投資學10版習題答案CH18

CHAPTER18:EQUITYVALUATIONMODELS

PROBLEMSETS

1.Theoretically,dividenddiscountmodelscanbeusedtovaluethe

stockofrapidlygrowingcompaniesthatdonotcurrentlypay

dividends;inthisscenario,wewouldbevaluingexpected

dividendsintherelativelymoredistantfuture.However,asa

practicalmatter,suchestimatesofpaymentstobemadeinthe

moredistantfuturearenotoriouslyinaccurate,rendering

dividenddiscountmodelsproblematicforvaluationofsuch

companies;freecashflowmodelsaremorelikelytobe

appropriate.Attheotherextreme,onewouldbemorelikelyto

chooseadividenddiscountmodeltovalueamaturefirmpayinga

relativelystabledividend.

2.Itismostimportanttousemultistagedividenddiscountmodels

whenvaluingcompanieswithtemporarilyhighgrowthrates.These

companiestendtobecompaniesintheearlyphasesoftheirlife

cycles,whentheyhavenumerousopportunitiesforreinvestment,

resultinginrelativelyrapidgrowthandrelativelylowdividends

(or,inmanycases,nodividendsatall).Asthesefirmsmature,

attractiveinvestmentopportunitiesarelessnumeroussothat

growthratesslow.

3.Theintrinsicvalueofashareofstockistheindividual

investor’sassessmentofthetrueworthofthestock.Themarket

capitalizationrateisthemarketconsensusfortherequiredrate

ofreturnforthestock.Iftheintrinsicvalueofthestockis

equaltoitsprice,thenthemarketcapitalizationrateisequal

totheexpectedrateofreturn.Ontheotherhand,ifthe

individualinvestorbelievesthestockisunderpriced.,

intrinsicvalue>price),thenthatinvestor’sexpectedrateof

returnisgreaterthanthemarketcapitalizationrate.

4.Firstestimatetheamountofeachofthenexttwodividendsand

theterminalvalue.Thecurrentvalueisthesumofthepresent

valueofthesecashflows,discountedat%.

5.Therequiredreturnis9%.k$1.22(1.05)0.05.09,or9%

$32.03

t

6.TheGordonDDMusesthedividendforperiod(+1)whichwouldbe.

$1.05

$35(k0.05)

$1.05

k

$350.050.088%

7.ThePVGOis$:

PVGO$41$3.640.09$0.56

D

k1g

P

0

0.16$2gg0.12,or12%

$50

b.P

0

kg0.160.05$18.18

Thepricefallsinresponsetothemorepessimistic

dividendforecast.Theforecastforcurrentyearearnings,

however,isunchanged.Therefore,theP/Eratiofalls.The

lowerP/Eratioisevidenceofthediminishedoptimism

concerningthefirm'sgrowthprospects.

gb

9.a.=ROE=16%=8%

D

1

b

=$2(1–)=$2(1–=$1

$1

D

1

P

0

kg0.120.08$25.00

PPg

b.=(1+)3=$253=$

3

0

kr[E(r)r]6%1.25(14%6%)16%

10.a.

f

m

f

2

g9%6%

3

DE(1g)(1b)$3(1.06)1$1.06

3

1

0

$1.06

D

1

P

0

kg0.160.06$10.60

PE

b.Leading/=$$=

0

1

c.PVGOPE1$10.60$30.16.18$9.275

0

k

ThelowP/EratiosandnegativePVGOareduetoapoorROE

(9%)thatislessthanthemarketcapitalizationrate(16%).

b

g

D

d.Now,youreviseto1/3,to1/39%=3%,andto:

1

E

0

g

(1+)

(2/3)

$3

(2/3)=$

Thus:

V

0

=$–=$

V

0

increasesbecausethefirmpaysoutmoreearningsinstead

ofreinvestingapoorROE.Thisinformationisnotyetknown

totherestofthemarket.

D

$8

11.a.P

0

kg0.100.05$160

1

b.Thedividendpayoutratiois8/12=2/3,sotheplowback

b

ratiois=1/3.TheimpliedvalueofROEonfuture

investmentsisfoundbysolving:

g=b

gb

ROEwith=5%and=1/3

ROE=15%

k

c.AssumingROE=,priceisequalto:

E$12$120

P

1

k0.10

0

Therefore,themarketispaying$40pershare($160–$120)

forgrowthopportunities.

PVGOPE1$10$100

0

k

c.Since=ROE,thestockpricewouldbeunaffectedby

k

cuttingthedividendandinvestingtheadditionalearnings.

kr

Err

=+β[()–]=8%+(15%–8%)=%

Mf

13.a.

f

g=b

ROE=

20%=12%

D(1g)$41.12

V

0

0.1640.12$101.82

0

kg

P=VVg

b.

=(1+)=$

=$

1

1

0

DPP

$4.48$114.04$100

$100

E(r)

0

0.1852,or18.52%

1

1

P

0

14.

Time:0

1

5

6

$

E

$

$

$

t

D

$

$

$

$

%

t

b

g

%

%

%

Theyear-6earningsestimateisbasedongrowthrateof×=.

D

6

$10.85

a.V

5

kg0.150.09$180.82

V

$180.82$89.90

1.155

V

0

5

(1k)5

b.Thepriceshouldriseby15%peryearuntilyear6:because

thereisnodividend,theentirereturnmustbeincapital

gains.

c.Thepayoutratiowouldhavenoeffectonintrinsicvalue

k

becauseROE=.

15.a.ThesolutionisshownintheExcelspreadsheetbelow:

Year

DividendDivgrowthTermvalueInvestorCF

Inputs

beta

mkt_prem

rf

k_equity0.0960

plowback

roe

0.95

0.08

0.02

2012

2013

2014

2015

2016

2017

2018

2019

2020

2021

2022

2023

2024

2025

2026

2027

0.78

0.85

0.93

1.00

1.09

1.18

1.28

1.38

1.49

1.60

1.72

1.85

1.98

2.12

2.26

2.41

0.78

0.85

0.93

1.00

1.09

1.18

1.28

1.38

1.49

1.60

1.72

1.85

1.98

2.12

2.26

92.75

0.75

0.09

0.068

0.0863

0.0845

0.0826

0.0807

0.0788

0.0769

0.0750

0.0732

0.0713

0.0694

0.0675

0.0675

term_gwth

Valueline

forecastsof

annualdividends

Transitionalperiod

withslowingdividend

growth

90.33

31.21=PVofCF

Beginningofconstant

E17*(1+F17)/(B5-F17)

growthperiodNPV(B5,H2:H17)

b.,c.UsingtheExcelspreadsheet,wefindthattheintrinsic

valuesare$and$,respectively.

16.ThesolutionsderivedfromSpreadsheetareasfollows:

IntrinsicIntrinsic

IntrinsicIntrinsic

ValueperValueper

Share:FCFFShare:FCFE

Value:

FCFF

Value:

FCFE

a.100,000

b.109,422

c.89,693

75,128

81,795

66,014

17.

Time:0

1

2

3

D

gt

$

%

$

%

$

%

$

%

a.Thedividendtobepaidattheendofyear3isthefirst

installmentofadividendstreamthatwillincrease

indefinitelyattheconstantgrowthrateof5%.Therefore,we

canusetheconstantgrowthmodelasoftheendofyear2in

ordertocalculateintrinsicvaluebyaddingthepresentvalue

ofthefirsttwodividendsplusthepresentvalueoftheprice

ofthestockattheendofyear2.

Theexpectedprice2yearsfromnowis:

PDk

2

g

=/(–)=$–=$

3

ThePVofthisexpectedpriceis$=$

ThePVofexpecteddividendsinyears1and2is

$1.25$1.5625$2.13

1.201.20

2

Thusthecurrentpriceshouldbe:$+$=$

DP

b.Expecteddividendyield=/=$$=,or%

1

0

c.TheexpectedpriceoneyearfromnowisthePVatthattime

ofP

D

and:

2

2

P

1

DP

=(+)/=($+$/=$

22

Theimpliedcapitalgainis

P

PP

(–)/=($–$/$==%

00

1

Thesumoftheimpliedcapitalgainsyieldandtheexpected

dividendyieldisequaltothemarketcapitalizationrate.

ThisisconsistentwiththeDDM.

18.

Time:

0

1

4

5

E

t

D

$

$

$

$

$

$

$

$

t

b

Dividends=0forthenextfouryears,so=(100%plowback

ratio).

a.PD

5$10.368$69.12

k

0.15

4

k

(Since=ROE,knowingtheplowbackrateisunnecessary)

$69.12$39.52

P

4

V

0

(1k)41.154

b.Priceshouldincreaseatarateof15%overthenextyear,

k

sothattheHPRwillequal.

19.Before-taxcashflowfromoperations

Depreciation

$2,100,000

210,000

TaxableIncome

1,890,000

661,500

Taxes(@35%)

After-taxunleveragedincome

After-taxcashflowfromoperations

1,228,500

(After-taxunleveragedincome+depreciation)1,438,500

(After-taxcashflowfromoperations–newinvestment)

$1,018,500

Thevalueofthefirm.,debtplusequity)is:

$1,018,500$14,550,000

C

kg0.120.05

V

0

1

Sincethevalueofthedebtis$4million,thevalueofthe

equityis$10,550,000.

20.a.g=ROE

b=20%=10%

D

kg

D(1g)$0.501.10$11

P

0

1

0

kg

0.150.10

b.Time

EPS

DividenComment

d

$

0

1

$

g=10%,plowback=

2

EPShasgrownby10%basedon

lastyear’searningsplowback

andROE;thisyear’searnings

plowbackrationowfallstoand

payoutratio=

3

$

$

EPSgrowsby(15%)=6%and

payoutratio=

$0.7696

D

3

Attime2:P

2

kg0.150.06$8.551

$0.55$0.726$8.551$7.493

Attime0:V

0

1.15

(1.15)2

P

0

PPg

=$11and=(1+)=$

10

c.

(Becausethemarketisunawareofthechangedcompetitive

situation,itbelievesthestockpriceshouldgrowat10%per

year.)

P

2

=$afterthemarketbecomesawareofthechangedcompetitive

situation.

P

=$

=$(Thenewgrowthrateis6%.)

Return

3

Year

1

($12.10$11)$0.55

0.150,or15.0%

$11

($8.551$12.10)$0.726

0.233,or23.3%

2

3

$12.10

($9.064$8.551)$0.7696

0.150,or15.0%

$8.551

Moral:Innormalperiodswhenthereisnospecialinformation,

k

thestockreturn==15%.Whenspecialinformationarrives,

alltheabnormalreturnaccruesinthatperiod

,asonewould

expectinanefficientmarket.

CFAPROBLEMS

Thisdirectorisconfused.Inthecontextofthe

g

growthratewillfall,andstockpricewillnotnecessarily

k

rise.Infact,ifROE>,pricewillfall.

.ROE

plowback)willfallasplowbackratiofalls.

(ii)Theincreaseddividendpayoutratewillreducethe

growthrateofbookvalueforthesamereason--lessfunds

arereinvestedinthefirm.

2.Usingatwo-stagedividenddiscountmodel,thecurrentvalueofa

shareofSundanciiscalculatedasfollows.

D

3

(kg)

D

1

D

2

V

0

(1k)1(1k)2(1k)2

$0.5623

$0.3770$0.4976(0.140.13)$43.98

1.1411.1421.142

where:

E

0

D

0

=$

=$

FCFE=Earnings+DepreciationCapitalexpenditures

IncreaseinNWC

=$80million+$23million$38million$41million=$24

million

FCFE

$0.286

#ofsharesoutstanding84millionshares

$24million

FCFEpershare=

Atthispayoutratio,Sundanci'sFCFEpershareequals

dividendspershare.

b.TheFCFEmodelrequiresforecastsofFCFEforthehigh

growthyears(2012and2013)plusaforecastforthefirst

yearofstablegrowth(2014)inordertoallowforan

estimateoftheterminalvaluein2013basedonperpetual

growth.BecauseallofthecomponentsofFCFEareexpected

togrowatthesamerate,thevaluescanbeobtainedby

projectingtheFCFEatthecommonrate.(Alternatively,the

componentsofFCFEcanbeprojectedandaggregatedforeach

year.)

FCFEBaseAssumptions

ActualProjectedProjected

2012

27%

2013

27%

2014

13%

Growthrate(g)

TotalPerShare

Earningsaftertax

$80

23

$

$

$

$

$

$

Plus:Depreciationexpense

Less:Capitalexpenditures

Less:Increaseinnetworking

capital

38

41

24

$

$

Equals:FCFE

Terminalvalue

$*

Totalcashflowstoequity

Discountedvalue

$

$?

$?

$?

Currentvaluepershare

$§

*Projected2013terminalvalue=(Projected2014FCFE)/(r

g)

?Projected2013Totalcashflowstoequity=

Projected2013FCFE+Projected2013terminalvalue

?Discountedvaluesobtainedusing=14%

k

§Currentvaluepershare=Sumofdiscountedprojected2012and2013

totalFCFE

c.i.TheDDMusesastrictdefinitionofcashflowstoequity,.

theexpecteddividendsonthecommonstock.Infact,takento

itsextreme,theDDMcannotbeusedtoestimatethevalueofa

stockthatpaysnodividends.TheFCFEmodelexpandsthe

definitionofcashflowstoincludethebalanceofresidualcash

flowsafterallfinancialobligationsandinvestmentneedshave

beenmet.ThustheFCFEmodelexplicitlyrecognizesthefirm’s

investmentandfinancingpoliciesaswellasitsdividendpolicy.

Ininstancesofachangeofcorporatecontrol,andthereforethe

possibilityofchangingdividendpolicy,theFCFEmodelprovides

abetterestimateofvalue.TheDDMisbiasedtowardfindinglow

P/Eratiostockswithhighdividendyieldstobeundervaluedand

conversely,highP/Eratiostockswithlowdividendyieldstobe

overvalued.Itisconsideredaconservativemodelinthatit

tendstoidentifyfewerundervaluedfirmsasmarketpricesrise

relativetofundamentals.TheDDMdoesnotallowforthe

potentialtaxdisadvantageofhighdividendsrelativetothe

capitalgainsachievablefromretentionofearnings.

ii.Bothtwo-stagevaluationmodelsallowfortwodistinct

phasesofgrowth,aninitialfiniteperiodwherethegrowthrate

isabnormal,followedbyastablegrowthperiodthatisexpected

tolastindefinitely.Thesetwo-stagemodelssharethesame

limitationswithrespecttothegrowthassumptions.First,there

isthedifficultyofdefiningthedurationoftheextraordinary

growthperiod.Forexample,alongerperiodofhighgrowthwill

leadtoahighervaluation,andthereisthetemptationto

assumeanunrealisticallylongperiodofextraordinarygrowth.

Second,theassumptionofasuddenshiftfromhighgrowthto

lower,stablegrowthisunrealistic.Thetransformationismore

likelytooccurgradually,overaperiodoftime.Giventhatthe

assumedtotalhorizondoesnotshift.,isinfinite),thetiming

oftheshiftfromhightostablegrowthisacritical

determinantofthevaluationestimate.Third,becausethevalue

isquitesensitivetothesteady-stategrowthassumption,over-

orunderestimatingthisratecanleadtolargeerrorsinvalue.

Thetwomodelsshareotherlimitationsaswell,notably

difficultiesinaccuratelyforecastingrequiredratesofreturn,

indealingwiththedistortionsthatresultfromsubstantial

and/orvolatiledebtratios,andinaccuratelyvaluingassets

thatdonotgenerateanycashflows.

4.a.Theformulaforcalculatingapriceearningsratio(P/E)for

astablegrowthfirmisthedividendpayoutratiodividedby

thedifferencebetweentherequiredrateofreturnandthe

growthrateofdividends.IftheP/Eiscalculatedbasedon

trailingearnings(year0),thepayoutratioisincreasedby

thegrowthrate.IftheP/Eiscalculatedbasedonnext

year’searnings(year1),thenumeratoristhepayoutratio.

P/Eontrailingearnings:

P/E=[payoutratio(1+)]/(

P/Eonnextyear'searnings:

P/E=payoutratio/(kg)=

=

b.TheP/Eratioisadecreasingfunctionofriskiness;asrisk

increases,theP/Eratiodecreases.Increasesintheriskiness

ofSundancistockwouldbeexpectedtolowertheP/Eratio.

TheP/Eratioisanincreasingfunctionofthegrowthrateof

thefirm;thehighertheexpectedgrowth,thehighertheP/E

ratio.SundanciwouldcommandahigherP/Eifanalystsincrease

theexpectedgrowthrate.

TheP/Eratioisadecreasingfunctionofthemarketrisk

premium.Anincreasedmarketriskpremiumincreasesthe

requiredrateofreturn,loweringthepriceofastockrelative

toitsearnings.Ahighermarketriskpremiumwouldbeexpected

tolowerSundanci'sP/Eratio.

5.a.Thesustainablegrowthrateisequalto:

b

Plowbackratio×Returnonequity=×ROE

wherebNetincome-(DividendspershareSharesoutstanding)

Netincome

ROE=Netincome/Beginningofyearequity

In2010:

b=[208–×100)]/208=

ROE=208/1380=

Sustainablegrowthrate=×=%

In2013:

b=[275–×100)]/275=

ROE=275/1836=

Sustainablegrowthrate=×=%

b.i.Theincreasedretentionratioincreasedthesustainable

growthrate.

[Netincome-(DividendpershareSharesoutstanding)]

Retentionratio=

Netincome

Retentionratioincreasedfromin2010toin2013.

Thisincreaseintheretentionratiodirectlyincreasedthe

sustainablegrowthratebecausetheretentionratioisoneof

thetwofactorsdeterminingthesustainablegrowthrate.

ii.Thedecreaseinleveragereducedthesustainablegrowth

rate.

Financialleverage=(Totalassets/Beginningofyearequity)

Financialleveragedecreasedfrom(3230/1380)atthebeginning

of2010toatthebeginningof2013(3856/1836)

ThisdecreaseinleveragedirectlydecreasedROE(andthusthe

sustainablegrowthrate)becausefinancialleverageisoneofthe

factorsdeterminingROE(andROEisoneofthetwofactors

determiningthesustainablegrowthrate).

D(1g)

kg

V

0

0

where:

D

0

=Dividendpaidattimeofvaluation

g=Annualgrowthrateofdividends

k=Requiredrateofreturnforequity

P

0

Intheaboveformula,

,themarketpriceofthecommon

V

0

g

andbecomesthedividendgrowth

stock,substitutesfor

rateimpliedbythemarket:

P

0

D

g

k

=[×(1+)]/(–)

g

0

Substituting,wehave:

gg

)]/–)

g=%

=[×(1+

b.UseoftheGordongrowthmodelwouldbeinappropriateto

valueDynamic’scommonstock,forthefollowingreasons:

i.TheGordongrowthmodelassumesasetofrelationships

aboutthegrowthratefordividends,earnings,andstock

values.Specifically,themodelassumesthatdividends,

earnings,andstockvalueswillgrowatthesameconstantrate.

InvaluingDynamic’scommonstock,theGordongrowthmodelis

inappropriatebecausemanagement’sdividendpolicyhasheld

dividendsconstantindollaramountalthoughearningshave

grown,thusreducingthepayoutratio.Thispolicyis

ii.ItcouldalsobearguedthatuseoftheGordonmodel,given

Dynamic’scurrentdividendpolicy,violatesoneofthegeneral

conditionsforsuitabilityofthemodel,namelythatthe

company’sdividendpolicybearsanunderstandableand

consistentrelationshipwiththecompany’sprofitability.

P0Payoutratio

kg

becomputedusingthefollowingformulas:

g

ind

=ROE

Retentionrate=

=

k

ind

=Governmentbondyield+(Industrybeta

Equity

riskpremium)

=+

=

0.60

P

30.0

Therefore:

0

E0.120.10

1

b.i.ForecastgrowthinrealGDPwouldcauseP/Eratiostobe

generallyhigherforCountryA.Higherexpectedgrowthin

GDPimplieshigherearningsgrowthandahigherP/E.

r

f+β(M–f)=%+%

k

r

%)=16%

b.Year

2009

Dividend

$

2010$

=

$

$

$

$

2011$

2012$

2013

=

=

$

=

Presentvalueofdividendspaidin2010–2012:

Year

PVofDividend

2010

2011

$=

$=

$

$

2012

$=

$

$

Total=

$2.63

Priceatyear-end2012D2013

kg0.160.09$37.57

$37.57$24.07

PVin2009ofthisstockprice1.163

Intrinsicvalueofstock=$+$=$

c.ThedataintheproblemindicatethatQuickBrushisselling

atapricesubstantiallybelowitsintrinsicvalue,while

thecalculationsabovedemonstratethatSmileWhiteis

sellingatapricesomewhatabovetheestimateofits

intrinsicvalue.Basedonthisanalysis,QuickBrushoffers

thepotentialforconsiderableabnormalreturns,while

SmileWhiteoffersslightlybelow-marketrisk-adjusted

returns.

Strengthsoftwo-stageversusconstantgrowthDDM:

Two-stagemodelallowsforseparatevaluationoftwo

accommodatelife-cycleeffects.Italsocanavoidthe

difficultiesposedbyinitialgrowththatishigherthan

thediscountrate.

Two-stagemodelallowsforinitialperiodofabove-

sustainablegrowth.Itallowstheanalysttomakeuseof

herexpectationsregardingwhengrowthmightshiftfrom

off-trendtoamoresustainablelevel.

AweaknessofallDDMsisthattheyareverysensitiveto

k

g

inputvalues.Smallchangesinorcanimplylarge

changesinestimatedintrinsicvalue.Theseinputsare

difficulttomeasure.

9.a.ThevalueofashareofRioNationalequityusingtheGordon

growthmodelandthecapitalassetpricingmodelis$,as

shownbelow.

Calculatetherequiredrateofreturnusingthecapital

assetpricingmodel:

k=r

k

r

f+β×(M–f)=4%+×(9%–4%)=13%

CalculatethesharevalueusingtheGordongrowthmodel:

D(1g)$0.20(10.12)$22.40

P

0

o

kg

0.130.12

b.ThesustainablegrowthrateofRioNationalis%,calculatedas

follows:

g=b×ROE=Earningsretentionrate×ROE=(1–Payoutratio)×

ROE=

1Dividends

Netincome

$3.20$30.160.09979.97%

1

NetincomeBeginningequity$30.16$270.35

10.a.Toobtainfreecashflowtoequity(FCFE),thetwo

adjustmentsthatShaarshouldmaketocashflowfrom

operations(CFO)are:

1.Subtractinvestmentinfixedcapital:CFOdoesnottake

intoaccounttheinvestingactivitiesinlong-termassets,

particularlyplantandequipment.Thecashflows

correspondingtothosenecessaryexpendituresarenot

availabletoequityholdersandthereforeshouldbe

subtractedfromCFOtoobtainFCFE.

2.Addnetborrowing:CFOdoesnottakeintoaccountthe

amountofcapitalsuppliedtothefirmbylenders.,

bondholders).Thenewborrowings,netofdebtrepayment,

arecashflowsavailabletoequityholdersandshouldbe

addedtoCFOtoobtainFCFE.

b.

Note1:RioNationalhad$75millionincapitalexpenditures

duringtheyear.

Adjustment:negative$75million

Thecashflowsrequiredforthosecapitalexpenditures(–

$75million)arenolongeravailabletotheequityholders

andshouldbesubtractedfromnetincometoobtainFCFE.

Note2:Apieceofequipmentthatwasoriginallypurchasedfor

$10millionwassoldfor$7millionatyear-end,whenithada

netbookvalueof$3million.Equipmentsalesareunusualfor

RioNational.

Adjustment:positive$3million

IncalculatingFCFE,onlycashflowinvestmentsinfixed

capitalshouldbeconsidered.The$7millionsalepriceof

equipmentisacashinflownowavailabletoequityholdersand

shouldbeaddedtonetincome.However,thegainoverbook

valuethatwasrealizedwhensellingtheequipment($4million)

isalreadyincludedinnetincome.Becausethetotalsaleis

cash,notjustthegain,the$3millionnetbookvaluemustbe

addedtonetincome.Therefore,theadjustmentcalculationis:

$7millionincashreceived–$4millionofgainrecorded

innetincome=

$3millionadditionalcashreceivedaddedtonetincometo

obtainFCFE.

Note3:Thedecreaseinlong-termdebtrepr