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Investment Analysis and

Portfolio Management

by Frank K. Reilly & Keith C. Brown

Ch

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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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' 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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' 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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' 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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' 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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"#hi$it 7%&

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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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"#hi$it 7%'

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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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"#hi$it 7%2

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"#hi$it 7%7

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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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"#hi$it 7%3

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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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' 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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' 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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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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"#hi$it 7%&5

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"#hi$it 7%&'

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' 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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"#hi$it 7%&(

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' 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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"#hi$it 7%&2

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"#hi$it 7%&

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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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"#hi$it 7%&9

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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
http://www.pionlie.com/http://www.investmentnews.com/http://www.ibbotson.com/http://www.styleadvisor.com/http://www.wagner.com/http://www.effisols.com/http://www.efficientfrontier.com/http://www.efficientfrontier.com/http://www.effisols.com/http://www.wagner.com/http://www.styleadvisor.com/http://www.ibbotson.com/http://www.investmentnews.com/http://www.pionlie.com/

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