This document provides an overview of the single-factor index model. It discusses how the single-factor model decomposes risk into systematic and firm-specific components. The model uses a common factor, often a stock market index like the S&P 500, to measure systematic risk through securities' betas. It explains how the single-factor model leads to a regression equation relating individual security returns to expected returns and the common factor. The covariance between securities can then be expressed as the product of their betas and the variance of the common factor. The document concludes by noting that as a portfolio becomes more diversified, its total variance approaches only the systematic variance determined by the portfolio's beta and the variance of the market factor.

1. CHAPTER 8
INDEX
MODEL
MARIBEL V. VILLAVERDE
MBA 511 Investment Management

2. CHAPTER OVERVIEW
• Advantages of Single-factor model
• Risk decomposition
• Systematic vs firm specific
• Single Index & its estimation
• Optimal Risky Portfolio in the index
model
• Index model vs Markowitz procedure

3. EMERGENCE TO SINGLE INDEX MODEL
• 1952 “harry markowitz” published portfolio selection model that
maximized a portfolio’s return for a given level of risk
• This model required the estimation of:
Expected returns for each security
Variances for each security
A covariance matrix (calculated the covariance between each possible
pair or securities w/in the portfolio based on historical data through a
scenario analysis.

4.  The Input List of 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 co variances
 A 50 asset portfolio requires:
(N2 - N)/2
=2500-50
=2,450/2
=1,225 estimate of covariances
=1,325 total estimates

5. A Single factor model
Model
ßi- response of an individual securities return to the common
factor, m- measures systematic risk
m- common macroeconomic factor(systematic risk) S & P500 often
used as proxy
ei−firm specific susprises
ri=(E(ri)+unacticipated surprise
ri=(E(ri)+ßim(ei)
Ri-rate if returns
Variances = systematic risk + firm specific risk
σi
2= ß2
i σ 2
m+ σ2(ei)
Covariances-product of betas x market risk
Cov (ri rj)=ß1ßj σ2M

6. Single index models-returns
Approach leads to an equation similar to the single factor,
which uses the market index to proxy for the common
factor.
• Regression equation
• Ri(t) = αi + ßiRM(t) + ei(t)
• Expected return Beta-Relationship
• E(Ri) = ßiE(RM)

7. Correlation
Product of correlations with the market
index

8. Answer the following
questions
1. What is the mean index excess
return of the portfolio?
2. What is the covariance between
A & B, B & C, C& D and A & C, A
& D, B & D?
3. What is the covariance
between the stock A & the
index B, C, & D ?
4. Breakdown the variance of the
4 assets
The standard deviation of
market portfolio is 35%

12. variance of an equally
weighted portfolio with risk
coefficient ßp in the single-
factor economy
It summarize as
diversification increases; the
total variance of portfolio
approaches the systematic
variance, defined as the
variance of the market factor
multiplied by square of the
portfolio sensitivity
coefficient, ß2/

13. THANK YOU