A New Variance Bound on the Stochastic Discount Factor*

Transcription

1 Raymond Kan Univrsity of Toronto Guofu Zhou Washington Univrsity in St. Louis A Nw Varianc Bound on th Stochastic Discount Factor* I. Introduction Hansn and Jagannathan (1991) provid a lowr bound on th varianc of a stochastic discount factor (SDF). As many asst pricing modls can b rprsntd by using an SDF (s,.g., Cochran [001] and rfrncs thrin), this bound bcam instantly known as th Hansn-Jagannathan bound and has bn applid widly in a varity of financ problms. On dvloping rlatd bounds, Snow (1991) drivs a bound in trms of highr momnts, Stutzr (1995) obtains a bound using Baysian information critrion, Bansal and Lhmann (1997) invstigat a growth form of th bound, Balduzzi and Kallal (1997) rlat th bound to risk prmia, and Chrétin (003) drivs a bound on th autocorrlation of SDFs. Morovr, Brnardo and Ldoit (000) and Cochran and Saá- Rqujo (000) driv similar bounds in incomplt * W ar gratful to Krry Back, Tim Bollrslv, Douglas Brdn, John Cochran, Phil Dybvig, Hbr Farnsworth, Christophr Gadarowski, Robrt Goldstin, Rick Grn, Yaniv Grinstin, Campbll Harvy, Yongmiao Hong, Ravi Jagannathan, Pt Kyl, Haitao Li, Hong Liu, Ludan Liu, Gorg Tauchn, S. Viswanathan, Kvin Wang, Zhnyu Wang, sminar participants at Cornll Univrsity, Duk Univrsity, Syracus Univrsity, Washington Univrsity in St. Louis, and York Univrsity, and spcially to two anonymous rfrs for hlpful discussions and commnts. Kan gratfully acknowldgs financial support from th Social Scincs and Humanitis Rsarch Council of Canada. Contact th corrsponding author, Guofo Zhou, at zhou@olin.wustl.du. In this papr, w construct a nw varianc bound on any stochastic discount factor (SDF) of th form m p m(x), with x bing a vctor of stat variabls, which tightns th wll-known Hansn- Jagannathan bound by a ratio of on ovr th multipl corrlation cofficint btwn x and th standard minimum varianc SDF, m 0. In many applications, th corrlation is small, and hnc th bound is much improvd. For xampl, whn x is th growth rat of consumption, th nw varianc bound can b 5 tims gratr than th Hansn-Jagannathan bound, making it much mor difficult to xplain th quity-prmium puzzl. [Journal of Businss, 006, vol. 79, no. ] 006 by Th Univrsity of Chicago. All rights rsrvd /006/ $

2 94 Journal of Businss markts. Th rol of conditional information was first xplord by Hansn and Richard (1987) and furthr invstigatd by Gallant, Hansn, and Tauchn (1990). Rcntly, Frson and Sigl (003) and Bkart and Liu (004) hav shown how conditional information might b usd to optimally tightn th original Hansn-Jagannathan bound. Rosnbrg and Engl (00) and rfrncs thrin provid mpirical stimats for th rlatd SDF. Howvr, non of ths studis hav analyzd th rol of stat variabls in th dtrmination of th bound, although most SDFs ar functions of som obsrvabl stat variabls. This papr studis th rol of stat variabls in th dtrmination of th Hansn-Jagannathan bound. W show that th Hansn-Jagannathan bound can b improvd by a factor of 1/rx,m, whr rx,m is th multipl corrlation 0 0 cofficint btwn th stat variabls and th standard minimum varianc SDF m 0. In many applications, th corrlations btwn th stat variabls and th rturns ar small, and hnc our bound is substantially tightr than Hansn-Jagannathan s. For xampl, whn x is th gross growth rat of consumption, th corrlation is usually lss than 30% and our bound is mor than 10 tims largr. Notic that our bound, lik th original Hansn-Jagannathan on, is still an unconditional bound and hnc is asily stimatd in practic. In contrast, stimation of th conditional bounds of Frson and Sigl (003) and Bkart and Liu (004) is mor difficult, and ths bounds oftn offr vry small improvmnts ovr th original Hansn-Jagannathan on. W also apply th nw bound to xamin consumption-basd asst pricing modls. In gnral, it offrs a much sharpr bound on th varianc of th marginal rat of substitution. As a rsult, it maks th quity prmium and corrlation puzzls mor difficult to xplain. Th rst of th papr is organizd as follows. Th bound is prsntd in Sction II, applications of th bound to consumption-basd asst pricing modls ar providd in Sction III, and Sction IV prsnts conclusions. II. An Improvd Bound on th Stochastic Discount Factor Undr th law of on pric, it is wll known (s,.g., Cochran 001) that thr xists a random variabl m t 1, calld th stochastic discount factor, th stat pric dnsity, or th pricing krnl, such that E[Rt 1mt 1FI t] p 1 N, (1) whr 1N is an N-vctor of ons, Rt 1 is th gross rturns on N assts at tim t 1, and I t is th information availabl at tim t. As conditional momnts ar vry difficult to stimat in practic, on is oftn intrstd in th unconditional form of (1). If th th tim subscript is supprssd, th unconditional pricing quation is givn by E[Rm] p 1. () N

3 Stochastic Discount Factor 943 Whil () is th rstriction on th SDF of an asst pricing modl, it is wll known that th rturn on a particular portfolio can also srv as an SDF: 1 m 0 p mm (1N mmm) S (R m), (3) whr mm p E[m] is th man of m that can b st as an arbitrary valu, and m and S ar th man and th covarianc matrix of th asst rturns. W assum that m is not proportional to 1 N in ordr to avoid th trivial cas. Th N assts ar risky and assumd to b nonrdundant hr so that S is nonsingular. For asir rfrnc, w call m 0 th dfault SDF sinc it always prics th N assts corrctly (satisfying q. [1] rgardlss of th validity of any asst pricing modl). If thr is a risk-fr asst with constant gross rturn R f, quation () implis that mm p 1/Rf. This puts a rstriction on th man of all SDFs. Howvr, in th prsnc of a risk-fr asst, it is asy to s that th dfault SDF is still dfind in th sam way as abov in trms of th risky assts, xcpt for rquiring furthr mm p 1/Rf. Bsids m 0, thr is a countlss numbr of SDFs that satisfy (). Th clbratd Hansn-Jagannathan bound placs a lowr bound on th varianc of all such SDFs, with man E[m] p m m : 1 Var [m] Var [m 0] p (1N mmm) S (1N mmm), (4) whr m0 is as dfind in (3). As m0 is an SDF and it attains th minimum, th Hansn-Jagannathan bound is optimal in th sns that on cannot find a bttr lowr bound for all th SDFs. How can on improv on th Hansn-Jagannathan bound? Th ida is to put a crtain structur on th SDFs. A good structur will rstrict th class of SDFs and yt rmain gnral nough to includ many intrsting SDFs. Th structur w impos is m p m(x), (5) whr x p (x 1,,x K) is a vctor of K stat variabls. Th SDFs of many wll-known thortical asst pricing modls hav such a form. For xampl, factor modls, such as th capital asst pricing modl (CAPM) and Fama and Frnch s (1993) thr-factor modl, all spcify m as a linar function of factors. In nonlinar modls, Bansal and Viswanathan (1993) spcify m as a nonlinar function of th quity markt rturn, th Trasury bill yild, and th trm sprad (th x hr), and Dittmar (00) spcifis m(x) as a cubic function of aggrgat walth. If on taks a stand that th stat variabl x is unobsrvabl or unknown, a projction of th pricing krnl on known variabls may b don to yild a nw krnl in trms of obsrvabls. For instanc, Aït-Sahalia and Lo (000) projct th pricing krnl onto quity rturns, avoiding th us of aggrgat consumption data, and Rosnbrg and Engl (00) xpand furthr on both th projction and th associatd stimation mthodology. Th qustion w ask is whthr thr xists such a constant c p c(x,

4 944 Journal of Businss m 0), which dpnds only on x and m 0 (and hnc is stimabl in mpirical studis), but is indpndnt of th particular functional form of m and satisfis Var [m(x)] c(x, m ) # Var [m ], (6) 0 0 whr c p c(x, m 0) 1. If so, this clarly offrs an improvmnt ovr th Hansn-Jagannathan bound. As it turns out, w can find such a constant c p c(x, m 0) 1 as follows. Considr th linar rgrssion of on x: m 0 m 0 p a b x 0. (7) It is wll known by construction that E[ 0] p 0 and Cov [ 0, x] p 0. To obtain th nw bound, w impos a slightly strongr assumption of E[ 0Fx] p 0. Undr this rgrssion condition, w prsnt th ky rsult of this papr in our first proposition. Proposition 1. Suppos that a stochastic discount factor m p m(x) is afunctionofkstat variabls x and w hav E[ 0Fx] p 0 in th rgrssion 1 of m 0 p a b x 0, whr m 0 p mm (1N mmm) S (R m) is a linar combination of asst rturns. Thn, for all m(x) with E[m(x)] p m m, w hav 1 Var [m(x)] Var [m ], (8) r x,m0 whr rx,m is th multipl corrlation cofficint btwn x and m 0, and th 0 quality holds if and only if m(x) p a b x. Proof. First, it is important to not that th SDF placs a strong rstriction on th covarianc btwn m and so that m 0 0 Cov [m, m ] p Var [m ]. (9) 0 0 This follows (s,.g., Frson and Sigl 003) through simpl algbra: E[mm ] p m E[m] (1 m m) S 0 m N m 1 E[m(R m)] 1 p mm (1N mmm) S (1N mmm) p E[m 0] (10) and th fact that both m and m 0 hav th sam man mm. Undr th assumption that E[ Fx] p 0, w hav 0 and hnc Cov [, m(x)] p E[ m(x)] p E[E[ Fx]m(x)] p 0 (11) Var [m 0] p Cov [m 0, m(x)] p Cov [b x, m(x)] p bsxm 1/ 1/ p bs S S, (1) xx xx xm

5 Stochastic Discount Factor 945 whr Sxm p Cov [x, m(x)] and Sxx p Var [x]. Applying th Cauchy-Schwarz 1/ 1/ inquality to th vctors Sxx b and Sxx Sxm, w hav 1/ 1/ 1 Var [m 0] p (b Sxx Sxx S xm) (b Sxxb)(SxmSxx S xm). (13) Now, from th rgrssion of m(x) on x, w hav 1 Var [m(x)] SxmSxx S xm. (14) A combination of (13) and (14) and using th xprssion bsxxb r p (15) x,m 0 Var [m ] 0 yilds th dsird inquality on Var [m(x)]. For (8) to b an quality, w nd both (13) and (14) to b qualitis. Exprssion (13) is an quality if and only 1 if Sxx Sxm is proportional to b. Exprssion (14) is an quality if and only if m(x) is a linar function of x. Togthr with th fact that m and m 0 hav th sam man, ths two conditions ar satisfid if and only if m(x) p a b x. QED Bfor w analyz th implications of proposition 1, it is usful to discuss its assumptions. First, in th spirit of th original Hansn-Jagannathan bound, m hrisanarbitrary function of stat variabls. Similarly to th popular Hansn-Jagannathan bound of (4), which is drivd undr th law of on pric, w do not rstrict m to b strictly positiv, although our bound also works for positiv m. It should b notd that Hansn and Jagannathan (1991) also provid a tightr bound on Var [m] by imposing an additional assumption of no arbitrag that m 1 0. Although th tightr bound is not analytically availabl, Hansn and Jagannathan (1991) find that it is clos to th standard Hansn-Jagannathan bound in thir applications. Thrfor, to th xtnt that our nw bound can substantially improv on th standard Hansn-Jagannathan bound, it will also b tightr than th Hansn-Jagannathan no-arbitrag bound. Scond, in comparison with th assumptions undrlying th Hansn- Jagannathan bound, th only additional on that w impos is th rgrssion assumption that E[ 0Fx] p 0. A sufficint condition for E[ 0Fx] p 0 to hold is whn th rturns and th stat variabls ar jointly lliptically distributd (s,.g., Muirhad 198, 36). So, w hav th following corollary. Corollary 1. Suppos that a stochastic discount factor m p m(x) is a function of K stat variabls x, and x and th asst rturns ar jointly lliptically distributd. Thn, for all m(x) with E[m(x)] p m m, w hav 1 Var [m(x)] Var [m ], (16) r x,m0 whr rx,m is th multipl corrlation cofficint btwn x and m 0. 0 Th usual multivariat normality assumption is a spcial cas of th lliptical assumption. Th normality assumption is common in both thory and m- 0

6 946 Journal of Businss pirical studis. For xampl, many asst pricing tsts assum that stock rturns and factors ar jointly normal. Thortically, diffusion modls imply locally lognormal distributions that ar wll approximatd by normal ons. Hnc, th corollary covrs many cass of practical rlvanc. Howvr, th lliptical assumption is far mor gnral than th normality assumption. It contains multivariat t, Kotz, mixtur normal, and many othr usful distributions that may provid for a bttr dscription of th rturn data. Whn on is intrstd in th consumption CAPM or in SDFs that ar basd on th Fama and Frnch (1993) factors, th multivariat lliptical distribution sms to b a good firstordr approximation of th data. For xampl, Zhou (1993) shows that th multivariat t-distribution is a good modl for th siz and industry portfolios, and Kan and Zhou (003b) and Tu and Zhou (004) dmonstrat that it also modls th Fama and Frnch portfolios and factors wll. It should b mphasizd that vn though corollary 1 maks th multivariat lliptical distribution assumption on x, it dos not imply that m has an lliptical distribution. In fact, m can b an arbitrary function of x, and thr is no distributional assumption imposd on m. In particular, m can b strictly positiv for all valus of x. Finally, if th stat variabls x ar not lliptically distributd, but if a suitabl transformation of y p g(x) and th asst rturns ar jointly lliptically dis- tributd, proposition 1 still applis to m(y) p m(g(x)) to yild an improvd bound by rplacing th arlir multipl corrlation of x with m 0 with th multipl corrlation of y with m 0. A rlatd point is that th projction of m 0 on x is not ncssarily linar as long as th rsidual has xpctation zro conditional on x. 1 Thortically, th condition E[ 0Fx] p 0 might not b sat- isfid in th linar rgrssion of m0 on x, m0 p a b x 0, but might b so in a crtain nonlinar rgrssion m 0 p f(x) 0. Thn, following th proof of proposition 1, w hav th following corollary. Corollary. Suppos that a stochastic discount factor m p m(x) is a function of K stat variabls x and w hav E[ 0Fx] p 0 in th nonlinar rgrssion of m p f(x). Thn, for all m(x) with E[m(x)] p m, w hav 0 0 m 1 Var [m(x)] Var [m ], (17) r f(x),m0 whr rf(x),m is th multipl corrlation cofficint btwn f(x) and m 0. 0 Proposition 1 looks amazingly simpl. Lik th Hansn-Jagannathan bound, it placs an (oftn much strictr) rstriction on th varianc of th SDF with th minimum knowldg of th functional form of th SDF. Bcaus th bound is formd with momnts of only obsrvabls, it has th sam appaling faturs of th Hansn-Jagannathan bound. In particular, it can oftn shd light on why a particular class of asst pricing modls fails to xplain asst rturns and indicat what stps may b takn to improv thm. As rx,m 0 1, th bound must b no wors than th Hansn-Jagannathan bound. In fact, 1. W thank an anonymous rfr for this intrsting point. 0

7 Stochastic Discount Factor 947 r x,m0 is oftn small in practic, so th bound can b much sharpr than th Hansn-Jagannathan bound. Howvr, it is important to not that our improvd bound coms at a cost. Unlik th Hansn-Jagannathan bound, which works for all SDFs, our bound is not univrsal and works only for a class of asst pricing modl that is in th form of m p m(x). Thrfor, for a diffrnt choic of stat variabls, w nd a diffrnt bound. Nvrthlss, th fact that our bound is spcializd to a givn class of asst pricing modls dos not prvnt us from using it as a tool for modl diagnostic. In almost vry application in th litratur in which on uss th Hansn- Jagannathan bound, on nds to spcify x and chck whthr an SDF m(x) violats th Hansn-Jagannathan bound. Our point is that if on is willing to spcify x to chck th Hansn-Jagannathan bound, on can b bttr off by comparing th varianc of m(x) with our tightr nw bound instad of th Hansn-Jagannathan bound. Although th us of our nw bound rquirs additional computational cost sinc w cannot us th sam bound on all SDFs, th advantag is that w ar abl to dtct som invalid SDFs that pass th tst of th Hansn-Jagannathan bound. Whn a proposd m fails our nw bound, th intrprtation is th sam as whn it fails th Hansn-Jagannathan bound. W can conclud that ithr th choic of th st of stat variabls or th functional form is wrong. Our bound, howvr, allows us to focus on th qustion of what functional form is ndd to mak th SDF fasibl givn a choic of th stat variabls. For xampl, if on blivs that th SDF is a polynomial of th markt rturn, on can us th nw bound to find out what ordr of th polynomial is ncssary for th SDF to b accptabl. On may suggst that givn th choic of x, it may b possibl to us a nonparamtric tchniqu to com up with an stimat of th functional form m(x) and dirctly tst th momnt condition of E[m(x)R] p 1 N instad of using our bound. Th problm is that it is unclar how a nonparamtric mthod can b usd to stimat th functional form m(x). Furthrmor, vn if a nonparamtric stimat of th SDF is availabl, it is a difficult task to stablish th distribution thory for th spcification tst. As a rsult, in th spirit of th original Hansn-Jagannathan bound, th us of our nw bound provids a simpl and fast spcification tst for dtcting invalid SDFs. By spcifying a paramtric functional form and th stat variabls for an SDF, th traditional spcification tst allows us to dirctly tst th validity of th SDF using rturn data. This approach imposs stringnt limits on th class of asst pricing modls, but it can rsult in a vry sharp prdiction on th validity of th SDF whn w hav sufficint data. On anothr xtrm, th Hansn-Jagannathan bound imposs almost no structur on th SDF othr than th law of on pric. Th rsult is that it can dlivr a varianc bound. Shankn (1987) drivs a bound similar to th Hansn-Jagannathan bound with th us of a multipl corrlation cofficint. Howvr, in that cas, th corrlation cofficint is btwn m and a proxy. In contrast, our corrlation cofficint hr is btwn x and m 0. Hnc, our bound diffrs from Shankn s bound.

8 948 Journal of Businss that is applicabl for all SDFs. Th pric to pay for this gnrality is that th bound may not b vry tight and informativ. Our approach stands btwn ths two xtrms. W limit th class of SDF to a function of a st of stat variabls x, but yt w do not nd to spcify its paramtric functional form. Th rsult is that w can dlivr a tightr bound than th Hansn-Jagannathan bound. Thr is always a trad-off btwn th broadnss of th class of asst pricing modls and th tightnss of th bound. W considr all thr approachs to hav thir rspctiv mrits, and on is dfinitly not suprior to th othr. Which approach is mor appropriat dpnds on th contxt of th problm. For xampl, if a proposd m fails th Hansn-Jagannathan bound, thr is no nd to us our nw bound. Howvr, if th proposd m passs th Hansn- Jagannathan bound, on may lik to compar th varianc of th proposd m with our nw bound to gathr mor information about its validity. In comparison with th Hansn-Jagannathan bound, our proposd nw bound has an additional advantag of bing robust to masurmnt rrors in th stat variabls. Intuitivly, if th tru stat variabls ar masurd with rrors, this will incras th varianc of th SDFs that ar basd on th noisy proxy of th stat variabls. In fact, th largr th masurmnt rrors, th largr th varianc of th proposd SDF and, hnc, th asir for th proposd SDF to pass th Hansn-Jagannathan bound that is compltly indpndnt of th stat variabls and thir masurmnt rrors. This obsrvation suggsts that, with othr things hld constant, a wrong SDF that is basd on a noisy stat variabl stands a bttr chanc of satisfying th Hansn-Jagannathan bound. In contrast, our nw bound dos not rward noisy stat variabls bcaus if a stat variabl x is masurd with a lot of nois, th rsulting r x,m0 is small and our nw bound for such an SDF will b tightr. As a rsult, it is not any asir for a wrong SDF to pass our nw bound by simply introducing a noisy stat variabl. Finally, it is important to not that whil th Hansn-Jagannathan bound is a quadratic function of m m, this is not th cas for our nw bound. This is so bcaus m 0 is a function of m m, so rx,m 0 is also a function of m m.inth following corollary, w giv an xplicit xprssion of our nw bound as a function of m m. Corollary 3. For a stochastic discount factor of th form m p m(x) with man, w hav whr m m (amm bmm c) Var [m(x)], (18) a m b m c 1 m 1 m 1 1 a p ms m, (19) 1 b p ms 1, (0) N

9 Stochastic Discount Factor c p 1NS 1 N, (1) a1 p ms SxRSxx SxRS m, () b1 p ms SxRSxx SxRS 1 N, (3) and SxR p Cov [x, R ]. Proof. From (3), w hav and c1 p 1NS SxRSxx SxRS 1, N (4) 1 Var [m 0] p (1N mmm) S (1N mmm) p amm bmm c (5) 1 1 Cov [x, m 0] p Cov [x,(r m) ]S (1N mmm) p SxRS (1N mmm). (6) Thn using (15), w hav bsxxb rx,m 0 p Var [m 0] 1 Cov [x, m 0] Sxx Cov [x, m 0] p amm bmm c (1N mmm) S SxRSxx SxRS (1N mmm) p a1mm b1mm c1 amm bmm c p. (7) am bm c m m Dividing (5) by (7), w prov th corollary. QED Th corollary shows that th lowr bound of th varianc of an SDF with th form of m(x) is actually a fourth-ordr polynomial of m m ovr a scond- ordr polynomial of m m. Thr ar two cass in which w can rul out th validity of m(x) as an SDF. In th first cas, S xr is a zro matrix. In this cas, w hav rx,m 0 p 0 for any valu of m m and th lowr bound on Var[m(x)] is infinity, so thr is no fasibl SDF of th form m(x) that can pric all th N assts corrctly. This suggsts that for x to b valid stat variabls in an SDF, it cannot b uncorrlatd with rturns on all th assts. In th scond cas, K p 1 and mm p b 1/a1. Whn K p 1, w hav ac 1 1 p b1, and as a rsult rx,m p 0 for mm p b 1/a1, which implis that th 0 lowr bound on Var [m(x)] is infinity. Thrfor, thr is no m(x) with man b 1/a1 that can pric all th N assts corrctly. Not that th minimum-varianc

10 950 Journal of Businss SDF m0 is a function of m m. Whn m is not proportional to 1 N, m0 for diffrnt valus of m m ar not prfctly corrlatd, so thr is no singl stat variabl that can b prfctly corrlatd with m 0 for vry choic of m m. Thrfor, for a givn stat variabl x, thr will always b on choic of m m such that m 0 is uncorrlatd with x. This suggsts that whn th SDF is a function of only on stat variabl, thr will always b a valu of m m such that m(x) is an infasibl SDF, rgardlss of th functional form of m(x). 3 Th K p 1 cas of proposition 1 is of particular intrst. In this cas, r x,m0 is th simpl corrlation cofficint btwn two univariat random var- iabls x and m 0. If rx,m 0 p 1, th abov bound rducs to th Hansn- Jagannathan bound. Morovr, if x p m p m 0, both our nw bound and th Hansn-Jagannathan on ar idntical. Howvr, our nw bound can in gnral b much tightr than th Hansn-Jagannathan bound. Considr two xampls. Th first is th xtrm cas in which x is uncorrlatd with m 0. Our nw bound says that it is impossibl to find such an SDF or its varianc must b infinity if found. Th Hansn-Jagannathan bound, howvr, still stats that Var [m 0] is th lowr bound with no us of th zro corrlation information. Thrfor, on may not b abl to dtct that m(x) is in fact an invalid SDF using th Hansn-Jagannathan bound alon. In th scond xampl, m p m(x), whr x is th growth rat of con- sumption. If x has a corrlation of 30% with m 0, thn th nw bound is mor than 10 tims highr than th Hansn-Jagannathan bound! Th 30% corrlation is in fact an optimistic assumption. Frson and Harvy (199) rport sampl corrlations of various consumption growth masurs and th stock rturns, and thy find that non of thm xcds 30%. Furthr applications of proposition 1 to consumption-basd asst pricing modls ar dtaild in Sction III. Som numrical illustrations may b illuminating. Considr th wll-known 5 siz and book-to-markt sortd portfolios usd by Fama and Frnch (1993). 4 In figur 1, w plot th standard Hansn-Jagannathan bound for any m that prics th 5 assts corrctly using a solid lin. Th bound is stimatd as ( ) N N ˆ 1 j ˆ 0(m m) p 1 (1N mm) mˆ S (1N mm) mˆ m m, (8) T T whr ˆm and S ˆ ar th sampl man and varianc of th rturns on th 5 portfolios, stimatd using monthly data ovr th priod 195/1 00/1. Undr normality assumption, Frson and Sigl (003, proposition 4) show that ĵ 0(m m) is an unbiasd stimator of Var[m 0], and it is suprior to th unadjustd bound, spcially whn N is larg rlativ to T. Now, suppos that w propos a class of asst pricing modls in which m is a (possibly nonlinar) function of th xcss rturn on th markt portfolio, 3. Whn K 1 1 and m is not proportional to 1 N, w hav ac b1 in gnral, so th dnominator of (18) will not b qual to zro for any choic of m m. 4. W ar gratful to Kn Frnch for making ths data availabl on his Wb sit.

11 Stochastic Discount Factor 951 Fig. 1. Varianc bounds of stochastic discount factors. Th figur plots thr varianc bounds on th stochastic discount factors whn th tst assts ar 5 siz and book-to-markt rankd portfolios. Th solid lin is th Hansn-Jagannathan bound for all stochastic discount factors. Th dashd lin is th varianc bound for m(x), whr x is th xcss rturn on th valu-wightd markt portfolio. Th dottd lin is th varianc bound for m(x), whr x is th thr Fama-Frnch factors (xcss rturn on th valu-wightd markt portfolio, rturn diffrnc btwn larg and small siz portfolios, and rturn diffrnc btwn high and low book-to-markt portfolios). Th thr varianc bounds ar stimatd using monthly data ovr th priod 195/ 1 00/1. R M, a particular cas of which is th wll-known CAPM. For this choic of stat variabl RM, w plot th lowr bound of Var [m(r M)] using a dashd lin in figur 1. This varianc bound is stimatd on th basis of ĵ 0(m m) m(x) m ˆr x,m0 ĵ (m ) p, (9) whr ˆr x,m is th sampl multipl corrlation cofficint btwn x and m 0. 0 Not that sinc w hav only on stat variabl, for som choic of m m (0.9901, corrsponding to a monthly intrst rat of 1%) w hav th lowr bound of Var [m(r M)] qual to infinity. With this valu of m m, w can s that our nw bound provids a substantial improvmnt ovr th standard Hansn-Jagannathan bound. Outsid this valu, is in gnral fairly highly corrlatd R M

12 95 Journal of Businss with m 0, so our nw bound provids lss an improvmnt, though still substantial. Incrasing th numbr of stat variabls will in gnral rduc th varianc bound bcaus r x,m0 can only incras with a largr st of stat variabls. W illustrat this by xpanding th st of stat variabls to th Fama-Frnch thr factors x p (R M, R SMB, R HML), whr RSMB is th rturn diffrnc btwn small and larg siz portfolios, and R HML is th rturn diffrnc btwn high and low book-to-markt portfolios. In figur 1, w plot our nw bound on Var [m(r M, R SMB, R HML)] using a dottd lin. Comparing this bound with th dashd on for th cas of x p R M, w can s that with mor stat variabls includd in th SDF, th nw bound is closr to th Hansn-Jagannathan on. Nvrthlss, th nw bound can still b substantially highr than th Hansn- Jagannathan bound. Ovr th rang of valus of m m that w plot in figur 1, our nw bound offrs at last a 34% incras ovr th Hansn-Jagannathan bound and as much as a 646% incras for som valus of m m. III. Impact on Consumption-Basd Modls Cochran (001) provids an xcllnt survy of th standard consumptionbasd asst pricing modls originatd by Brdn (1979). Th wll-known first-ordr condition (Eulr quation) for an invstor s xpctd utility maximization problm is u (C t) p E t[du (C t 1)R t 1], (30) whr u is th utility function, d is th subjctiv tim discount factor of th invstor, Ct is th consumption at tim t, and Rt 1 is th gross rturn of an asst at tim t 1. So th basic asst pricing quation is u (C t 1) 1 p E t[mr t 1 ], m p d, (31) u (C ) whr m is th wll-known SDF or th intrtmporal marginal rat of substitution. Applying proposition 1 to som wll-known utility functions is straightforward. For xampl, considr th powr utility ) 1 g g Ct Ct 1 gx u(c t ) p, m(x) p d( p d, (3) 1 g C t whr x p ln (C t 1/C t) is consumption growth. If w ar sur that th utility function is indd a powr utility function and th tru valus of d and g ar known, w can dirctly tst (31). Howvr, rsarchrs ar oftn not quippd with th knowldg of th xact functional form of th utility function. In that cas, if w ar willing to assum joint lliptical distribution of x and t

13 Stochastic Discount Factor 953 Fig.. Varianc bound of stochastic discount factors that ar functions of growth rat of consumption. Th figur plots th varianc bound on th stochastic discount factors whn th tst assts ar 5 siz and book-to-markt rankd portfolios and th stochastic discount factor is a function of th continuously compoundd growth rat of durabl consumption. Th varianc bound is stimatd using monthly data ovr th priod 1959/ 00/1. rturns and assum that th intrtmporal marginal rat of substitution can b writtn as a function of x, thn proposition 1 says Var [m 0] Var [m(x)]. (33) r x,m0 In figur, w plot this lowr bound for Var [m(x)] using th sam 5 portfolio rturns as bfor, whr th consumption growth pr capita is masurd using nondurabl consumption data from th Citibas availabl from Fbruary 195 to Dcmbr 00. Comparing this with th Hansn-Jagannathan bound in figur 1, w find that th bound for Var [m(x)] in figur is much highr, in fact, at last 18 tims highr than th Hansn-Jagannathan bound. Th rason is that th highst r x,m0 that w can find within th rang of m m that w plot is Thrfor, in ordr for m(x) to pric th 5 siz and book-to-markt rankd portfolios corrctly, it has to b xtrmly volatil. This high rquird volatility on m(x) hr also has important implications

14 954 Journal of Businss on th paramtrs of th utility function. For xampl, substituting (31) into (3), w obsrv that for a fixd valu of g, th invstor s subjctiv tim discount factor, d, must satisfy 1 Var[m 0] d. (34) gx Fr F Var [ ] x,m 0 As d discounts th futur utility to prsnt, it masurs th invstor s impatinc. Th smallr th d is, th mor impatint th invstor. Using th Hansn- gx Jagannathan bound, on considrs a valu of d p Var [m 0]/ Var [ ] to b accptabl. Howvr, vn at a vry high corrlation lvl of th consumption growth with th asst rturns that rsults in Fr x,m 0 F p 0.3, quation (34) suggsts that th invstor has to b at last 3.33 tims mor patint in ordr for m(x) to b a valid SDF. Applying proposition 1 is straightforward if th marginal rat of substitution can b writtn as a function of th ratio of consumption C t 1/Ct or th first diffrnc of consumption Ct 1 Ct, sinc ths two trms can b rasonably assumd to hav an lliptical distribution. Howvr, not vry utility function has such a simpl rprsntation. Nvrthlss, as long as w ar willing to mak an lliptical distribution on C t 1/Ct, which can b justifid thortically undr constant absolut risk avrsion (CARA) utility, 5 and its conditional man and varianc ar constant ovr tim, th following proposition shows that th bound in proposition 1 continus to hold. Proposition. Suppos that a stochastic discount factor m p m(c t, C t 1) p du (C t 1)/u (C t). Lt x p ln (C t 1/C t). Suppos, conditional on Ct, that x and m 0 ar multivariat lliptically distributd with constant man and var- ianc. Thn Var [m 0] Var [m(x)], (35) r x,m0 whr rx,m is th corrlation btwn x and m 0. 0 x Proof. Writ m p m(c t, C t ). Conditional on Ct, proposition 1 can b applid to yild quation (35), xcpt th trms on both sids ar conditional on C t : Var [m FC ] 0 t Var [m(x)fc ]. (36) t r x,m FC 0 t Howvr, undr th assumption that th conditional man and varianc of x and ar constant, th conditional momnts on th right-hand sid ar m 0 5. For xampl, Mrton (1973) and Cochran (001) show that th optimal consumption growth for a CARA invstor is lognormal in th standard diffusion stup for th asst rturns.

15 Stochastic Discount Factor 955 th sam as th unconditional momnts. As for th lft-hand sid, using th itratd law of xpctations, w hav Var [m(x)] p E[Var [m(x)fc t]] Var [E[m(x)FC t]] E[[Var [m(x)fc ]] t [ ] Var [m 0] E r x,m0 Var [m 0] p. (37) r x,m0 This complts th proof. QED Now lt us xamin th implications of proposition 1 on th quity prmium and corrlation puzzls. Sinc Mhra and Prscott (1985), th quity prmium puzzl bcam wll known: th consumption-basd SDF is not volatil nough to xplain th risk prmium of quity. As put by Cochran (001, 456), it follows from th dfinition of an SDF that j(m) FE[R ]F, (38) E[m] j(r ) whr j is th standard dviation oprator and is th xcss rturn on th markt indx. Altrnativly, (38) is th rsult of applying th Hansn-Jagannathan bound to th two-asst cas: th risk-fr asst and th markt indx. Using data that th postwar xcss rturn on th Nw York Stock Exchang (NYSE) valu-wightd indx is, on avrag, approximatly 8% pr yar and th standard dviation is approximatly 16% pr yar, and assuming E[m] p 1/Rf p 0.99, Cochran shows that j(m) To justify this, a vry larg risk avrsion paramtr is rquird. Undr ithr powr utility or xponntial utility, th stat variabl of th SDF can b takn as ithr th consumption growth or th chang of consumption, and it is rasonabl, to at last a first-ordr approximation, to assum that x and R t hav a multivariat lliptical distribution. Thn, on th basis of Cochran s (001, 457) stimat of a valu of rx,r p 0., proposition 1, togthr with th fact that m 0 is a linar function of R, implis that R 1 FE[R ]F 1 FE[R ]F j(m) E[m] p E[m] Fr F j(r ) Fr F j(r ) x,m0 x,r p 5 # 0.5 p.5, (39) which dmands an vn gratr risk avrsion paramtr (in trms of varianc, this bound is 5 tims gratr than th Hansn-Jagannathan bound). Furthr mpirical study of this and rlatd modls basd on rcnt data is providd latr in this sction.

16 956 Journal of Businss Aftr manipulation of E(mR ) p 0, it is simpl to show that 1 FE[R ]F j(m) p E[m]. (40) Fr F j(r ) m,r Th ky diffrnc btwn this bound and (39) is that rm,r in gnral dpnds on th choic of a utility function, but rx,r of (39) is known or is asily stimatd from th data (indpndnt of th spcial functional form of m). In th spcial cas in which m(x) is a linar function of x, (39) and (40) ar th sam. Th much strictr bound (40) is trmd as th corrlation puzzl by Cochran (001, 457). Th proof thr applis only to th cas in which m(x) is a linar function of x. In contrast, m(x) hr can b an arbitrary nonlinar function of th stat variabl. Thrfor, it gnralizs th corrlation puzzl to potntially many utility functions. In a stting with multipl assts, Cochran and Hansn (199) show that Var [m 0] Var [m] p, (41) r m,m0 which follows dirctly from (9). Howvr, this bound (which is actually an idntity) can b calculatd only if w know r m,m0. This in turn rquirs us to spcify m xplicitly, which oftn is difficult bcaus w may hav doubts on its functional form. In addition, th rsulting varianc bound is applicabl only to that particular choic of m. In contrast, our varianc bound maks us of th multipl corrlation cofficint of th dfault SDF with th stat variabls and studis its impact on th varianc bound of an arbitrary m(x). Finally, lt us xamin applications of proposition 1 to som rcnt asst pricing modls. Owing to th failur to xplain th quity prmium puzzl, modls of SDFs with multipl stat variabls hav bn dvlopd. Th addition of mor variabls in gnral should incras th multipl corrlation btwn x and m 0, making th nw bound closr to th Hansn-Jagannathan on. Abl (1990), for xampl, provids a modl in which th invstor s powr utility dpnds not only on th consumption but also on a tim-varying bnchmark. Undr som simplifying assumptions (s,.g., Kirby 1998), this rsults in an SDF of d(c /C ) g t 1 t m p. (4) (C /C ) 1 g t 1 t In this cas, w can tak x1 p ln (C t 1/C t) and x p ln (C t 1/C t ). Th in- novations of consumption growth can b assumd to b multivariat lliptically distributd, and thn proposition 1 asily applis to yild a bound on j(m). Out of th modls with multipl stat variabls, th Campbll and Cochran

17 Stochastic Discount Factor 957 (1999, 000) modl sms gtting th most attntion. Thy propos a modl with an SDF ) g St 1 Ct 1 mcc p d (, (43) S C whr S t is th surplus consumption ratio. In thir modl, thy assum that th two ratios in m ar conditionally lognormal, and w can tak x p (ln (C t 1/C t), ln (S t 1/S t)) as th stat variabls in th modl. Thrfor, w can apply proposition 1 to yild a bound on j(m CC). In what follows, w will focus our mpirical study on this modl. At th outst, it should b notd that St p (Ct X t)/ct is unobsrvabl sinc th lvl of habit X t is latnt. Following Li (001) as wll as Liu (003), w xtract S t from a modl and thn comput th momnts and bounds on th basis of th xtractd sris. Th undrlying data-gnrating procss for S t is th nonlinar squar root modl of Campbll and Cochran (1999, 000). Thy assum that th log surplus consumption ratio volvs according to t t s p (1 f)s fs l(s )(c c g), (44) t 1 t t t 1 t whr s,, and f, g, and t p log (S t) c t p log (C t) s ar paramtrs. Th snsitivity function l(s t ) is givn by (s s) 1 if s! s t t (1 S ) S l(s t) p (45) 1 0 if s s t (1 S ), whr S p j c g/(1 f) is th stady-stat surplus consumption ratio and s p log (S). Notic that g and j c ar th man and standard dviation of th log consumption growth and hnc can b asily stimatd from th data (as th sampl man and standard dviation following Campbll and Cochran s indpndntly and idntically distributd assumption on th log consumption). Howvr, othr paramtrs (f, g, and d) hav to b spcifid xognously. In our applications, w choos paramtrs following th sam approach as in Campbll and Cochran (1999, 000). Th valu of f is chosn to b (0.87 annualizd). For g, w choos ovr a rang of two to 0. Finally, for ach valu of g, d is chosn such that th log risk-fr rat is % (0.94% annualizd), whr th log risk-fr rat is givn by g rf p ln (d) gg (1 f). (46) Campbll and Cochran show that thir modl with ths choics of paramtrs, vn for g as low as two, matchs a wid varity of phnomna including th prdictability of stock rturns from pric-dividnd ratios and th lvrag ffct by which low prics imply mor volatil rturns. Howvr, thy did not carry out a diagnostic tst using th Hansn-Jagannathan bound, nor has

18 958 Journal of Businss TABLE 1 g j(m C) j(m CC) Varianc Bound Tst of th Campbll and Cochran Habit Modl Markt Portfolio Fama-Frnch Portfolios ĵ 0 ˆr x,m0 ĵ m(x) ĵ 0 ˆr x,m0 ĵ m(x) Not. Th modls ar th standard consumption CAPM and Campbll and Cochran (1999, 000). Th varianc bound tst is basd on monthly data ovr th priod 1959/ 00/1. Th g column is th curvatur paramtr of th utility function; j(m C) is th standard dviation of th stochastic discount factor of th consumption CAPM, j(m is that of th Campbll and Cochran modl, ˆ CC) j0 is th Hansn-Jagannathan bound, r x,m0 is th multipl corrlation cofficint btwn th stat variabls and th dfault stochastic discount factor m, and ˆ 0 jm(x) is th nw bound, whn th valu-wightd markt portfolio of th NYSE is usd as th tst asst. Th last thr columns of th tabl rports th bounds and corrlation whn th Fama-Frnch 5 siz and book-to-markt rankd portfolios ar usd as th tst assts. this bn carrid out by othrs, spcially whn multipl portfolios ar usd as tst assts. As a rsult, it is of intrst hr to s how thir modl prforms in trms of th Hansn-Jagannathan and our nw bounds on th basis of th markt portfolio and th Fama-Frnch portfolios usd arlir. Tabl 1 provids th rsults using monthly data ovr th priod 1959/ 00/1. Th bounds ar computd using two diffrnt sts of tst assts. Th first st is a singl-asst cas, th valu-wightd markt indx of th NYSE. Th scond st is th Fama-Frnch 5 siz and book-to-markt rankd portfolios. W discuss th rsults using th markt portfolio first. Whn th utility curvatur paramtr g is qual to two, th standard dviation of th SDF of th consumption CAPM, j(m C), is only As a rsult, th traditional consumption CAPM has a hard tim satisfying th Hansn-Jagannathan bound. Evn if w rais g to as high as 0, j(m C) still cannot satisfy th Hansn- Jagannathan bound. In contrast to th consumption CAPM, th SDF of th Campbll and Cochran modl is much mor volatil. For xampl, whn g p, w find that th standard dviation of th SDF in th Campbll and Cochran modl has an imprssiv standard dviation of j(m CC) p , about 9. tims largr than that of th consumption CAPM. Although j(m CC) still cannot satisfy th Hansn-Jagannthan bound, which is stimatd to b ĵ0 p , th valus ar fairly clos to ach othr. So, it is not surprising that if w allow g to incras to four or abov, j(m CC) asily satisfis th Hansn-Jagannathan bound. Thrfor, on th basis of th Hansn- Jagannthan bound alon, on might conclud that th Campbll and Cochran modl is a suprb modl, vn with a rlativly small valu of g. Howvr, comparison of j(m CC) with our nw bound raiss nw issus on this modl. Bcaus th stat variabls (growths of consumption and surplus consumption

19 Stochastic Discount Factor 959 ratio) hav a fairly low multipl corrlation cofficint with th markt rturn (it rangs from to 0.146, dpnding on th valus of g), our nw bound is, on avrag, about six to svn tims largr than th Hansn-Jagannathan bound. Thrfor, vn if w incras g to 0, th Campbll and Cochran modl still cannot pass our nw bound. W now turn our attntion to th rsults using th Fama-Frnch 5 siz and book-to-markt sortd portfolios as th tst assts. It is apparnt from th tabl that all th bounds now hav gratr valus than bfor. This is intuitiv. Whn mor tst assts ar usd, it bcoms mor difficult for th modl to xplain th asst prics. Indd, whn g p, th modls fail mor signifi- cantly than bfor in passing th bounds. Intrstingly, dspit using mor assts, th Campbll and Cochran modl can still pass th Hansn-Jagannathan bound at th high nd of g ( g 16). Nvrthlss, our nw bound is in th rang of , making it almost impossibl for th consumption and Campbll and Cochran modls to satisfy whn g 0. Bcaus th krnl varianc is an incrasing function of g, th Campbll and Cochran modl can vntually satisfy our nw bound if g is larg nough. Th qustion is how larg it must b. It can b vrifid that, in ordr for th Campbll and Cochran modl to pass our nw bound, w will nd g to b 19 or abov, which is quit an unrasonably high valu. In summary, whil th Campbll and Cochran (1999, 000) modl has rmarkabl powr in xplaining th asst prics and can pass th Hansn- Jagannathan bound with a rasonably high risk vrsion paramtr, it still fails to pass th proposd nw bound of this papr. With data from 1959/ 00/ 1, th nw bound is at last mor than six tims highr than th Hansn- Jagannathan bound. Th rason for such a highr bound is that th stat variabls hav low corrlations with th asst rturns. This sms to suggst that futur asst pricing modls should focus on stat variabls that ar highly corrlatd with th rturns on th assts. An incras of th volatility of th pricing krnl alon may not b sufficint to xplain th xpctd rturns of th assts if th stat variabls hav low corrlations with th rturns of th assts. IV. Conclusions In this papr, w driv a nw varianc bound on any stochastic discount factor of th form m p m(x), whr x is a st of stat variabls. In contrast to th wll-known Hansn-Jagannathan bound, our bound tightns it by a ratio of 1/rx,m, whr rx,m is th multipl corrlation btwn x and th 0 0 standard minimum varianc SDF, m 0. In many applications, th corrlation is small, and hnc our bound is substantially tightr than Hansn and Jagannathan s. W show that if x is th gross growth rat of consumption and if w us Cochran s (001) stimats of markt volatility and r x,m0, th nw bound is 5 tims gratr, making it much mor difficult to xplain th quity prmium puzzl on th basis of xisting asst pricing modls. Morovr,

20 960 Journal of Businss applying th nw bound, with th growth rat of consumption as a stat variabl, to th 5 siz and book-to-markt sortd portfolios usd by Fama and Frnch (1993) can vn yild a varianc bound that is mor than 100 tims gratr than th Hansn-Jagannathan on. As th Hansn-Jagannathan bound poss significant challngs for xisting asst modls to mt, our nw sharply improvd bound sms to rais this challng onto a nw platau. In particular, w show that th rcnt modl of Campbll and Cochran (1999, 000) can pass th Hansn-Jagannathan bound asily whn th markt is th only tst asst and can also pass th bound for a rlativly high valu of th risk avrsion paramtr whn th Fama-Frnch 5 portfolios ar usd as th tst assts, but fails to do so for our nw bound with any rasonabl risk avrsion paramtr. Th ky insight of this papr is that in ordr for us to succssfully xplain asst prics using a thortical pricing krnl, th stat variabls must hav high corrlations with th asst rturns. This suggsts that a potntial dirction for improving modls, such as that of Campbll and Cochran (1999, 000), is to idntify stat variabls that ar highly corrlatd with th stock markt. In addition, motivatd by Frson and Sigl (003), Bkart and Liu (004), and othrs, it is of intrst to xamin how conditional information might b usd to tightn th bound vn furthr. Anothr important issu, inspird by Hansn and Jagannathan (1997), is to dvlop SDF-basd distanc masurs for compting misspcifid asst pricing modls. Hodrick and Zhang (001), Dittmar (00), and Kan and Zhou (003a), among othrs, show th wid usfulnss of th Hansn-Jagannathan distanc. In contrast to ths applications, th distanc masur can b rfind to b dpndnt on stat variabls. This would likly shd nw insights on th rols playd by th stat variabls in an asst pricing modl, a topic of intrst for futur rsarch. Rfrncs Abl, A Asst prics undr habit formation and catching up with th Jonss. Amrican Economic Rviw 80:38 4. Aït-Sahalia, Y., and A. W. Lo Nonparamtric risk managmnt and implid risk avrsion. Journal of Economtrics 94:9 51. Balduzzi, P., and H. Kallal Risk prmia and varianc bounds. Journal of Financ 5: Bansal, R., and B. N. Lhmann Growth-optimal portfolio rstrictions on asst pricing modls. Macroconomic Dynamics 1: Bansal, R., and S. Viswanathan No arbitrag and arbitrag pricing: A nw approach. Journal of Financ 48: Bkart, G., and J. Liu Conditional information and varianc bounds on pricing krnls. Rviw of Financial Studis 17: Brnardo, A. E., and O. Ldoit Gain, loss, and asst pricing. Journal of Political Economy 108: Brdn, D. T An intrtmporal asst pricing modl with stochastic consumption and invstmnt opportunitis. Journal of Financial Economics 7: Campbll, J. Y., and J. H. Cochran By forcs of habit: A consumption-basd xplanation of aggrgat stock markt bhavior. Journal of Political Economy 107: Explaining th poor prformanc of consumption-basd asst pricing modls. Journal of Financ 55:

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