# Index Models

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28 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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1

A single
-
factor security market

The single
-
index model

Estimating the single
-
index model

Topic 3
(Ch. 8)

Index Models

2

The success of a portfolio selection rule depends
on the quality of the input list (i.e. the estimates of
expected security returns and the covariance
matrix).

e.g.

To analyze 50 stocks, the input list includes:

n

= 50 estimates of expected returns

n

= 50 estimates of variances

(
n
2
-

n
)/
2

= 1,225 estimates of covariances

1,325 estimates

A Single
-
Factor Security Market

3

If

n

=

3
,
000

(roughly

the

number

of

NYSE

stocks),

we

need

more

than

4
.
5

million

estimates
.

Errors

in

the

assessment

or

estimation

of

correlation

coefficients

can

to

nonsensical

results
.

This

can

happen

because

some

sets

of

correlation

coefficients

are

mutually

inconsistent
.

e
.
g
.

Asset

Standard
Deviation

(%)

Correlation Matrix

A

B

C

A

20

1.0

0.9

0.9

B

20

0.9

1.0

0.0

C

20

0.9

0.0

1.0

4

Construct

a

portfolio

with

weights
:

-
1
.
00
;

1
.
00
;

1
.
00
,

for

assets

A
;

B
;

C,

respectively,

and

calculate

the

portfolio

variance
.

Portfolio

variance

=

-
㈴2
!

Covariances

between

security

returns

tend

to

be

positive

because

the

same

economic

forces

affect

the

fortunes

of

many

firms

(e
.
g
.

cycles,

interest

rates,

technological

changes,

etc
.
)
.

All

these

(interrelated)

factors

affect

almost

all

firms
.

Thus,

unexpected

changes

in

these

variables

cause,

simultaneously,

unexpected

changes

in

the

rates

of

return

on

the

entire

stock

market
.

5

Suppose

that

we

summarize

all

relevant

economic

factors

by

one

macroeconomic

indicator

and

assume

that

it

moves

the

security

market

as

a

whole
.

We

further

assume

that,

beyond

this

common

effect,

all

remaining

uncertainty

in

stock

returns

is

firm

specific

(i
.
e
.

there

is

no

other

source

of

correlation

between

securities)
.

Firm
-
specific

events

would

include

new

inventions,

deaths

of

key

employees,

and

other

factors

that

affect

the

fortune

of

the

individual

firm

without

affecting

the

economy

in

a

measurable

way
.

6

We

can

summarize

the

distinction

between

macroeconomic

and

firm
-
specific

factors

by

writing

the

holding
-
period

return

on

security

i

as
:

where

E
(
r
i
)
:

expected

return

on

the

security

i

as

of

the

beginning

of

the

holding

period
;

m

:

impact

of

unanticipated

macro

events

on

all

securities’

return

during

the

period
;

e
i
:

impact

of

unanticipated

firm
-
specific

events
.

Note
:

Both

m

and

e
i

have

0

expected

values

because

each

represents

the

impact

of

unanticipated

events,

which

by

definition

must

average

out

to

0
.

7

Since

m

and
e
i

are uncorrelated, the variance of
r
i

arises from two uncorrelated sources, systematic
and firm specific.

Since
m

is also uncorrelated with any of the firm
-
specific surprises, the covariance between any two
securities
i
and
j
is

8

Some

securities

will

be

more

sensitive

than

others

to

macroeconomic

shocks
.

We

can

capture

this

refinement

by

assigning

each

firm

a

sensitivity

coefficient

to

macro

conditions
.

Thus,

if

we

denote

the

sensitivity

coefficient

for

firm

i

by

i
,

we

have

the

following

single
-
factor

model
:

The

systematic

risk

of

security

i

is

determined

by

its

beta

coefficient

(

i
)
.

9

The variance of the rate of return on each
security
includes 2 components:

: variance attributable to the uncertainty
of the common macroeconomic factor (i.e.
systematic risk)

: variance attributable to firm
-
specific
uncertainty.

10

The covariance between any pair of securities
is determined by their betas:

11

To make the single
-
factor model operational,
we use the rate of return on a broad index of
securities (such as S&P 500) as a proxy for the
common macroeconomic factor.

This approach leads to an equation similar to
the single
-
factor model, which is called the
single
-
index model
, because it uses the market
index to proxy for the common factor.

The Single
-
Index Model

12

The regression equation of the
single
-
index model

Denote the market index by
M
, with excess
return of
R
M

= r
M

-

r
f

and standard
deviation of
σ
M
.

Excess return of a security:

R
i

= r
i

r
f

13

Collect a historical sample of paired observations
and regress
R
i
(t
) on
R
M
(t
), where
t

denotes the date
of each pair of observations.

The regression equation

is

Intercept
:

α
i
: the security

i
’s expected excess return when the

market excess return is zero.

14

Slope coefficient
:

β
i
: the security

i
’s sensitivity to the market index.

For every + (or
-
) 1% change in the market excess
return, the excess return on the security will
change by + (or
-

i
%.

Residual
:

e
i

is the zero
-
mean, firm specific surprise in the
security return in time
t
.

15

The expected return
-
beta
relationship

part of a security’s
of the market index

→ systematic risk

nonmarket

16

Risk and covariance in the
single
-
index model

Recall that we have the following equation:

The variance of the rate of return on each security
includes 2 components:

: variance attributable to the uncertainty of the

market index

17

:

v
ariance

attributable

to

firm
-
specific

uncertainty
.

(total risk = systematic risk + firm
-
specific risk)

Note
:

The covariance between
R
M

and
e
i

is zero because
e
i

is
defined as firm specific (i.e. independent of movements
in the market).

18

The

covariance

between

the

rates

of

return

on

2

securities
:

Note
:

Since

i

and

j

are

constants,

their

covariance

with

any

variable

is

zero
.

Further,

the

firm
-
specific

terms

(
e
i
,

e
j
)

are

assumed

uncorrelated

with

the

market

and

with

each

other
.

Covariance

=

Product

of

betas

×

Market

index

risk

19

The

covariance

between

the

return

on

stock

i

and

the

market

index
:

Notes
:

W
e

can

drop

i

from

the

covariance

terms

because

i

is

a

constant

and

thus

has

zero

covariance

with

all

variables
.

T
he

firm
-
specific

or

nonsystematic

component

is

independent

of

the

marketwide

or

systematic

component

(i
.
e
.

Cov(
e
i
,

R
M
)

=

0
)
.

20

The

correlation

coefficient

between

the

rates

of

return

on

2

securities
:

(product of correlations with the market index)

21

If

we

have
:

n

estimates

of

the

extra
-
market

expected

excess

returns,

α
i

n

estimates

of

the

sensitivity

coefficients,

β
i

n

estimates

of

the

firm
-
specific

variances,

σ
2
(e
i
)

1

estimate

for

the

market

risk

1 estimate for the variance of the (common)

macroeconomic factor, σ
M
2

then

these

(
3
n

+

2
)

estimates

will

enable

us

to

prepare

the

input

list

for

this

single
-
index

security

universe
.

The set of estimates needed
for the single
-
index model

22

For

n

=

50
:

need

152

estimates

(not

1
,
325

estimates)
.

n

=

3
,
000
:

need

9
,
002

estimates

(not

4
.
5

million)
.

23

S
uppose

that

we

choose

an

equally

weighted

portfolio

of

n

securities

(I
.
e
.

w
i

=

1
/
n
)
.

The

excess

rate

of

return

on

each

security

is
:

The

excess

retrn

the

portfolio

secrities
:

Note
:

The index model and diversification

24

The

portfolio

has

a

sensitivity

the

market

given

:



(
the

average

of

the

individual

i
s
)

has

a

nonmarket

retrn

component

a

constant

(intercept)
:

(the

average

of

the

individual

alphas)

has

a

zero

mean

variable
:

(the

average

of

the

firm
-
specific

components)

25

T

portfolio’s

variance

is
:

The systematic risk component of the portfolio variance
(the component that depends on marketwide
movements) is and depends on the sensitivity
coefficients of the individual securities.

This part of the risk depends on portfolio beta and ,
and will persist regardless of the extent of portfolio
diversification.

No matter how many stocks are held, their common
exposure to the market will be reflected in portfolio
systematic risk.

26

In contrast, the nonsystematic component of the
portfolio variance is

2
(
e
P
)

and is attributable to firm
-
specific components
e
i
.

Because the
e
i
s are uncorrelated, we have:

where

: the average of the firm
-
specific variances
.

Because this average is independent of
n,
when
n
gets large,

2
(
e
P
) becomes negligible.

Thus, as more and more securities are added to the
portfolio, the firm
-
specific components tend to cancel
out, resulting in ever
-
smaller nonmarket risk.

27

28

Summary
:

As more and more securities are combined into a
portfolio, the portfolio variance decreases because of
the diversification of firm
-
specific risk.

However, the power of diversification is limited.

Even for very large
n
, part of the risk remains because
of the exposure of virtually all assets to the common, or
market, factor.

Therefore, this systematic risk is said to be
nondiversifiable.

29

The

single
-
index

model

suggests

how

we

might

go

actually

measuring

market

and

firm
-
specific

risk
.

Suppose

that

we

observe

the

excess

return

on

the

market

index

and

a

specific

asset

over

a

number

of

holding

periods
.

We

use

as

an

example

monthly

excess

returns

on

the

S&P

500

index

and

GM

stock

for

a

one
-
year

period
.

Estimating the Single
-
Index Model

30

31

We

can

summarize

the

results

for

a

sample

period

in

a

scatter

diagram
:

32

The

single
-
index

model

states

that

the

relationship

between

the

excess

returns

on

GM

and

the

S&P

500

is

given

by

the

following

regression

equation
:

In

this

single
-
variable

regression

equation,

the

dependent

variable

plots

around

a

straight

line

with

an

intercept

and

a

slope

.

The

deviations

from

the

line

(
e
)

are

assumed

to

be

mutually

uncorrelated

and

uncorrelated

with

the

independent

variable
.

33

The

sensitivity

of

GM

to

the

market,

measured

by

GM
,

is

the

slope

of

the

regression

line
.

The

intercept

of

the

regression

line

is

GM
,

representing

the

average

firm
-
specific

return

when

the

market’s

excess

return

is

zero
.

Deviations

of

particular

observations

from

the

regression

line

in

any

period

are

denoted

e
GM
,

and

called

residuals

(i
.
e
.

each

of

these

residuals

is

the

difference

between

the

actual

security

return

and

the

return

that

would

be

predicted

from

the

regression

equation

describing

the

usual

relationship

between

the

security

and

the

market)
.

Thus,

residuals

measure

the

impact

of

firm
-
specific

events
.

34

Estimating

the

regression

equation

of

the

single
-
index

model

gives

us

the

security

characteristic

line

(SCL)
.

The

SCL

is

a

plot

of

the

typical

excess

return

on

a

security

as

a

function

of

the

excess

return

on

the

market
.

Compute

GM

and

GM
:

Let

y
t
:

excess

return

on

GM

in

month

t

x
t
:

excess

return

on

the

market

(S&P

500
)

in

month

t

n
:

the

total

number

of

months
.

35

The

estimate

of

beta

coefficient

(i
.
e
.

the

slope

of

the

regression

line

SCL)
:

The

intercept

of

the

regression

line
:

36

37

38

Compute

residuals
:

For

each

month

t
,

our

estimate

of

the

residual

is

the

deviation

of

GM’s

excess

return

from

the

prediction

of

the

SCL
:

Deviation

=

Actual

Predicted

Return

These

residuals

are

estimates

of

the

monthly

unexpected

firm
-
specific

component

of

the

rate

of

return

on

GM

stock
.

39

40

Hence,

we

can

estimate

the

firm
-
specific

variance
:

The

standard

deviation

of

the

firm
-
specific

component

of

GM’s

return
:

which

is

equal

to

the

standard

deviation

of

the

regression

residual
.

41

Practitioners often use a “modified”
index model that uses total rather than
excess returns (deviations from T
-
bill
rates) in the regressions:

The Industry Version of the Index Model

42

To see the impact of this departure:

If
r
f

is constant over the sample period, both
equations have the same independent variable
r
M

and residual
e
.

Thus, the slope coefficient will be the same in the
two equations.

43

However, the intercept is really an estimate of

The apparent justification for this procedure is
that, on a monthly basis,
r
f
(
1
-

) is small.

But, note that for
β≠1, the regression intercept will
not equal the index model alpha.

44

Betas estimated form past data may not be the
best estimates of future betas.

This suggests that we might want a forecasting
model for beta.

One simple approach would be to collect data on
beta in different periods and then estimate a
regression equation:

Current beta = a + b (Past beta)

Given estimates of a and b, we would then forecast
future betas using the rule:

Forecast beta = a + b (Current beta)

Predicting Betas

45

However, there is no reason to limit ourselves to
such simple forecasting rules.

Why not also investigate the predictive power of
other financial variables in forecasting beta?

Rosenberg and Guy find the following variables
help predict betas:

Variance of earnings.

Variance of cash flow.

Growth in earnings per share.

Market capitalization (firm size).

Dividend yield.

Debt
-
to
-
asset ratio.

46

Rosenberg and Guy also find that even after
controlling for a firm’s financial
characteristics, industry group helps to
predict beta.