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6
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Unformatted text preview: Chapters
6
 Efficient
Diversifica3on
and
Op3mal
 Por:olio

 Diversifica3on
and
Por:olio
Risk 
 •  Market
risk
 –  Systema3c
or
non‐diversifiable
risk
 –  Largely
macro‐economic
factors,
such
as
business 
cycle,
infla3on
rate,
interest
rate
…
 •  Firm‐specific
risk
 –  Diversifiable,
nonsystema3c,
or
residual
risk
 –  Subject
to
firm‐specific
impact,
e.g.
firm’s
earning 
report,
research
and
development,
management 
style
…
 Por:olio
risk
as
a
func3on
of
the 
number
of
stocks
in
the
por:olio 
 Por:olio
risk
decreases
as 
diversifica3on
increases,
1987 
 Why?
 Two‐Security
Por:olio:
Return 
 •  rp=W1
r1
+
W2
r2
,
with
W1
+
W2
=1,
 •  W1,
W2
:
%
of
total
money
invested
in
Security
 1
and
Security
2,
respec:vely.

 •  
 r1
,
r2
:
rate
of
return
on
Security
1
&
2
 •  E(rp)=W1
E(r1)+
W2
E(r2
)

 •  In
the
special
case
where
security
1
is
risky
 porAolio,
security
2
is
risk‐free
fund:

 –  
E(rp)=W1
E(r1)+
W2
rf


 Two‐Security
Por:olio:
risk
 •  rp=W1
r1
+
W2
r2
,
with
W1
+
W2
=1
 •  In
the
special
case
where
security
1
is
risky
 por:olio,
security
2
is
risk‐free
fund:

 –  σp=W1
σr1
<
σr1

(since
W1<1)
 •  What
if
both
assets
are
risky?
 –  σp2=W12
σr12+
W22
σr22+2W1W2Cov(r1,
r2)
 Por:olio
risk
 
 in
a
risky
universe
of
assets 
 •  Por:olio
risk
depends
on
the
correla3on 
between
the
returns
of
the
assets
in
the 
por:olio
 •  Covariance
and
the
correla3on
coefficient 
provide
a
measure
of
the
way
in
which
the 
returns
of
two
assets
vary
 Por:olio
risk
when
ρ1,2 = 1
 Por:olio
risk
when
ρ1,2 = -1 
 Two‐Security
Por:olio:
Risk 
 •  σ12 = Variance of Security 1 •  σ22 = Variance of Security 2 •  Cov(r1, r2) =
Covariance of returns for Security 1 and Security 2, –  Cov(r1, r2) =
E{[r1-E(r1)][r2‐E(r2)]} •  rp=W1
r1
+
W2
r2
,
with
W1
+
W2
=1
 •  E(rp)=W1
E(r1)+
W2
E(r2
)
 •  σp2=W12
σr12+
W22
σr22+2W1W2Cov(r1,
r2)
 Another
way
to
express
variance 
of
the
por:olio: 
 •  Cov(r1,r2) = ρ1,2σ1σ2 •  ρ1,2 = Correlation coefficient of returns •  σ1 = Standard deviation of returns for Security 1 •  σ2 = Standard deviation of returns for Security 2 •  σp2=W12
σr12+
W22
σr22+2(W1σr1)(W2σr2)ρ








(6.6) Covariance
in
scenario
analysis 
 S Cov ( rs, rB ) = ∑ p(i)[ rs (i) − rs ][ rB (i) − rB ] i=1 € Calculating Variance and Covariance Ex post σ2ABC = 1.37387 / (10-1) = 0.15265 σABC = 39.07% σ2XYZ = 1.57885 / (10-1) = 0.17543 σXYZ = 41.88% Cov(ABC,XYZ) = 0.533973 / (10-1) = 0.059330 ρABC,XYZ = Cov / (σABCσXYZ) = 0.059330 / (0.3907 x 0.4188) = 0.3626 n Cov(r1, r2 ) = ∑ t =1 (r1,t − r 1 ) × (r 2,t − r 2 ) n -1 Ex
post
Covariance
in
Excel 
 Correla3on
Coefficients:
Possible 
Values 
 •  Range of values for ρ1,2: + 1.0 > ρ > -1.0 •  If ρ = 1.0, the securities would be perfectly positively correlated •  If ρ = - 1.0, the securities would be perfectly negatively correlated •  What is portfolio risk when ρ=1, 0, -1? Background
knowledge
about 
covariance 
 •  Covariance
does
not
tell
us
the
intensity
of 
the
co‐movement
of
the
stock
returns,
only 
the
direc3on.
 •  We
can
standardize
the
covariance
however 
and
calculate
the
correla3on
coefficient 
which
will
tell
us
not
only
the
direc3on
but 
provides
a
scale
to
es3mate
the
degree
to 
which
the
stocks
move
together.
 Background
knowledge
about 
correla3on 
 •  ρ1,2
=
ρ2,1
and
the
same
is
true
for
the
COV
 •  The
covariance
between
any
stock
such
as 
Stock
1
and
itself
is
simply
the
variance
of 
Stock
1,

 •  ρ1,1
=
+1.0
by
defini3on
 •  Measures
for
how
three
or
more
stocks
move 
together?
 •  
co‐integra3on
 Two‐Security
Por:olio:
Risk 
 •  Investment
opportunity
set
 –  All
aiainable
combina3ons
of
risk
and
return 
offered
by
por:olios,
using
the
available
assets 
in
differing
propor3ons.

 Risk
and
return
in
a
por:olio
of
 
 
stock
and
bond
funds 
 The
investment
opportunity
set 
with
the
stock
and
bond
funds 
 Efficient
fron3er
of
risky
assets 
 •  The
curve
in
last
slide
is
called
“efficient 
fron3er”
 –  It
is
the
most
efficient
(best
possible)
risk‐return 
combina3ons
available
from
the
universe
of 
risky
assets
 Minimum
Variance
Por:olio
 
 for
a
fixed
ρ 
 •  Consider
a
por:olio
with
a
stock
fund
and
a
bond 
fund,

 •  The
minimum
variance
por:olio
has
weights 
(propor3on
of
money
invested):
 2 σ S − σ Bσ S ρ B ,S WB = 2 2 σ S + σ B − 2σ Bσ S ρ B ,S W S = 1 − WB Note:
Cov(WB,
WS)=σBσSρB,S
 Q:
What’s
the
weight
in
stock
fund
in
our
example?
 € The
impact
of
ρ 
 PorAolio
risk
with
various 
correla:on
coefficients 
 Investment
opportunity
sets
for 
bonds
and
stocks
with
various
ρ
 
 Correla3on
Effects 
 •  The
rela3onship
depends
on
the
correla3on 
coefficient
 •  ‐1.0
<
ρ
<
+1.0
 •  The
smaller
the
correla3on,
the
greater
the 
risk
reduc3on
poten3al
 •  If
ρ =
+1.0,
no
risk
reduc3on
is
possible
 –  σp=W1
σr1+
W2
σr2
 •  If
ρ =
‐1.0,
 –  σp=|W1
σr1
–
W2
σr2|
 Opportunity
set
of
stocks,
bonds,
and
 
a
risk‐free
asset,
with
two
CALs 
 Recall
re
CAL:

 Risk‐free
rate
=
intercept

 Sharpe
Ra:o=Slope


 The
Sharpe
Ra3o 
 •  Maximize
the
slope
of
the
CAL
for
any 
possible
por:olio,
p
 •  The
objec3ve
func3on
is
the
slope:
 E ( rP ) − rf Sharpe ratio = σP •  Sharpe
ra3o
of
the
minimum
variance 
por:olio=
(5.46
–
3)/7.8=0.32
 € The
Opportunity
Set
of
stocks
and 
bonds,
with
the
op3mal
CAL
and 
the
op3mal
risky
por:olio 
 Sharpe
Ra3o
of
op3mal

 Risky
por:olio=
 (7.17‐3)/10.15=0.41
 Op3mal
por:olio 
 •  The
op3mal
por:olio
weights
are:
 2 [ E ( rB ) − rf ]σ S − [ E ( rS ) − rf ]σ Bσ S ρ B ,S WB = 2 2 [ E (rB ) − rf ]σ S + [ E (rS ) − rf ]σ B − [ E (rB ) − rf + E ( rS ) − rf ]σ Bσ S ρ B ,S W S = 1 − WB •  For
ρ=0.2, WB(O)=0.568,
WS(O)=0.432
 •  So,
E(rO)=0.568(5%)
+
0.432(10%)=7.16%;
 
σ2=
0.5682(8%)2
+
0.4322
(19%)2
+
2 
x0.568x0.432(8%)(19%)(0.2)
 σO=
10.145%
 A
complete
por:olio 
 •  Recall:
a
preferred
complete
por:olio
formed 
from
a
risky
por:olio
and
a
risk‐free
asset 
depends
on
the
investor’s
risk
aversion
 –  More
risk‐averse
investors
prefer
low‐risk 
por:olios
despite
lower
expected
return
 •  For
example,
an
investor
would
like
to
put 
55%
of
wealth
in
por:olio
O
and
45%
in
T‐bill
 –  E(rC)=
0.45(3%)+0.55(7.16%)=5.29%
 –  σC=
0.55(10.15%)=5.58%
 A
complete
por:olio:
cont 
 Por:olio
construc3on
 
 1.  Iden3fy
the
best
possible
or
most
efficient 
risk‐return
combina3ons
available
from
the 
universe
of
risky
assets
 2.  Determine
the
op3mal
por:olio
of
risky 
assets
by
finding
the
por:olio
that
supports 
the
steepest
CAL
 3.  Choose
an
appropriate
complete
por:olio 
on
CALO
based
on
investor’s
risk
aversion
by 
mixing
risk‐free
asset
with
op3mal
risky 
por:olio.
 Efficient
diversifica:on
with
 
 many
risky
assets 
 A
Single‐Index
model 
 •  Index
model:
relate
stock
returns
to
returns 
on
both
a
market
index
and
firm‐specific 
influences
 •  excess
return
of
security
i:

Ri=
ri
–
rf
 •  Ri=
βi
RM
+
ei
+αi
 –  βi
RM
:
component
due
to
overall
market 
movement,
where
β
men3ons
stock’s
sensi3vity 
to
the
market
 –  ei:
component
aiributable
to
unexpected,
firm ‐specific
events.
 –  αi:
expected
excess
return
of
security
i
if
the 
market
factor
is
neutral.



 Alpha
and
Beta 
 •  Beta
:
 –  Market
index
has
β=1
 –  Aggressive
stock
has
β>1
 –  Defensive
stock
has
β<1
 •  Alpha:
 –  Alpha
chasing
 –  Is
posi3ve
or
nega3ve
α
more
airac3ve
to 
investor?
Why?

 Systema3c
vs.
firm‐specific
risk 
 •  Ri=
βi
RM
+
ei
+αi
 –  Ri
,
RM
:
excess
returns
of
security
i
and
of
the 
market,
respec3vely.
 •  Var(Ri)
=
βi2
σM2











+




σ2(ei)
 













=
systema3c
risk
+
firm‐specific
risk
 •  Total
risk
depends
on
 •  Uncertainty
common
to
the
market
 •  Variability
in
security
i
that
is
independent
of
the 
market.
 Sta:s:cal
representa:on
of
the 
Single‐Index
model 
 αD=4.5%
 βD=1.4
 E(RD|RM) = αD + βD RM T:
(17%,
27%)
 Firm‐specific
surprise:
the
devia:on
between







 the
scaTered
actual
returns
and
the
SCL.
 Size
of
firm‐specific
risk:
dispersion
of
the



 scaTered
actual
returns
 Rela:ve
importance
 
 of
systema:c
risk 
 •  Ra:o
of
systema:c
variance
to
total
variance
 22 22 β Dσ M β Dσ M ρ2 = =22 2 σD β Dσ M + σ 2 (eD ) € Single‐Index
Model 
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