THE CROSS-SECTIONAL VARIATION OF VOLATILITY RISK PREMIA

Transcription

1 THE CROSS-SECTIONAL VARIATION OF VOLATILITY RISK PREMIA Ana González-Ureaga Universidad Pública de Navarra Gonzalo Rubio Universidad CEU Cardenal Herrera Absrac This paper analyzes he deerminans of he cross-secional variaion of he average volailiy risk premia for a se of 0 porfolios sored by volailiy risk premium bea. The marke volailiy risk premium and, especially, he defaul premium are shown o be key deerminans risk facors in he cross-secional variaion of average volailiy risk premium payoffs. The cross-secional variaions of risk premia reflecs he differen uses of volailiy swaps in hedging defaul and he financial sress risks of he underlying componens of our sample porfolios. This version: Sepember, 04 Keywords: volailiy risk premia; sochasic discoun facor; consumpion-based models; linear facor models; defaul premium JEL classificaion: G, G3 We hank Javier Gil-Bazo, Belén Nieo, James Sinclair and seminar paricipans a Pompeu Fabra and Carlos III Universiies, and he Eighh Poruguese Finance Associaion Inernaional Conference. The auhors acknowledge financial suppor from he Minisry of Economics and Compeiiveness hrough gran ECO In addiion, Gonzalo Rubio acknowledges financial suppor from Generalia Valenciana gran PROMETEOII/03/05, and Ana González-Ureaga acknowledges financial suppor from ECO Corresponding auhor: Gonzalo Rubio (gonzalo.rubio@uch.ceu.es).

2 . Inroducion Since he seminal paper of Bakshi and Kapadia (003a), he marke variance risk premium has been repored o be negaive, on average, during alernaive sample periods. Since he payoff of a variance swap conrac is he difference beween he realized variance and he variance swap rae, negaive reurns o long posiions on variance swap conracs for all ime horizons mean ha invesors are willing o accep negaive reurns for purchasing realized variance. Equivalenly, invesors who are sellers of variance and are providing insurance o he marke, require subsanial posiive reurns. This may be raional, since he correlaion beween volailiy shocks and marke reurns is known o be srongly negaive and invesors wan proecion agains sock marke crashes. Along hese lines, Bakshi and Madan (006), and Chabi- Yo (0) heoreically show ha he skewness and kurosis of he underlying marke index are key deerminans of he marke variance risk premium. Indeed, Bakshi and Madan (006), Bollerslev, Gibson, and Zhou (0), Bekaer and Hoerova (03), and Bekaer, Hoerova and Lo Duca (03) argue ha he marke variance risk premium is an indicaor of aggregae risk aversion. 3 A relaed inerpreaion is due o Bollerslev, Tauchen, and Zhou (009) and Drechsler and Yaron (0), who inerpre he marke variance risk premium as a proxy of macroeconomic risk (consumpion uncerainy). They show ha ime-varying economic uncerainy and a preference for he early resoluion of uncerainy are required o generae a negaive marke variance risk premium. Zhou (00) shows ha he marke variance risk premium significanly predics shor-run equiy reurns, bond reurns, and credi spreads. Consequenly, he For empirical evidence of he negaive variance risk premium on he marke index, see Carr and Wu (009) and he papers cied in heir work. A variance swap is an over-he-couner derivaive conrac in which wo paries agree o buy or sell he realized variance of an index or single sock on a fuure dae. 3 More specifically, Bekaer, Hoerova, and Lo Duca (03) show he ineracions beween moneary policy and he marke variance risk premium, suggesing ha moneary policy may impac aggregae risk aversion.

3 auhor argues ha risk premia in major markes comove in he shor- run and ha such comovemen seems o be relaed o he marke variance risk premia. Campbell, Giglio, Polk, and Turley (04), using an ineremporal capial asse pricing model (CAPM) framework, argue ha covariaion wih aggregae volailiy news has a negaive premium. Finally, Nieo, Novales, and Rubio (04) show ha he uncerainy ha deermines he variance risk premium he invesors fear of deviaing from normaliy in reurns is also srongly relaed o a variey of macroeconomic and financial risks associaed wih defaul, employmen growh, consumpion growh, and sock marke and marke illiquidiy risks. A his poin, i is fair o argue ha we undersand he behavior of he marke variance risk premium and is implicaions for financial economics. However, i is surprising how lile we know abou he variance risk premium a he individual level. Bakshi and Kapadia (003b) show ha he variance risk premium is also negaive in individual equiy opions. However, Driessen, Maenhou, and Vilkov (009) show ha he variance risk premium for sock indices is sysemaically larger, ha is, more negaive, han for individual securiies. They argue ha he variance risk premium can, in fac, be inerpreed as he price of ime-varying correlaion risk. They show ha he marke variance risk is negaive only o he exen ha he price of he correlaion risk is negaive. In a relaed paper, Buraschi, Trojani, and Vedolin (04) argue ha he wedge beween index and volailiy risk premia is explained by invesor disagreemen. Hence, he greaer he differences in beliefs among invesors, he larger he marke volailiy risk relaive o he volailiy risk premium of individual opions. Even hese papers are paricularly concerned wih he behavior of he marke variance risk premium, despie employing daa a he individual level. 3

4 We argue ha an analysis and he undersanding of he ime-series and crosssecional behavior of he variance risk premium a he individual level is lacking in he previous lieraure. This paper parially covers his gap. More specifically, we analyze he cross-secional variaion of he volailiy risk premium (svrp) a he porfolio level. We employ daily daa from OpionMerics for he Sandard & Poor s (S&P) 00 Index opions and for individual opions on 8 socks included a some poin in he S&P 00 Index during he sample period from January 996 o February 0. We employ opions wih one monh o expiraion. We calculae svrp for each sock a he 30-day horizon as he difference beween he corresponding realized volailiy and he modelfree implied volailiy described by Jiang and Tian (005). Similarly, we esimae he marke volailiy risk premium using he S&P 00 Index as he underlying index. For each monh, using an individual svrp wih a leas 5 daily observaions, we consruc 0 equally weighed porfolios ranking he individual svrp values according o heir beas wih respec o he marke svrp. These volailiy risk premium beas are esimaed over he previous monh wih daily daa. Alhough we briefly describe he ime-varying behavior of volailiy risk premia for our 0 svrp bea-sored porfolios and heir beas wih respec o alernaive aggregae sources of risk, he main objecive of he paper is o analyze he deerminans of he cross-secional variaion of average volailiy risk premia across our sample of 0 porfolios. We find ha he beas of he svrp bea-sored porfolios esimaed wih respec o he marke svrp, obained from he S&P 00 Index opions, range from o 3.89, where he porfolio wih he mos negaive bea has he highes average svrp and he porfolio wih he mos posiive bea presens he mos negaive average svrp. Therefore, we find boh negaive and posiive average svrp values ranging from

5 o on an annual basis, while he average marke svrp is negaive, as in previous lieraure. Regarding he cross-secional variaion of he volailiy risk premia, we find ha, independenly of he preferences imposed, consumpion risk does no seem o explain he cross-secional behavior of svrp. Facor asse pricing models seem o be more useful in explaining svrp a he cross secion. The key facors explaining average svrp across our 0 porfolios are he marke volailiy risk premium and, especially, he defaul premium. The risk premia associaed wih he defaul premium beas are posiive and saisically significan even if we explicily recognize he poenial misspecificaion of he models. Moreover, we canno rejec he overall specificaion of he wo-facor model and he cross-secional R is equal o 0.54, wih an asympoic sandard error of 0.. Finally, our findings are relaed o credi risk and financial marke sress condiions. More precisely, he cross-secional variaions of risk premia reflecs he differen uses of volailiy swaps o hedge defaul and he financial sress risks of he underlying componens of our sample porfolios. This paper is organized as follows. Secion briefly describes variance swaps and volailiy swap conracs and presens he alernaive asse pricing models ha we employ in he sudy of he cross-secional variaion of average svrp. Secion 3 conains a descripion of he daa. Secion 4 discusses he model-free implied volailiy and he esimaion of svrp a he porfolio level. Secion 5 presens he basic characerisics of he 0 svrp bea-sored porfolios and some empirical resuls using uncondiional svrp bea esimaes. Secion 6 repors he main empirical findings of he paper and discusses he economeric sraegy. Secion 7 relaes our evidence o financial sress condiions. Secion 8 concludes he paper. 5

6 . Theoreical Framework In a variance swap, he buyer of his forward conrac receives a expiraion a payoff equals o he difference beween he annualized variance of sock reurns and he fixed swap rae. The swap rae is chosen such ha he conrac has zero presen value, which implies ha he variance swap rae represens he risk neural expeced value of he realized reurn variance: Q where ( ) measure Q, E Q a a ( RV, ) SW, + τ = + τ () E is he ime condiional expecaion operaor under some risk neural a RV, + τ is he realized variance of asse (or porfolio) a beween and + τ, and a SW, + τ is he delivery price for he variance or he variance swap rae on he underlying asse a. The variance risk premium of asse a is defined as VRP Q ( RV ) E ( RV ) a P a a, + τ E, + τ, + τ = () On he oher hand, a expiraion, a volailiy swap pays he holder he difference beween he annualized volailiy and he volailiy swap rae, where a srv, N vol ( srv ssw ) a a, τ, + τ + (3) + τ is he realized volailiy of asse a beween and + τ, a ssw, + τ is he fixed volailiy swap rae, and N vol denoes he volailiy noional. This paper analyzes he deerminans of he cross-secional variaion of volailiy risk premia. We herefore define he volailiy risk premium of asse a as follows, svrp Q ( srv ) E ( srv ) a P a a, + τ E, + τ, + τ = (4) Using he fundamenal asse pricing equaion, we know ha he risk premium of any asse a wih rae of reurn a R is given by 6

7 ( M,R ) P a a Cov, + τ, + τ RP, + τ = (5) P E ( M, + τ ) where M, + τ is he sochasic discoun facor (SDF). Therefore, given he definiion of he volailiy risk premium, he following expression holds: P a a P a Cov ( ) ( ) (, + τ + τ ) = M,sRV srv E srv, Q E + τ + τ + (6) P E ( M, + τ ) Thus, using he payoff of a volailiy swap, he fundamenal pricing framework implies ha E P [ M ( srv ssw )] = E [ M ( svrp )] 0 P a a a, + τ, + τ, + τ, + τ, + τ = (7) In his paper, he SDF, M, + τ, is allowed o be based on eiher power, recursive, and habi preferences or on alernaive linear SDF specificaions based on sae variables poenially capable of explaining he cross-secional variaion of volailiy swaps. In paricular, we es he following models: a) Model, C, power uiliy wih aggregae consumpion: γ U ( C + τ ) C + τ + τ ρ ρ ( ) = = M, (8a) U C C where C is he aggregae consumpion of non-durable goods and services, γ > 0 represens he degree of risk aversion, and ρ is he subjecive discoun facor. b) Model, C, power uiliy wih sockholder consumpion, denoedc SHC : γ SHC + τ + τ ρ C M, = (8b) SHC C c) Model, C, recursive uiliy wih aggregae consumpion: 7

8 8 ( ) ( ) ( ) κ γ κ γ τ κ ρ ρ + + = U E C U (9) where he non-observable coninuaion value is approximaed, as for Epsein and Zin (99), by he reurn on he marke porfolio or marke wealh so ha he corresponding SDF becomes κ κ γ τ η κ τ τ ρ = m, R C C M (0a) where κ γ η = and κ is he inverse of he elasiciy of ineremporal subsiuion. d) Model, C, recursive uiliy wih sockholder consumpion: κ κ γ τ η κ τ τ ρ = m SHC SHC, R C C M (0b) e) Model 3, C, exernal habi preferences, as for Campbell and Cochrane (999): ( ) γ γ = X C U () where X is he level of habi and he SDF is given by γ τ τ τ ρ =, C C S S M () where γ is a parameer of uiliy curvaure, C X C S = is he surplus consumpion raio, and he couner-cyclical ime-varying risk aversion is given by S γ. The aggregae consumpion follows a random walk and he surplus consumpion process is ( ) ( )( ) g c c s s s s + + = + + λ φ φ (3)

9 where g is he mean rae of consumpion growh, φ is he persisence of he habi shock, and he response or sensiiviy coefficien λ ( s ) is given by where ( s ) = ( σ γ φ ) ( s s ) λ (4) c σ c is he volailiy of he consumpion growh rae and lower capial leers denoe variables in logarihms. f) Model 3, C, exernal habi wih sockholder consumpion: γ SHC SHC + + τ + τ τ ρ S C M, = (5) SHC SHC S C g) Model 4, C, recursive preferences wih he marke volailiy risk premium as he coninuaion value: θ γ κ κ C κ + τ + τ ρ M, = (6) C m svrp + τ where m svrp τ + is he marke volailiy risk premium. h) Model 4, C, recursive preferences wih he marke volailiy risk premium, and sockholder consumpion: θ γ κ κ SHC κ + τ + τ ρ C M, = (7) SHC m + τ C svrp i) Model 5: linear SDF for boh he marke reurn and he squared of aggregae wealh: M, + τ = a + brm+ τ + crm+ τ (8) As previously discussed, recen empirical work has consisenly shown ha risk neural volailiy is higher, on average, han physical reurn volailiy. Lile work has been done on heoreically characerizing he disance beween boh ypes of volailiy, wih Bakshi and Madan (006) and Chabi-Yo (0) being wo excepions. In boh cases, he marke 9

10 variance risk premium is derived as a funcion of he sandard deviaion, skewness, and kurosis of equiy reurns. Therefore, he magniude and behaviour over ime of he marke variance risk premium may also be empirically relaed o higher -order momens of he equiy reurn disribuion. This suggess ha a poenially relevan model o explain he cross-secional variaion of volailiy risk premia should explicily recognize higher-order momens of he underlying marke porfolio reurn. In paricular, Bakshi and Madan (006) show ha, when he SDF is a linear funcion on boh he marke reurn and he squared of marke reurn, as in expression (8), hen he variance risk premium is a funcion of boh he skewness and kurosis of he marke and m M R m < 0 and M R > 0. j) Model 6: CAPM wih he marke volailiy risk premium: M m, + τ = a + bsvrp + τ (9) This may be jusified by noing ha Bali and Zhou (0) show ha he cross- secion of equiy reurns porfolios is explained by he marke, and also by economic uncerainy proxied by he marke variance risk premium. k) Model 7: muli-facor SDF wih he marke volailiy risk premium and he defaul premium as he difference beween he Moody s yield on Baa corporae bonds and he 0-year governmen bond yield, denoed DEF + τ : m M, + τ = a + bsvrp + τ + cdef + τ (0) The economic raionale of his model comes from he findings of Zhou (00) and Wang, Zhou, and Zhou (03), who show ha he firm-level variance risk premium has significan explanaory power for credi defaul swap spreads over and above he marke variance risk premium and he VIX. The predicive abiliy increases as he credi qualiy of he credi defaul swap underlying companies deerioraes. 0

11 All hese SDF specificaions will be esed using a generalized mehod of momens (GMM) framework wih he same weighing marix across all es porfolios o compare he performance of he models by he Hansen Jagannahan (997, henceforh HJ) disance. Addiionally, we employ he wo-pass cross-secional regression approach of Fama and MacBeh (973). In his case, we use he linear versions of all previous discussed models and also include he simple CAPM wih he marke porfolio reurn and exended models using he marke porfolio reurn, he marke volailiy risk premium, he Fama French HML facor, and he defaul premium as addiional pricing facors. 3. Daa We employ daily daa from OpionMerics for he S&P 00 Index opions and for individual opions on all socks included in he S&P 00 Index a some poin during he sample period from January 996 o February 0. This yields a oal of 8 socks used in our esimaions. From he OpionMerics daabase, we obain all pu and call opions on he individual socks and on he index wih ime o mauriy beween six days and 90 days. Given ha he opions are American syle, i is convenien o work wih shor-erm mauriy opions, for which he early exercise premium ends o be negligible. 4 We selec he bes bid and ask closing quoes o calculae he mid-quoes as he average of bid and ask prices, raher han acual ransacion prices, o avoid he well known bid ask bounce problem described by Bakshi, Cao, and Chen (997). In selecing our final opion sample, we apply he usual filers. We discard opions wih zero open ineres, zero bid prices, missing dela or implied volailiy, and negaive implied volailiy. We also ignore opions wih exreme moneyness, ha is, pus wih a 4 See he evidence repored by Driessen, Maenhou, and Vilkov (009) who employ a similar daabase.

12 Black Scholes dela above and calls wih a dela below Finally, regarding he exercise level, we employ ou-of-he-money opions using pus wih a dela above -0.5 and calls wih a dela below 0.5. I seems reasonable o expec ha aggregae macroeconomic variables and markewide porfolios exensively used by researchers when explaining he ime series and cross-secional behavior of excess equiy reurns should also be relevan facors in explaining variance risk premia across asses. This is he main crierion we follow when collecing our daa. As our opion daa, he marke reurn for he S&P 00 Index and individual sock reurns and dividends are also obained from OpionMerics, while porfolio reurn daa are from Kenneh French s websie. In paricular, we collec monhly daa on he value-weighed sock marke porfolio reurn, he risk-free rae, he SMB and HML Fama French risk facors, and he momenum facor denoed MOM. Addiionally, yields for 0-year governmen bonds, -monh T-bills, and Moody s Baa corporae bonds are obained from he Federal Reserve Saisical Release. The defaul premium, denoed DEF, is he difference beween Moody s yield on Baa corporae bonds and he 0-year governmen bond yield. We obain nominal consumpion expendiures on nondurable goods and services from Table.8.5 of he Naional Income and Produc Accouns (NIPA), available a he Bureau of Economic Analysis. Populaion daa are from NIPA s Table.6 and he price deflaor is compued using prices from NIPA s Table.8.4, wih he year 000 as is basis. All his informaion is used o consruc monhly raes of growh of seasonally adjused real per capia consumpion expendiures on nondurable goods and services from January 959 o Sepember 0. We also use aggregae per capia sockholder consumpion growh raes. Exploiing micro-level household consumpion daa, Malloy, Moskowiz, and Vissing-Jorgensen (0) show ha long-run sockholder consumpion

13 risk explains he cross-secional variaion in average sock reurns beer han he aggregae consumpion risk obained from nondurable goods and services. In addiion, hey repor plausible risk aversion esimaes. They employ daa from he Consumer Expendiure Survey (CEX) for he period March 98 o November 004 o exrac consumpion growh raes for sockholders, he wealhies hird of sockholders, and non-sockholders. To exend heir available ime period for hese series, he auhors consruc facor-mimicking porfolios by projecing he sockholder consumpion growh rae series from March 98 o November 004 ono a se of insrumens and use he esimaed coefficiens o obain a longer ime series of insrumened sockholder consumpion growh. In his paper, we employ he repored esimaed coefficiens of Malloy, Moskowiz, and Vissing-Jorgensen (0) o obain a facor-mimicking porfolio wih he same se of insrumens for sockholder consumpion from January 960 o Sepember Model-Free Implied Volailiy and Esimaion of he Volailiy Risk Premia Brien-Jones and Neuberger (00) firs derived he model-free implied volailiy under diffusion assumpions. They obain he risk neural expeced inegraed variance over he life of he opion conrac when prices are coninuous and volailiy is sochasic. Jiang and Tian (005) exend heir paper o show ha heir mehod is also valid in a jump- diffusion framework and, herefore, heir mehodology is considered o be a model-free procedure. We calculae he model-free implied variance denoed a MFIV, + τ by he following inegral over a coninuum of srikes: MFIV a, ( K,0 ) dk a ( K ) B(, + τ ) max S B(, + τ ) C, + τ τ = () K + 0 a 3

14 where ( K ) C a, + τ is he spo price a ime of a τ-mauriy call opion on eiher an asse or index a wih srike K, (, +τ ) $ a ime + τ, and B is he ime price of a zero-coupon bond ha pays a S is he spo price of asse a a ime minus he presen value of all expeced fuure dividends o be paid before he opion mauriy. Expression () can be accuraely approximaed by he following sum over a finie number of srikes: where [ g ( K ) g ( K )] K m a a a MFIV, + τ, + τ j +, + τ j () j = ( K K ) max min K =, K j = Kmin + j K for j = 0,, K, m m and a g, + τ ( K ) j a C, + τ = ( K,0 ) a ( K ) B(, + τ ) max S B(, + τ ) j j K j For each ime -o- mauriy from six days o 60 days, we calculae he model-free implied variance each day for each underlying asse ha has a leas hree available opions ousanding, using all he available opions a ime. 5 For he risk-free rae, we use he T-bill rae of appropriae mauriy (inerpolaed when necessary) from OpionMerics, namely, he zero-coupon curve. For he dividend rae for he index we employ he daily daa on he index dividend yield from OpionMerics. To infer he coninuously compounded dividend rae for each individual asse, we combine he forward price wih he spo rae used for he forward price calculaions. We obain he mean coninuously compounded dividend rae by averaging he implied OpionMerics 5 The window from six days o 60 days corresponds o he maximum range of ime o mauriy we allow in he necessary inerpolaion o have enough opions every day in he sample wih 30 days o mauriy. See he discussion below. 4

15 dividends. Finally, we annualize he model-free implied variance using 5 rading days in a calendar day. The specific implemenaion follows he approach of Jiang and Tian (005). I is well known ha opions are raded only over a limied number of srikes. In principle, expression () requires he prices of opions wih srikes K j for j = 0,, K, m. However, he corresponding opion prices are no observable because hese opions are no lised. We apply he curve-fiing mehod o Black Scholes implied volailiies insead of opion prices. The prices of lised calls (and pus wih differen srikes) are firs ransformed ino implied volailiies using he Black Scholes model and a smooh funcion is fied o he implied volailiies using cubic splines. 6 Then, we exrac implied volailiies a srikes K j from he fied funcion. Finally, we employ equaion () o calculae he model-free implied variance using he exraced opion prices. I is someimes he case ha he range of available srikes is no sufficienly large. For opion prices ouside he range beween he maximum and minimum available srikes, we also follow Jiang and Tian (005) and use he endpoin implied volailiy o exrapolae heir opion prices. This implies ha he volailiy funcion is assumed o be consan beyond he maximum and minimum srikes. 7 Finally, discreizaion errors are unlikely o have any effec on he model-free implied variance if a sufficienly large m, beyond 0, is chosen. In our case, we employ an m ha equals 00. A each ime, we focus on a 30-day horizon mauriy, inerpolaed when necessary using he neares mauriies τ and τ following he procedure of Carr and Wu (009). The inerpolaion is linear in oal variance: 6 As poined ou by Jiang and Tian (005), he curve-fiing procedure does no assume ha he Black Scholes model holds. I is a ool o provide a one-o-one mapping beween prices and implied volailiies. 7 Jiang and Tian (005) discuss his approximaion error and he (differen) runcaion error ha arise when we ignore he ails of he disribuion across srikes. In our case and o avoid he runcaion error, we use 3.5 sandard deviaions from he spo underlying price as runcaion poins. 5

16 a ( τ τ ) + MFIV τ ( ), + τ τ τ ( τ τ ) a a MFIV τ, + τ MFIV =, + τ (3) τ We use he square roo of he model-free implied variance o approximae he modelfree annualized implied volailiy as: smfiv a a, + τ = MFIV, + τ (4) For each day in he sample period, we also calculae he realized variance over he same period as ha for which implied variance is obained for ha day, ha is, for 30 days, requiring ha no more han 4 reurns be missing from he sample: τ a, + τ = R + s τ s= RV (5) where R denoes he rae of reurn adjused by dividends and splis. As before, we annualized he realized variance and ake he square roo o obain he realized volailiy: srv a, a τ RV, + τ + = (6) Finally, for each asse and he index, we calculae he volailiy risk premium, svrp, a he 30-day horizon as he difference beween he corresponding realized and model-free implied volailiy: svrp a a a, + τ = srv, + τ smfiv, + τ (7) We nex consruc 0 svrp bea-sored porfolios using he following procedure. We esimae rolling svrp beas for each monh using daily daa over he previous monh on he individual svrp and he marke svrp. Each monh, we rank all svrp beas and consruc 0 equally weighed svrp bea-sored porfolios. Porfolio conains he mos negaive svrp beas, while Porfolio 0 includes he mos posiive svrp beas. The componens of all porfolios are updaed every monh during he sample period. All 6

17 porfolios have approximaely he same number of securiies, wih an average of 5.3 securiies per porfolio, and he asse mus have a leas 5 daily observaions o be included in he porfolios. Figure displays he behavior of porfolios, 0, and 0 sored by svrp bea, as well as he marke svrp. Noe ha we display he svrp of he marke using opions wrien on he S&P 00 Index, so ha he series conained in Figure is no he crosssecional average of he individual svrp. For he porfolios P0B and P0B and he marke, he posiive peaks coincide wih periods of high realized volailiy. Porfolio PB ends o have a posiive svrp even during normal economic imes, while porfolio P0B presens a negaive svrp during normal and expansion monhs and a posiive svrp during bad economic imes. As expeced, given ha he svrp bea of porfolio P0B is as high as 3.89, is behavior closely follows he marke svrp, bu wih more exreme peaks. In any case, his figure suggess ha he ranking procedure generaes sufficienly differen cross-secional behaviour o jusify he analysis of he crosssecional empirical resuls under his soring characerisic Volailiy Risk Premium Characerisics a he Porfolio Level Table repors he basic characerisics of our 0 svrp bea-sored porfolios. The average svrp values are 0.3% and -3.4% for porfolios PB and P0B, respecively. All of hese figures are given in annualized erms. As expeced, given he well-known evidence provided, among ohers, by Carr and Wu (009), he marke svrp is, on average, negaive and equal o -.4%. The average annualized svrp obained direcly from daily daa presen a very similar paern, wih he range going from 0.% o 8 We also consruc an alernaive se of 0 porfolios based on he svrp level. Using he svrp on he las day of he previous monh, we rank all svrp values from he lowes (more negaive) o he highes. Porfolio conains he asses wih he lowes svrp, while porfolio 0 includes securiies wih he highes svrp. Our main empirical resuls and conclusions will be checked employing his alernaive ranking o analyze he robusness of our resuls. 7

18 -4.5%. The magniude of he svrp cross-secional differences is large and seems o jusify he sudy of heir deerminans. These averages indicae ha invesors may have very differen volailiy invesmen vehicles depending on wheher hey go long or shor on volailiy. We end o idenify he purchase of volailiy as a hedging insrumen agains poenially large sock marke declines. The evidence repored in Table suggess ha, on average, going long on volailiy can also lead o subsanial gains, depending on he porfolio for which invesors buy volailiy. 9 The sandard deviaions of he svrp values of hese porfolios sugges ha porfolios wih a higher average svrp and, especially, hose wih a more negaive average svrp are he mos volaile porfolios in erms of svrp payoffs. As poined ou before, Figure also reflecs he highly volaile behavior of he svrp of P0B, followed by he relaively smooher behavior of PB. The fifh column of Table conains he svrp beas of each of he porfolios relaive o he svrp of he marke index. Using monhly daa, we esimae a marke model ype of ordinary leas squares (OLS) regression of he following form: p m svrp, + τ = a + β svrp, + τ + ε, + τ, (8) where p svrp, + τ is he volailiy risk premium of each of he 0 porfolios, and m svrp, + τ is he volailiy risk premium of he marke index from January 996 o February 0. The svrp beas reflec he consrucion crierion, wih uncondiional svrp beas of for PB and 3.89 for P0B. As in he case of average volailiy risk premia, he cross-secional differences in svrp beas are large. 9 As discussed by Carr and Lee (007, 009), due o he concaviy s price impac associaed wih Jensen s inequaliy, he difference beween he value of a variance swap and he value of a volailiy swap depends on he volailiy of volailiy of he underlying asse. If we recognize his poenial bias and adjus our esimaed volailiy risk premia accordingly, he dispersion beween he volailiy risk premia across porfolios remains. See Burashi, Trojani, and Vedolin (04) for a similar approximaion. 8

19 Given ha, for each monh during he sample period, we can idenify he underlying componens of he 0 porfolios, we calculae he porfolio reurns of he 0 svrp bea-sored porfolios. In Table, we also display he marke beas of he 0 porfolios wih respec o he US marke porfolio index and he S&P 00 Index. As wih he sandard deviaion, he cross-secional behavior of marke beas presens a U- shaped paern, wih marke beas being especially high for porfolios wih a more negaive average svrp. Porfolio P0B has he highes reurn bea, wih a value as high as.5 when measured relaive o he S&P 00 Index reurn. Finally, he las column of Table conains he average relaive bid ask spread of he opions associaed wih he componens of he 0 porfolios. The opions raded on he componens of porfolios wih posiive and high average svrp values may be exremely illiquid. If his is he case, he replicaing sraegy employed o obain synheic variance swaps associaed wih illiquid opions may be more cosly han in oher cases. However, he average bid ask spreads reflecs precisely he opposie. The porfolio PB conains, on average, he mos liquid opions, while P0B presens he highes relaive bid ask spread across he 0 porfolios. Therefore, on average, marke reurn beas and bid ask spreads are higher for he wo porfolios wih he highes svrp beas. Table conains he correlaion coefficiens beween represenaive porfolios sored by svrp beas and he marke svrp. Panel A employs monhly daa, while Panel B displays he resuls wih daily daa. As expeced, given is highly negaive svrp bea, he correlaion beween porfolio PB and he res of he porfolios becomes increasingly negaive. No surprisingly, he correlaion of hese porfolios wih he marke svrp displays an increasingly monoonic relaion going from a negaive 9

20 correlaion of for PB o a posiive correlaion of for P0B. A similar paern is found when using daily daa. Table 3 repors he correlaion beween he marke svrp and several macroeconomic and financial indicaors. The correlaion beween he excess marke reurn and he marke svrp is negaive and equals This is well known and implies a negaive correlaion beween marke reurns and realized marke volailiies. Thus, going long on he marke volailiy swap provides a hedging invesmen vehicle for momens of exremely high marke volailiy. However, he compensaion for his hedging sraegy is, on average, negaive. The resuls also show a negaive correlaion of he marke svrp wih consumpion growh, alhough he correlaion is more negaive for aggregae consumpion han for sockholder consumpion. The correlaion wih he HML and momenum facors is posiive, while he correlaion wih he defaul premium is also posiive and equals As expeced, he correlaion beween he defaul premium and eiher he excess marke reurn or consumpion growh is negaive, being especially negaive wih respec o aggregae consumpion growh. Panels A and B of Table 4 conain he full-sample svrp beas for five represenaive svrp bea-sored porfolios conrolling for well-known aggregae risk facors. The robusness of he magniudes of he svrp beas, repored again in he firs column of Table 4, is clear across all porfolios. Independenly of he facors employed in he regressions, porfolio PB has a negaive bea, while P0B has a very high bu posiive volailiy risk premium bea. In all cases, we employ heeroskedasiciyauocorrelaion (HAC) robus sandard errors. The relaion beween he svrp beas and he average volailiy risk premia of all porfolios is mainained across all aggregae facors. We may conclude ha, for svrp bea-sored porfolios, he volailiy risk premia are especially explained by he marke svrp, he excess marke reurn, he 0

21 defaul premium, and consumpion growh. However, svrp beas do no seem o be significanly differen from zero when sockholder consumpion growh is used. Overall, we conclude ha he uncondiional beas of hese sae variables are, in mos cases, saisically differen from zero, even when we employ all hree explanaory variables simulaneously. 6. Cross-Secional Variaion of Porfolio Volailiy Risk Premia 6. GMM Esimaion and Tess We nex es he compeing specificaions given by models hrough 7 described in Secion using he GMM esimaion procedure and our se of 0 porfolios as es asses. Given he heoreical framework of Secion, we work wih he volailiy risk premia of he 0 svrp bea-sored porfolios. We define an (N+) x vecor conaining he pricing errors generaed by he model a ime. The firs N condiions are he pricing errors of he model when explaining he volailiy risk premia of N porfolios. The las condiion forces he SDF o go o is mean value µ. More precisely and using he fundamenal pricing equaion given by (7), he following vecor defines he momen resricions: ( M µ ) ( ) α + svrp svrp θ α µ = f N µ,, E (9) M µ where s VRP is he N x vecor of volailiy risk premia of he N porfolios a ime, denoes an N x vecor of ones, ( θ ) N M is one ou of he seven specificaions of equaions (8) o (0), and θ is he vecor of he preference parameers for each

22 paricular specificaion. 0 The inclusion of he parameer α enables he separae evaluaion of he model s abiliy o explain he emporal pricing behavior of he compeing specificaions and he cross secion of volailiy risk premia. So, if α is zero, we can conclude ha he model presens a zero average pricing error over he sample period. We define a vecor conaining he sample averages corresponding o he elemens of f as gt T f ( θ, α, µ ) θ = = T (30) (, α, µ ) and he GMM minimizes he quadraic form, gt ( θ, α, µ ) W g ( θ, α, µ ) T T (3) where W T is a weighing (N+) x (N+) marix. For he GMM esimaion and o compare he performance of he models, we employ he pre-specified weighing marix ha conains he marix proposed by HJ. I weighs he momen condiions for he N esing porfolios using he (inverse) marix of second momens of he volailiy risk premia of our se of 0 porfolios. Moreover, as for Parker and Julliard (005), he weigh of he las momen condiion is chosen large enough o ensure ha significan changes in ha weigh have no effecs on he parameer esimaes. A weigh of 000 for he las momen condiion ensures he sabiliy of he esimaor for he mean of he SDF wih respec o differen iniial condiions. Hence, he pre-specified weighing marix is where HJ 0N WT = (3) 0N See Parker and Julliard (005), and Yogo (006) for examples of GMM esimaion using he same esimaion sraegy. In he empirical esimaion, we ake he subjecive discoun rae as a fixed parameer ha is equal o he inverse of he risk-free rae over he sample period.

23 HJ T ( T ) svrp svrp = = (33) and 0 N is an N-dimensional vecor of zeros. Given he unknown disribuion of he performance es, we follow Jagannahan and Wang (996), HJ and Parker and Julliard (005) o infer he p-value of he es. The evaluaion of he model performance is carried ou by esing he following null hypohesis: where he HJ disance is defined as [ ( θ, α, µ )] 0 H0 : T δ = (34) δ = g, (35) T ( θ, α, µ ) W g ( θ, α µ ) I is well known ha a limiaion of he HJ disance in comparing asse pricing models is ha i does no allow for saisical comparison among compeing models. Chen and Ludvigson (009) propose a procedure ha can be used o compare any number of muliple compeing models, some of hem possibly non-linear. The benchmark model is he model wih smalles squared HJ disances among compeing models. The auhors are able o compue he disribuion of he differences beween squared HJ disances via a block boosrap, where he reference disance corresponds o ha wih he smalles HJ disance among all models. Kan and Roboi (009, KR hereafer) also develop a mehodology o es wheher wo compeing models have he same HJ disance and hey show ha he asympoic disribuion of he es saisic depends on wheher he models are correcly specified or no. In his paper, we apply he KR es of he comparison of he HJ disances of wo alernaive specificaions under poenially misspecified models. T T We employ a version of heir es for which he SDF does no have o be necessarily linear. 3

24 We briefly described heir comparison es which amouns o obaining he asympoic disribuion of ˆ δ ˆ δ. Le d = s s, where s s φ ( φ svrp ) φ N δ φ = Wg = svrp M, where φ ( φ svrp ) φ N δ = svrp M, where = Wg Under he null hypohesis of ˆ ˆ δ = δ 0, τ = where v = E( d d ) d A [ ˆ ˆ δ ( δ δ )] N( 0, ) T δ vd (36) τ. In he empirical applicaion, his expression can be approximaed using he well-known Newey Wes (987) esimaor given by, φ vˆ d k k τ T = = k k T τ = ( d d τ ) 6. GMM Empirical Resuls The empirical resuls using he GMM framework described above and he 0 svrp bea-sored porfolios are repored in Table 5. Panel A conains he resuls of he SDF specificaions given by models o 4 under boh he aggregae consumpion growh of non-durable goods and services (NDC) and sockholder growh consumpion growh (SHC). The las column of Table 5 displays he HJ disance given by expression (35) wih he corresponding p-value in parenheses. All alernaive specificaions are rejeced. A he same, he esimaors of he preference parameers across models end o be esimaed wih a lo of noise. For all preference esimaors, sandard errors are repored in parenheses. In all cases, we check he shape of he objecive funcion when we minimize he weighed average pricing errors according o expression (3) for he parameers esimaed under he power, recursive, and 4

25 Regarding recursive preferences and power uiliy and for sockholder consumpion, wih he excepion of recursive preferences wih he marke reurn as he proxy for coninuaion value, he magniudes and he sign of he risk aversion coefficiens are sysemaically reasonable. For recursive preferences wih he marke svrp as he coninuaion value, he risk aversion coefficien is equal o 0.4. Unforunaely, in his case, he sign of he elasiciy of ineremporal subsiuion is negaive. A sysemaic difference when using one approximaion of he coninuaion value or anoher relies on he sign of he elasiciy of ineremporal subsiuion. When we employ eiher aggregae consumpion growh or sockholder consumpion growh and marke wealh, he signs of he elasiciy of ineremporal subsiuion are posiive and less han one. However, when we use marke volailiy swaps, he elasiciy of ineremporal subsiuion becomes negaive for boh ypes of consumpion growh. We also repor he resuls using he habi preferences for boh ypes of consumpion. I is imporan o noice ha he empirical implemenaion of he model described by equaions () o (5) simulaneously esimaes all preference parameers and he surplus consumpion process. To provide some inuiion abou he behavior of he resuling ime-varying risk aversion given by ˆγ S, where he curvaure parameer esimaor is repored in Table 5 and he surplus consumpion is obained using equaions (3) and (4), Figure displays he marke volailiy risk premium and he wo-monh lagged changes of risk aversion. 3 We observe how he behavior of risk aversion changes follows he previously available payoffs of volailiy swaps. Indeed, he correlaion coefficien beween boh series is as large as In any case, under he habi preference models, risk aversion esimaes are.46 and. for aggregae habi preference specificaions. The minimum value of he funcions corresponds o he parameer esimaors repored in Panel A of Table 5. The resuls srongly sugges ha he numbers repored are robus o a large number of alernaive iniial condiions. 3 This figure is consruced using sockholder consumpion growh. 5

26 consumpion growh and sockholder consumpion growh, respecively, bu he esimaed coefficiens are no saisically differen from zero. In addiion, he average pricing errors are saisically differen from zero and he pricing specificaion is rejeced wih a p-value of he HJ disance of Panel B of Table 5 conains he resuls of he linear SDF specificaions given by models 5 o 7. As in all previously analyzed models, he linear specificaions are rejeced. The parameers across he specificaions using eiher he marke svrp as a facor or he SDF wih skewness and kurosis are esimaed wih low precision. The average pricing errors are all negaive and saisically differen from zero. Ineresingly, he slope parameers of he wo-facor model wih he marke svrp and defaul are negaive and saisically differen from zero, which suggess ha he risk premia associaed wih boh risk facors are posiive and saisically significan. We nex empirically invesigae wheher compeing models exhibi significanly differen sample HJ disances. If our alernaive specificaions fail o find differences in significance across models, i would imply ha he proposed facors are oo noisy o explain he cross-secional differences and o conclude ha one model is superior o he ohers. We herefore employ he es saisic given by equaion (36) based on he differences beween he square of he HJ disances for wo given models. Table 6 repors he empirical resuls. The numbers in his able represen pairwise ess of equaliy of he squared HJ disances for all alernaive specificaions of SDF linear and non-linear models. We repor he differences beween he sample squared HJ disances of he models in row i and column j, or ˆ δ ˆ δ. For example, given ha he HJ disance of he power aggregae consumpion model from Panel A of Table 5 is and he HJ disance for he same model wih sockholder consumpion is , he firs number in he firs row of Table 6, which is equal o 0.006, is obained as i j 6

27 As discussed above, he asympoic disribuion of his es saisic allows for misspecificaion of he models. The associaed p-values are provided in parenheses. The resuls sugges ha, generally, here is no saisical significance beween he compeing models when we employ he HJ disance. The only imporan excepion is he model ha combines he marke volailiy risk premium and he defaul premium as facors. 4 The linear SDF on he marke volailiy risk premium and defaul premium is saisically superior o all he oher models, wih he excepion of he recursive preference specificaion using aggregae consumpion growh and wih eiher marke wealh or he marke svrp as coninuaion values. 6.3 Two-Pass Cross-Secional Esimaion and Tess A es of he compeing asse pricing models of he deerminans of he cross secion of volailiy risk premia using he models bea specificaions may help clarify maers. In paricular, we now es he models described below using our 0 svrp bea-sored porfolios. In all cases, λ 0 is he zero-bea rae and λ k for k =,., K are he risk premia associaed wih he K aggregae risk facors ha drive he cross-secional variaion among volailiy swap payoffs for our se of 0 porfolios, p =,,0, as follows, a) Model : power uiliy wih boh aggregae consumpion and sockholder consumpion: p ( ) = λ λ β E svrp, p + τ 0 + ndc c (37a) p ( ) = λ λ β E svrp, p + τ 0 + shc sc (37b) 4 A model ha recognizes he skewness and kurosis of he underlying marke reurn is also saisically superior o he one-facor model wih he marke svrp. 7

28 b) Model : recursive uiliy wih boh aggregae consumpion and sockholder consumpion and marke wealh and he marke volailiy risk premium: p p ( ) = λ0 + λndcβc λ m β m E svrp, p + τ + (38a) p p ( ) = λ0 + λshcβ sc λ m β m E svrp, p + τ + (38b) p m p ( ) = λ0 + λndcβc λ svrp β msvrp p E svrp, + τ + (38c) p m p ( ) = λ0 + λshcβ sc λ svrp β msvrp p E svrp, + τ + (38d) c) Model 3: habi preferences wih ime-varying risk aversion: Using he expression of risk aversion under he habi preference model, we can wrie he consumpion surplus as S = γ RA, where RA is he ime-varying risk aversion. Then, by aking logarihms in expression (), he SDF can wrien as ln ln ρ γ c γ ( τ ) τ + τ + = + ra M +, e + ln ρ γ c + τ + γ ra + τ which we wrie as a bea facor model, (39) p p ( ) = λ + λ β λ β E svrp, p + τ 0 ndc c + ra ra (40a) p p ( ) = λ + λ β λ β E svrp, d) Model 4: he CAPM wih marke wealh: p + τ 0 shc sc + ra ra (40b) p ( ) = λ 0 λ m β m p E svrp, + τ + (4) e) Model 5: he Bakshi Madan (006) model wih higher-order momens: p p ( ) = λ + λ β λ β p E svrp, τ 0 m m + skku m + (4) d) Model 6: he CAPM wih he marke volailiy risk premium as he only risk facor: m p ( ) = λ 0 λ svrp β msvrp p E svrp, + τ + (43) e) Model 7: a wo-facor model wih he marke volailiy risk premium and he defaul premium: 8

29 p p ( ) = λ + λ β λ β p m E svrp, + τ 0 svrp msvrp + def def (44) f) Model 8: a hree-facor model wih he marke volailiy risk premium, he defaul premium, and he HML Fama French facor: p p p ( ) = λ + λ β + λ β λ β p m E svrp, + τ 0 svrp msvrp def def + hml hml (45) g) Model 9: a four-facor model wih he marke volailiy risk premium, he defaul premium, he HML Fama French facor, and marke wealh: m p p p p ( ) = λ 0 + λ svrp βmsvrp + λdef β + λhml β λ m β m p E svrp, τ def hml + + (46) Therefore, we now es he linear versions of he models using he alernaive K-facor bea specificaions described above in which he volailiy risk premia are linear in he volailiy risk premium beas, ha is, E ( svrp) = Xλ, where X [,β ] [ λ ] 0,λ λ = is a vecor consising of he zero-bea rae, 0 K facors, λ. The pricing errors of he N porfolios are given by ( svrp) Xλ = and λ, and he risk premia on he e = E (47) As a goodness- of- fi measure of he compeing models, we employ he cross-secional R defined by Kan, Roboi, and Shanken (03, KRS hereafer) as N R Q Q0 = (48) where he Q saisic given by Q = e V e = E ( svrp) V E( svrp) E( svrp) V X ( X V X ) X V E( svrp) represens he aggregae pricing errors and Q0 = e0 V e0 denoes he deviaions of he mean reurns from heir cross-secional average, wih ( V ) V E( svrp) e0 = I N N N N N 9

30 and V is he variance covariance marix of he porfolio volailiy risk premia. As KRS poin ou, he R saisics given by (48) is a relaive measure of he goodness- of- fi since i compares he magniude of he model s expeced reurn deviaions o ha of ypical deviaions from he average expeced reurn. Moreover, 0 R and R is a decreasing funcion of he aggregae pricing errors Q. Thus, R given by (48) is a reasonable and well-defined measure of goodness- of- fi. Noe ha, in fac, we employ R for average reurns raher han he average of monhly R values. In addiion, KRS show how o perform a es of wheher he model has any explanaory power for pricing asses cross-secionally. In oher words, hey es wheher we can rejec he null hypohesis of R = 0. The asympoic es is given by A K ξi TR = xi Q i= 0 (49) where he x i s are independen χ ( ) random variables and he ξi s are he K nonzero eingenvalues of where ( ˆ ) ( V ) V β Var( λ ) β V β β V ˆ N N N N Var λ is he expression adjused by errors-in-he-variable and misspecificaion of he model. 5 In paricular, he asympoic disribuion of λˆ under he misspecified models is Var τ and τ = where ( ˆ λ) = E( v v ) A ( ˆ λ) N ( 0,Var( ˆ λ )) T λ K + (50) 5 The p-values o es he null H 0 : R = 0 are calculaed as before, using he procedure of Jagannahan and Wang (996), HJ and Parker and Julliard (005). 30

31 ( λ) ( η η) v = λ ω + H z 3 u (5) var when rue beas EIV adjusmen misspecificaion erm wih η = [ λ0, ( λ, f ) ] ; η = λ0,( λ f ) [ ] ;u e = V ( svrp E( svrp )) ( ) [ ( ) f f ; z =, f f ], = ω λω 0 Ω H = ( ) X V X where Ω is he variance covariance marix of he facors denoed by f. Finally, we presen he es for comparing wo compeing models. Suppose M M and 0 < R = R <. Then A ( Rˆ Rˆ ) N 0, E( d ) T d τ (5) τ = where ( u u M u u M ) d = Q0 + u e = V ( svrp E( svrp )) and u e = V ( svrp E( svrp )) 6.4 Two-Pass Cross-Secional Empirical Resuls As in Secion 6.3, Panel A of Table 7 conains he resuls of he wo-pass cross-secional regressions using consumpion-based facors, while Panel B of Table 7 displays he resuls concerning facor-based models. In all cases, we adap he esing framework discussed above o he Fama MacBeh (973) wo-pass cross-secional mehodology, where we esimae rolling beas using he firs 60 monhs of he sample as a fixed esimaion period and hen use a rolling window of 59 monhs of pas daa plus he monh in which we perform he cross-secional regression wih he 0 porfolios. Hence, for each monh we always 3

32 employ a bea esimaed wih 60 observaions. Moreover, below all risk premia esimaors, we repor he p-values associaed wih he radiional Fama MacBeh sandard error in parenheses and in brackes, wih he sandard error adjused for errors in variables, and he poenial misspecificaion of he model as capured by expression (5). We also provide wo measures of goodness- of- fi. We repor he mean absolue pricing error (MAE) calculaed as where 0 ê p p= MAE = (53) 0 ê p is he mean pricing error associaed wih each of he 0 porfolios. The las column of Table 7 repors he Rˆ value given by equaion (48), where below we display he p-value for he es of he null hypohesis given by R = 0 from expression (49) and in brackes we repor he sandard error of Rˆ under he assumpion ha 0 R. Regarding consumpion models, he resuls sugges ha he sandard errors of he risk premia esimaors are very sensiive o poenial model misspecificaion. A he same ime, in mos cases, he esimaor of he zero-bea rae is saisically differen from zero independenly of he adjusmen. These resuls already pu ino quesion he validiy of he models. Indeed, all risk premia associaed wih consumpion growh, eiher aggregae consumpion or sockholder consumpion, are no saisically differen from zero. Consumpion risk does no seem o be priced in he cross secion of he volailiy risk premia. The only saisically significan risk premia are he marke porfolio reurn in he case of he recursive preference model wih aggregae consumpion growh and ha relaed o he marke volailiy risk premium in he recursive model when we approximae he coninuaion value wih volailiy swaps raher han wih he marke porfolio reurn. As heory suggess, he sign of he saisically significan risk premium associaed wih marke wealh is posiive and i 3