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8.3 HOWTOMEASURERISKCONTRIBUTION
215
1. Justasyoudidforyourvariancecalculations,firsttranslateallreturnsintodevi-
First,de-meaneachrateof
return.
Variancecalculations,
Section6.1B,p.141
ationsfromthemean.Thatis,forM,C,andD,subtracttheirownmeansfrom
everyrealization.
(Howdemeaning!)
OriginalRatesofReturn
Net-of-MeanRatesofReturn
Future
PfioM
PfioC
PfioD
PfioM
PfioC
PfioD
InScenarioS1
−1.0%
−2.0%
+14.0%
−5.0%
−7.0%
+12.0%
InScenarioS2
+2.0%
+3.0%
+6.0%
−2.0%
−2.0%
+4.0%
InScenarioS3
+4.0%
+7.0%
0.0%
0.0%
+2.0%
−2.0%
InScenarioS4
+11.0%
+12.0%
−12.0%
+7.0%
+7.0%
−14.0%
“Reward”(E(˜r))
4.00%
5.00%
2.00%
0.00%
0.00%
0.00%
2. Computethevarianceoftheseriesonthex-axis.Thisisthevarianceoftherates
Takesquaresandthen
average.Thisisthevariance.
ofreturnonM.Withthenet-of-meanMreturns,thisiseasy:
Var(˜r
M
)=
(−5%)
2
+(−2%)
2
+(0)
2
+(7%)
2
4
=19.5%%
=
SumoverAllScenariosS:[˜r
M
inScenarioS−AverageE(˜r
M
)]
2
N
Because1% is“multiplyby0.01,” 19.5%% couldberewrittenas0.195%or
0.00195.(NotealsothatyoudonotneedtocomputethevariancesofeitherC
orDtoobtaintheirmarketbetas.)
3. Computetheaverageproductofthenet-of-meanvariables.Inthiscase,youwant
Forcovariances,multiply
net-of-meanreturns,then
average.
tocomputethemarketbetaforC,soyouworkwiththeratesofreturnonM
andC.
Cov(˜r
M
, ˜r
C
)=
(−5%)
.
(−7%)+(−2%)
.
(−2%)+(0)
.
(2%)+(7%)
.
(7%)
4
=
22%%=0.22%
(8.4)
=
SumoverAllScenariosS:[˜r
M
inScenarioS−E(˜r
M
)]
.
[˜r
C
inScenarioS−E(˜r
C
)]
N
ThisstatisticiscalledthecovariancebetweentheratesofreturnonMandC.
4. ThebetaofCwithrespecttothemarketM,formallyβ
C,M
butoftenabbreviated
Thebetaisthecovariance
dividedbythevariance.
asβ
C
,istheratioofthesetwoquantities,
β
C
C,M
=
22%%
19.5%%
≈1.128
(8.5)
=
Cov(˜r
M
, ˜r
C
)
Var(˜r
M
)
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216
CHAPTER8
INVESTORCHOICE:RISKANDREWARD
Thisslopeof1.128(alittlemorethanaperfect45
diagonal)isexactlythemar-
Youcanconfirmour
calculationsusinga
spreadsheet.
ketbetawedrewinFigure8.4.Manyspreadsheetsandallstatisticalprogramscan
computeitforyou:Theycalltheroutinethatdoesthisalinearregression.
YoushouldalwaysthinkofthebetaofanassetiwithrespecttoaportfolioPas
Thinkofmarketbetaasthe
characteristicofanasset.
acharacteristicmeasureofyourassetirelativetoanunderlyingbaseportfolioP.The
rateofreturnonPisonthex-axis;therateofreturnoniisonthey-axis.Aswe
statedearlier,mostoften—butnotalways—theportfolioPisthemarketportfolio,
M,soβ
i,M
isoftenjustcalledthemarketbeta,orevenjustthebeta(andthesecond
subscriptisomitted).
Nowthinkforamoment.Whatshouldtheaveragebetaofastockintheeconomy
Theaveragebetaofthe
market(allstocks)is1,not0. be?Equivalently, whatisthebetaofthemarketportfolioitself?ReplacetheCin
Formula8.5withM:
β
M
=
Cov(˜r
M
, ˜r
M
)
Var(˜r
M
)
Ifyoulookatthedefinitionofcovariance,youcanseethatthecovarianceofavariable
withitselfisthevariance.(Thecovarianceisageneralizationofthevarianceconcept
fromonevariabletotwo variables.)Therefore, Cov(˜r
M
, ˜r
M
)=Var(˜r
M
), and the
marketbetaofthemarketitselfis1.Graphically,ifboththex-axisandthey-axis
aregraphingthesamevalues,everypointmustlieonthediagonal.Economically,this
shouldnotbesurprising,either:themarketgoesupone-to-onewiththemarket.
Nowthatyouknowhowtocomputebetasandcovariances,youcanconsider
Whytortureyouwith
computations?Soyoucanplay
withscenarios.
scenariosforyourproject.Forexample,youmighthaveanewprojectforwhichyou
wouldguessthatitwillhavearateofreturnof−5%ifthemarketreturns−10%;
arateofreturnof+5%ifthemarketreturns+5%;andarateofreturnof30%if
themarketreturns10%.Knowinghowtocomputeamarketbetathereforemakes
itusefultothinkofsuchscenarios.(Youcanalsousethistechniquetoexplorethe
relationshipbetweenyourprojectsandsomeotherfactors.Forexample,youcould
Anoil-pricebeta,Section
9.8A,p.292
determinehowyourprojectscovarywiththepriceofoiltolearnaboutyourproject’s
oilriskexposure.)
Intherealworld,youwillsometimesthinkintermsofsuchscenarios,butmore
Practicaladvicetohelpyou
estimatemarketbetainthe
realworld:Use3–5yearsof
dailyobservationsandthen
adjust.
oftenyouhavetocomputeamarket betafromhistorical ratesofreturnfor the
overallstockmarketandforyourproject(orsimilarprojects).Fortunately,aswe
notedupfront,thebetacomputationsthemselvesareexactlythesame.Ineffect,
whenyouusehistorical data, , yousimplyassumethat t each time period wasone
representativescenarioandproceedfromthere.Nevertheless,therearesomereal-
worldcomplicationsyoushouldthinkabout:
1. Shouldyouusedaily,weekly,monthly,orannualratesofreturn?Theansweris
thatthebestmarketbetaestimatescomefromdailyorweeklydata.Annualdata
shouldbeavoided(exceptinatextbookinwhichspaceislimited).Monthlydata
canbeusedifneedbe.
2. Howmuchdatashouldyouuse?Mostresearcherstendtouse3–5yearsofhistor-
icalrateofreturndata.Thisreflectsatrade-offbetweenhavingenoughdataand
notgoingtoofarbackintoancienthistory,whichmaybelessrelevant.Ifyouhave
dailydata,3yearsworksquitewell.
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8.3 HOWTOMEASURERISKCONTRIBUTION
217
3. Isthehistoricalbetaagoodestimateofthefuturebeta?Itturnsoutthathistory
cansometimesbedeceptive,especiallyifyourestimatedhistoricalbetaisfaraway
fromthemarket’sbetaaverageof1.Fortunately,thereareatleasttwomethodsto
helpadjusthistoricalbetassothatyougetbetterestimatesoffuturebetas:
(A)Averaging:Youcouldrelynotjustonthehistoricalbetacomputedfromyour
ownproject’sreturns.Instead,youcouldusetheaveragehistoricalbetasfor
manyotherprojectsthataresimilartoyourown(forexample,projectsfrom
thesameindustryorinthesamesizeclass).Suchaveragesareusuallyless
noisy.
(B) Shrinking:Youcould“shrink”yourhistoricalbetatowardtheoverallmarket
betaof1.Forexample,inthesimplestsuchshrinker,youwouldsimplycom-
puteanaverageoftheoverallmarketbetaof1andyourhistoricalmarketbeta
estimate.Ifyoucomputedahistoricalmarketbetaof4foryourproject,you
wouldworkwithapredictionoffuturemarketbetaofabout(4+1)/2=2.5
foryourproject.
Manysmartexecutivesstartwithastatisticalbetaestimatedfromhistoricaldata
andthenusetheirintuitivejudgmenttoadjustit.
solvenow!
Q8.6
Returntoyourcomputationofmarketbetaof1.128inFormula8.5.We
calleditβ
C,M
,orβ
C
forshort.Istheorderofthesubscriptsimportant?
Thatis,pleasecomputeβ
M,C
andseewhetheritisalso1.128.
8.3C WHYNOTCORRELATIONORCOVARIANCE?
Thereisaclosefamilyrelationshipbetweencovariance,beta,andcorrelation.The
Covarianceandbeta(and
correlation)alwayshavethe
samesign.
betaisthecovariancedividedbyoneofthevariances.Thecorrelationisthecovariance
dividedbybothstandarddeviations.Thedenominatorsarealwayspositive.Thus,
ifthecovarianceispositive,soarethebetaandthecorrelation;ifthecovarianceis
negative,soarethebetaandthecorrelation;andifthecovarianceiszero,soarethe
betaandthecorrelation.Thenicethingaboutthecorrelation,whichmakesituseful
inmanycontextsoutsidefinance,isthatithasnoscaleandisalwaysbetween−100%
and+100%:
.
Twovariablesthatalwaysmoveperfectlyinthesamedirectionhaveacorrelationof
100%.
.
Twovariablesthatalwaysmoveperfectlyinoppositedirectionshaveacorrelationof
−100%.
.
Twovariablesthatareindependenthaveacorrelationof0%.
Thismakescorrelationsveryeasytointerpret.Thenot-so-nicethingaboutcorrela-
tionisthatithasnoscaleandisalwaysbetween−100%and+100%.Thismeansthat
twoinvestments,thesecondbeingamilliontimesbiggerthanthefirst(allproject
ratesofreturnmultipliedbyamillion),havethesamecorrelationwiththestockmar-
ket.Yet,thesecondinvestmentwouldgoupordownwithanyslighttremorinthe
marketbyamilliontimesmore,whichwouldofcoursemeanthatitwouldcontribute
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218
CHAPTER8
INVESTORCHOICE:RISKANDREWARD
TABLE8.2 SomeMarketBetasandCapitalizationsonMay10,2008
Mkt-
MarketBeta
Mkt-
MarketBeta
Company
Ticker Cap
a
Yahoo AOL L Company
Ticker
Cap
a
Yahoo
AOL
AMD
AMD
4
1.96
2.67 Intel
INTC
124
1.73
1.85
Coca-Cola
KO
130
0.52
0.78 PepsiCo
PEP
107
0.24
0.22
Citigroup
C
124
1.71
1.31 J.P.Morgan
JPM
158
0.85
0.91
GoldmanSachs
GS
74
2.24
1.84 MorganStanley
MS
51
1.75
1.71
IBM
IBM
170
0.95
0.94 Hewlett-Packard
HPQ
121
1.09
1.46
Dell
DELL
39
1.53
1.36 Sun
JAVA
10
1.01
2.13
AppleInc
AAPL
162
2.86
2.57 Sony
SNE
45
0.97
1.08
Google
GOOG
180
2.60
2.17 Yahoo
YHOO
36
0.39
1.03
Ford
F
16
2.11
2.13 GeneralMotors
GM
12
1.50
1.64
AmericanAirlines
AMR
2
1.71
2.69 Southwest
LUV
9.5
−0.12
0.13
ExxonMobil
XOM
469
1.14
1.04 BarrickGold
ABX
34
−0.20
0.49
PhilipMorris
PM
109
0.00
NA
Procter&Gamble
PG
199
0.63
0.57
Textron
TXT
15
1.46
1.81 Boeing
BA
63
1.08
1.22
a.“Mkt-Cap”istheequitymarketvalueinbillionsofdollars.Yahooexplaineditsbetasasfollows:
TheBetaisbetaofequity.BetaisthemonthlypricechangeofaparticularcompanyrelativetothemonthlypricechangeoftheS&P500.
ThetimeperiodforBetais5yearswhenavailable,andnotlessthan2.5years.Thisvalueisupdatedmonthly.
NotethatYahoo!Financeseemstoignoredividends,butthisusuallymakeslittledifference.Icouldnotfindanexplanationforthemarket
betasprovidedbyAOL.Google’smarketbetaswerethesameasAOL’s,buttherewasnoexplanationforthem,either.
muchmorerisk.Thecorrelationignoresthis,whichdisqualifiesitasaseriouscandi-
dateforaprojectriskmeasure.Fortunately,betatakescareofscale—indeed,thebeta
forthesecondprojectwouldbeamilliontimeslarger.Thisiswhywepreferbetaover
correlationasameasureofriskcontributiontoaportfolio.
8.3D INTERPRETINGTYPICALSTOCKMARKETBETAS
Themarketbetaisthebestmeasureof“diversificationhelp”foraninvestorwhoholds
Marketbetaworkswellwhen
investorsareholdingthe
marketandaddingonlyalittle
ofyourproject.
thestockmarketportfolioandconsidersaddingjustalittleofyourfirm’sproject.
Fromyourperspectiveasamanagerseekingtoattractinvestors,thisisnotaperfect,
necessarilytrueassumption—butitisareasonableone.Recallthatweassumethat
investorsaresmart,sopresumablytheyareholdinghighlydiversifiedportfolios.To
convinceyourmarketinvestorstolikeyour$10millionproject,youjustneedthe
averageinvestortowanttobuy$10milliondividedbyabout$20trillion(thestock
marketcapitalization),whichis1/2,000,000oftheirportfolios.Foryourinvestors,
yourcorporateprojectsarejusttinyadditionstotheirmarketportfolios.
Youcanlookupthemarketbetasofpubliclytradedstocksonmanyfinancial
Mostfinancialwebsitespublish
marketbetaestimates.
websites.Table8.2liststhebetasofsomerandomlychosencompaniesinMay2008
fromYahoo!Finance andfromAOL’sfinancesite.Mostcompanybetasareinthe
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8.4 EXPECTEDRATESOFRETURNANDMARKETBETASFOR(WEIGHTED)PORTFOLIOSANDFIRMS
219
rangeofaround0toabout2.5.Abetaabove1isconsideredrisk-increasingforan
investorholdingtheoverallstockmarket(itisriskierthanthestockmarketitself),
whileabetabelow1isconsideredrisk-reducing.Betasthatarenegativearequite
rare—inTable8.2,therehappentobeonlytwosuchstocks,SouthwestandBarrick
Gold,accordingtotheYahoobetas,andnoneaccordingtotheAOLbetas.
Marketbetahasyetanotherniceintuitiveinterpretation:Itisthedegreetowhich
Betacanbeviewedasthe
marginalchangeofyour
projectwithrespecttothe
market.
thefirm’svaluetendstochangeifthestockmarketchanges.Forexample,Apple’smar-
ketbetaofapproximately2.7(somewherebetween2.57and2.86)saysthatifthestock
marketwillreturnanextra5%nextyear(aboveandbeyonditsexpectations),Apple
willreturnanextra2.7
.
5%=13.5%(aboveandbeyondApple’sexpectations).Of
course,marketbetaisnotameasureofhowgoodaninvestmentAppleis.(Thismea-
sureisthealpha[whichcanbeinterpretedasanexpectedrateofreturn].Inthenext
section,youwilllearnamodelthatrelatesmarketbetatotheexpectedrateofre-
turn.Fornowlet’sassumeforillustrationthatthereisnoreasonablerelationship.)
Let’smaketheabsurdassumptionthatApple’sexpectedrateofreturnis−30%and
themorereasonableassumptionthatthemarket’sexpectedrateofreturnis10%.All
thatApple’smarketbetathentellsyouisthatwhenthemarketdoes1%betterthan
expected(i.e.,(10%+1%=11%)),thenApplewoulddo2.7%betterthanexpected
(i.e.,(−30%+2.7
.
1%=−27.3%)).Ifthemarketdoes0%(i.e.,10%worsethan
expected),Applewouldbeexpectedtodo−57%(i.e.,27%worsethanexpected).And
soon.Apple’shighmarketbetaisusefulbecauseitinformsyouthatifyouholdthe
stockmarket,addingApplestockwouldnothelpyoudiversifyyourmarketriskvery
much.HoldingApplestockwouldamplifyanymarketswings,notreducethem.In
anycase,Apple’smarketbetadoesnottellyouwhetherAppleispricedsohighthatit
isaninvestmentwithanegativeexpectedrateofreturnthatyoushouldavoidinthe
firstplace.
solvenow!
Q8.7
Youestimateyourprojectxtoreturn−5%ifthestockmarketreturns
−10%,and+5%ifthestockmarketreturns+10%.Whatwouldyou
useasthemarketbetaestimateforyourproject?
Q8.8
Youestimateyourprojecttoreturn+5%ifthestockmarketreturns
−10%,and−5%ifthestockmarketreturns+10%.Whatwouldyou
useasthemarketbetaestimateforyourproject?
8.4 EXPECTEDRATESOFRETURNAND
MARKETBETASFOR(WEIGHTED)
PORTFOLIOSANDFIRMS
Let’sgobacktoyourmanagerialperspectiveoffiguringouttheriskandreturnofyour
Portfoliosconsistofmultiple
assets(orofotherportfolios).
Value-weightedandequal-
weightedportfoliosare
defined.
corporateprojects.Manysmallprojectsarebundledtogether,soitisverycommonfor
managerstoconsidermultipleprojectsalreadypackagedtogetherasoneportfolio.
Forexample,youcanthinkofyourfirmasacollectionofdivisionsthathavebeen
packagedtogether.IfdivisionCisworth$1millionanddivisionDisworth$2million,
thenafirmconsistingofCandDisworth$3million.Cconstitutes1/3oftheportfolio
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220
CHAPTER8
INVESTORCHOICE:RISKANDREWARD
“Firm”andDconstitutes2/3oftheportfolio“Firm.”Thiskindofportfolioiscalleda
value-weightedportfoliobecausetheweightscorrespondtothemarketvaluesofthe
components.(Aportfoliothatinvests$100inCand$200inDwouldalsobevalue-
weighted.Aportfoliothatinvestsequalamountsintheconstituents(forexample,
$500ineach)iscalledanequal-weightedportfolio.)
Thus,asamanager,youhavetoknowhowtoworkwithaportfolio(firm)when
Whataretheexpectedrateof
returnandmarketbetaofa
portfolio?
youhavealltheinformationaboutallofitsunderlyingcomponentstocks(projects).If
Itellyouwhattheexpectedrateofreturnoneachprojectis,andwhatthemarketbeta
ofeachprojectis,canyoutellmewhatthefirm’soverallexpectedrateofreturnand
overallmarketbetaare?Let’stryit.UsetheCandDstocksfromTable8.1onpage202,
andcallCDDtheportfolio(orfirm)thatconsistsof1/3investmentindivisionCand
2/3investmentindivisionD.
Actuallyyoualreadyknowthatindividualportfolioratesofreturncanbeaver-
Youcanaverageactualrates
ofreturn.
aged.Forexample,inscenarioS4(
),investmentChasarateofreturnof+12%,and
investmentDhasarateofreturnof−12%.Consequently,theoverallinvestmentCDD
hasarateofreturnof
r
CDD, inS4(
)
=1/3
.
(+12%)+2/3
.
(−12%)=−4%
= w
C
.
r
C, inS4
+ w
D
.
r
D, inS4
Letusverifythis:Put$100intoCand$200intoD.Cturnsinto1.12
.
$100=$112.
Dturnsinto(1−12%)
.
$200=$176.Thetotalportfolioturnsinto$288,whichis
arateofreturnof$288/$300−1=−4%ona$300investment.
Itisalsointuitivethatexpectedratesofreturncanbeaveraged.Inourexample,
Youcanaverageexpected
ratesofreturn.
Chasanexpectedrateofreturnof5%,andDhasanexpectedrateofreturnof2%.
Consequently,youroverallfirmCDDhasanexpectedrateofreturnof
E(˜r
CDD
)=1/3
.
(+5%)+2/3
.
(+2%)=3%
= w
C
.
E(˜r
C
) + + w
D
.
E(˜r
D
)
Letusverifythis,too.Therearefourpossibleoutcomes:InS1,youractualrateof
returnis8.67%;inS2,itis5%;inS3,itis2.33%;andinS4,itis−4%.Theaverageof
thesefouroutcomesisindeed3%.
Buthereisaremarkableandlessintuitivefact:Marketbetas—thatis,theprojects’
Newsflash:Youcanalso
averagemarketbetas.
riskcontributionstoyourinvestors’marketportfolios—canbeaveraged,too.Thatis,
IclaimthatthebetaofCDDistheweightedaverageofthebetasofCandD.You
alreadycomputedthelatterinFormula8.3as+1.128and−2.128,respectively.Their
MarketbetasofCandD,
Formula8.3,p.213
value-weightedaverageis
β
CDD
=1/3
.
(+1.128)+2/3
.
(−2.128)≈−1.043
(8.6)
=
w
C
.
β
C
+
w
D
.
β
D
YouwillbeaskedinQ8.9toconfirmthis.However,donotthinkforamomentthat
(Butyoucannotaverage
variancesorstandard
deviations!)
youcancomputevalue-weightedaveragesforallstatistics.Forexample,variancesand
standarddeviationscannotbeaveraged.
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8.4 EXPECTEDRATESOFRETURNANDMARKETBETASFOR(WEIGHTED)PORTFOLIOSANDFIRMS
221
IMPORTANT:
.
Youcanthinkofthefirmasaweightedinvestmentportfolioofcomponents,
suchasindividualdivisionsorprojects.Forexample,ifafirmnamedab
consistsonlyoftwodivisions,aandb,thenitsrateofreturnisalways
˜r
ab
=w
a
.
˜r
a
+w
b
.
˜r
b
wheretheweightsaretherelativevaluesofthetwodivisions.(Youcanalso
thinkofthisonefirmasa“subportfolio”withinalargeroverallportfolio,
suchasthemarketportfolio.)
.
Theexpectedrateofreturn(“reward”)ofaportfolioistheweightedaverage
expectedrateofreturnofitscomponents,
E(˜r
ab
)=w
a
.
E(˜r
a
)+w
b
.
E(˜r
b
)
Therefore,theexpectedrateofreturnofafirmistheweightedaveragerate
ofreturnofitsdivisions.
.
Likeexpectedratesofreturn,betascanbeweightedandaveraged.Thebeta
ofafirm—i.e.,thefirm’s“riskcontribution”totheoverallmarketportfolio—is
theweightedaverageofthebetasofitscomponents,
β
ab
=w
a
.
β
a
+w
b
.
β
b
Therefore,themarketbetaofafirmistheweightedaveragemarketbetaof
itsdivisions.
.
Youcannotdoanalogousweightedaveragingwithvariancesorstandard
deviations.
Youcanthinkofthefirmnotonlyasconsistingofdivisions,butalsoasconsisting
Afirmisaportfolioofdebt
andequity.Thus,theportfolio
formulasapplytothefirm
(withdebtandequityasits
components),too!
ofdebtandequity.Forexample,sayyour$400millionfirmisfinancedwithdebt
worth$100millionandequityworth$300million.Ifyouownalldebtandequity,
youownthefirm.Whatisthemarketbetaofyourfirm’sassets?Well,thebetaofyour
overallfirmmustbetheweightedaveragebetaofitsdebtandequity.Ifyour$100
millionindebthasamarketbetaof,say,0.4andyour$300millionofequityhasa
marketbetaof,say,2.0,thenyourfirmhasamarketbetaof
1/4
.
(0.4)
+
3/4
.
(2.0)
=1.6
β
Firm
=
Debtvalue
Firmvalue
.
β
Debt
+
Equityvalue
Firmvalue
.
β
Equity
(8.7)
This1.6iscalledtheassetbetatodistinguishitfromtheequitybetaof2.0that
financialwebsitesreport.Putdifferently,ifyourfirmrefinancesitselfto100%equity
(i.e.,$400millionworth),thenthereportedmarketbetaofyourequityonYahoo!
222
CHAPTER8
INVESTORCHOICE:RISKANDREWARD
Financewouldfallto1.6.Theassetbetaisthemeasureofyourfirm’sprojects’risk
contributiontotheportfolioofyourinvestors.Itistherelevantmeasurethatwill
determinethecostofcapitalthatyoushoulduseasthehurdlerateforprojectsthat
areliketheaverageprojectinyourfirm.
solvenow!
Q8.9
Let’scheckthatthebetacombinationformula(Formula8.6onpage
220)iscorrect.Letmeleadyoualong:
(a)Writedownatablewiththerateofreturnonthemarketandon
portfolioCDDineachofthefourpossiblestates.(Hint:Inscenario
S1[
],therateofreturnonCDDis8.67%.)Thenforgetabout
CandDaltogether.(Inthisquestion,youwillworkonlywiththe
marketandCDD.)
(b)ComputetheaveragerateofreturnonthemarketandonCDD.
(c)Writedownatablewiththede-meanedmarketrateofreturnand
CDDrateofreturnineachofthefourpossiblestates.(Themeanof
thede-meanedreturnsmustbezero.)
(d)Multiplythede-meanedratesofreturnineachscenario.Thisgives
youfourcross-products,eachhavingunitsof%%.(Hint:Insce-
narioS1[
],itisabout−28.35%%.)
(e)Computetheaverageofthesecross-products.Thisisthecovariance
betweenCDDandM.
(f)DividethecovariancebetweenCDDandMbythevarianceofthe
market.Isitequaltothe−1.04fromFormula8.6?
(g)Whichisfaster—thisrouteorFormula8.6?
Q8.10 Let’sconfirmthatyoucannottakeavalue-weightedaverageofcom-
ponentvariances(andthusofstandarddeviations)thesamewaythat
youcantakevalue-weightedaverageexpectedratesofreturnandvalue-
weightedaveragemarketbetas.
(a)Whatwouldthevalue-weightedaveragevarianceofCDDbe?
(b)WhatistheactualvarianceofCDD?
Q8.11 Consideraninvestmentof2/3inCand1/3inD.Callthisnewportfolio
CCD.Computethevariance,standarddeviation,andmarketbetaof
CCD.Dothistwoways:firstfromthefourindividualscenarioratesof
returnofCCD,andthenfromthestatisticalpropertiesofCandDitself.
Q8.12 Assumethatafirmwillalwayshaveenoughmoneytopayoffitsbonds,
sothebeta ofitsbondsis0.(Being riskfree, therateofreturnon
thebondsisobviouslyindependentoftherateofreturnonthestock
market.)Assumethatthebetaoftheunderlyingassetsis2.Whatwould
financialwebsitesreportforthebetaofthefirm’sequityifitchangesits
currentcapitalstructurefromallequitytohalfdebtandhalfequity?To
90%debtand10%equity?
8.5 SPREADSHEETCALCULATIONSFORRISKANDREWARD
223
8.5 SPREADSHEETCALCULATIONSFORRISK
ANDREWARD
Doingallthesecalculationsbyhandistedious.Wecomputedthesestatisticswithin
Inreallife,youcando
calculationsfasterwitha
spreadsheet.
thecontextofjustfourscenarios,sothatyouwouldunderstandtheirmeaningsbetter.
However, youcandothisfasterinthereal world.Usually, youwould download
reamsofrealhistoricalratesofreturndataintoacomputerspreadsheet,likeExcel
orOpenOffice.Spreadsheetshaveallthefunctionalityyouneedalreadybuiltin—and
younowunderstandwhattheirfunctionsactuallycalculate.Inpractice,youwould
usethefollowingfunctions:
.
average(range)computestheaverage(rateofreturn).
.
varp(range)computesthe(population)variance.Ifyouworkedwithhistoricaldata
insteadofknownscenarios,youwouldinsteadusethevar(range)formula.(The
latterdividesbyN −1ratherthanbyN,whichIwillexplaininamoment.)
.
stdevp(range)computesthe(population)standarddeviation.Ifyouusedhistorical
datainsteadofknownscenarios,youwouldinsteadusethestdev(range)formula.
.
covar(range-1,range-2)computesthepopulationcovariancebetweentwoseries.If
Excelwasconsistent,thisfunctionshouldbecalledcovarpratherthancovar.
.
correl(range-1,range-2)computesthecorrelationbetweentwoseries.
.
slope(range-Y,range-X)computesabeta.Ifrange-Ycontainstheratesofreturnofan
investmentandrange-Xcontainstheratesofreturnonthemarket,thenthisformula
computesthemarketbeta.
Table8.3showsacomputerspreadsheetthatcomputeseverythingthatyoudidinthis
chapter.
8.5A STATISTICALNUANCES
Inthischapter,wehavecontinuedtopresume(justaswedidinSection7.1E)that
Willhistoryrepeatitself?,
Section7.1E,p.189
historicaldatagivesusanunbiasedguidetothefuturewhenitcomestomeans,
variances,covariances,andbetas.Ofcourse,thisisasimplification—andremember
thatitcanbeaproblematicone.Ialreadynotedthatthisislessofaproblemfor
covariances,variances,andbetasthanitisformeans.Relyonhistoricalmeansas
predictorsoffutureexpectedratesofreturnonlyatyourownrisk!
Thereisasecond,smallerstatisticalissuethatyoushouldbeawareof.Statisti-
Whenworkingwithasample,
the(co)varianceformula
dividesbyN−1.When
workingwiththepopulation,
the(co)varianceformula
dividesbyN.
ciansoftenuseacovarianceformulathatdividesbyN −1,notN.Strictlyspeaking,
dividingbyN −1isappropriateifyouworkwithhistoricaldata.Thesearejustsam-
pledrawsandnotthefullpopulationofpossibleoutcomes.Withasample,youdo
notreallyknowthetruemeanwhenyoude-meanyourobservations.Thedivisionby
asmallernumber,N −1,givesalargerbutunbiasedcovarianceestimate.Itisalso
oftencalledthesamplecovariance.Incontrast,dividingbyNisappropriateifyou
workwith“scenarios”thatyouknowtobetrueandequallylikely.Inthiscase,the
statisticisoftencalledthepopulationcovariance.Thedifferencerarelymattersinfi-
nance,whereyouusuallyhavealotofobservations—exceptinourbookexamples
whereyouhaveonlyfourscenarios.(Forexample,dividingbyN =1,000andby
N =1,001givesalmostthesamenumber.)
TABLE8.3
TheComputerSpreadsheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
S1
S2
S3
S4
Average
Variance
Risk
Market Beta
Alpha
Correlation
Covariance
A
Investor Choice, Sample Spreadsheet
B
C
D
E
F
G
H
I
J
K
L
M
–1.0%
2.0%
4.0%
11.0%
2.0%
11.0%
–1.0%
4.0%
–2.0%
3.0%
7.0%
12.0%
14.0%
6.0%
0.0%
–12.0%
1.0%
1.0%
1.0%
1.0%
4.0%
0.1950%
4.42%
4.0%
0.1950%
4.42%
5.0%
0.2650%
5.15%
2.0%
0.9000%
9.49%
1.000
Base Portfolios
Combinations with A (M)
MC
MB
MD
wB
1/2
0.5%
6.5%
1.5%
7.5%
4.0%
0.093%
3.041%
0.474
2.10%
68.9%
0.09%
1/2
–1.5%
2.5%
5.5%
11.5%
4.5%
0.225%
4.743%
1.064
0.24%
99.1%
0.21%
1/2
6.5%
4.0%
2.0%
–0.5%
3.0%
0.066%
2.574%
–0.564
5.26%
–96.8%
–0.11%
(1/3, 2/3)
8.7%
5.0%
2.3%
–4.0%
3.0%
0.214%
4.625%
–1.043
7.17%
–99.6%
–0.20%
←=average(×5:×8)
←=varp(×5:×8)
←=stdevp(×5:×8)
←=slope(×5:×8,B5:B8)
←=intercept(×5:×8,B5:B8)
←=correl(×5:×8,B5:B8)
←=covar(×5:×8,B5:B8)
Formula
wC
wD
wCDD
A (M)
B
C
D
F
0.00%
100.0%
0.20%
–0.051
4.21%
–5.1%
–0.01%
1.128
0.49%
96.8%
0.22%
–2.128
10.51%
–99.1%
–0.42%
1.0%
0.0000%
0.000%
0.000
1.00%
#DIV/0!
0.00%
Thisspreadsheet(alsoavailableonthebookwebsite)demonstratesthemainstatisticalcalculationsthatareperformedinthischapter.Pleasenotethatweareusingthepopulationvarianceand
populationstandarddeviationformulas,notthesamplevarianceandsamplestandarddeviationformulas.Spreadsheetcellsthatareformulascontainan’=’.Inrows5–8,columnsI–Kareequal
combinationsofM(columnB)andoneotherportfolio(B–D),whichareincolumnsC–E,respectively.ColumnMisaweightedaverageofcolumnsJandK.Formulasinrows11–20aregivenon
therightside.
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