Research Division Federal Reserve Bank of St. Louis Working Paper Series

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1 Research Division Federal Reserve Bank of S. Louis Working Paper Series Is he Value Premium a Proxy for Time-Varying Invesmen Opporuniies: Some Time Series Evidence Hui Guo Rober Savickas Zijun Wang and Jian Yang Working Paper C hp://research.slouisfed.org/wp/005/ pdf April 005 Revised Ocober 006 FEDERAL RESERVE BANK OF ST. LOUIS Research Division P.O. Box 44 S. Louis, MO The views expressed are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis, he Federal Reserve Sysem, or he Board of Governors. Federal Reserve Bank of S. Louis Working Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Federal Reserve Bank of S. Louis Working Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors.

2 Is he Value Premium a Proxy for Time-Varying Invesmen Opporuniies: Some Time Series Evidence Hui Guo Federal Reserve Bank of S. Louis Rober Savickas Deparmen of Finance, George Washingon Universiy Zijun Wang Privae Enerprise Research Cener, Texas A&M Universiy Jian Yang Deparmen of Accouning, Finance and MIS, Prairie View A&M Universiy Firs Version: April 004 This Version: Ocober 006 Absrac We uncover a posiive, empirical risk-reurn radeoff in he sock marke afer conrolling for he covariance of sock marke reurns wih he value premium. The underlying premise is ha, as conjecured by Fama and French (1996), he value premium is a proxy for ime-varying invesmen opporuniies. By ignoring he value premium, early specificaions suffer from an omied variable problem ha leads o a downward bias in he esimae of he risk-reurn radeoff. The paper also documens a new finding on a significanly posiive relaion beween he value premium and is condiional variance. Keywords: ICAPM, value premium, sock reurn predicabiliy, realized variance, and GARCH. JEL number: G1. Corresponding auhor: Hui Guo, Research Division, Federal Reserve Bank of S. Louis, P.O. Box 44 Sain Louis, MO 63166; Phone: (314) ; Fax: (314) ; hui.guo@sls.frb.org. We hank Andrew Ang, Joe Chen, Long Chen, Amar Gande, John Scruggs, Lu Zhang, and paricipans a he 005 Easern Finance Associaion meeing in Norfolk, he 006 FMA European meeing in Sockholm, and he 006 FMA meeing in Sal Lake Ciy for useful commens and discussion. We also hank Ken French for making he daa available hrough his websie. Jason Higbee provided able research assisance. The views expressed in his paper are hose of he auhors and do no necessarily reflec he official posiions of he Federal Reserve Bank of S. Louis or he Federal Reserve Sysem.

3 Is he Value Premium a Proxy for Time-Varying Invesmen Opporuniies: Some Time Series Evidence Absrac We uncover a posiive, empirical risk-reurn radeoff in he sock marke afer conrolling for he covariance of sock marke reurns wih he value premium. The underlying premise is ha, as conjecured by Fama and French (1996), he value premium is a proxy for ime-varying invesmen opporuniies. By ignoring he value premium, early specificaions suffer from an omied variable problem ha leads o a downward bias in he esimae of he risk-reurn radeoff. The paper also documens a new finding on a significanly posiive relaion beween he value premium and is condiional variance. Keywords: ICAPM, value premium, sock reurn predicabiliy, realized variance, and GARCH. JEL number: G1.

4 1. Inroducion The capial asse pricing model (CAPM) developed by Sharpe (1964) and Linner (1965) fails o explain he sock reurn daa along wo imporan dimensions. Firs, Fama and French (1993), for example, show ha he CAPM doesn accoun for he cross-secion of sock reurns, e.g., he value premium and he size premium. 1 Second, many auhors, e.g., Campbell (1987), Glosen, Jagannahan, and Runkle (1993), Whielaw (1994), and Brand and Kang (004), find a weak or negaive risk-reurn radeoff in he sock marke across ime, in conras wih he posiive relaion sipulaed by he CAPM. The CAPM-relaed anomalies sugges ha he sock marke migh ac as a hedge agains changes in invesmen opporuniies, as illusraed in Meron s (1973) ineremporal CAPM (ICAPM). In paricular, Fama and French (1996) argue ha he value and size premia move closely wih invesmen opporuniies and include hem as addiional risk facors in heir hreefacor model perhaps one of he mos influenial and successful empirical asse pricing models. Consisen wih Fama and French s conjecure, Liew and Vassalou (000) find ha he value premium forecass oupu growh in many indusrial counries. Campbell and Vuoleenaho (004), Brennan, Wang, and Xia (004), Hahn and Lee (006), and Pekova (006) show ha he value premium is correlaed wih innovaions in heir measures of invesmen opporuniies. Also, Gomes, Kogan, and Zhang (003), Zhang (005), and Leau and Wacher (006) develop equilibrium models o esablish a link beween he value premium and invesmen opporuniies. Moivaed from Fama s (1991) conjecure of an explici link beween he cross-secional and ime-series sock reurn predicabiliy, we invesigae in his paper wheher he value 1 The value premium is he reurn on a porfolio ha is long in socks wih a high book-o-marke value raio (value socks) and shor in socks wih a low book-o-marke value raio (growh socks). The size premium is he reurn on a porfolio ha is long in socks wih small capializaions and shor in socks wih big capializaions. 1

5 premium consruced from he cross-secion of socks sheds ligh on he on-going debae abou he ineremporal relaion beween sock marke risk and reurn. Our ime-series es also provides a robusness check for he cross-secional evidence of he empirical ICAPM, which is poenially sensiive o he choice of priced sae variables (e.g., Chen and Zhao [005]) and alernaive economeric specificaions (e.g., Lewellen, Nagel, and Shanken [006]). If he value premium is a proxy for invesmen opporuniies, Meron s (1973) ICAPM indicaes ha he condiional excess sock marke reurn, E ( R + 1), is deermined by is condiional variance, σ M,, and is condiional covariance wih he value premium, σ : (1) E ( R + ) = γ σ + γ σ. 1 M M, H The parameer γ M is he coefficien of relaive risk aversion and should be posiive. The coefficien JWF γ H is equal o J W, where J(W(), F(), ) is he indirec uiliy funcion of he represenaive agen wih subscrips denoing parial derivaives, W() is wealh, and F() is a vecor of sae variables ha describe invesmen opporuniies. Similarly, he condiional value premium, E ( HML + 1), is deermined by is condiional variance, σ,, and is condiional covariance wih he sock marke reurn, σ : H () E ( HML + ) = γ σ + γ σ. 1 M H H, For robusness, as in French, Schwer, and Sambaugh (1987), we esimae equaions (1) and () using boh he realized variance model advocaed by Meron (1980) and he ARCH model advanced by Engle (198). We obain qualiaively similar resuls using boh echniques, and our main findings can be summarized as follows. Firs, over he period 1963 o 005, here is

6 a weak risk-reurn relaion in he U.S. sock marke. However, i becomes significanly posiive afer we conrol for he covariance of sock marke reurns wih he value premium; condiional sock marke reurns are posiively relaed o he covariance as well. Second, we documen a new finding on a significanly posiive relaion beween he value premium and is condiional variance afer conrolling for is covariance wih sock marke reurns. Many auhors, e.g., Fama and French (1996), Leau and Ludvigson (001a), and Zhang (005), sugges ha value socks are riskier han growh socks especially during business downurns. Consisen wih his hypohesis, we find ha he condiional value premium ends o move counercyclically. Lasly, o address he poenial concern over he daa miming, we esimae he ICAPM using Fama and French s (1998) inernaional daa, and find qualiaively similar resuls for he world marke as well as mos of he oher G7 counries. Overall, our resuls are consisen wih he conjecure ha he value premium is a proxy for invesmen opporuniies. Scruggs (1998) esimaes a bivariae GARCH model using he long-erm ineres rae as a proxy for invesmen opporuniies. However, his resuls are somewha sensiive o he assumpion of a consan correlaion coefficien beween sock marke reurns and he long-erm ineres rae (e.g., Scruggs and Glabadanidis [003]). Guo and Whielaw (006) use he consumpion-wealh raio proposed by Leau and Ludvigson (001b) as a proxy for invesmen opporuniies and find resuls very similar o ours. 3 Guo and Whielaw (006) focus on he sock Recen sudies, e.g., Campbell and Vuoleenaho (004), Ang and Chen (005), Pekova and Zhang (005), and Fama and French (005), find ha he CAPM explains he value premium in he early period 196 o 196. One possible explanaion is ha Campbell and Vuoleenaho (004) find ha he value premium is a poor proxy for changes in invesmen opporuniies. Consisen wih heir evidence, we find ha he value premium doesn help uncover he posiive risk-reurn radeoff in he early period. 3 We can use Campbell and Shiller s (1988) log-linearizaion mehod o show ha he scaled sock price, e.g., he consumpion-wealh raio, is a linear funcion of condiional sock marke variance and condiional covariance of sock marke reurns wih he shock o invesmen opporuniies. Consisen wih he hypohesis ha he value premium is a proxy for invesmen opporuniies, we find ha he predicive power of he value premium for sock marke reurns is very similar o ha of he consumpion-wealh raio. 3

7 marke risk-reurn radeoff; by conras, our main moivaion is o es he hypohesis of wheher he value premium proxies for invesmen opporuniies. 4 Moreover, Scruggs (1998) and Guo and Whielaw (006) use only U.S. daa, while we provide inernaional evidence as well. In a paper circulaed afer he firs version of his paper, Brand and Wang (006) use he value premium as a proxy for invesmen opporuniies o invesigae he ime-varying risk-aversion. The value premium is an empirically moivaed risk facor and has limiaions, for example, i has some difficulies in explaining he dynamic of sock reurns (Ferson and Harvey [1999]). Neverheless, our evidence raises he bar for some alernaive hypoheses by uncovering a close link beween ime-series and cross-secional sock reurn predicabiliy. Such a link is well esablished in Meron s (1973) ICAPM; however, i poses a challenge o he irraional pricing (e.g., Lakonishok, Shleifer, and Vishny [1994]) and daa mining (e.g., MacKinlay [1995]) explanaions for he value premium. The remainder of he paper is organized as follows. We presen he esimaion resuls of he realized variance model in Secion and he bivariae GARCH model in Secion 3. We provide he inernaional evidence in Secion 4 and discuss he main findings in Secion 5. We offer some concluding remarks in Secion 6.. The Realized Variance Model.1. Daa Descripions We obain daily and monhly daa of he Fama and French hree facors from Ken French a Darmouh College. Daily daa are available over he period July, 1963, o December 31, 4 A few recen sudies also uncover a posiive risk-reurn radeoff by using (1) alernaive measures of he condiional sock marke variance (Ghysels, Sana-Clara, and Valkanov [005]); () alernaive measures of he condiional sock marke reurn (e.g., Graham and Harvey [003] and Pasor, Sinha, and Swaminahan [006]); (3) 4

8 005, and monhly daa are available over he period July 196 o December 005. Following Meron (1980) and Andersen, Bollerslev, Diebold, and Labys (003), among many ohers, we use he sum of he squared daily reurns in a quarer as a measure of realized variance for boh sock marke reurns and he value premium. 5 Realized covariance is measured as he sum of he cross-produc of daily excess sock marke reurns wih he daily value premium. We also consruc quarerly reurn daa by aggregaing monhly reurns hrough simple compounding. Figure 1 plos realized sock marke variance, v M, (dashed line), along wih realized covariance beween he sock marke reurn and he value premium, v (solid line). The variable v rose dramaically during he 1987 sock marke crash and revered o he normal M, level shorly afer. Because many auhors, e.g., Schwer (1990), argue ha he 1987 crash is unusual in many ways, we follow Campbell, Leau, Malkiel, and Xu (001) and replace realized variance for 1987:Q4 wih he second larges observaion in he sample. The variable v is almos always negaive, suggesing ha he marke provides a hedge for changes in invesmen opporuniies, given he premise ha hey are proxied by he value premium. The absolue value of v ends o be relaively high jus before or during business recessions (daed by he Naional Bureau of Economic Research (NBER)), as denoed by he shaded areas. The wo variables in Figure 1 usually move in opposie direcions. Figure shows ha realized variance of he value premium, v H, (solid line), is also negaively relaed o v (dashed line); and Figure 3 shows ha realized variance of he sock marke reurn (dashed line) is closely relaed o longer hisorical sock reurn daa (Lundblad [006]); and (4) condiioning variables exraced from a large se of macroeconomic variables (Ludvigson and Ng [006]). 5 We focus on quarerly daa raher han monhly daa because Ghysels, Sana-Clara, and Valkanov (005) argue ha realized variance is a funcion of long disribued lags of squared pas reurns. Also, as in French, Schwer, and Sambaugh (1987), we find essenially he same resuls by correcing he serial correlaion in daily reurn daa. For breviy, hese resuls are no repored here bu are available on reques. 5

9 he realized variance of he value premium (solid line). As we show in he nex subsecion, hese paerns help explain why ignoring he hedge for changes in invesmen opporuniies leads o a downward bias in he esimaion of risk-reurn radeoff. Table 1 presens summary saisics for he excess sock marke reurn and he value premium as well as heir realized variances and covariance over he period 1963:Q4 o 005:Q4. Panel A shows ha he excess sock marke reurn, R, is negaively relaed o he value premium, HML, wih a correlaion coefficien of Also, consisen wih Figures 1 o 3, he variables v, M, v H,, and v are closely relaed o each oher; however, he correlaion is far from perfec. Panel B shows ha he realized second momens are relaively persisen: The auocorrelaion coefficiens are 0.53, 0.7, and 0.56 for v M,, v v H,, and, respecively. Therefore, realized variances and covariance are good predicors of heir fuure levels... Esimaion Resuls of Meron s (1973) ICAPM We can rewrie equaions (1) and () in he realized reurn form and use realized variances and covariance as proxies for heir condiional values: (3) R = α + γ v + γ v + ε + 1 M MM M, HM MH, M, + 1 HML = α + γ v + γ v + ε + 1 H MH MH, HH H, H, + 1, where ε and M, + 1 ε H, + 1 are shocks o he marke reurn and he value premium, respecively. Meron s (1973) ICAPM also imposes resricions on he coefficiens in equaion (3): α M = α H = 0, γ MM = γmh = γm, and γ HM = γhh = γh. We esimae equaion (3) using he GMM (generalized mehods of momens) advanced by Hansen (198) and repor he esimaion resuls in Table. 6

10 Row 1 of panel A, Table replicaes he familiar resul ha realized sock marke variance, v, has weak forecasing power for he excess sock marke reurn, R + 1: Is M, coefficien is posiive bu only marginally significan, wih an adjused R-squared of 1.6%. However, i remains posiive and becomes significan a he 1% level afer we conrol for realized covariance of sock marke reurns wih he value premium, v (row ). Ineresingly, he effec of v is also significanly posiive, and he adjused R-squared increases o 4.8% from 1.6% in row 1. Because v M, and v are negaively correlaed (as shown in Figure 1), our resuls sugges ha he specificaion in row 1, panel A, suffers from a classic omied variable problem, which leads o a downward bias in he esimae of he risk-reurn radeoff. 6 Row 1 of panel B, Table, shows ha he relaion beween realized value premium variance, v, and he one-quarer-ahead value premium, HML + 1, is posiive bu saisically H, insignifican. However, he coefficien of vh, becomes marginally significan afer we conrol for realized covariance of he value premium wih sock marke reurns, v (row ). Because and v are negaively correlaed wih each oher (as shown in Figure ), hese resuls sugges ha he specificaion in row 1, panel B, also suffers from an omied variable problem. In row 3 of Table we esimae he wo equaions joinly. We use a consan, v M,, and v H, v as insrumenal variables for he sock reurn equaion and a consan, v H,, and v for he value premium equaion. Thus he equaion sysem is jus-idenified and he poin esimaes 6 Because of he correlaion beween v M, and v, here is a poenial concern over mulicollineariy. However, mulicollineariy canno explain our resuls because i usually leads o low -saisics, in conras wih he increase of -saisics when boh variables are included. Moreover, he characerisic-roo-raio es proposed by Belsley, Kuh, and Welsch (1980) confirms ha mulicollineariy is unlikely o plague our resuls. 7

11 are idenical o hose repored in row. Noe ha from row 3 on, we repor he R-squared raher han he adjused R-squared (as in rows 1 and ) in he column under R. In row 4 we impose he ICAPM resricions ha he consan erms are zero in boh equaions. The resricions can be esed using Hansen s (198) J-es, which has a chi-squared disribuion wih degrees of freedom. The J-es saisic is essenially zero, indicaing ha he resricions canno be rejeced a any convenional significance level. Row 5 shows ha we canno rejec he resricions ha he risk prices are equal across asses, and row 6 shows ha we canno rejec he resricions of no inerceps and he equal risk prices across asses. As expeced, imposing he ICAPM resricions improves he esimaion efficiency and he sandard errors in he resriced specificaions are subsanially smaller han hose repored in row 3. Afer imposing all he ICAPM resricions, row 6 shows ha he slope coefficiens are significan a he 1% level. Our resuls provide srong suppor for a posiive risk-reurn radeoff in he sock marke afer conrolling for changes in invesmen opporuniies, as proxied by he value premium. Early auhors, e.g., Fama and French (1989) and Campbell (1987), find ha he dividend yield, he defaul premium, he erm premium, and he sochasically derended risk-free rae forecas sock marke reurns. Ferson and Harvey (1999) show ha hese variables also have predicive power for he value premium. One possibiliy is ha hese variables comove wih he variance and covariance erms in equaion (3) a he business-cycle frequency. To address his issue, we include hem as insrumenal variables, in addiion o hose used in row 6 of Table. Row 7 shows ha he model is no rejeced a he 0% significance level, suggesing ha he sock reurn predicabiliy documened by early auhors is indeed consisen wih he ICAPM. Leau and Ludvigson (001b) argue ha he consumpion-wealh raio, predicor of sock marke reurns. If we also add CAY, is a srong CAY o he insrumenal variable se (row 8, 8

12 Table ), only a he 5% significance level is he model no rejeced; however, he oher resuls are very similar o hose repored in rows 6 and 7. Therefore, again, our resuls sugges ha he value premium reflecs ineremporal pricing, alhough i migh be a noisier measure of invesmen opporuniies han some oher sock reurn predicors proposed in he lieraure. In Figures 1 o 3, realized variances and covariance exhibi a big spike around he laes recession in our sample, during which sock prices firs increased sharply and hen collapsed wih he burs of he echnology bubble. To invesigae wheher his seemingly unusual episode has any special effec on our inference, we analyze a shorer sample spanning he period 1963:Q4 o 1997:Q4 and repor he resuls in rows 9 and 10 of Table, which have he same specificaions as hose in rows 7 and 8, respecively. We find ha he resuls are very similar o hose obained using he full sample..3. The Value Premium and Oher Proxies of Invesmen Opporuniies Guo and Whielaw (006) use he consumpion-wealh raio, CAY, and he sochasically derended risk-free rae, RREL, as proxies for invesmen opporuniies. Guo and Savickas (006) find ha, when combined wih sock marke variance, a measure of value-weighed idiosyncraic variance, IV, forecass sock marke reurns possibly because i is a proxy for realized variance of a risk facor omied from he CAPM. Table 3 invesigaes wheher he predicive power of he value premium for sock reurns is relaed o ha of hose variables. Row 1 shows ha he forecasing power of vm, and v is qualiaively unchanged in he presence of RREL, of which he coefficien is negaive and marginally significan. By conras, v loses 9

13 he predicive power afer we conrol for CAY (row ) or expeced sock reurns remains posiive and highly significan. IV (row 3), while he effec of v on One can show ha under some condiions he expeced sock marke reurn is a linear funcion of v M, and v, H, and such a specificaion holds even if he value premium is no perfecly correced wih he shock o invesmen opporuniies. For breviy, we do no provide he derivaion here bu i is available on reques. Row 4, Table 3 shows ha, as expeced, he coefficiens of v M, and v H, are boh significan, wih an adjused R-squared of 7.7%. 7 Neverheless, row 7 shows ha he effec of v H, becomes insignifican a he 10% level afer M, conrolling for IV. To summarize, our resuls sugges ha he value premium is relaed o he alernaive measures of invesmen opporuniies..4. Condiional Value Premium Consisen wih equaion (), v H, has some forecasing power for he value premium when combined wih v (row of Table ). This resul suggess ha predicable variaion in he value premium documened in some early sudies (e.g., Ferson and Harvey [1999]) migh be consisen wih ineremporal pricing. To address his issue, in Table 4 we compare he forecasing power of v H, wih alernaive measures of invesmen opporuniies, namely, 7 Some auhors, e.g., Bossaers and Hillion (1999), Goyal and Welch (006), Ferson, Sarkissian, and Simin (003), have challenged he robusness of he in-sample evidence of sock marke reurn predicabiliy. To address his issue, we use hree saisics o compare he ou-of-sample performance of he model using v H, and v M, as predicors wih a benchmark model of consan excess sock reurns: The mean-squared forecasing error raio; Clark and McCracken s (001) encompassing es; and McCracken s (1999) es of equal forecas accuracy. In he earlier versions of his paper, we show ha hese wo variables have significan ou-of-sample predicive power. For breviy, we do no repor hese resuls here bu hey are available on reques. 10

14 RREL, CAY, and IV. 8 The effec of H, v remains posiive and marginally significan afer conrolling for RREL (row 1) and CAY (row ). However, i becomes insignifican when combined wih IV (row 3). Row 4 of Table 4 presens he regression resuls using v M, insead of v in he forecasing equaion. Consisen wih he resuls repored in Table 3 for sock marke reurns, he alernaive specificaion appears o provide a beer fi for he value premium as well. Now he effec of v H, is posiive and significan a he 5% level; and he effec of v M, is negaive and significan a he 5% level. Also, he adjused R-squared is 4.9%, which is noiceably higher han he 3.9% repored in row of Table. The coefficien of v is negaive because of is negaive M, correlaion wih v (Table 1), which in urn is posiively correlaed wih he value premium. The forecasing power of v H, (as in row 4 of Table 4) is very similar o ha of IV, as repored by Guo and Savickas (006). These auhors show ha IV and v joinly have srong M, predicive power for he value premium; moreover, while v M, is negaively correlaed wih he one-quarer-ahead value premium, he relaion is posiive for IV. To formally address his issue, we also include IV in he forecasing equaion, ogeher wih H, v and v M,. Row 7 shows ha, while he coefficien of v M, remains significanly negaive, he coefficiens of boh IV and vh, become insignifican, indicaing ha he wo variables indeed capure common variaions in he value premium. This resul should no be oo surprising because Guo and Savickas (006) 8 The erm premium, he defaul premium, and he dividend yield (as used by Ferson and Harvey [1999]) do no provide addiional informaion abou he fuure value premium, and including hem does no change our resuls in any qualiaive manner. To conserve space, hese resuls are no repored here bu are available on reques. 11

15 poin ou ha, by consrucion, IV is a proxy for realized variance of a risk facor omied from he CAPM, which could be he value premium. However, by conras wih IV, conrolling for RREL (row 5) or CAY (row 6) does no affec our resuls in any qualiaive manner. Lasly, Figure 4 plos he fied expeced value premium (as based on he esimaion resuls of he benchmark ICAPM repored in row 6, Table ). I moves counercyclically and ends o increase sharply during he business recessions daed by NBER, as denoed by shaded areas. As we will discuss in Secion 5, his paern is consisen wih he conjecure (e.g., Fama and French [1996], Leau and Ludvigson [001a], and Zhang [005]) ha value socks are riskier han growh socks especially during he economic downurn..5. The Value Premium Consruced wih Small and Big Socks If he value anomaly reflecs ineremporal pricing, we expec o find very similar resuls using he value premium consruced wih boh small and big socks. To invesigae his issue, we obain from Kenneh French he daily reurn daa for six porfolios, which are he inersecions of wo independen sors size (small and big) and he book-o-marke value raio (high, median, and low). Table 5 shows ha we find qualiaively similar resuls using realized variance of he value premium consruced from small and big socks. 3. Bivariae GARCH Model 3.1. Empirical Specificaions Several sudies, e.g., Chrisensen and Prabhala (1998) and Fleming (1998), find ha realized variance is no an efficien measure of condiional variance. To address his issue, in his secion we esimae equaions (1) and () using he more elaborae bivariae GARCH models, 1

16 which migh provide a beer measure for he condiional second momens han he simple realized variance model. 9 Again, we rewrie equaions (1) and () in he realized reurn form: (4) R + 1 R MM M, HM M, + 1 HML = α + γ σ + γ σ + ε = α + γ σ + γ σ + ε + 1 H MH MH, HH H, H, + 1, where ε and M, + 1 ε H, + 1 are shocks o sock marke reurns and he value premium, respecively. We use he asymmeric dynamic covariance (ADC) model proposed by Kroner and Ng (1998). These auhors show ha i is very flexible in describing he dynamic of covariance erms because i ness several commonly used mulivariae GARCH models. In he ADC model, he dynamic of variances and covariances is governed by he following equaion sysem: (5) σ σ = θ M, MM, + 1 = θ H, HH, + 1 σ = ρ θ θ + φ θ MH, MH MM, + 1 HH, + 1 MH MH, + 1 εm, ηm, θij, + 1 = ωij + bhb i j + a i εm, εh, aj gi ηm, ηh, gj,, i j ( H, M) ε + H, η H,, where H is he condiional variance-covariance marix: (6) H hmm, hmh, σm, 1 σ MH, 1 = hmh, h =. HH, σmh, 1 σh, 1 Glosen, Jagannahan, and Runkle (1993), among many ohers, find ha a negaive reurn shock leads o a higher subsequen volailiy han does a posiive reurn shock of he same magniude. 9 In an earlier version of he paper, we formally invesigae he relaive performance of he realized variance model and he GARCH model using he Mone Carlo simulaion. In paricular, we firs esimae he bivariae GARCH model using daily reurn daa. We hen use he esimaed GARCH model o generae simulaed daily daa, which are used o esimae he ICAPM. We find ha boh he quarerly realized variance model and he monhly GARCH model provide reliable inference for he risk-reurn radeoff, while he GARCH model performs somewha beer. For breviy, we do no repor hese resuls here bu hey are available on reques. 13

17 This asymmeric effec can be capured by he erm ηm, max[0, εm, ] η = max[0, ε ] H, H, in equaion (5). ρ MH and φ MH are scalar parameers and he oher parameers can be wrien in marix forms: (7) ωmm ωmh amm amh W = CC =, A { am, ah} ωmh ω = = HH ahm a HH, bmm bmh gmm gmh B= { bm, bh} =, G { gm, gh} bmh b = = HH ghm g HH where W is posiive definie and C is a symmeric marix. Our noaions in equaion (7) reflec he fac ha marixes W and B are symmeric bu marixes A and G are no. Kroner and Ng (1998) show ha, if marixes A and B are diagonal and φ MH is equal o 0, he ADC model becomes he asymmeric version of he consan condiional correlaion model, as used by Scruggs (1998), for example. Also, if ρ MH is equal o 0 and φ MH is equal o 1, hen he ADC model reduces o he asymmeric version of he popular BEKK model proposed by Engle and Kroner (1995), which, as we show below, seems o apply in his sudy. We esimae he GARCH model using he quasi-maximum likelihood (QML) mehod. Bollerslev and Woodridge (199) show ha QML parameer esimaes can be consisen, even hough he condiional log-likelihood funcion assumes normaliy while sock reurns are known o be skewed and lepokuric. Neverheless, we find qualiaively he same resuls using he maximum likelihood esimaion (MLE) mehod by assuming a disribuion or a normal disribuion. Given a sample of T observaions of he reurn vecor, he parameers of he bivariae GARCH model are esimaed by maximizing he condiional log-likelihood funcion: (8) T T ' 1 ( ) ( log( π ) 0.5log 0.5 ε ε) = 1 = 1, L= l P = H H 14

18 where P denoes he vecor of all he parameers o be esimaed. Nonlinear opimizaion echniques are used o calculae he maximum likelihood esimaes based on he Broyden, Flecher, Goldfarb, and Shanno (BFGS) algorihm. The ADC model should be esimaed under some parameer resricions o ensure he posiive definie covariance marix. I is possible o impose he consrain ρ φ < 1 in he mv + mv model. To serve a similar purpose, Scruggs and Glabadanidis (003) propose o penalize he likelihood funcion whenever he covariance marix is no posiive definie, which we followed in his sudy. While such reamen migh lose he coninuiy of he likelihood funcion, i gains he abiliy o impose a less resricive consrain and avoid he possibiliy of a non-posiive definie covariance marix. Also, imposing a penaly in he likelihood funcion migh resul in a funcion wih muliple local opima. In his case, i is imporan o resar he opimizaion rouine a several differen saring poins o ensure ha he esimaed parameers correspond o he global maximum of he likelihood funcion. All our resuls are esed for robusness using differen saring values in he maximizaion of he likelihood funcion. We focus mainly on he modern period January 1963 o December 005 because, as menioned in foonoe and confirmed in his sudy, he value premium is a poor proxy for invesmen opporuniies in he pre-1963 sample. Table 6 provides summary saisics of he excess sock marke reurn and he value premium (in percenages) for he modern sample. Consisen wih quarerly daa in Table 1, he wo variables are negaively correlaed, wih a correlaion coefficien of 0.3. The Ljung-Box es indicaes ha he value premium is serially correlaed. 15

19 3.. Model Selecion Tess Kroner and Ng (1998), among ohers, argue ha choosing a parsimonious GARCH specificaion is imporan for he asse pricing ess because hey criically depend on he covariance marix esimaes. In fac, heir ADC model was originally proposed o faciliae he model selecion (Kroner and Ng, 1998, p. 833). A parsimonious daa-deermined model is desirable also because he number of observaions is limied, while a large amoun of he daa is required o yield precise esimaes of GARCH models. Hence, i is imporan in his sudy o impose saisically accepable consrains and reduce he redundan parameers. The model selecion es follows he general-o-specific approach. Similar o Scruggs (1998) and Scruggs and Glabadanidis (003), we firs look a he second-momen modeling. The resuls, which are repored in Table 7, can be easily summarized as follows. Using he fullfledged bivariae ADC model as he alernaive hypohesis, we overwhelmingly rejec he null model of he pooling of wo univariae GARCH specificaions (panel A). By conras, we fail o rejec he more resricive, and ye quie general, ABEKK model a he 10% level (panel B). Also, for he BEKK model, panel C shows ha he null hypohesis of symmery is srongly rejeced. Because he ADC model involves more parameers and hus has poorer convergence properies, we hereafer focus on he ABEKK model in he remaining discussion, alhough we find qualiaively he same resuls using he ADC model. We hen urn o he model selecion es on he firs-momen modeling for he ABEKK model. We firs es he null hypohesis ha he slope parameers are joinly zero in equaion (4) or γ = γ = γ = γ = 0. Panel D of Table 7 shows ha hese resricions are rejeced a MM MH HM HH he 1% significance level, indicaing ha condiional variance and covariance erms are significan deerminans of he excess sock marke reurn and he value premium. However, 16

20 consisen wih he resuls obained from he realized variance model, panels E and F show ha we fail o rejec he ICAPM resricions a he convenional significance level Esimaion Resuls Table 8 presens he esimaion resuls of he mean equaions. We use he percenage reurn in he esimaion; o make hem comparable wih he resuls in Table, we scale he consan erms by 1/100 and he slope parameers by 100. For comparison wih early sudies, we firs repor in panel A of Table 8 he esimaion resuls of he pooling univariae asymmeric GARCH model i.e., we resric he ineracion erms beween he sock marke reurn and he value premium o be zero in equaions (4) and (5). For he excess sock marke reurn equaion, he condiional reurn is posiively relaed o he condiional variance wih a poin esimae of 0.87; however, he relaion is saisically insignifican a he 10% level. Similarly, we find a posiive bu insignifican risk-reurn relaion for he value premium. Neverheless, such a resul should be inerpreed wih cauion because he specificaion poenially suffers from an omied variable problem, which we discuss nex. Panel B of Table 8 presens he esimaion resuls using he ABEKK model. In he unresriced specificaion (row ), only he slope parameers in he value premium equaion are significan a he 10% level. Because he slope parameers are joinly significan (panel D of Table 7), his resul suggess ha our esimaion is no efficien. One way o address his issue, as we have learned from he realized variance model repored in Table, is o impose he resricions dicaed by Meron s (1973) ICAPM. As expeced, row 4 shows ha he slope parameers in he mean equaions are saisically significan a he 1% level afer we impose he ICAPM resricions of zero consan erms and he same risk prices across asses. 17

21 The price of sock marke risk, γ M, has a poin esimae of 4.74 and a sandard error of 1.1. I appears o be quie reasonable because Mehra and Presco (1985), for example, sugges a plausible range 1 o 10. Ineresingly, i is also srikingly similar o he poin esimae of 4.93 repored by Guo and Whielaw (006), who use CAY as a proxy for invesmen opporuniies. This is mainly because, as shown in row of Table 3, v and CAY appear o capure he common variaions of sock marke reurns. Figure 5 plos he fied values of condiional sock marke variance (dashed line) and covariance beween he sock marke reurn and he value premium (solid line) from he benchmark esimaion repored in row 4 of Table 8. The paern is very similar o ha presened in Figure 1. The paerns documened in Figures 6 and 7 are qualiaively he same as hose in Figures and 3, respecively. Also, Figure 8 shows ha here is subsanial variaion in he coefficien of condiional correlaion beween he sock marke reurn and he value premium. This resul confirms he finding of Scruggs and Glabadanidis (003) ha i is imporan o allow for a ime-varying correlaion coefficien in he ICAPM esimaion. Table 9 presens he parameer esimaes of he benchmark ABEKK model. Panels A and B repor he esimaes of he mean equaions, which are he same as hose in row 4 of Table 8. Panels C, D, and E show ha mos parameers in he marices W, A, B, and G are saisically significan. This resul highlighs he imporance of allowing for a ime-varying variancecovariance marix Robusness Checks Panel C of Table 6 repors he mean of fied values of condiional variances and covariance based on he esimaion resuls of he benchmark specificaion repored in Table 9. 18

22 They are very similar o he uncondiional variance-covariance marix of he excess sock marke reurn and he value premium, as repored in panel B of Table 6. Row 5 of Table 8 repors he esimaion resuls of he ABEKK model for he early period July 196 o December 196. The risk price associaed wih he value premium has a negligible poin esimae of -0.00, which is saisically insignifican a any convenional level. The price of sock marke risk is again saisically significan; neverheless, is poin esimae of.0 is subsanially smaller han he poin esimae of 4.74 obained from he modern period, as repored in row 4 of Table 8. These resuls confirm ha in he early period he value premium is a poor proxy for invesmen opporuniies and can be explained by he CAPM. Row 6 shows ha in he full sample spanning he period July 197 o December 005, he value premium risk is no priced bu he price of he marke risk is significanly posiive. However, because of he likely srucural break in he value premium, we should inerpre his resul wih cauion. Alhough we concenrae on a resriced ABEKK specificaion in he previous discussion, i is worh noing ha we find similar resuls using he ADC model, as shown in panel C of Table 8. In he unresriced model (row 7), we find ha he risk prices are all posiive, alhough mos of hem are saisically insignifican. By conras, row 8 shows ha he risk prices again become significan a he 1% level afer imposing he ICAPM resricions, which canno be rejeced a he convenional significance level. Moreover, he poin esimaes are very similar o hose obained using he benchmark ABEKK model, as shown in row 4 of Table 8. We also esimae he resriced ABEKK model using he MLE mehod by assuming a disribuion and a normal disribuion for he modern sample and repor he main resuls in rows 9 and 10, respecively, of Table 8. For he disribuion, he degree of freedom of he disribuion has a poin esimae of 9.14 and a sandard error of This resul is consisen wih he general 19

23 belief ha he disribuion of sock reurns is characerized by fa ails. Neverheless, he oher resuls are essenially he same as he benchmark ABEKK model. We reach he same conclusion for he normal disribuion as well. Lasly, we repea he above analysis using daily and weekly daa. Again, our main finding ha he loadings on he sock marke reurn and he value premium carry a posiive and significan risk premium holds well in he modern period. For breviy, hese resuls are no repored here bu are available on reques Diagnosics Tess To evaluae he adequacy of he benchmark ABEKK model repored in Table 9, we conduc several specificaion ess on he sandardized residuals ( ε i, = εi, / hii,, i= M, H ) and sandardized producs of residuals ( ε i, ε j, = εi, ε j, / hij,, i = M, H ). Specifically, we examine some momen condiions required for he consisency of QML esimaes. Panel A of Table 10 shows ha he wo mean sandardized residuals are no significanly differen from zero. However, he evidence is somewha mixed for esing he null hypohesis ha he mean of he producs of he residuals is 1. The null canno be rejeced for ε M, ε, M and ε H, ε, H bu can be rejeced for he cross-produc, ε M, ε H,. We also noe ha he skewness and kurosis for he sandardized residuals is much lower han he skweness and kurosis for he value premium bu no for he sock marke reurn. Panel B of Table 10 summarizes he Ljung-Box es for auocorrelaion in he esimaed residual series. The auocorrelaion is sill presen in he residuals of he HML equaion. (Recall ha he original HML series conains auocorrelaion.) 0

24 Overall, hese resuls indicae ha, while he model provides a reasonable descripion of he daa, here is sill room for improvemen. 4. Inernaional Evidence To address he quesion of daa snooping, Fama and French (1998) invesigae he value premium for major inernaional equiy markes consruced from MSCI (he Morgan Sanley Capial Inernaional) daa. They have wo main findings. Firs, he value premium is pervasive in major inernaional equiy markes. Second, he value premium appears o be a priced risk facor omied from he CAPM. In his secion we esimae he bivariae GARCH model using he Fama and French inernaional daa for he period January 1975 o December 005. Wihou he loss of generaliy, we focus on he world marke as well as he oher G7 counries, namely, Canada, France, Germany, Ialy, Japan, and he U.K. The world marke porfolios are especially relevan because hey represen he mos diversified porfolios: For example, Fama and French (1998) use he world sock marke reurn and value premium as risk facors in heir inernaional ICAPM. We also expec o uncover qualiaively similar paerns for each of he oher G7 counries because Fama and French (1998) find ha he counry-specific sock marke reurn and value premium move closely o heir world marke s counerpars. For breviy, we consider only he ABEKK model because, consisen wih U.S. evidence, i also provides a good descripion for all he inernaional markes ha we considered. In he esimaion we also impose he ICAPM resricions: γ MM = γ MH, γ HM γ HH =, and α = α = 0, R H which we fail o rejec using he log likelihood raio es. Table 11 shows ha inernaional evidence is quie consisen wih ha documened in U.S. daa. For he world marke, he price of marke risk, γ M, is significanly posiive, wih a poin esimae of Similarly, he risk price 1

25 for he value premium, γ H, is significanly posiive, wih a poin esimae of We also find qualiaively he same resuls for he individual markes. Excep for Ialy, he parameer γ H is posiive and saisically significan a leas a he 10% level for all he oher G7 counries. Similarly, he parameer γ M is always posiive, and i is significan a leas a he 10% level for France, Germany, Japan, and he U.K. Thus, he inernaional evidence provides furher suppor for he conjecure ha he value premium is a proxy for invesmen opporuniies. 5. Some Discussions In he pos-1963 sample, he CAPM fails o explain he value premium. Lakonishok, Shleifer, and Vishny (1994) argue ha he value premium reflecs mispricing: Invesors end o overesimae fuure earnings of growh socks bu underesimae fuure earnings of value socks. MacKinlay (1995) aribues he value premium o daa snooping. By conras, Fama and French (1996, 1998) advocae for a sysemaic risk explanaion for he value premium because i is a pervasive phenomenon in boh he U.S. and inernaional sock markes. One well-known raional-pricing explanaion is ha, as poined ou by Fama and French (1996), he value premium reflecs a disress risk. Fama and French explain he poin as follows: Why is relaive disress a sae variable of special hedging concern o invesors? One possible explanaion is linked o human capial, an imporan asse for mos invesors. Consider an invesor wih specialized human capial ied o a growh firm (or indusry or echnology). A negaive shock o he firm s prospecs probably does no reduce he value of he invesor s human capial; i may jus mean ha employmen in he firm will grow less rapidly. In conras, a negaive shock o a disressed firm more likely implies a negaive shock o he value of human capial since employmen in he firm is more likely o conrac. Thus, workers wih specialized

26 human capial in disressed firms have an incenive o avoid holding heir firms socks. If variaion in disress is correlaed across firms, workers in disressed firms have an incenive o avoid he socks of all disressed firms. The resul can be a sae-variable risk premium in he expeced reurns of disressed socks. (p.77). Fama and French (1995) and Liew and Vassalou (000) find ha he effec of he disress risk is more pronounced during business recessions han during business expansions. Thus he disress risk hypohesis helps explain he evidence ha he value premium has a posiive mean, alhough is uncondiional marke bea is negaive, as repored in Table 1. Consisen wih his conjecure, Leau and Ludvigson (001a) esimae a condiional consumpion-based CAPM, and find ha value socks are subsanially riskier han growh socks during economic recessions, when he condiional risk premium is high. Zhang (005) develops a parial equilibrium model, in which he marke bea of he value premium moves counercyclically; and Pekova and Zhang (005) find some empirical suppor for his predicion, especially in he early period 197 o 196. Noe ha hese explanaions also predic counercyclical movemen in he condiional value premium, as we documen in Figure 4. Alernaively and complemenarily, Campbell (1993) emphasizes ha he hedging demand associaed wih he ime-varying cos of capial has imporan effecs on expeced sock reurns. In his model, here are wo ypes of shocks he cash-flow shock and he discoun-rae shock. The cash-flow shock has a permanen effec on sock prices, while he effec of he discoun-rae shock is only emporary. Therefore, he cash-flow shock is riskier han he discoun-rae shock, and carries a higher risk price. Campbell and Vuoleenaho (004) find ha in he pos-1963 sample, he value premium has a negaive marke bea because of is large 3

27 posiive loadings on he discoun-rae shock. 10 However, he sample average of he value premium is posiive because of is posiive loadings on he cash-flow shock, which carries a much higher risk price han does he discoun-rae shock. In Campbell s (1993) ICAPM, he discoun-rae shock is negaively correlaed wih sock marke reurns. The value premium is a poenially good proxy for he discoun-rae shock because Figures 1 and 6 show ha he covariance beween he value premium and sock marke reurns is almos always negaive. This inerpreaion is also consisen wih recen empirical sudies by Cornell (1999) and Dechow, Sloan, and Soliman (004), who find ha growh socks are more vulnerable o he discoun-rae shock because hey have a higher duraion. Campbell s (1993) ICAPM also appears o provide a coheren explanaion for our main empirical findings. 11 For example, he discoun-rae shock is overpriced in he CAPM because invesors require a higher risk price for he cash-flow shock han he discoun-rae shock. The second righ-hand-size erm in equaion () serve as a correcion for he mispricing of he CAPM for he value premium. Overall, Figure 4 shows ha he expeced value premium is mosly posiive mainly because of is posiive loading on economic fundamenals, e.g., cash flows. As menioned above, his resul is in general consisen wih inuiion of Fama and French s (1996) disress risk hypohesis. I is also consisen wih he recen sudies by Bansal, Dimar, and Lundblad (005), Cohen, Polk, and Vuoleenaho (003), and Hansen, Heaon, and Li (004), who show ha he cash flows of value socks covary more wih aggregae cash flows han do hose of growh socks. Therefore, he CAPM fails o explain he cross-secion of sock reurns 10 The discoun-rae shock is negaively relaed o sock marke reurns because an increase in he discoun rae leads o an immediae fall in sock prices. 11 Leau and Wacher (006) propose a parsimonious model o explain he sylized fac ha he value premium has a posiive mean bu a negaive marke bea. Their main economic inuiion is similar o ha of Campbell (1993). 4

28 (e.g., Pekova and Zhang [005], Lewellen and Nagel [005] and Fama and French [005]) possibly because i provides a poor measure of sysemaic risk. However, Fama (1998) has poined ou ha he empirical ICAPM is also vulnerable o he criicism of daa snooping. In paricular, he empirical ICAPM sudied by Campbell and Vuoleenaho (004), for example, is poenially sensiive o wo ypes of misspecificaions (e.g., Chen [003] and Chen and Zhao [005]). Firs, Campbell (1993) suggess ha we should use variables ha forecas sock marke reurns as proxies for invesmen opporuniies; however, he provides lile guidance for he choice of he sock reurn predicors. Second, innovaions in he sae variables are no direcly observable and Campbell and Vuoleenaho, for example, mus rely on some ad hoc assumpions o idenify hem. In his paper we avoid hese wo idenificaion issues by direcly using he value premium as a proxy for invesmen opporuniies and hen invesigaing is asse pricing implicaions. We canno rule ou he daa mining; neverheless, i is unlikely o be he only explanaion of our main findings because hey hold up well in boh he U.S. and inernaional markes and are consisen wih numerous cross-secional sudies. 6. Conclusion This paper esimaes a varian of Meron s (1973) ICAPM using he value premium as a proxy for ime-varying invesmen opporuniies. In conras wih many early auhors, we uncover a posiive and significan risk-reurn radeoff afer conrolling for covariance of he sock marke reurn wih he value premium. We also documen a new finding on a significanly posiive relaion beween he value premium and is condiional variance. These resuls sugges ha we canno fully aribue he value premium o irraional pricing or daa mining. 5

29 Our resuls also shed ligh on ime-series sock marke reurn predicabiliy. We find ha i canno by fully aribued o irraional pricing or daa mining for hree reasons. Firs, exising economic heories have provided guidance for idenifying predicive variables, i.e., condiional variances and covariances of he risk facors in Meron s (1973) ICAPM. Second, despie is simpliciy, our analysis shows ha he heoreically moivaed variables forecas sock marke reurns in sample and ou of sample. Third, many financial variables forecas sock reurns mainly because of heir close correlaion wih condiional variances and covariances of sock marke reurns and oher risk facors. Wih few noable excepions, e.g., Campbell and Cochrane (1999) and Bansal and Yaron (004), he exising lieraure provides lile guidance for he fundamenal economic sources of variaion in he risk premium. A furher invesigaion of he link beween macroeconomy and financial markes should improve our undersanding of risk-reurn radeoff, and we leave i for fuure research. 6

30 Reference: Andersen, T., T. Bollerslev, F. Diebold, and P. Labys, 003, Modeling and Forecasing Realized Volailiy, Economerica, 71, Ang, A., and J. Chen, 005, CAPM Over he Long Run: , Journal of Empirical Finance, Forhcoming. Bansal, R., R. Dimar, and C. Lundblad, 005, Consumpion, Dividends, and he Cross-Secion of Equiy Reurns, Journal of Finance, 60, Bansal R. and A. Yaron, 004, Risks For The Long Run: A Poenial Resoluion of Asse Pricing Puzzles, Journal of Finance, 59, Belsley, D., E. Kuh, and R. Welsch, 1980, Regression Diagnosics: Idenifying Influenial Daa and Sources of Collineariy, John Wiley and Sons, New York, NY. Bollerslev, T., and J. Wooldridge, 199, Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time Varying Covariances, Economeric Reviews,11, Bossaers, P., and P. Hillion, 1999, Implemening Saisical Crieria o Selec Reurn Forecasing Models: Wha Do We Learn? Review of Financial Sudies, 1, Brand, M., and Q. Kang, 004, On he Relaionship beween he Condiional Mean and Volailiy of Sock Reurns: A Laen VAR Approach, Journal of Financial Economics, 7, Brand, M., and L. Wang, 006, Measuring he Time-Varying Risk-Reurn Relaion from he Cross-Secion of Equiy Reurns, Unpublished Working Paper, Duke Universiy. Brennan, M., A. Wang, and Y. Xia, 004, Esimaion and Tes of a Simple Model of Ineremporal Asse Pricing, Journal of Finance, 59,