCL)
Formula for the SCL
Ri  i  i * RM  ei
Variables for the Characteristic Line
Beta factor
indicator of the degree to which the stock responds to changes in the return in the
market
slope of the characteristic line
Note the similarity between the formulas for Beta and the correlation coefficient
The denominator is different


 j  rj   j * rm  cons tan t
Alpha
This term represents the return on the stock if the market had a return of zero
Residual
e j,t  rj,t   j   j * rmt   rj,t  E(rj,t )
since we will seldom if ever see perfect relationships, there will always be some “error”
in the representation of the line of best fit
It is a line of “best” fit not “perfect” fit
Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard
Multiple R = correlation = .7238
Adjusted R is a better measure – adjusts for upward bias 51.57%
Standard Error = impact of firm specific variables
Regression SS = % of variance explained by the mkt return
Residual M = % variance from unexplained factors
P value = % chance that the coefficient = zero;
if < .05 then coefficient is the “correct “ number and not 0
Table 8.1 Interpretation
Correlation of HP with the S&P 500 is 0.7238.
The model explains about 52% of the variation in HP.
HP’s alpha is 0.86% per month(10.32% annually) but it is not statistically significant.
HP’s beta is 2.0348 and has a p value of .0000.
This is less than .05 so we say it is important or signficant
The formula for predicting returns would be:
E(RHP )  0  2.0348 * E(RS&P )
C
or
re
la
ti
o
n
Coefficient of Determination
Standard Error
Regression MS
Model significance
Intercept value
Beta
Alpha and Security Analysis
The single index model separates some of the information
Risk premiums come from Macro-economic data
Betas and residuals come from statistical properties
We use the 2 pieces of data to establish a benchmark return
Any return beyond the benchmark (ALPHA) must come from some non-market factor
Positive alphas reflect a premium over the index based returns (under-priced)
GOAl
If only interested in diversification; hold the index
Security analysis then provides some additional potential returns by uncovering positive
alpha securities
Optimal risky portfolio
An active component of “analyzed” securities
A passive investment in the market index
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
Combination of:
Active portfolio denoted by A
Market-index portfolio, the passive portfolio denoted by M
The Information Ratio
The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index
portfolio (the passive strategy):
The contribution of the active portfolio depends on the ratio of its alpha to its residual
standard deviation.
The information ratio measures the extra return we can obtain from security analysis.
Is the Index Model Inferior to the Full-Covariance Model?
Full Markowitz model may be better in principle, but
Using the full-covariance matrix invokes estimation risk of thousands of terms.
Cumulative errors may result in a portfolio that is actually inferior to that derived from
the single-index model.
The single-index model is practical and decentralizes macro and security analysis.