NBER WORKING PAPER SERIES CAPM FOR ESTIMATING THE COST OF EQUITY CAPITAL: INTERPRETING THE EMPIRICAL EVIDENCE. Zhi Da Re-Jin Guo Ravi Jagannathan

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1 NBER WORKING PAPER SERIES CAPM FOR ESTIMATING THE COST OF EQUITY CAPITAL: INTERPRETING THE EMPIRICAL EVIDENCE Zh Da Re-Jn Guo Rav Jagannathan Workng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA Aprl 2009 We thank for helpful comments Bob Chrnko, Bob Dttmar, Bruce Grundy, Dermot Murphy, Ernst Schaumburg, Bll Schwert (the Edtor), Wllam Sharpe, Mke Sher, Mtchell Warachka, Hong Yan, We Yang, Janfeng Yu, an anonymous referee, and 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, 2009 FIRS Conference, Unversty of Mnnesota macro-fnance conference, 2009 WFA, and the 20th conference on Fnancal Economcs and Accountng. The vews expressed heren are those of the author(s) and do not necessarly reflect the vews of the Natonal Bureau of Economc Research. NBER workng papers are crculated for dscusson and comment purposes. They have not been peerrevewed or been subject to the revew by the NBER Board of Drectors that accompanes offcal NBER publcatons by Zh Da, Re-Jn Guo, and Rav Jagannathan. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 CAPM for Estmatng the Cost of Equty Captal: Interpretng the Emprcal Evdence Zh Da, Re-Jn Guo, and Rav Jagannathan NBER Workng Paper No Aprl 2009, Revsed June 2011 JEL No. G00,G11,G12,G31 ABSTRACT We argue that the emprcal evdence aganst the Captal Asset Prcng Model (CAPM) based on stock returns does not nvaldate ts use for estmatng the cost of captal for projects n makng captal budgetng decsons. Snce stocks are backed not only by projects n place, but also the optons to modfy current projects and undertake new ones, the expected returns on stocks need not satsfy the CAPM even when expected returns of projects do. We provde emprcal support for our arguments by developng a method for estmatng frms' project CAPM-betas and project returns. Our fndngs justfy the contnued use of the CAPM by frms n spte of the mountng evdence aganst t based on the cross-secton of stock returns. Zh Da Unversty of Notre Dame 239 Mendoza College of Busness Notre Dame, Indana zda@nd.edu Re-Jn Guo Department of Fnance Unversty of Illnos at Chcago Unversty Hall 2431, M/C South Morgan Chcago, IL rguo@uc.edu Rav Jagannathan Kellogg Graduate School of Management Northwestern Unversty 2001 Sherdan Road Leverone/Anderson Complex Evanston, IL and NBER rjaganna@northwestern.edu

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. Whatever the crtcsm n the academc lterature, t contnues to be the preferred model n manageral fnance courses, and managers contnue to use t. Welch (2008) fnds that about 75.0% of fnance professors recommend usng the CAPM to estmate the cost of captal for captal budgetng. A survey of CFOs by Graham and Harvey (2001) ndcates that 73.5% of the respondents use the CAPM. The prmary emprcal challenge to the CAPM comes from several well-documented anomales. A varety of managed portfolos constructed usng varous frm characterstcs earn very dfferent returns on average from those predcted by the CAPM. 2 Fama and French (1993) conjecture that two addtonal rsk factors beyond the stock market factor used n emprcal mplementatons of the CAPM are necessary to fully characterze economywde pervasve rsk n stocks. Ther three-factor model has receved wde attenton and has become the standard model for computng rsk-adjusted returns n the emprcal fnance lterature. Almost all the anomales documented apply to stock returns. Should that be a reason to refran from usng the CAPM to calculate the cost of captal for a project? We revew the lterature and provde new emprcal evdence to argue that there s lttle drect evdence aganst usng the CAPM to estmate a project s cost of captal. The partcular model we 2 Notable among the anomales that challenge the valdty of the CAPM are the fndngs that average returns on stocks are related to frm sze (Banz, 1981), the earnngs-to-prce rato (Basu, 1983), the book-to-market value of equty (BM) (Rosenberg, Red, and Lansten, 1985), the 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). Many other studes confrm smlar patterns n dfferent datasets, ncludng n nternatonal markets. 3

4 consder s Ross's (1976) sngle-factor lnear beta prcng model based on the stock ndex portfolo. We refer to ths as the CAPM for convenence, followng conventon. Most frms have the opton to undertake, reject, or defer a new project, as well as the opton to modfy or termnate a current project. Therefore, we can look at a frm as a collecton of current and future projects and complex optons on those projects. McDonald and Segel (1985) observe that managers should optmally exercse these real optons to maxmze a frm s total value. The resultng frm value wll consst of both the net present values of the projects and the value of assocated real optons, whch s determned by how the frm expects to exercse those optons. Berk, Green, and Nak (1999) buld on ths nsght to develop a model where the expected returns on all projects satsfy the CAPM, but the expected returns on the frm s stock do not. 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), and Cooper (2006) provde several addtonal nsghts n buldng on the Berk, Green, and Nak (1999) framework. 3 We brng out the general ntuton behnd the falure of the CAPM n prcng optons on prmtve assets. Ths ntuton comes from Dybvg and Ingersoll (1982) and Hansen and Rchard (1987) n whch a gven stochastc dscount factor, lke the one correspondng to the CAPM, whle assgnng the rght prces to a subset of assets, may assgn the wrong prces to other assets. We frst llustrate ths ntuton n a factor prcng example smlar to that n Connor (1984). We then llustrate the mpact of optons n a numercal example that can be 3 In related work, Bernardo, Chowdhry, and Goyal (2007) hghlght the mportance of separatng the growth optons 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. 4

5 nterpreted usng the condtonal CAPM of Jagannathan and Wang (1996). An econometrcan usng standard tme seres methods may conclude that the CAPM does not hold for frms wth real optons, even when the returns on such frms satsfy the CAPM n a condtonal sense. When the senstvty of frms stock returns to economywde rsk factors changes n nonlnear ways because of the presence of such real optons, even when returns on ndvdual prmtve projects satsfy the CAPM, t may be necessary to use excess returns on certan managed portfolos (lke the Fama and French (1993) SMB and HML factors) as addtonal rsk factors to explan the cross-secton of stock returns. If that s the case, t would be justfable to use the CAPM for estmatng the cost of captal for projects, even f the CAPM cannot explan the cross-secton of average returns on varous managed portfolos. In general, both the equty rsk premum and the equty beta of a frm wll be complex functons of the frm s project beta and real opton characterstcs. If we project them on a set of varables capturng the features of real optons usng lnear regressons, the resdual rsk premum and the resdual beta wll be opton-adjusted, and wll more closely resemble the underlyng project rsk premum and project beta. Consequently, the CAPM may work well on the opton-adjusted rsk premum and beta. 4 We frst provde support for the opton-adjustment procedure and the CAPM wth respect to the opton-adjusted return and beta. We smulate a large cross-secton of all-equtyfnanced frms, each as a portfolo of a prmtve asset (project) and a call opton on the asset. Whle the CAPM works for the asset, n the presence of the opton, t does not work for the 4 Jagannathan and Meer (2002) dscuss another reason why the CAPM may be useful for captal budgetng. They argue that organzatonal captal may be n lmted supply n frms wth talented managers who generate postve net present value (NPV) projects. Such frms wll choose to mplement only those projects wth suffcently large NPVs, as f they use a hgh hurdle rate by addng a large hurdle premum to ther CAPM-based weghted average cost of captal. Jagananthan, Meer, and Tarhan (2011) fnd that whle managers do use a sgnfcant hurdle premum, the CAPM-based cost of captal s also an mportant determnant of the hurdle rate they use for makng captal budgetng decsons. 5

6 frm as a whole. A cross-sectonal regresson of the frm rsk premum on the frm beta produces a large ntercept term, a very small slope coeffcent, and an R-square close to zero, just as we would fnd n data. Once we opton-adjust the frm s beta by makng t orthogonal to a set of real opton proxes (opton moneyness, frm book-to-market rato, and asset dosyncratc volatlty), however, the opton-adjusted beta matches 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 evdence supportng the use of the CAPM for calculatng the cost of captal of a project for a full sample of stocks n the US from 1970 through Although real optons are not drectly observable, we proxy them usng three emprcal varables. The frst varable s the frm s book-to-market rato (BM thereafter), a common proxy for growth optons n the fnance lterature (see Smth and Watts, 1992, among others). Berk, Green, and Nak (1999) explctly lnks BM to growth optons. The second s the dosyncratc volatlty (Ivol). Cao, Smn, and Zhao (2008) and Bekaert, Hodrck, and Zhang (2010) establsh a theoretcal lnk between the growth optons avalable to managers and the dosyncratc rsk of equty. The thrd s the frm s return on asset (ROA). Recently, Chen, Novy-Marx, and Zhang (2010) argue that ROA s a good emprcal proxy for the margnal product of captal, whch s related to the margnal expanson opton as n the real opton model of Abel, Dxt, Eberly, and Pndyck (1996). We examne the performance of the CAPM for project cost of captal calculaton usng a two-stage cross-sectonal regresson. In the frst stage, we regress both the stock excess return and the stock beta on the three real opton proxes. The real optons proxes are measured n excess of these measures of the market, and the regresson has no ntercept terms; such 6

7 procedures ensure that the CAPM holds for the market exactly. The resdual excess returns and betas are opton-adjusted. In the second stage, we regress the opton-adjusted excess return on the opton-adjusted beta. Whle the stock beta s not sgnfcant n explanng the cross-sectonal varaton n average excess returns, the opton-adjusted beta s very sgnfcant n explanng the cross-sectonal varaton n average opton-adjusted excess returns. We correct the errors-n-varables problem n our cross-sectonal regresson that arses from estmaton errors assocated wth the rollng-wndow betas (Jagannathan, Km, and Skoulaks, 2010). After ths correcton, we fnd the regresson slope coeffcent on the opton-adjusted beta to be closer to the actual market rsk premum and the regresson ntercept to be much closer to zero, consstent wth the predcton of the CAPM. The opton-adjusted beta s related to but not exactly equal to the beta on the frm s asset-n-place. Ths s because a frm s beta s n general a complcated functon of the asset-n-place beta and the beta of embedded optons. A lnear regresson procedure wll not do a perfect job n solatng the asset-n-place beta. Fnally, we nvestgate the mpact of real opton adjustment on several well-known crosssectonal expected return anomales. We fnd that real opton adjustment allevates or even drves out several anomales related to long-term stock prce mean reverson. These anomales nclude the asset growth anomaly of Cooper, Gulen, and Schll (2008), the nvestment-related anomaly of Anderson and Garca-Fejoo (2006), Xng (2008), and Chen, Novy-Marx, and Zhang (2010), and the long-term return reversal of DeBondt and Thaler (1985). The real opton adjustment, however, has lttle mpact on anomales that are related to short-term return contnuaton such as prce momentum (Jegadeesh and Ttman, 1993) and earnngs momentum (Chan, Jegadeesh, and Lakonshok, 1996). To the extent that such short-term prce contnuaton typcally does not persst beyond a few quarters and requres frequent portfolo 7

8 rebalancng, t s probably less relevant for the cost of captal calculaton for a typcal project whose lfe usually extends beyond fve years. When we confne the analyss to a subsample of stocks after excludng stocks whose betas are lkely to be measured wth large errors, support for use of the CAPM beta n project cost of captal calculaton becomes even stronger. For nstance, the slope coeffcent n the crosssectonal regressons s almost the same as the hstorcal average excess return on the stock market ndex, and the ntercept term s nsgnfcantly dfferent from zero. Determnaton of the cost of captal has been an mportant focus n fnance. Fama and French make a convncng case that the CAPM fals to descrbe the cross-secton of stock returns (Fama and French, 1992, 1996, 1997, 1999, 2004, and 2006). 5 Indeed, most of the research n the asset prcng lterature focuses on understandng the determnants of expected returns on stocks. 6 Our prmary nterest, however, s n evaluatng the emprcal evdence aganst the use of the CAPM for project cost of captal calculatons n makng captal budgetng decsons. We llustrate the mpact of real optons through two examples n Secton 2, and also provde a more detaled revew of the related lterature. Secton 3 descrbes a smple regresson procedure to allevate the effect of real optons and a smulaton example. Secton 4 demonstrates the effectveness of opton-adjusted beta and presents evdence that supports use of the CAPM n project cost of captal estmaton usng emprcal analyss. Secton 5 concludes. 5 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. 6 We refer readers nterested n the broader asset prcng lterature to the excellent surveys by Campbell (2003), Duffe (2003), Ferson (2003), and Mehra and Prescott (2003). 8

9 2. Examples Dybvg and Ingersoll (1982) and Hansen and Rchard (1987) pont out that a gven stochastc dscount factor (SDF thereafter), lke the one correspondng to the CAPM, whle assgnng the rght prces to a subset of assets, may assgn the wrong prces to other assets. Treynor and Mazuy (1966), Henrksson and Merton (1981), Merton (1981), Dybvg and Ross (1985), Jagannathan and Korajczyk (1986), and Glosten and Jagannathan (1994) make related observatons that n an economy where the CAPM holds for stock returns, the returns on managed portfolos that have opton-lke features may not satsfy the CAPM. We use two examples to llustrate the ntuton behnd those observatons,.e., why the CAPM may prce the expected returns on prmtve projects but not those on optons. We frst explan ths ntuton n a factor prcng example smlar to that n Connor (1984). In ths example, the CAPM may hold condtonally as well as uncondtonally for a subset of assets but need not hold ether condtonally or uncondtonally for other assets. In the second numercal example, we show that ths ntuton can also be made clear usng the observatons n Jagannathan and Wang (1996), Gomes, Kogan, and Zhang (2003), and Zhang (2005), who show that the CAPM may not hold uncondtonally even when t holds n a condtonal sense The CAPM n a factor economy of Connor (1984) Followng Connor (1984), who derves an equlbrum verson of Ross s (1976) Arbtrage Prcng Theory, consder an economy wth K economywde pervasve factors, where the representatve nvestor s margnal utlty for end-of-perod wealth s a functon of only those pervasve factors. In such an economy, Connor (1984) shows that there s a unque SDF, that 9

10 s, some nonlnear functon of the pervasve factors that assgns the rght prces to all assets. We wll consder the specal case where K = 1;.e., there s only one pervasve factor, and that s the return on aggregate wealth portfolo (or the market portfolo). Now, followng Connor (1984), consder a subset of assets (denoted as projects) whose returns have the lnear factor structure: where r m s the market return (all returns n ths subsecton are gross returns), and E[ε r m ] = 0. As a result, the slope coeffcent b (whch we call the CAPM beta for convenence) can be computed by regressng r on r m. Let M denote the unque SDF that prces all assets correctly. Recall that,.e., M s some nonlnear functon of only r m. The functon f s lnear only under specal crcumstances (such as when nvestors have quadratc utlty as shown n Dybvg and Ingersoll, 1982). The economy may also nclude optons whose returns do not have the lnear factor structure. Nevertheless, we can stll regress the opton return on the market return and wrte: where ε o satsfes E[ε o r m ] = 0 by defnton of regresson, but E[ε 0 r m ] s strctly dfferent from 0. In that case, wthout loss of generalty, t should be possble to choose the opton n such a way (.e., there exsts such an opton snce we are n a complete market economy) that E[ε 0 f(r m )]= E[ε 0 M] 0. Snce M s a vald prcng kernel, we have:, The assumpton E[ε r m ] = 0 mples E[f(r m )ε ] = 0 or E[Mε ] = 0. Therefore we have: 10

11 If a rsk-free asset exsts, then the rsk-free rate satsfes: Substtutng ths expresson for nto gves: Note that by substtutng ths expresson for a nto the expresson for expected returns on any project whose return has a lnear factor structure, we get: whch gves the CAPM lnear-beta-prcng relaton for all assets,, wth returns that have a strct lnear factor structure. The expected return on the opton, however, does not satsfy ths CAPM relaton. Ths s because when we prce the opton return usng M, we have: Snce E[Mε o ] =E[f(r m )ε o ] 0, we have: Thus the CAPM relaton wll not hold for the opton expected return. Note, however, that a CAPM-lke sngle-beta relaton wll hold for all assets, ncludng optons that do not have lnear factor structure when r m s replaced by the return on the asset whose payoff s dentcal to that of M (such asset exsts when the market s complete). However, the CAPM sngle beta relaton (where beta s the regresson slope coeffcent of the 11

12 return on the asset on the return on the market portfolo) wll only hold for those assets that have a lnear factor structure A numercal example and the condtonal CAPM Now we present a numercal example n whch the CAPM correctly prces all prmtve projects but not the stock expected returns, and asset prcng anomales such as sze and bookto-market effects wll arse. The example s consstent wth several emprcal regulartes: (1) value stocks have hgher expected returns than the market and have postve CAPM alphas; (2) growth stocks have lower expected returns than the market and have negatve CAPM alphas; (3) value stocks have lower CAPM betas than growth stocks; (4) the equty rsk premum s countercyclcal; (5) value stocks are rsker than growth stocks when the expected rsk premum s hgh; and (6) sze and BM can descrbe cross-sectonal varaton n expected returns on stocks. The ntuton behnd these patterns can be understood n the context of the condtonal CAPM. Ths example and the frst one llustrate that when a frm s endowed wth real optons, the CAPM wll not explan ts equty expected return, and anomales such as sze and BM effects can arse. The CAPM could stll hold, however, for the prmtve projects, and can be used to compute the costs of captal for projects The economy For purpose of llustraton, we consder an economy wth a market 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: 7 To see ths more clearly, let M* denote the lnear projecton of M on the set of returns that have a lnear factor structure and let P(M*) denote the prce of M*. Then M*/P(M*) r f = K(r m r f ) where K s a constant, snce M*/P(M*) s the mnmum second moment return on the mean varance return fronter generated by the returns that have a lnear factor structure (see Hansen and Rchard, 1987). 12

13 25%). The returns on the market portfolo n these three states are: 40.4%, 8.0%, and -16.3%, respectvely, translatng to an expected return of 10.0% for the market portfolo. A frm n ths economy conssts of multple projects and one opton. All projects are dentcal wth an ntal cost of $1.00 whch can be vewed as the book value. Once undertaken, each project pays out an expected perpetual annual cash flow of $0.20. By assumpton, the CAPM prces these projects, whch all have a CAPM beta of 1, wth an approprate dscount rate of 10% (5% + 1 5% = 10% as predcted by the CAPM). The market value of each project s therefore 0.20/10% = $2.00. The opton can be one of two types: a value opton (VO) or a growth opton (GO) wth the state-contngent payoffs: State Probablty VO Payoff GO Payoff Up 0.25 $0.949 $1.808 Md 0.50 $0.949 $1.095 Down 0.25 $0.456 $ Prces and expected returns We assume no arbtrage opportuntes exst,.e., all SDFs that assgn the rght prces are strctly postve. Consder one such vald SDF, M = [0.7313, , ] across Up, Md, and Down states. 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 SDF, we can prce the two optons usng E[M payoff]. The results are summarzed as follows: 13

14 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 $0.726 $1.173 ER 5.0% 10.0% 13.8% 8.5% CAPM Beta CAPM ER 10.0% 10.5% 10.7% Gven the prces of these two optons, we can compute ther annual returns and expected returns. Wth the help of returns on any of the two optons, the market s now complete, and t can be verfed that M s the unque SDF that prces all the assets n ths economy. Snce there are three states, the opton payoffs cannot be replcated by tradng only the market portfolo and the rsk-free asset. The opton return therefore does not satsfy the factor structure assumpton (E[ε o r m ] = 0) dscussed earler so the CAPM relaton does not apply to ts expected return. Ths can be seen by computng optons covarances wth the market and ther CAPM betas. Although 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, the growth opton actually has a hgher CAPM beta. Because of the hgher CAPM beta, the CAPM wll predct a hgher expected return on the growth opton (10.7%) than on 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% % = 3.3%) whle the growth opton seems to underperform the market (t carres a negatve CAPM alpha of 8.5% % = - 2.2%). 14

15 The condtonal CAPM The ntuton behnd these results can be made clearer usng the condtonal CAPM of Jagannathan and Wang (1996) or, more broadly, the results n Hansen and Rchard (1987). The condtonal CAPM nterpretaton adds addtonal economc nsghts on why the value opton (VO) earns a hgher expected return than the growth opton (GO). When we add an ntermedate tme perod to the example, the one-perod trnomal tree s expanded to be a two-perod bnomal tree. The two states on each node are assocated wth equal probablty. The payoffs to an nvestment n the market portfolo (assumng an ntal nvestment of $1) are: T = 0 T = Sx Months 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%. The mmedate consequence of addng the ntermedate state s that the market s now dynamcally complete wth only two assets: the market portfolo, and the rsk-free asset, snce the opton payoff can now be replcated by tradng these two assets dynamcally. Both the value opton and the growth opton can now be prced usng the standard no-arbtrage replcaton argument, whch justfes the SDF gven n Secton (see Rubnsten, 1976, among others). 15

16 Value Opton (VO) Growth Opton (GO) T = 0 T = Sx T = Sx T = One Year T = 0 Months Months T = One Year $0.949 $1.808 $0.926 $1.390 $0.726 $0.949 $1.173 $1.095 $0.614 $1.069 $0.456 $1.095 Gven the payoffs (and the mpled returns) of both the market and the optons, we can compute the values on both nodes U and D for the perod from sx months 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 on the market gong forward s hgher followng a negatve market return n state (D), consstent wth the fact that the rsk premum s counter-cyclcal. In addton, the CAPM works for both optons condtonally (on each node). Ths s not surprsng, as the opton can be replcated by both the market and the bond, and the CAPM prces the expected returns on both assets (see Dybvg and Ross, 1985). The value opton has a hgher expected return uncondtonally because t has a 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 rsker when the rsk premum gong forward s hgh? Some emprcal evdence n the lterature suggests the answer s 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, whch offers at least a partal explanaton 16

17 for the value premum. 8 Why are value stocks more rsky when the expected rsk premum s hgh? Zhang (2005) provdes an elegant explanaton wthn the framework of the neoclasscal theory of nvestment. It s more costly for value frms to downsze ther captal assets snce they are typcally burdened wth more unproductve captal. As a result, value stocks returns covary more wth economc downturns when the expected rsk premum s hgh Stock characterstcs Despte the falure of the CAPM n prcng optons, the book-to-market rato and the sze of the frm serve as two suffcent statstcs for descrbng the expected returns of all frms n the economy. To see ths, note that all frms n the economy have two components: (1) the assets-n-place component, whch ncludes I projects, and (2) the 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. It can easly be verfed that, whether a frm s endowed wth the value opton or the growth opton, ts expected return can be expressed as 20% BM + 10%/Sze. Therefore the expected return ncreases wth BM and declnes wth Sze. In addton, BM and Sze explan the expected returns on all frms. 8 Other recent studes on the condtonal CAPM nclude Wang (2003), Ang and Chen (2007) and Gulen, Xng, and Zhang (2010). Smlar evdence s provded n the context of the consumpton CAPM by Lettau and Ludvgson (2001), Santos and Verones (2006), and Lustg and Van Neuwerburgh (2005). 9 Lewellen and Nagel (2006), however, argue that the varaton n betas and the equty premum would have to be mplausbly great for the condtonal CAPM to explan the sze of the value premum. Lewellen and Nagel (2006) use hgh-frequency returns n ther emprcal analyss. Chan, Hameed, and Lau (2003) demonstrate that prce and return may be drven n part 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 crosssectonal performance of the condtonal CAPM by usng more effcent estmaton technques. 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 cross-sectonal explanatory power and may explan the value premum after all. 17

18 Frms wth the value opton resemble value stocks. These frms have more assets-n-place, and because the value opton s cheaper, value stocks are assocated wth hgher BM. Because the value opton has a hgher expected return and postve CAPM alpha, so wll the value stocks. Frms wth the growth opton resemble growth stocks. Unlke 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 expectatons about future payoffs, market value must be correlated wth systematc rsk across stocks. In our numercal example, BM summarzes the frm s rsk relatve to the scale of ts asset base, and sze descrbes the relatve mportance of assets-n-place and the opton. Other work combnes ths ntuton wth key nsghts from the real optons lterature poneered by McDonald and Segel (1985) n lnkng frm-specfc nvestment patterns, valuaton, and expected returns. A 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; 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) show that these results contnue to hold n a general equlbrum settng as well. 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. Carlson, Fsher, and Gammarno (2004) model the optmal dynamc nvestment behavor of monopolstc frms facng stochastc product market condtons. Ther approach s smlar 18

19 n sprt to Berk, Green, and Nak (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 the counter-cyclcal 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 that allows the BM to be 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) together provde supportng evdence that the nvestment dynamcs of a frm drve the BM effect. More recently, Lu, Whted, and Zhang (2009) show that the dsperson n nvestment-to-captal between value and growth frms s the man drvng force of the BM effect. 3. The CAPM and the real opton adjustment More generally, both the equty rsk premum and beta wll be functons of the project beta and other varables capturng real opton effects. Assume that these functons can be lnearzed around the values of the market portfolo wth small errors: 19

20 p f (, OP ) p f (, OP p p g(, OP ) g( M M, OP M M ) f ) g 1 1 p p M f 2 ' OP OPM, p p g ' OP OP, M 2 M where and are the equty rsk premum and the equty CAPM beta on stock ; and p p denote the project rsk premum and the project CAPM beta on stock ; and OP represents the vector of varables that captures the effect of real optons. Subscrpt M denotes the market portfolo. By constructon, f (, OP ) and g( p, OP ) 1. Whle the CAPM p M M M M M M may fal to explan the equty rsk premum ( ) because of the presence of real p p optons, t may work for projects themselves ( ). The lnearzaton thus suggests that f we remove the real opton effects (terms nvolvng M M OP ) from and, the resdual (or real opton-adjusted) equty rsk premum and beta should generally satsfy a lnear relaton (subject to lnearzaton errors). Cross-sectonal regressons (wthout the constant) can be used to remove the effect of OP from and : a b OA ' OP OPM, OA ' OP OP. M The resduals, and OA, can be vewed as the opton-adjusted equty rsk premum and the OA OA OA equty CAPM beta on stock and n that case we should expect. Snce the market ndex portfolo s a value-weghted average of all stocks, f we take the cross-sectonal (value-weghted) average of the opton-adjusted equty rsk premum and beta across all stocks, we have: 20

21 M M OA OA,. Snce OA OA = 1 by constructon,, whch n turn mples: M M OA OA M, Ths consttutes the man testable hypothess of our work: the opton-adjusted CAPM beta explans the cross secton of opton-adjusted equty returns. We test ths hypothess frst usng smulaton analyss Smulaton evdence In a smulaton exercse to evaluate the effect of real optons on expected frm stock returns and the performance of the CAPM, we consder a large cross-secton of all-equtyfnanced frms. Each frm conssts of one prmtve asset (project) and a call opton on the asset. The assets dffer n ther CAPM beta ( p ) and dosyncratc volatlty (Ivol ). For smplcty, we assume that all assets are assocated wth a book value of $1 and a market value of $1 and that the expected return on the prmtve asset satsfes the CAPM. All call optons expre n one year and dffer n ther moneyness ( k ). We assume that the call opton can be prced usng the standard Black-Scholes (1973) model. The cross-sectonal varaton n the call prces s therefore drven by three parameters: k, Ivol, and p, and the last two parameters jontly determne the volatlty of the asset ( ). The call prce then determnes w, or the weght of the opton as a percentage of the total value of the frm. More precsely,, where I s equal to 1 (-1) f frm s takng a 21

22 long (short) poston n the call opton, and O denotes the opton value. In ths smple economy, the book-to-market rato (BM) of the frm perfectly reveals ts opton weght ( ). The expected return on the frm does not satsfy the CAPM because there s a call opton. Overall, the rsk premum on frm ( ) s a value-weghted average of the rsk premum on the project ( ) and the opton ( ): p O p (1 w ) w O p (1 w ) M O w. We generate the cross-secton of frms by choosng the parameter values as follows (on an annual bass): a market rsk premum ( ) of 8.00%; a market rsk-free rate of 1.00%; a M market volatlty of 0.20 (on the log market return); values for k from 0.75 to 1.25 n ncrements of 0.05 at a tme, values for Ivol from 0.10 to 0.70 n ncrements of 0.10 at a tme; and values of the prmtve project beta ( ) from 0 to 2 n ncrements of 0.10 at a tme. p Fnally, for each frm wth a short poston n a call opton, we also generate two otherwse dentcal frms, each wth a long poston n the same opton so optons are n postve net supply overall. For each frm we can compute the rsk premum ( ) and the CAPM beta ( ) of the stock numercally. 10 Fnally, the CAPM beta s measured wth errors, and the measurement error s drawn from a normal dstrbuton of zero mean and a varance of As the CAPM cannot prce the expected return on the 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 s expected return vs. ts CAPM betas, we do not observe a sgnfcant 10 Specfcally, for each frm, we use Monte Carlo smulaton to smulate 10,000 possble project payoffs at the end of the year. For each project payoff, we compute the correspondng opton payoff. The expected payoffs are computed as averages across all 10,000 paths. The covarance between the frm return and the market return and the beta can also be computed usng these 10,000 realzatons. 22

23 postve relaton as would be predcted by the CAPM. In fact, f we run a cross-sectonal regresson of frm expected return on frm CAPM betas, we confrm the well-documented falure of the CAPM to explan the cross-sectonal varaton n equty rsk premum. Frst, the slope coeffcent, whch can be nterpreted as the market rsk premum, s only 2.04%, whch s way below the assumed value of 8.00%. Second, when the rsk premum estmate s based toward zero, the ntercept term s lkely to be postve and sgnfcant. Ths s exactly what we fnd. The ntercept term s 7.00% wth a t-value of about Fnally, the regresson R- square s lower than 2.00%. As the CAPM holds on the prmtve projects by constructon n our smulaton, the falure of the CAPM to predct frm returns should not nvaldate ts use for the purpose of project cost of captal calculaton. Our man hypothess posts that the CAPM should hold approxmately for the opton-adjusted equty rsk premum and beta, and we test that n our smulated sample. The cross-sectonal varaton n real opton effects n our example can be fully captured by three parameters: the book-to-market rato (BM ), the moneyness of the call opton ( k ), and the dosyncratc volatlty (Ivol ). Therefore, the opton proxes ( OP ) n ths example are a vector of cross-sectonally demeaned [ BM, k, Ivol ]. We remove the opton effects from the frm CAPM beta by regressng t on the opton proxes n a cross-sectonal regresson wth no constant. The resdual or the opton-adjusted beta ( OA ) should therefore capture the project CAPM beta. We verfy ths n Panel B of Fgure 1. When plotted together, the project beta ( ) lnes up qute well wth the optonadjusted beta ( OA ). When we regress p p OA on, we get an R-square almost of 90.00% and 23

24 a slope of Despte the nonlnear nature of the opton effect, a smple lnear regresson seems to be qute successful n removng opton effects from the frm beta and recoverng the project beta. In addton, we regress the frm rsk premum ( ) on the opton proxes n the crosssecton wth no constant to obtan the opton-adjusted frm rsk premum ( OA ). The optonadjusted rsk premum should capture the rsk premum on the project. Our man hypothess OA predcts that wll be equal to the product of the opton-adjusted beta and the market rsk premum, or. Panel C of Fgure 1 provdes supportng evdence for the OA OA M hypothess. We fnd that OA s clearly postvely related to OA. When we regress OA on OA, we fnd a slope coeffcent of 8.10%, qute close to the assumed market rsk premum of 8.00%. At the same tme, the ntercept term s -0.04%, not sgnfcantly dfferent from zero. In other words, the CAPM performs better once the opton effects are removed wth a smple regresson procedure. Fnally, the R-square s also much hgher at 50.5%. The R-square s not OA OA equal to 1 for several reasons. Frst, lnear approxmaton errors affect both and. Second, beta s assumed to be measured wth error. 4. The CAPM and project cost of captal: emprcal analyss For our emprcal analyss, we conduct monthly Fama-MacBeth (1973) cross-sectonal regressons at the ndvdual stock level wth the Newey-West (1987) correcton usng a lag of 36. Each month, we regress monthly stock excess return (over the rsk-free rate) on betas and 24

25 stock characterstcs that are measured usng the most recent return and accountng data avalable to nvestors Sample and varable defntons We start wth frms covered by CRSP wth common shares outstandng over , wth the excluson of penny stocks (wth prces lower than $5), and frms lsted for less than three years. For each sample frm, the beta estmates are calculated as the slope coeffcents of the CAPM regressons; whle Beta_MKT, Beta_SMB, and Beta_HML are computed as the slope coeffcents of the Fama-French three-factor regressons, both usng the pror 60 months of return records from CRSP. 11 Two return anomaly varables are constructed usng the CRSP data: medum-term prce momentum (Momt) as the cumulatve monthly stock return of [t - 13, t - 2], and long-term return reversal (Lret) as the cumulatve monthly stock returns of [t - 60, t - 13], pror to month t. Monthly dosyncratc volatlty (Ivol) n month t s computed followng the procedure n Ang, Hodrck, Xng, and Zhang (2006). Our CRSP sample s further ntersected wth COMPUSTAT data where accountng nformaton s avalable. We requre a mnmum sx-month gap n matchng the accountng data of calendar year t - 1 to monthly return data of calendar year t to ensure that the accountng nformaton s avalable to market nvestors. We construct a number of varables usng nformaton avalable from COMPUSTAT fles. BE s book value (n mllons) as the 11 Hoberg and Welch (2007) argue that nvestors may be slow n adjustng to recent changes n market rsk, and recommend the use of aged beta. For ths reason, we also examne aged betas (the CAPM beta estmated usng [t - 85, t - 25] fve-year rollng wndows after skppng the most recent two years) and fnd them to produce smlar results n the cross-sectonal regressons. 25

26 sum of stockholders equty, deferred tax, nvestment tax credts, and convertble debt, mnus the lqudaton value of preferred stocks (Fama and French, 1992, 1993). BM s the rato of BE over frm market captalzaton measured as of the most recent June. ROA s the rato of quarterly earnngs scaled by the one-quarter-lagged asset n the pror quarter. The asset growth rate (Ast_gw) s calculated as the year-on-year percentage change n total assets (Cooper, Gulen, and Schll, 2008), whle nvestment-to-captal rato (Inv) s the annual change n gross property, plant, and equpment scaled by the lagged book value of assets (Chen, Novy-Marx, and Zhang, 2010). Earnngs surprse (Sue) s computed usng the dfference n quarterly earnngs n the [t - 3, t - 6] wndow and the correspondng value announced 4 quarters ago scaled by the standard devaton of the correspondng earnngs change over the prevous 8 quarters pror to return measurement n month t (Chan, Jegadeesh, and Lakonshok, 1996). Table 1 defnes all the varables. We focus on BM, Ivol, and ROA as our emprcal real opton proxes. BM s a common proxy for growth optons n the fnance lterature (see Smth and Watts,1992, among others). The choce of Ivol s motvated by Cao, Smn, and Zhao (2008) and Bekaert, Hodrck, and Zhang (2010), whch establsh a theoretcal lnk between growth optons avalable to managers and the dosyncratc rsk of equty. Fnally, Chen, Novy-Marx, and Zhang (2010) argue that ROA s a good emprcal proxy for the margnal product of captal, whch s related to the margnal expanson opton as n the real opton model of Abel, Dxt, Eberly, and Pndyck (1996). The choces of these three real opton proxes are consstent wth the smulaton example n Secton 3.1. Table 2 reports parwse correlatons among our emprcal varables. Several nterestng patterns emerge. Frst, asset growth (Ast_gw), nvestment-to-captal rato (Inv), and past long- 26

27 term return (Lret) are hghly correlated wth one another, suggestng that those anomales are related, and all seem to capture a long-term return reversal pattern. These three varables are also sgnfcantly correlated wth BM. To the extent that BM proxes for the real opton, these correlatons suggest that asset growth, nvestment-related, and long-term reversal anomales could be related to real opton features and thus may be partally allevated f we control for BM n our opton adjustment. Second, not surprsngly, earnngs surprse (Sue) and medumterm prce momentum (Momt) are postvely correlated, and they are both correlated wth ROA, suggestng that earnngs and prce momentum may be allevated f we control for ROA n our opton adjustment. Our fnal sample conssts of a panel of monthly stock observatons wth non-mssng real opton proxes (BM, Ivol, ROA) from July 1970 through June There are on average 2,087 stocks each month. We denote ths as our full sample. We also consder a subsample where we further elmnate stocks whose CAPM betas are lkely to be estmated wth large errors. We use two flters for ths purpose. Frst, we exclude stocks that do not have complete fve-year data to estmate betas. Second, we exclude stocks whose CAPM betas are extreme (below the 5th percentle or above the 95th percentle n the cross-secton). The second flter s smlar to what Mamaysky, Spegel, and Zhang (2008) use n ther analyss of mutual fund performance persstence. Altogether, these two flters remove about 14.6% (or 5.3% by market captalzaton) of stocks from our full sample. Our subsample thus ncludes about 1,783 stocks per month on average Cross-sectonal regresson analyss: full sample 27

28 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. Table 3 confrms the well-documented falure of the CAPM beta to explan the cross-sectonal varaton n returns. In Model 1, when we regress monthly ndvdual stock excess return (n excess of the rsk-free rate) on the CAPM beta, we get a slope coeffcent very close to zero and a huge ntercept term (78 bass ponts per month or 9.36% per year). Models 2-6 confrm the presence of several well-known asset prcng anomales n our sample. For example, frms wth hgher asset growth rates are assocated wth lower stock returns (asset growth anomaly, see Cooper, Gulen, and Schll, 2008); frms wth hgher nvestments are assocated wth lower stock returns (nvestment-related anomaly; see Anderson and Garca-Fejoo, 2006; Xng, 2008; and Chen, Novy-Marx, and Zhang, 2010); stocks wth hgher long-term past returns are assocated wth lower returns (long-term reversal; see DeBondt and Thaler, 1985); stocks wth hgher returns n the last year have hgher current returns (prce momentum; see Jegadeesh and Ttman, 1993); and stocks wth postve earnngs surprses have hgher returns (earnngs momentum; see Bernard and Thomas,1990). The CAPM beta s not sgnfcant at all n the presence of these anomales. Models 7-12 add the Fama-French three factor betas estmated usng the standard fveyear [t - 60, t - 1] rollng wndow. Whle the HML factor beta becomes sgnfcant, the MKT factor beta remans nsgnfcant and s assocated wth a close-to-zero rsk premum. In addton, all fve anomales reman to be sgnfcant after the ncluson of two addtonal SMB and HML factor betas. 28

29 Under our man hypothess, f the falure of the CAPM on equty return s drven by real optons, then removng the effect of real optons from equty returns and betas should mprove the performance of the CAPM. We follow the procedure descrbed n Secton 3 to conduct real opton adjustment usng a cross-sectonal regresson. In the frst-stage regresson, we regress the monthly (excess) stock return or the factor loadngs (ncludng the CAPM beta) on the three real opton proxes (BM, Ivol, and ROA). All three varables are measured n terms of the excess over ther counterparts for the market portfolo, and the regresson has no ntercept term. These procedures ensure that the CAPM relaton holds for the market portfolo. The resduals from these regressons are the opton-adjusted (excess) returns and optonadjusted betas. Gven the fact that true real optons, whch are not drectly observable, affect both returns and betas n hghly complcated and nonlnear fashon, our smple regresson approach may not fully remove real opton effects. Yet to the extent that our real opton proxes correlate wth the true real optons, the opton-adjusted (excess) returns and optonadjusted betas from the 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 real opton adjustments. Table 4 reports results of the regresson analyss n Table 3 but after the frst-stage real opton adjustments. In Model 1, we fnd the opton-adjusted CAPM beta becomes sgnfcantly and postvely related to the opton-adjusted (excess) stock return. The slope coeffcent s 42 bps per month (or 5.04% per year wth a t-value of 3.11). In addton, the ntercept term drops from 78 bps per month n Table 3 to 20 bps per month (or 2.40% per year), whch s stll sgnfcant (t-value = 3.14). 29

30 Models 2-6 fnd the opton-adjusted CAPM beta to be stll hghly sgnfcant (wth t- values usually above 2.6) even n the presence of the anomaly varables. Interestngly, the opton-adjusted CAPM beta helps to weaken or even drve out several anomales that are assocated wth long-term stock prce mean reverson (Ast_gw, Inv, and Lret). For nstance, after real opton adjustment, the t-value assocated wth asset growth (Ast_gw) drops from (n Table 3) to Furthermore, nvestment-to-captal rato (Inv) and past long-term return (Lret) become nsgnfcant wth t-values of and -0.94, respectvely. Ths result s consstent wth recent fndngs by Cooper and Prestley (2010), whch suggest that frms real optons are lkely drvng the asset growth and nvestment-related anomales. On the other hand, the real opton adjustment has lttle mpact on anomales that are related to short-term return contnuaton. Both prce and earnngs momentum (Momt and Sue) reman hghly sgnfcant. Addtonal unreported dagnostc tests confrm that both anomales, at monthly frequency, are drven by components n Momt and Sue that are orthogonal to ROA. To the extent that short-term prce contnuaton typcally does not persst beyond a few quarters and requres frequent portfolo rebalancng, t s probably less relevant for the cost of captal calculaton for a project whose lfe usually goes beyond fve years. Models 7-12 agan nclude the Fama-French three factor betas after the real opton adjustment. The CAPM beta now drves out the factor loadng on HML n explanng opton-adjusted stock returns. Our analyss so far suffers from the standard errors-n-varables problem as the tme-seres estmaton of betas ntroduces measurement errors. Km (1995) shows that the problem leads to a lower rsk premum estmate and a hgher ntercept estmate n the cross-sectonal regresson, potentally explanng why we fnd that the slope coeffcent s smaller than the market rsk premum, and the ntercept term s sgnfcantly dfferent from zero. Followng 30