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Investment Analysis and
Portfolio Management
by Frank K. Reilly & Keith C. Brown
Ch
apte
r7
Ch
apte
r7
An Introduction to
Portfolio Management !ome Ba"kgro#nd Ass#m$tions
Markowit% Portfolio heory
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Some Background Assumptions
' As an investor yo# want to ma(imi%e theret#rns for a given level of risk.
' )o#r $ortfolio in"l#des all of yo#r assets and
liabilities.
' he relationshi$ between the ret#rns for
assets in the $ortfolio is im$ortant.
' A good $ortfolio is not sim$ly a "olle"tion of
individ#ally good investments.
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Some Background Assumptions
' Risk Aversion *iven a "hoi"e between two assets with e+#al
rates of ret#rn, risk-averse investors will sele"t the
asset with the lower level of risk
viden"e' Many investors $#r"hase ins#ran"e for/ 0ife,
A#tomobile, 1ealth, and 2isability In"ome.
' )ield on bonds in"reases with risk "lassifi"ations
from AAA to AA to A, et".
3ot all Investors are risk averse
' It may de$ends on the amo#nt of money involved/
Risking small amo#nts, b#t ins#ring large losses
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Some Background Assumptions
' 2efinition of Risk 4n"ertainty/ Risk means the #n"ertainty of f#t#re
o#t"omes. For instan"e, the f#t#re val#e of an
investment in *oogle5s sto"k is #n"ertain6 so the
investment is risky. 7n the other hand, the$#r"hase of a si(-month C2 has a "ertain f#t#re
val#e6 the investment is not risky.
Probability/ Risk is meas#red by the $robability of
an adverse o#t"ome. For instan"e, there is 89:
"han"e yo# will re"eive a ret#rn less than ;:.
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Markowitz Portfolio Theory
' Main Res#lts
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Markowitz Portfolio Theory
' Ass#m$tions for Investors Consider investments as $robability distrib#tions of
e($e"ted ret#rns over some holding $eriod
Ma(imi%e one-$eriod e($e"ted #tility, whi"h
demonstrate diminishing marginal #tility of wealth stimate the risk of the $ortfolio on the basis of the
variability of e($e"ted ret#rns
Base de"isions solely on e($e"ted ret#rn and risk
Prefer higher ret#rns for a given risk level.!imilarly, for a given level of e($e"ted ret#rns,
investors $refer less risk to more risk
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Markowitz Portfolio Theory
' 4sing these five ass#m$tions, a single assetor $ortfolio of assets is "onsidered to be
effi"ient if no other asset or $ortfolio of assets
offers higher e($e"ted ret#rn with the same =or
lower> risk, or lower risk with the same =orhigher> e($e"ted ret#rn.
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Alternatie Measures of !isk
' ?arian"e or standard deviation of e($e"tedret#rn
' Range of ret#rns
'Ret#rns below e($e"tations !emivarian"e a meas#re that only "onsiders
deviations below the mean
hese meas#res of risk im$li"itly ass#me that
investors want to minimi%e the damage fromret#rns less than some target rate
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Alternatie Measures of !isk
' he Advantages of 4sing !tandard 2eviationof Ret#rns
his meas#re is somewhat int#itive
It is a "orre"t and widely re"ogni%ed risk meas#re
It has been #sed in most of the theoreti"al asset
$ri"ing models
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"#pected !ates of !eturn
If yo# want to "onstr#"t a $ortfolio of n risky assets,what will be the e($e"ted rate of ret#rn on the
$ortfolio is yo# know the e($e"ted rates of ret#rn on
ea"h individ#al assets
he form#la
!ee (hibit @.
iassetforreturnof rateexpectedthe)i
E(R iassetin portfoiotheof percentthei!"#here
1R !)
portE(R
==
∑=
= n
i ii
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Indiidual Inestment !isk Measure
' ?arian"e It is a meas#re of the variation of $ossible rates of
ret#rn Ri, from the e($e"ted rate of ret#rn D=Ri>E
∑==n
i 1
i
2
ii
2
$)%E(R -R &)('ariance σ
where Pi is the $robability of the $ossible rate of
ret#rn, Ri
' !tandard 2eviation =>
It is sim$ly the s+#are root of the varian"e
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Indiidual Inestment !isk Measure
"#hi$it 7%(
?arian"e = > G 9.9998H
!tandard 2eviation = > G 9.9@
σ
σ
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Coariance of !eturns
' A meas#re of the degree to whi"h twovariables Jmove together relative to theirindivid#al mean val#es over time
' For two assets, i and L, the "ovarian"e of rates
of ret#rn is defined as/
Coi) * "+,!i - ".!i/0 ,! ) - ".! )/01
' (am$le
he ilshire H999 !to"k Inde( and 0ehmanBrothers reas#ry Bond Inde( d#ring 99@
!ee (hibits @.8 and @.@
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Coariance and Correlation
' he "orrelation "oeffi"ient is obtained bystandardi%ing =dividing> the "ovarian"e by the
$rod#"t of the individ#al standard deviations
' Com$#ting "orrelation from "ovarian"e
(tR of de)iationstandardthe
itR of de)iationstandardthe
i
returnsof tcoefficienncorreatiothei(r
*o)r
=
=
=
=
j
ji
ijij
σ
σ
σ σ
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Correlation Coefficient
' he "oeffi"ient "an vary in the range N to -.' A val#e of N wo#ld indi"ate $erfe"t $ositive
"orrelation. his means that ret#rns for the
two assets move together in a $ositively and
"om$letely linear manner.
' A val#e of wo#ld indi"ate $erfe"t negative
"orrelation. his means that the ret#rns for two
assets move together in a "om$letely linearmanner, b#t in o$$osite dire"tions.
' !ee (hibit @.;
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Standard 4eiation of a Portfolio
' he Form#la
jiσ σ
σ
σ
σ σ
ii
i
2i
i
port
r *o#here
+andiassetsforreturnof ratese ,et#een thcoariancethe*o
iassetforreturnof ratesof ariancethe
portfoioin theaueof proportion , thedeter.inedare#ei/hts
#here portfoio+in theassetsindiiduatheof #ei/htsthe!
portfoiotheof deiationstandardthe
"#here
n
1i
n
1i i*o
#
n
1i i#2
i2i
# port
=
=
=
=
=
∑=
∑=∑=
+=
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Standard 4eiation of a Portfolio
' Com$#tations with A wo-!to"k Portfolio Any asset of a $ortfolio may be des"ribed by two
"hara"teristi"s/
' he e($e"ted rate of ret#rn
' he e($e"ted standard deviations of ret#rns
he "orrelation, meas#red by "ovarian"e, affe"ts
the $ortfolio standard deviation
0ow "orrelation red#"es $ortfolio risk while notaffe"ting the e($e"ted ret#rn
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Standard 4eiation of a Portfolio
' wo !to"ks with 2ifferent Ret#rns and Risk
.9 .H9 .998O .9@
.9 .H9 .999 .9
>=R Asset ii
ii ss
Case Correlation Coeffi"ient Covarian"e
a N.99 .99@9
b N9.H9 .99H
" 9.99 .9999
d -9.H9 -.99H
e -.99 -.99@9
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Standard 4eiation of a Portfolio
Assets may differ in e($e"ted rates of ret#rn andindivid#al standard deviations
3egative "orrelation red#"es $ortfolio risk
Combining two assets with N.9 "orrelation will not
red#"es the $ortfolio standard deviation
Combining two assets with -.9 "orrelation may
red#"es the $ortfolio standard deviation to %ero
!ee (hibits @.9 and @.
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Standard 4eiation of a Portfolio
' Constant Correlation with Changing eights Ass#me the "orrelation is 9 in the earlier e(am$le
and let the weight vary as shown below.
Portfolio ret#rn and risk are =!ee (hibit @.>/
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Standard 4eiation of a Portfolio
' A hree-Asset Portfolio he res#lts $resented earlier for the two-asset
$ortfolio "an e(tended to a $ortfolio of n assets
As more assets are added to the $ortfolio, more
risk will be red#"ed everything else being the same
he general "om$#ting $ro"ed#re is still the same,
b#t the amo#nt of "om$#tation has in"rease ra$idly
For the three-asset $ortfolio, the "om$#tation hasdo#bled in "om$arison with the two-asset $ortfolio
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"stimation Issues
' Res#lts of $ortfolio allo"ation de$end ona""#rate statisti"al in$#ts
' stimates of
($e"ted ret#rns
!tandard deviation
Correlation "oeffi"ient
' Among entire set of assets
' ith 99 assets, 8,OH9 "orrelation estimates
' stimation risk refers to $otential errors
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"stimation Issues
' ith the ass#m$tion that sto"k ret#rns "an bedes"ribed by a single market model, the
n#mber of "orrelations re+#ired red#"es to the
n#mber of assets
' !ingle inde( market model/
i.iii R ,aR ε ++=
,i G the slo$e "oeffi"ient that relates the ret#rns forse"#rity i to the ret#rns for the aggregate market
R . G the ret#rns for the aggregate sto"k market
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"stimation Issues
' If all the se"#rities are similarly related tothe market and a bi derived for ea"h one, it
"an be shown that the "orrelation "oeffi"ient
between two se"#rities i and L is given as/
marketsto"kaggregate
thefor ret#rnsof varian"ethewhere
i
m LiiL bbr
=
=
2
m
j
σ
σ σ
σ
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The "fficient 6rontier
' he effi"ient frontier re$resents that set of$ortfolios with the ma(im#m rate of ret#rn forevery given level of risk, or the minim#m riskfor every level of ret#rn
' ffi"ient frontier are $ortfolios of investmentsrather than individ#al se"#rities e("e$t theassets with the highest ret#rn and the assetwith the lowest risk
' he effi"ient frontier "#rves (hibit @.8 shows the $ro"ess of deriving the
effi"ient frontier "#rve
(hibit @.H shows the final effi"ient frontier "#rve
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"fficient 6rontier and Inestor 8tility
' An individ#al investor5s #tility "#rve s$e"ifiesthe trade-offs he is willing to make between
e($e"ted ret#rn and risk
' he slo$e of the effi"ient frontier "#rve
de"reases steadily as yo# move #$ward
' he intera"tions of these two "#rves will
determine the $arti"#lar $ortfolio sele"ted by
an individ#al investor ' he o$timal $ortfolio has the highest #tility for
a given investor
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"fficient 6rontier and Inestor 8tility
' he o$timal lies at the $oint of tangen"ybetween the effi"ient frontier and the #tility
"#rve with the highest $ossible #tility
' As shown in (hibit @., Investor Q with the
set of #tility "#rves will a"hieve the highest#tility by investing the $ortfolio at Q
' As shown in (hibit @., with a different set of
#tility "#rves, Investor ) will a"hieve thehighest #tility by investing the $ortfolio at )
' hi"h investor is more risk averse
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The Internet Inestments :nline
' htt$/www.$ionlie."om' htt$/www.investmentnews."om
' htt$/www.ibbotson."om
' htt$/www.styleadvisor."om
' htt$/www.wagner."om
' htt$/www.effisols."om
' htt$/www.effi"ientfrontier."om
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