Valuation, Linear Information Dynamic, and Stochastic Discount rates. Dan Gode Stern School of Business New York University

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1 Vluion, Line Infomion Dynmic, nd Sochsic Discoun es Dn Gode Sen School of Business New Yo Univesiy Jmes Ohlson Sen School of Business New Yo Univesiy Mch 22, 2000

2 Asc We genelize Ohlson 995 o sochsic inees es. Ou nlysis povides fou insighs. Fis, he enings cpilizion muliple depends on he lgged e, no he cuen e. Second, he noml enings pesisence pmee inceses in he cuen e nd deceses in he lgged e. Thid, i is no necessy o specify he sochsic pocess undelying inees es o ele soc pices nd ccouning numes. Finlly, only he lgged e is needed o cpilize cuen enings o deemine cuen soc pice, while oh he lgged nd cuen es e needed o foecs ne-peiod enings sed on cuen enings.

3 . Inoducion Ohlson 995 eles ccouning numes nd soc pices unde is neuliy nd non-sochsic discoun es. The model specifies noml enings s fis-ode uoegessive pocess. Thee e wo eeme enchm vluions. In m-ome ccouning, oo vlues e equl o pices nd noml enings hve no pesisence; in pemnen-enings ccouning, pices e equl o cpilized enings ne of dividends nd noml enings hve pesisence of one. The enings cpilizion muliple equls /- whee denoes he is-fee discoun e. The model lso llows fo conve cominions of he wo eemes such h pice is weighed vege of oo vlue nd cpilized enings ne of dividends. The weigh depends on he pesisence of noml enings. We genelize Ohlson 995 o sochsic discoun es. The nul quesions e: How does he enings cpilizion muliple depend on he inees es? Wh line infomion dynmic susins he picing equion unde sochsic inees es? Does one need o specify he sochsic pocess undelying inees es? How do he inees es ffec he cuen enings nd pice elion, nd cuen enings nd nepeiod epeced enings elion? Ou nlysis povides he following nswes. Fis, he enings cpilizion muliple depends on he lgged e, no he cuen e. Second, he noml enings pesisence pmee in he line infomion dynmic inceses in he cuen e nd deceses in he lgged e. Thid, one need no specify he sochsic pocess undelying inees es o model he elionship eween soc pices nd ccouning numes. Finlly, only he lgged e is needed o cpilize cuen enings o deemine cuen soc pice. The lgged e is needed ecuse he enings e fo he cuen peiod is he e peviling he eginning of he peiod. In cons, oh he lgged nd cuen es e needed o foecs ne-peiod enings sed on cuen enings. Cuen enings e divided y he lgged e o ive he cuen pice, which is hen muliplied y he cuen e o ive he foecs of ne-peiod enings.

4 We uild ou nlysis of he fou issues ove y nlyzing models wih incesing geneliy nd compleiy. Secion 2 descies he noion nd ssumpions. Secion 3 nlyzes he pue m-o-me model. Secion 4 nlyzes he pue pemnen-enings model. Secion 5 nlyzes he weighed vege of he wo models. Secion 6 nlyzes he weighed vege model wih ohe infomion. Secion 7 summizes nd concludes he ppe. 2. Noion nd Assumpions A de, he peceding peiod efes o he peiod fom de - o de, nd he fohcoming peiod efes o he peiod fom de o de. enings fo he peiod - o, i.e., he peceding peiod d dividends, ne of cpil coniuions, de e-dividend me pice of equiy, de oo vlue, de g - goodwill, de is fee inees e fo he peiod o. A de, is he cuen e nd - is he lgged e noml o esidul enings fo he peceding peiod. Assumpions:. is neuliy, 2 which yields: d N Noe h is ndom efoe. 2. Clen suplus elion: - d CS See Fnel nd Lee 998, Dechow, Huon, nd Slon 999, nd Lo nd Lys 2000 fo n empiicl ssessmen of he Ohlson 995 model. 2 Fo is vesion, one cn eplce he epecion opeo y he * h eflecs is-djused poiliies. See Hung nd Lizenege

5 Susequen deivions e sed on he following goodwill equion G, which holds if nd only if one ssumes is neuliy nd CS: g g G 3. The M-o-Me Model We s wih he simple u impon enchm -- he pue m-o-me model. As descied in he inoducion, we now emine he following fou specs of he mo-me model:. The ehvio of noml enings: Since hee is no goodwill, he goodwill equion G yields The picing equion:. 3. The ole of he sochsic pocess undelying inees es: Inees es ply no ole hee ecuse he oo vlue susumes infomion ou inees es. An nlogy o n invesmen fund is helpful. The pices of secuiies held y he fund will genelly depend on inees es, u since m-o-me ccouning ses he oo vlue of ech secuiy o is me pice, he oo vlue will viions in me vlue due o inees es wihou hving o model sochsic inees es. 4. The ole of cuen nd lgged es: In m-o-me ccouning, goodwill nd epeced noml enings e zeo. I lso follows h. Thus, he epeced fohcoming enings depend only on he cuen e. 3 The lgged e plys no diec ole in he nlysis ecuse he oo vlue cpues ll he infomion. 4. The emnen-nings Model We now nlyze he pemnen-enings model long hese fou dimensions. In cons o he m-o-me model, he pemnen-enings model is moe sule nd comple ecuse eling enings o pices equies specificion of he enings 3 See Nissim nd enmn 2000 fo n empiicl elionship eween inees es nd ccouning es of eun. 3

6 cpilizion muliple. 4 Ohlson 995 specifies he pemnen-enings model wih nonsochsic inees es s: d -. The enings cpilizion muliple equls / whee is he is-fee e nd 4. The icing quion unde Sochsic Discoun es In he pemnen-enings model, pice equls cpilized enings minus dividends. The min quesion is: When inees es e sochsic, should he enings cpilizion muliple e defined s - / - o /? I is impon h he choice lso pply o he cse of ceiny, i.e., svings ccoun. We show h only he fome sisfies his cieion. If de we oseve s he enings fo he peiod - o, we cn infe h he svings ccoun lnce - ws / -. By, he lnce gows o. The lnce fe he wihdwl d is he pice. The enings e fo he peiod - o is he e peviling -, no, so he cpilizion fco used o inepe enings fo he peceding peiod depends on he lgged e no he cuen e. Theefoe, one oins he following picing equion unde ceiny: d Fom he pespecive of ou nlysis, he mjo diffeence eween ceiny nd unceiny is h noml enings e zeo unde ceiny, u no unde unceiny. I emins o e seen whehe he ove enings cpilizion muliple eends o sochsic discoun es in he spii of he Ohlson 995 model. 4.2 The Behvio of Anoml nings nd nings Ohlson 995 shows h in pemnen-enings model unde consn inees es he noml enings pesisence pmee is consn. 4 See yn

7 ε, whee ε 0. We now specify he line infomion dynmic h susins he picing equion unde sochsic inees es o see how he noml enings pesisence pmee depends on inees es. We hypohesize he following line infomion dynmic: ω ε, whee ω cn depend only on he hisoy of inees es. The wo min issues e: Does ω depend on he enie hisoy of inees es o is smlle suse sufficien? Does ω oscille ound, which is is vlue when inees es do no chnge coss ime? oposiion : Given is neuliy nd clen suplus, ω. oof: See Appendi I. d implies The noml enings pesisence pmee depends only on he lgged nd cuen e, no he enie hisoy of inees es. I deceses in he lgged e nd inceses in he cuen e. If he disiuion of inees es sisfies esonle eguliy condiions, hen he medin noml enings pesisence pmee is, which is is vlue when he inees es e consn. The inuiion undelying he funcionl fom of he enings pesisence pmee cn e iefly sed s follows. The cuen noml enings e fis divided y he lgged e s cpilizion fco nd e hen muliplied y he cuen e o compue foecsed fohcoming noml enings. Fuhe deils e in secion 4.4. The ndom Wl of nings Ohlson 995 implies he following sochsic pocess fo enings: The fis em epesens he sndd ndom wl model of enings nd is vlid only if hee is no new invesmen nd hee e no chnges in inees es. The second em epesens he djusmen o epeced enings due o chnges in invesmen levels 5

8 . I is esy o see h will e eplced y when inees es e sochsic ecuse epeced enings depends on he cuen e pplied o new invesmens. The following coolly evels chnging o is no enough; sochsic inees es inoduce n ddiionl em in he sndd ndom wl model. Coolly : oof: See Appendi I. % The hid em, which hs no een ecognized in pio esech, shows he diecion of chnge in inees e, no jus he level of inees es, ffecs enings foecss; n upic in inees es led o highe enings foecss, nd vice ves. 4.3 The Lc of Need To Specify he Sochsic ocess Undelying Inees es The pemnen-enings model does no equie specificion of he sochsic pocess undelying inees es ecuse enings susume infomion ou inees es. In he cse of svings ccoun discussed in secion 4., he lgged e is sufficien o infe he svings ccoun lnce fom oseved enings nd he cuen e is sufficien o compue he gowh in he lnce ove he fohcoming peiod. pecion of fuue inees es is no needed. 4.4 The ole of Cuen nd Lgged es A ey insigh of he ppe is h only he lgged e is needed o cpilize cuen noml enings nd only he cuen e is needed o cpilize epeced fohcoming noml enings. Coolly 2: g nd g. oof: See Appendi I. The coolly ings ou he cucil inuiion h he enings e fo peiod is he inees e peviling he eginning of h peiod. 6

9 Fom he coolly, we ge pesisence pmee ω g, i.e., he noml enings. Given cuen noml enings, he highe he lgged e, he lowe he cuen goodwill; he highe he cuen e, he highe he noml enings h his goodwill is epeced o genee. 5. A Weighed-Avege of he Two Models We now eend he weighed vege of he pemnen-enings model nd he m-o-me model pesened in Ohlson 995 o sochsic inees es. To fcilie compison, we coninue o sudy he fou specs lised in he inoducion. 5. The icing quion Ohlson 995 epesses pice s weighed vege of he wo models s follows: d We specify he picing equion s weighed vege of he pemnen-enings model nd he m-o-me model unde sochsic inees es s follows: d whee [0,]. Ou ojecive is o deive he line infomion dynmic nd he modificion o he ndom wl of enings h e implied y such epesenion. 5.2 The Behvio of Anoml nings Ohlson 995 shows h he ove picing equion unde non-sochsic es implies he following line infomion dynmic: ω ε, whee ε 0 nd ω. 7

10 We hypohesize he following line infomion dynmic: ω ε. As efoe, ω cn depend only on he hisoy of inees es. One cn s whehe ω coninue o incese in he cuen e nd decese in he lgged e, s in he pemnen enings model. oposiion 2: Given is neuliy nd clen suplus, implies ω. oof: See Appendi I. d Simil o he pemnen-enings model, he noml enings pesisence pmee deceses in he lgged e nd inceses in he cuen e Fo > 0, ω > 0 nd ω < 0. As he weigh ssigned o enings in he picing equion inceses, he ω noml enings pesisence pmee inceses > 0. In he m-o-me model 0, ω 0, while in he pemnen enings model, ω / -. Alhough he sensiiviy of he noml enings pesisence pmee o inees es my e epeced, is funcionl fom is no ovious. enging he ems in ω highlighs he impc of chnging inees es on ω. ω The fis em eflecs he coecion due o he chnging inees es while he second em equls ω unde consn inees es. A fuhe undesnding of his elionship equies specificion of how cuen goodwill eles o cuen enings nd epeced fohcoming enings. These elionships e emined in Secion 5.4. So f, we hve ssumed h, he weigh ssigned o pemnen-enings model, is consn. One cn quesion he een o which ou esuls depend on his ssumpion. 8

11 The ousness of ou esuls is emined in Appendi II, which llows o vy coss ime. I shows h ω coninues o incese in he cuen e nd decese in he lgged e when vies ove ime u is nown he eginning of peiod. The ndom Wl of nings Ohlson 995 implies he following epession fo epeced fohcoming enings in he weighed-vege model: ω ω Two feues of he epession ove e noewohy. Fis, epeced fohcoming enings e weighed vege of he epeced fohcoming enings unde he wo models. Second, he weigh ssigned o pemnen enings equls he noml enings pesisence pmee ω. The coolly elow shows h unde sochsic inees es he epeced enings coninue o e weighed vege of enings unde he pemnen-enings model nd he m-o-me model. I uns ou, howeve, h he weigh is no longe equl o he noml enings pesisence. Coolly 3: θ % θ whee θ. oof: See Appendi I. In cons o he non-sochsic cse, now he weigh, θ, ssigned o pemnen enings in he ndom wl equion diffes fom he noml enings pesisence pmee, ω. In fc, θ ω, nd θ depends only on he cuen e while ω depends on oh he cuen nd he lgged e. Thee is, howeve, ey similiy eween he non-sochsic nd sochsic cse. In oh cses, he weigh ssigned o pemnen enings in he epeced enings equion inceses wih. Boh ω nd θ incese in The Lc of Need o Specify he Sochsic ocess of Inees es 9

12 The weighed-vege model does no equie h we specify he sochsic pocess undelying inees es ecuse he enings nd oo vlue susume his infomion. This is no ecuse is ime independen in ou model. Appendi II shows h we do no need specificion of he sochsic pocess even if vies hough ime u is nown he eginning of peiod. 5.4 The ole of Cuen nd Lgged es The pemnen-enings model showed h one needs only he lgged e o cpilize cuen noml enings nd only he cuen e o cpilize epeced fohcoming noml enings. The coolly elow shows h his inuiion eends o he weighed-vege model. Coolly 4: g oof: See Appendi I. nd g. Thee is ey diffeence eween he weighed-vege model nd is wo eemes he pemnen-enings model nd he m-o-me model. A oh eemes, g nd. This, howeve, is no longe ue in he weighed vege of he wo models. The following esemen of he elionship eween epeced fohcoming noml enings nd cuen goodwill evels why his is so: g. is he enings e fom he cuen goodwill ove he fohcoming peiod. Since,. When he pemnen-enings model, g. When < he weighed-vege model, g in ddiion o he enings fom he cuen goodwill, p of he cuen goodwill iself epeced o e ooed s enings, i.e., he cuen goodwill is epeced o decy ove ime s i is gdully nsfomed ino oo vlue hough enings. In he m-ome model, goodwill is ideniclly zeo nd so e epeced noml enings. is 0

13 6. The ole of Ohe Vlue elevn Infomion So f, we hve genelized he Ohlson 995 model wihou ohe infomion. We hve eslished how soc pices nd foecss of fohcoming enings depend on ccouning numes lone when inees es e sochsic. The min insigh fom he peceding nlysis is h oh he lgged nd cuen es e needed o foecs fohcoming enings sed on cuen enings. Cuen enings e fis divided y he lgged e o cpilize hem nd e hen muliplied y he cuen e o ive he foecs of fohcoming enings. We now eend ou nlysis o include he Ohlson 995 model wih ohe infomion. I is ineesing o deemine whehe he cuen nd fuue es coninue o ply he sme ole in he pesence of such ohe vlue elevn infomion. Ohlson 995 llows fo non-ccouning vlue-elevn infomion nd epesses pice s follows d β υ The line infomion dynmic is specified s follows ω υ υ ε, γ υ ε 2, whee ε, 0, ε 2, 0. Ohlson 995 hen deives γ. β We llow sochsic inees es nd specify pice s follows: d β υ The line infomion dynmic is s follows: ω υ υ ε, γ υ ε 2, ω nd whee ε, 0 nd ε 2, 0. We hypohesize h ω nd γ depend only on he hisoy of inees es.

14 One cn s whehe he inoducion of ohe infomion chnge he funcionl fom of ω, nd whehe γ depends on he lgged e. Noe h ω depends on he lgged e ecuse he lgged e is needed o inepe cuen enings. γ is, howeve, no epeced o depend on he lgged e. oposiion 3: Given is neuliy nd clen suplus, d βυ implies ω oof: See Appendi I. ndγ. β The poposiion shows h he funcionl fom of noml enings pesisence ω is unffeced y he inoducion of ohe infomion. The pesisence of ohe infomion γ depends only on he cuen e, no he lgged e. Thus, his ppe genelizes Ohlson 995 o sochsic inees es nd highlighs he ole of cuen nd lgged es in vluion nd foecsing. The ne secion descies he empiicl implicions of ou esuls. 7. Summy nd Implicions The nlysis in his ppe yields nume of siing osevions. Fis, he genelizion of Ohlson [995] hinges on hoough undesnding of how he enchm seings m-o-me nd pemnen-enings ccouning cn llow fo sochsic inees es. Neihe of hese wo cses leves ny choice s o how one models vlue s i eles o oo vlue nd enings, especively, when inees es chnge. In picul, wih espec pemnen enings i is cle h he cpilizion depends solely on he lgged inees e. Second, given he wo enchms i is esonly sighfowd o epnd he modeling o weighed-vege seings, nd o include so-clled ohe infomion. Thid, in ll of hese cses he lgged inees es seves he ciicl ole of scling cuen enings so one cn infe how cuen vlue eles o cuen enings. Fouh, cuen inees es ene he nlysis y influencing he foecs of ne-peiod s epeced enings. Whehe one consides cuen oo 2

15 vlue o cuen cpilized enings, he cuen inees e hus deemines he enings e in diionl sense. Fom n empiicl pespecive, i my seem unsisfcoy h cuen es do no show up eplicily in he vluion funcion. I is, fe ll, well nown h unepeced chnges in inees es coele wih me euns. Bu his osevion is cully eniely consisen wih his ppe s nlysis. Inees e chnges e elevn ecuse hey modify pecepions ou long un enings elive o he cuen inees e. The mos genel vesion of he Ohlson [95] model hee sunmes his cse. Simply conside he possiiliy of hving ohe infomion ν depend on he cuen inees e; h is, he innovion ε 2 my coele negively wih unepeced chnges in inees es. This spec of he model complees he nlysis in h he model developed is fully consisen wih he ide h cuen es should influence cuen me vlues. 3

16 4 Appendi I: oofs oof of oposiion We cn ese he epession fo s: Th is: g Fom he goodwill equion G we ge,, which simplifies o. Thus, ω QD. oof of Coolly Fom oposiion we ge,. Susiuing he epession fo noml enings, we ge, which simplifies o,o % oof of Coolly 2 Fom oposiion we ge,. Fom he poof of poposiion, we ge g. Susiuing, we ge g. QD

17 5 I is ineesing o emine he elionship eween epeced fohcoming enings nd cuen soc pice. Susiuing he epession fo noml enings in we ge,. Using CS, we cn ese his s d d. An nlogy o he svings ccoun ings ou he elionship eween pices nd epeced enings. The enings fo he peiod -, imply h he svings ccoun lnce - ws / -. The lnce equls he lnce - plus he enings ove he peiod -, minus he wihdwls ove h peiod -d. The enings e fo he peiod, is. oof of oposiion 2 The picing equion d cn e esed s follows: d. Fom he clen suplus elion, we ge d -. Susiuing fo d in he epession ove, we ge Susiuing fo he epession of noml enings, we ge, which implies g. Using he goodwill equion G we ge,, which implies ω QD

18 6 oof of Coolly 3 Fom oposiion 2 we ge,. Susiuing fo noml enings we ge, Define θ Thus, θ θ, which cn e esed s follows: % θ θ QD. oof of Coolly 4 Fom oposiion 2 we ge,. Fom he poof of poposiion 2, we ge g. Susiuing we ge, g. QD Susiuing fo noml enings nd goodwill in he equion ove, we ge: Upon simplificion, we ge: oof of oposiion 3 υ β d g Susiuing fo fom he clen suplus elion, d -, nd using he definiion of noml enings we ge: υ β g Using he goodwill equion G we ge,

19 7 υ υ β β Since ε υ γ υ 2,, we ge υ γ β This implies, γ β β γ QD Appendi II: The Weighed Avege Model wih Vile u Known Weighs We now emine seing whee he weighs cn vy ove ime, u e nown he eginning of peiod. Thus, pice is epessed s follows: d Fom he ove equion, i is cle h d g Using CS, d -, we ge g Using he goodwill equion G we ge, Since is nown ime, we ge: 5 Thus, he noml enings pesisence pmee is epesened y ω 5 If is no nown ime, we would need o now he covince of nd.

20 As discussed elie, ω coninues o depend on he cuen nd lgged es. Specificlly, i inceses in he cuen e nd deceses in he lgged e. We do no peceive consn ω o e plusile scenio. This cn e seen y esing he epession ove in ems of. ω A consn ω implies h he epession fo weighs used in he picing equion is ecusive depends on, which implies h genelly he weighs depend on he enie hisoy of inees es. Only in he specil cse of m -, whee m is consn we ge he following epession whee he pesisence pmee depends only on he cuen e. ω m m Alhough, his esuls in simple specificion of he line infomion dynmic, hee is no sighfowd economic inepeion of his scenio. The nlysis ove shows h Ohlson 995 cn e genelized o llow fo vile weighs in he picing equion. 8

21 Biliogphy Dechow, ici M., Huon, Amy., nd Slon, ichd G An empiicl ssessmen of he esidul income vluion model. Jounl of Accouning nd conomics, -34. Fnel, ichd M., nd Lee, Chles M. C Accouning vluion, me epecion nd coss-secionl euns. Jounl of Accouning nd conomics, Hung, C. nd. H. Lizenege. Foundions of Finncil conomics, Noh Hollnd, 988. Lo, Kin nd Thoms Lys 999, The Ohlson Model: Coniuion o Vluion Theoy, Limiions, nd mpiicl Applicions, Jounl of Accouning, Audiing nd Finnce. Nissim, Doon, nd Sephen enmn, 2000, The mpiicl elionship Beween Inees es nd Accouning es of eun, Columi Univesiy Woing pe. Ohlson, Jmes A., 995, nings, Boo Vlues, nd Dividends in quiy Vluion, Conempoy Accouning esech Vol. No. 2 Sping 995 pp yn, S., 988, Sucul Models of he Accouning ocess nd nings, h.d. Disseion, Snfod Univesiy. 9