﻿

# c# code to convert pdf file to tiff : Add text to pdf using preview Library software class asp.net windows winforms ajax Welch-ed125-part1812

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
)
Add text to pdf using preview - C# PDF Annotate Library: Draw, edit PDF annotation, markups in C#.net, ASP.NET, MVC, Ajax, WPF
Draw, Add and Edit Various Annotations on PDF File in C# Programming
Add text to pdf using preview - VB.NET PDF Annotate Library: Draw, edit PDF annotation, markups in vb.net, ASP.NET, MVC, Ajax, WPF
Guide to Draw, Add and Edit Various Annotations on PDF File in VB.NET Programming
216
CHAPTER8
INVESTORCHOICE:RISKANDREWARD
Thisslopeof1.128(alittlemorethanaperfect45
diagonal)isexactlythemar-
Youcanconﬁrmour
calculationsusinga
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
oilriskexposure.)
Intherealworld,youwillsometimesthinkintermsofsuchscenarios,butmore
estimatemarketbetainthe
realworld:Use3–5yearsof
dailyobservationsandthen
oftenyouhavetocomputeamarket betafromhistorical ratesofreturnfor the
overallstockmarketandforyourproject(orsimilarprojects).Fortunately,aswe
notedupfront,thebetacomputationsthemselvesareexactlythesame.Ineffect,
whenyouusehistorical data, , yousimplyassumethat t each time period wasone
representativescenarioandproceedfromthere.Nevertheless,therearesomereal-
thatthebestmarketbetaestimatescomefromdailyorweeklydata.Annualdata
shouldbeavoided(exceptinatextbookinwhichspaceislimited).Monthlydata
canbeusedifneedbe.
2. Howmuchdatashouldyouuse?Mostresearcherstendtouse3–5yearsofhistor-
notgoingtoofarbackintoancienthistory,whichmaybelessrelevant.Ifyouhave
dailydata,3yearsworksquitewell.
How to C#: Preview Document Content Using XDoc.Word
C# DLLs for Word File Preview. Add references: RasterEdge.Imaging.Basic.dll. using RasterEdge.Imaging.Basic; using RasterEdge.XDoc.Word; Get Preview From File.
add text field to pdf acrobat; adding text to a pdf form
How to C#: Preview Document Content Using XDoc.PowerPoint
Add necessary XDoc.PowerPoint DLL libraries into your created C# application as references. using RasterEdge.XDoc.PowerPoint; Get Preview From File.
8.3 HOWTOMEASURERISKCONTRIBUTION
217
3. Isthehistoricalbetaagoodestimateofthefuturebeta?Itturnsoutthathistory
cansometimesbedeceptive,especiallyifyourestimatedhistoricalbetaisfaraway
fromthemarket’sbetaaverageof1.Fortunately,thereareatleasttwomethodsto
(A)Averaging:Youcouldrelynotjustonthehistoricalbetacomputedfromyour
manyotherprojectsthataresimilartoyourown(forexample,projectsfrom
thesameindustryorinthesamesizeclass).Suchaveragesareusuallyless
noisy.
(B) Shrinking:Youcould“shrink”yourhistoricalbetatowardtheoverallmarket
betaof1.Forexample,inthesimplestsuchshrinker,youwouldsimplycom-
puteanaverageoftheoverallmarketbetaof1andyourhistoricalmarketbeta
estimate.Ifyoucomputedahistoricalmarketbetaof4foryourproject,you
foryourproject.
Manysmartexecutivesstartwithastatisticalbetaestimatedfromhistoricaldata
solvenow!
Q8.6
Returntoyourcomputationofmarketbetaof1.128inFormula8.5.We
calleditβ
C,M
,orβ
C
forshort.Istheorderofthesubscriptsimportant?
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
inmanycontextsoutsideﬁnance,isthatithasnoscaleandisalwaysbetween−100%
and+100%:
.
Twovariablesthatalwaysmoveperfectlyinthesamedirectionhaveacorrelationof
100%.
.
Twovariablesthatalwaysmoveperfectlyinoppositedirectionshaveacorrelationof
−100%.
.
Twovariablesthatareindependenthaveacorrelationof0%.
tionisthatithasnoscaleandisalwaysbetween−100%and+100%.Thismeansthat
twoinvestments,thesecondbeingamilliontimesbiggerthantheﬁrst(allproject
ratesofreturnmultipliedbyamillion),havethesamecorrelationwiththestockmar-
ket.Yet,thesecondinvestmentwouldgoupordownwithanyslighttremorinthe
marketbyamilliontimesmore,whichwouldofcoursemeanthatitwouldcontribute
C# WinForms Viewer: Load, View, Convert, Annotate and Edit PDF
Supported PDF Processing Features by Using RasterEdge WinForms Viewer for C#.NET. Overview. Highlight PDF text. • Add text to PDF document in preview.
C# WPF Viewer: Load, View, Convert, Annotate and Edit PDF
Features about PDF Processing Features by Using RasterEdge WPF Viewer for C#.NET. Overview. Highlight PDF text in preview. • Add text to PDF document.
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
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
muchmorerisk.Thecorrelationignoresthis,whichdisqualiﬁesitasaseriouscandi-
dateforaprojectriskmeasure.Fortunately,betatakescareofscale—indeed,thebeta
forthesecondprojectwouldbeamilliontimeslarger.Thisiswhywepreferbetaover
correlationasameasureofriskcontributiontoaportfolio.
8.3D INTERPRETINGTYPICALSTOCKMARKETBETAS
Themarketbetaisthebestmeasureof“diversiﬁcationhelp”foraninvestorwhoholds
Marketbetaworkswellwhen
investorsareholdingthe
ofyourproject.
Fromyourperspectiveasamanagerseekingtoattractinvestors,thisisnotaperfect,
necessarilytrueassumption—butitisareasonableone.Recallthatweassumethat
investorsaresmart,sopresumablytheyareholdinghighlydiversiﬁedportfolios.To
convinceyourmarketinvestorstolikeyour\$10millionproject,youjustneedthe
marketcapitalization),whichis1/2,000,000oftheirportfolios.Foryourinvestors,
Mostﬁnancialwebsitespublish
marketbetaestimates.
websites.Table8.2liststhebetasofsomerandomlychosencompaniesinMay2008
fromYahoo!Finance andfromAOL’sﬁnancesite.Mostcompanybetasareinthe
C# PDF insert image Library: insert images into PDF in C#.net, ASP
Using this C# .NET image adding library control for PDF document, you can easily and quickly add an image, picture or logo to any position of
C# PDF insert text Library: insert text into PDF content in C#.net
Able to add a single text character and text string to PDF files using online source codes in C#.NET class program. Supports adding text to PDF in preview
8.4 EXPECTEDRATESOFRETURNANDMARKETBETASFOR(WEIGHTED)PORTFOLIOSANDFIRMS
219
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
much.HoldingApplestockwouldamplifyanymarketswings,notreducethem.In
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
Forexample,youcanthinkofyourﬁrmasacollectionofdivisionsthathavebeen
packagedtogether.IfdivisionCisworth\$1millionanddivisionDisworth\$2million,
thenaﬁrmconsistingofCandDisworth\$3million.Cconstitutes1/3oftheportfolio
VB.NET PDF insert image library: insert images into PDF in vb.net
try with this sample VB.NET code to add an image As String = Program.RootPath + "\\" 1.pdf" Dim doc New PDFDocument(inputFilePath) ' Get a text manager from
How to C#: Preview Document Content Using XDoc.excel
C# DLLs: Preview Excel Document without Microsoft Office Installed. Add necessary references: using RasterEdge.XDoc.Excel; Get Preview From File.
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?
Itellyouwhattheexpectedrateofreturnoneachprojectis,andwhatthemarketbeta
ofeachprojectis,canyoutellmewhattheﬁrm’soverallexpectedrateofreturnand
overallmarketbetaare?Let’stryit.UsetheCandDstocksfromTable8.1onpage202,
andcallCDDtheportfolio(orﬁrm)thatconsistsof1/3investmentindivisionCand
2/3investmentindivisionD.
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
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
(Butyoucannotaverage
variancesorstandard
deviations!)
youcancomputevalue-weightedaveragesforallstatistics.Forexample,variancesand
standarddeviationscannotbeaveraged.
C# PDF Page Insert Library: insert pages into PDF file in C#.net
XDoc.PDF, offers easy & mature APIs for developers to add & insert after the last page or after any desired page of current PDF document) using C# .NET
C# PDF replace text Library: replace text in PDF content in C#.net
C#.NET DLLs: Replace Text in PDF in C#.NET. Add necessary references: RasterEdge. Imaging.Basic.dll. RasterEdge.XDoc.Raster.Core.dll. RasterEdge.XDoc.PDF.dll.
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
(a)Writedownatablewiththerateofreturnonthemarketandon
portfolioCDDineachofthefourpossiblestates.(Hint:Inscenario
S1[
CandDaltogether.(Inthisquestion,youwillworkonlywiththe
marketandCDD.)
(b)ComputetheaveragerateofreturnonthemarketandonCDD.
(c)Writedownatablewiththede-meanedmarketrateofreturnand
CDDrateofreturnineachofthefourpossiblestates.(Themeanof
thede-meanedreturnsmustbezero.)
(d)Multiplythede-meanedratesofreturnineachscenario.Thisgives
youfourcross-products,eachhavingunitsof%%.(Hint:Insce-
narioS1[
(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?
223
ANDREWARD
Doingallthesecalculationsbyhandistedious.Wecomputedthesestatisticswithin
Inreallife,youcando
calculationsfasterwitha
thecontextofjustfourscenarios,sothatyouwouldunderstandtheirmeaningsbetter.
younowunderstandwhattheirfunctionsactuallycalculate.Inpractice,youwould
usethefollowingfunctions:
.
average(range)computestheaverage(rateofreturn).
.
varp(range)computesthe(population)variance.Ifyouworkedwithhistoricaldata
latterdividesbyN −1ratherthanbyN,whichIwillexplaininamoment.)
.
stdevp(range)computesthe(population)standarddeviation.Ifyouusedhistorical
.
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.
chapter.
8.5A STATISTICALNUANCES
Inthischapter,wehavecontinuedtopresume(justaswedidinSection7.1E)that
Willhistoryrepeatitself?,
Section7.1E,p.189
historicaldatagivesusanunbiasedguidetothefuturewhenitcomestomeans,
variances,covariances,andbetas.Ofcourse,thisisasimpliﬁcation—andremember
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