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8/9/2019 BF2201 Chap006
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hapter 6
Efficient
Diversificatio
n
Copyright 2010 by The McGraw-Hill Companies, Inc. ll rights reser!e".McGraw-Hill#Irwin
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Outline
Diversification and Portfolio Risk
Asset Allocation With Two Risky
Assets The Optimal Risky Portfolio With A
Risk-Free Asset
Sinle !nde" #odel
$-2
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$%& Diversification and Portfolio Risk
'Review% (overed in F#)
$-%
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6.1 Diversification (Review)
When we put stocks in a
portfolio* p+ ,ecause
the firm-specific risk or uniue
risk is diversified away%
The risk that remains is called
systematic risk or market risk%
'Wii)
n = # securities in the portfolio
$-&
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$%. Asset Allocation With Two Risky
Assets
$-'
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E(rp) = W1r1 +W2r2
W1=
W2=
=
=
Two!ecurit" ortfolio Return
E(rp) = $.6(%.2&') + $.(11.%') = 1$.*6'
Wi = ' of total one"
investe, in securit" i
$.6
$.
%.2&'
11.%'
r1
r2
$-$
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ortfolio -ariance an, !tan,ar,
Deviation
-ariance of a Two !toc ortfolio/
$-(
= =
=/
&!
/
&0
0!0!.
p )1r*(ov'rW2W3
portfoliotheinstocksofnum,ertotalThe/
lyrespective0and!stockininvestedportfoliototaltheofPercentaeW*W 0!
=
=
0Stockand!Stockofreturnstheof(ovariance)r*(ov'r 0! =
)r*r'(ov)r*(ov'r43)r*'r(ovthen0!!f !00!.!0! ===
22
222121
21
21
22 ++= W)r*r'(ovWWWp
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0easurin the correlation coefficient
(ovariance does not tell us the intensityof thecomovement of the stock returns* only the
direction%
We can standardi5e the covariance and
calculate the correlation coefficientwhich tellsus the direction and provides a scaleto estimate
the deree to which the stocks move toether%
For Stock & and Stock .
$-)
.&
.&'&*.)
33
)r*(ov'r6
=
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an, ,iversification in a 2stoc portfolio
is always in the rane 7777777777 inclusive% What ,oes (12) = +1.$ ipl"8
The two areperfectly positivelycorrelated%
!f '&*.) 9 :&* then '&*.) 9 W&&: W..
What ,oes (12) = 1.$ ipl"3
The two are perfectly neativelycorrelated%
!f '&*.) 9 -&* then '&*.) 9 ;'W&&< W..)
!t is possi,le to choose W&and W.such that
'&*.) 9 =%
1.$ to +1.$
$-*
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an, ,iversification in a 2stoc portfolio
!f -& + '&*.) + & then
There are some diversification ,enefits from com,inin
stocks & and . into a portfolio%
p2
= W121
2+ W222
2+ 2W1W2 4ov(r1r2)
And since (ov'r&r
.) 9
1,2
&
.
p2
= W121
2+ W222
2+ 2W1W21212
'&*.) 9 '.*&)
'&*&) 9 :&%= ,y definition The covariance ,etween any stock such as Stock & and
itself is the variance of Stock &
$-10
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p2 =W1
21
2+ 2W1W2 4ov(r1r2) + W222
2
5et W1= 6$' an, W2= $' (!toc 1 = 748 !toc 2 = 9:;)
p2
= $.*6($.1
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The effects of correlation 4
covariance on diversification
sset an, 7 > ortfolio 7
sset 4 an, D > ortfolio 4D
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= +1
= .*
E(r)
1*'
&'
12' 2$'!t. Dev
TW?!E4@RAT: ?RTB?5A?! WATC
DABBERET 4?RRE5TA?!
!toc !toc 7
W= $'
W7= 1$$'
W= 1$$'
W7= $'
= $
= 1
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Summary> Portfolio Risk?Return
Two Security Portfolio
Amount of risk reduction depends criticallyon 7777777777777777777777777%
Addin securities with correlations 77777will result in risk reduction%
!f risk is reduced ,y more than e"pectedreturn* what happens to the return per unit
of risk 'the Sharpe ratio)8
correlations covariances
1
$-1&
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6. Efficient Diversification With
0an" Ris" ssets
$-1'
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@"tendin (oncepts to All Securities
4onsi,er all possiFle coFinations of securitieswith all possiFle ,ifferent weihtins an, eep trac
of coFinations that provi,e ore return for less
ris or the least ris for a iven level of return an,
raph the result.
The set of portfolios that provi,e the optial tra,e
offs are ,escriFe, as the efficient frontier.
The efficient frontier portfolios are ,oinantor the
Fest ,iversifie, possiFle coFinations.
ll investors shoul, want a portfolio on the efficient
frontierG until we a,, the risless asset
$-1$
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E(r)
The iniuvariance frontier of
ris" assets Efficient Brontier is the Fest ,iversifie, set ofinvestents with the hihest returns
$-1(
HloFaliniu
variance
portfolio
Efficientfrontier
An,ivi,ual
assets
0iniu
variancefrontier
!t. Dev.
Boun, F" forinportfolios of securities
with the lowest
covariances at a iven
E(r) level.
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E(r)The EB an, asset allocation
Efficient
frontier
!t. Dev.
2$' !tocs
&$' 7on,s
1$$' !tocs
EB inclu,in
international Ialternative
investents&$' !tocs
2$' 7on,s6$' !tocs
$' 7on,s
$' !tocs
6$' 7on,s
$-1)
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Recap with a -i,eo
u55 word> @fficient Frontier http>??www%,loom,er%com?video?,o,-s-daily-,u55word-
efficient-frontier-BC@EwGRCuAympdfhnHw%html
http://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.htmlhttp://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.htmlhttp://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.htmlhttp://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.html8/9/2019 BF2201 Chap006
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$% The Optimal Risky Portfolio With
A Risk-Free Asset
$-20
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E(r)
otential 4apital llocation 5ines
E(r)
45 ()
B
Ris Bree
EfficientBrontier
H
E(r)
45 ()
IB
E(rIB)
$-22
45 (H)
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The 405 I !harpe Ratio
I The 4apital 0aret 5ine (405)is the dominant (A,ecause it has the larest slope%
I !t provides the ,est rewar,tovolatilit"trade off%
I The reward-to-volatility trade off is called the !harpe
0easure!harpe Ratio>
!harpe Ratio = (E(rp) J rf) p
I Reardless of risk preferences* some com,inations of P 4 Fhas an eual or hiher Sharpe ratio than P alone%
$-2&
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ractical Aplications for sset
llocation
A financial planner identifies the optial ris portfolioon the efficient frontier 'call it P)%
I The optial ris portfolio () is the tanency portfolio on
the efficient frontier that connects the risk-free rate%
I may include funds* stocks* ,onds* international and otheralternative investments%
This tanenc" portfolio will serve as the startin point
for alltheir clients%
I The planner varies the asset allocation ,etween the risky
portfolio and risk-free investment accordin to risk tolerance
of client to create a complete portfolio%
$-2'
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$%G A Sinle !nde" Asset #arket
Prelude to (hapter J% (AP#%
$-2$
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An,ivi,ual !ecurities
!t is ,eneficial to hold individual securities in aportfolio to diversify away idiosyncratic risk%
A well,iversifie, portfolio has no idiosyncratic
risk% The remainin risk is called s"steatic
risaret ris%
!"steatic ris arises from events that effect
the entire economy% These include chanes in
interest rates?EDP or financial crises% Kow do we measure a stockLs systematic risk8
$-2(
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!ndividual securities
Kow do we measure a stockLs systematicrisk8
Returns
!toc
Returns
well
,iversifie,
portfolio
K interest rates
K HD
K consuer spen,in
etc.
!"steatic Bactors
$-2)
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An,eL 0o,els/ ?verview
An,eL o,els are statistical models desined toestimate the two coponents of risk for a
particular security or portfolio%
Separates the actual return of a security into macro
's"steatic) and micro 'fir-specific) components%
,vantaes/ Practicality> Reduces the num,er of inputs needed to
account for diversification ,enefits%
(onvenient> @asy reference point for understandin stock
risk usin the measure M%
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!inle Bactor An,eL 0o,el
$-%0
Ri9 MiRm: Ni : iRi9Actuale"cessreturn9 ri< rf
Rm 9 rm< rf 'e"cess return)% Systematic factor or pro"y
in this case # is unanticipated movement in a well-diversified,road market inde" like the S4PG==
Mi9 sensitivity of a securityLs particular return to the
systematic factor
Ni 9 unanticipated firm specific events
i9 @"pected return ,eyond any return induced ,y
movements in the market inde"% Typically 9=%
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An,eL 0o,el/ !catter lot
ELcess Returns (i)
!ecurit"
4haracteristic
5ine.. ..
..
.. ..
..
.. ..
.. ....
.. .. .... ..
.. ....
....
..
.. ..
.. ....
......
.. ..
....
..
.. .. .... ..
..
.. ...... .... .... ..
ELcess returns
on aret in,eL
-ariation in Ri eLplaine, F" the line isthe stocMs NNNNNNNNNNNNN
-ariation in Riunrelate, to the aret
(the line) is NNNNNNNNNNNNNNNNuns"steatic ris
Ri= i+ OiR+ ei
s"steatic ris
$-%1
!lope of !45 = Feta"intercept = alpha
..
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@n,erstan,in 7eta (P)
7etais the sensitivity of a securityLs returns to the
systematic or market factor%
I P > 1 indicates a stock with reater sensitivity to the
economy that the averae stock in the market%I P 1indicates a stock with ,elow-averae sensitivityto
the economy%
I P = 1. y definition* the market portfolio has a ,eta of &%
Qideo> http>??www%youtu,e%com?watch8v9&($A@u,Ew
$-%2
http://www.youtube.com/watch?v=14C6AEubGNwhttp://www.youtube.com/watch?v=14C6AEubGNwhttp://www.youtube.com/watch?v=14C6AEubGNw8/9/2019 BF2201 Chap006
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4oponents of RisDecompose stock variance into two components%
Qariance 'Ri) 9 Qariance 'MiRm: i: i)
9 Qariance 'Mi Rm) : Qariance'i)
9 M.i3.
m : 3.'
i)
9 Systematic risk : Firm-specific risk
!"steatic ris of each stock depends on ,oth Q2and the
sensitivity Pi.
Birspecific ris is measured ,y
i. !t has an e"pected valueof 5ero as the impact of unanticipated events must averae
out to 5ero%
-ariance(i) is 5ero%
$-%%
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( i S it
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(omparin Security
(haracteristic ines
Descri,e
e
for each%
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