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:aa610@columbia.edu,ph: (212)854-9154.�AndersonSchoolC509,UCLA, CA 90095.jliu@anderson.ucla.edu, 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

Ang, A., andJ.Liu. (1998).“A GeneralizedEarningsModelof StockValuation,” Stanford

ResearchPaper1491.

Bakshi,G., andZ. Chen.(2001).“StockValuationin DynamicEconomies,” forthcoming

Journal of Finance

Bekaert,G., andS.Grenadier. (2001).“StockandBondPricingin anAffineEconomy,”

workingpaper, ColumbiaUniversity.

Black,F., andM. Scholes.(1973).“The Pricingof OptionsandCorporateLiabilities,” Journal

of Political Economy81,637-654.

Cox,J.S.,J. Ingersoll,andS.Ross.(1985).“A Theoryof theTermStructureof Interest

Rates,” Econometrica53,385-408.

Duffie, D., andR. Kan.(1996).“A Yield-FactorModel of InterestRates,” Mathematical

Finance6, 4, 379-406.

Duffie, D., andJ.Liu. (2001).“Floating-FixedCreditSpreads,” FinancialAnalystsJournal57,

3, 76-87.

Fama,E. F., andK. R. French.(2000).“ForecastingProfitabilityandEarnings,” Journal of

Business73,2, 161-175.

Feltham,G. A., andJ.A. Ohlson.(1995).“ValuationandCleanSurplusaccountingfor

OperatingandFinancialActivities,” Contemporary AccountingResearch 11,2, 689-731.

Feltham,G. A., andJ.A. Ohlson.(1999).“ResidualEarningsValuationwith Risk and

StochasticInterestRates,” TheAccountingReview 74,2, 165-183.

Harrison,J.M., andD. M. Kreps.(1979).“MartingalesandArbitragein Multiperiod

SecuritiesMarkets,” Journalof EconomicTheory2, 3, 381-408.

37

Lucas,R. (1978).”AssetPricesin anExchangeEconomy”,Econometrica46,6, 1429-45.

Miller, M., andF. Modigliani. (1961).“Di videndPolicy, Growth andtheValuationof Shares,”

Journal of Business34,4, 411-433.

Nissim,D., andS.H. Penman.(2000).“The EmpiricalRelationshipBetweenInterestRates

andAccountingRatesof Return,” workingpaper, ColumbiaUniversity.

Ohlson,J.A. (1990).“A Synthesisof SecurityValuationTheoryandtheRoleof Dividends,

CashFlows,andEarnings,” Journal of Contemporary AccountingResearch 6, 2, 648-676.

Ohlson,J.A. (1995).“Earnings,Book ValuesandDividendsin EquityValuation,” Journal of

Contemporary AccountingResearch 11,2, 661-687.

Ou,J.A., andS.H. Penman.(1989).“AccountingMeasurement,Price-EarningsRatio,andthe

InformationContentof SecurityPrices,” Journalof AccountingResearch 27,0, 111-44.

Penman,S.H. (1991).“An Evaluationof AccountingRate-of-Return,” Journalof Accounting,

AuditingandFinance6, 2, 233-255.

Zhang,G. (1998).“AccountingInformation,CapitalInvestmentDecisions,andEquity

Valuation:TheoryandEmpiricalImplications,” workingpaper, HongKongUniversityof

ScienceandTechnology.

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