Accounting-Based Valuation With Changing Interest Rates

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1 Accouning-Bsed Vluion Wih Chnging Inees Res Dn Gode Sen School of Business New Yok Univesiy hp:// Jmes Ohlson Sen School of Business New Yok Univesiy June 29, 2002 We cknowledge he helpful commens of n nonymous efeee, Suesh Govindj, Sefn Reichelsein (edio), Mev Rom, Sephen Ryn, Ken Skogsvik, nd semin picipns he Chinese Univesiy of Hong Kong, Nionl Univesiy of Singpoe, Snfod Univesiy, Sen School, Sockholm School of Economics, nd he Whon School.

2 Accouning-Bsed Vluion Wih Chnging Inees Res Absc We genelize Ohlson s (995) model o sochsic inees es while mking no specific ssumpions bou he sochsic pocess of inees es. Ou nlysis of he cse when enings suffice fo vluion yields hee insighs. () In he vluion funcion, he muliplie fo fohcoming enings depends on he cuen e, bu he muliplie fo cuen enings depends on he lgged e. (2) In he esidul enings dynmic, he pesisence of esidul enings inceses in he cuen e nd deceses in he lgged e. (3) In he enings dynmic, he diionl ndom wlk equies n ddiionl em, cuen enings muliplied by he pecenge chnge in inees es. Keywods: Sochsic Inees Res, Vluion, Ohlson Model, Rndom Wlk Model of Enings, Pemnen Enings JEL Clssificion: M4, G2 2

3 Inoducion Dynmic models eling ccouning d o equiy vlue need discoun fco o compue pesen vlues nd pice-enings muliples. Ohlson (995) ssumes isk-neuliy nd shows h he muliple fo cuen enings equls R/ whee is he isk-fee e nd R +. The model, howeve, ssumes h he discoun e emins fixed ove ime. Relxing his ssumpion is of obvious inees in wold of chnging inees es. This ppe develops clss of vluion funcions nd deives he implied infomion dynmics when inees es e sochsic. Like Ohlson (995), wo benchmk seings undepin his clss: mk-o-mke ccouning o he blnce-shee ppoch whee book vlue suffices fo vluion, nd, enings-sufficiency ccouning o he income-semen ppoch whee enings suffice fo vluion. A weighed vege of he wo benchmks combined wih infomion besides cuen ccouning d genelizes he model. Thee specs of he model dmi geneliy nd vesiliy. Fis, he nlysis ssumes no picul sochsic pocess of inees es. Second, elizions of ccouning d nd ohe infomion cn depend on hisoicl nd expeced inees es. Thid, unexpeced chnges in inees es cn coele wih unexpeced enings nd gowh in expeced enings. While exending he mk-o-mke model o sochsic es is sighfowd, exending he enings-sufficiency model is no. Such n exension mus confon he key issue of how one cpilizes enings. Much of he nlysis heefoe peins o he enings-sufficiency model. The hee min insighs e s follows: Beve (999), p. 37, quesions he ssumpion of consn discoun es in empiicl sudies. Felhm nd Ohlson (999) show how he esidul income vluion cn be exended o sochsic es. They do no, howeve, conside closed-fom vluion funcions, PE muliples, nd suppoing infomion dynmics exmined hee. 3

4 . Vluion funcion (Poposiions nd 2): Using Modiglini-Mille dividend policy ielevncy, we show h if enings x fo he peiod (-, ) suffice fo he cum-dividend vlue P +d, hen P+ d = ( R ) x whee - is he spo inees e de - fo he peiod (-, ). The use of - insed of in he muliplie ess on simple ye poweful insigh: The expeced enings e fo peiod (-,) is -, he e he beginning of he peiod, no, he e he end of he peiod. 2. Residul enings dynmic (Poposiion 3): The pesisence of esidul enings equls, which oscilles ound he hn equl s in Ohlson (995). 3. Enings dynmic (Coolly ): The diionl ndom wlk of enings equies n ddiionl em he pecenge chnge in inees es muliplied by cuen enings. We hen llow fo sochsic inees es in he Ohlson s (995) weighed-vege model wih ohe infomion. Deivion of he implied infomion dynmic is sighfowd (Poposiion 4), bu i bings ou new spec of enings foecsing. Wih sochsic es, he elive imponce of cuen enings vis-à-vis book vlue in foecsing fohcoming enings inceses s he cuen e deceses (Coolly 2). Anlysis of he moe genel model povides wo esble hypoheses in ddiion o he modified ndom wlk of enings menioned bove:. Chnges egession: In euns-on-enings egession, he coefficien of unexpeced enings (ERC) should be lge when lgged inees es e low (nd convesely). 2 2 A compive sic nlysis of Ohlson s (995) model does no ell us whehe he cuen e o he lgged e should ffec ERC. Thee is lso no gunee h compive sics will hold in dynmic seings. Ou dynmic nlysis fomlizes nd shpens he empiicl hypohesis. 4

5 2. Levels egession: In egession of pice on book vlue, cpilized cuen enings, nd cpilized expeced enings, he fis wo coefficiens decese in he cuen e while he hid inceses in he cuen e. Th is, s he cuen e ises, he elive infomion conen of cuen ccouning d declines vis-à-vis expeced enings. (The lgged e hs no impc on he elive infomion conen of he hee vibles.) Afe inoducing noion nd coe ssumpions in secion 2, he ppe nlyzes incesingly complex seings. Secion 3 nlyzes he mk-o-mke model. Secion 4 nlyzes he enings-sufficiency model. Secion 5 nlyzes he weighed vege model wih ohe infomion. Secion 6 summizes nd concludes he ppe. Noion nd Bsic 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 +. x = d = P = b = g = = enings fo he peiod - o, i.e., he peceding peiod dividends, ne of cpil conibuions, de ex-dividend mke pice of equiy, de book vlue, de P - b = goodwill, de isk-fee inees e fo he peiod o +. (A de, is he cuen e nd - is he lgged e.) R = + x = x - - b - = esidul enings fo he peceding peiod 5

6 Two ssumpions will be minined houghou he ppe:. Risk neuliy nd no bige: E ( P + + d + ) P = (RNNA) R Noe h R is obseved only de ; i is ndom fom he pespecive of pio des. 3,4,5 2. Clen suplus elion: b + = b + x + - d + (CSR) The bove ssumpions imply he following goodwill equion h we use in ou nllysis: E ( g ) g + + x + = R (GE) Subsequen secions inoduce ddiionl ssumpions. 2 The Mk-o-Mke Model: The Blnce-Shee Appoch We s wih simple bu impon benchmk -- he mk-o-mke model. In his cse, he blnce shee (i.e., he book vlue, b ) povides sufficien vlue-elevn infomion. Since sochsic inees es pose few poblems in his seing, i povides useful pespecive befoe one consides moe compliced seings. The hee peinen poins e s follows:. Pice depends only on book vlue nd goodwill is zeo. P = b. 3 I is esy o consuc n economy such h in equilibium he expeced eun on n sse is he isk-fee e, nd ye he e chnges sochsiclly ove ime. (The consucion is picully esy if one llows fo unconsined, even-coningen pefeences.) 4 Fo isk vesion, one cn eplce he expecion opeo E by E*, which eflecs isk-djused pobbiliies. [Hung nd Lizenbege (988)]. 5 τ Ohlson (995), nd ohes, ssume consn inees es nd PVED ( P = R Eτ[ d + τ] ). As is well known, PVED is equivlen o RNNA when es e consn; hus RNNA does no devie fom pio esech. 6

7 2. Seing goodwill o zeo in he goodwill equion (GE) yields E [ x + ] = 0 nd E [ x + ] = b = P, so h fohcoming expeced enings depend on he cuen e. 3. Applying mk-o-mke ccouning does no equie one o model he sochsic pocess of inees es becuse pices ledy impound such infomion. Fo exmple, mking n invesmen fund o mke simply equies seing he book vlue of ech secuiy o is mke vlue. I does no equie one o know he sochsic pocess of inees es h he mke is using o pice he secuiies. 3 The Enings-Sufficiency Model: The Income-Semen Appoch We now un o he second benchmk: The income semen (i.e., enings x ) povides sufficien vlue-elevn infomion. 6 As poin of efeence, conside enings sufficiency in consn e seing [Ohlson (995) nd Ryn (988)]: R P = x d whee R/ is he pice-enings muliple. Given RNNA nd consn inees es, i follows h P [ = E x + ]. Though enings cuen o expeced nd cum-dividend pice boh epesen he sme undelying infomion, one hinks of enings s sufficien sisic wihou specificiy s o he ccouning ules. 7 The subsequen secions exend he vluion funcions o llow fo sochsic inees es by specifying how he P-E muliples depend on inees es. 6 Secion 4.2 illuses he inuiion nd pcicl pplicion of he enings-sufficiency ccouning. 7 Ohlson nd Zhng (998) consuc ccouning ules h mp nscions ino enings sufficiency mesuemens. 7

8 3. The Vluion Funcion unde Sochsic Discoun Res One my be emped o exend he consn-e seing o sochsic-e seing by simply eplcing by in he vluion funcion so h P = ( R ) x d. The nlysis below shows, howeve, h unde ny esonble condiions, he muliple mus be bsed on he lgged e so h R P = x d. Befoe deducing his elion, we conside svings ccoun o elucide he inuiion. In svings ccoun, he pice P is he ccoun blnce, d is he wihdwl, enings e he inees ened fo he peiod (-,), i.e., x = P. Now RNNA implies P R P =. d Subsiuing =, we ge = ( ). The enings e fo he peiod (-,) is he P x P R x d e peviling -, no, so he muliple fo enings fo he peiod (-, ) depends on he e -, no. A svings ccoun cn be hough of s specil cse of ceiny. Unceiny equies moe sophisiced nlysis becuse unde unceiny x P nd P d R P +. Theefoe, we cn no longe deive he vluion funcion by simply subsiuing = in RNNA. We show below, howeve, h one cn sill deive = ( ) by using Modiglini-Mille (MM) dividend policy ielevncy pecep. P R x d P x We dp n MM condiion in Yee (200) o llow fo sochsic es. Specificlly, fim s vlue de should be he sme if he fim hd loweed is dividends by $z de -, invesed $z in zeo NPV invesmen such s Tesuy bill, nd ised de dividends by R - z. The inees ened on he invesmen would hve ised enings fo he peiod (-,) by - z. Yee s 8

9 MM condiion heefoe coesponds o equiemen h P(x,d ;,, -, ) = P(x + - z, d +R - z;, -, ) fo ny vlue of z. We nex show h his MM condiion implies h he muliplie equls R - / - s befoe. Poposiion : Assume h: (i) P +d = f(, -, )x whee f(.) depends only on he hisoy of inees es 8, nd (ii) P sisfies Modiglini-Mille condiion s in P(x,d ;, -, )=P(x + - z, d +R - z;, -, ), whee z is ny numbe. Then f(.) = R - / -. Poof: See Appendix I. Thee specs of poposiion e noewohy. Fis, poposiion does no explicily ssume RNNA. The MM condiion, howeve, implies h RNNA holds he mgin fo chnge in dividends. Second, poposiion mkes no ssumpions bou he sochsic pocess of inees es. One cn sill deive he muliple becuse in he MM condiion he enings e fo he peiod (-,) is he -, no. Hd he enings e been, which is no known de -, one would hve hd o specify he sochsic pocess of inees es o pply he MM condiion de -. Thid, he inuiion behind he use of he lgged e fo cpilizion exends o he cpilizion of expeced enings s well. Poposiion nd RNNA imply cpilizion consisency in h expeced enings ( [ x ] ) e cpilized by he cuen e ( ) while E + cuen enings (x ) e cpilized by he lgged e ( - ). RNNA implies P = E[ P + + d + ] R. Subsiuing fom poposiion, we ge P = E[( R ) x + ] R = E [ x + ]. Noe h he ls equliy would no hold if he muliple fo x wee ( R+ + ), since + is no known de. 8 This epesenion llows he muliple f(.) o depend on expeced inees es s long s such expecions e bsed solely on he hisoy of inees es. Allowing he muliple o depend on ccouning o non-ccouning infomion ohe hn inees es is inconsisen wih he noion h enings suffice fo vluion. 9

10 In fc, s he nex poposiion shows, ssuming cpilizion muliple fo expeced enings povides n lenive wy of deiving he muliple fo cuen enings. Poposiion 2: Assume (i) isk neuliy nd no bige (RNNA); (ii) P = E[ x + ] ; (iii) P +d = f(, -, )x whee f(.) depends only on he hisoy of inees es; nd (iv) hee e no picul esicions on he sochsic pocess eled o given -, -2,. Then f(.) = R - / -. Poof: In Appendix I. Poposiions nd 2 highligh wo wys of expessing enings sufficiency. One eles cuen pice o cuen enings P = ( R ) x d, whees he ohe eles cuen pice o expeced enings, P = E[ x + ]. Boh elions use he e he beginning of he especive enings mesuemen peiods s he bsis fo enings muliples. We now descibe pcicl pplicion of enings sufficiency nd show how i eles o he concep of smoohing. 3.2 Undesnding Smoohing --The Coe Concep Undelying Enings- Sufficiency Enings sufficiency fo vluion s defined bove my ppe bsc nd conived, especilly in seings such s n invesmen fund, fo which, fis glnce, he mk-o-mke model ppes o be he only coec model. We now show h he enings-sufficiency model yields vlid nd inuiive ccouning mesuemens even fo n invesmen fund. The implemenion undescoes he coe concep of smoohing h undelies he eningssufficiency model. This concep is lso used in pcice o ccoun fo pension sses pe GAAP. 0

11 Le us compe he ccouning lenives fo simple invesmen fund h holds one she of he sme equiy secuiy ll imes. Wihou loss of geneliy, ssume h dividends eceived by he fund e pssed on o is invesos. Then RNNA implies he following equion: P + d = R P + ε whee E [ ε ]. = 0 Mk-o-mke ccouning: One ses b = P. CSR yields x = P + ε. Enings-sufficiency ccouning: One ses x = P + ( R ) ε. The book vlue is mesued by pplying CSR ecusively wih he iniil condiion of b 0 =0. I is esy o veify h ( / ) P + d = R x The diffeence beween he wo models is siking. In mk-o-mke ccouning, he enie vlue shock ε is booked ino enings immediely. In cons, in enings-sufficiency ccouning, only p of he vlue shockε is booked ino enings in he cuen peiod. GAAP ccouning fo pension sses ppoximes he smoohing inheen in he equion fo mesuing x. Pension sses e, of couse, no mked o mke. Insed, ecognized enings eled o pension sses consis of wo ps (i) expeced enings fom pln sses, P, nd (ii) pil ecogniion of he diffeence beween cul nd expeced eun on pln sses,( / R ) ε The definiion of x unde enings sufficiency highlighs s o how such enings genelly depend on boh cuen nd lgged es. x depends on due o he componen P. Bu x depends on s well s long s ε coeles wih elizions of, which hppens when P

12 depends on. Becuse enings, x, e lwys concepulized s flow vible peining o he peiod (-, ), he ide h x eles o boh of nd hs much ppel. I emins o be seen whehe he bove enings cpilizion muliple cn be incopoed ino full-fledged vluion fmewok in he spii of Ohlson s (995) model. Befoe developing his genel model, we conside he implicions of he enings-sufficiency fmewok fo he susining enings dynmic. 3.3 Residul Enings Dynmic: Inees-e Dependen Pesisence Enings sufficiency unde consn inees es esuls in esidul enings pesisence being equl o one [Ohlson (995)]. Specificlly, esidul enings follow sic ndom wlk: x + = x + ε +, whee E [ ε + ] = 0. We genelize he equion o sochsic inees es by hypohesizing he following infomion dynmic: x + = ω x + ε +, whee ω cn depend only on he hisoy of inees es. I emins o be seen whehe he bove epesenion is vlid when inees es e sochsic nd enings sufficiency s peviously defined pplies. If i is vlid epesenion, hen he wo min quesions e: Does ω depend on he enie hisoy of inees es o is smlle subse sufficien? Does ω oscille ound, is vlue wih consn inees es? 2

13 Poposiion 3: Given isk neuliy nd no bige (RNNA), nd clen suplus (CSR), R P = x d implies ω =. Poof: In Appendix I. Poposiion 3 shows h one cn genelize he Ohlson (995) epesenion o sochsic inees es in psimonious wy. The esidul enings pesisence pmee (ω ) 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 disibuion of inees es sisfies esonble eguliy condiions, hen he medin esidul enings pesisence is, which is is vlue wih consn inees es. The inuiion behind poposiion 3 is s follows. As shown in he ppendix, cpilized esidul enings equl goodwill. x g = P b = x + ( x ) ( ) ( ) E + E g + = E = Noe h he ls equliy would no hold if he muliple wee bsed on he cuen e s he denomino of would be +, which is no known de. The use of he lgged e x + llows us o evlue he expeced goodwill wihou knowing he covince beween expeced esidul enings nd inees es. Poposiion 3, heefoe, does no ely on specific sochsic pocess of inees es. As shown in he ppendix, RNNA implies h goodwill follows ndom wlk. Subsiuing fo goodwill fom he bove equions, one ges: 3

14 x + x = + ε + Renging ems yields, ω = /. In he specil cse of consn inees es, he dynmic equion educes o ndom wlk of esidul enings. I is ppen h he convese of poposiion 3 holds: ω = /, RNNA, nd CSR imply enings sufficiency s defined. This obsevion shows h if one ssumes ω = long wih RNNA nd CSR, hen enings sufficiency cnno hold, i.e., sic ndom wlk (pemnen enings) nd enings sufficiency e muully inconsisen when inees es e sochsic. In fc, s Appendix II noes, ω = implies h P now lso depends on b. 3.4 Enings Dynmic: A Modified Rndom Wlk Reflecing Inees Re Chnges Ohlson (995) implies he following sochsic pocess fo enings (he hn esidul enings): E [ x + ] = x + b The fis em epesens he sndd ndom wlk model of enings, nd i equls expeced enings in he bsence of new ne invesmen ( b ) nd consn inees es. The second em epesens he djusmen o expeced enings due o new invesmen. Becuse expeced enings depend on he cuen e pplied o new ne invesmens, obviously eplces when inees es e sochsic. The following coolly evels h eplcing by is no enough; sochsic inees es inoduce n ddiionl em in he ndom wlk model of expeced enings. Coolly : E x + = x+ b+ % x whee % ( )/ 4

15 Poof: In Appendix I. The hid em, which pio esech hs no ecognized, shows h he pecenge chnge in inees e, no jus he level of inees es, ffecs enings foecss; n up ick in inees es led o highe enings foecss, nd vice ves. 4 A Weighed-Avege of he Two Models wih Ohe infomion The pevious secions exend he mk-o-mke nd enings-sufficiency benchmks o sochsic inees es. Ohlson (995) lso povides moe genel model which is weighed vege of he wo benchmks nd which lso llows ohe vlue elevn infomion besides ccouning d. Thee is n obvious moivion fo exmining weighed-vege model wih ohe infomion. In he el wold, boh blnce shee nd income semens e used fo vluion nd model h incopoes boh by king hei weighed vege is ppeling. Ohe infomion lso hs cucil ole. Wihou ohe infomion, he ccouning d mus pick up ll vlue elevn infomion including expecions of fuue inees es. Fo exmple, suppose he cuen e dops nd sock pice ises becuse he mke expecs incese in fuue sles. If ccouning semens wee o suffice fo vluion, cuen book vlue nd enings mus eflec such highe expecions of fuue sles. This plces high buden on nscions-bsed ccouning sysem. Inoducing ohe infomion llows us o pick up he effec of cuen es on vlue while llowing he ccouning d o be impced gdully. 9 We now exend his genel model o sochsic es nd exc he following implicions: (i) The pice cn now depend on he cuen e even if ccouning d is held fixed ( P 0, x, b, nd d fixed). Pevious secions uled ou his possibiliy by ssuming h 9 Nissim nd Penmn (999) sudy empiiclly how inees es influence subsequen ccouning d. 5

16 ccouning d subsumed ll vlue elevn infomion, including he cuen e. (ii) In ddiion o he specificion fo pice levels, we lso deive one fo euns s funcion of unexpeced enings nd unexpeced ohe infomion. This specificion leds o n empiiclly esble implicion: The coefficien fo unexpeced enings (ERC) inceses in he lgged e. (iii) We expess he vluion funcion s weighed vege of hee benchmk models: book vlue, cpilized enings, nd cpilized expeced enings. On he bsis of his expession we se ye nohe empiicl implicion: An incese in inees es educes he infomion conen of cuen ccouning d elive o h of expeced enings. 4. The Weighed Avege Vluion Funcion wih Ohe Infomion We fis descibe Ohlson s (995) model wih consn es nd hen inoduce sochsic es. Wih fixed es, Ohlson (995) model specifies he vluion funcion s: P = k ( R x d) + ( k ) b+ β υ whee k [0,] nd he suppoing dynmics s x + = ω x + υ + ε, + υ = γ υ + ε + 2, + whee E [ ε j, + τ] = 0 fo ll τ nd j =,2. The vible υ hus epesens ll ohe infomion influencing esidul enings foecss nd, o be consisen, he vluion funcion. kr Given he RNNA condiion, he pmees sisfy ω = k + nd k+ γ = R. β 6

17 Allowing () ω nd γ o depend on inees es while (b) keeping k nd β consn genelizes he model. We heefoe conside he vluion funcion: 0 P R = k ( x d) + ( k ) b+ β υ The genelized model does no equie specificion of he sochsic pocess of inees es. This conclusion follows becuse k nd β e consns, nd he muliple fo cuen enings is R, which ensues h one cn evlue E [ P + d ] (he numeo of RNNA) wihou knowing he disibuion of de -. This model cn be exended wihou specifying he sochsic pocess of inees es s long s he sucue of he vluion funcion de is known de -. Appendix II deils such genelizion in which k nd β depend on he lgged e i.e., k is eplced by k - nd β is eplced by β - in he vluion funcion. The infomion dynmics suppoing he vluion funcion e s follows: x + = ωx + υ+ ε, + υ = γ υ + ε + 2, + whee E [ ε j, + τ] = 0fo ll τ nd j =, 2. I is ssumed h he wo pesisence pmees ω nd γ depends mos on k, β, nd he hisoy of inees es. Poposiion 4: Given RNNA nd CSR, ( R P = k x d) + ( k ) b+ βυ implies ω = + + k k nd + k γ = R. β Poof: In Appendix I. 0 Appendix I povides he moivion fo his funcion using he Modiglini-Mille dividend policy ielevncy condiions. 7

18 The emining subsecions dw ou he implicions of he soluion wih no ddiionl ssumpions o modificions. We lso discuss he ciicl ole of ohe infomion, nd se some empiicl hypoheses h follow fom he model. 4.2 Residul Enings Dynmic: Robusness of Ohlson (995) This secion dws ou he implicions of he funcionl fom of ω nd γ deived in poposiion 4. Like he enings-sufficiency seing, ω depends on he lgged s well s he cuen e. In shp cons, γ depends only on he cuen e becuse ν epesens geneic ohe infomion, nd hee e no esons why γ should include cpilizion componen bsed on he lgged e. On he bsis of he enings-sufficiency seing, one would expec ω o incese in nd decese in -. Diffeeniing ω wih espec o nd - yields he niciped conclusion: Fo k > 0, ω inceses in ( ω > 0 ) nd deceses in - ( ω < 0 ). Alhough he sensiiviy of ω o inees es my be expeced, is funcionl fom is no obvious. The fis em ( / - ) eflecs he coecion due o he chnging inees es while he second em is he expession fo ω when inees es e no sochsic s in Ohlson (995). Diffeeniing ω wih espec o k yields ω k > 0 fo ny, - > 0. Th is, esidul enings e moe pesisen when enings hve moe weigh in he vluion funcion. We elboe on his consisency beween he elive imponce of cuen enings in foecsing nd in vluion in he nex secion. If one ssumes h P = α + α 2φ x+ α 3d+ α 4b+ α 5ν, hen one cn show h he dynmics imply he weighed-vege vluion funcion. 8

19 4.3 Enings Dynmic: Consisency beween Vluion nd Foecsing In Ohlson s (995) model, esidul enings follow n uo-egessive pocess wih pesisence pmee ω. One eses he esidul enings dynmic in ems of expeced fohcoming enings s follows: E [ x + ] = ω( x + b ) + ( ω) b Thee specs of he expession e noewohy. Fis, expeced fohcoming enings e weighed vege of he expeced fohcoming enings unde he wo benchmk models. Second, he esidul enings pesisence pmee (ω) deemines he weigh ssigned o he fis componen. Thid, since ω inceses in k, he weighs used in he enings dynmics nd he vluion funcion e muully consisen. These hee specs genelize o he seing wih sochsic inees es excep fo he qulificion h one cnno simply eplce ω by ω. Coolly 2: E ] ( [ x + = θ x+ b+ % x ) ( ) + + θ b+ υ whee θ = k k +. Poof: In Appendix I. The weigh θ hs sevel ineesing popeies. Fis, θ = ( ) ω, which equls ω when es e consn. Second, θ inceses s deceses; i.e., he enings-sufficiency componen is elively moe impon vis-à-vis mk-o-mke componen when cuen e is low. In cons, lgged es do no ffec θ. Thid, θ inceses in k, i.e., s in Ohlson (995), he weighs used in he enings dynmics nd he vluion funcion e muully consisen. 9

20 4.4 The Role of Ohe Infomion: Exmple of he Influence of Cuen Res on Vlue We now povide n exmple h shows how cuen es cn ffec vlue vi ohe infomion ν. Suppose ε 2, + = q( + E[ + ]) + u + whee E [ u + ] = 0, nd q is some fixed consn, which eflecs he fim s ype of business, ccouning ules, ec. Hence, + E[ + ] defines he unniciped chnge in inees es, E [ ε 2, + ] = 0 s equied, nd unexpeced chnges in inees es coele wih he eo em in he ν -dynmic. Recusive subsiuion hen yields ν s n explici funcion o, -, E [ ], E 2[ ] nd u, u,. This exmple undescoes wo impon poins. Fis, he vible ν cn depend explicily on he cuen e, nd, second, even hough ν depends on, he evluion of E [ ν + ] is independen of he sochsic pocess h deemines he disibuion of given ny de infomion. + A seing whee ν depends on mens h he ccouning ules h poduce x nd b do no fully eflec he vlue implicions of chnge in inees es fo he peiod (-,). Only wih he pssge of ime will he ccouning d eflec he pio hisoy of inees es. Bu his scenio lso equies h he enings foecss depend on he cuen (nd ps) es vi ν. In nlyicl ems, x ] depends on ν, which in un depends on he hisoy of inees es. E [ + Inoducing ν heeby llows consideble geneliy s o how inees es cn influence vlue nd enings foecss. 20

21 4.5 Two ddiionl implicions: A euns model nd n expeced enings model Two ddiionl implicions of he bove model genelize Ohlson (995). Fis, one cn explin he unexpeced euns in ems of he supise in esidul enings ε, nd he supise in ohe infomion ε 2,. Specificlly, s shown in Appendix I, P + d ε ε = ( + ) + P P P, 2, R α, α 2, whee 2 α α = k, 2, = β. The pice-nomlized esponse coefficien, α,- is known he beginning of he eun inevl (-,). One inepes +α,- s he esponse coefficien ssocied wih unexpeced enings. An ineesing empiicl hypohesis follows: The esponse coefficien should be lge when inees es e lowe (fo fixed k, α,- inceses s - deceses). Second, following Ohlson (200) 3, one cn subsiue expeced nex-peiod enings in lieu of υ. As shown in he Appendix I, R P = + + E [ x + w, b w2, x d w3, ] whee w j, =. j 2 Becuse of RNNA, he expeced vlue of he lef-hnd side equls zeo. 3 Fo discussion of he model see Hnd (200). 2

22 w, ω w2, ω w 3, = ( k β + β ) = ( k) β ( ) + = k β = k β k + = β + k Expessing vlue s funcion of b, x, nd [ x ] hs n inuiive inepeion. Vlue E + deives fom weighed vege of hee benchmk models: (i) mk-o-mke [b ], (ii) enings-sufficiency [ ( R ) x d ], nd (iii) cpilized expeced enings [ E [ ] x + ]. Fuhe noe h componens (ii) nd (iii) einfoce he key ide of cpilizion consisency : The muliple fo cuen enings is bsed on he lgged e, while he muliple fo expeced enings depends on he cuen e. I is esy o veify h he weighs ssocied wih book vlue nd cpilized enings decese s he inees e ises while he weigh ssocied wih cpilized expeced fohcoming enings inceses (wih k > 0, w, < 0 nd w2, < 0 whees w3, > 0 ). A shp empiicl poposiion hus follows: As inees es ise, cuen ccouning d hs less infomion conen s comped o expeced enings. 5 Summy nd Implicions The ppe povides coheen fmewok fo eling vlue o ccouning d nd ohe infomion when inees es e sochsic. The ppe ss wih n exminion of wo benchmk models -- mk-o-mke ccouning whee he blnce shee suffices fo vluion 22

23 nd enings-sufficiency ccouning whee he income semen suffices fo vluion -- nd hen exmines weighed vege of he wo models wih ohe infomion. 4 Mk-o-mke ccouning poses few poblems s he book vlue is se equl o he pice, which impounds expecion of fuue inees es. Theefoe, much of he nlysis evolves ound he exc ole of he cuen e (he e peviling he end of he enings mesuemen peiod) vesus he lgged e - (he e peviling he beginning of he enings mesuemen peiod) in n enings-sufficiency seing. Ou dynmic nlysis clifies he ole of hese es in ccouning-bsed vluion nd foecsing, nd povides insighs h cnno be obined fom compive sic nlysis of Ohlson (995). The sucue of he vluion funcion in enings-sufficiency ccouning hinges on he following cucil inuiion: The enings e fo given peiod is he e peviling he beginning of he enings mesuemen peiod he hn he e peviling he end of he enings mesuemen peiod. This inuiion nd Modiglini-Mille dividend policy ielevncy condiions imply he following cpilizion consisency elion (poposiions nd 2): he cpilizion muliple fo cuen enings is bsed on he lgged e (+/ - ) while he cpilizion muliple fo expeced fohcoming enings is bsed on he cuen e (/ ). Deivion of he esidul enings dynmic suppoing enings-sufficiency ccouning shows h he esidul enings no longe follow ndom wlk. Thei pesisence is no longe ; insed i now equls / - (poposiion 3). The inuiion is s follows: he sndd iskneuliy-no-bige condiion nd he clen suplus elion imply h he goodwill ( g = P ) ( b = x = E x ) mus follow ndom wlk. + 4 I is uncle how sochsic inees es will ffec vluion unde consevive ccouning s exmined in Felhm nd Ohlson (995) nd Zhng (2000). No is i cle wh he implicions would be in cse of non- 23

24 Deivion of he eled enings dynmic shows h he diionl ndom wlk of expeced enings now equies n ddiionl em eflecing chnges in inees es. This em hs no been ecognized peviously; i ises becuse he lgged e is needed o scle cuen enings while he cuen e is needed o foecs fohcoming enings. Thus, he pecenge chnge in inees e, no jus he level of inees es, ffecs enings foecss; n up ick in inees es led o highe enings foecss, nd vice ves. Inoducing ohe infomion poses few poblems (poposiion 4 nd coolly 2) nd mkes he model moe flexible s cuen ccouning d is no longe equied o fully eflec ll vlue elevn infomion bou inees es. Insed, ohe infomion picks up vlue elevn infomion bou expeced fuue evens h is no cpued by he cul-nscionsbsed ccouning sysem. Allowing expeced enings o poxy fo ohe infomion yields nohe impon empiicl pedicion: n incese in inees es educes he infomion conen of cuen ccouning d elive o h of expeced enings. A euns specificion shows h unexpeced eun coeles wih unexpeced enings nd evisions in enings expecions, consisen wih diionl empiicl specificions. Howeve, new empiicl implicion of he model is h he enings-esponse coefficien depends on he lgged e, no he cuen e. Imponly, unexpeced chnges in inees es will genelly influence he euns specificion becuse he model llows fo ohe infomion o be coeled wih cuen es. Ovell, ou nlysis genelizes he Ohlson (995) model o sochsic inees es. Allowing he inees es o be sochsic povides bee undesnding of he key dynmic specs ccouning-bsed vluion fmewok h wee obscued by he ssumpion of consn lineiies due o poenil bnkupcies [Bh, Beve, nd Lndsmn (998)]. 24

25 inees es in Ohlson (995). Specificlly, he lgged e, no he cuen e, seems o ply he key ole in eling pices o conemponeous enings, while he cuen e plys key ole in enings foecsing s well s in eling pices o expeced enings. Ou nlysis fomlizes nd shpens elie empiicl hypoheses nd lso povides new empiicl hypoheses egding he elionship beween pices nd ccouning vibles. 25

26 Appendix I: Poofs Poof of Poposiion Suppose h he fim wihholds $z in dividends de -. Since he one-peiod inees e ime - is -, wihholding $z de - will incese enings fo he peiod (-, ) by - z nd incese dividends de by (+ - )z. Such wihholding of dividends should hve no impc on de vlue if dividend policy ielevncy pplies. Th is, P = f(.)x d = f(.)[x + - z] [d + (+ - )z]. This implies f(.) - z - (+ - )z = 0 fo ll z. Thus, f(.) = (+ - )/ -. Remk: The poof woks only becuse f(.) is ssumed no o depend on enings o dividends. Howeve, if f(.) wee dependen on enings, hen P +d would be non-line in x. Poof of Poposiion 2 Define f(.)= f(, -, ); RNNA combined wih ssumpion (ii) implies: R - P - = E - [P +d ] = E - [f(.)x ] Due o ssumpion (i), R - P - = R - E - [x ]/ - so h (R - / - )E - [x ] = E - [f(.)x ] This equion mus be sisfied fo ll fesible sochsic pocesses of inees es. In picul, conside he cse when hppens o be known de - fo some hisoy of inees es. I hen follows h E - [f(.)x [ = f(.)e - [x ], so h (R - / - )E - [x ] = f(.)e - [x ]. Hence, f(.)= R - / - s sseed. Poof of Poposiion 3 We cn ese he expession fo P s: 26

27 x P = b + Th is: g = x Fom he goodwill equion (GE) we ge, + ( x+ ) R x x = E +. Since is known de, he bove equion simplifies o E =. Thus, ω = QED. x+ x Poof of Coolly Fom Poposiion 3 we ge, E x+ = x. Subsiuing he expession fo esidul enings, we ge Ex+ b = ( x b ), which simplifies o x E x+ = x+ ( b b ) + ( ),o E x + = x + b + x % Modiglini-Mille esicions on he weighed vege vluion funcion Le P = α x + α2 d + ( k) b + βυ Dividend policy ielevncy implies he following condiions: 27

28 . P d = nd whee x d = 0, b d =, nd υ d = 0 [fom Ohlson (995)] 2. P d = ( + ) nd whee x d =, b d = ( + ) [fom Yee (200)], nd υ d = 0 [fom Ohlson (995)] These wo condiions yield he following: α 2 ( k) = α ( k)( + ) = ( + ) Solving he bove equion, we ge he following: α 2 = k k( + ) α = Poof of Poposiion 4 R g = P b = k x d b + ( ) βυ Subsiuing fo b fom he clen suplus elion, b + d = x + b -, nd using he definiion of esidul enings we ge: x g = k + βυ Using he goodwill equion (GE) we ge, x x+ k + βυ = ( k + βυ+ + x+ R R E ) Since k nd β e fixed nd υ+ = υ+ ε 2, +, we ge γ E + x x+ = k + β ( R γ ) υ + + k k 28

29 This implies, ω = + + k k nd + k β ( R γ ) = Thus, γ + k = R QED β Poof of Coolly 2 + k Fom Poposiion 4 we ge, Ex+ = x + υ. Subsiuing fo esidul enings we + k ge, + ( x b ) E x+ = k + b + + k υ Define θ = + k k + E x+ = θ x b + b + θ b+ υ, which cn be esed s follows: Thus, ( ) ( ) E x = θ ( x + b + + % x) + ( θ ) b + υqed. 29

30 Deivion of he Coefficiens in he Reuns Specificion k P = b + x + βν 2 k P = b+ x + βν k P + d = b+ d+ x + βν Fom CSR b + d = b + x k P + d = b + x+ x + βν k = b + b + x b + x + βν k + = R b + x + βν k + = R b + ( ω x, ) ( ν ε β γ ν , ) ε Subsiuing fo ω - nd γ -, we ge, R k k+ [ k+ ( k+ = )] ( R β + ) + β R k k+ = R b + x + βr ν + ε, + βε 2, 2 k + = R P + ε, + βε 2, b x R ν ε, ε 2, 2 β P + d R P = ( + α ) ε / P + α / P k α, = α 2, = β,, 2, 30

31 Deivion of he Tiple Weighed Avege ( ) R P = k b+ k x d + βν We cn expess ν in ems of expeced enings s follows: + E x = ω x + ν ν = E x+ ω x = Ex+ b ω ( x b ) = Ex+ b ω ( x ( b x+ d)) = Ex+ b ω ( R x ( b+ d)) = Ex+ b ω ( R x ( b+ d)) Subsiuing fo ν in he vluion funcion we ge he following: k k R P E R ( + ω ) = ( ) b+ x d + β x b [ x ( b+ d)] R = ( k β + β ) + ( k β ) + β E x + ω b ω x d Fom Poposiion 4, we know + ω = + k k. Thus, + ω = + k k, which depends only on. Thus, we cn expess pice s follows: P = + R + w j, = j + w, b w2, x d w3, w, w w 2, 3, = ( k β + βω ) = ( k βω ) = β E x 3

32 Appendix II: The Weighed Avege Model wih Vible bu Known Weighs We now exmine seing whee he weighs cn vy ove ime, bu e known he beginning of peiod. Fo beviy, we conside he seing wihou ohe infomion. The conceps illused in his ppendix will emin unchnged if we inoduce ohe infomion. Thus, pice is expessed s follows: ( R P = ) + ( ) k x d k b Renging he ems in he bove equion nd pplying CSR, i follows h: x x+ = E + + Rk ( k x ) Inseing he ls equion ino he goodwill equion (GE) yields, x x+ = E + + Rk ( k x ) k Since k nd e known ime, he RHS equls Ex+ + Ex+ nd one obins he following: 5 Ex + + k = + k x The esidul enings pesisence pmee heefoe is epesened by 5 If k, insed of k -, is he weigh in he picing equion, i.e., he weigh is deemined he end of he peiod he hn he beginning, hen we would need o know he covince of k + nd +. x 32

33 + k ω = + k In his expession, one cn hink of ω s being he endogenous esul of elizion of inees es nd k s whee he k s follow some exogenous sochsic pocess (hough, s noed, he weighs e deemined he beginning of peiod). Moe impon, s shown below, we cn enge he ems o se k in ems ω, k -,, nd -, i.e., we cn hink of ω s being exogenous. k k = R ω If one subsiues ecusively, i follows h k is some funcion of he hisoy of inees es nd he hisoy of ω ; i.e., one cn wie k + = f(, -,, ω, ω -, ) whee he ω e deemined by some exogenous pocess. One cn now sk he following quesion: Wh hppens if ω is simply consn, such s ω =? Th is, wh hppens if esidul enings follow ndom wlk nd inees es cn chnge ove ime? The nswe is cle: P will genelly depend on book vlue s well s cpilized enings (djused fo dividends). This is becuse, in cons o he seing in which is consn, k need no be when ω =, i.e., he weigh on book vlue (-k ) cn be non-zeo even when ω =. One cnno, heefoe, view enings s sufficien sisic when esidul enings follow ndom wlk s book vlue sill genelly enes he vluion funcion. 33

34 Bibliogphy Bh, My E., Willim H. Beve, nd Willim R. Lndsmn, (998) Relive Vluion Roles of Equiy Book Vlue nd Ne Income s Funcion of Finncil Helh, Jounl of Accouning nd Economics 25 (): -34. Beve, Willim H., 999, Commens on An Empiicl Assessmen of he Residul Income Vluion, Jounl of Accouning nd Economics, 26 (999), pp Felhm, Geld A., nd Jmes A. Ohlson, 995, Vluion nd Clen Suplus Accouning fo Opeing nd Finncil Aciviies. Conempoy Accouning Resech (Sping): Felhm, Geld A., nd Jmes A. Ohlson, 999, Residul Enings Vluion Wih Risk nd Sochsic Inees Res, Volume 74, No. 2 - Apil 999. Hnd, John, 200, Discussion of Enings, Book Vlues, nd Dividends in Equiy Vluion: An Empiicl Pespecive., Conempoy Accouning Resech Vol. 8 No., Sping 200. Hung, C. nd R. H. Lizenbege. Foundions of Finncil Economics, Noh Hollnd, 988.Ohlson, Jmes A., 995, Enings, Book Vlues, nd Dividends in Equiy Vluion, Conempoy Accouning Resech Vol. No. 2 (Sping 995) pp Nissim, Doon, nd Sephen Penmn, 2000, The Empiicl Relionship Beween Inees Res nd Accouning Res of Reun, Columbi Univesiy Woking Ppe. Ohlson, Jmes A., 995, Enings, Book Vlues, nd Dividends in Equiy Vluion, Conempoy Accouning Resech Vol. No. 2 (Sping 995) pp Ohlson, Jmes A. nd Xio-Jun Zhng, 998, Accul Accouning nd Equiy Vluion Jounlof-Accouning-Resech;36(0), Supplemen 998, pges 85-. Ohlson, Jmes A., 200, Enings, Book Vlues, nd Dividends in Equiy Vluion: An Empiicl Pespecive., Conempoy Accouning Resech Vol. 8 No., Sping 200. Ryn, S., 988, Sucul Models of he Accouning Pocess nd Enings, Ph.D. Disseion, Snfod Univesiy. 34

35 Yee, Kenon (200), Ph.D. Thesis, Gdue School of Business, Snfod Univesiy. Zhng, Xio-Jun, 2000, Consevive Accouning nd Equiy Vluion, Jounl of Accouning nd Economics (Febuy 2000):