Disclosure Quality, Diversification and the Cost of Capital. Greg Clinch University of Melbourne June 2013

Transcription

1 Dsclosure Qualy, Dversfcaon and he Cos of Capal Greg Clnch Unversy of Melbourne June 03 I hank Cynha Ca, Kevn L, and Sorabh Tomar for helpful commens and suggesons on an earler (ncomplee) draf of hs paper.

2 . Inroducon In hs paper I dscuss several aspecs of he lnk beween dsclosure qualy and cos of capal, wh a parcular focus on how dversfcaon nfluences hs lnk. The exen o whch uncerany nheren n dsclosure abou a frm (somemes descrbed as nformaon rsk ) s refleced n a frm s cos of capal and wheher mgh be dversfable s, undersandably, a queson of neres o accounng researchers. I s also a queson I do no am o provde a defnve answer o. Raher, my approach s o employ a smple and hghly sylzed model o llusrae several relaed resuls from some relavely recen analycal research regardng dsclosure, cos of capal, and dversfcaon. My objecve s no o provde new resuls hough he model does, I hnk, yeld one or wo mnor new observaons bu raher o clarfy he role of dsclosure qualy for a frm s cos of capal n a large economy. Alhough my prmary focus s on he role of dversfcaon for dsclosure qualy and he cos of capal, I frs consder (n secon ) a smple sngle frm seng. My purposes here are wofold. Frs, he sngle frm seng provdes a poenally more accessble enrée no he deals of he model ha can be obscured n a more complex mul-frm seng. Second, I use he sngle frm seng o explore he mpac of dsclosure qualy on cos of capal n a seng where boh fundamenal rsk and prce rsk play mporan roles. Shor, fne-perod models generally employed n he relaed leraure are mperfec vehcles whn whch o capure prce rsk snce, by consrucon, he fnal payoff nvolves no prce rsk. Ths means ha an nvesor who holds shares for he enre perod spanned by he model wll face no prce rsk, poenally nfluencng model conclusons regardng dsclosure qualy and cos of capal. For example, Chrsensen, De la Rosa and Felham (00) employ such a fne perod model and show ha n her model ex-ane cos of capal ha s, cos of capal relang o he enre fne model perod bu assessed pror o ha perod s unaffeced by he qualy of fuure dsclosure qualy, and only reflecs underlyng cash flow (fundamenal) My paper and he paper by Terry Shevln (Shevln (03)) share a smlar opcal neres, hough hs paper focuses largely on emprcal research and a broader range of ssues, whle I concenrae on narrower and analycal-relaed quesons. Hs secon 5 provdes a good overvew and dscusson of he analycal leraure relaed o my paper (and beyond), so I do no replcae hs here. Also, I resrc my aenon o a seng where here s no nformaon asymmery among nvesors (n conras o, for example, Easley and O Hara (004) and Lamber, Leuz and Verreccha (0)) and no real effecs of dsclosure,.e., a pure exchange economy (n conras o, for example, he las secon n Lamber, Leuz, and Verreccha (007), Gao (00), and Ca (03)). For recen comprehensve surveys of analycal research regardng a broad range of dsclosure-relaed research ssues see Beyer, Cohen, Lys and Walher (00) and Berger (0), as well as Verreccha (00) regardng slghly earler research.

3 rsk. In conras, he sngle frm seng I employ here nvolves an nfnely-lved frm bu wh fnely-lved nvesors. Thus nvesors are unable o avod prce rsk. In hs seng, I show ha he resul of Chrsensen e al (00) does no hold, and ha (ex ane) cos of capal s decreasng n he qualy of dsclosure. In he man par of he paper (secon 3) I exend he nfne perod model o nclude mulple frms. Thus he mul frm seng slghly exends he exsng relaed research (e.g., Easley and O Hara (004), Lamber, Leuz and Verreccha (007), and Hughes, Lu and Lu (007), among ohers) o ncorporae boh fundamenal and prce rsk. I employ hs seng as he bass for dscussng and clarfyng several ssues rased n he pror research regardng dversfcaon and he lnk beween dsclosure and cos of capal. Specfcally, I focus on hree aspecs of a large economy ha nfluence how dsclosure qualy affecs cos of capal: () he number of frms across whch rsk s dsrbued; () he number of nvesors among whom hs rsk s shared; and (3) he number of nformaon sgnals (dsclosures) avalable o nvesors from whch o exrac nformaon. Conssen wh he famlar CAPM, he model llusraes ha n a large economy nvesors prce only sysemac (or covarance) rsk relang o her end of perod payoffs. Neverheless, as poned ou by Lamber, Leuz and Verreccha (007), dsclosure can affec nvesors assessmen of hs rsk and, as a resul, he qualy of dsclosure can maer for cos of capal. Ths occurs even hough he qualy of dsclosure s due only o dosyncrac, frm-specfc, nose n he dsclosure he dosyncrac nformaon rsk does no dversfy away. However, he model also ndcaes ha f here are many dsclosures n a large economy (for example, here s a dsclosure for each frm) hen he qualy of hose dsclosures no longer affecs cos of capal. Because here are many dsclosures avalable o nvesors ha asss n assessng a frm s end of perod payoff, he qualy of each ndvdual sgnal becomes mmaeral o rsk assessmen - he large number of avalable sgnals swamps he nfluence of dsclosure qualy. Fnally (n secon 4) I exend he model n a more speculave drecon o ncorporae some nvesors who adop non-raonal radng rules. Specfcally, hese nvesors follow a smple heursc (or rule of humb) radng rule ed o each frm s dsclosure whou reference o curren prce f he dsclosure has ncreased (decreased) compared wh he pror perod hey buy (sell) shares. I show n hs seng ha he exsence of such heursc raders resuls n an addonal dsclosure-conngen facor n equlbrum prce ha causes prce o devae from raonal nvesors expeced payoffs. Tha s, cos of capal n hs seng comprses wo facors: one relaed o covarance or sysemac rsk, and he oher

4 relaed o a heursc rader effec. Moreover, he heursc rader effec does no dsappear n a large economy (where he number of frms and raders - boh raonal and heursc - are large), ha s, does no dversfy away. Ths s despe he heursc rades beng ed o frm-specfc dosyncrac dsclosures. In summary, he model I employ, hough hghly sylzed and smplfed, does hghlgh some useful nsghs no he lnks beween dsclosure qualy, dversfcaon, and he cos of capal.. The sngle frm seng. The model The model I employ n hs secon s based on a very smple nfne perod, sngle frm seng. The frm generaes a payoff, x, whch follows a random walk, ha s: x = x +, where s normally dsrbued wh mean zero, varance σ, and ndependen over me. There s also a rsk-free secury whch pays + r each perod. Thus r represens he rskfree rae of reurn. In each perod here are n dencal and prce akng nvesors each wh negave exponenal uly wh rsk averson parameer ρ. Invesors are assumed o exs for only a sngle perod, a he end of whch hey lqudae her posons and consume he proceeds. Ther uly funcons are defned over hese proceeds whch comprse he frm s payoff x plus he prce, P, a whch hey sell her shares. Fnally, supply of he frm s secury s normalzed o one (and consan over me). Gven hs srucure, s sraghforward o derve he followng expresson for prce: ρ P = E ( P+ + x+ ) Var( P+ + x+ ) + r n, () where E( P+ + x+ ) and Var ( P+ + x+ ) denoe nvesors expeced proceeds and varance of proceeds condonal on whaever nformaon s avalable o hem a me. Equaon () All prmary dervaons relang o he paper are provded n he appendx. Oher deals are avalable on reques from he auhor. 3

5 ndcaes ha prce s equal o he presen value of nvesors end of perod expeced proceeds less a dscoun for he assessed rsk of, or uncerany aached o, he proceeds. I s naural o nerpre he dscoun as a cos of capal merc, and I use as such hroughou he paper. Equaon () llusraes ha he dscoun depends on nvesors rsk averson ( ρ ), he number of nvesors amongs whom rsk s shared ( n ), and nvesors assessed rsk ( Var ( P+ + x+ ) ). The frs wo facors ogeher reflec he marke s appee for rsk, whle he varance erm reflecs assessed rsk. I s only hs hrd facor ha mgh be affeced by dsclosure.. Dsclosure qualy and cos of capal To nvesgae he mpac of dsclosure on equaon () I assume ha nvesors observe a publc repor each perod: y = x+ + e, where e s normally dsrbued wh mean zero, varance σ, and ndependen boh over me and of. Gven hs nformaon srucure, and assumng lnear and sable prces over me, s possble o derve he followng equlbrum prce expresson: ρ P = x by ( x) s r + n () where σ b = σ + σe s P x r r b r, and = Var ( + ) = ( + ) (( + ) ) + + σ. Thus prce n perod s equal o a facor, /r, mulpled by wo erms. The frs erm, x+ by x), s 4 ( he expeced value of x + gven he sgnal y. The mulple mes hs expecaon represens he expeced presen value of he perpeuy of fuure paymens by he frm and s he prce ρ ha would preval n a rsk neural seng. The second erm, s, represens a combnaon n ρ of he economy s appee for rsk ( ) and nvesors assessed uncerany/rsk ( s ). As n noed prevously, he dscoun n prce capured by hs second erm represens he frm s cos of capal due o rsk/uncerany. Based on () s sraghforward o assess he mpac of dsclosure qualy on he frm s cos of capal n hs smple sngle frm economy. Dsclosure qualy n he model s represened by σ e, he varance of he error erm n he dsclosure y. Is mpac on he

6 dscoun n prce s va he b coeffcen, whch s decreasng n σ e. Snce s s also decreasng n b, he frm s cos of capal s unambguously decreasng n dsclosure qualy (.e., ncreasng n σ e ). As noed n he nroducon, hs conrass wh he resul n Chrsensen, De la Rosa and Felham (CDF) (00) where ex ane cos of capal s unaffeced by a change n dsclosure qualy. In CDF, a change n dsclosure qualy has wo effecs: () decreases nvesors assessed uncerany/rsk regardng he ermnal paymen by he frm, and () ncreases he volaly of prce a he forhcomng release of he dsclosure. Ths second effec arses because hgher qualy dsclosure causes nvesors o change her expecaons o a greaer exen whch flows hrough o greaer magnude prce movemens. In CDF hese sources of (ex ane) uncerany exacly offse, resulng n no change n he ex ane cos of capal. In conras, n he nfne perod seng employed here alhough he frs of hese effecs s presen he second s no. Specfcally, an mprovemen n dsclosure qualy decreases he nvesors assessed varance of he frm payoff a he end of he perod, x +, whch acs o decrease he frm s cos of capal. However, he assessed varance of end of perod prce s no affeced by dsclosure qualy: Var ( P+ ) = Var [ x+ + by ( + x+ ) ] = σ. The r r reason why s because prces are affeced n wo offseng ways by dsclosure qualy mproved dsclosure qualy causes end of perod prces o be more volale as nvesors reac o beer nformaon, bu also mproves nvesors curren nformaon abou hose lkely fuure prce movemens whch reduces nvesors curren assessmen of fuure prce volaly. The wo effecs exacly offse causng prce volaly o be unaffeced by dsclosure qualy. 3 As a resul, he ne effec of dsclosure qualy s drven solely by s effec on nvesors assessmen of he uncerany regardng x +, he end of perod cash paymen by he frm. To summarze, alhough he fne perod model employed n CDF suggess no role for dsclosure qualy for ex ane cos of capal, he nfne perod model I employ here does adm such a role, wh hgher dsclosure qualy leadng o lower cos of capal. 3 Noe ha hs also mples ha he uncondonal varance of he change n prce also s unaffeced by dsclosure qualy. Ths can easly be verfed usng equaon (). 5

7 3. The mul frm seng 4 In hs secon I exend he sngle frm, nfne perod model o ncorporae mulple frms. Specfcally here are m frms who each generae a perodc cash flow o nvesors of x = x +, where each s mean zero normally dsrbued wh varance σ, and s ndependen over me bu may be correlaed across frms wh covarance σ j. Wh hese assumpons ogeher wh he same assumpons for nvesors as n secon, s sraghforward o show ha he expresson for equlbrum prce n he mul-frm seng ha corresponds o equaon () (n he sngle frm seng) s: ρ P = E ( P + + x + ) Cov ( P + + x +, PM + + xm+ ) + r n, (3) m where P = M+ P j + and x M + = x j + represen he aggregae prce and cash paymen j= j= m from he marke porfolo of all frms n he economy. Agan, s naural o nerpre he dscoun erm n equaon (3) as represenng cos of capal. In hs case, and conssen wh he famlar CAPM, he relevan rsk measure s he covarance erm beween he proceeds from holdng frm and from holdng he marke porfolo, M : Cov ( P + x, P + x ), gven all avalable nformaon o nvesors a me. + + M+ M+ Equaon (3) s que general, allowng each frm s perodc cash paymens o have dfferen varances, and for covarances across frms o also dffer. I smplfy hs n order o focus on ssues relang o dversfcaon by assumng ha varances and covarances are he same across all frms, ha s σ = σ and σ j = τσ for all frms, where τ represens he correlaon beween he cash paymens of each par of frms n he economy. Fnally, smlar o he sngle frm seng, he lnk beween dsclosure and cos of capal occurs va s effec on nvesors assessed covarance beween he proceeds from holdng a frm s shares and from holdng he marke porfolo of all frms, Cov ( P + + x +, PM+ + xm+ ). I consder wo cases n he followng subsecons: () frs where here s only a sngle dsclosure regardng one frm s lkely fuure cash paymen, and () where here s a dsclosure regardng each frms fuure cash paymens. Ths allows a consderaon of how he lnk beween dsclosure qualy and cos of capal s nfluenced by hree facors: () he number of nvesors among 4 Many of he resuls dscussed n hs secon mrror smlar resuls n Lamber, Leuz and Verreccha (007). 6

8 whom rsk s shared, () he number of frms across whch he economy s rsk s spread, and (3) he number of dsclosures avalable o nvesors upon whch hey base her rsk assessmens. 3. Dsclosure by a sngle frm In hs subsecon I assume only a sngle frm, frm, provdes a perodc dsclosure, y = x + + e, regardng s end of perod cash paymen. e s normally dsrbued wh mean zero and varance σ e, and ndependen over me and of for all and. Wh hs sgnal avalable o all nvesors he equlbrum prce expresson n (3) becomes for he dsclosng frm (frm = ): ρ P = x by ( x ) s r + n (4) σ where b = σ + σ e and s = Cov ( P + + x +, PM+ + xm+ ) = ( + r) (( + r) ) b σ ( + ( m ) τ) r. Equaon (4) has he same form as n he sngle frm seng (equaon ()) bu he relevan rsk measure, s, represens he frm s assessed covarance wh proceeds from holdng he marke porfolo (sysemac rsk), raher han he frm s varance. Moreover, hs assessed covarance erm conans wo prmary componens. One componen, σ ( ( m ) τ) 7 +, s equal o Cov ( x +, xm+ ) n he absence of any dsclosure and so represens he covarance or sysemac rsk assocaed wh he frm s underlyng cash paymens. The oher componen, ( ) ( + r) ( + r) b, reflecs he mpac of he dsclosure on nvesors assessmen of rsk. As n he sngle frm seng hs componen s decreasng n b, whch self s ncreasng n dsclosure qualy (.e., decreasng n σ e), and so he dscoun, or cos of capal, s decreasng n dsclosure qualy. Tha s, an mprovemen n dsclosure qualy causes nvesors o assess he frm s sysemac rsk o be lower and resuls n a lower cos of capal. An mporan queson s o wha exen hs effec would reman n a large economy. In a large economy n, he number of nvesors, and/or m, he number of frms, are

9 lkely o also be large. A larger n allows for more rsk sharng among nvesors, whle a larger m allows for greaer dversfcaon opporunes o nvesors bu also greaer aggregae economy-wde rsk o be shared among nvesors. 5 However, hese effecs have no mpac on he componen of rsk ha reflecs he qualy of dsclosure by frm, ( ) r r b. Thus he exen o whch dsclosure qualy effecs (or nformaon ( + ) ( + ) rsk) wll be dversfed away n a large economy wll smply mrror he exen o whch rsk n general s dversfed away. If rsk, n general, s deecable n prces va a non-zero dscoun, hen dsclosure qualy wll also be deecable o he exen here s suffcen varaon n dsclosure qualy. Moreover, as equaon (4) makes clear, s he effec of an ncreased n (number of nvesors) ha causes he dscoun o decrease n a large economy, no he ncreased number of frms, m. Ths can mos easly be seen by consderng an economy where all frms are ndependen,.e., τ = 0. In hs case, s, he assessed covarance rsk ha s prced for frm =, collapses o smply ( + r) ( ( + r) ) b σ r, rrespecve of he number of frms n he economy. Thus despe a large number of ndependen frms across whch nvesors are able o (and n equlbrum do) dversfy, he assessed rsk componen ha s prced does no change. Ths s because alhough he greaer number of frms offer ncreased dversfcaon opporunes, here s also greaer economy wde rsk (due o more frms) ha mus be absorbed by nvesors. These wo effecs offse each oher n equlbrum. I s only f here s also a greaer number of nvesors amongs whom o share hs rsk wll he dscoun n prce decrease. Thus, n effec, he dversfcaon effec n a large economy mgh more accuraely be descrbed as a rsk-sharng effec. ρ nr A furher mplcaon of he defnon of he prce dscoun n equaon (4), ( + r) ( + r) b + ( m ) ( ) ( ) σ τ, s ha n a large economy where boh n and ( ) σ + ( m ) τ m are large, he underlyng sysemac rsk per nvesor, n, approaches m τσ. Thus, as observed n Lamber, Leuz and Verreccha (007), as long as τ s posve n 5 Ths (reasonably) assumes ha he average covarance beween frms cash paymens, represened by τ, s non-negave. 8

10 (.e., on average here s posve correlaon among frms cash paymens), and m n approaches a fne consan, he dscoun wll be posve and he mpac of frm s dsclosure qualy on cos of capal wll no dversfy away. Ths s despe dsclosure qualy beng solely relaed o dsclosure nose, σ e, ha s dosyncrac o frm =. Tha s, he dsclosure nose self has no sysemac componen and ye he degree of hs nose (.e., dsclosure qualy) mpacs he dscoun n prce n a large economy even when any unsysemac componen of a frm s underlyng cash paymens does no. 6 The reason hs occurs s ha he dsclosure s beng used by nvesors o assess underlyng sysemac rsk ha s mporan o prcng. Wheher nose n a frm s dsclosure s dosyncrac or no does no affec hs. Fnally, equaon (4) descrbes equlbrum prce for he dsclosng frm. I s also possble o show ha equlbrum prce for non-dsclosng frms n he economy s: ρ Pj = xj τby ( x ) sj r + n (5) where j, b s he same as n equaon (4), and sj = ( + r ) ( + r ) b + ( m ) r ( ) τ σ ( τ). Thus frm s dsclosure s used by nvesors n prcng oher frms as long as he frms cash paymens are correlaed,.e., τ 0. In parcular, dsclosure qualy for he dsclosng frm s negavely assocaed wh nondsclosng frms cos of capal. 7 Moreover, all he mplcaons dscussed above regardng dsclosure qualy and cos of capal connue o hold for he non-dsclosng frms. To summarze he resuls of hs subsecon, n a large economy where here s a sngle dsclosure regardng one frm he qualy of ha dsclosure wll nfluence cos of 9 ( + ( m ) τ) σ 6 The oal underlyng per-nvesor sysemac rsk erm, represens an dosyncrac rsk n σ σ ( m ) τ componen,, and a componen ha reflecs correlaon wh oher frms n he economy,. In a n n large economy where boh n and m are large, he frs of hese wll be mmaeral,.e., dversfed away, whle he second wll no. 7 Noe ha he defnon of s ndcaes ha cos of capal for non-dsclosng frms wll be hgher han for he j dsclosng frm, as mgh be expeced gven ha wh less han perfec correlaon, he dsclosure provdes less nformaon relevan for assessng non-dsclosng frms fuure cash paymens han for he dsclosng frm s fuure cash paymen.

11 capal as long as underlyng (fundamenal) rsk s no dversfed away. Ths occurs even hough dsclosure qualy relaes o nose n he dsclosure ha s dosyncrac o he sngle frm. 3. Dsclosure by all frms Followng he prevous subsecon, I connue o assume ha he erms (ha s, he perodc nnovaons n each frm s cash paymen) are dencally dsrbued across frms wh covarance equal o τσ. However, n hs subsecon I assume ha each of he m frms dscloses a sgnal, y = x + + e, where e s normally dsrbued wh mean zero and varance σ e. The e s are ndependen across frms and perods, and also ndependen of across frms and perods. 8 In hs seng nvesors wll use he dsclosures of all frms o assess E( P + + x + ) and Cov ( P + + x +, PM+ + xm+ ) n formng prces accordng o equaon (3). In parcular, s sraghforward o show ha n hs case he prcng equaon (3) becomes: m ρ P = x + bj ( yj xj ) s (6) r j= n where he b j s represen he coeffcens for frm aached o he dsclosures by each frm, y, n deermnng E( x + ), and j s = Cov ( P + + x +, PM + + xm+ ) = ( r) ( r) b ( ) + + j + m r j= m ( ) σ ( τ). As n he prevous secons, he effec of dsclosure qualy on he dscoun n prce s capured va he m bj erm n he expresson for j= s, whch s generally a que complcaed expresson of he model parameers. 9 For hs reason deermnng he effec of an ncrease n dsclosure qualy s mpraccal excep under very src resrcons on he parameers n he model. For 8 I s possble, a some cos n erms of algebrac complexy, o allow for correlaon across frms n he dsclosure nose erms, e, whou changng he resuls dscussed n hs subsecon. 9 As shown n he appendx, f B denoes he m x m marx of b coeffcens, hen B = Σ ( Σ + Σ ), j e where Σ and Σ are he varance-covarance marces of and e respecvely. The qualy of dsclosures s e m capured by Σ. The e b erm s hen obaned by summng across he h row n B. j j = 0

12 example, s possble for m bj (and hus cos of capal) o be ncreasng n he qualy of j= dsclosure by frm for specfc parameer values. Of parcular neres here s he behavour of m bj as m becomes large. Agan, o j= smplfy maers I focus on wo hghly sylzed suaons: () where one frm (frm ) has a dsclosure qualy dfferen o ha of all oher companes who have equal dsclosure qualy; and () where all frms have he same dsclosure qualy. The frs suaon allows varyng dsclosure qualy for a sngle frm whle holdng he dsclosure qualy of oher frms consan, as would be he case for example where a specfc frm changed s dsclosure qualy. The second suaon allows varyng all frms dsclosure quales smulaneously, for example by a change n accounng sandards appled o all frms (e.g., he adopon of IFRS). In he appendx I show ha n boh of hese sengs m bj approaches as m becomes large. j= An mmedae mplcaon of hs s ha for large m dsclosure qualy (for eher a sngle frm, or for all frms) becomes rrelevan o rsk assessmen, and hus cos of capal. Tha s, for large m he collecon of dsclosures for all frms provdes maxmal nformaon ( m j= b j = ) regardng nvesors end of perod payoffs rrespecve of he qualy of he ndvdual dsclosures nvolved. The nuon s sraghforward, n a large economy each ndvdual sgnal provdes lle ncremenal nformaon regardng relevan sysemac rsk beyond wha s avalable from he oal collecon of dsclosures across all frms, and so changng he qualy of each ndvdual dsclosure maers lle for rsk assessmen. 0 Fnally, s neresng o noe ha hs resul s no due o dversfcaon of porfolos by nvesors across many frms, bu raher by dversfcaon of nformaon sources for nvesors n a large economy. The same effec would occur f here were only a sngle frm bu many dsclosures. 0 Noe ha hs does no mean ha dsclosure qualy does no affec each ndvdual dsclosure response coeffcen, b. In fac, s sraghforward o show ha hese vary wh dsclosure qualy even for large m. j However, for large m hese effecs offse for he purposes of rsk assessmen by nvesors ha s refleced n he dscoun n prce. Thus changng dsclosure qualy affecs how each frms dsclosure moves prce n equaon (6), bu has an mmaeral effec on nvesors assessmen of rsk.

13 To summarze he resuls from hs secon, n a large economy hree hngs occur: () here are more nvesors among whom o share he economy s rsk; () here are more frms generang economy-wde rsk and across whch he economy s rsk s spread (and can be dversfed); and (3) here are more dsclosures/sgnals from whch nvesors can oban nformaon o help assess ha rsk. Each of hese affecs he level and assessmen of rsk ha s refleced n prces and hus cos of capal. In an economy where nvesors payoffs are correlaed across frms, he frs wo facors resul n cos of capal reflecng sysemac rsk conssen wh he CAPM. If only a lmed number of dsclosures are avalable, hen he qualy of hose dsclosures wll be refleced n nvesors rsk assessmens (and hus n cos of capal) even hough qualy relaes o frm-specfc dosyncrac nose n he dsclosures. Bu f many dsclosures are avalable, dsclosure qualy has a mnmal mpac on rsk assessmens and cos of capal. 4. The mpac of heursc raders The analyss above s based on a seng where all nvesors are assumed o be fully raonal, ha s hey correcly process all nformaon avalable n makng her demand decsons o maxmze expeced uly. In hs secon, I modfy he model slghly o also nclude nvesors who are no fully raonal, bu nsead follow a smple heursc radng rule lnked o frm dsclosure. My objecve s o develop some nal houghs on how he lnk beween dsclosure, dversfcaon and he cos of capal mgh be affeced by he presence of less han fully raonal nvesors, albe n a very smple and sylzed seng. The model s he same as n he prevous secon excep ha n addon o n fully raonal raders each perod, here are also λ n heursc raders, where λ 0. Thus when λ = 0 (.e., here are no heursc raders) he model revers o ha used n he prevous secon. Lke he raonal nvesors, heursc raders exs for a sngle perod. However heursc raders se her demand for each frm s shares by followng a smple radng heursc, or rule of humb, and se demand equal o h y, where h s a posve mulple and y = y y, he change n frm s dsclosure from he prevous perod. Ths smple The analyss n hs secon draws upon ongong research wh Ma Pnnuck. A large and growng body of research smlarly consders he mpac of less han fully raonal radng n a varey of capal marke sengs. See, for example, Kohar (00), Lee (00), Rchardson, Tuna and Wysock (00), Shlefer (000), and Thaler (005) for some recen surveys ha dscuss relevan research.

14 radng rule represens a seng where heursc raders e her radng decsons o wheher a frm s dsclosed y has ncreased or decreased relave o he pror perod (.e., represens eher good or bad news.) The parameer h capures how aggressve heursc raders are n followng hs radng rule. Fnally, I only nvesgae he smples seng where all frms cash paymens and dsclosures are ndependen,.e., and e are ndependen across frms (as well as over me). Wh hs assumpon he mul-frm seng smplfes consderably o one where each frm s prce s ndependen of all oher frms prces and dsclosures. Alhough hs s a consderable smplfcaon urns ou ha he resulng analyss sll yelds several poenally neresng nsghs. s: In hs seng s possble o show (see he appendx) ha equlbrum prce for frm ρ P = x ( b γ ( b ))( y x ) ( r) γ y s r n (7) where b σ = σ + σe r, γ = ( + r) λhρs, and rγ Var ( ( ( s ) ) = P + + x + ) = ( ) ( ) ( ) ( ) + γ + r + r b + rγ + + γ σ. r b There are several aspecs of equaon (7) ha are noable. Frs, as would be expeced, f here are no heursc raders,.e., λ = 0, he equaon collapses o he correspondng prcng equaon from he prevous secon (equaon (6)) n he uncorrelaed frm case (.e., when τ = 0 ). Second, as s he case n he prevous secons, he effec of dsclosure qualy s capured va b, he coeffcen ha raonal nvesors place on he frm s dsclosure when seng her expecaons of he end of perod frm paymen; a hgher b represens hgher dsclosure qualy. Thrd, snce he defnon of s ncludes γ whch self depends on s deermned mplcly by hs expresson. Because he expresson s quadrac n s s sraghforward o solve bu yelds an algebracally messy soluon. 3 Neverheless s s, 3 Because he expresson s quadrac s possble ha no, one or wo soluons exs. I s possble o show ha no soluon exss (and hus no equlbrum occurs n he marke) f here are oo many heursc raders, or here s oo much rsk. Also, when wo soluons exs only one s economcally plausble. Tha s, only one 3

15 possble o show ha he resulng value for s s ncreasng n he fracon of heursc raders ( λ ), and he underlyng rsk assocaed wh he frm s cash paymens ( σ ), as would be expeced. I s also decreasng n dsclosure qualy (.e., decreasng nσ ), ndcang ha, as n prevous secons, for a gven sze of he economy (.e., n and m ) greaer dsclosure qualy resuls n a lower assessed rsk by raonal nvesors. I s sraghforward o rearrange he prce expresson n (7) o he followng: e E( P + + x + ) + r ρ P = + γ y s. (8) + r r + rn Ths ndcaes ha prce s equal o he presen value of he raonal nvesors expeced value of her end of perod payoff from holdng frm plus a facor relang o he effec of heursc raders on prce less a second facor relang o raonal nvesors rsk assessmen assocaed wh her end of perod payoff. These wo facors combned represen he dvergence of prce from he presen value of nvesors payoffs, and so also represen he frm s cos of capal. The second facor mrrors he rsk-relaed dscoun n he prevous secons, alhough as ndcaed above, he assessed rsk wll be greaer n he presence of heursc raders. The frs facor ndcaes ha heursc raders generae addonal prce pressure whch causes prces o move away from he dscouned expeced value, and ha hs prce pressure s conngen on he dsclosure realsaon each perod, reflecng he heursc raders rule of humb radng rule. Tha s, he dscoun, or cos of capal has a perod and dsclosure specfc componen whch s no due o rsk bu raher o he prce pressure brough abou by he heursc raders. Imporanly, he wo componens of cos of capal are affeced dfferenly n a large ρ economy,.e., as m and n become large. The rsk-relaed componen, s, behaves n + rn he same manner as n he prevous secons. Specfcally, because frms are assumed ndependen here, assessed rsk ( s ) does no change wh eher m or n, bu because n s large ges shared among more nvesors and so has less effec on prce (or s dversfed away ). In conras, he dsclosure-conngen componen of he dscoun, + r γ y, s no r soluon converges o he value ha would be obaned when no heursc raders are presen as he fracon of heursc raders ( λ ) goes o zero. Deals are avalable from he auhor. I s hs soluon I focus on here. 4

16 dversfed away n a large economy because s unaffeced by eher m or n. Despe ncreased frms across whch o dversfy and ncreased numbers of raonal nvesors among whch o share rsk, heursc raders generae prce pressure whch s refleced n equlbrum prces. Ths occurs due o he assumpon ha when he number of raonal nvesors ( n ) s large, so oo s he number of heursc raders ( λ n ). Only f heursc raders are presumed o become less prevalen n a large economy would her nfluence on prce be dversfable. Ths s despe her acons beng dosyncrac,.e., uncorrelaed across frms. 5. Concluson Based on a smplfed and modfed verson of models ypcally employed n relaed research I hghlgh several observaons concernng dsclosure qualy, dversfcaon, and he cos of capal. Frs, n a seng where prce rsk canno be avoded by nvesors, (ex ane) cos of capal s decreasng n dsclosure qualy. Ths conrass wh Chrsensen, de la Rosa and Felham (00). Second, as n Lamber, Leuz and Verreccha (007), dsclosure qualy connues o play a role n a large economy (.e., does no dversfy away) f he number of avalable dsclosures s no also large. However, f he number of dsclosures s also large, hen dsclosure qualy ceases o be mporan. Fnally, when he model ncludes nvesors who follow a heursc radng rule lnked o frms dsclosures whou reference o prce, cos of capal ncludes an addonal dsclosure-conngen facor beyond he effec of rsk. So long as he relave numbers of raonal and heursc raders reman unchanged n a large economy hs facor does no dversfy away. Ths s despe he heursc rader facor beng ed o frm-specfc dosyncrac dsclosures. 5

17 References Berger, P. 0, Challenges and opporunes n dsclosure research A dscusson of The fnancal reporng envronmen: Revew of he recen leraure, Journal of Accounng and Economcs, vol. 5, pp Beyer, A., D. Cohen, T. Lys & B. Walher, 00, The fnancal reporng envronmen: revew of he recen leraure, Journal of Accounng and Economcs, vol. 50, pp Ca, C. 03, Dsclosure qualy, cos of capal and real effecs of dsclosure. PhD hess, Unversy of Melbourne. Chrsensen, P., L. de la Rosa, & G. Felham 00, Informaon and he cos of capal: an ex ane perspecve, The Accounng Revew, vol. 83, no. 3, pp Easley, D. & M. O Hara 004, Informaon and he cos of capal, The Journal of Fnance, vol. 59, no. 4, pp Gao, P. 00, Dsclosure qualy, cos of capal and nvesors, The Accounng Revew, Vol. 85, No., pp. -9. Hughes, J., J. Lu, and J. Lu 007, Informaon asymmery, dversfcaon, and cos of capal, The Accounng Revew, Vol. 8, No. 3, pp Kohar, S.P. 00, Capal markes research n accounng, Journal of Accounng and Economcs, Vol. 3, pp Lamber, R., C. Leuz & R. Verreccha 007, Accounng nformaon, dsclosure, and he cos of capal, Journal of Accounng Research, vol. 45, no., pp Lamber, R., C. Leuz & R. Verreccha 0, Informaon Precson, Informaon Asymmery, and he Cos of Capal, Revew of Fnance, vol. 6, pp. -9. Lee, C. 00, Marke effcency and accounng research: a dscusson of capal marke research n accounng by S.P. Kohar, Journal of Accounng and Economcs, Vol. 3, pp

18 Rchardson, S., I. Tuna & P. Wysock 00, Accounng anomales and fundamenal analyss: A revew of recen research advances, Journal of Accounng and Economcs, Vol. 50, pp Shevln, T. 03, Some personal observaons on he debae on he lnk beween fnancal reporng qualy and he cos of equy capal, unpublshed paper. Shlefer, A. 000, Ineffcen markes: An nroducon o behavoral fnance, Oxford Unversy Press, Oxford. Thaler, R. (ed) 005, Advances n behavoral fnance II, Russell Sage Foundaon, New York. Verreccha, R. 00, Essays on dsclosure, Journal of Accounng and Economcs, vol. 3, pp

19 Appendx Some Dervaons In hs appendx I oulne a que general verson of he model employed n he varous secons of he paper and hen employ smplfed versons of he model o provde dervaons of resuls dscussed n he paper. Le x denoe he m x vecor of frms paymens whch follow he process x = x +, where s dsrbued mulvarae normal wh mean zero and varancecovarance marx Σ. Also le = + + y Cx e be he k x vecor of k dsclosures or sgnals avalable o nvesors (n addon o x ) a me, where e s dsrbued mulvarae normal wh mean zero and varance-covarance marx Σ e. and e are assumed o be ndependen of each oher and over me. The marx C allows for que general suaons. For example, f here s a dsclosure avalable only abou frm (as n secon 3. of he paper) C wll be a x m vecor wh a as he frs elemen and zeroes elsewhere. Alernavely, f all frms provde a sgnal equal o y = x+ + e (as n secon 3. of he paper) hen C s an (+r) a he end of each perod. m x m deny marx. There s also a rsk free secury whch pays There are n dencal raonal nvesors each wh negave exponenal uly wh rsk averson coeffcen ρ. Each nvesor lves for one perod only, and chooses her demand for secures o maxmze expeced uly over end of perod payoffs: P+ + x +, where P s he n x vecor of prces. Gven hese assumpons s well known ha he demands DRaonal S P+ x+ P, where ρ raonal raders wll choose are gven by = ( E ( + ) ( + r) ) E( P + x ) denoes expeced value of he end of perod payoff gven all nformaon + + avalable o nvesors a me, and S= Var ( P+ + x + ) s he varance-covarance marx of end of perod paymens assessed by nvesors gven nformaon avalable a me. 4 are also λ n heursc raders whose demands are se accordng o D = Hy ( y ) = H y, where H s an m x k dagonal marx of mulples ha Heursc There 4 Because of he normaly and lneary assumpons S does no vary across dfferen perods as confrmed below. 8

20 heursc raders apply o he componens n assume supply s equal o, he un vecor, each perod. y when seng her demands. 5 Fnally, I Gven hese assumpons he marke clearng requremen for equlbrum each perod s: n Raonal λn Heursc D + D =. Subsung n for raonal and heursc nvesors demands and rearrangng yelds he followng expresson ha mus be sasfed by equlbrum prce: ρ ( + r) P = E ( P+ + x+ ) + λρsh y S. (9) n Equaon (9) collapses o equaon () n secon. of he paper when here s only one frm ( m = ) and here are no heursc raders ( λ = 0 ). I becomes equaon (3) n secon 3 when here are m frms and no heursc raders, and equaon (8) n secon 4 when here are heursc raders bu when frms and sgnals are uncorrelaed (.e., Σ and dagonal). Σ e are boh Equaon (9) defnes equlbrum prce recursvely. To provde a closed-form soluon I assume ha prce s lnear n he nformaon avalable o nvesors a me. In parcular, I assume prce can be expressed as follows: P= a + AE( x ) + Ay + Ay. (0) Gven hs assumpon I subsue hs no equaon (9) and hen equae coeffcens across equaons (9) and (0) o deermne a seady sae equlbrum prce expresson. In oher words, when he coeffcens are equaed, f nvesors ancpae ha end of perod (me + ) prces are as n equaon (0), her resulng demands and marke clearng a me wll resul n perod prces ha also accord wh equaon (0). The resulng expresson for equlbrum prce s gven by subsung he followng coeffcens no equaon (0): 6 5 The model can be made slghly more general by, for example, allowng he λ s o be dfferen for each frm, and/or allowng H o have non-zero off dagonal elemens. 6 Deals of he dervaon are avalable from he auhor. 9

21 ρ a0 = S rn A = I + λρshc r ( + r) A r = ( + r) A3 = λρsh + r where S = Var ( P + x ) = ( + r) A Var ( x ) A' + ( AB+ A )( CΣ C' +Σ )( AB+ A )' λρsh e '( ' e). B =Σ C CΣ C +Σ When here are heursc raders, as noed n secon 4 of he paper, S appears n he rgh hand sde of S= Var ( P+ + x + ) and so s defned mplcly. 7 However, when here are no heursc raders A = I, A = A 3 = 0, and S does no appear n he rgh hand sde of r S= Var ( P + x ), and so s more easly deermned. + + In each case n he body of he paper (n secons, 3 and 4) he expresson for equlbrum prce s smply a specal case of hs general soluon bu also usng he followng well known expressons based on he mulvarae normaly assumpon: E ( x ) = E( x y, x ) = x + B( y Cx ) + + Var ( x ) = Var( x y, x ) = ( I BC) Σ + + B=Σ C CΣ C +Σ '( ' e). 7 The rgh hand sde of S = Var ( P + x ) s quadrac n he m x m S marx when here are heursc + + raders (.e., when λ s no zero), whch makes dffcul o solve excep n smplfed sengs such as under he assumpon n secon 4 ha each frm s ndependen. 0