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
lead
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
.
business
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
broad
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

)β
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
risk premium is due
to the risk premium
of the market index
→ systematic risk
premium
nonmarket
premium
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
premium,
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
retrn
潮
the
portfolio
潦
secrities
:
Note
:
The index model and diversification
24
The
portfolio
has
a
sensitivity
瑯
the
market
given
批
:
(
the
average
of
the
individual
i
s
)
䥴
has
a
nonmarket
retrn
component
潦
a
constant
†
(intercept)
:
(the
average
of
the
individual
alphas)
䥴
has
a
zero
mean
variable
:
(the
average
of
the
firm

specific
components)
25
T
桥
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
about
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:
instead of
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.
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