CAPM for Estimating the Cost of Equity Capital: Interpreting the Empirical Evidence 1

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1 CAPM for Estmatng the Cost of Equty Captal: Interpretng the Emprcal Evdence 1 Frst Draft: Apr 29, 2008 Ths Draft: Oct 14, 2009 Zh Da Mendoza College of Busness Unversty of Notre Dame zda@nd.edu (574) Re-Jn Guo Department of Fnance Unversty of Illnos at Chcago rguo@uc.edu (312) Rav Jagannathan Kellogg School of Management Northwestern Unversty and NBER rjaganna@kellogg.northwestern.edu (847) We thank Bob Dttmar, Bruce Grundy, Dermot Murphy, Ernst Schaumburg, Mke Sher, Mtchell Warachka, Hong Yan and the semnar partcpants at Sngapore Management Unversty, Natonal Unversty of Sngapore, Nanyang Technologcal Unversty, Unversty of Illnos - Chcago, Unversty of Brtsh Columba, 2008 Australan Competton and Consumer Commsson Regulatory Conference, 2008 FIRS Conference, Unversty of Mnnesota macro-fnance conference, 2009 WFA for helpful comments. 1

2 Abstract We argue that the CAPM may be a reasonable model for estmatng the cost of captal for projects n spte of ncreasng emprcal evdence n the lterature aganst the CAPM based on stock returns. As McDonald and Segel (1985) and Berk, Green and Nak (1999) pont out, stocks are backed by projects n place as well as optons to modfy exstng projects and undertake new projects. In that case, the expected return on stocks need not satsfy the CAPM even when project expected returns do. We propose a method for estmatng frms' project betas and project returns and fnd that there s a lnear relaton between the two. Our fndngs support the use of the CAPM along wth real opton valuaton models n project evaluaton. 2

3 1. Introducton The Sharpe (1964) and Lntner (1965) Captal Asset Prcng Model (CAPM) s the workhorse of fnance for estmatng the cost of captal for project selecton. In spte of ncreasng crtcsm n the emprcal academc lterature, the CAPM contnues to be the preferred model for classroom use n manageral fnance courses n busness schools, and managers contnue to use t. Welch (2008) fnds that about 75% of fnance professors recommend usng the CAPM for estmatng the cost of captal for captal budgetng purposes. A survey of CFOs by Graham and Harvey (2001) ndcates that 73.5% of the respondng fnancal executves use the CAPM. The prmary emprcal challenge to the CAPM comes from several well-documented anomales: several managed portfolos constructed usng varous frm characterstcs earn very dfferent returns on average from those predcted by the CAPM. The queston we want to examne s whether these anomales should stop us from usng the CAPM for estmatng the cost of captal for undertakng a project. Notable among the anomales that challenge the valdty of the CAPM are the fndngs that the average returns on stocks s related to frm sze (Banz (1981)), earnngs to prce rato (Basu (1983)), bookto-market value of equty (BM) (Rosenberg, Red, and Lansten (1985)), cash flow to prce rato, sales growth (Lakonshok, Shlefer and Vshny (1994)), past returns (DeBondt and Thaler (1985) and Jegadeesh and Ttman (1993)), and past earnngs announcement surprse (Ball and Brown (1968)). Numerous subsequent studes confrm the presence of smlar patterns n dfferent datasets, ncludng those of nternatonal markets. Fama and French (1993) conjecture that two addtonal rsk factors, n addton to the stock market factor used n emprcal mplementatons of the CAPM, are necessary to fully characterze economy-wde pervasve rsk n stocks. The Fama and French (1993) threefactor model has receved wde attenton and has become the standard model for computng rsk-adjusted returns n the emprcal fnance lterature. Almost all the exstng anomales apply to the stock return of a frm. Should such anomales prevent one from usng CAPM n calculatng the cost of captal for a project? In ths paper, we revew the related lterature and provde new emprcal evdence to 3

4 argue that there s lttle evdence aganst the use of the CAPM for estmatng the cost of captal for projects. Most frms have the opton to turn down, undertake, or defer a new project, n addton to the opton to modfy or termnate an exstng project. Therefore, a frm can be vewed as a collecton of exstng and future projects and complex optons on those projects. McDonald and Segel (1985) observe that a frm should optmally exercse these real optons to maxmze ts total value. The resultng frm value wll consst of both the NPVs of the projects and the value of assocated real optons whch s determned by how those optons are expected to be exercsed by the frm. Berk, Green and Nak (1999) buld on that nsght and present a model where the expected returns on all projects satsfy the CAPM but the expected returns on the frm s stocks do not. That s because, as Dybvg and Ingersoll (1982) and Hansen and Rchard (1987) show, whle the CAPM wll assgn the rght expected returns to the prmtve assets (projects), t wll n general assgn the wrong expected returns to optons on those prmtve assets. Gomes, Kogan and Zhang (2003), Carlson, Fsher and Gammarno (2004), Cooper (2006), and Zhang (2005) provde several addtonal nsghts by buldng on the Berk, Green and Nak (1999) framework. Anderson and Garca-Fejoo (2006) and Xng (2008) fnd that the BM effect dsappears when one controls for proxes for frms nvestment actvtes. Bernardo, Chowdhry and Goyal (2007) hghlght the mportance of separatng out the growth opton from equty beta. Jagannathan and Wang (1996) argue that because of the nature of the real optons vested wth frms, the systematc rsk of frms wll vary dependng on economc condtons, and the stock returns of such frms wll exhbt opton lke behavor. An econometrcan usng standard tme seres methods may conclude that the CAPM does not hold for such frms, even when the returns on such frms satsfy the CAPM n a condtonal sense. We llustrate the mpact of real optons through a stylzed numercal example n the next secton. When the senstvty of frms stock returns to economy wde rsk factors changes n nonlnear ways due to the presence of such real optons, t may be necessary to use excess returns on certan cleverly managed portfolos (lke the Fama and French 4

5 (1993) SMB and HML factors) as addtonal rsk factors to explan the cross secton of stock returns, even when returns on ndvdual prmtve projects satsfy the CAPM. If that were the case, the contnued use of the CAPM for estmatng the cost of captal for projects would be justfed, n spte of the nablty of the CAPM to explan the crosssecton of average returns on the 25 sze and book-to-market sorted benchmark stock portfolos. In general, both equty rsk premum and equty beta of the frm wll be a complex functon of frm s project beta and real optons. If we orthogonalze them on a set of varables capturng the real optons usng lnear regressons, the resdual rsk premum and resdual beta are opton-adjusted and wll more closely resemble the underlyng project rsk premum and project beta, respectvely. Consequently, the CAPM may work well on the opton-adjusted rsk premum and beta. We frst provde support for the CAPM wth respect to the opton-adjusted beta n a smple smulaton exercse. We smulate a large cross-secton of all equty-fnanced frms, each as a portfolo of prmtve projects and optons on them. Whle the CAPM works for the prmtve projects, due to the exstence of optons, t does not work for the frm. A cross-sectonal regresson of frm rsk premum on the frm beta produces a large ntercept term, a very small slope coeffcent and a R-square close to zero, just lke what we would fnd n data. However, once we orthogonalze the frm beta on a set of real opton proxes (strke prce, mpled volatlty and ts portfolo weght), the optonadjusted beta resembles the underlyng project beta very well and explans a large porton of the cross-sectonal varaton n opton-adjusted frm rsk premum. We also provde emprcal observatons supportng the use of the CAPM for calculatng the cost of captal of a project. We frst document stronger emprcal support for the CAPM when we reduce the nose assocated wth the measurement of beta. The nose reducton s acheved n two ways. Frst, we exclude extremely small stocks, extreme past wnner and losers from our sample. Chopra, Lakonshok and Rtter (1992) note that the beta of stocks experencng recent extreme returns wll be measured poorly. Second, we 5

6 skp two years after portfolo formaton followng the suggeston of Hoberg and Welch (2007) and use aged betas. In partcular, we fnd that the CAPM performs well n prcng the thrd-year average returns on the ten CAPM-beta-sorted portfolos durng the perod The CAPM cannot be rejected usng the Gbbons, Ross, and Shanken (1989) GRS test. The CAPM beta explans 81% of the cross-sectonal varaton n average returns across the ten portfolos. The addtonal explanatory power of the Fama-French three-factor model that uses two addtonal pervasve rsk factors s small. We then focus on the book-to-market (BM) anomaly. Schwert (2003) argues that most of the anomales are more apparent than real and often dsappear after they have been notced and publczed. 2 As we are examnng the use of the CAPM for project cost of captal calculaton, we lmt attenton to anomales that are (1) pervasve and not drven by stocks of very small frms; (2) persstent over longer horzons; and (3) robust n the sense that the anomaly does not dsappear soon after ts dscovery. The BM effect s probably the most mportant anomaly satsfyng these crtera (see Fama and French (2006)). Lakonshok, Shlefer and Vshny (1994) argue that hgh book-to-market stocks earn a hgher return because they are underprced to start wth, and not because they have hgher exposure to systematc rsk. Consstent wth that pont of vew, Danel and Ttman (1997) fnd that frms characterstcs help explan the cross-secton of returns. Potrosk (2000) provdes evdence consstent wth the presence of msprcng by showng that among hgh book-to-market stocks frms wth better fundamentals outperform the rest. Mohanram (2005) reaches a smlar concluson for low book-to-market stocks. Fnally, Ferson, Sarkssan, and Smn (1999) argue that returns on portfolos constructed usng stock attrbutes may appear to be useful rsk factors, even when the attrbutes are completely unrelated to rsk. Clearly, f the BM effect s ndeed due to msprcng and unrelated to rsk, t should not nvaldate the use of the CAPM n cost of captal calculatons as argued by Sten (1996). 2 The nterested reader s referred to Schwert (2003) for an excellent and comprehensve survey of the fnancal markets anomales lterature. 6

7 We provde addtonal evdence that the exstence of the BM effect, even f due to systematc rsk, should not prevent the use of CAPM for calculatng project cost of captal, snce the systematc rsk may arse from the real optons assocated wth the project. Although real optons are not drectly observable, we proxy them wth four emprcal varables. The frst two varables --- fnancal leverage (DE) and nverse nterest coverage (Icov) --- are proxes for fnancal dstress whch capture frm s opton to declare bankruptcy and abandon exstng projects. In addton, DE captures the leverage effect resultng from frm s captal structure decson. The next two varables --- growth rate n captal expendture (Capex) and dosyncratc volatlty (Ivol) --- are proxes for growth optons whch capture frm s opton to expand ts current projects or undertake new projects. Consstent wth many earler results n the lterature, we fnd that BM effect s only present among frms wth a lot of real optons (hgh DE, Icov, Capex and Ivol). The fndng s consstent wth BM effect beng due to a frm s real opton to termnate or modfy exstng projects and to undertake or defer new projects. Fnally, we provde some drect emprcal evdence supportng the use of the CAPM for the project cost of captal calculaton. We conduct a two-stage cross-sectonal regresson. In the frst stage, we regress both the stock return and the stock beta (aged beta) on the four real opton proxes. The real optons proxes are measured n excess of those of the market and the regresson does not have ntercept terms. These procedures ensure that the CAPM holds on the market exactly. The resdual returns and betas are opton-adjusted. In the second stage, we then regress the opton-adjusted return on the opton-adjusted beta. Whle stock beta s not sgnfcant n explanng the cross-sectonal varaton n average returns, opton-adjusted beta s very sgnfcant n explanng the cross-sectonal varaton n average opton-adjusted returns. In addton, BM, once opton-adjusted, does not provde addtonal explanatory power. The determnaton of cost of captal has been an mportant and frutful area of research n fnance. Fama and French, n a seres of papers, make a convncng case that CAPM fals to descrbe the cross-secton of stock returns (Fama and French (1992, 1996, 1997, 1999, 7

8 2004 and 2006). Among many other related works, Ferson and Locke (1998) fnd that the great majorty of the error n estmatng the cost of equty captal usng the CAPM s due to the rsk premum estmate; Pastor and Stambaugh (1999) show that the cost of equty estmaton can be mproved n a Bayesan framework; Ang and Lu (2004) dscuss a general approach for dscountng cashflows wth tme-varyng expected returns. In ths paper, our prmary nterest s n evaluatng the emprcal evdence aganst the use of the CAPM based estmates of costs of captal for elementary projects for makng captal budgetng decsons. In contrast, the focus of most of the studes n the asset prcng lterature s n understandng the determnants of expected returns on stocks. In vew of that, we refer readers nterested n the broader asset prcng lterature to the excellent surveys by Campbell (2003), Ferson (2003), Mehra and Prescott (2003), and Duffe (2003). Followng ths ntroducton, we llustrate the mpact of real optons through a stylzed numercal example n Secton 2. Secton 3 descrbes a smple regresson procedure to remove the effect of real optons. Secton 4 demonstrates the effectveness of optonadjusted beta usng smulaton evdence. Secton 5 presents emprcal evdence that performance of the CAPM n explanng returns mproves when we follow the suggestons of Hoberg and Welch (2007). In addton, the opton-adjusted beta explans the opton-adjusted return well and opton-adjusted BM does not add ncremental explanatory value. Secton 6 concludes. 2. A Smple Real Opton Example In ths secton, we present a smple example to llustrate our man pont n an economy where CAPM correctly prces all prmtve projects, stock returns can exhbt sze and BM effects. Further, the example s consstent wth the followng addtonal emprcal regulartes: Value stocks have hgher expected returns than the market and have postve CAPM alphas; 8

9 Growth stocks have lower expected returns than the market and have negatve CAPM alphas; Value stocks have lower CAPM betas than growth stocks; Equty rsk premum s countercyclcal; Value stocks are rsker than growth stocks when the expected rsk premum s hgh; Sze and book-to-market rato can descrbe cross-sectonal varaton n expected returns on stocks. 2.1 The Economy For llustratve purposes, we consder an economy wth a rsk premum of 5% per year, an annual rsk free rate of 5%, and a flat yeld curve. There are three possble states at the end of the year: Up (probablty: 25%), Md (probablty: 50%) and Down (probablty: 25%). The returns on the market portfolos n these three states are: 40.4%, 8.0% and -16.3%, respectvely, translatng to an expected return of 10.0%. All exstng projects n ths economy are dentcal, wth an ntal cost of $1. Once undertaken, each project pays out an expected perpetual cash flow of $0.2. All projects have a CAPM beta of 1 (as the market conssts of those dentcal projects), wth an approprate dscount rate of 10% (5% + 1 5% = 10% as predcted by CAPM). The market value of each project s therefore 0.2/10% = $2. Consder a frm whch has undertaken I projects. The book value of the frm s $I and ts market value s $2I. Note that the frm, whch s a portfolo of I projects each wth a CAPM beta of 1, also has a CAPM beta of 1. Therefore, the expected return for the frm would be 10% per year, as descrbed by the CAPM. Note also that the book-to-market rato ($I/$2I = 0.5) does not have addtonal predctve power of the frm s expected return. 9

10 We now ntroduce two types of optons n the economy: a value opton (VO) and a growth opton (GO). Each frm n the economy s randomly endowed wth one of the two optons. Wth ether opton, the frm has the capacty for nvestng n at most one more new project, wth an ntal cost of $1, ether now or one year later. If the frm chooses to nvest now, t wll get a project dentcal to ts exstng project (wth an expected perpetual annual cash flow of $0.2). However, f the frm chooses to wat a year, the expected perpetual annual cash flow of the project wll change. In the case of the value opton, the expected annual cash flow wll be $ n the Up state, $ n the Md state and $ n the Down state. In the case of the growth opton, the expected annual cash flow wll be $ n the Up state, $ n the Md state and $ n the Down state. Wth both optons, the nvestment opportunty dsappears after one year. As such, the optons exst only for the frst year after whch the economy wll consst of only prmtve projects. Ths s clearly a smplfed assumpton used for llustratve purpose only. Berk, Green and Nak (1999) present a much more realstc model where new projects arrve and old projects de on a dynamc bass. The frm s opton to nvest n ths case resembles the opton-to-wat analyzed n McDonald and Segel (1985) and dscussed n Jagannathan and Meer (2002). We summarze the nformaton about these two optons n the followng table: State Probablty Expected annual value opton (VO) cash flow (1) NPV VO cash flow =(1)/0.1-1 Expected annual growth opton (GO) cash flow (2) NPV GO cash flow =(2)/0.1-1 Up 0.25 $ $1.423 $ $2.711 Md 0.50 $ $1.423 $ $1.643 Down 0.25 $ $0.685 $ $1.643 Snce all project cashflows are assocated wth CAPM betas of 1 and costs of $1, ther NPVs can be computed by dscountng the expected annual cashflows at 10% and subtractng $1. Snce the NPVs are all postve, the projects wll always be undertaken at the end of the year (f they have not been undertaken at the begnnng of the year). 10

11 2.2 Prces and Expected Returns For prcng purpose, we assume a state prce vector (or the Stochastc Dscount Factor, SDF for short) M = [0.7313, , ] across Up, Md and Down states, respectvely. It can be verfed that E[M(1+R)] = 1 for the rsk free rate and the market return, meanng the SDF can prce the rsk free asset and the market portfolo. Wth the state prce vector, we can prce the two optons usng E[M*payoff]. The results are summarzed n the followng table: State Prob State Prce (M) Rskfree Rate Market Return VO payoff VO Return GO payoff Go Return Up % 40.4% $ % $ % Md % 8.0% $ % $ % Down % -16.3% $ % $ % prce $1.088 $1.760 ER 5.0% 10.0% 13.8% 8.5% CAPM beta CAPM ER 10.0% 10.5% 10.7% The value opton has a value of $1.088 today, hgher than the payoff f the project were to be taken today ($0.2/0.1-1 = $1), whch means the frm wll choose to wat. The growth opton has a value of $1.760 today, also hgher than the payoff f the projects were to be taken today, so the frm wll choose to wat too. Gven the prces of these two optons, we can compute ther annual returns and expected returns. In addton, we can compute ther covarances wth the market and therefore ther CAPM betas. We fnd that the value opton (VO) has a hgher expected return (13.8%) than the market whle the growth opton (GO) has a lower expected return (8.5%) than the market. Interestngly, the growth opton has a hgher CAPM beta. Because of the hgher CAPM beta, CAPM wll predct a hgher expected return on the growth opton (10.7%) than the value opton (10.5%). In other words, the CAPM, although perfectly explanng the expected returns on prmtve projects n the economy, fals to explan the expected returns on these two optons. As a result, the value opton seems to outperform the market (t carres a postve CAPM alpha of 13.8%-10.5% = 11

12 3.3%) whle the growth opton seems to underperform the market (t carres a negatve CAPM alpha of 8.5%-10.7% = - 2.2%). 2.3 Some Intuton based on the condtonal CAPM Why does the value opton (VO) earn a hgher expected return than the growth opton (GO) n the smple economy? One way to gan some ntuton s based on the condtonal CAPM of Jagannathan and Wang (1996) although we must emphasze that the real opton argument n general does not hnge on the valdty of the condtonal CAPM. We nsert an ntermedate tme perod nto our example. Consequently, the one-perod trnomal tree s expanded to be a two-perod bnomal tree. The payoffs to an nvestment n the market portfolo (assumng an ntal nvestment of $1) are: T = 0 T = Sx month T = One year Up (UU): $1.404 U: $1.200 $1.000 Md (UD, DU): $1.080 D: $0.900 Down (DD): $0.837 The rsk free rate n each sx-month perod s = 2.47%. Snce there are two states (wth equal probablty) assocated wth each node on the bnomal tree, the market s complete f both the stock and the bond can be traded on each node. 3 Both the value opton and the growth opton can be prced usng the standard noarbtrage replcaton argument (see Rubnsten (1976) among others): 3 It can be easly verfed that M = [0.7313, , ] s the unque arbtrage-free SDF n ths economy. 12

13 Value Opton (VO) Growth Opton (GO) T = 0 T = Sx T = Sx T = One year T = 0 month month T = One year $1.423 $2.711 $1.389 $2.085 $1.088 $1.423 $1.760 $1.643 $0.921 $1.604 $0.685 $1.643 Gven the payoffs (and the mpled returns) of both the market and the optons, we can compute the followng on both node U and D for the perod from sx month to one year: State ER (Market) Beta (VO) ER (VO) Beta (GO) ER (GO) U 3.5% % % D 6.5% % % Note frst that the expected return gong forward on the market s hgher followng a negatve market return n state (D), consstent wth the emprcal fact that rsk premum s counter-cyclcal. In addton, CAPM works for both optons condtonally (on each node). Ths s not surprsng snce the opton can be replcated by both the market and the bond and CAPM prces the expected returns on both assets (see Dybvg and Ross, 1985). The value opton s more rsky uncondtonally because t has hgher beta n state D, precsely when the market rsk premum s hgh. Ths s hghlghted by Jagannathan and Korajczyk (1986) and s the key nsght of the condtonal CAPM of Jagannathan and Wang (1996). Are value stocks ndeed more rsky when the rsk premum gong forward s hgh? Several emprcal evdences are provded n the lterature suggestng the answer to be yes. For example, Petkova and Zhang (2005) fnd that value betas tend to covary postvely, and growth betas tend to covary negatvely wth the expected market rsk premum, thus 13

14 offerng at least a partal explanaton to the value premum. 4 Why are value stocks more rsky when expected rsk premum s hgh? Zhang (2005) provdes one explanaton. It s more costly for value frms to downsze ther captal stocks snce they are typcally burdened wth more unproductve captal. As a result, the value stocks returns covary more wth economc downturns when the expected rsk premum s hgh. Lewellen and Nagel (2006), however, argue that the varaton n betas and the equty premum would have to be mplausbly large for the condtonal CAPM to explan the magntude of the value premum. Lewellen and Nagel (2006) make use of hgh-frequency returns n ther emprcal analyss. Chan, Hameed and Lau (2003) demonstrate that prce and return may be n part drven by factors unrelated to fundamental cash flow rsk. Such factors, together wth lqudty events, may contamnate the estmaton of beta at hgher frequences (see Pastor and Stambaugh (2003)). Bal, Cakc, and Tang (2009) and Bauer, Cosemans, Frehen and Schotman (2009) mprove the cross-sectonal performance of the condtonal CAPM by usng more effcent estmaton technques. In addton, recent studes by Kumar, Srescu, Boehme and Danelsen (2008) and Adran and Franzon (2008) demonstrate that once the estmaton rsk or parameter uncertanty assocated wth beta and rsk premum are accounted for, the condtonal CAPM wll have sgnfcantly more explanatory power n the cross-secton and may explan the value premum after all. 2.4 Stock Characterstcs Despte the falure of CAPM n prcng optons, book-to-market rato and sze of frm wll serve as two suffcent statstcs n descrbng the expected returns of all frms n the economy. To see that, note that all frms n the economy have two components: (1) the asset-n-place component that ncludes I exstng projects, and (2) an opton component (O = VO or GO). The market value or sze of each frm s: V = 2I + O. The expected return of the frm s a weghted-average of expected returns on these two components: ER = 2I /V *10% + O /V *ER O. 4 Other recent studes on the condtonal CAPM nclude Wang (2003), Ang and Chen (2007). Smlar evdences are provded n the context of consumpton-capm by Lettau and Ludvgson (2001), Santos and Verones (2006), and Lustg and Van Neuwerburgh (2005). 14

15 In the case that the frm has a value opton, ER = 2I /V *10% /V *13.8% = 20%*BM + 15%/Sze. In the case that the frm has a growth opton, ER = 2I /V *10% /V *8.5% = 20%*BM + 15%/Sze. In both cases, the expected return can be expressed as 20%*BM+15%/Sze. Therefore the expected return s ncreasng wth BM and decreasng wth Sze. In addton, BM and Sze explan the expected returns on all frms. Frms wth the value opton resemble value stocks. These frms have more assets-nplace, and because the value opton s cheaper, value stocks are assocated wth hgher BM. Because the value opton has hgher expected return and postve CAPM alpha, so wll the value stocks. Frms wth the growth opton resemble growth stocks. In contrast to value stocks, growth stocks have lower BMs, lower expected returns, and negatve CAPM alphas. Why do characterstcs such as BM and sze descrbe cross-sectonal return varatons? The key ntuton follows from Berk (1995). Gven expectaton about future payoffs, market value must be correlated wth systematc rsk across stocks. In the smple economy, BM summarzes the frm s rsk relatve to the scale of ts asset base, and sze descrbes the relatve mportance of exstng assets and the opton. Recent papers combne the above ntuton wth key nsghts from the real opton lterature poneered by McDonald and Segel (1985) n lnkng frm-specfc nvestment patterns, valuaton, and expected returns. The semnal paper by Berk, Green, and Nak (1999) studes the mplcatons of the optmal exercse of real nvestment optons. In ther model, nvestment opportuntes wth low systematc rsk are attractve to the frm, and makng such nvestments ncreases frm value and reduces the average rsk of the frm. Consequently, the expected return of the frm s dynamcally lnked to prce-based characterstcs such as BM and sze. Gomes, Kogan, and Zhang (2003) extend the model 15

16 to a general equlbrum settng. Sze and BM, correlated wth true condtonal betas n ther model, help to explan stock returns n the cross secton, especally when true betas are measured wth error emprcally. Carlson, Fsher, and Gammarno (2004) model the optmal dynamc nvestment behavor of monopolstc frms facng stochastc product market condtons. Ther approach s smlar n sprt to Berk et al. (1999) except that they also ntroduce operatng leverage, reversble real optons, fxed adjustment costs, and fnte growth opportuntes. They show that the BM effect can arse even f there s no cross-sectonal dsperson n project rsk as BM summarzes market demand condtons relatve to nvested captal. Zhang (2005) demonstrates n an ndustry equlbrum model that the frm s optmal nvestments, together wth asymmetry n captal adjustment costs and countercyclcal prce of rsk, can generate the BM effect. Ths s because value frms have dffculty dsnvestng, makng them more rsky n bad tmes when the market rsk premum s hgh. On the other hand, Cooper (2006) develops a dynamc model n whch BM s nformatve of the devaton of a frm s actual captal stock from ts target. As a frm becomes dstressed, book-value remans constant but market value falls, resultng n hgher BM. Gong forward, ts extra nstalled capacty allows t to expand producton easly wthout new nvestment, makng ts payoff more senstve to aggregate shocks and ts equty more rsky. Emprcally, Anderson and Garca-Fejoo (2006) and Xng (2008) both provde supportng evdence that the nvestment dynamcs of a frm s drvng the BM effect. In summary, our example llustrates that when prmtve projects are assocated wth real optons, the sze and BM effects can arse. However, the sze and BM effects do not mply that t s ncorrect to use CAPM n calculatng the costs of captal for prmtve projects. 3. Opton Adjustment and the CAPM In a generalzed model, wth smlar ntuton presented n the real opton example, both 16

17 the equty rsk premum and beta wll be functons of the project beta and other varables capturng real opton effects. Further assume that these functons can be lnearzed wth small errors: where p μ and μ and β are the equty rsk premum and the equty CAPM beta on stock ; p β denote the project rsk premum and the project CAPM beta on stock ; OP represents the vector of varables that capture the effect of real optons. As OP s measured n excess of the correspondng characterstcs of the market portfolo, by constructon, OP = 0. M μ = p β = g( β, OP ) = g p f ( μ, OP ) = f + f μ + f ' OP + ε, p + g β p 2 + g ' OP 2 + η, Even when the CAPM captures the correlaton between rsk and return for the projects μ = β μ ), t may fal n capturng such correlaton for the stocks ( μ β μ M ) due p p ( M to the exstence of real opton.. To removng the real-opton effects, the followng crosssectonal regresson (wthout the constant) s formulated to remove effects of μ and β : μ = a' OP β = b' OP + μ + β,. OP from The resduals, μ and β, are the opton-adjusted equty rsk premum and the equty CAPM beta on stock. The resdual varables, beng orthogonal to the real opton proxes, wll more closely capturng the project rsk premum and project beta. As the CAPM predcts, μ β. Take the cross-sectonal average: μ β M M = μ = a' OP = β = b' OP + μ + β = a' OP = b' OP M M + μ + β = μ = β,. Snce β =1 by constructon, μ = β μ, whch n turn mples: M M 17

18 μ = β or the CAPM holds for the opton-adjusted equty rsk premum and beta approxmately (subject to the lnear approxmaton error). We put ths hypothess to test n our smulaton exercse (Secton 4) and n the two-stage regresson analyss on data (secton 5). μ M, 4. The CAPM and Project Cost of Captal: Smulaton Evdence In ths secton, we conduct a smulaton exercse to evaluate the effect of real optons on expected stock returns. We smulate a large cross-secton of frm. Each frm conssts of a prmtve project wth attached call optons, wth those call optons exprng n one year. The value of those attached call optons, w, as percentage of the total value of the frm, vares across frms. In addton, the call opton also dffers across frms n ts moneyness ( k ) and mpled volatlty ( σ ). We assume that the call opton can be prced correctly usng the standard Black-Scholes model. We do not model fnancal leverage n ths smulaton. The expected return on the prmtve project s prced by the CAPM wth a frm-specfc p CAPM beta of β. The CAPM cannot capture the expected return on the call opton due to the nonlnear nature of opton payoff. Overall, the expected excess return or the rsk premum on the frm s: p O p O μ = (1 w ) μ + w μ = (1 w ) β μ + w μ. M In the smulaton, we choose the followng parameter values: a market rsk premum ( μ ) of 7% (contnuously compounded), market rsk free rate of 1% (contnuously compounded), a market volatlty of 0.2, w from [-0.5, 0.5] wth an ncrement of 0.1 at a tme, k from [0.75,1.25] wth an ncrement of 0.05 at a tme, σ from [0.3,0.7] wth an ncrement of 0.1 at a tme, and the prmtve project beta ( β ) from [0,2] wth an p M 18

19 ncrement of 0.1 at a tme. For each vector of [ w, k, σ, p β ], we can compute the rsk premum ( μ ) and the CAPM beta ( β ) of the stock numercally. Fnally, we add a measurement error to the CAPM beta. The measurement error s drawn from a normal dstrbuton of zero mean and a varance of Snce the CAPM cannot prce the expected return on opton component, t wll not be able to prce the expected return on the frm ether. Panel A of Fgure 1 confrms ths result. When we plot the frm expected return vs. the frm CAPM betas, we do not observe the postve relaton as would be predcted by the CAPM. In fact, f we run a cross-secton regresson of frm expected return on the frm CAPM betas, we confrm the well-documented falure of the CAPM n explanng the cross-sectonal varaton n equty rsk prema. Frst, the slope coeffcent, whch can be nterpreted as the market rsk premum, s only 2.02%, whch s way below the assumed value of 9.51% (gven contnuously compounded market rsk premum of 7%, rsk free rate of 1% and a market volatlty of 20%). Second, when the rsk premum estmate s based towards zero, the ntercept term s lkely to be postve and sgnfcant. Ths s exactly what we fnd. The ntercept term s 7.40% wth a t-value about 24. Fnally, the regresson R-square s only 1.24%. As the CAPM holds on the prmtve projects by constructon n our smulaton, the falure of the CAPM n predctng the frm returns should not nvaldate the use of the CAPM for the purpose of the project cost of captal calculaton. A natural queston that arses s how to recover the project beta from the emprcally observable frm beta. The prevous secton dscusses a smple method of removng the effect of real optons from both frm beta and frm rsk premum. We frst evaluate the effectveness of such smple regresson approach n the context of our smulated example. The mpact of optons on expected returns n the smulated example s fully determned by three parameters: portfolo weght on the call opton ( w ), the moneyness of the call opton ( k ) and the mpled volatlty ( σ ). Therefore, the opton proxes ( OP ) n ths 19

20 example, s a vector of cross-sectonally demeaned [ w, k, σ ]. Although ther mpact on frm rsk premum and beta enters n a nonlnear fashon, we are hopng to capture to the frst order usng a lnear regresson. We remove the opton effects from the frm CAPM beta by regressng t on the opton proxes n the cross-secton wth no constant. The resdual or the opton-adjusted beta ( β ) should therefore capture the true project CAPM beta. We verfy ths n Panel B of Fgure 1. When plotted together, the true project beta ( β ) lnes up qute well wth the opton-adjusted beta ( β p ). When we regress β on p β, we get a R-square above 90% and a slope of Despte the nonlnear nature of the opton effect, a smple lnear regresson seems to be qute successful n removng opton effect from the frm beta and recoverng the true project beta. In addton, we regress the frm rsk premum ( μ ) on the opton proxes n the crosssecton wth no constant to obtan the opton-adjusted frm rsk premum ( μ ). The opton-adjusted rsk premum should capture the rsk premum on the project. Snce the CAPM s assumed to hold for the project, we expect μ to be equal to the product of the opton-adjusted beta and the market rsk premum, or μ = β μ. Panel C of Fgure 1 M confrms ths. We fnd that μ μ on s clearly postvely related to β. When we regress β, we fnd a slope coeffcent of 9.33% whch s very close to the true market rsk premum of 9.51%. At the same tme, the ntercept term s 0.02% and s not sgnfcantly dfferent from zero. In another words, the CAPM holds once the opton effects are removed wth a smple regresson procedure. Fnally, the R-square s also much hgher at 53.2%. The R-square s not equal to 1 for several reasons. Frst, lnear approxmaton errors are affectng both μ measured wth error. and β. Second, beta s assumed to be 20

21 In realty, nvestors do not drectly observe opton-related parameters [ w, k, σ ]. They observe emprcal proxes that are hghly correlated wth real opton effects. We analyze these real opton proxes and ther effectveness n removng real opton effects n the next secton. 5. The CAPM and Project Cost of Captal: Emprcal Evdence 5.1 Sample and Data Constructon We start wth frms covered by CRSP wth common shares outstandng over the perod of We focus on large and mature frms. As many market consttuents actvely produce and gather nformaton for large frms, nformaton asymmetry s mnmal, and market nvestors are more lkely to share common belefs on frm prospects. Smlarly, large frms are less lkely to experence an extended perod of msprcng. At the tme we form portfolos of stocks, we therefore exclude frms wth market captalzaton less than the NYSE 10 th percentle breakpont, frms wth a prce of less than $5, as well as frms that are lsted for less than 3 years. Chopra, Lakonshok and Rtter (1992) note that the beta of stocks experencng recent extreme returns wll be measured poorly. To mnmze the effects of temporary prce movements, we further exclude frms n the decles wth the hghest/lowest pror 12-month stock returns (momentum stocks) n each June of the sample perod. All the appled flters make use of nformaton that s avalable at the tme of portfolo formaton, thus should not ntroduce any look-ahead bas. After applyng these flters, our sample stll covers 75% of the entre unverse of CRSP stocks n terms of market captalzatons. In computng portfolo returns, we use CRSP delstng returns whenever approprate. For stocks that dsappear from the dataset due to delstng, merger or acquston, we assume that we nvest the proceeds from such events n the remanng portfolo. For each sample frm n June of a gven sample year, the followng varables are constructed: Beta estmate (Beta) s obtaned as the slope of the CAPM regresson, usng the pror 60 months of return records from CRSP. Sze s measured as the market 21

22 captalzaton (n mllon) for a sample frm on the last tradng day of June each year. Detaled varable defntons can be found n Table 1. Our CRSP sample s further ntersected wth COMPUSTAT data where the book values n June of the portfolo formaton year are avalable. We requre a mnmum 3-month gap n matchng the accountng data of calendar year t-1 and return data of calendar year t to ensure that the accountng nformaton s known for the use of portfolo constructon n June. In our study, we match accountng data of frms wth fscal year end n or before March of calendar year t wth returns for July of year t to June of year t+1. The followng varables are constructed usng nformaton avalable from annual COMPUSTAT fles: BE s the book value (n mllons) as the sum of stockholders equty, deferred tax, and nvestment tax credt, and convertble debt, mnus the lqudaton value of preferred stocks 5 (Kayhan and Ttman (2007)). The data s further supplemented wth hstorcal equty data (Davs, Fama, and French (2000)) avalable from Ken French s webste. BM s the book-to-market rato (BE/SIZE), whle DE s the debt-to-equty rato, calculated as the dfference of total assets and book value of equty, scaled by Sze. We measure a sample frm s nvestment growth by the percentage dfference n captal expendtures (Data128) at the end of fscal year t 1 relatve to captal expendtures at the end of fscal year t 3 (Capex). Icov s the nverse nterest coverage rato gven by nterest expense dvded by operatng ncome before deprecaton. Ivol measures the average monthly dosyncratc volatlty (as n Ang et al, 2006). The majorty of our emprcal analyss s conducted usng our CRSP/COMPUSTAT sample as descrbed above. 5.2 Aged Beta Some of the poor emprcal performance of the CAPM may be due to measurement ssues. The CAPM s a statc model, so frms have constant betas. The real world s dynamc, and n practce the systematc rsk, as measured by the CAPM betas of frms, wll change over tme. An unantcpated ncrease n beta wll cause the frm s stock prce to fall due to the ncrease n the return nvestors requre to bear the ncreased rsk. Whereas n a 5 If the value of stockholder s equty s mssng, we replace the value of BE wth (total assets total labltes lqudaton value of preferred stocks, deferred tax and nvestment tax credt, and convertble debt). 22

23 hypothetcal deal world, prces wll adjust nstantaneously, n the real world prce dscovery nvolves aggregatng dfferent vews about the systematc rsk n a stock through tradng among nvestors. Thus, convergence s lkely to be slow and takes place over tme. Illusory proftable tradng opportuntes based observed transactons prces may appear, even though they may not be realzable gven the prce mpact of trades through whch prce dscovery takes place. In that case an ncrease n the systematc rsk of a stock wll lead to lower returns n the mmedately followng months. In order to take nto account such possble slow dffuson of nformaton, Hoberg and Welch (2007) suggest usng aged betas (betas estmated usng data from two to ten years back) n evaluatng the emprcal support for the CAPM. When we follow ther suggeston, we fnd stronger emprcal support for the CAPM n explanng the average returns on large and mature stocks, whch presumably are less prone to msprcng. We work wth the beta-sorted portfolos. Specfcally, we sort stocks n our sample, whch conssts mostly of large and mature frms, nto ten portfolos by Beta n each June of Agan, Beta s estmated as the slope of the CAPM regresson, usng monthly return records n the prevous fve years of tradng. On average, each of our ten beta-sorted portfolos conssts of approxmately 108 frms. The descrptve statstcs of characterstcs for the Beta-sorted portfolos are presented n Panel A of Table 2. The average beta vares from a hgh of 2.09 for portfolo 1 (hgh Beta), to 0.36 for portfolo 10 (low Beta). Among the hgh Beta frms, analysts are more lkely to provde hgher growth estmates. Hgh Beta frms also tend to have lower bookto-market (BM) and leverage (DE) ratos, wth the average DE ncreasng monotoncally from portfolo 2 to portfolo 9. There s no notceable trend n Sze and the number of analysts provdng earnngs forecasts (Nayst) across the ten portfolos. The value-weghted monthly returns of the 10 Beta-sorted portfolos are used n our asset prcng tests. The average portfolo returns are computed up to the thrd year subsequent to portfolo formaton. Our frst-year average return (Vwret 1 ) corresponds to an average of monthly returns n the perod of July (t) to June (t+1); smlarly, Vwret 3 corresponds to 23

24 the average return of July (t+2) to June (t+3). For the Beta-sorted portfolos, the frstyear average return (Vwret 1 ) ranges from 1.25% for portfolo 6 to 0.99% for portfolo 10, whle the thrd-year average return (Vwret 3 ) ranges from 1.17% for portfolo 1 to 0.90% for portfolo 10. Whle the portfolo returns postvely correlate wth Beta n the frst year subsequent to portfolo formaton, the correlaton becomes stronger n the thrd year. We compare the standard CAPM and the Fama-French three-factor model n our tme seres regresson analyss on the 10 Beta-sorted portfolos. For each model, we compute the average prcng error as well as the Gbbons-Ross-Shanken (GRS) statstcs for testng whether all the ntercepts for the 10 portfolos are jontly zero. We analyze the correlaton between the frst-year portfolo returns and rsk factors (as presented n Panel B of Table 2). There exsts sgnfcant prcng error of 17 bass ponts per month wth the standard CAPM, and the GRS statstcs reject that all 10 ntercepts are jontly zero. Includng two more factors (SMB and HML) does not reduce the prcng error. Hoberg and Welch (2007) argue that nvestors may be slow n adjustng to the recent change n market rsk, and recommend the use of aged beta (beta from 2 to 10 years ago). Table 2 gves the relatve performance of the CAPM and the Fama and French threefactor model usng aged betas. The descrptve statstcs n Panel A of Table 2 s consstent wth Hoberg and Welch (2007). There s stronger relaton between Beta and Vwret 3. As can be seen from Panel C of Table 2, when we use aged betas (.e., skp two years before formng portfolos based on ther hstorcal betas), the average absolute alpha drops to 10bp/month for both models; there s no evdence aganst ether model accordng to the GRS statstc. Whle Hoberg and Welch s suggeston to skp two years certanly mprove the performance of the CAPM, we also note that the sze and momentum flters appled to our sample are also mportant n mnmzng the nose assocated wth the beta and the return. In Panel D of Table 2, we present the tme-seres regresson results wthout applyng the sze and momentum flters. GRS test stll reject both the CAPM and the three-factor model despte usng the thrd-year returns. 24

25 Panel E of Table 2 reports the results obtaned usng the cross sectonal regresson method. Once we appled the sze and momentum flters and used the thrd-year return, the cross sectonal adjusted R-Square ncreases from 39% to 81% for the CAPM. There s not much gan from movng to the Fama and French three-factor model for these ten portfolos. Why do the thrd-year returns produce smaller CAPM alphas? We take a closer look at ths queston n Panel F of Table 2. We decompose the dfference between the thrd-year alpha ( α y3 ) and the frst-year alpha ( α y1) nto a component due to the return dfference and a component due to the change n beta: where α e Ry 1, e e e e e e ( ) ( ) ( ) ( ) e y β y RM Ry β y RM = Ry Ry β y y RM y3 α y1 = R β 1 e Ry 3 and e R M denote the frst-year portfolo, thrd-year portfolo and the market excess returns; β y1 and β y3denote the CAPM beta estmated usng the frst- and thrd-year returns respectvely. The result n Panel F suggests that the dfference n alpha s manly due to the return dfference rather than a change n rsk measured by beta. Consequently, the larger alphas (n absolute terms) assocated wth frst-year returns lkely reflect abnormal returns that are transtory n nature. 6, 5.3 BM Effect and Real Optons In ths secton, we focus on the most promnent emprcal challenge to the CAPM, the book-to-market (value) effect, n that those stocks wth hgher book-to-market ratos tend to earn hgher returns. We examne whether the BM effect s more pronounced among frms wth certan realopton-related characterstcs usng the Fama-MacBeth (1973) cross-sectonal regresson 6 We fnd (not reported, avalable on request) that stocks mgratng across beta-decles recently pose the most challenge to the CAPM. Ther realzed return durng the two years followng a large ncrease (decrease) n beta tends to be relatvely low (hgh) on a rsk adjusted bass. In contrast, the thrd-year returns are less affected by such transtory component n return and thus are more approprate to use for asset prcng tests. 25

26 analyss. The ndependent varable s the annual ndvdual stock excess return (n excess of rsk-free rate). As explanatory varables, we nclude the BM rato multpled by ndcator dummy varables for Hgh/Low Capex stocks, Hgh/Low DE stocks, Hgh/Low Icov stocks and Hgh/Low Ivol stocks. As control varables, we also nclude sze, past returns and turnovers. Regresson model 1 confrms the BM effect. The BM characterstc s sgnfcantly postvely related to stock return. Confrmng the results n Anderson and Garca-Fejoo (2006) and Xng (2008), regresson model 2 suggests that BM effect s only sgnfcant among stocks experencng hgher growth rates n ther captal expendture. In addton, regresson model 4 shows that BM effect s only sgnfcant among stocks wth hgh dosyncratc volatlty whch also proxes for more growth optons as argued by Cao, Smn and Zhao (2008). Regresson model 3 suggests that BM effect s only sgnfcant among stocks experencng fnancal dstress as measured by hgh fnancal leverages, confrmng the results n Grffn and Lemmon (2002). Fnally, ths pattern s confrmed when fnancal dstress rsk s measured by a hgh nverse nterest coverage rato (Icov). Overall, our results ndcate that BM effects are lkely drven by real optons on the prmtve projects, whch are frm-specfc characterstcs, and not by characterstcs of the underlyng prmtve projects. 5.3 Real Opton Adjustment When there are real optons assocated wth the prmtve projects undertaken by the frm, Both equty returns and equty betas are complcated functons of these real optons and the CAPM may not hold. Panel A of Table 4 confrms the well-documented falure of the CAPM beta n explanng the cross-sectonal varaton n returns. When we regress the annual ndvdual stock excess return (n excess of rsk-free rate) on the CAPM beta (aged) n a Fama-MacBeth (1973) cross-sectonal regresson, we get a slope coeffcent close to zero and a huge ntercept term. When we nclude the BM characterstc, t becomes sgnfcant (t-value = 2.07). Cohen and Polk (1998) document that the BM effect s stronger wthn ndustry 26

27 rather than across ndustres. When we frst demean the BM characterstc wthn each of the 10 ndustry to obtan BM_nd, we ndeed fnd BM_nd to be more sgnfcant than BM n explanng cross-sectonal varaton n stock returns. When we nclude the Fama- French factors HML and SMB n the regresson, we fnd HML factor loadngs to be more sgnfcant. However, at ndvdual stock level, nether factor loadng s sgnfcant. Confrmng the result n Danel and Ttman (1997), BM characterstc remans to be sgnfcant even after ncludng the Fama-French factor loadngs. Another reason why BM s always sgnfcant s that BM s a functon of stock prce and s measured wthout error. As ponted out by Berk (1995), the CAPM beta and other factor loadngs, measured wth error, may not drve out BM even n an economy where lnear factor prcng model holds. If the falure of the CAPM on equty return s drven by real optons, then removng the effect of real optons from return and beta should mprove the performance of the CAPM. We follow the procedure descrbed n Secton 3 to conduct real opton adjustments usng a cross-sectonal regresson. Specfcally, n the frst-stage regresson, we regress the annual (excess) stock return or the factor loadngs (ncludng the CAPM beta) on our four real opton proxes (Capex, Icov, Ivol and DE). The real opton proxes are measured n excess of those of the market portfolo and the regresson has no ntercept term. The procedures ensure that the CAPM relaton holds for the market portfolo. The resduals from these regressons are the opton-adjusted (excess) returns and opton-adjusted betas. Gven the fact that the true real optons, whch are not drectly observable, affect both returns and betas n hghly complcated and nonlnear fashons, our smple regresson approach may not fully remove real opton effects. However, to the extent our real opton proxes correlate wth the true real optons, the opton-adjusted (excess) returns and opton-adjusted betas from our frst-stage regresson should more closely resemble the rsk premum and betas of the underlyng prmtve projects. If so, the CAPM should perform better after the real opton-adjustment. When we regress the opton-adjusted (excess) returns on the opton adjusted betas n the second-stage Fama-MacBeth cross-sectonal regresson, we fnd the opton-adjusted 27

28 CAPM beta becomes sgnfcantly and postvely related to opton-adjusted (excess) stock return. The results are reported n Panel B of Table 4. The slope coeffcent s 5.41% (tvalue = 2.70). We expect effects of the standard error-n-varable problem n our analyss, as both the tme-seres estmaton of beta and the frst-stage opton adjustment ntroduce measurement errors. We fnd that the slope coeffcent s smaller than the market rsk premum, and the ntercept term s sgnfcantly dfferent from zero, lkely results of such error-n-varable problem. However, the ntercept term of 2.66% s much smaller compared to the correspondng ntercept n Panel A (when no real opton adjustment s made). Interestngly, once real opton adjustments are performed, the CAPM beta becomes the most useful beta among all three Fama-French factor loadngs. The CAPM beta drves out the factor loadngs on both HML and SMB n explanng opton-adjusted stock returns. Fnally, to the extent that BM s capturng the mpact of real optons, a realopton-adjusted BM ceases to be sgnfcant n explanng stock returns. Ths s true even when we examne the ndustry-demeaned verson of BM. To summarze, we provde drect evdence suggestng that the falure of the CAPM on stock returns s lkely due to the real opton effect. Once these real optons are removed, the CAPM works reasonably well n explanng the rsk-premum on the prmtve projects and BM characterstc provdes no addtonal explanatory power. 6. Concluson In ths paper, we evaluate the emprcal evdence aganst the standard CAPM from the perspectve of a person who beleves that t provdes a reasonable estmate of a project s cost of captal. For that we dfferentate the requred expected return on potental elementary projects avalable to a frm from the requred expected return on a frm s stocks. The real opton to modfy exstng projects and undertake new projects avalable to frms may be an mportant reason for the poor performance of the CAPM n explanng the 28

29 cross secton of returns on sze and book-to-market sorted stock portfolos. That lends support to the McDonald and Segel (1985) and Berk, Green and Nak (1999) observaton that stock returns need not satsfy the CAPM even when the expected returns on all ndvdual projects do. We propose a smple regresson procedure to estmate the project beta of a frm and demonstrate that the CAPM works well on capturng the correlaton between project beta and rsk premum. Our smulaton evdence suggests that the CAPM performs reasonably n computng the project cost of captal. Emprcally, we frst show that there s more support for the CAPM than has been prevously reported n explanng cross-sectonal varaton n stock returns. We document that the CAPM beta does a reasonable job n explanng the returns on CAPM-beta-sorted portfolos once we apply several data flters. We then examne the book-to-market effect whch poses one of the greatest challenges to the CAPM. We fnd that the BM effect s only present among frms assocated wth lots of real optons. These fndngs are consstent wth the vew that the BM effect may n a large part be due to the opton a frm has to modfy/abandon exstng projects and/or undertake new projects. Fnally, when we compute project betas from a smple regresson procedure, we fnd that t does a reasonable job n explanng the opton-adjusted stock returns even at the ndvdual stock level. Ths drect emprcal evdence further supports the vew that the falure of CAPM n explanng the cross-secton of stock returns s due to real optons avalable to a frm. Overall, there s lttle evdence n the data to change one s pror belefs that project cost of captal estmates provded by the CAPM are satsfactory. Our evdence suggests that the CAPM may stll be used to calculate the project cost of captal and the real optons assocated wth the project should be prced separately for the purpose of captal budgetng. 29

30 References: Anderson, C., and L. Garca-Fejoo, 2006, Emprcal Evdence on Captal Investment, Growth Opton, and Securty Returns, Journal of Fnance 61, Ang, A., and J. Chen, 2007, CAPM Over the Long Run: , Journal of Emprcal Fnance 14, Ang,A.,R. Hodrck, Y.Xng,and X. Zhang, 2005, The Cross-Secton of Volatlty and Expected Returns, Journal of Fnance 51: Ang, A., and J. Lu, 2004, How to Dscount Cashflows wth Tme-Varyng Expected Returns, Journal of Fnance 59, Adran, T. and F., Franzon, 2008, Learnng about Beta: Tme-Varyng Factor Loadngs, Expected Returns, and the Condtonal CAPM, workng paper, Federal Reserve Bank of New York. Bal, T., N., Cakc, and Y., Tang, 2009, The Condtonal Beta and the Cross-Secton of Expected Returns, Fnancal Management, forthcomng. Ball, R., and P., Brown, 1968, An Emprcal Evaluaton of Accountng Income Numbers, Journal of Accountng Research 6, Banz, R. W., 1981, The Relatonshp between Return and Market Value of Common Stocks, Journal of Fnancal Economcs 9, Basu, S., 1983, The Relatonshp between Earnngs Yeld, Market Value, and Return for NYSE Common Stocks: Further Evdence, Journal of Fnancal Economcs 12, Bauer, R., M., Cosemans, R., Frehen,and P., Schotman, 2009, Effcent Estmaton of Frm-Specfc Betas and ts Benefts for Asset Prcng and Portfolo Choce, workng paper. Berk, J., 1995, A crtque of sze-related anomales, Revew of Fnancal Studes 8, Berk, J., R. Green, and V. Nak, 1999, Optmal Investment, Growth Optons, and Securty Returns, Journal of Fnance 54, Bernardo, A., B., Chowdhry, and A., Goyal, 2007, Growth Optons, Beta, and the Cost of Captal, Fnancal Management 36(2), Cao, C., T. Smn, and J. Zhao, 2008, Can Growth Optons Explan the Trend n Idosyncratc Rsk? Revew of Fnancal Studes 21, 2008,

31 Chan, K., A., Hameed, and S., Lau, 2003, What If Tradng Locaton s Dfferent from Busness Locaton? Evdence from the Jardne Group, Journal of Fnance 58, Campbell, J., 2003, Consumpton-Based Asset Prcng, Chapter 13 n George Constantndes, Mlton Harrs, and Rene Stulz eds. Handbook of the Economcs of Fnance Vol IB, Noth-Holland, Amsterdam, Carlson M., A. Fsher, and R. Gammarno, 2004, Corporate Investment and Asset Prce Dynamcs: Implcaton for the Cross-secton of Returns, Journal of Fnance 59, Chopra, N., J. Lakonshok and J. R. Rtter, 1992, Measurng Abnormal Performance, Journal of Fnancal Economcs 31, Cohen, R., and C. K. Polk, 1998, The Impact of Industry Factors n Asset-Prcng Tests, workng paper, Harvard Unversty. Cooper, I, 2006, Asset Prcng Implcaton of Nonconvex Adjustment Costs and Irreversblty of Investment, Journal of Fnance 61, Danel, K., and S. Ttman, 1997, Evdence on the Characterstcs of Cross-Sectonal Varaton n Stock Returns, Journal of Fnance 52, Davs, J., E. Fama, and K. French, 2000, Characterstcs, Covarances, and Average Returns: , Journal of Fnance 55, DeBondt, W., and R. Thaler, 1985, Does the Stock Market Overreact? Journal of Fnance 40, Duffe, D., 2003, Intertemporal Asset Prcng Theory, Chapter 11 n George Constantndes, Mlton Harrs, and Rene Stulz eds. Handbook of the Economcs of Fnance Vol IB, North-Holland, Amsterdam, Dybvg, P., and J. Ingersoll, 1982, Mean-varance Theory n Complete Markets, Journal of Busness 55, Dybvg,P., and S. Ross, 1985, Dfferental Informaton and Performance Measurement Usng a Securty Market Lne, Journal of Fnance 40, 1985, Fama, E., and K. French, 1992, The Cross Secton of Expected Stock Returns, Journal of Fnance 47, Fama, E., and K. French, 1993, Common Rsk Factors n Stock and Bond Returns, Journal of Fnancal Economcs, 33,

32 Fama, E. and K. French, 1996, The CAPM s Wanted, Dead or Alve, Journal of Fnance 51, Fama, E. and K. French, 1997, Industry Cost of Captal, Journal of Fnancal Economcs 43, Fama, E. and K. French, 1999, The Corporate Cost of Captal and the Return on Corporate Investment, Journal of Fnance 54, Fama, E. and K. French, 2004, The CAPM: Theory and Evdence, Journal of Economc Perspectves 18, Fama, E., and K. French, 2006, The Value Premum and the CAPM, Journal of Fnance, 61, Fama, E. F., and J. MacBeth, 1973, Rsk, return and equlbrum: Emprcal tests, Journal of Poltcal Economy 81, Ferson, W., 2003, Tests of Multfactor Prcng Models, Volatlty Bounds and Portfolo Performance, In George M. Constantndes, Mlton Harrs and Rene M. Stultz, Edtors: Handbook of the Economcs of Fnance, Elsever Scence Publshers, North Holland, pp Ferson, W. and D. H. Locke, 1998, Estmatng the Cost of Captal Through Tme: An Analyss of the Sources of Error, Management Scence 44, Ferson, W., S. Sarkssan, and T. Smn, 1999, The Alpha Factor Asset Prcng Model: A Parable, Journal of Fnancal Markets 2, Gbbons, M., S., Ross and J. Shanken, 1989, A Test of the Effcency of a Gven Portfolo, Econometrca 57, Gomes J. F., Leond Kogan, and Lu Zhang, 2003, Equlbrum cross secton of returns Journal of Poltcal Economy 111, Graham, John R., and Campbell Harvey, 2001, The Theory and Practce of Corporate Fnance: Evdence from the Feld, Journal of Fnancal Economcs 60, Grffn, J.M., and M.L. Lemmon, 2002, Book-to-Market Equty, Dstress Rsk, and Stock Returns, Journal of Fnance 57, Hansen, L. P., and S. Rchard, 1987, The Role of Condtonng Informaton n Deducng Testable Restrctons Impled by Dynamc Asset Prcng Models, Econometrca 55,

33 Hoberg, G. and I. Welch, 2007, Aged and Recent Market Betas n Securtes Prcng, workng paper, Brown Unversty. Jegadeesh, N., and S. Ttman, 1993, Returns to Buyng Wnners and Sellng Losers: Implcatons for Stock Market Effcency, Journal of Fnance 48, Jagannathan, R., and R. A. Korajczyk, 1986, Assessng the Market Tmng Performance of Managed Portfolos, Journal of Busness 59, Jagannathan, R., and I., Meer, 2002, Do We Need CAPM for Captal Budgetng? Fnancal Management 31, Jagannathan, R., and Z. Wang, 1996, The Condtonal CAPM and the Cross-Secton of Expected Returns, Journal of Fnance 51, Kayhan and Ttman, 2007, Frms Hstory and Ther Captal Structure, Journal of Fnancal Economcs 83, Kumar, P., S. Srescu, R. Boehme and B. Danelsen, 2008, Estmaton Rsk, Informaton, and the Condtonal CAPM: Theory and Evdence, Revew of Fnancal Studes 21, Lakonshok, J., A. Shlefer, and R. Vshny, 1994, Contraran Investment, Extrapolaton, and Rsk, Journal of Fnance 49, Lettau, M., and S., Ludvgson, 2001, Resurrectng the (C)CAPM: A Cross-Sectonal Test When Rsk Prema Are Tme-Varyng, Journal of Poltcal Economy 109, Lewellen, J. and S., Nagel, 2006, The Condtonal CAPM Does Not Explan Asset- Prcng Anomales, Journal of Fnancal Economcs 82, Lntner, J., 1965, The Valuaton of Rsky Assets and the Selecton of Rsky Investments n Stock Portfolos and Captal Budgets, The Revew of Economcs and Statstcs 47, Lustg, H., and S.,Van Neuwerburgh, 2005, Housng Collateral, Consumpton Insurance, and Rsk Prema: An Emprcal Perspectve, Journal of Fnance 60, McDonald R. L., and D. Segel, 1985, Investment and the Valuaton of Frms When There s an Opton to Shut Down, Internatonal Economc Revew 26, Mehra and Prescott, 2003, The Equty Premum n Retrospect, George Constantndes, Mlton Harrs, and Rene Stulz eds. Handbook of the Economcs of Fnance Vol IB, North-Holland, Amsterdam. 33

34 Mohanram, P., 2005, Separatng Wnners from Losers among Low Book-to-Market Stocks Usng Fnancal Statement Analyss, Revew of Accountng Studes 10, Pastor, L. and R. F. Stambaugh, 1999, Costs of Equty Captal and Model Msprcng, Journal of Fnance 54, Pastor, L. and R. F. Stambaugh, 2003, Lqudty rsk and expected stock returns, Journal of Poltcal Economy 111, Petkova, P. and L. Zhang,2005, Is Value Rsker than Growth? Journal of Fnancal Economcs 78 (1), Potrosk, J., 2000, Value Investng: The Use of Hstorcal Fnancal Statement Informaton to Separate Wnners from Losers, Journal of Accountng Research 38, Rosenberg, B., J., Red, and R., Lansten, 1985, Persuasve evdence of market neffcency, Journal of Portfolo Management 11, Rubnsten, M., 1976, The Valuaton of Uncertan Income Streams and the Prcng of Optons, Bell Journal of Economcs and Management Scence 7, Santos, T., and P., Verones, 2006, Habt Formaton, the Cross Secton of Stock Returns and the Cash-Flow Rsk Puzzle,workng paper, Unversty of Chcago. Schwart, W., 2003, Anomales and Market Effcency, Chapter 15 n George Constantndes, Mlton Harrs, and Rene Stulz eds. Handbook of the Economcs of Fnance Vol IB, North-Holland, Amsterdam, Sharpe,W., 1964, Captal Asset Prces: A Theory of Market Equlbrum under Condtons of Rsk, Journal of Fnance 19, Sten, J. C., 1996, Ratonal Captal Budgetng n an Irratonal World, Journal of Busness 69, Welch, I., 2008, The Consensus Estmate for the Equty Premum by Academc Fnancal Economsts n December 2007, workng paper, Brown Unversty. Wang, K. Q., 2003, Asset Prcng wth Condtonng Informaton: A New Test, Journal of Fnance 58, Xng, Y., 2008, Interpretng the Value Effect Through the Q-theory: An Emprcal Investgaton, Revew of Fnancal Studes 21, Zhang, L., 2005, The Value Premum, Journal of Fnance 60,

35 Fgure 1 Opton-adjusted Betas and the CAPM: Smulaton Evdence We smulate a large cross-secton of frm. Each frm s a portfolo of a prmtve project and some call optons on t. The call optons expre n one year. We assume a market rsk premum ( μ M ) of 7% and market rsk free rate of 1%, both contnuously compounded. We also assume a market volatlty of 0.2. We vary portfolo weght on the call optons ( w ) from -0.5 to 0.5 wth an ncrement of 0.1. We vary opton moneyness ( k ) from 0.75 to 1.25 wth an ncrement of We vary opton mpled volatlty ( σ ) from 0.3 to 0.7 p wth an ncrement of 0.1. We vary the prmtve project beta ( β ) from 0 to 2 wth an ncrement of 0.1. For each vector of [ w, k, σ, p β ], we can compute the rsk premum ( μ ) and the CAPM beta ( β ) of the stock numercally. Fnally, we add a measurement error to the CAPM beta. The measurement error s drawn from a normal dstrbuton of zero mean and a varance of Panel A plots the frm rsk premum aganst frm CAPM beta. Panel B plots the project beta aganst opton-adjusted beta computed usng lnear regresson. Panel C plots opton-adjusted rsk premum aganst opton adjusted beta. 35