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A GeneralAffineEarnings
ValuationModel
Andrew Ang�
ColumbiaUniversityandNBER
JunLiu�
UCLA
First Version:April 21,1998
This Version:July 1, 2001
JEL ClassificationCodes:G12,M41.
Keywords: stockvaluation,earnings,residualincomemodel,
asset-pricing,affine model,linearinformationdynamics
�Thispaperwasoriginally circulatedastheworkingpaper, “A GeneralizedEarningsModelof Stock
Valuation”. We thankGeertBekaert,Michael Brennan,Ron Kasznik,CharlesLee, Jing Liu, StephenPenman,Stefan Reichelstein,Ken Singleton,RamuThiagarajanandseminarparticipantsat StanfordUniversity andMellon Capital for comments.We especiallythankJamesOhlsonfor encouragement,mentorshipandvaluableadvice.�
3022Broadway, 805Uris, New York, NY 10027.email:[email protected],ph: (212)854-9154.�AndersonSchoolC509,UCLA, CA [email protected], ph: (310)825-7132.
Abstract
We introducea methodology, with two applications,that incorporatesstochasticinterest
rates,heteroskedasticityandrisk aversioninto the residualincomemodel. In the first appli-
cation,goodwill is anaffine (constantplus linear term) functionwheretheconstantandlinear
coefficientsaretime-varying. Homoskedasticrisk givesrise to a constantrisk premium,while
heteroskedasticrisk givesrise to linearstate-dependentrisk premiums.In thesecondapplica-
tion, we presenta classof modelswherea non-linearfunction for theprice-to-bookratio can
bederived. We show how interestrates,risk, profitability andgrowth affect theprice-to-book
ratio.
This paperprovides a parametricclassof modelsthat shows how a firm’s market value
relatesto accountingdataunderstochasticinterestrates,heteroskedasticityandadjustmentsfor
risk aversion. We usethe framework of the ResidualIncomeModel (RIM), which expresses
thevalueof a stockasthe firm’s book valueplus the expectedfuture discountedvalueof the
firm’s abnormal(or residual)earnings.Our methodologybuilds on the framework of Feltham
andOhlson(1999),whoextendtheRIM to ano-arbitragesettingto accommodatetime-varying
interestratesandrisk aversion. FelthamandOhlsongive a partial parametricmodelof stock
valuationusingaccountinginformationin this setting. In a heteroskedasticenvironmentwith
stochasticinterestratesandrisk aversion,we extendthis analysisin several ways. First, we
apply this methodologyto thecasewherethedynamicsof accountingvariablesareexpressed
in dollar amountsasin FelthamandOhlson(1995).Second,wederivea solutionfor theprice-
to-bookratio of a firm asa functionof stochasticinterestrates,accountingratesof returnand
growth in book.
Our first resultextendsthe Linear InformationModel (LIM) developedin Ohlson(1995)
andFelthamandOhlson(1995). The LIM presentsfirm valueasa linear function of current
observableaccountinginformationandis derivedunderconstantdiscountrates.This assump-
tion leadsto astandardsimplificationwhereasinglediscountfactorcanbeappliedto all future
periods.The discountfactorcanincorporateanad hocadjustmentfor risk. Therearecertain
questions,however, whichcannotbeaddressedunderthisassumption.For instance,is it always
possibleto incorporaterisk aversionasa spreadin a constantdiscountfactor?Cana linearso-
lution be found underthe addition of time-varying interestrates,heteroskedasticityand risk
aversion?FelthamandOhlson(1999)show how to adjusttheRIM for risk but do not provide
a completeparametricmodel to answerthesequestionsdirectly. They hint, however, that a
modelastractableastheLIM might exist undermoregeneralconditions.We proposea class
of modelswherean extendedFeltham-Ohlsonlinear form canbe preserved understochastic
interestratesandtime-varyingrisk premiums.Underrisk neutralityandconstantinterestrates,
ourmodelreducesto theFeltham-OhlsonLIM.
Ourextensionto theLIM expressesfirm valueasanaffinecombination(constantpluslinear
1
form) of abnormalearningsandbookvalue.We show thatit is possibleto choosea parameter-
izationsothatanaffine form for goodwill (thedifferencebetweenpriceandbookvalue)exists
understochasticinterestratesandrisk aversion,provided that the interestrateprocessis un-
correlatedwith accountingvariables.Thecoefficientsin theaffine form aretime-varying,and
reflect the dependenceon currentzerocouponbondprices. Risk aversionpotentiallyaffects
boththeconstantandthelinearterms.With risk aversion,homoskedasticityis capturedconve-
niently only in theconstantterm,while risk aversioncombinedwith heteroskedasticrisk leads
to state-dependentrisk premiumswhichappearin thelinearterms.
UndertheLIM, dollaramountsof residualearningsareassumedto follow astationarypro-
cessandfirm valueis a linear function of contemporaneousearningsinformation. Questions
aboutthe rateof earningsgrowth ratherthan the dollar amountof earnings,or firm growth,
cannoteasilybeaddressedin this setting.Our secondapplicationfocuseson theprice-to-book
ratio,ratheronthanthedollardifferencein thepriceandbook,asin theLIM. Weapplyourpric-
ing methodologyto theRIM framework of FelthamandOhlson(1999),with a normalization
by book-value.� This framework modelstheprice-to-bookasa function of stochasticinterest
rates,arateof returnmeasurebasedonprofitability (accountingreturnsof earningsin excessof
therisk-freerate),andfirm growth. Risk aversionandheteroskedasticityof all threevariables
areexplicitly modeled.
Ratioanalysishighlightstheeffect of therateof profitability andgrowth on valuation.We
might anticipatethathigherprofitability of a firm would leadto higherprice-to-bookratios,as
higherprofitability increasesfirm valuerelative to currentbookvalue. However, theeffect of
growth in bookontheprice-to-bookis notsoclear. � By directlyparameterizinggrowth in book
wecandeterminehow theprice-to-bookratiobehaveswhenparametersunderlyingtheprocess
for growth in book arechanged.Anothercomparative staticof interestis themean-reversion
of profitability andgrowth. Standardeconomicargumentsin a competitiveenvironmentargue
thatthesevariablesaremean-reverting.� Doeshighermean-reversionin profitability or growth
lead to higher or lower price-to-bookratios? Finally, risk-averseagentsare affectedby the
volatility of interestrates,profitability andgrowth. Hence,underrisk aversion,volatility of
2
thesevariablesaffectstheprice-to-book.
We presenta closed-formnon-linearsolutionof theprice-to-bookratio. Thesolutionfor-
mulaallowscharacteristicsof thebehavior of theprice-to-bookto beexaminedby comparative
statics. As expected,increasingprofitability increasesthe price-to-bookratio. Increasingthe
conditionalmeanof bookgrowth unambiguouslyincreasestheprice-to-bookratio. Wefind that
increasingthe autocorrelationof profitability or growth in book increasesthe price-to-book.
That is, ceterisparibus, firms with highly mean-reverting profitability or growth have lower
price-to-bookratios.
Conductingcomparative staticswith respectto thedynamicsof theinterestrateyieldsone
surprisingbehavior of theprice-to-bookwhichseemscounter-intuitive. Wemayexpectthatin-
creasingthevolatility of interestrateswoulddecreasetheprice-to-book,sincenominalinterest
ratesarepositive (which is thecaseundera Cox-Ingersoll-Ross(1985)termstructuremodel)
andincreasingthevolatility of interestratesimplieshigherdiscountfactorsin futureperiods.
However, underconservative accountingandrisk neutrality, theprice-to-bookis an increasing
functionof thevolatility of all statevariablesincluding,surprisingly, thevolatility of theinter-
estrate. This counter-intuitivebehavior is dueto a Jensen’s inequalityeffect which dominates
underrisk neutrality. Whenrisk aversionis introduced,theprice-to-bookratio mayfall asthe
volatility of theinterestrateincreases.
Ourmethodologyincorporatesseveralstylizedfactsof interestratesandrisk aversionwhich
bearon accountingvaluation.First, interestratesaretime-varying,which affectsthediscount
factorsusedin future periods. This is a “denominator”effect but interestratesalso predict
future abnormalearnings,which is a “numerator” effect.� Our formal methodologysimul-
taneouslyhandlesboth the denominatorand numeratoreffect of time-varying interestrates.
Moreover, predictabilityof accountinginformationby any variable,not just interestrates,can
beaccommodated.Second,risk-averseagentsin theeconomyneedto becompensatedfor the
uncertaintyin theevolution of financialstatementinformationbecauseaccountinginformation
is a driving factorof prices.Viewedanotherway, therisk premiumsassociatedwith theuncer-
taintiesof accountinginformationarereflectedin discountfactors,asdiscussedby Felthamand
3
Ohlson(1999).Our methodologytractablyincorporatesrisk aversionandshowshow it affects
theLIM andbook-to-market ratio dynamics.
Our methodologytractablyandparsimonouslyincorporatesrich dynamicsof interestrates
andaccountingvariablesby using“affine” processes(Duffie andKan (1996)),whereboththe
conditionalmeanandconditionalvolatility take on affine forms (constantplus linear terms).
Ohlson(1995)andFelthamandOhlson(1995)rely on simpleAR(1) or modifiedAR(1) pro-
cessesto derive theLIM. Thesearespecialcasesof theaffine set-up.Theaffineprocessesalso
formally encompassFelthamand Ohlson(1999)’s partial model, sincethey can incorporate
bothheteroskedasticityof thedriving variablesandrisk adjustments.
Therestof thepaperis organizedasfollows. In Section1 wedescribetheroleof a “pricing
kernel” in no-arbitragevaluation.Section2 presentstheaffineextensionof theLIM andshows
thatlinearitycansurvivetheintroductionof stochasticinterestratesandrisk aversion.Section3
appliesthemethodologyto thecaseof ratiodynamics,andpresentsaclosed-formmodelof the
price-to-bookratio. Comparativestaticexercisesshow how changesin theinterestrateprocess,
profitability, growth andrisk aversionaffect theprice-to-bookof thefirm. Section4 concludes.
1 A Parameterization of the No-Arbitrage RIM
This sectionintroducesnotationto interpretno-arbitragevaluation. This is accomplishedby
specifyinga tractable“pricing kernel,” which we parameterizeasa log-affine form in Section
1.1.Similar to FelthamandOhlson(1999)’sanalysis,webring theRIM into this framework in
Section1.2.
1.1 A Log-Affine Pricing Kernel
The assumptionof no-arbitrage,togetherwith sometechnicalconditions(seeHarrisonand
Kreps(1979)),guaranteestheexistenceof arandomprocesswhichpricesall assetsin theecon-
omy. This randomprocessis calleda “pricing kernel,” which we denoteby � � � . Thepricing
kernelis uniqueif marketsarecomplete,meaningthat thenumberof securitiesis sufficient to
4
insureagainstall possiblesourcesof risk in theeconomy. While theassumptionof no-arbitrage
guaranteesthe existenceof � � � , no-arbitragegivesno information,however, aboutthe true
functionalform of � � � , only thatsome� � � exists.�The pricing kernel � � � relatesthe price of a securitytoday with its payoffs in the next
period.For any asset:�� ������� � � ��� � � ��� (1)
where� is thepriceof anasset,and � � � areits payoffs at time ��� � . Notethatif � � � �!� , a
unit payoff, then� is thepriceof a oneperiodbond.In thecaseof a stock "# , thepriceof the
stockis relatedto its dividends$% by:
"#&�'��(� � � � ) "# � � �*$% � � + ��, (2)
Time-varying interestratesandrisk aversionarecapturedby the pricing kernel � � � . To
separateout therole of theshortrate -. andrisk in � � � , we introduceanotherrandomvariable/ � � whichwedefineas:
/ � � � � � ��� ) � � � +,
(3)
Recallthat �� ) � � � + �10%2(3 )54 -� + is the price of theone-periodrisk-freebondat time � . This
enablesusto rewrite equation(2) as:
"#&�'��(�60%2(3 )74 -� + / � � ) "# � � �*$% � � + �8� (4)
where� � � � 0%2(3 )74 -� + / � � . Equation(4) separatelydecomposestherole of thepricingkernel
into theshortrateprocess-� , and/ � � , which allows for explicit adjustmentsfor risk aversion
in theno-arbitrageenvironment.
The mostgeneralparameterizationof no-arbitrageis determinedby (i)/ � �:9<; and(ii)
�� ) / � � + �=� . We now assumea parameterizationfor/ � � . Specifically, we assumethat
/ � � is
5
log-normallydistributed:
/ � � �'0>2(3 4 �?A@CBED DAB @ � @CBED >FG � � , (5)
where FG � � is a HJIK� vectorIID N(0,I), @ is a HLIK� vectorand D is a HJIMH matrix. The
subscript� on D indicatesthat D may be a function of time � information, andhencemay
vary throughtime. Theerrors FG � � representall shocksto H driving variablesin theeconomy.
For now, thesedriving variablesremainunspecified,but they canbeany variablewhich affects
pricesin theeconomy. In Section2 wespecifythedriving variablesto beaccountingvariables
in levelsandin Section3 wespecifythedriving variablesto beratios.
Thelog-normalpricingkernelin equation(5) is avalid pricingkernel.First,it satisfiesstrict
positivity as 0>2�3 )5NO+ 9P; . Second,by propertyof the log-normaldistribution (seeAppendix),
��Q� / � � � �R� . SWe referto @ asthepriceof risk which capturestherisk aversionof agents.Thefollowing
exampleillustratesthe role of @ . Let time �T� ; , andsuppose,without lossof generality, the
prevailing shortrate-.UV� ; . Thereis only onesourceof risk in theeconomy, sayfrom earnings,
so HW�=� , and F �YX N(0,1). We would like to pricea claim which hasa payoff D UZF � , thatis the
security’spayoff is theunanticipatedfactorshock.Thepriceof this security� U is givenby:
� U[���\U]� / � D U^F � ����\U 0%2(3 4 �? @ � D �U � @&D UZF � D UZF �� @&D �U , (6)
Thelastequalitycanbederivedusinga lemmain theAppendix.Underthecaseof risk neutral-
ity, @ � ; andthepriceof thesecurityis zero. This is expected,becauseunderrisk neutrality
thepriceof thesecurityis just theexpectedvalueof thesecurity’spayoffs,whichis zero.Under
risk aversion@'_� ; , andrisk-averseagentsmustbecompensatedto take on a risk with a zero
expectedvaluepayoff. If @ ` ; , the price of the securityis negative, which is lessthanthe
risk-neutralpriceof zero.Thatis, risk-averseagentsmustbepaidto bearthis risk. Thegreater
6
thedegreeof risk aversion,themorenegative thevalueof @ , andthemorerisk averseagents
mustbecompensatedfor bearingrisk.
Another interpretationof the role of @ is to look at the role risk plays in factor pricing.
Risk is relatedto thedegreeof correlationbetweenshocksin thedriving variables(factors)and
thenegative of thepricing kernel. To make this concrete,againsupposethat thereis only one
driving variablein theeconomy, Ha�b� , time �[� ; , and F �cX N(0,1). Usingthelemmastated
in theAppendix,theconditionalcovarianceof F � with4 / � at time �Y� ; is givenby:
covU ) F � � 4 / � + � covU F � � 4 0>2(3 4 �? @ � D �U � @&D UZF �� 4 @&D �U , (7)
If factorshockshave a correlationof zero,thentheir pricesof risk arezero. If factorshocks
have non-zerocorrelationwith the pricing kernel,thenthe degreeof correlationis a measure
of thedegreeof risk associatedwith thatvariable.Only if @ is non-zerowill thecovarianceof
thefactorshockwith thesignednormalizedpricing kernel/ � benon-zero.In particular, if @ is
negative,thecovariancewith4 / � is positive,which translatesto non-zerorisk aversion.As the
degreeof risk aversionbecomesgreater, @ becomesmorenegative,andthecorrelationbetween
thedriving factorand4 / � becomesmorepositive.
1.2 The RIM and No-Arbitrage Valuation
Wenow introducenotationto beusedfor applyingthelog-normalparameterizationof thepric-
ing kernelto level andratio analysis.This notationenablestherole of/ � � to beusedin evalu-
atingtheexpectationsof futureperiods.Westartby workingwith pricing kernel� � � notation,
andthenwe definenotationsothatexpectationsof futureperiodscanbetakenwith respectto/ � � . Thesetransformedexpectationsareequivalentto thepricingkernelexpectations.
Thepricingkernel� � � relatesthepriceof astockto its futurepayoffs. Repeatingequation
(2) wehave:
"#&�'��(� � � � ) "# � � �*$% � � + ��,
7
Iteratingthis forwardandassumingtransversality �� ) �de"#d +Vf ; as g f h , we get theDivi-
dendDiscountModel (DDM):
"#��'��ijlk �� � j $m � j , (8)
To relatethedividendprocessbackto accountingvariables,residualaccountingmakesthe
cleansurplusaccountingassumption:
n �� n 5o � �qpr 4 $% (9)
wheren is thebookvalueof equityand pr representsnetearningsat time � . This saysthatthe
increasein thebookvalueof equitycomesfrom earningslessdividendspaid.
FelthamandOhlson(1999)develop a generalizedRIM undertime-varying interestrates
andrisk aversion.They show thatby usingcleansurplusaccountingandtheDDM in equation
(8), theequityvalueof afirm canbewrittenas:s
"#t� n (�u��ijlk �� � j prv � j (10)
wherep v areabnormalearnings:
prv ��pr 4 0%2(3 ) -.5o � +�4 � n 5o � (11)
where-�5o � is therisk-freeshortratefrom time � 4 � to time � , which canbestochastic.In the
basicRIM with constantshort rates,the appropriatecapitalcharge is the (constant)risk-free
rate.In a settingwith stochasticshortrates,therelevantcapitalchargecomponentof abnormal
earningsis the risklessone-periodinterestrate appliedto the book value at the start of the
period. Note that risk is embeddedin theFelthamandOhlson(1999)framework by usingthe
pricingkernelto take thefutureexpectationsof equation(10).
We shall now re-write equation(10) so that the expectationis taken with respectto/ � � ,
8
calledthe“risk-neutralmeasure”.Partof therole of/ � � in thepricingkernelis to separatethe
effectsof therisk-freerateandrisk aversionin � � � . Repeatingequation(4) wehave:
"#�����w�60%2(3 )54 -� + / � � ) "# � � �q$m � � + ��,
This expectationis takenunderthe“real measure”(theprobabilitydensityfunctionexisting in
therealworld). It canberewrittenas:
"#t�K��x �60%2(3 )74 -� +.) "# � � �*$m � � + �., (12)
This expectationis takenunderthemeasure,or probabilitydensityfunction, y . Themeasure
y is calledthe risk-neutralmeasurebecauseunder y the price of equity is just the expected
discountedpayoffs of thesecurity. This is alsocalledtheequivalentmartingalemeasure.zThepricingkernel� � � and
/ � � arerelatedrecursively by:
�{U �
|}k U/ | 0%2(3 )74 - | o � + , (13)
Using the risk-neutralmeasurewe canequivalently rewrite thevalueof equityusingtheRIM
in equation(10),by substitutingfor thepricingkernelin theinfinite sum,as:
"#�� n (�ijlk �� x
j o �|}k U 0%2(3
)74 -� � | + p v � j (14)
This is anequivalentrepresentationof theRIM valuationequation.
Theadvantageof theparameterizationof equation(14) is that it explicitly shows theinter-
actionof stochasticshortrates(throughthestochasticdiscountfactor 0%2(3 )74 -� � | + ) andrisk
aversion(throughthedensity/ usedto evaluatetheexpectation).If thereis norisk, then@ � ;
so/ ��~� andtherealandrisk-neutralmeasurecoincide.
The specialcaseof the RIM presentedin Ohlson(1995)shows the simplificationswhich
ariseusingconstantdiscountratesin thevaluationproblem. The following claim shows how
9
theRIM with constantinterestratesanda constantad hoc risk adjustmentis a specialcaseof
ourgeneralvaluationmethodology.
Claim 1.1 Assumethat
1. shortratesareconstant,so 0%2(3 ) -� + � ���2. the risk premiumis constant,cov ) -�� � �
� 4 / + ���D , where - � is the return on the stock
- � � ���) "# � � �q$m � � +�� "#
ThenthetraditionalRIM of stock valuationholds:
"#&� n ��ijlk �� o j �� prv � j (15)
where "# is the value of firm equity, ��� �����b�D , n is book value, and p v are abnormal
earnings:
p v � pr 4�) � 4 � + n 5o �where pr representsfirm earnings.
Comparedto the traditional RIM, a pricing kernelmethodologymust be employed once
interestratesarestochasticandrisk premiumsaretime-varying.Underthis generalizedsetting
the valuationproblembecomesmuchmorecomplex. Sincethe discountfactor � is constant
in Claim 1.1, it passesthroughtheexpectationoperatorin equation(14). Whenshortratesare
stochastic,the discountfactor 0%2(3 )74 -� � | + is time-varyingandit cannotbe passedthrough
the expectationoperator. In Claim 1.1, risk aversionis capturedby a constantrisk premium
� 4 ��� 9~; . If risk premiumsaretime-varying,cov ) -�� � �� 4 / + is no longerconstantand
/ playsa role in valuation.In a generalizedsettingof stochasticinterestratesandrisk aversion,
wemustvalueequityusingequation(14)where-� and/ needto beparameterized.
10
2 An Affine Inf ormation Model
2.1 Discrete-Time Affine Processes
We now specifywhatarethedriving variablesin theeconomyandhow they evolveover time.
ThegeneralizedRIM in equation(14) is ableto take into accountstochasticshortratesandrisk
aversion. However, equation(14) doesnot relaterealizedfinancialnumberswith firm value,
sincethe valueson the right handsideof equation(14) areforecasts.In this form, equation
(14) givesus a theoreticalframework to study risk andreturn, but not a parametricform to
relatefirm valuewith realizedfinancialaccountingreports.This sectiondevelopsa parametric
no-arbitragemodelof firm valuewhenaccountinginformationis givenin dollaramounts.
TheLIM of Ohlson(1995)andFelthamandOhlson(1995)is a parametricformulationof
theRIM thatexpressesthevalueof a stockasa linearfunctionof currentaccountingvariables
specifiedin dollarsratherthanin termsof abstractfutureexpectations.However, it is developed
underconstantinterestratesandrisk-neutrality. Our aim is to extendthe LIM to accountfor
stochasticinterestratesand risk aversion. We specificallyask underwhat assumptionsand
parameterizationslinearitycansurviveundermoregeneralno-arbitrageconditions.Weshow it
is notalwayspossibleto incorporaterisk aversionby adjustingaconstantdiscountfactor.
For concreteness,we assumethe samedriving variablesin the economyasFelthamand
Ohlson(1995).Specifically, supposethedriving variablesaredenotedby �� , andlet:
���� ) prv���� �� � ^� � + B
where p v denotingabnormalearnings,��� operatingassets,� � and � � “other” informationat
time � . We alsoassumethat �: followsa discrete-timeaffine process(Duffie andKan (1996)).
(Theterm“affine” refersto aconstantpluslinearterm.)
Definition 2.1 A H<Iq� vector�� is saidto follow a discrete-timeaffineprocessif:
�: � � �q���*����(� D >FG � � � (16)
11
where � is an H�I'� vector, � and D are H�IuH matricesand F7 X N(0,I). Theconditional
mean�� ) �� � � + �K�������� is affinein �� , andtheconditionalcovarianceis alsoaffinein ��andis givenby:
D DAB �����*� N �� , (17)
Thenotation“N” representsa tensorproduct,andis interpretedas:
� N �� ��|}k ��� | ��� |} (18)
where �� | refers to the ¡ th elementof �� . The H¢I£H matrices� and � � |� aresymmetric.
Discrete-timeaffine processesare an attractive parsimoniousclassof modelswhich can
capturefeedback(mean-reversion)andstochasticvolatility. They capturefeedbackthroughthe
companionmatrix � in theconditionalmean.Stochasticvolatility dependson the level of the
variablesthroughthe tensorproductin the conditionalvolatility. Hence,the variablesin ��maybeheteroskedastic( � � |� _� ; for some¡ ). Homoskedasticityoccursasaspecialcasewhen
� � |� � ;\¤ ¡ and � _� ; .Wenow givetwo examplesof affineprocesses.First, if thereis noheteroskedasticityin the
conditionalcovariances( � � |� � ;\¤ ¡ ), theprocessreducesto a VectorAutoregressionof first
order(VAR(1)). This is the processusedin the Feltham-OhlsonLIM, where �: follows the
following VAR(1):
�� � � ������(� D FG � � (19)
where�:t� ) p v¥�Z� �� � ^� � + B , and DwD B � � , where � is a constantsymmetricmatrix. In Feltham
12
andOhlson(1995)thecompanionmatrix � takestheform:
���¦ �§� ¦ �§� � ;; ¦ �§� ; �; ; @ � ;; ; ; @ �
,(20)
Note that the LIM extendsto othermoregeneralforms of � , ratherthan to just this special
form. This is a specialcaseof the discrete-timeaffine process,formedby setting ��� ; and
specifyingthecovariancehaveno heteroskedasticity, so � � |} � ;c¤ ¡ .A secondexampleof anaffine processhaving heteroskedasticityis theCox, Ingersolland
Ross(CIR) (1985)modelof termstructure.If ��[�¨-. , theunivariateshortrate,thensetting
��� ; givesa discretizedCIR model,wherethe varianceis proportionalto the level of the
interestrate:
�� � � �q���*����C� D >FG � �where D � �L� � � �� , and � � � is a positive scalar. Undera CIR model, the yield curve can
assumea variety of shapesincluding upward sloping, humpedand downward sloping yield
curves. Thestochasticmovementsof all interestratesareinferredfrom theshortrate -� , once
the dynamicof -� in the CIR model is specified. We usethe CIR term structuremodel to
incorporatetime-varyinginterestrates.
2.2 An Affine Inf ormation Model (AIM)
UndertheLIM, firm valueis a linearfunctionof accountinginformation.Westateassumptions
underwhich linearity, or an affine form, canbe maintainedundera more generalsettingof
heteroskedasticity, risk aversionandtime-varyinginterestrates.
For mostof thissectionweassumethatspotinterestratesareindependentof theaccounting
variables.Theinterestrateprocesscanbevery general.At theendof thesectionwe comment
on thecasewhereinterestratesandaccountingvariablesarecorrelated.
13
2.2.1 An AIM under IndependentInterestRates
We let the H driving variables���� ) p v � B + B , with p v denotingabnormalearnings,and ��a vectorrepresenting“other” informationat time � . This formulationsubsumesFelthamand
Ohlson(1995)’s parameterization,by letting ��©� ) �Z� �� � ^� � + B . We usethe affine processof
Section2.1:
Assumption2.1 �� is a HJI'� vectorwith first elementabnormalearnings p v andother el-
ementsrepresentingother informationat time � . �� follows a discrete-timeaffine processas
definedbyDefinition2.1.
Next we assumethat interestratesfollow a processthat is consistentwith no-arbitragebut
independentof accountinginformation:
Assumption2.2 The economyis arbitrage-free, and spot interest rates -� follow a process
independentof �: . The randomvariable definedin equation(3)/ � � is the productof two
factors/ �«ª � � and
/ �¬ � � :/ � � � / �ª
� �N / �¬ � �
�(21)
where/ �«ª � � and
/ �¬ � � are independentand
/ �«¬ � � ��0>2�34 �? @CBlD DAB @ � @CBED >FG � � � (22)
where FG � � are theshocksof theprocessof �� . Theonly requirementfor thefactor/ �ª � � is that
it remainsarbitrage-free.
Theprocessfor shortratesleadsto zerocouponbondprices®c¯±°r² for period³ :
®c¯±°e² ����x °o �|�k U 0>2�3
)54 -.5o | + , (23)
Oneexampleof anadmissibleprocessfor -. is aCIR modeluncorrelatedwith �: , but any term
structureuncorrelatedwith �� is possible. In the caseof the CIR model,zerocouponbond
14
pricesareexponentialaffine functionsof -� :
® ¯±°e² �'0>2(3 ) � ) ³ + �q´ ) ³ + -� + �
wherethe � ) ³ + and ´ ) ³ + coefficientsareclosed-form(seeCox, IngersollandRoss(1985)).
Usingthemethodologypresentedin Section1, anextendedversionof theFeltham-Ohlson
LIM holds,whereweextendtheLIM to anaffinesetting.
Proposition2.1 Under Assumptions2.1 and 2.2 the valuationfunction µ¶·�<"# 4 n can be
expressedas:
µ¶&�'¸CC�º¹ B �� � (24)
where theconstantcoefficient ¸� andthelinear coefficient ¹� of theaffineformaregivenby:
¸�&�i° k U
i» k �® � ° � » ¼ B �
) �½� �� + ° ) ���q� @ +
¹�&�i° k �®Y¯±°e² ¼ B �
) �½� �� + ° � (25)
where ¼ � is a H<I*� vectorwith first element1 andtherestzero, � is thecompanionmatrix of
theprocessfor �� in equation(16),and �� is a HaIºH matrixdefinedas:
�� j¾| ��» k �� � |� j » @ » �
where �� j¿| is theelementin the À -th row, ¡ -th columnof �� , � � |� j » is theelementin the À -th row,Á-th columnof � � |� (the H¢IºH matrix in equation(18))and @ » is the
Á-th elementof @ .
Wemakeseveralcommentson theaffine form of valuationin Proposition2.1.First, theen-
vironmentis verygeneral,andProposition2.1showsthatanaffine form surviveswith stochas-
tic interestrates,risk aversionandheteroskedasticityin accountinginformation.However, this
affine form in accountingvariables�: dependscrucially on the assumptionthat the interest
15
rateis orthogonalto accountinginformation.Giventhis restriction,spotinterestratesmaytake
on any dynamicconsistentwith no-arbitrage. Second,when interestratesare not constant,
theaffine coefficients ¸� and ¹� dependon time � throughtheir dependenceon the time � zero
couponbondprices ®c¯±°r² . In particular, ¹� canbe interpretedasthe valuationof a perpetuity
whosepaymentsarerisk-adjustedfutureresidualearnings.Thediscountfactorson theperpe-
tuity are ®c¯±°r² . Without risk, �� � ; and ¼�B � � ° �� is the future expectedabnormalearnings³periodsinto thefuture,assumingabnormalearningshave zeromeanasin theLIM. Underrisk
aversion, �� _� ; , andthefutureexpectedabnormalearningsin period ���£³ incorporatea risk
adjustment.
Finally, risk aversioncontributesto both ¸C and¹� . In thecaseof homoskedasticity( � � |� �;\¤ ¡ and � _� ; ), risk aversiongivesriseonly to state-independentrisk premiumsin ¸C . In this
case, ��¢� ; sotheeffect of homoskedasticityentersthroughtheactionof the � @ term. Under
heteroskedasticity( � � |� _� ; for some¡ ), risk aversionentersthe ¹� terms.In thiscase,therisk
premiumis state-dependent,throughthenon-zero �� term.
2.2.2 An AIM Under Constant InterestRates
To focusontheeffectof risk aversionandheteroskedasticity, weanalyzethecasewhereinterest
ratesareconstant(-�Â�Ã-]� ¤ � , ���Ä��0%2(3 ) -]� + and ®Y¯±°e² �J� o °� ). In this casethe valuation
formulain Proposition2.1canbefurthersimplifiedbecauseall bondpriciesaregeometrically
related. We look at several examples,including the original LIM. We also determinewhen
linearity canbemaintainedunderrisk aversionandheteroskedasticity.
Corollary 2.1 In thecaseof constantinterestrates,®c¯±°r² �'� o °� , thecoefficientsareconstant:
¸�&��¸ and ¹�&�u¹ andaregivenby:
¸�� �Å��Å� 4 � ¼ B �
) ��� 4 � + o � ) ���*� @ +
¹½� ) ��� 4�) �½� �� + B + o � ) �½� �� + B ¼ � , (26)
16
Notetheconstantterm ¸ canarisethrougheithera non-zero� , or asa constantrisk premium
throughhomoskedasticrisk ( � _� ; ).Previously, applicationsof the RIM accountfor risk by usingan ad hoc adjustmentto a
constantdiscountfactor. Corollary 2.1 shows that with constantinterestrates,it may not be
possibleto incorporatethe effectsof risk aversionin this way. The traditionalRIM model in
Claim1.1 incorporatesrisk aversionby settingtheconstantdiscountrate � to be �K� �����'�D ,wherethespread�D over therisk-freeratetakesrisk aversioninto account:
µ cdf �ijEk �� o j �� prv � j � (27)
where“cdf” denotesconstantdiscountfactor. Thefollowing lemmashowsthatunderaconstant
discountrate � , thevaluationfunction µ cdf is linearin �� anddoesnothaveaconstantterm:
Lemma 2.1 UnderAssumption2.1and �£� ; , µ cdf is a linear functionof �� .Corollary2.1showsthattheconstanttermis zeroonly if �Æ� ; and@ � ; , or �º� ; and �Ç� ; .If eitherconditionis not satisfied,thevaluefunction µ¶ cannotbedescribedasin equation(27)
usinga constantdiscountfactor � , sowehave thefollowing corollary:
Corollary 2.2 It is not alwayspossible, evenunderconstantinterestrates,to havea constant
discountfactor.
Note that FelthamandOhlson(1995) parameterizep v to have zero mean,but if somestate
variablesin �� have non-zeromeanthen ¸ is no longerzero.In this case,only theassumption
of risk neutralityguaranteestheabsenceof aconstantrisk premium.
The following exampleshows how theAIM neststheLIM of Ohlson(1995)andFeltham
andOhlson(1995)asaspecialcase.
17
Example2.1 The Feltham-OhlsonLIM . In thecaseof �º� ; , homoskedasticity( � � |� � ; so
��P� ; ) andrisk neutrality, @ � ; , theAIM reducesto theFeltham-OhlsonLIM, where:
¸ � ;¹ � ) ��� 4 � B + o � � B ¼ � ,
We commentthat underhomoskedasticity, if � _� ; then ¸ is no longer zero but the linear
coefficient ¹ in thetraditionalLIM remainsunchanged.
In the next examplewe statean alternative setof conditionsunderwhich linearity canbe
maintainedunderrisk aversion.
Example2.2 In thecase�Æ� ; , �Ç� ; andrisk aversion(@È_� ; ), thevaluationfunctionÉ� has
a linear form,where:
¸ � ;¹ � ) ��� 4 ) �Ê� �� + B + o � ) �½� �� + B ¼ � ,
Sincewe assumeno homoskedasticrisk ( �Ë� ; ), �� mustexhibit heteroskedasticity( � � |� _� ;for some¡ ) to benon-degenerate.In thiscase,theeffectof risk aversionis absorbedinto thelin-
earcoefficient ¹ (throughthe �� term).Thatis, state-dependentrisk (throughheteroskedasticity
andrisk aversion)givesriseto state-dependentrisk premiums.
2.2.3 The Caseof Corr elatedInterestRates
When interestratesarecorrelatedwith the accountingvariables,they mustbe includedasa
statevariablein �: . Thatis, were-define�� as��&� ) -�]p v � B + B . Now, therandomvariable/ � �
canno longerbefactoredinto two independentterms,onedependingon -. andonedepending
only onaccountingvariables.In thiscase,anaffinesolutionfor µ¶ is no longerpossible,but we
18
canstill find a functionalform to relateµ¶ with contemporaneous�: . It canbeshown that:
µA��ijEk �¼ vÌ� jE �rÍ � jl ÏÎ ¬tÐ )7Ñ�) À + �qÒ ) À + B �� + �
where � ) À + andÑ�) À + arescalars,and ´ ) À + and Ò ) À + arevectors.Thecoefficients � ) À + , ´ ) À + , Ñ�) À +
and Ò ) À + areconstantandcanbederivedsimilarly to Proposition3.1.� U Thisformulastill relates
µ¶ to observedvaluesof �� but therelationis now non-linear.
3 A Parametric GeneralizedEarningsModel
While theAIM or LIM presentsfirm valueasa linearfunctionof earnings,it is not convenient
for determininghow growth in earningsor growth of the firm affectsvaluation. This section
usesa measureof firm profitability andgrowth in book,togetherwith thestochasticshortrate,
asdriving factorsto build a modelof the price-to-bookratio. This allows direct inferenceof
how therateof profitability andfirm growth affect ratiovaluation.We introducetheframework
in Section3.1.Section3.2presentsanexampleof aspecializedprocessfor thedriving variables
to motivatehow themodelcanbeusedin comparativestatics.We derive themodelin Section
3.3.Finally, Section3.3conductscomparativestaticsusingtheprice-to-bookformula.
3.1 Normalizing the RIM by Book Value
Westartby repeatingtheno-arbitrageRIM in equation(14):
"#�� n ��ijEk ���x
j o �|�k U 0%2(3
)74 -� � | + p v � j ,
To dealwith accountingratios,wenormalizeanddivideeachsideby bookvaluen :"#n �!���q��x
ijlk �
j o �|�k U 0>2(3
)74 -� � | + pr � jn � j o �4 ) ¼ ª пÓZÔÖÕ�× 4 � + n � j o �n (28)
We introducesomedefinitions to convert the accountingvariablesin levels to ratios or
19
growth rates:
Ø &��"# � n ¼�Ù Ð\� n � n 5o �-�Ú ��pr � n 5o �Û &�u- Ú 4�) 0>2(3 ) -�5o � +�4 � + (29)
In equation(29),Ø is theprice-to-bookratio and É� is thegrowth ratein bookvalue. The
next equationfor - Ú is theaccountingreturnon equity (earningsto book). Higheraccounting
returnsdenotehigherprofitability. ThevariableÛ is theaccountingreturnin excessof therisk-
freerate,which we defineas“the abnormalaccountingreturn”. It is derivedusingtheFeltham
andOhlson(1999)definitionof abnormalearnings,andthennormalizingtheabnormalearnings
by bookvalue:
prv ��pr 4�) 0%2(3 ) -�5o � +�4 � + n 5o �Û � p vn 5o � �prn 5o �4�) 0>2�3 ) -�5o � +�4 � +
n 5o �n 5o � (30)
Undermark-to-market accountingØ Y�W� and Û c� ; . Underconservative accountingÛ may
benon-zero.
Usingtheabovedefinitionswecanrewrite equation(28)as:
Ø &�!��� ¼ o ª Ð � x Û � � �ijlk �
j o �|�k �¼ o �«ª ÐÖÓ.Ü o Ù ÐÖÓ.Ü Û � j
�!��� ¼ o ª Ð � xijlk �
j o �|�k �¼ o �«ª ÐÖÓ.Ü o ٠пÓ�Ü Û � j (31)
wherewe assumethat theproductterm is equalto 1 if the index is negative. We assumethat
thevariablesØ , -� , Û andÉ� arestationary. �§�
Equation(31) rewrites the RIM, but expressingthe price-to-bookratio asdiscountedab-
normal returns. It still remainsin the no-arbitragesettingof the RIM. We remarkthat the
20
normalizedsettingof equation(31) is still Miller-Modigliani (1961) consistent,so dividend
policy doesnot influencevaluebecausetheoriginalRIM in levelsis Miller-Modigliani consis-
tent(seeOhlson(1995)).Thedriving variablesbehindtheprice-to-bookareaccountingreturns
of earnings(ratherthanearningsfor pricelevels),growth in bookvalue(ratherthanbookvalue
in levels),andaccountingabnormalreturnsof earnings(ratherthanabnormalearnings).This
movesusfrom thesettingof dollaramountsor pricelevelsto ratiosor growth rates.
Althoughequation(31)showstheprice-to-bookratioto beafunctionof spotrates,abnormal
returnsand growth, the numberson the right handside of equation(31) are forecasts. We
needaparameterizationof thedriving variablesto relatetheprice-to-bookto contemporaneous
accountinginformation.Wedo in this in thefollowing section.
3.2 A SpecializedProcessfor Inter estRates,Profitability and Growth
Wespecifythedriving variablesof theeconomyastherisk-freerate,abnormalreturnsof earn-
ingsandgrowth in bookvalue.Denote���� ) -� Û ^É� + B andassumethat �� follows a discrete-
timeaffineprocess,asdefinedin equations(16)and(17). To giveaconcreteexample,suppose
�� is givenby:
-� � � �q� ª �ÆÝ ª -��� D ªÞ -�§F � � �Û � � �q�\ß��*¸ � -���*¸ � Û (� D ß^F � � �
É� � � �q� Ù �*¸ � -�C�*¸ � Û (�q¸ � É��� D Ù F � � � (32)
with FG�� ) F�� F�� F � + B X N(0,I).
This structuremay seemoverly restrictive, but it is only intendedasa simpleexampleof
how feedbackdynamicscanbe accomplishedin the affine system. Assumingthe errorsare
independentimplies that interestrates,abnormalearningsand growth in equity are subject
to independentshocks. We make this assumptionso that we cananalyzethe effect of each
of the variablesin �� separately. The first equationis a discretizedsquareroot process(the
workhorseCIR modelof termstructureassetpricing) of theshortrate. Throughthevariance
21
term,conditionalvolatility increasesproportionallywith thelevel of theinterestrate.
Thesecondequationis aGaussianprocesswhichsaysthatabnormalearningsareautocorre-
lated,andthattheshortrateGranger-causesabnormalreturns.As interestratesgoup,weexpect
abnormalearningsto decrease(seeNissimandPenman(2000)).This is capturedby anegative
¸ � coefficient. Increasinginterestratesdecreasethediscountfactorsapplyingin futureperiods
anddecreasetheprice-to-bookthrougha “denominator”effect. Thepredictabilityof account-
ing returnsby interestratesis a “numerator”effect becauseit decreasescashflows in future
periods.Both thedenominatorandnumeratoreffectarehandledsimultaneouslyby thedynam-
ics of companionmatrix � in thediscrete-timeaffine process(equation(16)). Thecoefficient
¸ � capturesthemean-reversionof profitability. As profitability becomesmoremean-reverting
(or lesspersistent), � decreases.
The last equationparameterizesgrowth asa Gaussianprocess.The conditionalmeanof
growth in equity is a predictablefunction of pastgrowth in equity, laggedshort ratesand
abnormalearnings. In particular, we would expect firm growth andprofitability to be posi-
tively related;this would be capturedby a positive ¸ � coefficient. The ¸ � coefficient reflects
mean-reversionor persistenceof growth in book. As mean-reversionincreases(or persistent
decreases),¸ � decreases.
In termsof the notationof Section1.1, this imposesthe following structureon the com-
panionmatrix � , and the matrices� and � driving the covariances: �<�Ý ª ; ;¸ � ¸ � ;¸ � ¸ � ¸ �
,
��� ; ; ;; D �ß ;; ; D �Ù
, � � � �D �ª ; ;; ; ;; ; ;
, and � � � � � � � � ; .
We examineseveral interestingeffectson the price-to-bookratio from this parameteriza-
tion. In particular, we separatelydeterminetheeffect of changingtheparametersof theshort
rate,abnormalreturnor growth in book on the price-to-book. Changingthe meanreversion
of profitability or growth is accomplishedby changing � and ¸ � . Risk-averseagentsareaf-
fectedby changesin volatility, so D ª , D ß and D Ù affectvaluationunderrisk aversion.To conduct
22
comparative statics,however, we needto derive ananalyticalexpressionfor theprice-to-book
ratio.
3.3 The Parametric EarningsModel
We now develop a non-linearformula for the price-to-bookratio. The casepresentedin the
previoussectionis only anexampleof a particularaffine system,but our derivationspresented
hereapply to the most generalaffine model. This formula is like the LIM in that it relates
the price-to-bookto observed accountingdatarather than to forecastsin equation(31), but
with abnormalreturnsandgrowth ratesa linear solutionis no longerpossible.The price-to-
book valuationformula is given in the following proposition,which evaluatesthe conditional
expectationof theinfinite sumin equation(31)asa functiononly of time � information.
Proposition3.1 Supposethevariables���� ) -. Û ^É� + B follow a discrete-affineprocessin Defi-
nition 2.1andthedefault-adjustedpricing kernel � � � takesthelog-linear formin equation(5).
Thentheprice-to-bookratio canbewritten onlyasa functionof time � information:
Ø ��~��� ¼ o ª Ð5��xijlk �
j o �|}k U ¼o �«ª ÐÖÓ^Ô o Ù ÐÖÓ^Ô Û � j
�~��� ¼ o ª ÐijEk �¼ vÌ� jE �rÍ � jl Î ¬ Ð )7Ñ�) À + �qÒ ) À + B �� + � (33)
where � ) À + andÑ�) À + are scalars,and ´ ) À + and Ò ) À + are àÂI�� vectors. Theconstantcoefficients
23
� ) À + , ´ ) À + , Ñ�) À + and Ò ) À + aregivenby:
� ) À + � � ) À 4 � + � )54 ¼ � � ¼ � �*´ ) À 4 � +}+ B ) ���q� @ +� �? )54 ¼ � � ¼ � �*´ ) À 4 � +m+ B � )54 ¼ � � ¼ � �*´ ) À 4 � +}+
´ ) À + B ��&� )74 ¼ � � ¼ � �q´ ) À 4 � +}+ B ) ����(�q� N �: @ +� �? )54 ¼ � � ¼ � �*´ ) À 4 � + B ) � N �: +.)74 ¼ � � ¼ � �q´ ) À 4 � +m+Ñ�) À + � Ñ�) À 4 � + �*Ò ) À 4 � + B ) ���*� ) @ 4 ¼ � � ¼ � �q´ ) À 4 � +m+}+
Ò ) À + B ��&��Ò ) À 4 � + B ) ���:�� ) � N �� +�) @ 4 ¼ � � ¼ � �*´ ) À 4 � +m+}+ � (34)
where ¼ | denotesa vectorof zeroswith a 1 in the ¡ th place. Theinitial conditionsaregivenby:
� ) � + � ;´ ) � + B ���� ;Ñ�) � + � ¼ B �
) �Ë�q� @ +Ò ) � + B ���� ¼ B �
)}) � N �� + @ �q���� + (35)
Let us interpretthevaluationmodelof Proposition3.1by comparingit to theLIM andthe
AIM in Section2. Although the LIM is given in dollar amountsand Proposition3.1 gives
a model in ratio termsthe two approachesaresimilar. First, the LIM relateshow pricesare
relatedto currentobservableaccountinginformationratherthanto forecasts.Proposition3.1
doesthesamething. It assumesaparameterizationof accountingratiosandgrowth rates(����) -� Û ^É� + B ) thatenablestheinfinite suminvolving expectationsin equation(31) to bea function
only of observable�� information.Second,theAIM presentedin Section2 incorporatestime-
varying interestratesandrisk aversionunderheteroskedasticity. There,to maintainan affine
form -� mustbeuncorrelatedwith accountingvariables.Here,-. is correlatedwith accounting
ratiosand includedasa statevariable. Third, both the AIM andProposition3.1 arederived
underno-arbitrageframework. Finally, the LIM is closed-formand gives price as a linear
function of accountinginformation. Proposition3.1 is also closed-form,but the solution is
24
non-linear.
We interpretthecoefficients � ) À + , ´ ) À + , Ñ�) À + and Ò ) À + in theclosed-formsolutionasfollows.
As long astransversality is satisfied,� ) À +:f 4�h when À f h so the exponentialtendsto
zero,andtheindividual termsin thesumquickly becomesmall. Practially, this meansthatthe
sumin equation(33)canbeevaluatedveryquickly withoutmany terms.
The � ) À + and ´ ) À + coefficients result from the effect of the - 4 É discountingterms. In
equation(34),thetermsquadraticin4 ¼ � � ¼ � �á´ ) À 4 � + areJensen’sinequalityterms.Theterms
linearin4 ¼ � � ¼ � ��´ ) À 4 � + involving @ resultfrom risk aversion.TheJensen’s inequalityterms
arealwayspositive, while therisk premiumtermscanbenegative if @ is negative. Increasing
thevolatility of a factorincreasestheJensen’s inequalityterms,unlesstherisk premiumterms
in the � ) À + and ´ ) À + recursionsoutweightheeffectof theJensen’s inequalityterms.This implies
that in a risk-neutralsetting,increasingthevolatility of a factorincreases,ceterisparibus, the
termsin the exponential.This increasesthe price-to-book.However, in a risk-aversesetting,
thatis when@£` ; , theprice-to-bookcandecreasewith volatility. We illustratethis in thenext
section.
TheÑ�) À + and Ò ) À + coefficients value the abnormalreturn stream. Notice from the initial
conditionsin equation(35) that the ¼ � vectorpulls out only the abnormalreturnsterms. The
termsinvolving � and ���� resultfrom theactionof theconditionalmeanof theprocessof �� ,andthetermsinvolving @ aretherisk premiumswhichacton thecovariances.In therecursions
forÑ�) À + and Ò ) À + in equation(34) theJensenterm
4 ¼ � � ¼ � �*´ ) À 4 � + alsoenters,alongwith a
risk premiumeffect.
Equations(34) and(35) completelydeterminetheresponseof theprice-to-book(equation
(33)) in termsof parametersto theunderlyingprocess�� . However, theactionof theprice to
theparametersis not immediatelytransparentdueto therecursive natureof thecoefficientsin
thevaluationequation.We now conducta seriesof exercisesin comparative staticsto further
analyzetheeffectsof parameterchangeson theprice-to-book.
25
3.4 ComparativeStaticsof the Price-to-Book
In thissectionweshow how varyingtheparametersof theprocessesof theshortrate,abnormal
returnsandgrowth in bookaffecttheprice-to-bookratioof afirm, usingthemotivatingexample
in equation(32). In a previousversionof this paper(Ang andLiu (1998)),we calibratedthis
modelto severalindividualstocks.Weusetheestimatedparametersof Intel, from January1975
to June1997,asa basisfor illustrating thecomparative statics.Theseparametersarelisted in
Table1. We conductour comparative staticsat thebasecaseof thesamplemeanfor �� over
thesample.�§� In our plots,weshow this baselinecaseasa circle.
In our comparative staticexerciseswe wish to clarify the role of risk aversion.To do this,
only onefactor, the growth in equity � , is priced,andwe set the pricesof risk for the short
rate -� andabnormalreturns Û to zero. The first assumptionis closeto reality, becauseterm
structureestimationshavefoundinsignificantpricesof interestraterisk. Theactionof theprice
of risk of Û is verysimilar to thepriceof risk of É� , soweconcentrateonly on theactionof one
priceof risk. Henceweset@ � ) ;�; @ � + B where@ � is thepriceof risk of growth in book.
For reference,we repeatthesystem��&� ) -� Û �É� + B here:
-� � � �u� ª �ºÝ ª -�(� D ªÞ -�>F � � �Û � � �u�\ßâ�q¸ � -�(�q¸ � Û (� D ß�F � � �
É� � � �u� Ù �q¸ � -�(�q¸ � Û C�*¸ � É�(� D Ù F � � �,
(36)
3.4.1 Effect of the Short Rateon the Price-to-Book
We first examinethe effect of the short-rateparameters.As � ª increases,short rate levels
increaseandfutureabnormalearningsarediscountedbackat higherrates.This decreasesthe
price-to-book. In Figure1 we seethe price-to-bookasa function of the persistenceÝ ª , and
the volatility D ª . We first discussthe effect of Ý ª . As interestratesbecomemorepersistent,
theprice-to-bookdecreases.Intuitively, increasingthepersistenceincreasestheunconditional
meanof theshortrate.As cashflows aregenerallyvaluedbackat a higherdiscountfactor, the
price-to-bookfalls.
26
In Figure1 theprice-to-bookincreaseswhen D ª increases.At first glance,this would seem
counter-intuitive, for two reasons.First, we would expectagentsto dislike volatility, so that
price-to-bookwould decreasewhenvolatility increases.Second,theinterestrateis boundedat
zeroin theCIR model,andincreasingthevolatility of theshortrateincreasestheunconditional
meanof interestrates.This impliesthatthediscountfactorswhich valuebackfutureabnormal
returnsarehigher. However, in our base-linecase,agentsarerisk-neutralwith respectto the
interestraterisk. Theincreasein theprice-to-bookwhen D ª increasesis purelydueto aJensen’s
inequalityeffect. Only if interestraterisk is pricedis it possibleto canceltheJensen’sinequality
effect.
3.4.2 Effect of Profitability on the Price-to-Book
We turn now to the effect of profitability on the price-to-book. As expected,increasing�\ßincreasestheprice-to-bookasa higher �\ß impliesgreaterprofitability. Theprice-to-bookratio
also increasesas the predictability coefficient ¸ � increases,if Û is positive. Economically,
this occursbecauseincreasingabnormalreturnscausescashflows in futureperiodsto increase,
and hencethis increasesthe price-to-bookratio. Correspondingly, decreasing � decreases
abnormalreturns.Theeffect is oppositefor negative levelsof Û .The top panelof Figure2 shows theeffect of alteringthepersistence( ¸ � ) of abnormalre-
turns.Increasingthepersistenceof Û , or decreasingthemean-reversion,increasestheprice-to-
book. Thestatisticalinterpretationis asfollows. Theunconditionalmeanof abnormalreturns
risesasthepersistencerises.Higheraverageabnormalreturnsthenimply higherprice-to-book.
Economically, we expecta firm’s profitability to mean-revert within andacrossindustries(see
FamaandFrench(2000)). Highermeanreversion(or lower persistenceof abnormalearnings)
meansthathigh relative earningsin theshorttermpersistfor fewer periods.This lower prof-
itability decreasestheprice-to-bookratio.
The bottompanelof Figure2 presentsthe price-to-bookasa function of abnormalreturn
volatility D ß . In our parameterization,increasingthe volatility of abnormalreturnsincreases
the price-to-book.This is becausethe price of risk of abnormalreturns@ � is zero,so agents
27
are risk-neutralwith respectto profitability. The Jensen’s effect causesthe price-to-bookto
increasewhen D ß increases.However, if @ � is negativeandagentsarerisk-aversewith respectto
abnormalreturns,thenit maybepossiblefor theprice-to-bookto fall asvolatility of profitability
increases.
3.4.3 Effect of Growth on the Price-to-Book
Any parameterswhich increasethegrowth in bookincreasetheprice-to-bookratio. For exam-
ple, increasing� Ù , ¸ � or ¸ � when Û is positive increasetheprice-to-bookbecauseeachparam-
eterraisesgrowth in book. The intuition is that increasinggrowth increasesthe likelihoodof
highercashflows in futureperiods.
Theparameter � capturesthepersistence,or mean-reversion,of firm growth. Thetoppanel
of Figure3 shows that increasingthepersistenceof thegrowth in book increasestheprice-to-
book. This effect is similar to increasingthe persistenceof Û ( ¸ � ), sincethe unconditional
meanof � risesasthepersistenceof � increases.Higherpersistenceimpliesthatfirm growth
mean-revertsto anindustryor marketaverageata fasterrate.Hencethefirm hasfewerperiods
to enjoy thebenefitsof relatively highergrowth.
Thebottompanelof Figure3 shows theeffect of a decreasingprice-to-bookwith increas-
ing volatility in growth in equity ( D Ù ). This is in line with intuition: we would expect, ce-
teris paribus, normalrisk-averseinvestorsto lower their valuationsthegreaterthevolatility in
growth. Weobtainthis resultbecausethereis a largenon-zeropriceof risk ongrowth in equity
@ � andthis causestheprice-to-bookto decreaseasvolatility increases.In this case,in addition
to the Jensen’s inequalityeffect (which increaseswith D Ù ), thereis alsoa risk aversioneffect
whichcounteractstheJensen’s inequalityeffect.
Finally, Figure4 showstheeffectof risk aversionon theprice-to-book.In Figure4, for risk
aversionlevelsbelow @ � � 4 �Qã the price-to-bookis very flat, but asinvestorsapproachrisk
neutralitytheprice-to-bookbecomesvery large.
28
4 Conclusion
This paperintroducesa methodologythat incorporatesstochasticinterestrates,risk aversion
andheteroskedasticityinto the ResidualIncomeModel (RIM). We provide two applications
of the methodology. First, in applyingthe methodologyto dollar amounts,we show that the
Ohlson(1995) and Felthamand Ohlson(1995) Linear Information Model generalizesto an
affine (constantplus linear terms)modelundertime-varying interestratesandrisk aversion.
Theprocessesfor accountinginformationmaybeheteroskedastic.Theinterestrateprocessis
very generalbut is assumedto be uncorrelatedwith the processesgoverningthe evolution of
accountinginformation.
Second,in applying the methodologyto ratio dynamics,we provide a non-linearclosed-
form formulafor theprice-to-bookratio in termsof stochasticshortrates,profitability andfirm
growth. In comparative staticexercises,increasingthe growth in book increasestheprice-to-
bookratio andincreasingthemean-reversionof profitability or growth decreasestheprice-to-
book. Theeffect of interestrateson theprice-to-bookdependson thedegreeof risk aversion.
Undersufficientlyhighrisk aversion,increasingthemeanor volatility of theshortratedecreases
theprice-to-bookratio.
29
Appendix Proofs
Westartby statingthefollowing Lemmawhichgivestheexpectationof theproductof anormal
with theexponentialof a normal,which canbe provedby evaluatingtheexpectation.This is
usedin someof theproofsbelow.
A Lemma
Lemma A.1 If ä is distributedas a H -variate normal with ä X å ) ; �8æ + , and @ and $ are
H¢I*� constantvectors then
� ) $ B ä ¼�ç Ϋè + � $ B æ @ ¼ ×é ç Îëê ç (A-1)
B Proof of Claim 1.1
Startingfrom the relation "#��¢��Q� � � � ) "# � � � $m � � + � we cansubstitutefor � � � �¢� o �� / � �usingtheassumptionsaboutaflat termstructureto get:
"#t�'� o �� �� / � � ) "# � � �*$% � � + , (B-1)
Usingthedefinitionof thereturnof thestock-�� � � �) "# � � �½$% � � +�� "# andcov ) -�� � �
� 4 / � � + ��D , we canwrite �� ) - � � �
+ ������'�D � � , with � aconstant.
In this case:
"#&� �� ��) "# � � �q$m � � + � (B-2)
anditeratingthis equationforwardandassumingtransversalityweobtain:
"#&�ijlk �� o j �� ) $m � j + , (B-3)
30
Then,asin Ohlson(1995),abnormalearningsbecomep v �¨pr 4ì) � 4 � + n 5o � , andsubsti-
tuting for dividendsin thepreviousequationyields:
"#&� n ��q��ijlk �� o j prv � j � n ��
ijlk �� o j �� ) prv � j + (B-4)
becausethetelescopingsumcollapses,assuming� o�d �� ) n �rd +�f ; asg f h .
C Proof of Proposition2.1
Fromequation(14)wecanwrite:
µ¶t�ijlk ���x
j o �|}k U 0%2(3
)54 -� � | + p v � j (C-1)
whereµA�� "# 4 n . An equivalentstatementis:
µ¶t�'�� 0%2(3 )74 -� + / � � ) µ¶ � � �*p v � �+
0>2(3 ) -� + µ¶t�'�� ) µA � � �*prv � �+ � cov ) µA � � �qprv � �
� / � � + � (C-2)
notingthat �� ) / � � + �~� .Next, conjectureanaffinesolutionfor µA whichhastheform:
µ¶�� ¸�(�ƹ�±�� �
where ¸� and ¹� candependon zerocouponbondprices ® ¯±°r² , but not the statevariables�� .Wecanrewrite equation(C-2)as:
) ® ¯ � ² + o � ) ¸CC�º¹ B �� + ���� ) ¸C � � � ) ¹� � � � ¼ � + B �: � � + cov )m) ¹� � � � ¼ � + B D >FG � � � / � � + , (C-3)
Herenotethat 0%2(3 ) -. + � ) ®Y¯ � ² + o � . UsingLemmaA.1 andevaluatingtheconditionalmeanof
31
�� � � gives:
) ® ¯ � ² + o � ¸CC� ) ® ¯ � ² + o � ¹ B ������� ) ¸C � � + � ) �� ) ¹� � � + � ¼ � + B ��� ) �� ) ¹� � � + � ¼ � + B ����� ) �� ) ¹� � � + � ¼ � + B ) ���q� N �� + @ � (C-4)
aftersubstitutingD D B �'�í�*� N �� . Equatingthecoefficientsof �� wehave:
) ®c¯ � ² + o � ¸�¢� �� ) ¸C � � + � ) �� ) ¹� � � + � ¼ � + B ) ���*� @ +) ®c¯ � ² + o � ¹�¢� ) �Ê� �� + B ) �� ) ¹� � � + � ¼ � + , (C-5)
Onecanshow by substitutionthattheaboveequationsaresolvedby setting:
¸�&�i° k U
i» k �® � ° � » ¼ B �
) �½� �� + ° ) ���q� @ +
¹�&�i
° k �® ¯±°e²�¼ B �
) �½� �� + ° , (C-6)
D Proof of Lemma 2.1
Since� is aconstant,p v �'pr 4') � 4 � + n 5o � is a linearfunctionof thestatevariables�� . That
is, p v �ìî B �: for someconstantvector î . Then, �� ) p v � j + ��î B � j �� is a linearfunctionof �� .Hence:
µ¶&�ijlk �� o j î B � j �:�� î B � ) � 4 � + o � �: � (D-1)
which is a linearfunctionof �� , thatis, thereis noconstantterm.
32
E Proof of Proposition3.1
To proveProposition3.1we show thata singletermin theinfinite sumat horizonï sumtakes
theform:
��xð o �jlk �¼ o �«ª ÐÖÓ^Ô o Ù ÐÖÓ^Ô Û � ð � ¼ vÌ� ð �rÍ � ð ÏÎ ¬Y� )5Ñ�) ï + �*Ò ) ï + B � ) � +m+ (E-1)
We show that the coefficients � ) ï + , ´ ) ï + , Ñ�) ï + , and Ò ) ï + aregiven by the Ricatti differ-
enceequationsin equation(34) with initial conditionsin equation(35). Oncethis is shown,
Proposition3.1follows immediatelyfrom evaluatingeachindividual termin theinfinite sum.
We prove equation(E-1) by induction. Assumevalidity of equation(E-1) for ï . We show
thattheequationholdsfor ïq� � . Usingiterativeexpectationswecanwrite:
� xðjlk �¼ o �ª ÐÖÓZÔ o ٠пÓZÔ Û � ð � � �·� x ¼ o �ª ÐÖÓr× o ٠пÓe×
� x � �ð o �jlk �¼ o �«ª ÐÖÓr×OÓZÔ o ٠пÓe×OÓ^Ô Û � ð � �
�·��x ¼ o � Ú × o Ú}ñ Î ¬ пÓe× ¼ v8� ð �rÍ � ð Î ¬ ÐÖÓe× )7Ñ�) ï + �*Ò ) ï + B �� � � + ,(E-2)
The induction assumptionis usedin the last equality. We then observe that �: � � under ysatisfies(this is thediscrete-timeversionof Girsanov’s theorem):
�� � � �q���*����C� D DAB @ � D >FGx � ��
(E-3)
33
whereF x � � is a (mean-zero)standardnormalrandomvariableunder y . Then:
� xðjlk �¼ o �«ª ÐÖÓ^Ô o Ù ÐÖÓZÔ Û � ð � �
� ¼ vÌ� ð � � o Ú × � Ú�ñ �rÍ � ð O Î �ò �ró ¬tÐ �rô Ð ô Î Ð ç Iõ� x ¼ � ç o � Ú × o Ú�ñ �rÍ � ð O Î ô оö§÷ÐÖÓe× )7Ñ�) ï + �*Ò ) ï + B ) ���*����C� D DAB @ + �*Ò ) ï + BED >F x � �
+� ¼ vÌ� ð � � o Ú × � Ú�ñ �rÍ � ð O Î � ô Ð ô Î Ð ç � ò �ró ¬&Ð ¼ ×é � o Ú × � Ú�ñ �rÍ � ð O Î ôQô Î Ð � o Ú × � Ú�ñ �rÍ � ð ë I Ñ�) ï + �qÒ ) ï + B ) ���*����(� D D B @ + �qÒ ) ï + B D D B )74 ¼ � � ¼ � �q´ ) ï +}+ ,
(E-4)
The last equality is obtainedby employing LemmaA.1. Equatingcoefficients givesus the
result.To obtaintheinitial conditionswe directlyevaluate� x ) Û � � + usingLemmaA.1.
34
Notes
� Penman(1991)arguesthattheRIM hasimplicationsfor ratio analysis,particularlyfor the
price-to-bookandprice-to-earningsratios.
� Zhang(1998)demonstratesthatfor agivenlevel of earningstherelationshipbetweenbook
andequityvalueis ambiguous.
� SeeFamaandFrench(2000),andOu andPenman(1989),amongothers.
� NissimandPenman(2000)demonstratethatinterestrateshavepredictivepowerto forecast
futureaccountingreturns.
� FelthamandOhlson(1999)usethe notation/ � � for the pricing kernel. We usethe no-
tation � � � aswe save/ � � to describea componentof � � � morein line with standardasset
pricing notation.In certainsituationsthepricing kernelhasa known form, suchasthecaseof
consumption-basedassetpricing (Lucas(1978)),or with completemarkets(BlackandScholes
(1973)).Bothof thesefunctionalformsarenotvalid herebecausewedonotspecifyarepresen-
tativeagentwhohasutility overconsumptionandbecausemarketsareincompletewith respect
to accountinginformation(market pricesarenotavailablefor earningsor bookvalues).
� An equivalentrepresentationof thepricing kernelis:
ø � ����� ) ø � ��� � � +
whereø � � satisfiestheequation:
� � � �ø � �ø
,
SeeHarrisonandKreps(1979).
S Althoughthisparameterizationmaylook restrictive,it is veryflexible. Thestrictpositivity
requirementis almostequivalentto specifying/ � � ��0%2(3 )7ù�) FG � � +m+ for somefunction
ù¥)7Në+. The
sourcesof risk arisefrom theshocksof thedriving variablesin theeconomy, which are FG � � .35
For smallvariability in thedriving factors,wecanapproximatethevariablepartofù¥) FG � � + by a
first-orderapproximation@ B D %FG � � . This leadsto theform of thepricing kernelin equation(5).
This form of pricing kernelhasbeenusedin many financialapplicationsincludingDuffie and
Liu (2001),BakshiandChen(2001),BekaertandGrenadier(2001),andothers.
s Strictly speakingFelthamandOhlson(1999)considera countablestatespaceanda finite
horizon. This can be generalizedto most uncountablestatespaceswith sometechnicalas-
sumptions(seeHarrisonandKreps(1979)),andappliedto aninfinite horizonsumby assuming
transversality.
z Theprocess/ � � convertsthe risk-neutralmeasureto the real measure.By definition the
following relationshipholdsbetweenthe real measureand y : � x � � � � �a�� / � ��� � � � for
any �Y�¨� measurablerandomvariable � � � . In technicalterms,the process/ � � is a Radon-
Nikodymderivativeof therealmeasurewith respectto therisk-neutralmeasurey . SeeHarrison
andKreps(1979).
� U Weomit this proofasthederivationis verysimilar to thederivationof Proposition3.1.
�§� Notethat É� doesnot appearin thefirst termof theinfinite sum.Thereasonis thatat time
�c�!� , the opportunitycostsof the firm are) ¼ ª%Ð 4 � + n , which areknown at time � . At time
��� ? theopportunitycostsof thefirm arebasedon n � � � n ¼ Ù Ð . This lag in the timing of É�disappearsif themodelis formulatedin continuoustime. SeeAng andLiu (1998).
�§� The samplemeanof �� is ���Ç� ) ; , ;rú �Ìû ��ü � ; , �Ìàrý�ý � ; , ? ûrûrà + B wherethe interestrate is
annualized,but usedasquarterly.
36
References
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ResearchPaper1491.
Bakshi,G., andZ. Chen.(2001).“StockValuationin DynamicEconomies,” forthcoming
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Bekaert,G., andS.Grenadier. (2001).“StockandBondPricingin anAffineEconomy,”
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Lucas,R. (1978).”AssetPricesin anExchangeEconomy”,Econometrica46,6, 1429-45.
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38
Table1: Parametersfor ComparativeStatics
BaselineParameters� ª 0.0054Ý ª 0.7548D ª 0.0374�\ß 0.0263¸ � -0.8695¸ � 0.7720D ß 0.0176� Ù 0.0247¸ � 1.1829¸ � 0.6008¸ � -0.0105D Ù 0.0698@ � -13.1982
Theseparametersarethebase-linecasefor thecomparativestaticsexercises.The equationsfor the processesaregivenby equation(32). Weset@ � � @ � � ; .
39
0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9
2
4
6
8
10
12
Changing mean−reversion of interest rate
ρr
pric
e to
boo
k
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
4.6
4.7
4.8
4.9
5
5.1
Changing volatility of interest rate
σr
pric
e to
boo
k
Figure1: ComparativeStaticsfor ShortRateParameters
40
0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8
3.5
4
4.5
5
5.5
6
6.5
Changing α2 of abnormal returns
α2
pric
e to
boo
k
0.01 0.015 0.02 0.025 0.03 0.035 0.04
5
5.5
6
6.5
7
Changing σz of abnormal returns
σz
pric
e to
boo
k
Figure2: ComparativeStaticsfor AbnormalReturnParameters
41
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1
4.6
4.8
5
5.2
5.4
5.6
Changing α5 of growth in equity
α5
pric
e to
boo
k
0.065 0.07 0.075 0.08
3
4
5
6
7
8
9
Changing σg of growth in equity
σg
pric
e to
boo
k
Figure3: ComparativeStaticsfor Growth in Equity Parameters
42
−20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10
5
10
15
20
25
30
35
Changing γ3 growth in equity risk
γ3
pric
e to
boo
k
Figure4: ComparativeStaticsfor Priceof Risk of Growth in Equity
43
FigureLegends
Figure1 plotstheprice-to-bookasa functionof persistenceof theshortrateÝ ª (topplot), and
asafunctionof theshort-ratevolatility D ª (bottomplot). Thebase-linecaseis shown asacircle.
Figure 2 plots the price-to-bookasa function of thepersistenceof abnormalreturns ¸ � (top
plot), andthevolatility of abnormalreturnsD ß (bottomplot). Thebase-linecaseis shown asa
circle.
Figure 3 plots the price-to-bookasa function of tje persistenceof growth in equity ¸ � (top
plot), andthevolatility of growth in equity D Ù (bottomplot). Thebase-linecaseis shown asa
circle.
Figure4 plotstheprice-to-bookasa functionof thepriceof risk of growth in equity(@ � ). The
base-linecaseis shown asacircle.
44