Single Index Models
Chapter 6
McGraw-Hill/Irwin
1
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
WARNING! DANGER!
To date we have been mostly solving problems
 We are now beginning the VERY
CONCEPTUAL component of the class

 Instead
of simple being able to solve the problems
you need to understand the theoretical concepts
and their implications
 In general this will require a much deeper
understanding of the material
 Please be prepared
2
6-2
Learning Objectives

Describe the advantages of a single-factor
model

Define systematic risk and firm-specific risk
and estimate the contribution of each to a
firm’s total risk

Describe the relationship between firmspecific risk and portfolio diversification

Identify the inputs of the single index model
and describe the security characteristic line
6-3
return
Where we left off
Optimal Risky
Portfolio
CAL
Efficient Portfolios
Risk
Free
P
4
6-4
Single Index Stock Market

In the real world constructing the efficient
frontier is practically impossible
 Too

many interactions
However, we can simplify by assuming that all
co-movement in returns results from a single
risk factor
 Idea
is that a single common risk factor
(systematic) is responsible for all the co-movement
in returns
5
6-5
Assumptions: Single Index Model

Returns are driven by a single, common systematic
factor
 Stock

The factor is systematic (Macroeconomic) affects
everything

measures a securities sensitivity to the factor


returns are joint normally distributed
Varies across securities
Firms are correlated with each other through their
correlation to the systematic (macroeconomic) factor
 Macroeconomic
surprises and firm-specific surprises are
not correlated
 Firm specific surprises are not correlated across firms
6-6
Single Index Stock Market
ri = βirM + ei + αi

A stocks excess return (ri)has three parts
1.
Return due to movements in the risk factor: βirM
is the market’s excess return
 βi is the sensitivity of the security’s returns to the
market factor; Systematic risk measure
 rM


Βi > 1: Cyclical stocks
Βi < 1: Defensive stocks
Return due to firm specific events: ei; Residual risk
3. Return beyond that induced by the market: αi
2.


Under or Over priced
Remember excess return → net of risk free
7
6-7
Where Does αi Come From?

Stocks have two values:
 Intrinsic
Value (IV) the present value of the
expected future cash flows

“True” Price according to a valuation model
 Market



Value (MV) is the consensus value of all
market participants
αi is postive when IV > MV, Under-Priced
αi is negative when IV < MV, Over-Priced
αi is 0 when IV = MV, Correctly Priced
8
6-8
Single Index Stock Market
ri = βirM + ei + αi
 ri is


the security’s excess return
rM is the market’s excess return
βi is the sensitivity of the security’s returns to the
market factor; Systematic risk measure
 Βi >
1: Cyclical stocks
 Βi < 1: Defensive stocks

ei is firm-specific or residual risk
 Surprises,

return independent of market factor
αi is the stock’s expected return beyond that induced
by market index; Under or Over priced
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Breaking Up Returns

A stocks excess return has three parts
Return due to movements in the risk factor: βirM
2. Firm Specific unexpected events: ei
3. Stock expected excess return: αi
1.

Is the stock under or over priced
10
6-10
Single Index Graph
ri = βirM + αi
Security
Characteristic
Line
11
6-11
Security Characteristic Line
Does NOT depict actual returns
 Does represent average tendencies

 Provides
 rD
the security’s expected return given rm
= βDRM + αD
 ei is
assumed to be 0
12
6-12
Example
If we expect the market to return to be 15%,
what return do we expect from a stock with a β
of 1.2?
 If its α is 3%?
 What about if the market only returned 13%?
 If e is -1%?

13
6-13
Expect versus Realized Returns
E(ri)
ri
ri  E (ri )  i F  ei






Actual return = Expected return + the effect of
surprises
ri = Actual return earn on the security
E(ri )= Expected return on the security
βi= Factor sensitivity or factor loading or factor beta
F = Surprise in macro-economic factor (+/-)
ei = Firm specific events
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6-14
Expect v Actual Return Example

If the market is expected to return 12% over
the next year, what is the expected return for a
stock with a β of 1.2? The risk free rate is 3%.

If the actual market return was 9%:
 What
is the market surprise?
 What was the actual return earned over the year?
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6-15
Breaking Down Variance

Variance is a measure of TOTAL risk
 Variance
= Systematic risk + Firm-specific risk
 Systematic risk = βi2σm2
 Firm-specific risk = σ(ei)2
σi2 = βi2σm2+ σ(ei)2

For a well diversified portfolio, what does
σ(ei)2 equal?
16
6-16
Variance Example
What is the variance of a stock with a beta of
0.9, if the standard deviation of the market is
25%, and its residual standard deviation is
30%?
 What if the market standard deviation
increases to 28%?

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6-17
How Important is the Market?

To determine the importance of systematic risk we
measure the ratio of systematic variance to total variance



This is the correlation coefficient squared
As ρ2 increases → the market is more important for
explaining firm returns
σ2(eD): variance of firm specific surprises


Determines spread of actual returns around SCL
Influences the importance of the market
18
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Single Index Graph
ri = βirM + ei + αi
β: Systematic risk
-Steepness
Spread for the SCL is
idiosyncratic risk
19
6-19
Diversification in a Single Index
World

All securities have systematic risk exposure, β
 Can’t

get rid of this
Portfolio β is just a weighted average of the
stock in the portfolio
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6-20
Diversification Continued

Variance of the equally weighted portfolio of
firm-specific components:
2
1 2
1 2
 (eP )      (ei )   (e)
n
i 1  n 
n
2

When n gets large, σ2(ep) becomes negligible and
firm specific risk is diversified away.
Questions
Which offers more diversification benefit?
 Which is riskier for an undiversified investor?
 Which is riskier for a diversified investor?

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6-22
Does investment horizon matter?
This is hotly debate
 Many belief that long term investors should
hold more stock (riskier assets) because they
become less risky over the long run

 Time

Diversification
The book and many academics argues against
this position
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