Accounting Information, Disclosure, and the Cost of Capital

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1 Unversty of Pennsylvana ScholarlyCommons Accountng Papers Wharton Faculty Research Accountng Informaton, Dsclosure, and the Cost of Captal Rchard A. Lambert Unversty of Pennsylvana Chrstan Leuz Robert E. Verreccha Unversty of Pennsylvana Follow ths and addtonal works at: Part of the Accountng Commons Recommended Ctaton Lambert, R. A., Leuz, C., & Verreccha, R. E. (2007). Accountng Informaton, Dsclosure, and the Cost of Captal. Journal of Accountng Research, 45 (2), Ths paper s posted at ScholarlyCommons. For more nformaton, please contact repostory@pobox.upenn.edu.

2 Accountng Informaton, Dsclosure, and the Cost of Captal Abstract In ths paper we examne whether and how accountng nformaton about a frm manfests n ts cost of captal, despte the forces of dversfcaton. We buld a model that s consstent wth the Captal Asset Prcng Model and explctly allows for multple securtes whose cash flows are correlated. We demonstrate that the qualty of accountng nformaton can nfluence the cost of captal, both drectly and ndrectly. The drect effect occurs because hgher qualty dsclosures affect the frm's assessed covarances wth other frms' cash flows, whch s nondversfable. The ndrect effect occurs because hgher qualty dsclosures affect a frm's real decsons, whch lkely changes the frm's rato of the expected future cash flows to the covarance of these cash flows wth the sum of all the cash flows n the market. We show that ths effect can go n ether drecton, but also derve condtons under whch an ncrease n nformaton qualty leads to an unambguous declne n the cost of captal. Keywords cost of captal, dsclosure, nformaton rsk, asset prcng Dscplnes Accountng Ths ournal artcle s avalable at ScholarlyCommons:

3 Accountng Informaton, Dsclosure, and the Cost of Captal Rchard Lambert* The Wharton School Unversty of Pennsylvana Chrstan Leuz The Wharton School Unversty of Pennsylvana Robert E. Verreccha The Wharton School Unversty of Pennsylvana September 2005 Revsed, March 2006 Abstract In ths paper we examne whether and how accountng nformaton about a frm manfests n ts cost of captal, despte the forces of dversfcaton. We buld a model that s consstent wth the CAPM and explctly allows for multple securtes whose cash flows are correlated. We demonstrate that the qualty of accountng nformaton can nfluence the cost of captal, both drectly and ndrectly. The drect effect occurs because hgher qualty dsclosures reduce the frm s assessed covarances wth other frms cash flows, whch s non-dversfable. The ndrect effect occurs because hgher qualty dsclosures affect a frm s real decsons, whch lkely changes the frm s rato of the expected future cash flows to the covarance of these cash flows wth the sum of all the cash flows n the market. We show that ths effect can go n ether drecton, but also derve condtons under whch an ncrease n nformaton qualty leads to an unambguous declne the cost of captal. JEL classfcaton: Key Words: G12, G14, G31, M41 Cost of captal, Dsclosure, Informaton rsk, Asset prcng *Correspondng Author. We thank the semnar partcpants at Oho State Unversty and an anonymous referee for ther helpful comments.

4 1. Introducton The lnk between accountng nformaton and the cost of captal of frms s one of the most fundamental ssues n accountng. Standard setters frequently refer to t. For example, Arthur Levtt (1998), the former charman of the Securtes and Exchange Commsson, suggests that hgh qualty accountng standards [ ] reduce captal costs. Smlarly, Neel Foster (2003), a former member of the Fnancal Accountng Standards Board (FASB) clams that More nformaton always equates to less uncertanty, and [ ] people pay more for certanty. In the context of fnancal nformaton, the end result s that better dsclosure results n a lower cost of captal. Whle these clams have ntutve appeal, there s surprsngly lttle theoretcal work on the hypotheszed lnk. In partcular, t s unclear to what extent accountng nformaton or frm dsclosures reduce non-dversfable rsks n economes wth multple securtes. Asset prcng models, such as the Captal Asset Prcng Model (CAPM), and portfolo theory emphasze the mportance of dstngushng between rsks that are dversfable and those that are not. Thus, the challenge for accountng researchers s to demonstrate whether and how frms accountng nformaton manfests n ther cost of captal, despte the forces of dversfcaton. Ths paper explores both of these questons. We defne the cost of captal as the expected return on a frm s stock. Ths defnton s consstent wth standard asset prcng models n fnance (e.g., Fama and Mller, 1972, p. 303), as well as numerous studes n accountng that use dscounted cash flow or abnormal earnngs models to nfer frms cost of captal (e.g., Botosan, 1997; Gebhardt et al., 2001). 1 In our model, we explctly allow for multple frms whose cash flows are correlated. In contrast, most analytcal models n accountng examne the role of 1 We also dscuss the mpact of nformaton on prce, as the latter s sometmes used as a measure of cost of captal. See, e.g., Easley and O Hara (2004) and Hughes et al. (2005).

5 nformaton n sngle-frm settngs (see Verreccha, 2001, for a survey). Whle ths lterature yelds many useful nsghts, ts applcablty to cost of captal ssues s lmted. In sngle-frm settngs, frm-specfc varance s prced because there are no alternatve securtes that allow nvestors to dversfy dosyncratc rsks. We begn wth a model of a mult-securty economy that s consstent wth the CAPM. We then recast the CAPM, whch s expressed n terms of returns, nto a more easly nterpreted formulaton that s expressed n terms of the expected values and covarances of future cash flows. We show that the rato of the expected future cash flow to the covarance of the frm s cash flow wth the sum of all cash flows n the market s a key determnant of the cost of captal. We demonstrate that accountng nformaton nfluences a frm s cost of captal n two ways: 1) drect effects where hgher qualty accountng nformaton does not affect cash flows per se, but affects the market partcpants assessments of the dstrbuton of future cash flows; and 2) ndrect effects where hgher qualty accountng nformaton affects a frm s real decsons, whch nfluences ts expected value and covarances of frm cash flows. In the frst category, we show (not surprsngly) that hgher qualty nformaton reduces the assessed varance of a frm s cash flows. Analogous to the sprt of the CAPM, however, we show ths effect s dversfable n a large economy. We dscuss what the concept of dversfcaton means, and show that an economcally sensble defnton requres more than smply examnng what happens when the number of securtes n the economy becomes large. More surprsngly, we demonstrate that an ncrease n the qualty of a frm s dsclosure about ts own future cash flows has a drect effect on the assessed covarances wth other frms cash flows. Ths result bulds on and extends the work on estmaton rsk n fnance. 2 In ths lterature, nformaton sgnals are typcally modeled as arsng from a hstorcal tme-seres of 2 See Brown (1979), Barry and Brown (1984 and 1985), Coles and Loewensten (1988), and Coles et al. (1995). 2

6 return observatons. In partcular, Barry and Brown (1985) and Coles et al. (1995) compare two nformaton envronments: n one envronment the same amount of nformaton (e.g., the same hstorcal tme-seres) s avalable for all frms n the economy, whereas n the other nformaton envronment there are more observatons for one group of frms than another. They fnd that the betas of the hgh nformaton securtes are lower than they would be n the equal nformaton case. They cannot unambguously sgn, however, the dfference n betas for the low nformaton securtes n the unequal- versus equal-nformaton envronments. Moreover, these studes do not address the queston of how an ndvdual frm s dsclosures can nfluence ts cost of captal wthn an unequal nformaton envronment. Rather than restrctng attenton to sgnals that are hstorcal observatons of returns, our paper uses a more conventonal nformaton-economcs approach n whch nformaton sgnals are related to realzed or future cash flows. Wth ths approach, we allow for more general changes n the nformaton envronment, and we are able to prove much stronger results. In partcular, we show that hgher qualty accountng nformaton and dsclosures affect the assessed covarances wth other frms, and ths effect unambguously moves a frm s cost of captal closer to the rsk-free rate. Moreover, ths effect s not dversfable because t s present for each of the frm s covarance terms and hence does not dsappear n large economes. Next, we dscuss the effects of dsclosure regulaton on the cost of captal of frms. Usng our framework, t s straghtforward to show that ncreasng the qualty of mandated dsclosures also reduces the cost of captal for frms. In addton to the effect of an ndvdual frm s dsclosures, there s an externalty from the dsclosures of other frms. We also argue that the magntude of the effect of mandated dsclosure on cost of captal wll be unequal across frms. Moreover, the reducton n the assessed covarances between frms and the market wll not 3

7 always result n a decrease n the beta coeffcent of each frm. After all, regardless of the structure of nformaton n the economy, the average beta across frms has to be 1.0. Therefore, even though frms cost of captal wll declne through mproved mandated dsclosure, ther beta coeffcents need not. In the ndrect effect category, we show that the qualty of accountng nformaton nfluences a frm s cost of captal through ts effect on a frm s real decsons. Frst, we demonstrate that f better nformaton reduces the amount of frm cash flows that managers approprate for themselves, the mprovements n dsclosure not only ncrease prce, but n general also reduce frms cost of captal. Second, we allow nformaton qualty to change a frm s real decsons, e.g., wth respect to producton or nvestment. In ths case, nformaton qualty changes the rato of expected cash flow to non-dversfable covarance rsk and hence nfluences a frm s cost of captal. We derve condtons under whch an ncrease n nformaton qualty results n an unambguous decrease n a frm s cost of captal. Our paper makes several contrbutons. Frst, we extend and generalze pror work on estmaton rsk. We show that nformaton qualty drectly nfluences frms cost of captal and that mprovements n nformaton qualty unambguously reduce non-dversfable rsks. Ths fndng s mportant as t suggests that a frm s beta factor s a functon of ts nformaton qualty and dsclosures. In ths sense, our study provdes theoretcal gudance to emprcal studes that examne the lnk between frms dsclosures and/or nformaton qualty, and frms cost of captal (e.g., Botosan, 1997; Botosan and Plumlee, 2002; Francs et al., 2004). 3 It s mportant to recognze, however, that our nformaton effects are fully captured by an approprately specfed, forward-lookng beta. Thus, our model does not provde support for an addtonal nformaton 3 In addton, our study provdes an explanaton for emprcal studes that fnd dfferences n dsclosure regulaton can explan dfferences n frms equty rsk premum (or the average cost of equty captal) across countres (e.g., Hal and Leuz, 2006). 4

8 rsk factor. One way to ustfy the ncluson of nformaton proxes n a cost of captal model would be to note that an estmate of beta, whch for nstance s based on hstorcal data alone, does generally not capture all nformaton effects. In ths case, however, t s ncumbent on accountng researchers to carefully specfy a measurement error model ustfyng the ncluson of nformaton proxes n the emprcal specfcaton. 4 A second contrbuton of our paper s that t provdes a drect lnk between nformaton qualty and frms cost of captal, wthout reference to market lqudty. Pror work suggests an ndrect lnk between dsclosure and frms cost of captal based on market lqudty and adverse selecton n secondary markets (e.g., Damond and Verreccha, 1991; Baman and Verreccha, 1996; Easley and O Hara, 2004). These studes, however, analyze settngs wth a sngle frm (or settngs where cash flows across frms are uncorrelated). Thus, t s unclear whether the effects demonstrated n these studes survve the forces of dversfcaton and extend to more general mult-securty settngs. We emphasze, however, that we do not dspute the possble role of market lqudty for frms cost of captal, as several emprcal studes suggest (e.g., Amhud and Mendelson, 1986; Chorda et al., 2001; Easley et al., 2002; Pastor and Stambaugh, 2003). Our paper focuses on an alternatve (and possbly more drect) explanaton as to how nformaton qualty nfluences non-dversfable rsks. Fnally, our paper contrbutes to the lterature by showng that nformaton qualty has ndrect effects on real decsons, whch n turn manfest n frms cost of captal. In ths sense, our study relates to work on real effects of accountng nformaton (e.g., Kanoda et al., 2000 and 2004). These studes, however, do not analyze the effects on frms cost of captal or nondversfable rsks. 4 See, e.g., Beaver et al. (1970) for an emprcal analyss that relates accountng nformaton to a frm s beta. 5

9 The remander of ths paper s organzed as follows. Secton 2 sets up the basc model n a world of homogeneous belefs, defnes terms, and derves the determnants of the cost of captal. Sectons 3 and 4 analyze the drect and ndrect effects of accountng nformaton on frms cost of captal, respectvely. Secton 5 summarzes our fndngs and concludes the paper. 2. Model and Cost of Captal Dervaton We defne cost of captal to be the expected return on the frm s stock. Consstent wth standard models of asset prcng, the expected rate of return on a frm s stock s the rate, R, that equates the stock prce at the begnnng of the perod, P, to the cash flow at the end of the perod, V : R ~ V ~ ~ ~ V P P (1 + ) =, or R =. Our analyss focuses on the expected rate of return, whch P ~ ~ E( V Φ) P s E( R Φ) =, where Φ s the nformaton avalable to market partcpants to P make ther assessments regardng the dstrbuton of future cash flows. We assume there are J securtes n the economy whose returns are correlated. The best known model of asset prcng n such a settng s the Captal Asset Prcng Model (CAPM) (Sharpe, 1964; Lnter, 1965). Therefore, we begn our analyss by presentng the conventonal formulaton of the CAPM, and then transform ths to show how nformaton affects expected returns. In the CAPM, the expected return on a frm s stock can be expressed as a functon of the rsk-free rate, R f, the expected return on the market, E R ~ ), and the frm s beta coeffcent, β : E(R ~ E(R [ ] ~ [ Cov(R ~, R ~ )] Var(R ~ ) R ) R E(R ~ ) R R M Φ Φ = f f + M Φ f β = f + m Φ. (1) Φ) ( m M 6

10 Eqn. (1) shows that the only frm-specfc parameter that affects the frm s cost of captal s ts beta coeffcent, or, more specfcally, the covarance of ts future return wth that of the market portfolo. Ths covarance s a forward-lookng parameter, and s based on the nformaton avalable to market partcpants. Consstent wth the conventonal formulaton of the CAPM, we assume market partcpants possess homogeneous belefs regardng the expected end-of-perod cash flows and covarances. Because the CAPM s expressed solely n terms of covarances, ths formulaton mght be nterpreted as mplyng that other factors, for example the expected cash flows, do not affect the frm s cost of captal. It s mportant to keep n mnd, however, that the covarance term n the CAPM s expressed n terms of returns, not n terms of cash flows. The two are related va the ~ ~ equaton ~ ~ V V 1 ~ ~ Cov ( R, ) (, M RM = Cov ) = Cov( V, VM ). Ths means that nformaton P PM PPM potentally affects the expected return on a frm s stock through ts effect on nferences about the covarances of future cash flows, or through the current perod stock prce, or both. Clearly the current stock prce s a functon of the expected-end-of-perod cash flow. In partcular, the CAPM can be re-expressed n terms of prces nstead of returns as follows (see Fama,1976, eqn. [83]): ~ ~ E( V M Φ) (1 + R f ) P M E( V Φ) ~ Var( VM Φ) P = 1 + R f J ~ ~ Cov( V, Vk Φ) k= 1, = 1,, J. (2) Eqn. (2) ndcates that the current prce of a frm can be expressed as the expected end-of-perod cash flow mnus a reducton for rsk. Ths rsk-adusted expected value s then dscounted to the begnnng of the perod at the rsk-free rate. The rsk reducton factor n the numerator of eqn. (2) 7

11 ~ E( VM Φ) (1 + R f ) PM has both a macro-economc factor, ~, and a frm-component, whch s Var( V Φ) M determned by the covarance of the frm s end-of-perod cash flows wth those of all other frms. As n Fama (1976), the term (V ~ J Cov, V ~ k= 1 k Φ) s a measure of the contrbuton of frm to ~ J ~ the overall varance of the market cash flows, V M = V k. k= 1 Eqns. (1) and (2) express expected returns and prcng on a relatve bass: that s, relatve to the market. If we make more specfc assumptons regardng nvestors preferences, we can express prces and returns on an absolute bass. 5 In partcular, f the economy conssts of N nvestors wth negatve exponental utlty wth rsk tolerance parameter τ, the begnnng-ofperod stock prce can be expressed as (detals n the Appendx): J ~ 1 ~ ~ E( V Φ) Cov( V, V k Φ) Nτ P k= = R f. (3) As n eqn. (2), prce n eqn. (3) s equal to the expected end-of-perod cash flow mnus a reducton for the rskness of frm, all dscounted back to the begnnng of the perod at the rskfree rate. The dscount for rsk s now smply the contrbuton of frm s cash flows to the aggregate rsk of the market dvded by the term Nτ, whch s the aggregate rsk tolerance of the marketplace. The prce of the market portfolo can be found by summng eqn. (3) across all ~ 1 ~ frms: ( 1+ R f ) PM = E( VM Φ) Var( VM Φ), whch can also be expressed as Nτ 5 More specfcally, the prcng and return formulas wll be expressed relatve to the rsk free rate, whch acts as the numerare n the economy. 8

12 1 = Nτ ~ E( VM Φ) (1 + R f ) P ~ Var( V Φ) M M. Therefore, the aggregate rsk tolerance of the market determnes the rsk premum for market-wde rsk. We can re-arrange eqn. (3) to express the expected return on the frm s stock as follows. Lemma 1. The cost of captal for frm s ~ 1 J ~ ~ ~ R f E( V Φ) + Cov( V, Vk Φ) ~ E( V Φ) P Nτ E( R Φ) = = k= 1. (4a) P ~ 1 J ~ ~ E( V ) Cov( V, Vk Φ) Nτ k= 1 If we further assume that (V ~ J Cov V ~, k Φ) 0, ths reduces to k= 1 ~ ~ R f H ( Φ) + 1 E( V Φ) E( R Φ) =, where H ( Φ) =. (4b) H ( Φ) 1 1 J ~ ~ Cov( V, Vk Φ) Nτ k= 1 Lemma 1 shows that the cost of captal of the frm depends on four factors: the rsk free rate, the aggregate rsk tolerance of the market, the expected cash flow of the frm, and the covarance of the frm s cash flow wth the sum of all the frms cash flows n the market. The latter three terms can be combned nto the rato of the frm s expected cash flows, to frm s contrbuton to aggregate rsk per-unt-of aggregate rsk tolerance. Note that the defnton of cost of captal n Lemma 1 does not requre that frm s expected cash flow, or the covarance of that cash flow wth the market, be of any partcular sgn. In the next result we show how a change n each of the four factors affects cost of captal. Proposton 1. Ceters parbus the cost of captal for frm, E (R ~ ), s: (a) ncreasng (decreasng) n the rsk free rate, R f, when the expected cash flow and the prce of the frm have the same (dfferent) sgn; 9

13 (b) decreasng (ncreasng) n the aggregate rsk tolerance of the market, Nτ, when the expected cash flow and covarance of that cash flow wth the market have the same (dfferent) sgn; (c) decreasng (ncreasng) n the expected end-of-perod cash flow, J Cov(V ~, V ~ k ) k= 1 s postve (negatve); and J (d) ncreasng (decreasng) n Cov(V ~, V ~ k ) when E(V ~ ) k= 1 E(V ~ ), when s postve (negatve). To make the ntuton that underles Proposton 1 as transparent as possble, consder the case n whch frm s expected end-of-perod cash flow, the covarance between ts end-of-perod cash flows and the market, and the frm s begnnng-of-perod stock prce are all postve. Here, the reason why the expected return on frm s ncreasng n the rsk-free rate s clear, because ths provdes the baselne return for all securtes. When Nτ ncreases, the aggregate rsk tolerance of the market ncreases; hence, the dscount appled to each frm s rskness decreases. 6 J the frm s expected rate of return closer to the rsk-free rate. When Cov(V ~, V ~ k ) k= 1 Ths moves ncreases, the contrbuton of the rskness of frm s cash flows to the overall rskness of the market goes up; hence, the expected return must ncrease to compensate nvestors for the ncrease n rsk. Ths s one of the key nsghts of the CAPM (Sharpe, 1964; Lntner, 1965). The most surprsng result s that an ncrease n the expected value of cash flows decreases the expected rate of return. The ntuton, however, s farly straghtforward. Consder a frm wth two components of cash flow: a rskless component ( V a ) and a rsky component ( V b ). Clearly the cost of captal for the frm wll be somewhere n between the cost of captal 6 Ths s analogous to the effect dscussed n Merton (1987). 10

14 for the rskless component and the cost of captal for the rsky component. But f the frm s expected cash flow ncreases wthout affectng the frm s varances or covarances, ths s exactly analogous to addng a new rskless component of cash flows to the frm s exstng cash flows. The frm s cost of captal therefore decreases. The results above vary one effect at a tme, holdng the others constant. But what f the expected cash flows and the covarance change smultaneously? For the specal case where expected cash flows and the covarance both change n exactly the same proporton, t s easy to see that the numerator and denomnator each change by that proporton n eqn. (4), and thus cancel out. In ths specal case, there s no effect on the cost of captal. There s an effect, however, for any smultaneous change that s not exactly proportonate on the two terms. Whle t s common n some corporate fnance and valuaton models to assume that the level of cash flow and the covarances move n exact proporton to each other (.e., all cash flows are from the same rsk class), we are unaware of any theoretcal results or emprcal evdence to suggest ths should be the case. On the contrary, the exstence of fxed costs n the producton functon, economes of scale, etc., generally make the expected values and covarances of frm s cash flows change n ways that are not exactly proportonal to each other. Moreover, there s ample emprcal evdence that betas vary over tme, whch mples the rato of expected cash flow to overall covarance vares, suggestng that new nformaton has an mpact as t becomes avalable. There s nothng n Proposton 1 that s specfc to accountng nformaton. Any shock new regulatons, taxes, nventons, etc. that affects the H term has a correspondng effect on the frm s expected return. In the followng two sectons we focus on how accountng nformaton mpacts the H(Φ) rato n the cost of captal equaton. In secton 3, we show how, holdng the 11

15 real decson of the frm fxed, accountng nformaton affects the assessments made by market partcpants of the dstrbuton of future cash flows, and how ths assessment mpacts the frm s cost of captal. In secton 4, we show that accountng nformaton affects real actons wthn the frm, and that ths naturally leads to changes n the rsk-return characterstcs of the frm, thereby affectng the frm s cost of captal. 3. Drect Effects of Informaton on the Cost of Captal In ths secton we hold constant the real (operatng, nvestng, and fnancng) decsons of the frm (we relax ths n Secton 4). Even though accountng and dsclosure polces do not affect the real cash flows of the frm here, they affect the assessments that market partcpants have regardng the dstrbuton of these future cash flows. As a result, they affect equlbrum stock prces and expected returns. In partcular, eqns. (3) and (4) show that stock prce and the expected return are, respectvely, decreasng and ncreasng functons of the covarance of a frm s end-of-perod cash flow wth the sum of all frms end-of perod cash flows. In the next two sub-sectons, we dscuss the two components of ths covarance: the frm s own varance and J k k= 1 k k ~ ~ ~ ~ ~ ~ the covarances wth other frms, Cov( V, V ) = Cov( V, V ) + Cov( V, V ). 3.1 Drect Effects Through the Varance of the Frm s Cash Flow The dea that better qualty accountng nformaton reduces the assessed varance of the frm s cash flow s well known. As an applcaton, consder the mpact on the cost of captal of frm f more nformaton becomes avalable (ether through more transparent accountng rules, addtonal frm dsclosure, or greater nformaton search by nvestors). Suppose the frm s nvestment decsons have been made; let V 0 and ω represent the ex-ante expected value and 12

16 ex-ante precson of the end-of-perod cash flow, respectvely. Suppose nvestors receve Q ndependently dstrbuted sgnals, z 1,...,z Q, about the ultmate realzaton of frm s cash flow, where each sgnal has precson cash flow has a normal dstrbuton wth mean γ. Then nvestors posteror dstrbuton for end-of-perod ~ ω γ Q E( V z 1,..., z Q) = + V0 + z + q = q, and precson ω + Qγ. ω Qγ ω Qγ 1 Therefore, usng part d of Proposton 1, the expected return s a decreasng functon of the precson, ω + Qγ. Ths mples that the frm s cost of captal declnes wth: 1) an ncrease n the pror precson, ω ; 2) the number of new sgnals, Q; or 3) the precson of these sgnals, γ. The frm-specfc varance reducton effect s an mportant factor n the cost of captal analyss of Easley and O Hara (2004). Whle ther paper models a mult-securty economy, the assumpton that all cash flows are ndependently dstrbuted mples that the prcng of each frm s also done ndependently. In partcular, f we smplfy ther model to remove the prvate nformaton component of ther model, ther prcng equaton reduces to: P ~ x 1 = E( V ), (5) Nτ ω + Qγ where x s the supply of the rsky asset (and the analyss assumes the rsk-free rate s zero). Therefore, the mpact of nformaton on the equlbrum prce s smlar to our eqn. (3). 7 As more publc sgnals are generated, the assessed varance of the frm s cash flows goes down. 7 In other work, we show that, contrary to ther clams, the other effect on the cost of captal n Easley and O Hara (2004) s not drven by the asymmetry of nformaton across nvestors. Instead, t s the average precson of nvestor s nformaton that s the determnant of the cost of captal n ther model. Moreover, the same argument we use here mples that ther nformaton effect, regardless of ther nterpretaton, s dversfable as the economy gets large. 13

17 The analyss above formalzes the noton that accountng nformaton and dsclosure reduce the assessed varance of the frm s end-of-perod cash flows. As eqn. (3) shows, ths s one of the components of the covarance of the frm s cash flow wth those of all frms. Therefore, ceters parbus, reducng the assessed varance of the frm s cash flows ncreases the frm s stock prce and reduces the frm s expected return. Moreover, because ths s an addtvely separate term n the overall covarance, the magntude of ths mpact on prce does not depend on how hghly the frm s cash flows co-vary wth those of other frms. For example, a decrease of, say, 10 percent n the assessed varance of frm cash flows has the same dollar effect on stock prce regardless of the degree of covarance wth other cash flows. Clearly, for a gven fnte value of N (the number of nvestors) and J (the number of frms n the economy), there s a non-zero effect on prce and on the cost of captal of reducng the assessment of frm-varance. Next we address the queston of the dversfablty (or magntude) of the effect of reducng the market s assessed varance of the frm s cash flows. Intutvely, the noton that a rsk s dversfable s usually expressed n terms of how t affects the varance of a portfolo as the number of frms n the portfolo gets large. 8 To examne ths more rgorously, we must ensure that economy-wde rsks are absorbed by the market partcpants collectvely, and economy-wde rsks are prced. Ths mples that J (the number of securtes) and N (the number of nvestors) must both get large. To see ths, consder as one polar case a stuaton n whch the number of frms n the economy ncreases, whle holdng the number of nvestors fxed. Ths does make the contrbuton of frm varance small relatve to the covarance wth all frms (assumng frms covarances tend to be postve). It also ncreases, however, the aggregate rsk 8 In partcular, the varance of an equally weghted portfolo of J securtes can be expressed as 1 1 J 1 R ~ AverageVar(R ~ Varance( AverageCov(R ~,R ~ ) = ) + k ). As J gets large ths converges to the average covarance J J J between the returns n the portfolo. The ndvdual varances of the frms returns asymptotcally dsappear. 14

18 n the economy: that s, J ~ ~ Cov( V, V k ) k= 1 ncreases wthout bound. Ths drves prces lower and results n an nfnte ncrease n the expected return requred to hold the stock (see eqns. [3] and [4]). On the other hand, consder as the other polar case a stuaton where N, the number of nvestors n the economy, spreads all rsks (not ust frms varances) over more nvestors, whch reduces all rsk premums and decreases all expected returns. In the lmt, J 1 ~ ~ Cov( V, V k ) approaches zero for each frm and even for the market; therefore, no rsks are N 1 k= prced. To avod these unnterestng, polar cases, J and N must both ncrease for the noton of dversfablty to be meanngful. When J and N both ncrease, the effect of frm-varance on the cost of captal, 1 ~ ~ Cov( V, V ), Nτ asymptotcally approaches zero, because ths term only appears once n the overall covarance for a frm. 9 1 ~ ~ The covarance wth other frms, however, Cov( V, V k ), Nτ k survves because the number of covarance terms (J-1) also ncreases as the economy gets large. In the next secton, we analyze how nformaton affects the covarance terms. 3.2 Drect Effect Through the Covarance wth other Frms Cash Flows In ths secton, we show that nformaton about a frm s future cash flows also affects the assessed covarance wth other frms. Our work n ths secton bulds on the estmaton rsk lterature n fnance (See Brown,1979; Barry and Brown;1984 and 1985; Coles and 9 In our smplfed verson of the Easley and O Hara result (our eqn. [5]), as N gets large, the last term on the rghthand sde of the equaton approaches zero. Therefore, the frm s prced as f t s rskless (recall that Easley and O Hara assume there are no covarances wth other frms). Smlarly, n ther full blown model (see ther proposton 2), as N gets large the per-capta supply of the frm s stock goes to zero, and agan the prcng equaton collapses to a rsk-neutral one. 15

19 Loewensten,1988; and Coles et al., 1995). Specfcally, our work dffers from ths lterature n three mportant ways. Frst, the estmaton rsk lterature generally focuses on the mpact of the nformaton envronment on the beta of the frm, whereas our focus s on the cost of captal. Because the nformaton structures analyzed n ths lterature generally affect all frms n the economy, the mpact on beta s confounded by the smultaneous mpact on the covarances between frms and the varance of the market portfolo. Ths s one reason why they obtan results that are mxed or dffcult to sgn. By focusng on the cost of captal, we can analyze the mpact of both effects. Second, the estmaton lterature focuses on very specfc changes n the nformaton envronment. Some papers examne the mpact of ncreasng equally the amount of nformaton for all frms. Other papers compare two nformaton envronments: an envronment where all frms have an equal amount of nformaton to an envronment where one subgroup of frms has more nformaton avalable than does a second group. Our framework allows us to analyze more general changes n nformaton structures: both mandatory and voluntary. In partcular, unlke the pror lterature, we are able to address the queston of how more nformaton about one frm affects ts own cost of captal wthn an unequal nformaton envronment. Fnally, our model represents the nformaton sgnals dfferently than n the estmaton rsk lterature. The estmaton rsk lterature assumes the nformaton sgnals avalable about the frms arse from hstorcal tme-seres observatons of the frms returns. Whle ths lterature clams that the ntuton behnd ther results apples to nformaton more generally, the assumed tme-seres nature of ther nformaton sgnals drves a substantal element of the covarance 16

20 structure n ther models. In partcular, new nformaton sgnals are condtonally correlated wth contemporaneously observed sgnals and condtonally ndependent of all other sgnals. 10 We model a more general nformaton structure that allows us to examne alternatve covarance structures of the sgnals. Specfcally, we model nformaton sgnals as representng nosy measures of the varables of nterest, whch are end-of-perod cash flows. That s, an nformaton sgnal, Z ~, about frm s cash flow, V ~, s modeled as ~ ~ Z ~ = V + ε, where ~ ε s the nose or measurement error n the nformaton sgnal. Dependng on the correlaton structure assumed about the cash flows and error terms, ~ Z, could also be nformatve about the cash flow of other frms, as well as nformatve n updatng the assessed varances and covarances of end-of-perod cash flows. Ths formulaton of nformaton s consstent wth the way nformaton s modeled n vrtually all conventonal statstcal nference problems (see DeGroot, 1970). It s also consstent wth vrtually all papers n the vast nosy ratonal expectatons lterature n accountng and fnance (see Verreccha, 2001, for a revew). Our characterzaton of dsclosures as nosy sgnals about frms future cash flows (or other performance measures) also corresponds to actual dsclosure practces. Frms dsclose nformaton about ther earnngs, whch s the sum of ther market, ndustry, and dosyncratc components. Smlarly, other dsclosures such as revenues or cash flow are also about the frm 10 See Kalymon (1971) for the orgnal dervaton of the covarance matrx used n much of ths lterature. 17

21 as whole. 11 Analysts forecasts of future earnngs are about the earnngs of the frm, not ust of the frm specfc component of future earnngs. 12 To provde a smple ntal example, suppose that the future cash flows for two frms have ~ ~ an ex-ante covarance of Cov ( V, Vk ), whch s non-zero. Suppose we observe an nformaton ~ ~ sgnal about frm, Z, whch s a nosy sgnal about frm s future cash flow, V. For convenence, assume that the error term ~ ε s uncorrelated wth ether of the true cash flows V ~ or V ~ k. As n the prevous secton, a greater precson of the nformaton sgnal Z ~ about frm ~ s cash flow leads to a smaller posteror varance of V. Moreover, we can also show that the nformaton sgnal Z ~ also leads to an updatng of the covarance between the two cash flows V ~ ~ and V k. In partcular, t s straghtforward to show the followng. Proposton 2. The covarance between the cash flows of frms and k condtonal upon a ~ ~ sgnal about frm s cash flow moves away from the uncondtonal Cov ( V, Vk ) and closer to zero as the precson of frm s sgnal ncreases. Specfcally, ~ ~ Cov( V, V k ~ ~ ~ Var( ε ) Z ) = Cov( V, V k ) ~. (6) Var( Z ) 11 In contrast, Hughes et al. (2005) model nformaton sgnals as beng artfcally decomposed nto market and dosyncratc factors. Moreover, n our model cash flows have a completely general varance-covarance structure, whereas the analyss n Hughes et al. assumes a very specfc factor structure. Smlarly, the betas and covarances that turn out to be relevant n our prcng equatons are relatve to the market portfolo (the sum of all frm s cash flows), whereas n Hughes et al. the betas and covarances are relatve to the exogenously specfed common factors. 12 Whle the dea that the earnngs of a frm can be useful n predctng future cash flows of the ndustry or the market as a whole mght seem counterntutve, there s emprcal evdence to support ths. See Potrosk and Roulstone (2004) about how the actvtes of market partcpants (analysts, nsttutonal traders, and nsders) mpacts the ncorporaton of frm specfc versus ndustry versus market components of future earnngs nto prces. See also the nformaton transfer lterature (e.g., Foster, 1981). As we show, t s not necessary that there be a large effect of frm s dsclosures on other frms. 18

22 Therefore, the condtonal covarance between V ~ and V ~ k s equal to the uncondtonal covarance, tmes a factor that can be nterpreted as the percentage of the varance of the nformaton sgnal that conssts of nose or measurement error. As the measurement error n Z ~ goes down, the assessed covarance between V ~ and V ~ k decreases (n absolute value). The ntuton s as follow. If there s nfnte measurement error n Z ~, then observng Z ~ does not communcate anythng. Therefore, there s no need to update an assessment of the uncondtonal varance of V ~, or the uncondtonal covarance between V ~ and V ~ k. At the other extreme, f there s no measurement error n Z ~, then observng Z ~ s the same as observng V ~. But f V ~ s observed, there s no further covarance between V ~ and V ~ k ; hence, the assessed covarance goes to zero. Note that Proposton 2 does not requre that the uncondtonal covarance between V ~ and V ~ k be postve; as the measurement error n Z ~ goes down, the assessed covarance between V ~ and V ~ k moves closer to zero, rrespectve of ts sgn. Proposton 2 apples equally to the condtonal covarances wth all other frms n the economy. Ths mples that J ~ ~ Cov( V, Vk k= 1 J Var( ~ ε ~ ) ~ ~ Var( ε ) ~ J ~ Z ) = ~ Cov( V, Vk ) = ~ Cov( V, Vk ). k= 1 Var( Z ) Var( Z ) k= 1 Therefore, the condtonal covarance between the cash flows of frm and those of the market as a whole s proportonal to the amount of measurement error n the nformaton sgnal about frm s cash flow. Moreover, ths effect does not dversfy away n large economes: the effect s present for each and every covarance term wth frm. 19

23 For example, suppose the dstrbuton of cash flows s represented as a sngle-factor ~ ~ ndex model, so that the cash flow for frm s V u~ = a + b θ +, where θ s a market factor and u s a frm specfc factor. For convenence, let all the u s be dstrbuted ndependently. Let the nformaton sgnal about frm be a nosy measure of ts cash flow, ~ ~ Z = V + ~ ε, where the error terms are dstrbuted ndependently of the true cash flows, as well as each other. Then the uncondtonal covarance between the cash flows of frms and k s b b k Varance( ~ θ ), and the condtonal covarance gven the sgnal Z s ~ ~ Cov( V, Vk ~ ~ Varance( ε ) Z ) = b bkvarance( θ ) ~, whch mples Varance( Z ) ~ ~ ~ ~ Varance( ε ) Cov( V, Vk Z ) = b Varance( θ ) ~ bk. Varance( Z ) k k As before, the posteror covarance between frm and the market decreases as the qualty of the nformaton about frm s future cash flow mproves. These fndngs are n contrast to those n a concurrent paper by Hughes et al. (2005), whch employs more restrctve and less natural nformaton structures. For example, Hughes et al. show that f the nformaton concerns exclusvely the dosyncratc component of a frm's cash flows, not the cash flows per se, and the nformaton sgnal matrx s exclusvely dagonal, then there s no covarance effect. That s, under these condtons, the nformaton sgnals are, by defnton, unrelated to the component of cash flows that vares across frms, so they cannot be useful n updatng the assessed covarance. Hughes et al. also consders an nformaton structure that relates only to the common factor porton of cash flows. In ths case, nformaton does affect the covarance between a frm s cash flows and the common factors. Smlarly, the covarance between the cash flows of 20

24 any two frms that are both affected by ths common factor wll also change. Whle ths result s smlar n some ways to ours, the nature of the cross-sectonal mpact on the covarance, and therefore the cost of captal, dffers n ther paper because of the dfferent nformaton structure assumed. When the nformaton s about the cash flow as a whole, we fnd that vrtually any more general representaton of V ~, V ~ k, Z ~ and/or Z ~ k results n Z ~ and Z ~ k changng the covarance of V ~ and V ~ k, whch s the effect that we obtan n our general varance-covarance framework. Thus, we clam that n general nformaton about frm cash flows (or other measures of frm performance) has a covarance effect, and hence leads to cross-sectonal dfferences n frms cost of captal. In partcular, at the extreme, f nformaton could fully reveal the frm s future cash flow, the cash flow would, by defnton, be rskless and the cost of captal on the frm s stock would converge to the rsk free rate. It s possble to expand the analyss n ths secton n several ways. For example, n Proposton 2, we refer to the covarance between the cash flows of frms and k condtonal on (a sngle) frm provdng nformaton about ts cash flow. In the proof of Proposton 2 n the appendx, we offer a more general result based on computng the covarance of cash flows condtonal on both frms and k provdng nformaton about ther respectve cash flows. The analyss could be generalzed further to allow all frms n the economy to provde nformaton about ther respectve cash flows. It s mportant to pont out that, even n the case where all frms provde nformaton, t s not necessarly the case that the uncertanty about the market cash 21

25 flow s elmnated. If each frm dscloses ts realzed cash flow wth nose, uncertanty about the market cash flow wll grow as the number of frms n the economy grows. 13 Another possble extenson s that n our paper (as s true n most of the ratonal expectatons lterature) we nterpret the sgnals as beng related to the realzed cash flow. We can also conduct the analyss by nterpretng them nstead as sgnals about the future expected cash flow (or parameters of the dstrbuton of future cash flows). In fact, most of the estmaton rsk lterature nterprets the sgnals n ths way. Of course, when learnng about the expected future cash flow, as opposed to the realzed future cash flow, a perfect sgnal does not resolve all uncertanty, so the asset does not become rskless. Nonetheless, better nformaton reduces the covarances. We can also repeat the analyss under alternatve assumptons regardng what parameters of the dstrbuton of future cash flows are uncertan: 1) the expected cash flows are unknown but the covarance matrx s known; or 2) the expected cash flows and the covarance matrx are both unknown. We could also allow the error terms ( ~ ε s) to be correlated across frms and examne the mplcatons of ths. 3.3 The Effects of Mandatory Dsclosures In the prevous sectons, we analyze the mpact of changng the qualty of accountng nformaton for a sngle frm on ts prce and cost of captal. We now brefly dscuss the effects of mandatory dsclosure of accountng nformaton. The man dfference s that dsclosure regulaton affects all frms. Therefore, n addton to the mpact of frm s dsclosure on, say, the covarance between the cash flows of frms and k, there s an addtonal mpact on ths covarance by frm k s dsclosures. In prncple, dsclosure by other frms can have a (small) 13 In contrast, f each frm dscloses the aggregate market cash flow wth dosyncratc nose, dsclosures by many frms reveals the market cash flow n the lmt; thus, there would be no aggregate uncertanty. Ths nformaton structure, however, s not very descrptve of what frms do. 22

26 mpact on the covarance wth frm. That s, each frm s dsclosure has an externalty on other frms cost of captal. Ths postve externalty provdes potentally a reason why there could be benefts to dsclosure regulaton, rather than relyng on voluntary dsclosures, because frms wll not take ths externalty nto account when decdng the optmal level of voluntary dsclosure. Whle ths effect s small ndvdually, t could become large collectvely. 14 Usng our framework, t s therefore straghtforward to show that ncreasng the qualty of mandated dsclosures reduces the cost of captal for all frms n the economy (assumng that the expected cash flow of each frm n the economy and the covarance of that frm s cash flow wth the market have the same sgn). The magntude of the mpact of mandatory dsclosure, however, on a partcular frm s cost of captal s less clear-cut. Even f mandatory dsclosures affect all elements of the covarance matrx of future cash flows by a scale factor, the effect on the cost of captal s not proportonate for all frms. It follows from the prcng formula n eqn. (2) that even f the mpact on the covarances of future cash flows s proportonate, the effect on the expected values of future cash flows s unlkely to have the same scale factor. Therefore, the prces of all frms wll not change proportonately. Thus, usng eqn. (4), the frms expected returns (and cost of captal) wll not change by the same proporton for all frms ether. 15 Moreover, t seems unlkely that mandated dsclosures alters the entre covarance matrx of all frms future cash flows by the same scale factor. The amount of new nformaton provded by a partcular mandated dsclosure depends on what other nformaton the frm already 14 See Admat and Pflederer (2000) and Fshman and Hagerty (1989) for other externalty-based explanatons of mandatory dsclosure. 15 Some early models n the estmaton rsk lterature n fnance (e.g., Kalymon, 1971; Brown, 1979) assumed that nformaton had a proportonate mpact on the covarance matrx of returns, and some found that nformaton would not affect the betas of frms. Ths result occurred because (by assumpton) all the covarances and varances of frms returns changed proportonately. Therefore, there was no effect on the beta coeffcent of returns, whch s the rato of the covarance of the frm s return to the market dvded by the varance of the market return, because the numerator and denomnator changed by the same scale factor. It s more natural to assume, however, that the mpact of nformaton s on the assessed dstrbuton of cash flows, not returns. See Coles and Lowensten (1988) for smlar dscusson. 23

27 dscloses. For some frms, ths dsclosure requrement duplcates other dsclosures, n whch case t provdes no addtonal nformaton; for others t may provde a small amount of ncremental nformaton, and for others stll t may be completely new. These nformaton effects mply an unequal mpact of dsclosure regulaton on the ndvdual elements of the covarance matrx of future cash flows. Thus, frms are lkely to beneft from mandatory dsclosures dfferentally. It s mportant to dstngush between the mpact of mandated dsclosure on the cost of captal and the mpact on the beta coeffcent. Our formulaton allows us to express the cost of captal n a reduced form that depends on assessed covarance between the cash flow of the frm and the cash flows of all frms n the market. Even though mandated dsclosure reduces the cost of captal for frms, ths does not mply that t reduces all beta coeffcents smlarly. Instead, the lowered covarance between end-of-perod cash flows mples a reducton n the product of the mpact on the market rsk premum and the beta coeffcent; t does not mply a reducton n the beta coeffcent separately. In fact, the average beta n the economy must stll be 1.0, regardless of the qualty of the nformaton envronment. Therefore, mandated dsclosure cannot reduce all frms betas. Ths suggests that researchers nterested n examnng the effect of mandated dsclosures on costs of captal must look to alternatve measures for cost of captal than the beta coeffcent. 4. Indrect Effects of Informaton on Cost of Captal In ths secton, we show how the qualty of the accountng and dsclosure system has an ndrect mpact the frm s cost of captal through ts effect on real decsons that mpact the expected cash flows and covarances of cash flows. Clearly, decson-makers n an economy make decsons on the bass of the nformaton they have avalable to them. If ths nformaton 24

28 changes, so wll ther decsons. To the extent these new decsons alter the dstrbuton of the frm s end-of-perod-cash flow, ths mpacts the frm s cost of captal. In partcular, as we showed n Proposton 1, f the rato of the frm s expected cash flow to the covarance between the frm s cash flow and the sum of all other frms cash flows changes (our parameter H n our Lemma 1), the expected rate of return on the frm s stock wll change, and by defnton, so wll ts cost of captal. The potental scope of the decsons that a frm s accountng and dsclosure systems may affect can be qute broad. In addton to the decsons of nvestors and credtors, a frm s accountng and dsclosure systems may affect manageral actons, as well as the potental actons of compettors, regulatory authortes, etc. The mpact on the frm s cost of captal can be ether postve or negatve. To be able to predct the ndrect effect of accountng nformaton on the frm s cost of captal therefore requres the researcher to carefully specfy: 1) the lnk between nformaton and these decsons; and 2) the mpact of these decsons on the dstrbuton of future cash flows. We provde two smple examples to llustrate these ssues. In one, accountng nformaton affects the amount of cash that s approprated from nvestors. In the second, accountng nformaton changes a manager s nvestment decsons. 4.1 Informaton, Governance, and Appropraton Many papers (e.g., n agency theory) have suggested that better fnancal reportng and/or corporate governance ncreases frm value by reducng the amount that managers approprate for themselves (e.g., LaPorta et. al., 1997; Lambert, 2001). These papers, however, do not dscuss the mpact that these systems have on the frm s cost of captal. In fact, some of them clam that these systems have a one-tme effect on prce, but do not affect the cost of captal as defned 25

29 by the dscount rate mplct n the frm s stock valuaton. An excepton s Lombardo and Pagano (2000) (LP), a paper that also analyzes the mpact of governance and appropraton on frm s cost of captal. We compare our model and results to LP below. ~ Represent the gross end-of-perod cash flow of frm before any appropraton by V *. As before, we assume ths cash flow has a normal dstrbuton. Managers approprate for themselves an amount A that decreases wth the qualty of the nformaton and/or governance systems. By accountng systems here, we do not mean smply the dsclosures the frm makes to outsders, but also nternal control systems, corporate governance polces, etc. 16 Clearly, a more sophstcated analyss could be conducted that endogenously derves the functonal form of A and/or determnes the types of nformaton that lead to the largest reductons n A, but these features are not necessary to make our pont. In partcular, the amount of the frm s end-of-perod cash flows that s approprated from shareholders s A Q = A Q + A Q V * ( ) 0 ( ) 1( ), where Q s the qualty of the accountng system, etc., and V* s the value of the frm gross of any appropraton. 17 A hgher qualty accountng system s assumed to reduce the amount of msappropraton: A 0 0 and A 1 0. Ths means that the net amount receved by the frm s shareholders s ~ V = V ~* ( ) (1 )) ~* A Q = A1( Q V A0( Q). 16 More generally, the amount a manager could approprate would also depend on dmensons of the legal system, such as the ablty to brng lawsuts aganst the frm and/or managers. 17 LP analyze a model that also has a parameter analogous to our A 1, whch reduces the cash flow avalable to shareholders by a fractonal amount. LP do not have a parameter, however, that corresponds exactly to our A 0, whch reduces the end-of-perod cash flow by a constant. Instead LP have a cost that shareholders pay out of ther own pocket for audtng and legal fees. Ths fee s proportonal to the share prce, whch obvously depends on the expected end-of-perod cash flow. Another dfference s that n LP appropraton parameters are economy-wde (or for segments of the market), whereas ours can be frm-specfc. Fnally, we explctly analyze the mpact the appropraton parameters have on the rskness of the end-of-perod cash flow (varances and covarances), whereas these are not ssues addressed n LP. 26