Estimating Beta For A Portfolio Of Stocks

Estimating
Beta
For
A
Portfolio
Of
Stocks

Geoffery
Mullings

March
29,
2016

Executive
Summary:

For
some
a
stock's
beta
can
be
used
as
a
proxy
for
the
asset's
systemic
risk
-‐
that
is,

how
vulnerable
the
stock
is
to
movements
in
the
market.
This
analysis
estimates

beta
coefficients
for
a
portfolio
of
stocks
while
interpreting
and
comparing
those

values
to
more
standard
measures
of
risk
such
as
variance
and
correlation.
While
to

a
certain
extent
beta
represents
systemic
risk
its
ability
to
proxy
for
risk
is
quite

limited.
The
difference
between
upside
and
downside
risk
is
difficult
to
attain
from

beta,
while
covariance
offers
much
more
flexibility
in
portfolio
risk
managment.

Introduction:

A
stock's
beta
can
be
a
convenient
proxy
for
risk
depending
on
many
factors.
Beta
is

a
measure
of
comovement
with
the
market,
often
interpreted
as
a
measure
of

systemic
risk.
If
a
stock
moves
more
than
the
market
it
will
have
a
higher
beta,
while

the
reverse
is
true
for
a
lower
beta.
The
risk-‐reward
tradeoff
framework
suggests

that
stocks
with
lower
betas
may
be
safer
but
should
provide
less
opportunity
for

large
gains.
That
will
be
tested
in
the
analysis
below.

Beta
can
be
estimated
theoretically
through
a
regression
model,
perhaps
even
a

linear
regression
depending
on
the
movement
of
the
stock.
It's
also
possible
to

observe
historical
data
and
assess
whether
beta
accurately
reflects
our
normal

understanding
of
"risk."

Data
Loading
and
Munging:

require(tseries)

require(zoo)

require(fBasics)

#
Retrieving
the
Market
Factor.
Using
the
S&P
500
as
a
proxy
for
the

market's
movements.

sp
<-‐
get.hist.quote("^GSPC",
start='1989-‐12-‐01',
quote="AdjClose",

compression="m",
quiet=TRUE)

mytick
=
c('ATI','TSN','GME','LB','PAYX')
#
Storing
all
of
the
stocks

in
a
variable
for
looping.

data
<-‐
get.hist.quote(mytick[1],
quote="AdjClose",
start="1989-‐12-‐01",

compression="m",
quiet=TRUE,
retclass="zoo")

for
(i
in
2:length(mytick))

{

temp
=
get.hist.quote(mytick[i],
quote="AdjClose",
start="1989-‐12-‐
01",
compression="m",
quiet=TRUE,
retclass="zoo")

data
=
cbind(data,
temp)

}

colnames(data)
=
mytick

#
Transforming
into
logarithmic
returns.
Log
returns
are
easier
to

compare
over
time,
and
accurate
as
long
as
returns
aren't
too
large.

ret
=
100
*
diff(log(data))

sp
=
100
*
diff(log(sp))

colnames(sp)
=
"SP.500"

StocksMarketRet
=
cbind(ret,sp)

sum(is.na(StocksMarketRet))
#
Any
missing
return
values?

##
[1]
295

There
are
some
NA
values
in
the
historical
data
because
each
stock
had
its
IPO
after

1990.
To
reduce
bias
in
our
analysis
we'll
eliminate
all
observations
preceding
the

IPO
of
the
youngest
stock
in
the
portfolio,
GME.

StocksMarketRetAdj
=
StocksMarketRet[151:318,]
#
Eliminating
the
first

149
observations
from
all
assets
to
reduce
bias.

sum(is.na(StocksMarketRetAdj))

##
[1]
0

Let's
first
analyze
some
of
the
basic
statistics
about
this
portfolio's
and
the
S&P

500's
logarithmic
return.

Analysis:

basicStats(StocksMarketRetAdj)

##

ATI

TSN

GME

LB

PAYX

##
nobs

168.000000
168.000000
168.000000
168.000000
168.000000

##
NAs

0.000000

0.000000

0.000000

0.000000

0.000000

##
Minimum

-‐50.573656
-‐31.198382
-‐64.077922
-‐36.862335
-‐16.913253

##
Maximum

47.759670

26.659710

24.203372

27.204306

17.582045

##
1.
Quartile

-‐8.765500

-‐4.404103

-‐5.359523

-‐3.520060

-‐3.848341

##
3.
Quartile

9.622867

6.204392

8.764377

7.757846

4.966828

##
Mean

0.160533

1.092622

0.737814

1.363810

0.436373

##
Median

-‐0.441562

1.976514

1.494889

1.851768

1.036216

##
Sum

26.969589
183.560521
123.952835
229.120095

73.310595

##
SE
Mean

1.267847

0.714102

0.954303

0.711278

0.498472

##
LCL
Mean

-‐2.342541

-‐0.317208

-‐1.146238

-‐0.040446

-‐0.547746

##
UCL
Mean

2.663608

2.502453

2.621867

2.768066

1.420491

##
Variance

270.049434

85.670170
152.996535

84.994034

41.743632

##
Stdev

16.433181

9.255818

12.369177

9.219221

6.460931

##
Skewness

0.011003

-‐0.400221

-‐1.015992

-‐0.601945

-‐0.147011

##
Kurtosis

1.055338

1.152953

3.521931

1.594190

-‐0.326907

##

SP.500

##
nobs

168.000000

##
NAs

0.000000

##
Minimum

-‐18.563647

##
Maximum

10.230659

##
1.
Quartile

-‐1.773925

##
3.
Quartile

2.984705

##
Mean

0.341674

##
Median

1.018362

##
Sum

57.401290

##
SE
Mean

0.331765

##
LCL
Mean

-‐0.313320

##
UCL
Mean

0.996669

##
Variance

18.491439

##
Stdev

4.300167

##
Skewness

-‐0.868752

##
Kurtosis

1.923297

There's
some
upside
risk
in
ATI
(a
small
cap
metals
and
mining
firm)
based
on
the

mean,
median,
and
high
variation,
if
not
the
maximum
observation.
This
could
be

due
to
its
relative
youth
on
the
market,
similar
to
GME
(the
infamous
small
cap

video
game
retailer,
GameStop),
which
also
exhibits
a
high
variance
and
many
low

returns,
evidenced
by
a
large
negative
skew
and
minimum
value.
High
variance
is

also
not
particularly
uncommon
for
technology
stocks.

All
but
ATI
and
PAYX
(a
large
cap
IT
firm)
seem
negatively
skewed
based
on
mean

and
median
values,
with
the
largest
tails
present
on
GME
(which
also
happens
to
be

the
second-‐highest
stock
in
variance)
based
on
its
kurtosis
value.

LB
(a
large
cap
retailer
of
women's
underwear)
and
TSN
(the
large
cap
food

producer
Tyson
Foods)
share
with
the
other
stocks
high
variances
relative
to
the

market,
accompanied
by
higher
mean
returns
-‐
except
for
ATI.

Every
stock
in
the
portfolio
has
a
higher
return
variance
than
the
market.
Alongside

variance,
comovement
between
assets
and
the
market
should
be
assessed
on
the

road
to
estimating
beta.

plot(as.zoo(StocksMarketRetAdj),
main="Portfolio
Stock
Market
Returns")

plot(as.data.frame(StocksMarketRetAdj),
main="Portfolio
Stock
Market

Returns
Plotted
Against
Each
Other")

Visually,
there's
evidence
of
comovement
between
all
the
stocks
and
the
market

factor.
Obviously
some
stock
return
data
exhibits
more
varied
comovement
than

others,
and
finding
these
differences
between
the
asset
and
the
market
factor's

comovement
will
give
us
the
sought-‐after
estimate
of
beta.

cor(StocksMarketRetAdj)
#
Would
have
included
use="pairwise.complete"

if
I
hadn't
eliminated
the
missing
observations
already.

##

ATI

TSN

GME

LB

PAYX

SP.500

##
ATI

1.0000000
0.2801193
0.2450891
0.3920166
0.2263508
0.5653850

##
TSN

0.2801193
1.0000000
0.1623045
0.3337894
0.2221864
0.3987431

##
GME

0.2450891
0.1623045
1.0000000
0.3515588
0.2799321
0.3822975

##
LB

0.3920166
0.3337894
0.3515588
1.0000000
0.4319816
0.6131435

##
PAYX

0.2263508
0.2221864
0.2799321
0.4319816
1.0000000
0.5867857

##
SP.500
0.5653850
0.3987431
0.3822975
0.6131435
0.5867857
1.0000000

fit
=
lm(StocksMarketRetAdj[,mytick]
~
StocksMarketRetAdj$SP.500)

coef(summary(fit))

##
Response
ATI
:

##

Estimate
Std.
Error

t
value

Pr(>|t|)

##
(Intercept)

-‐0.5776988

1.052226
-‐0.5490256

5.837256e-‐01

##
StocksMarketRetAdj$SP.500

2.1606307

0.244650

8.8315179

1.411880e-‐15

##

##
Response
TSN
:

##

Estimate
Std.
Error

t
value

Pr(>|t|)

##
(Intercept)

0.7993742

0.6589283
1.213143
2.267986e-‐01

##
0.8582674

0.1532055
5.602066
8.630383e-‐08

##

##
Response
GME
:

##

Estimate
Std.
Error

t
value

Pr(>|t|)

##
(Intercept)

0.3620902

0.8872695
0.408095
6.837298e-‐01

##
1.0996562

0.2062965
5.330465
3.158340e-‐07

##

##
Response
LB
:

##

Estimate
Std.
Error

t
value

Pr(>|t|)

##
(Intercept)

0.9146684

0.5653664

1.617833
1.075972e-‐
01

##
1.3145315

0.1314517
10.000109
1.006882e-‐
18

##

##
Response
PAYX
:

##

Estimate
Std.
Error

t
value

Pr(>|t|)

##
(Intercept)

0.1351402
0.40613107
0.3327502
7.397427e-‐
01

##
0.8816360
0.09442837
9.3365581
6.361038e-‐
17

The
p
and
t-‐values
of
all
the
coefficients
suggests
that
each
is
statistically
significant

and
most
likely
a
reliable
estimate
of
their
respective
betas.

Notably
even
though
TSN
and
GME
have
some
of
the
weakest
correlations
with
the

market
factor,
they
have
betas
that
are
very
close
to
1.
While
correlation
shows
the

strength
of
return
similarities
between
the
stock
and
the
market,
beta
really

estimates
the
effect
of
a
market
movement
on
each
asset.

Conclusion:

The
preceding
has
hopefully
made
it
clear
that
beta
provides
some
utility

accompanied
by
many
limits.
Beta
is
much
better
at
capturing
all
around
variance

within
a
stock's
returns,
relative
to
the
market,
than
specifically
downside
risk.
Take

for
example
that
ATI
had
the
highest
upside
risk
as
measured
by
its
maximum

observation
and
other
variables,
but
has
a
far
higher
estimated
beta
than
GME
-‐
the

stock
with
the
highest
downside
risk
in
our
portfolio
based
on
the
same
measures.

Similarly,
beta
does
not
communicate
well
correlation
or
covariances,
measures

with
the
potential
to
control
portfolio
risk
much
more
effectively.

It's
a
relatively
easy
to
estimate
the
measure,
but
beta
hardly
scratches
the
surface

in
relaibility
when
it
comes
to
assessing
portfolio
risk.

Estimating Beta For A Portfolio Of Stocks

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