Journal of Banking & Finance

Journal of Bankng & Fnance 35 (011) 1563 1580 Contents lsts avalable at ScenceDrect Journal of Bankng & Fnance journal omepage: www.elsever.com/locate/jbf Does more rmaton n stock prce lead to greater or smaller dosyncratc return volatlty? Dong Wook Lee a, Mark H. Lu b, a Korea Unversty Busness Scool, Seoul 136-701, Republc of Korea b Gatton College of Busness and Economcs, Unversty of Kentucky, Lexngton, KY 40506, USA artcle abstract Artcle story: Receved 3 February 010 Accepted 1 November 010 Avalable onlne 11 November 010 JEL classfcaton: G1 G14 Keywords: Idosyncratc volatlty Nosy ratonal expectatons equlbrum Prce rmatveness We nvestgate te relaton between prce rmatveness and dosyncratc return volatlty n a multasset, mult-perod nosy ratonal expectatons equlbrum. We sow tat te relaton between prce rmatveness and dosyncratc return volatlty s eter U-saped or negatve. Usng several prce rmatveness measures, we emprcally document a U-saped relaton between prce rmatveness and dosyncratc return volatlty. Our study terefore reconcles te opposng vews n te followng two strands of lterature: (1) te growng body of researc sowng tat frms wt more rmatve stock prces ave greater dosyncratc return volatlty, and () te studes argung tat more rmaton n prce reduces dosyncratc return volatlty. Ó 010 Elsever B.V. All rgts reserved. 1. Introducton Tere s a recent debate n te lterature about te relaton between prce rmatveness and dosyncratc return volatlty. A growng body of studes sow tat more rmatve stock prces are assocated wt greater dosyncratc return volatlty. For example, Morck et al. (000) fnd tat stocks n countres wt stronger property rgts ave ger dosyncratc volatlty. Tey argue tat strong property rgts promote rmed arbtrage, wc captalzes frm-specfc rmaton and ncreases dosyncratc return volatlty. 1 Many emprcal researcers use greater dosyncratc return volatlty as a measure of more rmatve stock prces (Brockman and Yan, 009). Anoter group of studes ave contradctng vews on ts ssue and argue tat ger dosyncratc return volatlty s an ndcaton of more nose and prcng errors n stock prces (.e., less rmatve stock prces). For example, West (1988) sows tat more rmaton n prce about future dvdends leads to lower dosyncratc volatlty. Kelly (005) fnds tat frms wt better rmaton envronments are assocated wt ger market-model R-square,.e., smaller dosyncratc return volatlty. Correspondng autor. Tel.: +1 859 57 984; fax: +1 859 57 9688. E-mal addresses: donglee@korea.ac.kr (D.W. Lee), mark.lu@uky.edu (M.H. Lu). 1 Jn and Myers (006), wo fnd tat more transparent frms ave ger dosyncratc return volatlty, argue tat poor nvestor protecton alone cannot explan te dfference n dosyncratc volatlty, and opaqueness of te frm also plays an mportant role. Many emprcal studes use greater dosyncratc return volatlty as a measure of less rmatve stock prces and greater asymmetrc rmaton between frm nsders and outsders (Krsnaswam and Subramanam, 1999). Understandng te true relaton between prce rmatveness and dosyncratc return volatlty s mportant, gven tat an ncreasng number of studes use dosyncratc volatlty as a prce rmatveness measure, and te contradctng assumptons tey make about te rmaton content of dosyncratc return volatlty. Furter, understandng te true rmaton content of dosyncratc volatlty s mportant for practtoners and polcy makers as well. For example, some crtcs of Regulaton FD argue tat te regulaton as caused less dsclosure by frms and as led to ncreases n stock return volatlty, mplctly assumng tat less rmaton n stock prce leads to ger dosyncratc return volatlty. If ger dosyncratc volatlty s not an ndcaton of less rmaton n stock prces, ten Regulaton FD may not ave te negatve mpact on corporate dsclosure as crtcs clam. 3 See, e.g., Dsclosure Rule Gets A Bad Rap by Jeff Opdyke and Mcael Scroeder (te Wall Street Journal, June 5, 001, page C1) and Deals & Deal Makers: Drect Effects of Dsclosure Rule Doubted by Pylls Pltc (te Wall Street Journal, July 4, 001, page C14). 3 We, owever, do not take any sde n te debate regardng te effectveness of Regulaton FD. We argue only tat greater dosyncratc volatlty may not be an ndcaton of less prce rmatveness. 0378-466/$ - see front matter Ó 010 Elsever B.V. All rgts reserved. do:10.1016/j.jbankfn.010.11.00

1564 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 Ts paper makes an attempt to understand te relaton between prce rmatveness and dosyncratc return volatlty. We examne dosyncratc return volatlty n a mult-asset, mult-perod nosy ratonal expectatons model. Stocks are traded among tree groups of nvestors: (1) lqudty traders, wose demand for a stock s exogenous and does not depend on te fundamental value of te stock; () rmed traders, wo ncur a cost and obtan a nosy sgnal about te value of te stock and trade based on ts prvate sgnal; (3) unrmed dscretonary traders (UDTs), wo do not ave prvate rmaton about te value of te stock, but nfer te rmaton conveyed by te stock prce, and wose demand for te stock depends on te prce. Te equlbrum stock prce s affected by bot nose and te fundamental value of te stock. We decompose dosyncratc return volatlty nto two parts: te nose component, wc s caused by te demand of lqudty traders, and te rmaton component, wc s drven by te rmaton regardng te fundamental value of te stock. We furter decompose te rmaton component of dosyncratc return volatlty nto two parts. Te frst part, wc we refer to as te rmaton updatng part, represents te fluctuaton n prce as prvate rmaton about te fundamental value of te stock s ncorporated nto prce. Te second part, wc we refer to as te uncertanty resolvng part, represents te fluctuaton n prce due to te resoluton of te remanng uncertanty n te stock value (.e., troug te realzaton of te fundamental stock value tat was not prevously reflected n prce). We sow tat te rmaton component of dosyncratc return volatlty frst decreases and ten ncreases wt prce rmatveness for te followng reason. As more nvestors coose to produce rmaton and more rmaton s ncorporated nto te stock prce, prce rmatveness ncreases. Ts, n turn, ncreases te rmaton updatng part of dosyncratc return volatlty. As more rmaton s reflected n prce, less uncertanty remans about te value of te stock, and te uncertanty resolvng part of dosyncratc return volatlty becomes smaller. Terefore, te rmaton updatng part ncreases wt prce rmatveness wereas te uncertanty resolvng part decreases wt prce rmatveness. Te rmaton component of dosyncratc return volatlty, wc s te sum of te rmaton updatng part and te uncertanty resolvng part, as a U-saped relaton wt prce rmatveness snce te average varance over tme s te lowest wen te uncertanty s resolved gradually. To use a numercal example to llustrate ts ntuton, suppose tat te value of a stock sould double from $1 to $ n two perods. Consder tree dfferent scenaros: (1) Te stock prce s extremely unrmatve and all rmaton wll be revealed only n te second perod. Te stock return wll be 0 n te frst perod and 100% n te second perod. Te varance of return n ts case s 1 ½ð0 50%Þ þ ð100% 50Þ Š¼0:5; () Informaton s ncorporated nto prce gradually so tat te stock prce rses to $1.5 at te end of te frst perod, ten stock return wll be 50% n te frst perod and 33.3% n te second perod. Te varance of return n ts case s 1 ½ð50% 41:65%Þ þð33:3% 41:65%Þ Š¼0:007; (3) Te stock prce s extremely rmatve and all rmaton wll be revealed n te frst perod, ten stock return wll be 100% n te frst perod and 0 n te second perod. Te varance of return n ts case s 1 ½ð100% 50Þ þð0 50%Þ Š¼0:5. Hence a U-saped relaton between prce rmatveness and te rmaton component of return volatlty. We also sow tat te nose component of dosyncratc return volatlty decreases monotoncally wt prce rmatveness. Ts s because as more nvestors coose to produce rmaton, lqudty tradng as a lower mpact on stock prce snce rmed nvestors can better absorb lqudty traders order flows, wc makes te stock prce less nosy. Ts, n turn, reduces te nose component of dosyncratc return volatlty. Terefore, te nose component of dosyncratc return volatlty decreases wt prce rmatveness. Our man teoretcal results on te relaton between prce rmatveness and dosyncratc volatlty are as follows. Frst, tere exst no parameter values suc tat dosyncratc return volatlty ncreases monotoncally wt prce rmatveness. Second, tere exst parameter values suc tat te relaton between prce rmatveness and dosyncratc return volatlty s U-saped. Ts appens wen te varance of te demand from lqudty traders s relatvely small compared to te varance n te frm s fundamental value. Fnally, tere exst parameter values suc tat dosyncratc return volatlty decreases monotoncally wt prce rmatveness. Ts appens wen te varance of te demand from lqudty traders s relatvely large compared to te varance n te frm s fundamental value. Emprcally, we fnd a U-saped relaton between prce rmatveness and dosyncratc volatlty. Our sample spans from 1983 to 004, wt slgtly more tan 3000 US stocks n a gven year. Te U-saped relaton s observed n vrtually every year, and t s robust to usng as many as sx dfferent measures of prce rmatveness (wc wll be detaled n Secton 4.1.3). Our results contrbute to te lterature n several ways. Frst, ts s te frst study to teoretcally model and emprcally document a U-saped relaton between prce rmatveness and dosyncratc volatlty. Prevous studes focus manly on a monotonc relaton between te two. Second, ts paper elps us better understand te source of dosyncratc return volatlty: bot nose and rmaton nfluence return volatlty, but wle te nose component decreases monotoncally wt prce rmatveness, te rmaton component frst decreases and ten ncreases wt prce rmatveness. Te rmaton component of dosyncratc return volatlty can be furter decomposed nto two parts: te part representng te fluctuaton n prce as prvate rmaton about te fundamental value of te stock s ncorporated nto prce, and te part representng te fluctuaton n prce due to te resoluton of te remanng uncertanty about te value of te stock wen te true frm value s revealed. Fnally, our results resonate wt some studes argung tat dosyncratc return volatlty s not a good measure of ow muc rmaton s reflected n stock prces; see, e.g., Asbaug-Skafe et al. (005). In oter words, our results sow tat researcers must be cautous wen usng dosyncratc return volatlty as a measure of prce rmatveness, snce te relaton between prce rmatveness and dosyncratc volatlty may not be monotonc. Te rest of te paper s structured as follows. We relate our work to te exstng lterature n Secton. Secton 3 develops te teoretcal model. Secton 4 reports emprcal results. We conclude n Secton 5. All proofs are confned to Appendx A.. Related lterature Ts paper s related to te teoretcal work on te relaton between prce rmatveness and dosyncratc return volatlty. Jn and Myers (006) develop a model n wc frm nsders can capture part of operatng cas flows, wc cannot be perfectly observed by outsders. Tey sow a postve relaton between prce rmatveness and dosyncratc volatlty. In ter model, te true frm value s never revealed to te market. 4 As a result, te uncertanty resolvng part of te dosyncratc return volatlty s 4 Wle some rmaton about frm value wll be revealed to te publc at te end of a perod (e.g., quarterly earnngs and cas flows), oter rmaton wll never be observed by nvestors (e.g., te ntrnsc value of te frm and management effort). Terefore, assumptons n bot Jn and Myers (006) and our model are consstent wt te realty to some degree. Our assumpton s sared by many oter studes n te lterature (Kyle, 1985).

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1565 not modeled n Jn and Myers (006), and ts explans te dfference n results between ter work and ts paper. In a one-perod nosy ratonal expectatons model wt multple assets, Ozoguz (005) sows a negatve relaton between prce rmatveness and dosyncratc return volatlty. Her defnton of dosyncratc return volatlty captures only te uncertanty resolvng part, but not te rmaton updatng part of dosyncratc return volatlty. In contrast, we use a dynamc model and capture bot parts of te dosyncratc return volatlty tat are caused by rmaton, and we sow tat te rmaton updatng part ncreases wt prce rmatveness, consstent wt Jn and Myers (006), and te uncertanty resolvng part decreases wt prce rmatveness, consstent wt Ozoguz (005). A growng number of studes focus on te propertes of te market-model R-square of common stocks and dosyncratc volatlty. Roll (1988) observes tat only a small proporton of te actual prce movements of ndvdual common stocks can be explaned by market and ndustry nfluences. Campbell et al. (001) fnd tat dosyncratc volatlty of common stocks n te Unted States as ncreased sgnfcantly durng te past few decades. Morck et al. (000) fnd tat frms n developed countres ave ger dosyncratc volatlty compared wt frms n developng countres, and tey argue tat ts s due to te poor protecton of nvestors property rgts n developng countres. Ts paper jons te current debate on weter ger dosyncratc volatlty means more or less rmaton n stock prces. Our results reconcle te opposng vews expressed n te two strands of lterature we ave mentoned n Secton 1. Unlke te exstng lterature, we fnd a non-monotonc relaton between stock prce rmatveness and dosyncratc volatlty. Exstng studes test only a monotonc relaton between te two, but ter results do not necessarly contradct ours. For example, Morck et al. (000) compare dosyncratc volatlty n dfferent countres. Snce rmaton s more effcent n te ndustry and country level tan n te frm-specfc level, ter results fall n te rgt alf of te U- sape we ave documented, and ts may explan te postve relaton between stock prce rmatveness and dosyncratc volatlty n ter studes. Kelly (005) uses frm-specfc data n te US, but e uses raw prce rmatveness measures nstead of resdual measures as we do. Ts explans wy Kelly (005) fnds a negatve relaton between prce rmatveness and dosyncratc volatlty, wle we fnd a U-saped relaton. 5 As we wll explan later n Secton 4.1.4, snce proftablty volatlty and frm sze produce a wde spread of dosyncratc volatlty and prce rmatveness, we need to control for tese two factors to solate te effect of prce rmatveness on dosyncratc return volatlty. Fnally, tere s an ongong debate n fnance on te relaton between dosyncratc volatlty and expected stock returns. Ang et al. (006) and Guo and Savckas (010) fnd a negatve cross-sectonal relaton between dosyncratc volatlty and subsequent stock returns. Guo and Savckas (008) fnd tat dosyncratc volatlty as negatve predctve power for aggregate stock market returns over tme n G7 countres. In contrast, Fu (009) fnds a postve relaton between dosyncratc volatlty and contemporaneous stock returns usng te exponental GARCH models. Goyal and Santa-Clara (003) fnd tat equal-wegted total volatlty (manly dosyncratc) s postvely related to future stock market returns. Even toug our focus s on ow dosyncratc volatlty s related to prce rmatveness nstead of expected stock returns, we contrbute to te lterature by elpng researcers better understand te sources of dosyncratc volatlty. Specfcally, we sow tat dosyncratc volatlty can be caused by eter nose or uncertanty 5 In unreported results, we also fnd a negatve relaton between raw prce rmatveness and dosyncratc volatlty, smlar to Kelly (005). n te fundamental value, and te latter can be furter decomposed nto rmaton updatng volatlty and uncertanty resolvng volatlty. 3. Te model Consder an economy wt one rskfree asset and N + 1 rsky assets. Assets 1 to N are ndvdual stocks and asset M s te market ndex. All assets lve for T perods. Te lqudaton value of asset n at tme T s V n;t ¼ V n þ XT d ; for n ¼ 1;...; N; M: V n s te expected lqudaton value of asset n, wc s announced at tme 0. d n,t s te nnovaton on te value of asset n n perod t, wc becomes known to te publc at tme t. We assume tat for te N ndvdual stocks, d n,t s nfluenced by a systematc component, m t, and an dosyncratc component, f n,t, d ¼ b n m t þ f ; for n ¼ 1;...; N; were b n s te senstvty of d n,t wt respect to m t, and m t and f n,t are ndependent of eac oter and over tme wt te followng dstrbutons: m t Nð0; r m Þ; and f Nð0; r f ;nþ: ðþ For te market ndex, we ave d M;t ¼ m t : Te rsky asset n {1,...,N,M} as a pyscal supply of Y n. For smplcty, we assume tat te rskfree asset s n perfectly elastc supply, and te net return on t s normalzed to zero. 3.1. An equvalent economy Gven te structure of te payoffs, we can consder an equvalent representaton of te orgnal economy, smlar to Ozoguz (005). In te equvalent economy, tere are N + 1 rsky assets. Te lqudaton value of asset n at tme T s u n;t ¼ u n þ XT g ; for n ¼ 1;...; N; N þ 1; were u n ¼ V n b n V M and g n,t = f n,t for n =1,...,N, and u Nþ1 ¼ V M and g N+1,t = m t. Tat s, asset n {1,...,N} n te equvalent economy s equvalent to a portfolo of one sare of stock n plus b n sares of te market ndex n te orgnal economy. Terefore, n te equvalent economy, te pyscal supples of te stocks are y n = Y n for n =1,...,N, and y Nþ1 ¼ Y M þ P N n¼1 b ny n. For smplcty, we call te lqudaton value of asset n at tme T as te fundamental value of asset n,.e., u n = u n,t. We wll frst focus our analyss on te equvalent economy and derve stock prces and return volatlty. Later, we wll go back to te orgnal economy and derve stock prces and dosyncratc return volatlty n te orgnal economy. 3.. Investors rmaton producton and utlty maxmzaton problem Tree types of traders (nvestors) are n te market: lqudty traders, rmed traders, and unrmed dscretonary traders (UDTs). Te aggregate demand (n terms of number of sares) from lqudty traders for asset n {1,...,N,N + 1} at tme t {0,1,..., T 1} s z n,t+1, wc as te followng dstrbuton: z þ1 Nð0; r z;n Þ: We furter assume tat z n,t+1 s ndependent across te N + 1 securtes and across tme. ð1þ ð3þ ð4þ

1566 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 Tere s a contnuum of UDTs over te nterval [0,1]. UDTs do not know te value of g n,t+1 at tme t. Tey can, owever, coose to eter reman unrmed or acqure a nosy sgnal about g n,t+1 at a cost and become rmed. 6 Specfcally, te value of g n,t+1 can be decomposed nto two parts g þ1 ¼ þ1 þ þ1 ; were n,t+1 and n,t+1 are ndependent of eac oter, across securtes, and across tme, 7 wt te followng dstrbutons: þ1 Nð0; r ;n Þ; and þ1 Nð0; r ;nþ: ð6þ By ncurrng a cost of C n at tme t, a UDT observes te value of n,t+1 and becomes an rmed trader. All nvestors ave te same exponental utlty functon of consumpton over te tme T wealt, W T : uðw T Þ¼ e aw T were a s te absolute rsk-averson coeffcent. At eac tme t {0,1,...,T 1}, te followng events occur sequentally: (1) nvestor j starts wt X j t ¼ðXj 1;t ;...; Xj Nþ1;t Þ0 sares of stocks and B j t dollars of cas (te rskfree asset); () te nvestor decdes weter to acqure rmaton on eac of te N + 1 stocks; (3) tradng takes place, and stock prces P t =(P 1,t,...,P N+1,t ) 0 are determned n equlbrum; (4) te nvestor carres X j tþ1 sares of stocks and Bj tþ1 dollars of cas to te next perod. We use I j to denote nvestor j s decson on weter to acqure rmaton about stock n at tme t, wt value 1 f yes and 0 oterwse. Te cas oldng of nvestor j canges over tme as follows: B j tþ1 ¼ Bj t þ P0 t ðxj t Xj tþ1 Þ ðij t Þ0 C; were I j t ¼ðIj 1;t ;...; Ij Nþ1;t Þ0 and C =(C 1,C,...,C N+1 ) 0. At tme T, te nvestor s fnal wealt s W j T ¼ Bj T þ u0 T Xj T : were u =(u 1,...,u N+1 ) 0 s te vector of te fundamental values. 3.3. Equlbrum n te last perod At tme T, all rmaton becomes publc, so stock prces are as follows: P n;t ¼ u n;t for n ¼ 1;...; N; N þ 1: At tme T 1, a number of l n,t 1 (0,1) UDTs acqure rmaton about stock n and become rmed. Te followng proposton summarzes te equlbrum stock prces at tme T 1: Proposton 1 (Stock prces n te last perod). Te prce for stock n {1,...,N,N+ 1} at tme T 1s P n;t 1 ¼ u n þ XT 1 g þ a n;t 1 n;t þ a z n;t 1 ðz n;t y n Þ; were a n;t 1 > 0 and az n;t 1 > 0 are gven n Eqs. (A.1) and (A.13). 6 Even toug we assume tat te stock value s affected by bot te market return and te dosyncratc component, our model setup s dfferent from CAPM n te sense tat CAPM assumes no asymmetrc rmaton among nvestors. In contrast, our model falls nto te lterature examnng te asset prcng mplcatons under asymmetrc rmaton. Weter CAPM olds under asymmetrc rmaton s an nterestng researc topc, but t s out of te scope of te current paper. We tank an anonymous referee for pontng ts out. 7 Snce g n,t+1 s are te dosyncratc components of te N securtes and te market factor n te orgnal economy, tey sould be ndependent of eac oter crosssectonally by defnton. It s not unduly restrctve to assume tat g n,t+1 s are ndependent over tme snce tey are te nnovatve parts of te fundamental values at tme t +1. ð5þ ð7þ ð8þ ð9þ Te equlbrum prce of stock n at tme T 1 s a lnear combnaton of te prvate rmaton eld by rmed traders, n,t, and te demand from lqudty traders, z n,t. Te prce partally ncorporates te prvate rmaton but does not fully reveal t, snce te prce s also nfluenced by z n,t. Te UDTs can only nfer part of te prvate rmaton troug prce. 3.4. Decomposton of return volatlty Te followng proposton summarzes te prce functon at tme t {0,1,...T }. Proposton (Equlbrum stock prces). Te prce of stock n {1,...,N,N + 1} at tme t {0,1,...T } s P ¼ u n þ Xt s¼1 g n;s þ a þ1 þ a z ðz þ1 y n Þ; ð10þ were a and az are caracterzed n Eqs. (A.4) and (A.5). At any tme from 0 to T, te stock prce s a lnear combnaton of te prvate rmaton eld by rmed traders, n,t+1, and te demand by lqudty traders, z n,t+1. Te prce partally ncorporates te prvate rmaton but does not fully reveal t. UDTs decde weter or not to produce rmaton on eac of te N + 1 stocks at tme t. Snce te margnal beneft from acqurng rmaton on stock n at tme t decreases wt te number of nvestors wo coose to acqure rmaton, equlbrum s reaced wen te margnal beneft equals te margnal cost of acqurng rmaton on stock n. Now, we defne te return of stock n n perod t as r ¼ P P 1 ¼ð1 a 1 Þ þ a þ1 þ þ a z ðz þ1 y n Þ þ a z 1 ðz y n Þ: Te varance of te stock return s terefore Varðr Þ¼ð1 a 1 Þ r ;n þða Þ r ;n þ r ;n þðaz Þ r z;n þða z 1 Þ r z;n : Te frst component, 1 a 1 r ;n, s caused by te realzaton of te rmaton about n,t tat s not prevously reflected n prce P n,t 1. Te second component, a r ;n, s caused by te ncorporaton of rmaton about n,t+1 n prce P n,t. Te trd component, r ;n, s caused by te mperfecton of rmaton producton about g n,t+1,.e., te nose n te rmaton producton process. Te fourt and fft components, ða z Þ r z;n and az 1 r z;n ; are caused by lqudty tradng at tme t and tme t 1, respectvely. Instead of focusng on te tme perod, we focus on te sources of te return volatltes, n,t+1, n,t+1, and z n,t+1. Te stock return from tme t to tme t +1,r n,t+1,s r þ1 ¼ 1 a þ1 þ a þ1 þ þ þ1 þ a z þ1 ðz þ y n Þ þ a z ðz þ1 y n Þ: In r n,t, a þ1 þ a z ðz þ1 y n Þ s caused by ( n,t+1,z n,t+1 ), and n r n,t+1, ð1 a Þ þ1 þ þ1 þ a z 1 ðz y n Þ s caused by ( n,t+1, n,t+1,z n,t+1 ). We defne nstead r 0 ¼ a þ1 þ a z ðz þ1 y n Þ þ ð1 a Þ þ1 þ þ1 þ a z 1 ðz y n Þ Ts newly defned return process reflects te return components caused by ( n,t+1, n,t+1,z n, t+1 ) only. Note tat te sum of r 0 over tme equals te sum of r n,t over tme. Terefore, we wll focus on ts return nstead. Te varance of te newly defned return s

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1567 Varðr 0 Þ¼ ð1 a Þ þða Þ r ;n þ r þ ðaz Þ r z;n Te frst part of te return volatlty, V Info ðr 0 Þ¼ 1 a þ a r ;n ; s te return volatlty caused by rmaton, correspondng to te rmaton component of return volatlty tat we refer to n te ntroducton. Te second part, V Nose ðr 0 Þ¼r þ ðaz Þ r z;n ; s te return volatlty caused by nose, correspondng to te nose component of dosyncratc return volatlty tat we refer to n te ntroducton. Furtermore, te rmaton component of return volatlty as two parts: te frst part, a r ;n, s te rmaton updatng part, wc s caused by te ncorporaton of prvate rmaton n te stock prce; te second part, 1 a r ;n, s te uncertanty resolvng part, wc s caused by te realzaton of te resdual prvate rmaton tat was not prevously reflected n te stock prce. Te followng proposton sows tat te rmaton component of return volatlty frst decreases wt te number of rmaton producers, l n,t, and ten ncreases wt l n,t. Tat s, V Info r 0 as a U-saped relaton wt respect to l n,t. Te nose component of return volatlty decreases monotoncally wt l n,t. Proposton 3 (Relaton between te number of rmed traders and te rmaton and nose components of return volatlty). () Tere s a U-saped relaton between te number of rmaton producers and te rmaton component of return volatlty. Tat s, for every stock n f1; ;...; N þ 1g; @V Info ðr 0 Þ @l < 0 for l ½0; ^l Þ and @VInfo ðr 0 Þ @l P 0 for l ½^l ; 1Š, were Fg. 1. Relaton between te fracton of rmed traders and rmaton component return volatlty. Example 1. Assume te followng parameter values: r,n =, r z,n =1,r,n = 0.5, and a = 1. We frst look at te relaton between te fracton of rmed traders and te rmaton component of return volatlty. Wen te fracton of rmed traders ncreases from 0% to around 40%, te rmaton component of return volatlty decreases monotoncally. Wen te fracton of rmed traders ncreases from 40% to 100%, te rmaton component of return volatlty ncreases monotoncally. Fg. 1 sows te U-saped relaton between te fracton of rmed traders and te rmaton component of return volatlty. We ten look at te relaton between te fracton of rmed traders and te nose component of return volatlty. Wen te fracton of rmed traders ncreases from 0% to 100%, te nose component of return volatlty decreases monotoncally. Fg. sows te relaton between te fracton of rmed traders and te nose component of return volatlty. 8 3.5. Relaton between prce rmatveness and return volatlty We defne prce rmatveness as te uncertanty reducton n te value of te stock due to te knowledge of te prce: ^l ¼ rffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff a r z;n R þ1 þ a r z;nr R ;n þ1 þ 4r ;n a r z;n R þ1 r ;n a r z;n R þ1 þ a r z;nr ;n R þ1 : ð11þ () Te nose component of return volatlty decreases monotoncally wt te number of rmaton producers. Tat s, for every stock n f1; ;...; N þ 1g; @VNose ðr 0 Þ @l < 0 for any l n,t [0,1]. Te rmaton updatng part of return volatlty, a r, ncreases wt te number of rmaton producers, l n,t. Te uncertanty resolvng part of return volatlty, 1 a r, decreases wt l n,t. Te sum of te two parts, wc s te rmaton component of return volatlty, s te lowest wen te number of rmaton producers, l n,t, s moderate so tat neter part s too large. In contrast, wen te number of rmaton producers s eter too large or too small, te sum s large. Terefore, we ave a U-saped relaton between te number of rmaton producers and rmaton component of return volatlty. We want to empasze tat te U-saped relaton olds for all parameter values, snce ^l s always between 0 and 1. Te nose component of return volatlty decreases monotoncally wt l n,t snce more rmaton producers reduce te mpact of nose on prce and reduce te nose component of return volatlty. We use a numercal example ere to sow our results n Proposton 3. W ¼ 1 Var tðu n jp Þ : ð1þ Var t ðu n Þ were Var t (u n jp n,t ) s te varance of stock n s fundamental value condtonal on te stock prce, P n,t, wle Var t (u n ) s te uncondtonal varance of stock n s value at tme t wtout observng te stock prce. 9 Ts measure captures te fracton of rmaton about te stock s fundamental value tat s ncorporated nto te prce. For example, wen te prce ncorporates no rmaton at all, ten Var t (u n jp n,t )=Var t (u n ) and W n,t = 0. In contrast, wen te prce s fully revealng, ten Var t (u n jp n,t ) = 0 and W n,t =1. Te followng proposton caracterzes te prce rmatveness of stock n, and ts relaton wt te number of rmaton producers n te economy and te rmaton producton cost. Proposton 4 (Relaton between te prce rmatveness and te number of rmed traders). Te prce rmatveness of stock n at tme t s 8 Because many of te parameters n our model (e.g., r ;n and r ;n ) are not drectly observable, our model predctons are mostly qualtatve nstead of quanttatve. 9 Kyle (1985) nterprets and defnes prce rmatveness n a smlar fason.

1568 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 Fg.. Relaton between te fracton of rmed traders and nose component return volatlty. Fg. 3. Relaton between prce rmatveness and return volatlty wen te varance of te nose s low. W ¼ a r 4 ðt tþ r þ r a r þ az r ; ð13þ z wc ncreases wt te number of rmed traders about stock n, l n,t, and decreases wt te rmaton producton cost for stock n, C n. Tat s, @W @l > 0 and @W < 0. @Cn As more UDTs coose to produce rmaton and become rmed traders, te prce becomes more rmatve. Ts s because te demand of te rmed traders reflects te fundamental values of te stock and ts rmaton s partally ncorporated nto te prce. Snce n equlbrum te fracton of rmed traders n te economy decreases wt te rmaton producton cost, te prce rmatveness decreases wt te rmaton producton cost. Te followng proposton summarzes te relaton between prce rmatveness and return volatlty. Proposton 5 (Relaton between prce rmatveness and return volatlty). () No parameter values exst suc tat te return volatlty ncreases monotoncally wt te prce rmatveness @Varðr 0 Þ P 0 for all W @W ; () Tere exst parameter values suc tat te relaton between te return volatlty and te prce @Varðr rmatveness s U-saped 0 Þ < 0 for W < W e and @Varðr 0 Þ @W @W P 0 for W P W e Þ ; () Tere exst parameter values suc tat te return volatlty decreases monotoncally wt te @Varðr prce rmatveness 0 Þ @l P 0 for all W. We use a numercal example ere to sow tat parameter values exst so tat tere s a U-saped relaton between prce rmatveness and return volatlty, as predcted by part () of Proposton 5. Example. Assume te followng parameter values: r,n =, r z,n =1,r,n = 0.5, T t = 1, and a = 1. Wen te prce rmatveness ncreases from 0% to around 70%, te return volatlty decreases monotoncally. Wen te prce rmatveness ncreases from 70% to 100%, te return volatlty ncreases monotoncally. Fg. 3 sows te U-saped relaton between te prce rmatveness and te return volatlty. In te above example, te varance of te value of rmed traders prvate rmaton s greater tan te varance on te demand of lqudty tradng ( r ;n ¼ 4 > r z;n ¼ 1), and we observe Fg. 4. Relaton between prce rmatveness and return volatlty wen te varance of te nose s g. a U-saped relaton between prce rmatveness and return volatlty. In te next example, we sow tat wen te varance of te value of rmed traders prvate rmaton s smaller tan te varance on te demand of lqudty tradng, te return volatlty decreases monotoncally wt te prce rmatveness. Ts next example also sows te exstence of parameter values suc tat tere s a negatve relaton between prce rmatveness and return volatlty, as predcted by part () of Proposton 5. Example 3. Assume te followng parameter values: r,n =1,r z,n = 1.,r,n = 0.5,T t = 1, and a = 1. Te return volatlty decreases monotoncally wt te prce rmatveness. Fg. 4 sows te negatve relaton between te prce rmatveness and te return volatlty. 3.6. Back to te orgnal economy A sare of stock n {1,,..., N} n te orgnal economy s equvalent to a portfolo of one sare of stock n and b n sares of stock N + 1 n te equvalent economy. We use te superscrpt O to denote prces n te orgnal economy. Te followng proposton summarzes te stock prces over tme n te orgnal economy. Proposton 6 (Stock prces n te orgnal economy). At tme t {0,1,...,T}, te prce of stock n {1,,...,N} n te orgnal economy s P O ¼ P þ b n P Nþ1;t ;

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1569 and te prce of te market ndex (.e., stock M) n te orgnal economy s P O M;t ¼ P Nþ1;t; were P n,t s gven n Eqs. (8) (10). Te stock prces of te N ndvdual stocks are correlated, snce tey are all affected by te market ndex. Te return volatlty of stock n {1,,...,N} at tme t s now Var O r 0 ¼ b n þ 1 a Nþ1;t 1 a þða Nþ1;t Þ r Nþ1; þr Nþ1; þ ðaz Þ r Nþ1;z þ a r n; þr n; þ az r n;z ; ð14þ Tat s, te return volatlty can be decomposed nto systematc return volatlty and dosyncratc return volatlty. Followng te emprcal lterature, we defne te (relatve) dosyncratc return volatlty as follows: 1 a þ a r n; þ r n; þ az r n;z IV ¼ ; Var O r 0 wc s essentally one mnus te market-model R-square. In te orgnal economy, te prce rmatveness of an ndvdual stock depends on te rmatveness of bot te systematc component and te dosyncratc component of te stock s fundamental value. However, snce all stocks are nfluenced by te same systematc factor, te cross-sectonal dfference n prce rmatveness s drven only by te dosyncratc component. For ts reason, we keep our defnton of prce rmatveness te same as tat n te equvalent economy. Tat s, we defne prce rmatveness of an ndvdual stock n te orgnal economy as te uncertanty reducton n te value of te stock s dosyncratc value due to te knowledge of te prce. Te followng proposton summarzes te relaton between te prce rmatveness and te (relatve) dosyncratc return volatlty. Te results follow naturally from te results n Proposton 5. Corollary 1 (Relaton between prce rmatveness and dosyncratc return volatlty). () No parameter values exst suc tat te dosyncratc return volatlty ncreases monotoncally wt te prce rmatveness O ðr 0 Þ P 0 for all W @Var @W ; () Tere exst parameter values suc tat te relaton between te dosyncratc return volatlty and te prce rmatveness s U-saped @Var O ðr 0 Þ < 0 for W < W e @Var and O ðr 0 Þ P 0 for @W @W W P W e ; () Tere exst parameter values suc tat te dosyncratc return volatlty decreases monotoncally wt te prce rmatveness P 0 for all W. 4. Emprcal tests @Var O ðrþ @l In ts secton, we emprcally test te relaton between prce rmatveness and dosyncratc return volatlty. We frst descrbe our sample and ow we measure prce rmatveness and dosyncratc return volatlty. Ten, we report our results on te relaton between te two. Fnally, we perform an array of robustness cecks to sow tat our emprcal results are robust to dfferent specfcatons. 4.1. Sample and data 4.1.1. Sample We construct te sample wt non-fnancal and non-utlty stocks wose CUSIP dentfer s eter 10 or 11. 10 We also exclude stocks wose ndustry classfcaton s not obvous (.e., SIC codes are mssng and tus ave a value of zero). We examne ter weekly stock returns and requre stocks to ave a full year of weekly return data. Te sample perod spans from 1983 to 004. 4.1.. Measurng dosyncratc volatlty We measure dosyncratc volatlty by estmatng te followng equaton: r ;w;t ¼ a ;t þ b ;t r m;w;t þ c ;t r ;w;t þ e ;w;t ; ð15þ were r,w,t s stock s smple return n week w (Wednesday close to te next Wednesday close) n year t, r m,w,t and r,w,t are, respectvely, te contemporaneous returns on te market portfolo and on te ndustry portfolo (based on te -dgt SIC). To avod any spurous results, bot te market and te ndustry portfolos are constructed wtout stock ; remanng stocks are ten valuewegted. Te frm-specfc return volatlty (FSRV ereafter) for frm n year t, FSRV t, s defned as one mnus te R-square from regresson (15). 11,1 4.1.3. Measurng prce rmatveness We use sx measures of prce rmatveness tat are wdely used n te lterature. Eac measure s transformed so tat a ger value of te measure corresponds to greater prce rmatveness. Snce we measure FSRV every year, te rmatveness measures are also calculated on an annual bass. 4.1.3.1. Informaton-based tradng (N_PIN). Our frst measure s based on te PIN n Easley et al. (010) and t s wdely used n te lterature as a measure of prce rmatveness (e.g., Kang, 010). It s te market maker s estmate of te probablty tat a certan trade s based on prvate rmaton about te stock. For frms wt rmatve stock prces, snce most frm-specfc rmaton s already ncorporated nto te prce, te probablty tat te market maker s tradng aganst an rmed trader s small. On te contrary, for stocks wt less rmatve stock prces, te probablty tat any gven trade s rmaton-based s ger; tus, te PIN wll also be ger. We make te followng transformaton to create an rmatveness measure based on PIN: N_PIN t = log(pin for year t). Our PIN measures cover all NYSE/ Amex common stocks from 1983 to 001 for wc PINs can be estmated. 4.1.3.. Prce mpact (N_PIM). Followng Amud (00) and smlar to Tapa and Posakwale (010), we measure prce mpact by te absolute daly return dvded by te daly dollar volume of trade (n 10 Bot ndustry and CUSIP rmaton s from CRSP. In partcular, we use te storcal SIC and CUSIP, not te ones n te eader fle. 11 Our defnton of FSRV can be vewed as relatve FSRV, and ts s te defnton used n most studes n te lterature (e.g., Morck et al., 000). A few studes use absolute FSRV, wc s te root mean square error of te regresson of te frm s stock return on te market (and ndustry) return. As we wll sow later n robustness ceck, we obtan broadly consstent results wen we use absolute FSRV. 1 Wle our model explctly predcts a U-saped relaton between absolute FSRV and prce rmatveness, t also mplctly predcts a U-saped relaton between relatve FSRV and prce rmatveness. We assume tat nvestors produce rmaton about te frm-specfc component but not about te systematc component. Terefore, wle te absolute dosyncratc volatlty canges wt ow muc rmaton s produced by nvestors (.e., prce rmatveness), te market volatlty remans constant wen prce rmatveness canges. Tat s, canges n relatve FSRV are drven by canges n absolute FSRV n our model. We tank te anonymous referee for pontng ts out.

1570 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 mllons), averaged over te year (namely, PIM). PIM measures ow easly nvestors can lqudate a stock wtout severely affectng te prce. A larger value of PIM means, upon a lqudty sock, one as to ncur a greater dollar loss to sell te stock. Effcently prced stocks tend to be more lqud. 13 Terefore, te prce mpact of any gven trade s greater for stocks wt less rmatve prces. We make te followng transformaton to create an rmaton measure based on te prce mpact: N_PIM t = log(0.0001 + PIM t ). 4.1.3.3. Analyst earnngs forecast error (N_ERR). Followng Krsnaswam and Subramanam (1999), we assume tat as analysts allocate more resources, ter researc wll produce a more precse forecast, and more rmaton about te fundamental value of te frm wll be ncorporated nto te prce. Followng ts logc, we use analyst earnngs forecast error as anoter prce rmatveness measure. Eac mont, we calculate forecast error as te absolute value of te dfference between te mean earnngs forecast for te next fscal year and te actual earnngs per sare (EPS), scaled by te stock prce n tat mont; ten, we average te montly forecast errors over te year. We make te followng transformaton: N_ERR t = log (0.0001 + average forecast error durng year t). 4.1.3.4. Analyst earnngs forecast dsperson (N_DSP). As analysts allocate more resources and ntensfy ter researc actvtes, t s also reasonable to assume tat ter opnons wll converge to a correct parameter value. Followng Krsnaswam and Subramanam (1999), we use analyst earnngs forecast dsperson as anoter prce rmatveness measure. Eac mont, we obtan forecast dsperson (.e., standard devaton) for EPS of te next fscal year, scaled by te mean forecast; we ten average te montly scaled forecast dspersons over te year. We make te followng transformaton: N_DSP t = log (0.0001 + average forecast dsperson durng year t). 4.1.3.5. Te lengt of te frm s publc tradng story (AGE). Avalablty of publc tradng story wll surely reduce te rmaton producton costs, and encourage te gaterng and ncorporaton of rmaton nto stock prces. Specfcally, we construct ts measure by countng te number of years durng wc te stock s publcly traded (.e., te number of days dvded by 365). Followng Pastor and Verones (003), we make te followng transformaton: AGE t = 1/(te number of publcly traded years as of te end of year t). 4.1.3.6. Insttutonal ownersp (IO). It s wdely accepted tat nsttutonal nvestors are more sopstcated tan retal nvestors (Rubn and Smt, 009). Terefore, nsttutonal ownersp for a stock can serve as a drect measure of te amount of rmaton ncorporated nto te stock. Insttutonal ownersp s measured as te fracton of sares tat are eld by nsttutons wo fle te 13F form wt te Securtes and Excange Commsson. If a stock s not eld by any of tose 13F-reportng nsttutons, we assume tat te nsttutonal ownersp of tat stock s zero. Snce ts s quarterly rmaton, we use te average over four quarters wtn a year. We make te followng transformaton: IO t = log(0.0001 + fracton of sares eld by nsttutons for year t). It s wort mentonng tat n te teoretcal part of te paper, we ave defned prce rmatveness as te uncertanty reducton n te value of te stock due to te knowledge of te prce, caracterzed by Eq. (1). Because we cannot drectly observe and measure te uncondtonal varance of te stock value Var t (u n ), 13 Wen a frm s rmatonally effcent, te market makers face less adverse selecton from potental rmed traders. Te probablty of any gven trade s rmaton-based s small, and te market maker tends to adjust te stock prce less wen tere s an mbalance of trade order flows. we can only use te above sx measures to proxy for prce rmatveness. Wen te stock prce becomes more rmatve, te value of Var t (u n jp n,t ) decreases, and te value of W n,t (our teoretcal defnton of prce rmatveness) ncreases. Emprcally, wen te prce becomes more rmatve, rmaton-based tradng sould decrease, prce mpact of tradng sould be smaller, and analyst earnngs forecasts sould be more accurate and less dspersed. As te frm as a longer tradng story and more ownersp by sopstcated nsttutonal nvestors, ts stock prce sould also become more rmatve. Tat s wy we coose tese sx wdely-used measures to proxy for our teoretcal defnton of prce rmatveness W n,t. 4.1.4. Controllng for te volatlty of proftablty and frm sze Frms wt more volatle cas flows and earnngs, ceters parbus, tend to ave ger dosyncratc stock return volatlty (e.g., Pastor and Verones, 003). It s also well documented n te lterature tat larger frms tend to ave lower dosyncratc volatlty (e.g., Roll, 1988; Kelly, 005). At te same tme, bot frm sze and cas flow volatlty greatly affect prce rmatveness. Wen estmatng te relatve mportance of cas-flow and expected-return news for frm-level stock returns, e fnds tat te varance of expected-return news s approxmately one-alf of te varance of cas-flow news for excess returns. Gven tat frm sze and cas flow volatlty greatly affect dosyncratc volatlty and prce rmatveness, we frst regress prce rmatveness on frm sze and cas flow volatlty and ten examne te relaton between resdual prce rmatveness and dosyncratc volatlty. 14 Ts way, we can solate te effects of prce rmatveness from te effects of frm sze and cas flow volatlty on dosyncratc return volatlty. Frm sze n year t s measured as te natural log of te frm s market captalzaton as of te end of year t 1(MVE). As a measure of te volatlty of corporate proftablty, we use te standard devaton of te frm s return on equty over te sample perod, wc we call SROE. We ten estmate te followng cross-sectonal regresson of one of our rmatveness measures on MVE and SROE, and use te resduals as te rmatveness measure n te analyss. INFO ;t ¼ a t þ b t SROE ;t þ c t MVE ;t þ e ;t ð16þ were INFO,t s one of our transformed rmatveness measures for stock n year t. We estmate ts regresson wtn te same NYSE sze quntle to avod mposng a smple lnear relatonsp between frm sze and a nose measure. We, owever, also report results based on one sngle cross-sectonal regresson as a robustness ceck. 15 4.. Emprcal results We report emprcal results n ts secton. We frst provde te summary statstcs for te varables used n ts paper. Ten, we document a U-saped relaton between prce rmatveness 14 Our teoretcal model predcts a U-saped relaton between prce rmatveness and dosyncratc volatlty under certan parameter values (see Fg. 3 and Proposton 5). However, tese results are based on comparatve statcs. Specfcally, t sows tat, oldng everytng else constant (ncludng frm sze and cas flow volatlty), dosyncratc volatlty frst decreases and ten ncreases wt prce rmatveness. Emprcally, because dfferent frms ave dfferent caracterstcs tat affect dosyncratc volatlty and prce rmatveness, we ave to control for tese frm caracterstcs frst before examnng te relaton between prce rmatveness and dosyncratc volatlty. 15 If we do not control for frm sze and proftablty volatlty n prce rmatveness, we fnd a monotonc negatve relaton between dosyncratc volatlty and raw prce rmatveness, smlar to results n Kelly (005). Alternatvely, f we frst regress dosyncratc volatlty on MVE and SROE, we fnd a monotonc negatve relaton between resdual dosyncratc volatlty and raw prce rmatveness.

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1571 and FSRV. Fnally, we provde some tests to sow tat our results are robust to dfferent specfcatons. 4..1. Summary statstcs Table 1 provdes te summary statstcs for te man varables used n ts paper. Panel A reports te summary statstcs of FSRV t. Specfcally, we estmate summary statstcs of FSRV eac year and ten average tem over te sample perod. Te mnmum FSRV n Panel A s tus te average mnmum over te sample perod. Consstent wt earler studes, our sample frms sow sgnfcant FSRV, meanng tat only a small porton of ter stock returns, on average about 16%, are explaned by te market or ndustry factors (as specfed n Eq. (15)). Panel B reports te correlaton coeffcents among frm sze, two earnngs volatlty measures, and te sx rmatveness measures. As we ave conjectured earler, large frms tend to ave smaller earnngs volatlty as s evdenced by te negatve correlaton between frm sze and te volatlty measures. Gven tat te sx rmatveness measures are avalable for samples wt dfferent sze (see te average number of stocks for wc te varables can be constructed), lookng for a general pattern across tem can furter assure robustness of our results. 4... Relatonsp between FSRV and resdual rmatveness measures We frst plot FSRV aganst te resdual rmatveness measures and present sx plots (one for eac rmatveness measure) n Fg. 5. Specfcally, n eac year, we frst estmate Eq. (16) to obtan te resdual rmatveness measures. We ten assgn stocks nto 10 groups by te value of ter resdual rmatveness measures. Fnally, we plot te mean and medan FSRV for eac of te 10 groups over te sample perod. All te sx rmatveness measures sow a U-saped relaton to FSRV. In eac plot, wen te resdual rmatveness measure s small, FSRV decreases n te resdual rmatveness measure. Ts suggests tat wen te rmaton envronment of te frm s relatvely poor, more rmaton n te stock prce leads to smaller FSRV, wc s consstent wt te argument of West (1988). Te rgt alf of te plot sows te opposte pattern as FSRV ncreases n te resdual rmatveness measure. Ts mples tat wen te rmaton envronment of te frm s relatvely good, more rmaton n stock prce leads to greater FSRV, wc s consstent wt te fndngs of Morck et al. (000). To formally test our ypotess and see f ts cange n te relaton between stock prce rmatveness and FSRV s statstcally sgnfcant, we estmate te followng year-by-year crosssectonal regresson: FSRV ;t ¼ a t þ X3 ðb d;t RG d;t Þþ X3 ðc d;t RG d;t R INFO ;t Þ d¼1 d¼1 þ e ;t ; s:t: : X3 b d;t ¼ 0 d¼1 ð17þ were FSRV,t s frm-specfc return varaton for stock n year t, R_INFO,t s te resdual rmatveness measure from regresson (16), and RG d,t s a 0/1 dummy varable for one of tree regons n te cross-secton tat s sorted by te resdual rmatveness Table 1 Sample and summary statstcs Ts table reports summary statstcs of frm-specfc return varaton and oter varables. Te sample ncludes all non-fnancal and non-utlty CRSP stocks wose CUSIP dentfer s eter 10 nor 11. Stocks wose SIC codes are mssng or wose weekly return data are not avalable for te full year are excluded. Frmspecfc return varaton of a stock s estmated by deductng from one te R of te year-by-year regresson of te stock s weekly return on te weekly returns for te market and ndustry portfolos. Bot portfolos are value-wegted and do not nclude te stock n queston. Industry s determned by two-dgt SIC codes. Frm sze s te market captalzaton as of te end of te prevous year. Informaton measures nclude: probablty of rmaton based tradng (PIN), prce mpact, analyst forecast error, analyst forecast dsperson, nsttutonal ownersp, and te number of publcly traded years. PIN s obtaned from Soeren Hvdkjaer s web ste. Prce mpact (PIM) s te absolute daly return over te daly dollar volume, averaged over te year and ten multpled by 10 6. Analyst forecast error (ERR) s measured as te absolute value of te dfference between te mean estmate and te actual EPS (scaled by te stock prce) for te next fscal year. Analyst forecast dsperson (DSP) s measured as te standard devaton of analyst forecasts. At two analysts are requred for forecast error and dsperson. Insttutonal ownersp s te fracton of sares eld by 13f-flng nsttutonal nvestors, averaged over te year. Te number of publcly traded years s te number of days from te frst publc tradng day to te last day of te year, dvded by 365. All te rmaton measures are transformed so tat a ger value corresponds to more rmaton: specfcally, N_PIN = log (PIN); N_ERR = log (0.0001 + ERR); N_DSP = log (0.0001 + DSP); N_PIM = log (0.0001 + PIM); AGE = 1/te number of publcly traded days dvded by 365; and IO = log (0.0001 + nsttutonal ownersp). MVE s te natural log of te market captalzaton at te end of te prevous year, SROE s te standard devaton of te return on equty (ROE) durng te sample perod. RROE s te mean square error from te regresson of ROE on ts 1-year lagged value, estmated over te sample perod. Panel A. Summary statstcs of frm-specfc return varaton n a gven year n Mean Stdev Mn q1 Medan q3 Max 3073 0.837 0.145 0.199 0.764 0.879 0.949 1.000 Panel B. Oter varables Average number of sample stocks n a gven year SROE RROE MVE N_PIN N_PIM N_ERR N_DSP AGE IO 3073 3073 3073 977 3073 1706 1648 3073 3073 Correlaton coeffcent wt one anoter RROE 0.847 (0.000) MVE 0.40 0.14 (0.000) (0.000) N_PIN 0.073 0.006 0.718 (0.000) (0.407) (0.000) N_PIM 0.1 0.110 0.91 0.715 (0.000) (0.000) (0.000) (0.000) N_ERR 0.05 0.13 0.49 0.77 0.45 (0.000) (0.000) (0.000) (0.000) (0.000) N_DSP 0.147 0.073 0.36 0.54 0.30 0.580 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) AGE 0.189 0.10 0.160 0.078 0.149 0.08 0.036 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) IO 0.7 0.153 0.613 0.439 0.606 0.81 0.59 0.193 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

157 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 dosyncratc volatlty 0.95 usng N_PIN dosyncratc volatlty 0.95 usng IO 0.93 0.93 0.91 0.91 0.89 0.89 0.87 0.87 0.85 0.85 0.83 0.83 0.81 0.81 0.79 0.79 0.77 0.77 0.75 0.75 d d3 d4 d5 d6 d7 d8 d9 most d d3 d4 d5 d6 d7 d8 d9 most dosyncratc volatlty usng N_ERR dosyncratc volatlty usng N_DSP 0.95 0.95 0.93 0.93 0.91 0.91 0.89 0.89 0.87 0.87 0.85 0.85 0.83 0.83 0.81 0.81 0.79 0.79 0.77 0.77 0.75 0.75 d d3 d4 d5 d6 d7 d8 d9 most d d3 d4 d5 d6 d7 d8 d9 most dosyncratc volatlty usng N_PIM dosyncratc volatlty usng AGE 0.95 0.95 0.93 0.93 0.91 0.91 0.89 0.89 0.87 0.87 0.85 0.85 0.83 0.83 0.81 0.81 0.79 0.79 0.77 0.77 0.75 0.75 d d3 d4 d5 d6 d7 d8 d9 most d d3 d4 d5 d6 d7 d8 d9 most Fg. 5. Idosyncratc volatlty and prce rmatveness after controllng for volatlty of proftablty and frm sze. Te above fgures are a plot of dosyncratc volatlty (.e., frm-specfc return varaton) aganst te resdual prce rmatveness measure. Resdual values are obtaned from a year-by-year cross-sectonal regresson of te orgnal prce rmatveness measure on cas flow volatlty and frm sze wtn te same sze quntle. Frm sze s te market captalzaton as of te end of te prevous year and volatlty of proftablty s te tme-seres standard devaton of te return on equty across all sample years. measure (RG 1,t : frst four decles; RG,t : mddle two decles; RG 3,t : last four decles). To avod lnear dependency, we mpose te restrcton tat coeffcents for te tree dummes sum to zero. Table reports te tme-seres average of te regresson coeffcents and ter Newey West t-statstcs, n te sprt of Fama and MacBet (1973). Te average coeffcent on te nteracton term between RG 1,t and R_INFO,t, c d,t, s negatve and statstcally sgnfcant for all sx rmatveness measures (te coeffcent on N_DSP s only margnally sgnfcant wt a t-statstc of 1.78). Ts means tat for frms wt poor rmaton envronments, more rmaton n te stock prce leads to smaller FSRV. Te average coeffcent on te nteracton term between RG,t and R_INFO,t s not statstcally sgnfcant for four of te sx rmatveness measures. It s postvely sgnfcant for te AGE measure, but negatvely sgnfcant for te N_PIM measure. Overall, te results ndcate tat for frms wt moderate rmaton envronments,

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1573 Table Fama MacBet regresson of frm-specfc return varaton on rmaton measure. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) from te followng year-by-year cross-sectonal regressons: FSRV ;t ¼ a t þ P 3 d¼1 b d;trg d;t þ P 3 d¼1c d;t ðrg d;t R INFO ;t Þþe ;t ; subject to P 3 d¼1 b d;t ¼ 0, were FSRV,t s frm-specfc return varaton for stock and year t, and RG d,t s a 0/1 dummy varable for one of tree regons n te cross-secton tat s sorted by R_INFO,t (RG 1 : frst four decles; RG : mddle two decles; RG 3 : last four decles). R_INFO,t s te resdual prce rmatveness measure from followng regresson: INFO ;t ¼ a t þ b t SROE ;t þ c t MVE ;t þ e ;t; were INFO,t s one of our transformed prce rmatveness measures (see Table 1 for detals) for stock and year t. We estmate ts regresson wt all stocks wtn te same NYSE sze quntle. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.770 0.003 0.009 0.01 0.168 0.001 0.157 0.35 (19 years) (4.16) (1.09) (3.67) (4.96) (5.9) (0.0) (10.71) (8.1) N_PIM 0.815 0.030 0.01 0.019 0.07 0.05 0.01 0.048 ( years) (55.35) (15.65) (5.57) (8.48) (7.54) (.03) (5.33) (6.57) N_ERR 0.771 0.003 0.006 0.009 0.011 0.03 0.014 0.05 ( years) (44.37) (1.10) (.79) (3.1) (3.00) (1.7) (4.84) (6.5) N_DSP 0.764 0.010 0.008 0.018 0.006 0.00 0.036 0.04 ( years) (4.01) (4.91) (3.15) (5.65) (1.78) (0.16) (8.35) (9.7) AGE 0.794 0.09 0.055 0.06 0.091 1.437 0.415 0.506 ( years) (55.56) (4.87) (4.67) (.19) (4.45) (4.14) (3.31) (4.07) IO 0.797 0.05 0.01 0.004 0.07 0.014 0.06 0.089 ( years) (5.37) (8.69) (6.59) (1.10) (10.35) (0.87) (14.37) (14.75) Table 3 F-tests from te year-by-year cross-sectonal regressons. Ts table reports p-values of te F-tests for te ypotess tat c 1,t s equal to c 3,t. n Eq. (17). AsnTable, we report results based on resdual rmaton measures. Tese F-tests use te eteroscedastcty-consstent covarance matrx. Resdual rmaton measure s from Eq. (16) usng all stocks wtn te same NYSE sze quntle. Year Informaton measure N_PIN N_PIM N_ERR N_DSP AGE N_SPRD 1983 (0.001) (0.001) (0.791) (0.147) (0.000) (0.000) 1984 (0.000) (0.000) (0.67) (0.000) (0.005) (0.000) 1985 (0.000) (0.000) (0.404) (0.17) (0.000) (0.000) 1986 (0.000) (0.000) (0.01) (0.016) (0.000) (0.000) 1987 (0.000) (0.000) (0.000) (0.006) (0.000) (0.000) 1988 (0.000) (0.000) (0.679) (0.000) (0.000) (0.000) 1989 (0.000) (0.000) (0.006) (0.000) (0.000) (0.000) 1990 (0.000) (0.000) (0.195) (0.003) (0.00) (0.000) 1991 (0.000) (0.000) (0.010) (0.451) (0.536) (0.000) 199 (0.000) (0.000) (0.08) (0.178) (0.018) (0.000) 1993 (0.000) (0.001) (0.463) (0.787) (0.749) (0.000) 1994 (0.000) (0.039) (0.001) (0.001) (0.639) (0.000) 1995 (0.000) (0.158) (0.000) (0.000) (0.003) (0.000) 1996 (0.000) (0.000) (0.545) (0.000) (0.006) (0.000) 1997 (0.186) (0.005) (0.00) (0.000) (0.000) (0.000) 1998 (0.000) (0.000) (0.111) (0.10) (0.000) (0.000) 1999 (0.017) (0.000) (0.019) (0.064) (0.000) (0.000) 000 (0.157) (0.199) (0.189) (0.003) (0.081) (0.000) 001 (0.000) (0.000) (0.000) (0.003) (0.077) (0.000) 00 (0.000) (0.001) (0.000) (0.000) (0.000) 003 (0.000) (0.005) (0.000) (0.000) (0.000) 004 (0.000) (0.510) (0.001) (0.000) (0.000) tere s no obvous relaton between te amount of rmaton n te stock prce and FSRV. Fnally, te average coeffcent on te nteracton term between RG 3,t and R_INFO,t, c 3,t, s postve and statstcally sgnfcant for all sx rmatveness measures. Ts suggests tat for frms wt good rmaton envronments, more rmaton n te stock prce leads to greater FSRV. Te last column, DIFF, reports te average dfference between c 1,t and c 3,t, along wt ts Newey West t-statstc. Te dfference s statstcally sgnfcant at te 1% level for all sx rmatveness measures, wc we nterpret as evdence tat rmaton n te stock prce affects FSRV dfferently dependng on te rmaton envronments of te stock. As anoter look at te dfference between c 1,t and c 3,t, Table 3 reports te p-value of te year-by-year F-test for te null ypotess tat c 1,t s equal to c 3,t. We reject ts null ypotess n most of te sample years. For example, f we use N_PIN as an rmatveness measure, out of te 19 years for wc PIN s avalable, we reject te ypotess at te 1% level n 16 years and at te 5% level n 17 years. Ts sows tat te U-saped relatonsp s present farly consstently over tme. Te rmatveness measure tat offers te weakest rejecton of te null ypotess s te one from te analyst forecast errors (N_ERR). Wt ts measure, we reject te null ypotess at te 5% level n 1 out of te sample years. Ts s probably due to tat fact tat te sample for ts rmatveness measure (N_ERR) s lmted to frms n te IBES database, wc ave a relatvely good rmaton envronment. Smlar (but better relatve to results based on N_ERR) results wt Table 4 Fama MacBet regresson results wtout truncatng extreme values. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on a dataset were extreme values are not truncated. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.770 0.004 0.008 0.01 0.165 0.018 0.157 0.3 (19 years) (4.34) (1.30) (.89) (4.94) (5.8) (0.) (10.51) (8.0) N_PIM 0.816 0.030 0.01 0.018 0.07 0.03 0.00 0.047 ( years) (55.75) (15.11) (6.1) (8.64) (7.66) (1.67) (5.38) (6.68) N_ERR 0.771 0.000 0.008 0.008 0.013 0.015 0.013 0.06 ( years) (44.33) (0.04) (4.7) (.7) (4.1) (0.76) (4.05) (6.69) N_DSP 0.769 0.007 0.004 0.010 0.007 0.009 0.0 0.08 ( years) (4.40) (.01) (1.57) (.90) (.06) (0.87) (14.50) (7.) AGE 0.797 0.07 0.054 0.07 0.089 1.361 0.371 0.461 ( years) (55.54) (4.35) (4.90) (.33) (4.4) (4.43) (3.16) (3.95) IO 0.799 0.04 0.01 0.003 0.07 0.0 0.061 0.088 ( years) (53.69) (9.55) (7.8) (0.9) (10.38) (1.51) (14.86) (14.7)

1574 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 Table 5 Fama MacBet regresson results wt alternatve measure of volatlty of proftablty. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on an alternatve measure of volatlty of corporate proftablty (.e., squared root of errors from a regresson of te annual return on equty on ts 1-year lagged value, estmated over te sample perod). Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.769 0.004 0.007 0.011 0.168 0.09 0.155 0.33 (19 years) (4.4) (1.7) (.53) (4.15) (5.88) (0.3) (10.13) (7.9) N_PIM 0.815 0.031 0.013 0.018 0.07 0.01 0.00 0.047 ( years) (55.59) (16.00) (6.9) (8.48) (7.79) (1.06) (5.39) (6.81) N_ERR 0.77 0.004 0.007 0.011 0.010 0.01 0.015 0.05 ( years) (44.75) (1.6) (3.64) (4.11) (.46) (0.48) (5.01) (5.74) N_DSP 0.763 0.010 0.008 0.019 0.006 0.000 0.037 0.043 ( years) (4.00) (3.90) (3.57) (5.68) (1.85) (0.04) (8.89) (9.45) AGE 0.783 0.04 0.059 0.017 0.081 1.735 0.575 0.656 ( years) (49.10) (6.83) (5.57) (1.56) (3.98) (6.00) (5.05) (5.57) IO 0.798 0.03 0.019 0.004 0.08 0.07 0.061 0.089 ( years) (53.78) (8.96) (6.08) (1.00) (10.54) (1.81) (15.1) (15.8) Table 6 Fama MacBet regresson results wt resdual rmaton measures obtaned from one sngle cross-sectonal regresson eac year. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on te resdual rmaton measures tat are obtaned from a sngle cross-sectonal regresson (nstead of fve regressons n eac sze quntle) eac year. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.77 0.007 0.009 0.016 0.145 0.03 0.138 0.83 (19 years) (4.98) (1.89) (3.58) (5.05) (5.40) (0.4) (11.95) (8.19) N_PIM 0.84 0.08 0.001 0.08 0.018 0.05 0.013 0.031 ( years) (57.36) (8.37) (0.39) (6.95) (5.11) (4.73) (.86) (4.05) N_ERR 0.77 0.006 0.005 0.011 0.009 0.015 0.015 0.04 ( years) (44.13) (1.76) (.60) (3.81) (.38) (1.13) (4.55) (6.19) N_DSP 0.765 0.007 0.008 0.015 0.007 0.00 0.03 0.039 ( years) (41.53) (3.81) (.98) (4.99) (.00) (0.14) (7.90) (8.13) AGE 0.795 0.043 0.05 0.068 0.044 0.33 1.0 1.45 ( years) (46.39) (6.83) (.00) (4.38) (.31) (1.13) (5.59) (5.80) IO 0.803 0.018 0.001 0.017 0.08 0.053 0.038 0.066 ( years) (54.31) (7.13) (0.7) (4.09) (11.9) (3.46) (11.0) (1.73) Table 7 Fama MacBet regresson results wt scaled rmaton measures. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on te scaled resdual rmaton measure tat ranges from 0.5 to 0.5 (see, e.g., Mendenall, 004). Specfcally, we frst determne te percentle rankng from 0 to 99 n a gven sample year, and deduct 49.5 from te rankng value and ten dvde by 99. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.753 0.004 0.05 0.01 0.199 0.008 0.197 0.396 (19 years) (38.7) (1.3) (6.61) (7.56) (7.0) (0.0) (8.13) (8.35) N_PIM 0.793 0.005 0.008 0.014 0.50 0.06 0.18 0.378 ( years) (45.83) (1.56) (.1) (4.6) (9.01) (.18) (4.96) (7.39) N_ERR 0.771 0.000 0.008 0.007 0.056 0.064 0.049 0.105 ( years) (43.81) (0.01) (.96) (.00) (.60) (.16) (.86) (4.63) N_DSP 0.760 0.003 0.01 0.016 0.055 0.003 0.18 0.18 ( years) (40.46) (0.95) (5.14) (5.16) (.11) (0.11) (6.57) (6.39) AGE 0.80 0.05 0.00 0.07 0.190 0.345 0.097 0.86 ( years) (55.36) (.49) (0.33) (3.00) (4.91) (4.36) (4.7) (6.81) IO 0.747 0.00 0.07 0.04 0.354 0.01 0.436 0.790 ( years) (38.97) (0.60) (8.17) (7.70) (10.8) (0.80) (13.46) (1.14) N_DSP furter confrm ts conjecture, snce N_DSP s also calculated usng te IBES database. 4..3. Robustness ceck In ts secton, we conduct a varety of robustness cecks to assure tat our earler results, te U-saped relaton between prce rmatveness and FSRV, are robust. Frst, we report results based on an untruncated dataset (recall tat tus far we ave treated te observaton as mssng f SROE, N_ERR, or N_DSP s eter below te 1st percentle or above te 99t percentle n a certan year). Table 4 reports results wen we keep all tose extreme values. Te results are vrtually dentcal: for all sx rmatveness measures, te average value of c 1,t s negatve and sgnfcant, te average value of c 3,t s postve and sgnfcant, and te dfference between c 1,t and c 3,t s statstcally sgnfcant. In Tables and 3, we used SROE, te standard devaton of te frm s return on equty over te sample perod, as a measure of

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1575 Table 8 Fama MacBet regresson results wt more varables controlled for. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te specfcaton for Table wt slgt modfcatons. Results n ts table are based on an alternatve specfcaton for Eq. (17) were sector dummes, te number of segments wtn te frm, leverage rato, and dvdend payment dummy are ncluded as control varables. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.890 0.004 0.008 0.01 0.149 0.034 0.16 0.76 (19 years) (37.76) (1.63) (3.96) (6.04) (6.13) (0.46) (1.46) (8.57) N_PIM 0.910 0.05 0.007 0.018 0.05 0.09 0.009 0.034 ( years) (47.4) (18.34) (4.67) (9.54) (8.05) (.76) (.34) (5.5) N_ERR 0.886 0.004 0.005 0.009 0.01 0.031 0.011 0.0 ( years) (35.64) (1.33) (3.07) (3.34) (3.64) (1.54) (3.07) (4.58) N_DSP 0.870 0.011 0.006 0.017 0.003 0.006 0.05 0.08 ( years) (34.8) (6.0) (.17) (6.19) (1.) (0.47) (6.87) (6.79) AGE 0.893 0.015 0.041 0.06 0.076 1.0 0.388 0.464 ( years) (46.8) (.9) (4.70) (.36) (5.18) (4.7) (.77) (3.8) IO 0.885 0.0 0.014 0.008 0.0 0.01 0.05 0.075 ( years) (44.67) (8.85) (4.74) (.40) (9.30) (1.31) (13.1) (1.85) Table 9 Fama MacBet regresson results wt FSRV estmated usng te market model wt four lagged market and ndustry returns. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on frm-specfc return varaton tat s obtaned from a modfed Eq. (15) were four lagged market returns and four lagged ndustry returns are ncluded, as well as te contemporaneous market and ndustry returns. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.636 0.001 0.008 0.009 0.134 0.03 0.11 0.55 (19 years) (39.1) (0.49) (3.50) (3.98) (5.86) (0.41) (8.55) (7.40) N_PIM 0.67 0.05 0.010 0.015 0.0 0.03 0.018 0.040 ( years) (50.8) (15.9) (5.36) (9.04) (7.30) (.0) (5.19) (6.54) N_ERR 0.636 0.00 0.006 0.008 0.009 0.09 0.013 0.0 ( years) (41.37) (0.81) (3.6) (3.13) (3.1) (1.67) (4.6) (6.65) N_DSP 0.69 0.007 0.009 0.016 0.006 0.00 0.030 0.036 ( years) (39.3) (4.40) (3.44) (5.74) (1.95) (0.18) (7.03) (7.81) AGE 0.653 0.05 0.046 0.01 0.071 1.07 0.360 0.431 ( years) (51.67) (5.36) (4.5) (.0) (4.15) (4.1) (3.36) (4.10) IO 0.657 0.01 0.017 0.004 0.03 0.007 0.053 0.075 ( years) (47.41) (8.88) (5.74) (1.18) (10.8) (0.53) (14.9) (14.85) Table 10 Fama MacBet regresson results wt alternatve resdual rmaton measures. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on alternatve resdual rmaton measures tat are obtaned by regressng eac of tem on earnngs volatlty alone (wtn te same sze quntle). Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.77 0.000 0.007 0.007 0.173 0.098 0.098 0.70 (19 years) (4.86) (0.06) (3.48) (1.73) (7.53) (1.5) (7.80) (9.85) N_PIM 0.807 0.05 0.004 0.0 0.03 0.08 0.016 0.048 ( years) (5.3) (1.89) (1.46) (9.8) (8.61) (4.4) (7.7) (9.47) N_ERR 0.771 0.00 0.007 0.009 0.013 0.007 0.01 0.05 ( years) (44.16) (0.81) (4.30) (3.5) (3.91) (0.56) (4.01) (6.51) N_DSP 0.764 0.009 0.008 0.017 0.008 0.007 0.031 0.039 ( years) (41.67) (4.05) (.59) (5.47) (.43) (0.43) (6.40) (7.45) AGE 0.79 0.031 0.053 0.0 0.084 1.39 0.48 0.566 ( years) (57.11) (4.16) (3.53) (1.57) (3.77) (3.1) (3.37) (4.04) IO 0.801 0.019 0.039 0.01 0.07 0.011 0.03 0.059 ( years) (53.17) (7.09) (5.91) (3.85) (10.19) (0.45) (7.48) (9.76) te volatlty of corporate proftablty. In Table 5, we obtan our results usng RROE, namely, te mean squared error from a regresson of te frm s return on equty on ts 1-year lag over te sample perod, as a measure of te volatlty of proftablty. Te results are also qualtatvely te same as tose n Table. So far, we ave calculated te resdual rmatveness measures by estmatng Eq. (16) wtn te same NYSE sze quntle n eac year. In Table 6, we report results wen we calculate te resdual rmatveness measures by estmatng Eq. (16) wt all sample frms n eac year. Agan, te results are qualtatvely te same as tose n Table. In Table 7, we report results wen we scale te resdual rmatveness varables nto a varable rangng from 0.5 to 0.5 (see, e.g., Mendenall, 004). It s bascally a percentle rankng

1576 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 dosyncratc volatlty 0.10 usng N_PIN dosyncratc volatlty 0.10 usng IO 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 d d3 d4 d5 d6 d7 d8 d9 most 0.04 d d3 d4 d5 d6 d7 d8 d9 most dosyncratc volatlty usng N_ERR dosyncratc volatlty usng N_DSP 0.10 0.10 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 d d3 d4 d5 d6 d7 d8 d9 most d d3 d4 d5 d6 d7 d8 d9 most dosyncratc volatlty usng N_PIM dosyncratc volatlty usng AGE 0.10 0.10 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 d d3 d4 d5 d6 d7 d8 d9 most d d3 d4 d5 d6 d7 d8 d9 most Fg. 6. Absolute measure of dosyncratc volatlty and prce rmatveness after controllng for volatlty of proftablty and frm sze. Te above fgures are a plot te absolute dosyncratc volatlty (.e., absolute FSRV) aganst te resdual prce rmatveness measure. Absolute dosyncratc volatlty s te mean root squared errors of te market model (Eq. (15)). Resdual value s obtaned from a year-by-year cross-sectonal regresson on earnngs volatlty and frm sze wtn te same sze quntle. Frm sze s te market captalzaton as of te end of te prevous year and volatlty of proftablty s te tme-seres standard devaton of te return on equty over te sample years. from 0 to 99 n a gven sample year; we deduct 49.5 from te rankng value and ten dvde by 99, so tat t ranges from 0.5 to 0.5. Ts addresses two potental problem of te orgnal approac. Frst, t mtgates te outler problem. Second, snce te resdual varables now ranges from 0.5 and 0.5, te coeffcent can be nterpreted as te cange n FSRV wen te resdual measure canges from te lowest percentle to te gest percentle. Results are consstent wt tose n Table. We furter control for oter factors wc may affect FSRV, suc as ndustry sectors, te number of segments n te frm, leverage, and dvdends. Specfcally, we nclude n Eq. (17) 14 ndustry sector dummes, te natural log of te number of segments, a

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1577 Table 11 Fama MacBet regresson results wt alternatve resdual rmaton measures. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on alternatve resdual rmaton measures tat are obtaned by regressng eac of te sx prce rmatveness measures on frm sze wtn te earnngs volatlty quntle. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.77 0.006 0.011 0.017 0.145 0.007 0.145 0.90 (19 years) (4.68) (1.61) (4.97) (4.60) (5.45) (0.10) (11.05) (8.06) N_PIM 0.84 0.07 0.001 0.08 0.019 0.051 0.015 0.033 ( years) (57.08) (8.56) (0.87) (8.0) (5.70) (5.44) (3.33) (4.6) N_ERR 0.771 0.003 0.008 0.010 0.01 0.00 0.015 0.08 ( years) (44.03) (0.84) (4.76) (4.36) (3.30) (0.18) (4.97) (6.93) N_DSP 0.764 0.010 0.009 0.019 0.007 0.01 0.035 0.04 ( years) (41.3) (3.78) (3.4) (6.06) (1.90) (0.85) (8.00) (7.91) AGE 0.800 0.03 0.014 0.018 0.06 0.4 0.713 0.775 ( years) (48.89) (5.00) (1.19) (1.1) (3.37) (1.63) (3.40) (3.65) IO 0.801 0.01 0.001 0.011 0.09 0.046 0.044 0.073 ( years) (54.05) (5.87) (0.9) (3.06) (11.19) (3.65) (1.17) (1.90) Table 1 Fama MacBet regresson results wt FSRV estmated usng te Fama Frenc 3-factor model. Ts table reports Fama MacBet coeffcents and Newey West t-statstcs (n parenteses) for Eq. (17) te same specfcaton as Table. Results n ts table are based on frm-specfc return varaton tat s obtaned from te Fama Frenc 3-factor model. Informaton measure Intercept Coeffcent for DIFF RG 1 RG RG 3 RG 1 R_INFO RG R_INFO RG 3 R_INFO N_PIN 0.763 0.00 0.008 0.011 0.143 0.06 0.131 0.73 (19 years) (39.35) (1.05) (3.84) (4.65) (7.01) (0.38) (9.50) (8.90) N_PIM 0.80 0.0 0.009 0.01 0.08 0.03 0.01 0.040 ( years) (53.3) (10.38) (4.76) (6.19) (8.8) (3.8) (.63) (5.31) N_ERR 0.759 0.003 0.005 0.009 0.015 0.06 0.009 0.04 ( years) (44.31) (1.36) (4.08) (3.15) (6.07) (1.48) (3.5) (9.77) N_DSP 0.753 0.014 0.006 0.019 0.013 0.016 0.07 0.040 ( years) (43.13) (7.78) (.44) (7.04) (3.51) (1.45) (6.48) (11.80) AGE 0.789 0.00 0.043 0.03 0.064 1.095 0.44 0.308 ( years) (53.7) (4.4) (4.49) (.80) (3.57) (3.76) (3.47) (3.99) IO 0.784 0.04 0.014 0.009 0.06 0.018 0.056 0.08 ( years) (51.14) (7.63) (6.10) (.60) (10.66) (1.3) (13.9) (13.9) dummy varable for frms wo pay dvdends durng te year, and te leverage rato. 16 Results are reported n Table 8. Te results are qualtatvely te same as tose n Table. Table 9 reports results based on an alternatve specfcaton for Eq. (15), te regresson for FSRV estmaton. Specfcally, to estmate FSRV, we use four lagged market portfolo returns and four lagged ndustry portfolo returns, as well as ter contemporaneous returns. Results are consstent wt tose n Table. Our defnton of te resdual rmatveness measure as been te resdual from te regresson of te raw rmatveness measures on te volatlty of corporate proftablty and frm sze. Ts robustness ceck, reported n Table 10, sows tat even f we control for te volatlty of proftablty alone n calculatng resdual rmatveness measures, te results stll old. So far, we ave been usng te relatve FSRV measure. Usng relatve FSRV s more approprate snce some busnesses can be more susceptble to systematc socks tan oters, and frm-specfc events n tese frms can be accordngly more ntense. In oter words, usng relatve FSRV elps control for suc envronmental volatlty. As a robustness ceck, we see f our results old f we use absolute FSRV nstead. Fg. 6 sows tat results based on te absolute measure are broadly consstent wt tose based on te relatve measure. 16 Te 14 ndustry sector dummes are based on -dgt SIC codes. Specfcally, sector 1 between 1 and 9; sector between 10 and 14; sector 3 between 15 and 19; sector 4 between 0 and 1; sector 5 between and 3; sector 6 between 4 and 7; sector 7 between 8 and 3; sector 8 between 33 and 34; sector 9 between 35 and 39; sector 10 between 40 and 48; sector 11 between 50 and 5; sector 1 between 53 and 59; sector 13 between 70 and 79; and sector 14 between 80 and ger. In Table, we calculate te resdual rmatveness measures by estmatng Eq. (16) wtn te same NYSE sze quntle eac year. In Table 11, we report results wen we calculate te resdual rmatveness measures by regressng INFO,t on MVE,t and a constant n eac SORE quntle eac year. Te results are qualtatvely te same as tose n Table. 17 Most studes examnng te relaton between dosyncratc volatlty and expected stock returns defne dosyncratc volatlty slgtly dfferently. Tey use te Fama and Frenc (1993) treefactor model nstead of CAPM. In Table 1, we report results wen we calculate FSRV by usng Fama and Frenc (1993) tree factors nstead of te market and ndustry returns n Eq. (15). We obtan results smlar to tose n Table. 5. Concluson Many recent studes use dosyncratc return volatlty as a measure of ow muc rmaton s ncorporated nto te stock prce. Alarmngly, wle some studes use ger dosyncratc volatlty as a measure of more rmatve prces, oters assume tat ger dosyncratc volatlty means less prce rmatveness. Emprcal evdence regardng te rmaton content of dosyncratc volatlty s also mxed. A growng body of researc sows tat frms wt more rmatve stock prces ave ger dosyncratc volatlty (Morck et al., 000). Anoter strand of studes fnd exactly te 17 If we nstead calculate te resdual rmatveness measures by regressng INFO,t on MVE,t, SORE,t, and a constant n eac SORE quntle eac year, te results are close to dentcal to tose n Table 11. Tese results are avalable upon request to nterested readers.

1578 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 opposte (e.g., Kelly, 005). Understandng te true relaton between prce rmatveness and dosyncratc volatlty s mportant, gven tat an ncreasng number of studes use dosyncratc volatlty as a measure of prce rmatveness or rmaton asymmetry. Furter, understandng te true rmaton content of dosyncratc volatlty s mportant for practtoners and polcy makers as well. Ts paper makes an attempt n ts drecton. We nvestgate te relaton between prce rmatveness and dosyncratc prce volatlty n a mult-asset, mult-perod nosy ratonal expectatons equlbrum. Idosyncratc return volatlty s decomposed nto two parts: (1) te part caused by nose, and () te part caused by rmaton regardng te frm s fundamental value. We sow tat te frst component decreases wt prce rmatveness, wle te second component frst decreases and ten ncreases wt prce rmatveness. Our man results are as follows. Frst, tere exst no parameter values suc tat dosyncratc return volatlty ncreases monotoncally wt prce rmatveness. Second, tere exst parameter values suc tat te relaton between prce rmatveness and dosyncratc return volatlty s U-saped. Fnally, tere exst parameter values suc tat dosyncratc return volatlty decreases monotoncally wt prce rmatveness. Usng several prce rmatveness measures, we emprcally document a U-saped relaton between prce rmatveness and dosyncratc return volatlty. Our study terefore reconcles te opposng vews expressed n te followng two strands of lterature: (1) te growng body of researc sowng tat frms wt more rmatve stock prces ave greater dosyncratc return volatlty (e.g., Morck et al., 000; Jn and Myers, 006), and () te studes argung tat more rmaton n prce reduces dosyncratc return volatlty (West, 1988; Kelly, 005). Acknowledgements We tank Azz Almov, Brent Ambrose, Paul Clds, Art Durnev, Brad Jordan, Ike Matur (te edtor), Randall Morck, Don Mullneaux, Joe Peek, We Xong, an anonymous referee, and semnar partcpants at te 006 Fnancal Management Assocaton Meetngs, te Frst Internatonal Conference on Asa Pacfc Fnancal Markets, te 007 Internatonal Fnance Conference at Queen s Unversty, Korea Unversty, Seoul Natonal Unversty, and Unversty of Kentucky for elpful comments. All errors and omssons are our own. Appendx A. Proofs of propostons and corollares Proof of proposton 1. Te rmaton producton and portfolo coce problem of nvestor j s max E T 1 e aw j T jx j T 1 ða:1þ X T s:t: : W j T ¼ Bj T 1 þ P0 T 1 ði j T 1 Þ0 C þ u 0 X j T : Xj T 1 Xj T Snce W j T s normally dstrbuted and te utlty functon s negatve exponental, t s well-known tat te soluton to te nvestor s portfolo coce problem s X j T ¼ 1 a ½VarðujXj T 1 ÞŠ 1 E T 1 ðujx j T 1 Þ P T 1 : ða:þ Gven tat te fundamental values of and te nosy demands for te N + 1 stocks are ndependent of eac oter, we ave 8 PT 1 >< u n þ g þ n;t f I j n;t 1 ¼ 1 E T 1 ðu n jx j T 1 Þ¼ >: u n þ PT 1 g þ E T 1 ðg n;t jp n;t 1 Þ f I j n;t 1 ¼ 0; ða:3þ and VarðujX j T 1Þ s an (N +1) (N + 1) dagonal varance covarance matrx wt te (n,n) t element as ( Varðu n jx j T 1 Þ¼ r ;n f I j n;t 1 ¼ 1 Var T 1 ðg n;t jp n;t 1 Þ f I j n;t 1 ¼ 0; ða:4þ were P n,t 1 s ndependent of eac oter across te stocks, as we wll sow later. Pluggng Eqs. (A.3) and (A.4) n Eq. (A.) yelds 8 P T 1 unþ g þ n;t P >< T 1 f I j X j n;t ¼ ar n;t 1 ¼ 1 ;n T 1 ða:5þ u nþp g þe T 1ðg n;t jp n;t 1 Þ P >: T 1 avar T 1 ðg n;t jp n;t 1 f I j Þ n;t 1 ¼ 0: Te value of X j n;t wen Ij n;t 1 ¼ 1 s te demand for stock nwen nvestor j produces rmaton about stock n, wle te value of X j n;t wen Ij n;t 1 ¼ 0 s te demand for stock n wen te nvestor s unrmed about n,t. From Eq. (A.5), we know tat for te fracton l n,t 1 of nvestors wo produce rmaton and observe te P value n,t, ter demand for stock n s unþ T 1 g þ n;t P T 1 ar, and for ;n te fracton 1 l n,t 1 of nvestors wo do not observe te value n,t, ter demand s E T 1ðunjP n;t 1 Þ P n;t 1 avar T 1 ðunjp n;t 1. Te market clearng condton Þ for stock n s terefore l n;t 1 u n þ P T 1 g þ n;t P T 1 ar ;n þð1 l n;t 1 Þ E T 1ðu n jp n;t 1 Þ P n;t 1 þ z n;t ¼ y avar T 1 ðu n jp n;t 1 Þ n : ða:6þ Tat s, te total demand for stock n from rmed nvestors, UDTs, and lqudty traders equals te total pyscal supply of te stock. If we rearrange Eq. (A.6), we ave u n XT 1 g þ P n;t 1 1 l n;t 1 r l E T 1 ðu n jp n;t 1 Þ P n;t 1 ;n n;t 1 Var T 1 ðu n jp n;t 1 Þ þ ar ;n l n;t 1 y n ¼ n;t þ ar ;n l n;t 1 z n;t : ða:7þ UDTs observe everytng at te left sde of te above equaton, wc s a nosy sgnal of te prvate rmaton eld by rmed traders, n,t. We defne te left sde of Eq. (A.7) as S(P n,t 1 ), wc s te set of rmaton tat s revealed to te UDTs by te prce, P n,t 1. If te belefs of UDTs are consstent, we ave E T 1 ðu n jp n;t 1 Þ¼E T 1 ðu n jsðp n;t 1 ÞÞ and Var T 1 ðu n jp n;t 1 Þ¼Var T 1 ðu n jsðp n;t 1 ÞÞ: ða:8þ ða:9þ Usng te propertes of condtonal normal dstrbutons, we can sow tat E T 1 ðu n jp n;t 1 Þ¼u n þ XT 1 and Var T 1 ðu n jp n;t 1 Þ¼r ;n þ g þ l n;t 1r ;n ðl n;t 1 n;t þ ar ;n z n;tþ l n;t 1r ;n þ a r 4 ;nr z;n r ;n a r 4 ;nr z;n l n;t 1r ;n þ a r 4 ;nr z;n ða:10þ : ða:11þ Pluggng Eqs. (A.10) and (A.11) nto Eq. (A.7), we obtan te equlbrum market prce of stock n n Eq. (9), were a n;t 1 ¼ 1 and ð1 l n;t 1 Þa r z;nr 4 ;n l n;t 1r ;n þ a r z;nr 4 ;n þ a l n;t 1 r z;nr ;nr ;n ða:1þ

D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 1579 a z n;t 1 ¼ ar ½l ;n n;t 1r ;n þ a r z;nr ;n ðr ;n þ r ;n ÞŠ l n;t 1r ;n þ a r z;nr 4 ;n þ a l n;t 1 r z;nr ;nr ;n : ða:13þ Proof of proposton. We start wt tme t = T. Defne J T 1 (B T 1,X n,t 1 ) as te expected payoff to an agent wo as a portfolo of (B T 1,X n,t 1 ) and observed prce P T 1 and a set of prvate sgnals S T 1. 18 Snce we ave proved tat UDTs wll be ndfferent between producng rmaton and not-producng rmaton on eac stock, and te rmaton producton decsons are ndependent of eac oter, we can assume te agent cooses to produce rmaton on all te N + 1 stocks wtout loss of generalty. J T 1 ðb T 1 ; X n;t 1 Þ¼max X T E T 1 ½ e aw T jx T 1 Š P Nþ1 P Nþ1 ¼ E T 1 ½ e aðb T 1þ P n;t 1 ðx n;t 1 X n;t Þ n¼1 n¼1 P Nþ1 P T 1 ¼ e aðb T 1þ fp n;t 1 ðx n;t 1 X n;t Þ C nþðu nþ g þ n;t ÞX n;t 0:5aX n;t r Þ n¼1 PNþ1 C nþ u n;t X n;t Þ n¼1 jxt 1 Š ða:14þ ða:15þ Te last equalty follows Eqs. (A.3) and (A.4) wen I n,t 1 =1. Terefore, te objectve of te nvestor at tme T s max E T ½J T 1 ðb T 1 ; X n;t 1 ÞjX T Š ða:16þ X T 1 s:t: : B T 1 ¼ B T þ XNþ1 P n;t ðx n;t X n;t 1 Þ XNþ1 I n;t C n : Or, equvalently, n¼1 n¼1 ða:17þ l n;t u n þ P T g þ n;t 1 P T ar ;n þð1 l n;t Þ E T ðu n jp n;t Þ P n;t þ z n;t 1 ¼ y avar T ðu n jp n;t Þ n : Followng steps smlar to Eqs. (A.8) troug (A.11), we ave P n;t ¼ u n þ XT g n;s þ a n;t n;t 1 þ a z n;t ðz n;t 1 y n Þ; were a n;t ¼ 1 s¼1 ð1 l n;t Þa r z;n R n;t 1 l n;t r ;n þ a r z;n R n;t 1 þ a l n;t r z;nr ;n R n;t 1 ar n;t 1 l a z n;t ¼ n;t r ;n þ a r R z;n n;t 1 R n;t 1 þ r ;n l n;t r ;n þ a r z;n R n;t 1 þ a l n;t r z;nr R ;n n;t 1 and R n;t 1 ¼ a n;t 1 r ;n þðaz n;t 1 Þ r z;n þ r ;n : It s stragtforward to use backward nducton to sow tat J t ðb t ; X Þ¼ max X tþ1 were P Nþ1 e aðbtþ fp ðx X þ1 ÞþE t½p þ1 ŠX þ1 0:5aVar t½p þ1 ŠX þ1 n¼1 ; ; ða:19þ ða:0þ ða:1þ ða:þ gþ ET ½e ak tþ1 Š ða:3þ max X T 1 were PNþ1 e aðbt þ n;t ðx n;t X n;t 1 ÞþE T ½P n;t 1 ŠX n;t 1 0:5aVar T ½P n;t 1 ŠX n¼1fp n;t 1 gþ E T ½e ak T 1 Š K T 1 ¼ XNþ1 n¼1 ða:18þ! P n;t 1 X n;t I n;t C n C n þðu n þ XT 1 g þ n;t ÞX n;t 0:5aX n;tr Þ: FOC for X n,t 1 leads to P n;t þ E T ½P n;t 1 Š avar T ½P n;t 1 ŠX n;t 1 ¼ 0; K tþ1 ¼ XNþ1 ð P þ1 X þ I C n ÞþK tþ n¼1 and te stock prce at tme t s gven by Eqs. (10), a ¼ 1 ð1 l Þa r z;n R þ1 l r ;n þ a r z;n R þ1 þ ; a l r z;nr ;nr þ1 a z ¼ ar þ1½l r ;n þ a r z;n R þ1ðr þ1 þ r ;n ÞŠ l r ;n þ a r z;n R þ1 þ a l r z;nr ;nr þ1 ; and ða:4þ ða:5þ or equvalently R þ1 ¼ða þ1 Þ r ;n þðaz þ1 Þ r z;n þ r ;n : ða:6þ X n;t 1 ¼ E T ½P n;t 1 Š P n;t : avar T ½P n;t 1 Š For te fracton l n,t of nvestors wo produce rmaton about Proof of proposton 3 () Te dervatve of V Info r 0 wt respect to l n,t ssnce @V Info ðr 0 Þ ¼ @l a R þ1r z;nr ;n l ð l Þr ;n þ a R þ1r z;n þ a r ;nr R z;n þ1 l r ;n þ a r z;n R þ1 þ a l r z;nr R 3 l r þðl ;n 1Þa r z;n R þ1 þ a l r z;nr R ;n þ1 ;n þ1 P stock n, ter demand for stock n s unþ T g þ n;t 1 P T, and for te ar ;n fracton 1 l n,t of nvestors wo do not observe te value n,t 1, ter demand s E T ðunjp n;t Þ P n;t avar T ðunjp n;t. Te market clearng condton for Þ stock n s terefore 18 We drop te subscrpt j snce all nvestors wt te same portfolo (B T 1,X n, T 1 ) wll ave te same expected payoff. a R þ1 r z;nr l ;n½ ð l Þr ;n þa R þ1 r z;n þa r ;n r z;n R þ1š 3 > 0 for all l r ;n þa r z;n R þ1 þa l r z;n r ;n R þ1 l n,t [0,1], wle l r ;n þðl 1Þa r z;n R þ1 þ a l r z;nr ;n R þ1š s negatve for l ½0; ^l Þ and postve for l ½^l ; 1Š, were we ave @VInfo ðr 0 Þ @l < 0 for l ½0; ^l Þ and @VInfo ðrþ @l P 0 for l ½^l ; 1Š. () Snce a z > 0, we ave

1580 D.W. Lee, M.H. Lu / Journal of Bankng & Fnance 35 (011) 1563 1580 @V Nose ðr 0 Þ @l @a ¼ a z r z z;n < 0: @l s domnated by Var Nose ðr 0 Þ, wc as a negatve relaton wt l n,t. Proof of proposton 4. We ave Var t ðu n Þ¼ðT tþðr þ r Þ; and Proof of proposton 6. Te proof follows drectly from te law of one prce and te fact tat a sare of stock n {1,,...,N} n te orgnal economy s equvalent to a portfolo of one sare of stock n and b n sares of stock N + 1 n te equvalent economy, and a sare of stock M n te orgnal economy s equvalent to a sare of stock N + 1 n te equvalent economy. Var t ðu n jp Þ¼r þ r Terefore, W ¼ 1 Var tðu n jp Þ Var t ðu n Þ a ða z Þ r z r þ þðt t 1Þðr az r þ r Þ: z a r 4 ¼ ðt tþ r þ r a r þ az r z r 4 ¼ " ðt tþ r þ r #: r r þ az a We ave proved tat @a @l > 0 and @az @l < 0. Terefore, az decreases a wt l n,t snce bot a and a z are postve, wc means tat W n,t ncreases wt l n,t ;.e., @W @l > 0. Te result @W < 0 follows drectly from @W @l > 0 and @l < 0. @Cn @Cn Proof of proposton 5. Te dervatve of return volatlty wt respect to l n,t s @Varðr 0 Þ @l a r z;nr ;n ¼ R þ1 3 l r ;n þ a r z;n R þ1 þ a l r z;nr ;nr þ1 n z l ð1 l Þr ;n l r ;n þðl 1Þa r z;n R þ1 þ a l r z;nr ;n R þ1 l r ;n þ a r z;n R þ1 þ a r z;nr ;n R þ1 l a r ;nr z;n R þ1 þ a 4 r ;nr 4 z;n R þ1 þ a4 r 4 z;n R3 þ1 To prove tat no parameter values exst suc tat @Varðr0 Þ P 0 for all @W W n,t, note tat wen l n,t =0, @Var r 0 ¼ a r z;nr ;n R þ1 @l 3 a r z;n R þ1 þ a r z;nr R ;n þ1 a r z;n R þ1 a 4 r ;nr 4 z;n R þ1 þ a4 r 4 z;n R3 þ1 wc s negatve. By te can rule, we ave @Varðr 0 Þ ¼ @Varðr0 Þ @W ; @l @W @l snce @W @l > 0 and wen l ¼ 0; @Varðr0 Þ @l < 0, we ave @Varðr0 Þ < 0 @W wen l n,t = 0. Ts proves part (). To prove part (), note tat wen r ;n s large enoug, Varðr r 0 Þ s domnated by VarInfo ðr 0 Þ, wc as a z;n U-saped relaton wt l n,t as we ave proved n Proposton 5. Smlarly, to prove part (), note tat wen r ;n s small enoug, Varðr r 0 Þ z;n Proof of corollary 1. Followng Eq. (14), we ave Var O ðr 0 Þ¼bVarðr0 Nþ1;t ÞþVar r0 for n =1,,...,N. Snce Var r 0 Nþ1;t s ndependent of W n,t, te relaton between Var O r 0 and W n,t depends only on te relaton between Var r 0 and W n,t, wc s summarzed n Proposton 5. References Amud, Y., 00. Illqudty and stock returns: cross-secton and tme-seres effects. Journal of Fnancal Markets 5, 31 56. Ang, A., Hodrck, R., Xng, Y., Zang, X., 006. Te cross-secton of volatlty and expected returns. Journal of Fnance 61, 59 99. Asbaug-Skafe, H., Gassen, J., LaFond, R., 005. Does stock prce syncroncty represent frm-specfc rmaton? Te nternatonal evdence. Workng Paper, Unversty of Wsconsn-Madson. Brockman, P., Yan, X., 009. Block ownersp and frm-specfc rmaton. Journal of Bankng and Fnance 33, 308 316. Campbell, J.Y., Lettau, M., Malkel, B.G., Xu, Y., 001. Have ndvdual stocks become more volatle? Am emprcal exploraton of dosyncratc rsk. Journal of Fnance 56, 1 43. Easley, D., Hvdkjaer, S., O Hara, M., 010. Factorng rmaton nto returns. Journal of Fnancal and Quanttatve Analyss 45, 93 309. Fama, E., Frenc, K., 1993. Common rsk factors n te returns on stocks and bonds. Journal of Fnancal Economcs 33, 3 56. Fama, E., MacBet, J., 1973. Rsk, return, and equlbrum: emprcal tests. Journal of Poltcal Economy 71, 607 636. Fu, F., 009. Idosyncratc rsk and te cross-secton of expected stock returns. Journal of Fnancal Economcs 91, 4 37. Goyal, A., Santa-Clara, P., 003. Idosyncratc rsk matters! Journal of Fnance 58, 975 1007. Guo, H., Savckas, R., 008. Average dosyncratc volatlty n G7 Countres. Revew of Fnancal Studes 1, 159 196. Guo, H., Savckas, R., 010. Relaton between tme-seres and cross-sectonal effects of dosyncratc varance on stock returns. Journal of Bankng and Fnance 34, 1637 1649. Jn, L., Myers, S., 006. R around te world: new teory and new tests. Journal of Fnancal Economcs 79, 57 9. Kang, M., 010. Probablty of rmaton-based tradng and te January effect. Journal of Bankng and Fnance 34, 985 994. Kelly, P.J., 005. Informaton Effcency and Frm-specfc Return Varaton. Workng Paper, Arzona State Unversty. Krsnaswam, S., Subramanam, V., 1999. Informaton asymmetry, valuaton, and te corporate spn-off decson. Journal of Fnancal Economcs 53, 73 11. Kyle, A., 1985. Contnuous auctons and nsder tradng. Econometrca 53, 1315 1335. Mendenall, R., 004. Arbtrage rsk and post-earnngs-announcement drft. Journal of Busness 77, 875 894. Morck, R., Yeung, B., Yu, W., 000. Te rmaton content of stock markets: wy do emergng markets ave syncronous stock prce movements? Journal of Fnancal Economcs 58, 15 60. Ozoguz, A., 005. Informed Tradng, Return Syncroncty and Equlbrum Asset Prcng. Workng Paper, Queen s Unversty. Pastor, L., Verones, P., 003. Stock valuaton and learnng about proftablty. Journal of Fnance 58, 1749 1789. Roll, R., 1988. R. Journal of Fnance 43, 541 566. Rubn, A., Smt, D., 009. Insttutonal ownersp, volatlty and dvdends. Journal of Bankng and Fnance 33, 67 639. Tapa, C., Posakwale, S., 010. Internatonal equty portfolo allocatons and transacton costs. Journal of Bankng and Fnance 34, 67 638. West, K., 1988. Dvdend nnovatons and stock prce volatlty. Econometrca 56, 37 61.