A Cross-Sectional Investigation of the Conditional ICAPM * ABSTRACT

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1 A Cross-Secional Invesigaion of he Condiional ICAPM * Turan G. Bali a and ober F. Engle b ABSTACT This paper provides a cross-secional invesigaion of he condiional and uncondiional ineremporal capial asse pricing model (ICAPM). The resuls indicae ha esimaing he condiional ICAPM wih a pooled panel of ime series and cross-secional daa in a mulivariae GACH-in-mean framework is crucial in idenifying he posiive risk-reurn radeoff. Differen from he radiional lieraure, he paper decomposes he aggregae sock marke porfolio ino en book-o-marke porfolios and hen esimaes a cross-secionally consisen slope coefficien on he condiional variance-covariance marix. The riskaversion coefficien, resriced o be he same across all porfolios, is esimaed o be posiive and highly significan. This is he firs sudy esing he cross-secional consisency of he ineremporal relaion by esimaing he mulivariae GACH-in-mean model wih differen slopes. The saisical resuls indicae he equaliy of slope coefficiens across all porfolios, supporing he empirical validiy and sufficiency of he condiional ICAPM. The paper also provides evidence ha he ime-varying condiional covariances can explain he value premium because he average risk-adjused reurn difference beween he value and growh porfolios is economically and saisically insignifican wihin he condiional ICAPM framework. JEL classificaions: G; G3; C5. Keywords: ICAPM; isk-reurn radeoff; isk aversion; Mulivariae GACH-in-mean. November 008 * We hank Sephen Brown, Sijn Van Nieuwerburgh, ober Whielaw, and seminar paricipans a Sern School of Business, NYU for heir exremely helpful commens and suggesions. We hank Kenneh French for making a large amoun of hisorical daa publicly available in his online daa library. a Turan G. Bali is he David Krell Chair Professor of Finance a he Deparmen of Economics and Finance, Zicklin School of Business, Baruch College, One Bernard Baruch Way, Box 0-5, New York, NY 000. Phone: (646) , Fax: (646) 3-345, uran_bali@baruch.cuny.edu. b ober F. Engle is he Michael Armellino Professor of Finance a New York Universiy Sern School of Business, 44 Wes Fourh Sree, Suie 9-6, New York, NY 00, Phone: () , Fax : () , rengle@sern.nyu.edu. Elecronic copy available a: hp://ssrn.com/absrac=98633

2 A Cross-Secional Invesigaion of he Condiional ICAPM ABSTACT This paper provides a cross-secional invesigaion of he condiional and uncondiional ineremporal capial asse pricing model (ICAPM). The resuls indicae ha esimaing he condiional ICAPM wih a pooled panel of ime series and cross-secional daa in a mulivariae GACH-in-mean framework is crucial in idenifying he posiive risk-reurn radeoff. Differen from he radiional lieraure, he paper decomposes he aggregae sock marke porfolio ino en book-o-marke porfolios and hen esimaes a cross-secionally consisen slope coefficien on he condiional variance-covariance marix. The riskaversion coefficien, resriced o be he same across all porfolios, is esimaed o be posiive and highly significan. This is he firs sudy esing he cross-secional consisency of he ineremporal relaion by esimaing he mulivariae GACH-in-mean model wih differen slopes. The saisical resuls indicae he equaliy of slope coefficiens across all porfolios, supporing he empirical validiy and sufficiency of he condiional ICAPM. The paper also provides evidence ha he ime-varying condiional covariances can explain he value premium because he average risk-adjused reurn difference beween he value and growh porfolios is economically and saisically insignifican wihin he condiional ICAPM framework. JEL classificaions: G; G3; C5. Keywords: ICAPM; isk-reurn radeoff; isk aversion; Mulivariae GACH-in-mean. Elecronic copy available a: hp://ssrn.com/absrac=98633

3 . Inroducion Meron s (973) ineremporal capial asse pricing model (ICAPM) provides a posiive equilibrium relaion beween he condiional mean and variance of excess reurns on he aggregae marke porfolio. However, Abel (988), Backus and Gregory (993), and Gennoe and Marsh (993) develop models in which a negaive relaion beween expeced reurn and volailiy is consisen wih equilibrium. Similarly, empirical sudies are sill no in agreemen on he direcion of a ime-series relaion beween expeced reurn and risk. The esimaes for he relaive risk aversion coefficien are mosly insignifican or even negaive. Due o he fac ha he condiional mean and volailiy of sock marke reurns are no observable, differen approaches and specificaions used by previous sudies in esimaing he wo condiional momens are largely responsible for he conflicing empirical evidence. Following he radiional lieraure, we esimae he risk-reurn radeoff using a single series of he U.S. equiy marke index. Three differen approaches are uilized when invesigaing he ICAPM based on he marke porfolio: GACH-in-mean, realized volailiy, and range volailiy models. Consisen wih earlier sudies, he risk aversion esimaes provide no evidence for a significan link beween he condiional mean and volailiy of excess reurns on he value-weighed NYSE/AMEX/NASDAQ index. These resuls indicae ha despie is fundamenal imporance, he ineremporal risk-reurn relaion is inherenly difficul o esimae. Mos sudies find low -squares on he risk-reurn regressions, showing he exisence of large noise in he realized marke reurns. Wihou a proper idenificaion scheme, hese large noises can compleely disguise he fundamenal relaion beween he expeced excess reurn and he condiional variance. In his paper, we move away from he narrow focus on he single series of a marke porfolio reurn. Insead, by using a cross-secion of equiy porfolios, we effecively enlarge our sample of observaions and gain saisical power in idenifying he risk-reurn radeoff. Meron s (973) ICAPM predics ha an asse s expeced reurn depends on is covariance wih he marke porfolio and wih sae variables ha proxy for invesmen opporuniies. In oher words, Meron s predicion ha expeced reurns should be relaed o condiional risk applies no only o he marke porfolio, bu also o individual socks and sock porfolios. Indeed, for he model o be cross-secionally consisen, he ineremporal relaion beween he expeced excess reurn and is covariance wih he marke porfolio should universally be he same across all sock porfolios. In his paper, we exploi his cross-secional universaliy of he risk-reurn relaion and find a posiive and significan ineremporal relaion for equiy porfolios. Differen from he radiional lieraure, we divide he aggregae sock marke porfolio ino en booko-marke porfolios over he sample period of July 96 o December 007. We hen esimae differen forms of a mulivariae GACH-in-mean model wih he consan condiional correlaion (CCC) model of Bollerslev (990) and he mean-revering dynamic condiional correlaion (DCC) model of Engle (00). Following he original heoreical work of Meron (973), we resric he relaive risk aversion coefficien o be he same across all porfolios and he common slope esimae urns ou o be posiive, highly significan,

4 and robus o variaions in he condiional covariance specificaions and including a wide variey of sae variables ha proxy for he ineremporal hedging demand. We invesigae wheher he power of our mehodology is coming from () he GACH-based imevarying condiional covariances, or () pooling he ime series and cross secion ogeher, or (3) boh. Consisen wih he exising lieraure, we find ha he GACH-in-mean model canno generae a posiive and significan relaion beween risk and reurn on he aggregae marke porfolio. Tha is, he GACH-based mehodology iself canno resolve he issue if one narrowly focuses on a single series of he marke porfolio reurn. When we pool he ime-series and cross-secion ogeher, we find ha he uncondiional measures of marke risk canno predic expeced fuure reurns on equiy porfolios, implying insufficiency of he uncondiional ICAPM. Pu differenly, pooling he ime series and cross secion ogeher wihou he imevarying condiional covariances canno help idenify a significan risk-reurn radeoff eiher. Our resuls indicae ha esimaing he condiional ICAPM wih a pooled panel of ime series and cross secional daa in a mulivariae GACH-in-mean framework is essenial in idenifying he posiive risk-reurn radeoff. This paper ess for he firs ime he cross-secional consisency of he ineremporal relaion. We esimae he mulivariae GACH-in-mean model wih a differen slope coefficien for each equiy porfolio. The maximum likelihood esimaes of he slope coefficiens are found o be posiive, similar in magniude, and saisically significan for all he en porfolios of book-o-marke raio. The saisical resuls indicae he equaliy of slope coefficiens on he condiional variance-covariance marix, supporing he empirical validiy and sufficiency of he condiional ICAPM wih a common slope. While esimaing he mulivariae GACH-in-mean models, we allow he inerceps o be differen across porfolios. The inerceps capure he monhly abnormal reurns on each porfolio ha canno be explained by he condiional measures of marke risk. One implicaion of he ICAPM is ha he inerceps should no be joinly differen from zero assuming ha he covariances of risky asses wih he marke porfolio have enough predicive power for expeced fuure reurns. To examine he empirical validiy of he condiional ICAPM, we es he join hypohesis ha all inerceps equal zero. The Wald saisics fail o rejec he null hypohesis, implying ha he condiional measures of marke risk have significan predicive power for he ime-series and cross-secional variaions in expeced reurns on he book-o-marke porfolios. The paper also ess wheher he reurn differences beween he value and growh porfolios (value premium) can be explained by he condiional ICAPM. The resuls clearly indicae ha he DCC- and CCCbased ime-varying condiional covariances can explain he value premium because he average risk-adjused reurn difference beween he value and growh porfolios is found o be economically and saisically insignifican wihin he condiional ICAPM framework. When invesigaing he ineremporal hedging demands and he associaed risk premiums induced by he condiional covariaion of porfolio reurns wih a se of macroeconomic variables, we find ha he common slope coefficiens on he condiional covariances wih he unexpeced news in he inflaion rae and

5 3 he aggregae dividend yield are saisically significan, implying ha he inflaion-relaed and dividendrelaed shocks conain sysemaic risks rewarded in he sock marke and hey can be viewed as a proxy for invesmen opporuniies. However, he innovaions in defaul spread, erm spread, shor-erm ineres rae, and he growh rae of indusrial producion do no play a significan role in ineremporal hedging demand. Incorporaing he condiional covariaion wih any of hese sae variables does no change he posiive risk premium induced by he condiional covariaion of porfolio reurns wih he marke porfolio. The paper is organized as follows. Secion briefly discusses he ineremporal relaion beween risk and reurn. Secion 3 describes he daa. Secion 4 presens resuls from esimaing he risk-reurn radeoff on he marke porfolio. Secion 5 provides a cross-secional invesigaion of he condiional ICAPM. Secion 6 esimaes he ineremporal relaion wih he uncondiional measures of marke risk. Secion 7 examines he risk-reurn radeoff by accouning for he ineremporal hedging demand. Secion 8 concludes he paper.. The ineremporal relaion beween expeced reurn and risk Meron s (973) ICAPM implies he following equilibrium relaion beween expeced reurn and risk for any risky asse i: where r is he risk-free ineres rae, μ, () i r = β im + λ ix μ i r denoes he expeced excess reurn on he risky asse σ im denoes he covariance beween he excess reurns on he risky asse i and he marke porfolio and denoes a ( k ) row of covariances beween he excess reurns on he risky asse i and he k sae variables x. β is he relaive risk aversion coefficien and λ measures he marke s aggregae reacion o shifs in a k- dimensional sae vecor ha governs he sochasic invesmen opporuniy se. Equaion () saes ha in equilibriu invesors are compensaed in erms of expeced reurn for bearing marke (sysemaic) risk and for bearing he risk of unfavorable shifs in he invesmen opporuniy se. Many empirical sudies focus on he ime-series implicaion of he equilibrium relaion in eq. () and apply i narrowly o he marke porfolio. Wihou he hedging demand componen ( σ = 0 ), his focus leads o he following risk-reurn relaion: μm r = β m. () When considering sochasic invesmen opporuniy, he lieraure ofen implicily or explicily projecs he covariance vecor σ ix linearly o he sae variables x o obain he following relaion: μ r = β m m ix σ ix + λ x. (3) Our work in his aricle differs from he above lieraure in wo major ways. Firs, we esimae he ineremporal relaion eq. () no on he single series of he marke porfolio, bu simulaneously on equiy porfolios, and consrain all hese porfolios o have he same cross-secionally consisen proporionaliy

6 4 coefficiens β and λ. Second, we direcly esimae he condiional covariances σ im and σ ix using he consan condiional correlaion model of Bollerslev (990) and he dynamic condiional correlaion model of Engle (00). We do no make any linear projecion assumpions on he sae variables. The second erm in eq. () reflecs he invesors demand for he asse as a vehicle o hedge agains unfavorable shifs in he invesmen opporuniy se. An unfavorable shif in he invesmen opporuniy se is defined as a change in x such ha fuure consumpion c will fall for a given level of fuure wealh. Tha is, an unfavorable shif is an increase in x if c/ x < 0 and a decrease in x if c/ x > 0. Meron (973) shows ha all risk-averse uiliy maximizers will aemp o hedge agains such shifs in he sense ha if c/ x < 0 ( c/ x > 0), hen, ceeris paribus, hey will demand more of an asse, he more posiively (negaively) correlaed he asse s reurn is wih changes in x. Thus, if he ex pos opporuniy se is less favorable han was anicipaed, invesors will expec o be compensaed by a higher level of wealh hrough he posiive correlaion of he reurns. Similarly, if he ex pos reurns are lower, hey will expec a more favorable invesmen environmen. In his paper, we focus on he sign and saisical significance of he common slope coefficien ( β ) on σ im in he following risk-reurn relaion: μ. (4) i r = αi + β im According o he original ICAPM of Meron (973), he relaive risk aversion coefficien β is resriced o be he same across all risky asses and i should be posiive and saisically significan, implying a posiive risk-reurn radeoff. We es wheher he slopes on σ im are differen across risky asses. Earlier sudies assume a common slope coefficien ( β ) following he original heoreical work of Meron (973) and do no quesion he validiy of his assumpion. In his paper, we examine he sign and saisical significance of differen slope coefficiens ( β i ) on σ im in he following risk-reurn relaion: μ. (5) i r = αi + βi im To deermine wheher here is a common slope coefficien ( β ) on σ im corresponding o he average relaive risk aversion, we examine he cross-secional consisency of he ineremporal relaion by esing he join hypohesis ha H 0 : β = β =... = βn assuming ha we have n risky asses. Anoher implicaion of he ICAPM is ha he inerceps ( α ) in eq. (4) should no be joinly differen from zero assuming ha he covariances of risky asses wih he marke porfolio have enough predicive power for he ime-series and cross-secional variaions in expeced reurns. To deermine if significan explanaory power, we es he join hypohesis ha H 0 : α α =... = α 0. i = n = σ im has

7 5 We hink ha macroeconomic variables such as he defaul spread, erm spread, relaive T-bill rae, aggregae dividend yield, inflaion rae, and economic growh can be viewed as poenial sae variables ha may affec he sochasic invesmen opporuniy se. Hence, we examine wheher he innovaions in hese sae variables are priced in he condiional ICAPM framework. In oher words, we es if he changes in hese macroeconomic variables are risks rewarded in he sock marke. Specifically, we firs es he saisical significance of he common slope coefficien ( λ ) on σ ix in equaion (6), μ, (6) i r = αi + β im + λ ix and hen examine wheher he common slope ( β ) on σ im remains posiive and significan afer including σ ix o he risk-reurn relaion. 3. Daa We use he monhly excess reurns on he value-weighed book-o-marke porfolios and he monhly excess reurns on he value-weighed sock marke porfolio. The aggregae marke porfolio is proxied by he value-weighed NYSE/AMEX/NASDAQ index, which is also known as he value-weighed index of he Cener for esearch in Securiy Prices (CSP). The CSP index conains all socks rading a NYSE, AMEX, and NASDAQ and hence i can be viewed as he broades possible index for he U.S. equiy marke. Excess reurns on porfolios are obained by subracing he reurns on he one-monh Treasury bill from he raw reurns on equiy porfolios. The daa are obained from Kenneh French s online daa library: hp://mba.uck.darmouh.edu/pages/faculy/ken.french/daa_library.hml. We use he longes sample period available, from July 96 o December 007, yielding a oal of 978 monhly observaions. 3.. Value-Weighed Book-o-Marke Porfolios As described in Fama and French (993), book-o-marke porfolios are formed on he raio of book value equiy (BE) o marke value of equiy (ME) a he end of each June using NYSE breakpoins. They use all NYSE, AMEX, and NASDAQ socks for which hey have marke equiy for December of year T- and June of year T, and book equiy for year T-. BE used in June of year T is he book equiy for he las fiscal year end in T-. ME is price imes shares ousanding a he end of December of year T-. In June of each year, Fama and French (993) rank all NYSE socks in CSP based on he raio of BE/ME o deermine he NYSE decile breakpoins according o he ranked values of BE/ME for NYSE socks. Then, hey break all NYSE, AMEX, and NASDAQ socks ino 0 book-o-marke groups based on he NYSE breakpoins. Appendix A presens summary saisics for he monhly excess reurns on he 0 value-weighed book-o-marke (BM) porfolios. Growh (Decile ) is he porfolio of growh socks wih he lowes booko-marke raios and Value (Decile 0) is he porfolio of value socks wih he highes book-o-marke

8 6 raios. Mean, median, maximu minimu and sandard deviaion are repored for each porfolio in Panel A. For he sample period from July 96 o December 007, he value-weighed average excess reurn increases from 0.57% per monh (growh porfolio) o.09% per monh (value porfolio). The average reurn difference beween he value and growh porfolios is 0.5% per monh wih he OLS -saisic of.46 and he Newey-Wes (987) adjused -saisic of.39, implying ha value socks on average generae higher reurns han growh socks. This indicaes economically and saisically significan value premium (5 basis poins per monh) over he sample period of As shown in Panel A, he value-weighed median excess reurn increases from 0.75% per monh (growh porfolio) o 0.96% per monh (value porfolio), indicaing value premium based on he median reurn differences. The las wo columns of Panel A sugges ha here can be a risk-based explanaion of he reurn differences beween growh and value socks. Specifically, he sandard deviaion of excess reurns increases from 5.75% o 9.34% per monh as we move from he growh o value porfolios. Tha is, value socks are expeced o generae higher reurns han growh socks because value socks are riskier. Furhermore, here is a significan difference beween he marke beas of he growh and value porfolios. As shown in he las column of Panel A, he marke bea of he growh porfolio is.0063 whereas he marke bea of he value porfolio is These resuls indicae ha value socks have higher sysemaic risk han growh socks over he sample period of Value-Weighed Marke Porfolio Panel B of Appendix A presens summary saisics for he monhly excess reurns on he valueweighed NYSE/AMEX/NASDAQ index. The average excess reurn on he marke porfolio is 0.65% per monh corresponding o he expeced marke risk premium of 7.8% per annum. The maximum excess reurn on he marke porfolio is 38.7% per monh observed in April 933, whereas he minimum excess reurn on he marke porfolio is 9.04% per monh observed in Sepember 93. The sandard deviaion of excess reurns on he marke porfolio is 5.4% per monh. The average Sharpe raio (or he expeced excess reurn per uni risk) is abou 0. (0.0065/0.054) for he aggregae marke over he sample period of Macroeconomic Variables Some sudies find ha macroeconomic variables associaed wih business cycle flucuaions can predic he sock marke. The commonly chosen variables include Treasury bill raes, defaul spread, erm spread, and dividend-price raios. We sudy how variaions in he defaul spread, erm spread, de-rended shor-erm ineres rae, and he aggregae dividend yield predic variaions in he invesmen opporuniy se

9 7 and how incorporaing he condiional covariances of porfolio reurns wih he innovaions in hese variables affecs he ineremporal risk-reurn relaion. We obain he monhly yields on he 3-monh Treasury bill, 0-year Treasury bond, BAA-raed and AAA-raed corporae bonds from he H.5 daabase of he Federal eserve Board. The defaul spread (DEF) is compued as he difference beween he yields on he BAA-raed and AAA-raed corporae bonds. The erm spread (TEM) is calculaed as he difference beween he yields on he 0-year Treasury bond and he 3-monh Treasury bill. The relaive T-bill rae (EL) is he de-rended shor-erm ineres rae, defined as he difference beween he 3-monh T-bill rae and is -monh backward moving average. We download he monhly aggregae dividend yield from ober Shiller s websie: hp://aida.econ.yale.edu/~shiller/. The aggregae dividend yield (DIV) is defined as he raio of he monhly dividends of he S&P 500 index o he curren level of he S&P 500 index. In addiion o he aforemenioned macroeconomic variables, we use he monhly inflaion rae and he monhly growh rae of indusrial producion proxying for economic growh. The inflaion rae (INF) is he monhly growh rae of he Consumer Price Index available a ober Shiller s websie. The economic or oupu growh (OUT) is defined as he monhly growh rae of he Indusrial Producion Index obained from he G.7 daabase of he Federal eserve Board. The sample period for all hese macroeconomic variables is from July 96 o December isk-reurn radeoff on he marke porfolio Meron s (973) ICAPM indicaes ha he condiional expeced excess reurn on a risky marke porfolio is a linear funcion of is condiional variance plus a hedging componen ha capures he invesor s moive o hedge for fuure invesmen opporuniies. Ignoring he hedging demand componen, he equilibrium relaion beween risk and reurn is defined as: E ( + + ) = β E ( σ ), (7) where E ) and E ( σ ) are, respecively, he condiional mean and variance of excess reurns on ( m, + + he marke porfolio, and β > 0 is he average risk aversion of marke invesors. Equaion (7) esablishes he dynamic relaion ha invesors require a larger risk premium a imes when he marke is riskier. A large number of sudies fail o idenify a robus and significan ineremporal relaion beween risk and reurn on he aggregae sock marke porfolio. French, Schwer, and Sambaugh (987) find ha he coefficien esimae ( β ) in eq. (7) is no significanly differen from zero when hey use pas daily reurns o esimae he monhly condiional variance. Follow-up sudies by Baillie and DeGennaro (990), Campbell A an earlier sage of he sudy, we also used he aggregae dividend yield from he CSP value-weighed index wih and wihou dividends. The resuls from he CSP daa are very similar o our findings repored in he paper.

10 8 and Henchel (99), Chan, Karoly and Sulz (99), Glosen, Jagannahan, and unkle (993), Harrison and Zhang (999), Goyal and Sana-Clara (003), and Bollerslev and Zhou (006) rely on he GACH-inmean and realized volailiy models ha provide no evidence for a robus, significan link beween expeced reurn and risk on he aggregae marke porfolio. Some sudies find ha he ineremporal relaion beween risk and reurn is negaive (e.g., Campbell (987), Breen, Glosen, and Jagannahan (989), Turner, Sarz, and Nelson (989), Nelson (99), Glosen, Jagannahan, and unkle (993), Whielaw (994, 000), Harvey (00), and Brand and Kang (004)). Some sudies do provide evidence supporing a posiive and significan relaion beween expeced reurn and risk on individual socks (e.g., Bali and Engle (008)) and equiy porfolios (e.g., Bollerslev, Engle, and Wooldridge (988), Scruggs (998), Ghysels, Sana-Clara, and Valkanov (005), Guo and Whielaw (006), Lundblad (007), and Bali (008)). For comparison, we follow he radiional lieraure and esimae he risk-reurn radeoff using a single series of he value-weighed NYSE/AMEX/NASDAQ index. We use hree differen approaches when esing ICAPM based on he marke porfolio: Generalized Auoregressive Condiional Heeroskedasiciy (GACH), ealized Volailiy, and ange Volailiy models. 4.. GACH-in-Mean Model The following GACH-in-mean model is used o esimae he ineremporal relaion beween he expeced excess reurn and risk on he marke porfolio: E + + β + + ε + α, (8) ( ε Ω ) = σ + = γ 0 + γε + γ σ +, (9) where + is he excess reurn on he marke porfolio for monh +, E ( m, + Ω ) + β + α is he condiional mean of excess marke reurns for monh + based on he informaion se up o ime denoed by Ω, ε + z + + = σ is he error erm wih E ( ε m, + ) = 0, σ m, + is he condiional sandard deviaion of monhly excess reurns on he marke porfolio, and z + ~ (0,) is a random variable drawn from he sandard normal densiy and can be viewed as informaion shocks in he equiy marke. σ m, + is he condiional variance of monhly excess reurns based on he informaion se up o ime. The condiional N variance, σ m, +, in equaion (9) follows a GACH(,) process as defined by Bollerslev (986) o be a funcion of he las period s unexpeced news (or informaion shocks), z, and he las period s variance, σ. The GACH-in-mean models are originally inroduced by Engle, Lilien, and obins (987) and hen used by a large number of sudies including Engle, Ng, and ohschild (990), Baillie and DeGennaro (990), Nelson (99), and many recen papers on risk-reurn radeoff.

11 9 Our focus is o examine he magniude and saisical significance of he risk aversion parameer β in equaion (8). Table presens he maximum likelihood parameer esimaes and he -saisics in parenheses for he GACH-in-mean model. The risk aversion parameer ( β ) is esimaed o be posiive,.44, bu saisically insignifican based on he normal and he Bollerslev-Wooldridge (99) robus -saisics. For a robusness check of his finding, we consider he condiional sandard deviaion and he condiional logvariance in he condiional mean equaion: α β σ ε, (0) β ln σ + + ε + α, () As presened in Table, he slope coefficien on σ + is posiive, 0.56, bu saisically insignifican wih he Bollerslev-Wooldridge -saisic of.04. Similarly, he slope coefficien on m, + lnσ is posiive, 0.003, bu insignifican wih -sa.= The resuls from alernaive specificaions of he GACH-in-mean model provide no evidence for a significan link beween expeced reurn and risk on he marke porfolio. Anoher noable poin in Table is he significance of volailiy clusering. For all specificaions of he volailiy process, he condiional variance parameers (γ, γ ) are posiive, beween zero and one, and highly significan. The resuls indicae he presence of raher exreme condiionally heeroskedasic volailiy effecs in he sock reurn process because he GACH parameers, γ and γ, are no only highly significan, bu also he sum (γ + γ ) is close o one. This implies he exisence of subsanial volailiy persisence in he sock marke. 4.. ealized Volailiy Model Earlier sudies ha invesigae he monhly risk-reurn radeoff generally rely on he GACH-inmean or ealized Volailiy models. Following earlier sudies, we calculae he monhly realized variance using he wihin-monh daily reurn daa: D D = d + d = d = where D is he number of rading days in monh and σ, () d d m, d is he daily reurn on he value-weighed marke porfolio on day d. The aggregae marke porfolio is measured by he value-weighed CSP index. The second erm on he righ hand side adjuss for he auocorrelaion in daily reurns using he approach of French, Schwer, and Sambaugh (987). 3 3 As discussed in French, Schwer, and Sambaugh (987), equaion () is no sricly speaking a variance measure because i does no demean reurns before aking he expecaion. However, for shor holding periods, he impac of subracing he means is rivial. Using daily daa, French, Schwer, and Sambaugh (987) and Schwer (989) find he

12 0 We firs generae he monhly realized variance based on equaion () and hen esimae he following risk-reurn regressions: + = α + β + ε + + = + β + ε + α (3) + = α + β ln σ + ε + where + is he one-monh ahead excess reurn on he value-weighed NYSE/AMEX/NASDAQ index and he expeced condiional variance of he marke porfolio, E ( σ ), is proxied by he lagged realized + variance measure, i.e., E ( σ ) = σ +. As shown in equaion (3), we consider he lagged realized sandard deviaion ( σ ) and he lagged log realized variance ( lnσ ) in he risk-reurn regressions as well. Table presens he parameer esimaes and heir OLS and Newey-Wes (987) adjused -saisics from he risk-reurn regressions wih monhly realized volailiy. The relaive risk aversion parameer ( β ) on σ is esimaed o be posiive, 0.0, bu saisically insignifican wih he Newey-Wes -saisic of 0.. Similar resuls are obained when he realized sandard deviaion and log-variance are used o es he significance of he ineremporal relaion. Specifically, he slope coefficien on σ is esimaed o be wih -sa.= 0.8 and he slope on lnσ is wih -sa.= 0.3. These resuls indicae ha he monhly realized volailiy canno predic fuure reurns on he marke porfolio, implying an insignifican ineremporal relaion beween expeced reurn and risk ange Volailiy Model I is no common o use a range volailiy esimaor when esing he significance of a risk-reurn radeoff. Following Alizadeh, Brand, and Diebold (00) and Brand and Diebold (006), monhly range volailiy is defined as he difference beween he logarihms of he highes and lowes index levels. where max P d, and range max min σ = ln( P ) ln( P ), (4) d, d, min P d, are he maximum and minimum daily index levels of he NYSE/AMEX/NASDAQ on day d in monh. 4 We firs generae he monhly range variance, sandard deviaion, and log-variance based on equaion (4) and hen esimae he risk-reurn regressions given in equaion (3). Table 3 presens he parameer esimaes and heir OLS and Newey-Wes correced -saisics from he risk-reurn regressions wih monhly squared mean erm is irrelevan o variance calculaions. A an earlier sage of he sudy, we used he demeaned daily reurns in equaion () and he qualiiaive resuls remained inac. We also eliminaed he second erm on he righ of equaion () and he qualiaive resuls urned ou o be very similar o hose repored in our ables. 4 Alizadeh e al. (00) and Brand and Diebold (006) indicae ha he range-based volailiy esimaor is highly efficien, approximaely Gaussian and robus o cerain ypes of microsrucure noise such as bid-ask bounce.

13 range volailiy. The relaive risk aversion parameer ( β ) on σ is esimaed o be posiive, 0.59, bu saisically insignifican wih he Newey-Wes -saisic of Similar resuls are obained when he range sandard deviaion and log-variance are used o es he significance of he ineremporal relaion. Specifically, he slope coefficien on σ is esimaed o be wih -sa. = 0.9 and he slope on lnσ is wih -sa.= 0.4. These resuls indicae ha he monhly range volailiy canno predic he ime-series variaion in excess marke reurns, implying an insignifican link beween risk and reurn. To make sure ha our resuls from esimaing equaion (7) are no due o model misspecificaion, we added o he regressions a se of conrol variables ha have been used in he lieraure o capure changes in he invesmen opporuniy se. As discussed in Appendix B, including a wide variey of macroeconomic variables does no affec he insignifican relaion beween risk and reurn on he aggregae marke porfolio. Among he conrol variables, we find a significanly negaive (posiive) relaion beween he excess marke reurn and he inflaion rae (he aggregae dividend yield), whereas he defaul spread, erm spread, relaive T-bill rae, and oupu growh canno predic fuure reurns on he marke porfolio. 5. A cross-secional invesigaion of he condiional ICAPM Consisen wih he exising lieraure, when we use a single series of he marke porfolio reurn, we find no evidence for a significan link beween risk and reurn in he aggregae sock marke. Differen from he radiional lieraure, in his secion, we firs divide he aggregae marke porfolio ino en book-omarke porfolios and hen esimae he cross-secionally consisen slope coefficien on he condiional variance-covariance marix. Second, we invesigae he cross-secional consisency of he ineremporal relaion by esing he equaliy of slope coefficiens in he mulivariae GACH-in-mean model. Finally, we es wheher he value premium can be explained wihin he condiional ICAPM framework Invesigaing ICAPM wih Consan Condiional Correlaions We esimae he risk aversion coefficien ( β ) based on he following mulivariae GACH-in-mean model wih consan condiional correlaions (CCC): E α β σ ε (5) + = i + i = m + β + + ε + α (6) i i i [ ε ] σ = γ + γ ε + γ σ (7) 5 In addiion o he book-o-marke porfolios, a an earlier sage of he sudy we used he en value-weighed momenum and indusry porfolios which are available a Kenneh French s online daa library. We also used he en equalweighed size porfolios from he CSP daabase. The qualiaive resuls from he size, momenum and indusry porfolios urn ou o be similar o our findings repored in he paper. They are available upon reques.

14 E E m m m [ ε ] σ + = γ 0 + γ ε + γ σ [ + ε + ] σ i + = ρim (8) ε (9) where + and + denoe he ime (+) excess reurn on porfolio i and he marke porfolio m over a risk-free rae, respecively, and E [.] denoes he expecaion operaor condiional on ime informaion. + σ is he ime- expeced condiional variance of +, σ is he ime- expeced condiional variance of +, and σ i + is he ime- expeced condiional covariance beween + and +. ρ im is he consan condiional correlaion beween beween + and +. In he condiional mean given by equaions (5)-(6), we have a common slope coefficien ( β ) bu differen inerceps for each book-omarke porfolio ( α ) and he sock marke porfolio ( α ). The condiional variance parameers are assumed i i i i ( 0 m + m m m ( 0 o be differen for each sock γ, γ, γ ) and he marke porfolio γ, γ, γ ). We use he maximum likelihood mehodology wih he mulivariae normal densiy o esimae equaions (5)-(9) in one sep. Using ε + and Σ + o denoe, respecively, he mulivariae residual vecor and he condiional variance-covariance marix, ε i =, + α β i αm β + i, Σ σ = σ σ + i + + i + σ +, we can wrie he condiional log-likelihood funcion as N T [ ln( ) + ln Σ + + ε + Σ + ] + L( Θ) = π ε, = where Θ denoes he vecor of parameers in he specificaions in (5) o (9), and N denoes he number of monhly observaions for each series. Since we have a oal of porfolios (including he marke porfolio), ε + is and Σ + is marix. The parameers are esimaed using he 0 value-weighed book-o-marke porfolios and he valueweighed NYSE/AMEX/NASDAQ index as a marke porfolio. Since we have a oal of porfolios, we esimae a oal of 00 parameers simulaneously. Specifically, we have condiional means wih a oal of parameers ( inerceps and a common slope), we have condiional variances wih a oal of 33 parameers, and we have a oal of 55 consan condiional correlaions. Table 4 repors he common slope esimae ( β ), he abnormal reurns ( α, α ) for each porfolio, he -saisics of he parameer esimaes, and he maximized log-likelihood value. Esimaion is based on he monhly excess reurns over he sample period of July 96 o December 007. The risk aversion coefficien i m

15 3 is esimaed o be posiive ( β = ), and highly significan wih he -saisic of This implies a posiive and significan ineremporal relaion beween expeced reurn and risk on equiy porfolios. When esimaing he mulivariae GACH-in-mean model in equaions (5)-(9), we allow he inerceps ( α, α ) o be differen across porfolios. These inerceps capure he monhly abnormal reurns i m on each porfolio ha canno be explained by he condiional covariances wih he marke porfolio. The firs column of Table 4 shows ha he abnormal reurn on he growh porfolio (wih he lowes book-o-marke raio) is α = 0.5% per monh wih he -saisic of 0.7, whereas he abnormal reurn on he value porfolio (wih he highes book-o-marke raio) is α 0 = 0.8% per monh wih he -saisic of 0.8. One implicaion of he ICAPM is ha he inerceps ( α, α ) should no be joinly differen from zero assuming ha he covariances of risky asses wih he marke porfolio have enough predicive power for expeced fuure reurns. To examine he empirical validiy of ICAPM, we es he join hypohesis ha H0 : α = α =... = α0 = αm = 0. As presened in Table 4, he Wald saisic is 5.84 wih a p-value of 4.7%, which fails o rejec he null hypohesis ha all inerceps equal zero. This resul indicaes ha he condiional covariances of he book-o-marke porfolios wih he aggregae marke porfolio have significan predicive power for he ime-series and cross-secional variaions in expeced reurns. According o he original ICAPM of Meron (973), he relaive risk aversion coefficien β is resriced o be he same across all risky asses and i should be posiive and saisically significan. The common slope esimae in Table 4 provides empirical suppor for he posiive risk-reurn radeoff. We now es wheher he slopes on ( σ, σ ) are differen across risky asses. Earlier sudies assume im m a common slope coefficien ( β ) following he original heoreical work of Meron (973) and do no quesion he validiy of his assumpion. In his paper, we examine he sign and saisical significance of differen slope coefficiens ( β, β ) on ( σ, σ ) in he following mulivariae GACH-in-mean model wih consan condiional correlaions: i m im m i m E E + = i + βi i + + ε + α, (0) + = m + βm + + ε + α, () i i i E [ ε ] σ + = γ 0 + γε + γ σ m m m [ ε ] σ = γ + γ ε + γ σ +, () + 0 +, (3) [ + ε + ] σ i + = ρim + + ε. (4)

16 4 To deermine wheher here is a common slope coefficien ( β ) on σ im corresponding o he average relaive risk aversion, we firs esimae he porfolio-specific slope coefficiens ( β, β ) and hen es he join hypohesis ha H0 : β = β =... = β0 = βm. Table 5 presens he maximum likelihood parameer esimaes based on he 0 value-weighed booko-marke porfolios and he value-weighed NYSE/AMEX/NASDAQ index. Compared o equaions (5)- (9), we have an addiional 0 slope coefficiens o esimae in equaions (0)-(4), yielding a oal of 0 parameers. As shown in Table 5, all of he slope coefficiens ( β, β ) are esimaed o be posiive and highly significan wihou any excepion. Specifically, he minimum slope is abou 3.57 wih -sa. =.8, and he maximum slope is abou 4.7 wih -sa. = We should noe ha he average of hese slope coefficiens is abou 4.3, which is close o he common slope esimae of 4.7 repored in Table 4. These resuls indicae a posiive and significan ineremporal relaion beween risk and reurn on equiy porfolios. We examine he cross-secional consisency of he ineremporal relaion by esing he equaliy of slope coefficiens based on he likelihood raio (L) es saisic. As shown in Tables 4 and 5, he maximized log-likelihood value of he resriced model (wih a common slope coefficien) is 7,798.8, whereas he maximized log-likelihood value of he unresriced model (wih differen slope coefficiens) is 7, These maximized log-likelihood values yield he L es saisic of 0.88 wih a p-value of 45.34%, which canno rejec he join hypohesis ha H0 : β = β =... = β0 = βm. 6 When esimaing he mulivariae GACH-in-mean model in equaions (0)-(4), we allow he inerceps ( α, α ) o be differen across porfolios. The firs column of Table 5 shows ha he abnormal i m reurn on he growh porfolio is α = 0.0% per monh wih he -saisic of 0.07, whereas he abnormal reurn on he value porfolio is α 0 = 0.5% per monh wih he -saisic of 0.8. We es wheher here is a significan average risk-adjused reurn difference beween he value and growh porfolios. Specifically, we es he null hypohesis ha H 0 : α = α0. The Wald saisic repored in Table 5 is abou 0.0 wih a p-value of 65.54%, indicaing insignifican value premium over he sample period As shown in Appendix A, he average excess reurn is 0.57% per monh for he growh porfolio and.09% per monh for he value porfolio. The average raw reurn difference beween he value and growh porfolios is 0.5% per monh wih he Newey-Wes -saisic of.39, implying ha value socks on average generae higher raw reurns han growh socks. However, afer conrolling for he condiional covariance risk, he average risk-adjused reurn difference beween he growh and value porfolios reduces o 0.7% i m i m 6 L saisic is calculaed as L = (LogL* LogL), where LogL* is he value of he log likelihood under he null hypohesis, and LogL is he log likelihood under he alernaive: L = (7, ,803.6) = The criical Chi-squared values wih degrees of freedom are χ (,0.0) = 7.8, χ (,0.05) = 9.68, and χ (,0.0) = 4.7 a he 0%, 5% and % level of significance, respecively.

17 5 per monh and i is saisically insignifican. In oher words, he condiional ICAPM explains he value premium for he sample period of July 96 December To furher examine he empirical validiy and sufficiency of he condiional ICAPM, we es wheher he abnormal reurns on equiy porfolios are joinly equal o zero. As shown in Table 5, he Wald saisic is 7. wih a p-value of 78.4%, which fails o rejec he join hypohesis ha H α α =... = α = α 0. 0 : = 0 m = This resul implies ha he condiional covariances of he book-o-marke porfolios wih he aggregae marke porfolio can capure he ime-series and cross-secional variaions in expeced reurns. 5.. Invesigaing ICAPM wih Dynamic Condiional Correlaions In his secion, we esimae he risk aversion coefficien ( β ) based on he mulivariae GACH-inmean model wih he mean-revering dynamic condiional correlaions (DCC) of Engle (00): q E E + = i + β i + + σ + u + α (5) + = m + β + + σ + u + α (6) i i i E [ ε ] σ + = γ 0 + γσ u + γ σ m m m [ ε ] σ = γ + γ σ u + γ σ + (7) (8) [ + ε + ] σ i + = ρ i ε (9) i + ρ i + =, qim + = ρim + a ( u u ρim ) + a ( qi ρim ) qi + qm +, (30) where ρ im is he uncondiional correlaion beween i u, and u m,. In he condiional mean given by eqs. (5)- (6), we have a common slope coefficien ( β ) bu differen inerceps for book-o-marke porfolios ( α i ) and he sock marke porfolio ( α ). The condiional variance parameers are assumed o be differen for i i i ( 0 m m m m ( 0 each sock γ, γ, γ ) and he marke porfolio γ, γ, γ ). Following Engle (00), we assume ha he correlaion parameers (a, a ) in he mean-revering DCC model are consan across porfolios. To ease he parameer convergence, we use correlaion argeing assuming ha he ime-varying correlaions mean rever o he sample correlaions ρ im. To reduce he overall ime of maximizing he condiional log-likelihood, following Engle (009), we firs esimae all pairs of he bivariae GACH-inmean model and hen use he median values of β, a and a as saring values along wih he bivariae GACH-in-mean esimaes of he variance parameers (γ 0, γ, γ ). Even afer going hrough hese seps o 7 This resul is consisen wih Ang and Chen (007) who esimaed a condiional CAPM specificaion wih sochasic marke bea.

18 6 increase he speed of parameer convergence, i akes long ime o obain he full se of parameers in he mulivariae GACH-in-mean model wih porfolios and 978 ime-series observaions. The common slope and he inerceps are esimaed using he monhly excess reurns on he 0 value-weighed book-omarke porfolios for he sample period from July 96 o December 007. The aggregae marke porfolio is proxied by he value-weighed NYSE/AMEX/NASDAQ index. Table 6 shows ha he risk-reurn coefficien ( β ) on he condiional variance-covariance marix is esimaed o be abou 5. wih he -saisic of The magniude and saisical significance of he common slope urns ou o be similar o our earlier findings from he consan condiional correlaions. We es he join hypohesis ha all inerceps equal zero and he Wald saisic is found o be 6.9 wih a p- value of 3.43%. Overall, he resuls in Table 6 provide similar evidence ha here is a significanly posiive relaion beween risk and reurn on equiy porfolios and he abnormal reurns are individually and joinly equal zero, suggesing validiy of he condiional ICAPM. Pu differenly, he DCC-based condiional covariances can explain he ime-series and cross-secional variaions in porfolio reurns. We now es wheher he slopes on he condiional variance-covariance marix are differen across sock porfolios. Specifically, we examine he sign and saisical significance of differen slope coefficiens ( i β, β ) on ( σ, σ ) in he mulivariae GACH-in-mean model wih DCC: m im m q E E + = i + βi i + + σ + ui, + α (3) + = m + βm + + σ + u + α (3) i i i E [ ε ] σ + = γ 0 + γσ u + γ σ m m m [ ε ] σ = γ + γ σ u + γ σ + (33) (34) [ + ε + ] σ i + = ρi ε (35) i + ρ i + =, qim + = ρim + a ( u u ρim ) + a ( qi ρim ) qi + qm +, (36) Table 7 shows ha all of he slope coefficiens ( β, β ) are esimaed o be posiive and highly significan wihou any excepion. Specifically, he minimum slope is abou 4. wih -sa.= 3.8, and he maximum slope is abou 5.63 wih -sa.= 5.7. We should noe ha he average of hese slope coefficiens is abou 4.98, which is close o he common slope esimae of 5. repored in Table 6. These resuls show a posiive and significan ineremporal relaion beween risk and reurn on equiy porfolios. We invesigae he cross-secional consisency of he ineremporal relaion by esing he equaliy of slope coefficiens based on he likelihood raio (L) es saisic. As shown in Tables 6 and 7, he maximized log-likelihood value of he resriced model (wih a common slope coefficien) is 7,874.77, whereas he i m

19 7 maximized log-likelihood value of he unresriced model (wih differen slopes) is 7, These maximized log-likelihood values yield he L saisic of 3. wih p-value = 7.9%, which canno rejec he join hypohesis H0 : β = β =... = β0 = βm. The firs column of Table 7 shows ha he abnormal reurn on he growh porfolio is α = 0.05% per monh wih -sa.= 0.0, whereas he abnormal reurn on he value porfolio is α 0 = 0.0% per monh wih -sa.= We es wheher here is a significan average risk-adjused reurn difference beween he value and growh porfolios. The Wald saisic from esing he null hypohesis H0 : α = α0 is abou 0.04 wih p-value = 83.36%, indicaing insignifican value premium over he sample period of Afer conrolling for he marke risk wih he DDC-based ime-varying condiional covariances, he average riskadjused reurn difference beween he growh and value porfolios becomes only 6 basis poins per monh and saisically insignifican. To furher examine he empirical validiy of he condiional ICAPM, we es wheher he abnormal reurns on equiy porfolios are joinly equal o zero. As shown in Table 7, he Wald saisic is 9.46 wih p-value = 57.98%, which fails o rejec he join hypohesis of zero inerceps. These resuls indicae ha he condiional ICAPM wih he dynamic condiional correlaions has significan predicive power for he ime-series and cross-secional variaions in expeced reurns and explains he value premium over he long sample period A cross-secional invesigaion of he uncondiional ICAPM I is well documened in he lieraure ha narrowly focusing on a single series of he marke porfolio, i is difficul o find a significan ineremporal relaion beween risk and reurn in he aggregae sock marke. Our resuls from he GACH-in-mean, realized, and range volailiy models are aligned wih he exising lieraure. However, by pooling he ime series and cross secion ogeher, we find ha he DCCand CCC-based ime-varying condiional covariances generae a significanly posiive ineremporal relaion beween expeced reurn and risk on book-o-marke porfolios. The significan, robus and sensible esimaes highligh he added benefis of using he GACH-based condiional measures of marke risk and by esimaing he condiional ICAPM wih a pooled panel of ime series and cross secional daa. 9 We invesigae wheher he power of our mehodology is coming from () he GACH-based imevarying condiional covariances, or () pooling he ime series and cross secion ogeher, or (3) boh. In his secion, we provide a cross-secional invesigaion of he uncondiional ICAPM. Specifically, we examine wheher he uncondiional measures of marke risk can predic expeced fuure reurns on equiy porfolios. 8 Appendix C presens descripive saisics of he DCC- and CCC-based condiional covariance esimaes which are key o our cross-secional invesigaion of he condiional ICAPM. 9 Alhough some of our findings are no presened in he paper o save space, he resuls are robus across differen equiy porfolios (book-o-marke, size, momenu indusry), differen specificaions of he condiional covariance process, and including a large se of conrol variables o he panel of mulivariae GACH-in-mean model.

20 8 We have already shown ha he GACH-in-mean model canno generae a posiive and significan relaion beween risk and reurn on he aggregae marke porfolio. In oher words, he GACH-based mehodology iself canno resolve he issue if one narrowly focuses on a single series of he marke porfolio reurn. We now ake a closer look a he effec of using a pooled panel of ime series and cross secional daa on he idenificaion of risk-reurn radeoff. Earlier sudies saring wih Fama and French (99) and French, Schwer, and Sambaugh (987) show ha he uncondiional bea (or he uncondiional covariance of risky asses wih he marke porfolio) canno predic he cross-secional variaion in sock reurns and ha he ineremporal risk-reurn relaion is no significanly posiive in ime-series regressions. 0 Based on he rolling regressions, Fama and French (99) firs esimae he uncondiional marke bea using he pas 4 o 60 monhs of reurns (as available) on individual socks rading a he NYSE, AMEX, and NASDAQ. Then, hey show ha here is no significan relaion beween he uncondiional measures of marke risk and he cross-secion of expeced reurns. In his paper, following Fama and French (99), we use he monhly rolling regressions and esimae he uncondiional variance of excess reurns on he marke porfolio as well as he uncondiional covariances beween excess reurns on he book-o-marke porfolios and he marke porfolio using he pas 4, 36, 48, and 60 monhs of daa. Given he uncondiional covariances and he uncondiional variance of he marke porfolio for each monh in our sample, we esimae he ineremporal relaion from he following sysem of equaions: + = i + β i + ε + + = m + β + ε + α, (37) α, (38) where he expeced condiional variance of he marke porfolio, E ( σ ), is proxied by he one-monh lagged realized variance, i.e., E ( σ ) = σ + +, and similarly he expeced condiional covariance of booko-marke porfolios wih he marke porfolio, E σ ), is proxied by he one-monh lagged realized ( i + covariance, i.e., E ( σ i + ) = σ i. We esimae he sysem of equaions (37)-(38) using a weighed leas square mehod ha allows us o place consrains on coefficiens across equaions. We consrain he slope coefficien ( β ) on he 0 Empirical sudies of he ICAPM diverge ino wo perpendicular dimensions. Sudies ha focus on he ineremporal risk-reurn relaion ofen choose o use merely one reurn series on he marke porfolio, ignoring he model s implicaion ha he same relaion beween excess reurns and heir condiional covariances wih he marke should hold across all sock porfolios o guaranee cross-secional consisency. However, some of he earlier sudies on condiional CAPM (e.g., Jagannahan and Wang (996)) deal wih an uncondiional implicaion of he condiional relaion by regressing excess reurns on he uncondiional bea and an uncondiional covariance erm ha accouns for he covariaion beween he condiional bea and he condiional marke risk premium. Wha his exercise ignores is ha a more direc es of he condiional relaion is o esimae he condiional relaion iself, insead of dealing wih he uncondiional implicaion of he condiional relaion. The narrow focus of boh srands of he lieraure ofen leads o insignifican esimaes on marke bea or marke variance.