The Impact of Separate Processes on Asset Pricing

Transcription

1 Th Impac o Spara Procsss o Ass Pricig DECISION SCIENCES INSTITUTE Th impac o spara procsss o aggrga dividds ad cosumpio o ass pricig wih a ails (Full Papr Submissio) Jacky So Uivrsiy o Macau Uivrsiy o Macau, Avida da Uivrsidad, Taipa, Macau, Chia, JackySo@umac.mo Qi Fu Uivrsiy o Macau Uivrsiy o Macau, Avida da Uivrsidad, Taipa, Macau, Chia UmacFuqi@gmail.com ABSTRACT This papr sudis a cosumpio-basd ass pricig modl i which aggrga dividds ad cosumpio ar modld as dir procsss wih sabl shocks. Th modl yilds ma quiy rurs largr o accord wih h hisorical daa ha h sadard ramwork wih h assumpio ha dividds ar idical o oal cosumpio. This improvs h poial o a ails o xplai h quiy prmium puzzl urhr. Wih mor ralisic assumpio o xogous dowm squcs, his w modl also lays a broadr oudaio or ohr ass pricig modls wih a ails. KEYWORDS: Equiy Prmium Puzzl, Cosumpio-basd Ass Pricig Modl, Sabl Paria Disribuio INTRODUCTION Th Capial Ass Pricig Modl (CAPM) aribus h larg dirc bw h avrag rurs o corpora quiy ad T-bills o a prmium or barig o-divrsiiabl aggrga risk. Howvr, Mhra ad Prsco id ha oly a small par o his dirc is a prmium or barig aggrga risk (Mhra & Prsco, 985). Thror, hy propos h quiy prmium puzzl. Isiv maisram xplaaios o h puzzl hav ocusd o mchaisms o icras h prmium or barig o-divrsiiabl risk. Th cosumpio-basd ass pricig modl proposd by Lucas is mployd wh Mhra ad Prsco iroduc h quiy prmium puzzl (Lucas, 978). I is possibl ha h lowr quiy

2 Th Impac o Spara Procsss o Ass Pricig prmium drivd rom h Lucas modl is du o a udrsimaio o h possibiliy o larg shocks ad hc a udrsimaio o rlad risk prmium. Thus, Bidakoa ad McCulloch chag h origial xogous dowm procss implmd by Mhra ad Prsco rom h wo-sa Markov procss o a irs-ordr auorgrssiv procss wih iovaios o h procss draw rom h amily o sabl paria disribuios (Bidakoa & McCulloch, 003). Thy amp o capur h igord big shocks wih such sabl procss ad icras h quiy risk prmium implid by h Lucas modl i ordr o mach h obsrvd mark daa. Th Bidakoa ad McCulloch s sabl modl capurs largr shocks ad yilds highr ma quiy rurs ha boh h Mhra ad Prsco s wo-sa Markov procss modl ad h Bursid s irs-ordr auorgrssiv procss modl wih Gaussia iovaios (Bursid, 998). Bu his sabl modl implis oly modraly grar ma quiy rurs, ragig rom.46 o 4., ar rom h hisorically obsrvd avrag i xcss o 7 prc pr aum (Bidakoa & McCulloch, 003). I sms ha his ucouragig rsul dmosras h limiaio o h abiliy o modls wih a ails o gra ralisic valus o obsrvd ma ras o rur. I accordac wih h iuiio o highr rurs or riskir asss, h Bidakoa ad McCulloch s sabl modl capurs a ail risks succssully bu h rurs ar o high ough. Thus, h issu o rsuls o h modl is jus as h quiy prmium puzzl isl as a quaiaiv issu. O probabl raso is ha Bidakoa ad McCulloch implm h Lucas modl wih idal coomic assumpio ha aggrga dividds qual cosumpio. Though hy propos a irs ordr auorgrssiv procss wih sabl shocks or h log dividd growh ra, h procss is simad wih h U.S. pr capia cosumpio daa ollowig Ccchi (Ccchi al, 000). A auorgrssiv dividd growh procss calibrad wih rlaivly smooh cosumpio daa lads o a procss as a cosa wih sabl shocks, whil losig h volailiy o dividds growh ra, hc gras mor modra rur ha i should b. Though h assumpio ha aggrga dividds qual cosumpio maks ss i h idal coomic modl, i is ar rom coomic ad mpirical raliy. Ecoomically, pricig bhavior o h claim o corpora dividds is o h sam hig as h claim o h whol dowm, ad sock s claims ar o corpora dividds, o cosumpio. Empirically, corpora dividds o avrag mak up abou 4% o cosumpio ach yar (Ccchi al, 993); i addiio, aual growh ras o dividds ad cosumpio ar wakly corrlad i h daa (Campbll, 999; Campbll & Cochra, 999). Th mai brakhrough o his papr is h rlaxaio o h abov assumpio wih which Bidakoa ad McCulloch mploy h Lucas modl ad driv soluios or ass prics ad rurs (Bidakoa & McCulloch, 003; Lucas, 978). David ad Vrosi id h mpirical ac ha h xpcd cosumpio growh ra is almos cosa (David & Vrosi, 000). Ad Bra ad Xia propos a rprsaiv ag modl whr cosumpio ad dividds ar

3 Th Impac o Spara Procsss o Ass Pricig modld as dir procsss, ad h xpcd cosumpio growh ra is a cosa (Bra & Xia, 00a). Rr o his liraur, h aggrga dividds ad cosumpio ar modld sparaly. I dail, h aggrga dividds growh ra is modld as a auorgrssiv procss wih sabl shocks, whil h xpcd cosumpio growh ra is modld as a cosa. Wih mor ralisic assumpio o xogous dowm squcs, h w modl improvs h xplaaio abiliy o a ails modls or quiy rurs ad h quiy prmium, hus improvs is poial o solv h quiy prmium puzzl urhr. Th rmaidr o his papr is orgaizd as ollows. W mak a summary dscripio o h ass pricig modl i h x scio, h Ass Pricig Modl. Th w driv h soluio or ass prics wh h dowm procsss o aggrga dividds ad cosumpio ar modld sparaly i Scio III, Soluio o Th Modl scio. I Scio IV, Evaluaig Ass Rurs, h soluio or ass rurs is drivd. I Scio V, Empirical Rsuls o h Modl, w compu h ass rurs implid by h Lucas modl. Th w compar our soluio o ha obaid by Bidarkoa ad McCulloch (003) udr h assumpio ha dividds qual cosumpio. W coclud i h las scio. THE ASSET PRICING MODEL Bidarkoa ad McCulloch iroduc h irs-ordr Rulr codiio i a sigl good Lucas coomy wih a rprsaiv ag ad a sigl ass ha pays xogous dividds o o-sorabl cosumpio goods (Bidarkoa & McCulloch, 003; Lucas, 978) as PU ( C ) = E U ( C )[ P + D ], () θ whr P is h ral pric o h sigl ass i rms o h cosumpio good, U ( C) is h margial uiliy o cosumpio C or h rprsaiv ag, θ is a subjciv discou acor, assumd o-sochasic ad cosa, D is h dividd rom h sigl produciv ui, ad E is h mahmaical xpcaio, codiiod o iormaio availabl a im. ( ) Assum a cosa rlaiv risk avrsio (CRRA) uiliy ucio: UC ( ) = ( γ ) C γ, γ 0. Wihou h assumpio ha cosumpio simply quals dividds i h modl, husc D vry priod, or spara aggrga dividds ad cosumpio, Equaio () rducs o PC = E θc [ P + D ]. () γ γ Ar rarragig, Equaio () yilds γ P C P D [ ] + = E θ C (3)

4 Th Impac o Spara Procsss o Ass Pricig As Bidarkoa ad McCulloch (003) procss, l v b h pric-dividd raio, ha is v = P/ D. Th Equaio (3) ca b rormd i rms o v as v γ C D [ v ] + + = Eθ + +. C D (4) L x = l( D / D ) b h log dividd growh ra, ad y = l( C / C ) b h log cosumpio growh ra. Th w ca rwri Equaio (4) i rms o x ad y as [ ] v = Eθxp γ y+ + x+ ( v+ + ). (5) I Vrosi (000), cosumpio ad dividds ar modld as h sam procss, ad h xpcd cosumpio growh ra is im varyig. Bidarkoa ad McCulloch (003) do h sam ad assum ha cosumpio simply quals dividds, i.. C= D vry priod. Hc hy rwri Equaio (4) i rms o v ' = Eθxp γ x + ( v + + ). Nohlss, lar i x as ( ) hir mpirical assssm scio, hy sima hir dowm procss wih obsrvd daa o U.S. pr capia cosumpio daa ollowig Ccchi al (Ccchi al, 000). This coicids wih hir assumpio ha cosumpio ad dividds ar idical. I ac, hy rwri Equaio (4) i rms o y as ( ) v ' = Eθxp γ y+ ( v+ + ). (6) SOLUTION TO THE MODEL I his scio, w discuss rasos or spara procsss o aggrga dividds ad cosumpio. Boh h log dividd growh ra x ad h log cosumpio growh ra y ar spciid as dir procsss. Th a aalyical soluio or v is obaid. Discussio o h Edowm Procsss Spciicaio Thr ar wo spcial cass or h xpcd cosumpio growh ra. O is ha h cosumpio growh ra is a cosa, such as h assumpio mad by David ad Vrosi (David & Vrosi, 000), ad Bra ad Xia (Bra & Xia, 00). Th ohr is ha h

5 Th Impac o Spara Procsss o Ass Pricig xpcd cosumpio growh ra is im varyig wih h assumpio ha cosumpio quals aggrga dividds. Ad his idal assumpio is xploid by Mhra ad Prsco (Mhra & Prsco, 985), Vrosi (Vrosi, 000) ad Bidarkoa ad McCulloch (Bidarkoa & McCulloch, 003). Though h idical cosumpio ad dividds assumpio simpliis h implmaio o Lucas modl (Lucas, 978), i is icosis wih mpirical acs. Cosumpio growh is cosidrably smoohr ha dividd growh. David ad Vrosi id h xpcd growh ra o cosumpio is almos cosa (David & Vrosi, 000). Ev Bidarkoa ad McCulloch sima h procss o dividds wih cosumpio daa ad g cosa growh ra as wll (Bidarkoa & McCulloch, 003). I addiio, aual growh ras o dividds ad cosumpio ar wakly corrlad i h daa (Campbll, 999; Campbll & Cochra, 999). Ecoomically spakig, h idical cosumpio ad dividds assumpio is also i coras wih h ralisic siuaio. Pricig bhavior o h claim o corpora dividds is o h sam hig as h claim o h ir dowm, bcaus sock s claims ar o corpora dividds, o cosumpio. Thror, sparaio procsss o cosumpio ad dividds ar mor ralisic, ad h xpcd cosumpio growh ra should b a cosa whil h dividds growh ra should b varyig. Accordig o boh h mpirical ad coomic rasos discussd abov, w assum h log dividd growh ra x ollowig h procss as x = ( ρµ ) + ρx + ε, ρ <, (7) ad h log cosumpio growh ra y is y = η + ε, (8) whr ε ~iid S( α, β, c,0). S( α, β, c,0) rprss a sabl paria disribuio wih characrisic xpo α, skwss paramr β, scal paramr c, ad locaio paramr s o zro. McCulloch dis h disribuio ad liss som o hir propris (McCulloch, 996). Aalyical Soluio o h Pric-dividd Raio L m = θxp[ γ y + x ], Equaio (5) ca b rducd o v = E m [ v + ]. (9) + +

6 Th Impac o Spara Procsss o Ass Pricig By orward iraig o Equaio (9), h soluio or v is giv by i i v = E m+ j + lime m+ jv+ i. i i= j= j= (0) As Bidakoa ad McCulloch (003), w ocus o h paricular soluio by imposig h rasvrsly codiio: i lime m v = 0. Soluios o h ass pricig modl ha implis i + j + i j= irisic bubbl is xcludd ou by h abov codiio (Froo & Obsld, 99). I his cas, Equaio (9) yilds i v = E m+ j. i= j= () EVALUATING ASSET RETURNS I his scio, h rur o risk r ass i our spara dowm coomy is drivd a irs. Th h rur o risky asss is giv by boh h Bidakoa ad McCulloch (003) coomy ad our w dowm coomy. Ad w mak h compariso bw hs wo ramworks. Rurs o Risk Fr Asss I Lucas dowm coomy (Lucas, 978), h pric o a risk r ass o h cosumpio good a mauriy. So w hav P U'( C ) = + θ E. U'( C) P surs o ui Wih h cosa rlaiv risk avrsio (CRRA) uiliy ucio, ad h log cosumpio growh ra y, Equaio () bcoms P [ y ] () = θe xp( γ + ). (3) Udr h procss did as h log cosumpio growh ra Equaio (8), wh y has h skwss paramr β =+, Appdix shows h pric o a risk r ass P is yildd as

7 Th Impac o Spara Procsss o Ass Pricig P xp ( c) α πα = θ γη γ sc. (4) Ad rurs o h risk r ass R udr gross quilibrium ar giv by R =. Thus risk P r rurs ar giv by R xp ( c) α πα = θ γη+ γ sc. (5) Sic h cosumpio growh ra y i Bidakoa ad McCulloch s papr is rgardd as h dividds growh ra x (Bidakoa & McCulloch, 003), hir risk r rurs ar dir rom Equaio (5) as α πα R = θ xp γη+ ( γc) sc + γρ( x η). For h risk r rur is h bchmark o h quiy prmium, hir rsuls udr h uralisic assumpio acs h calculaio o boh h risk r rur ad h quiy prmium. I h cas whr y has h skwss paramr β =, Appdix shows ha P is iii. Probably, hug ucraiy wih isiv shocks drmis h dowm procss, ad iii amou would b paid by ivsors who ar risk avrs o avoid such kid o xrm virom, hus risk r rurs bcom zro cosquly. Rurs o Risky Asss As h diiio i Bidakoa ad McCulloch s papr, h quilibrium gross quiy rurs R o asss hld rom priod hrough priod + ar giv by P+ + D + R =. P Subsiuig h pric-dividd raio v = P / D ad h log dividd growh ra ( ) x = l D / D, his rducs o R + v + = v xp[ x+ ]. (6)

8 Th Impac o Spara Procsss o Ass Pricig Sic v is a ucio o x ad y as Equaio () wih m = θxp[ γ y + x ], h ma o h implid quiy rurs is hard o driv. W ca xpad v ad gai isigh how complx i is. i v = E m+ j, i= j= ( ) ( ) ( ) ( ) v = E m + E m m + E m m m + + E m m m m. (7) Covily, Bidakoa ad McCulloch rgard x as y implid by h assumpio ha E, aggrga dividds qual cosumpio. Thror, h ims i Equaio (7) such as ( ) ( m m ) E m ad so o dduc o cosas. Cosquly h summaio o hs cosas bcoms h cosa v which is idpd o im. This coicids wih h xplaaio i Bidakoa ad McCulloch s papr ha i h cas o a radom walk or dividd growh ras, ρ = 0 ad v rducs o cosa (Bidakoa & McCulloch, 003). I such simplr cas, hy driv a xac aalyical xprssio or E ( R ) + v α πα E( R ) = xp µ c sc. v Ad hy mploy Equaio (8) o calcula rurs o risky asss. I ac, hir soluios cosis o wo major problms. Firs, hy rplac h dividd growh ra procss by h cosumpio growh ra procss, udr h rlaiv uralisic assumpio ha wo procsss ar idical. Scod, hy sima h dividd procss wih cosumpio daa ad g paramrs o sabl iovaio ε ~iid S( α, β, c,0). as (8) I ordr o avoid hs problms, udr h ralisic assumpio o spara aggrga dividds ad cosumpio procsss as x ad y, w g h soluio o Equaio (6) i Appdix. Wih sps giv i Appdix, h umrical soluio ca b obaid ad compard wih h rsul o Equaio (8) mployd by Bidakoa ad McCulloch i h x scio. EMPIRICAL RESULTS OF THE MODEL

9 Th Impac o Spara Procsss o Ass Pricig From h aalyical soluio giv i Appdix, w hav E R = E E[ xp( x+ ) + v+ xp( x+ ) ]. v Ad h abov quaio ca b viwd as (9.a) ( ) E x = ( x) pxdx, (9.b) whr px is h probabiliy disribuio ucio (pd) o x. Th w mploy Simpso s rul o compu h igral i Equaio (9. b). Ad h pd px ca b obaid by usig Zoloarv s propr igral rprsaios (Zoloarv, 986). Also w ca implm h compuaioal algorihm dvlopd by J.P. Nola. For covic o compariso bw rsuls o our modl ad h Bidakoa ad McCulloch s modl, w us h sam paramrs simad o drmi whhr our modl yilds highr implid ma quiy rurs ha hirs. Thy sima hir dowm procss wih h U.S. pr capia cosumpio daa xdig rom 890 hrough 987, ollowig Ccchi al (Ccchi al, 000). Ad i Tabl o Bidakoa ad McCulloch s papr (Bidakoa & McCulloch, 003), assumig x = µ + ε, ε ~iid S( α, β, c,0), wh β =, hy g α =.8703, c= 0.045, µ = Rcall ha our dowm procsss ar modld as Equaio (7) ad Equaio (8) as x = ( ρµ ) + ρx + ε, ρ <, y = η+ ε, ε ~iid S( αβ,, c,0). Ipu paramrs abov whil assumigη = µ, w hav our rsuls wih dir ρ s i Tabl. Tabl. Modl-implid ma quiy rurs Tabl provids ma quiy rurs o h modl wih spara procsss o aggrga dividds ad cosumpio. θ is h discou acor i h Lucas (978) modl. γ is h risk avrs paramr i h CRRA uiliy ucio ad γ > 0. auorgrssiv paramr which saisis ρ <. R is h modl-implid ma quiy rurs. Ad ρ is h

10 Th Impac o Spara Procsss o Ass Pricig θ γ 00( R ) ρ = 0 ρ = 0. ρ = 0.4 ρ = From Tabl w ca s ha, as ρ bcom largr, modl-implid ma quiy rurs R g highr. Prsumably, largr auorgrssiv paramrs would ampliy h impac o shocks o h dividds growh ra x hus rsul i highr ma quiy rurs. Irsigly, whil ρ quals zro, our modl rducs o h sadard sabl modl wih h assumpio ha aggrga dividds ad cosumpio ar idical. Ad our modl yilds h sam rsuls as h Bidakoa ad McCulloch modl. I h sam words, hir sabl modl mrgs as a spcial cas o our modl. Tabl. Compariso bw dir modl-implid ma quiy rurs Tabl provids ma quiy rurs o hr dir modls. From l o righ, rsuls o our w modl wih dir valus o ρ ar i h hird ad ourh colums. Rsuls o h Bidakoa ad McCulloch (003) modl ar i h ih colum. Ad rsuls o h Bursid (998) modl ar i h sixh colum. Th maig o paramrs ca b rrrd back o Tabl. γ 00( R ) θ ρ = 0.6 ρ = 0. 00( R ) Sabl modl 00( R ) Gaussia modl I is obvious rom Tabl ha or all combiaios o h idpd paramrs mployd, our w modl wih spara procsss o dividds ad cosumpio gras highr ma quiy rurs ha h corrspodig sadard sabl modl i Bidakoa ad McCulloch s papr which assums aggrga dividds quals cosumpio. This is i accordac wih h iuiio o highr rurs or riskir asss. Ad h rsuls o our w modl ar closr o h hisorically obsrvd avrag a 7 prc pr aum. Noic ha w hr us h sam paramrs i

11 Th Impac o Spara Procsss o Ass Pricig Bidakoa ad McCulloch s papr which ar simad by h cosumpio daa or h covic o comparisos. Rsuls o our modls would b highr wh paramrs simad by dividds daa ar implmd. CONCLUSION W sudy a cosumpio-basd ass pricig modl i which aggrga dividds ad cosumpio ar modld as dir procsss wih sabl shocks. Th w driv soluios o h modl. Th w modl iroducs spara procsss wih sabl shocks o aggrga dividds ad cosumpio o h Lucas (978) modl. By rlaxig h assumpio ha aggrga dividds qual cosumpio, h quiy rurs implid by h modl ar grar ha drivd i h Bidakoa ad McCulloch (003) sabl modl. This w ramwork wih such mor ralisic assumpio gras ma quiy ras o rur ha ar largr o accord wih hisorical daa, hus hacs h abiliy o h a ails as a risk acor ha ca xplai h quiy prmium puzzl. Wih mor ralisic assumpio o xogous dowm squcs, his w modl also lays a broadr oudaio or ohr ass pricig modls wih a ails. Ar Bidakoa ad McCulloch iroduc a ails o h Lucas modl wih rrc o Bursid (Bursid, 998), Bidarkoa ad Dupoy iroduc h habi ormaio o h sadard sabl modl (Bidarkoa ad Dupoy, 007). Lar Bidarkoa, Dupoy ad McCulloch add h icompl iormaio o h sadard sabl modl as wll (Bidarkoa al, 009). Whil hir ors or hacig h abiliy o h a ails modl o xplai h quiy prmium ar succssul o som x, hy all modl boh aggrga dividds ad cosumpio wih h sam procss ad mploy h cosumpio procss oly. Basd o hir ors, i is probabl ha rsuls o our w modl wih h habi ormaio ad icompl iormaio would prorm v br. APPENDIX. Drivaio o h Risk Fr Ass Prics From Scio IV subscio Rurs o Th Risk Fr Ass, h pric o h risk r ass is P U'( C ) = + θ E. U'( C) Wih h cosa rlaiv risk avrsio (CRRA) uiliy ucio i Scio II, his dducs o P C = θ γ + E. C (A.) Employig h log cosumpio growh ra y, Equaio (A.) bcoms P [ y ] = θe xp( γ + ).

12 Th Impac o Spara Procsss o Ass Pricig Subsiuig h diiio o h log cosumpio growh ra procss Equaio (8), w g ( ) P = θe xp γη γε +. (A.) Sic i Equaio (8) h sabl shocks mas ε ~iid S( αβ,, c,0), accordig o h sabl disribuio diiio o rasormaio i McCulloch (996) ha ax S( α,sig( a), aca, δ) hav :, w Sicγ > 0, γε : iid S( α,sig( γ) β, γ c,0). γε : iid S( α, βγ, c,0). Also accordig o h sabl disribuio aur i McCulloch (996) ha wh β =, ( πα ) X le = δ c α sc /, w hav cas ad cas. Cas : β =+, E xp xp sc. ( γε ) ( γc) α πα + = Thror, subsiuig io Equaio (A.), w hav This is Equaio (4) i h x. P a πα = θ xp γη ( γc) sc. Cas : β =. I McCulloch (996), wh, ( ) E xp γε + =. X β = w hav δ c α ( πα ) le = sc /, which implis Equaio (A.) h implis ha P =. APPENDIX. Drivaio o Rurs o h Risky Ass From h subscio Rurs o Risky Asss i Scio IV, h quilibrium gross quiy rurs R o asss hld rom priod hrough priod + ar giv by Equaio (6) as

13 Th Impac o Spara Procsss o Ass Pricig R + v xp[ x+ ]. + = v Ad v is did as i v = E m+ j, i= j= whr m = θxp[ γ y + x ]. Th w driv h populaio ma o h implid quiy rurs E R. Wih h oio ha giv iormaio s icludig boh x ad y as F σ { x, y} =, i.. h σ -algbra is grad by x ad, y w hav E [ X] = E XF. Th w obai: + v = + ER E xp[ x+ ] v + v = + ER E E xp[ x+ ] v E R = E E[ xp( x+ ) + v+ xp( x+ ) ]. v Accordig o h srucur o Equaio (A.), w solv i i hr sps. (A.) Sp : or h algbra rm v, For covic, w rorm x as + x+ = A+ Bx + ε+, hus x = A+ Bx + ε ( ε ) = A+ B A+ Bx + + ε ( ) = A + B+ B + + B + ε + Bε + + B ε + B ε Th w ca sum x as

14 Th Impac o Spara Procsss o Ass Pricig x + x + + x ( ( ) ( ) ( )) + ( + B) + ( + B+ B ) + + ( + B+ B + + B ) ( B+ B + + B ) x = A + + B + + B+ B B+ B + + B + ε ε ε ε , ad hav E [ m m m ] γη A( ( B) ( B B ) ( B B B )) (( γ) ε+ ) (( + B γ) ε+ ) (( + B+ B γ) ε+ ) (( + B+ B + + B γ) ε+ ) ( B+ B + + B ) x = θ xp E xp E xp E xp E xp ( ) xp. (A.) Th, wih Equaio (7), w ca g v. Sp : or h algbra rm v ( x ) xp, + + Accordig o Equaio (A.), w hav E [ m m m ] xp( x ) γη A( ( B) ( B B ) ( B B B )) (( γ) ε+ + ) (( + B γ) ε+ ) (( + B+ B γ) ε+ ) (( + B+ B + + B γ) ε+ ) ( B+ B + + B ) x+ = θ xp E xp E xp E xp E xp ( ) xp. Th w obai v xp( x ) + + wih Equaio (7) as wll. (A.3) Sp 3: or h algbra rm v ( x ) E + xp +, As Sp ad Sp 3, wih Equaio (A.3), w g

15 Th Impac o Spara Procsss o Ass Pricig [ m m m ] ( x ) E E xp γη A( ( B) ( B B ) ( B B B )) (( γ) ε+ + ) (( + B γ) ε+ ) ( + B+ B γ) ε+ + B+ B + + B γ ε+ = θ xp E xp E xp ( ) (( ) ) + (( + B+ B + + B ) ε+ ) xp (( B B B ) x ). E xp E xp E xp (A.4) By Equaio (7), h algbra rm E v xp( x ) + + ca b obaid. Fially, accordig o h aur o sabl disribuio, as ε ~iid S( αβ,, c, δ ), w g α E[ xp( k )] xp k k c α πα ε = δ sc. (A.5) Ad wih h rsuls o algbra rms rom Sp o Sp 3, w ca g h ma o h implid quiy rurs E R wih Equaio (A.). REFERENCES Bidarkoa, P.V., & McCulloch, J.H. (003). Cosumpio ass pricig wih sabl shock-xplorig a soluio ad is implicaios or ma quiy rurs. Joural o Ecoomic Dyamics & Corol, 7, Bidarkoa, P.V., & Dupoy, B.V. (007). Th impac o a ails o quilibrium ras o rur ad rm prmia. Joural o Ecoomic Dyamics & Corol, 3, Bidarkoa, P.V., Dupoy, B.V., ad McCulloch, J.H. (009). Ass pricig wih icompl iormaio ad a ails. Joural o Ecoomic Dyamics & Corol, 33, Bra, M.J., & Xia Y. (00). Sock rur volailiy ad h quiy prmium. Joural o Moary Ecoomics, 47, Bursid, C. (998). Solvig ass pricig modls wih Gaussia shocks. Joural o Ecoomic Dyamics ad Corol,, Ccchi, S.G., Lam, P.S., ad Mark, N.C. (993). Th quiy prmium ad h risk-r ra:

16 Th Impac o Spara Procsss o Ass Pricig Machig h moms. Joural o Moary Ecoomics, 3, -45. Ccchi, S.G., Lam, P.S., ad Mark, N.C. (000). Ass pricig wih disord blis: ar quiy rurs oo good o b ru? Th Amrica Ecoomic Rviw, 90, Campbll, J.Y. (999). Ass Prics, cosumpio, ad h busiss Cycl. Hadbook o Macrocoomics, Vol.. Amsrdam: Elsvir. Campbll, J.Y., & Cochra, J.H. (999). By orc o Habi: Cosumpio-basd xplaaio o aggrga sock mark bhavior. Joural o poliical Ecoomy, 07, David, A., & Vrosi, P. (000). Opio pricig wih ucrai udamals. Fdral Rsrv Board ad Uiv. o Chicago Workig Papr. Froo, K.A., & Obsld, M. (99). Irisic bubbls: h cas o sock prics. Th Amrica Ecoomic Rviw, 8(5), Lucas Jr., R.E. (978). Ass prics i a xchag coomy. Ecoomrica, 46, Mhra, R., & E. C. Prsco. (985). Th quiy prmium a puzzl. Joural o Moary Ecoomics, 5, McCulloch, J.H. (996). Fiacial applicaios o sabl disribuios. Hadbook o Saisics, Vol. 4. Elsvir, Vrosi, P. (000). How dos iormaio qualiy ac sock rurs? Joural o Fiac, 55, Zoloarv, V.M. (986). O-dimsioal sabl disribuios. Amrica Mahmaical Sociy, Providc, RI (Traslaio o Odomry Usoichivy Rasprdliia, Nauka, Moscow, 983).