The Skewness of the Stock Market at Long Horizons

Transcription

1 The Skewness of he Sock Marke a Long Horizons Anhony Neuberger and Richard Payne Cass Business School, Ciy, Universiy of London January 218 Absrac Momens of long-horizon reurns are imporan for asse pricing bu are hard o measure. Proxies for hese momens are ofen used bu none is wholly saisfacory. We show analyically ha shor-horizon (i.e. daily) reurns can be used o make more much precise esimaes of long-horizon (e.g. annual) momens wihou making srong assumpions abou he daa generaing process. Skewness comprises wo componens: he skewness of shor-horizon reurns, and a leverage effec, which is he covariance beween conemporaneous variance and lagged reurns. We provide similar resuls for kurosis. Applying he echnology o US sock index reurns, we show ha skew is large and negaive and does no significanly aenuae wih horizon as one goes from monhly o muli-year horizons

2 INTRODUCTION This paper makes wo conribuions: mehodological and empirical. The mehodological conribuion is o show how shor horizon reurns can be used o esimae he higher momens of long horizon reurns while making only weak assumpions abou he daa generaing process. The empirical conribuion is o show ha long horizon (muli-year) US equiy marke reurns are highly negaively skewed. The skew coefficien, a around -1.5, is economically significan. We also show ha his skew a long horizons is enirely aribuable o he leverage effec he negaive correlaion beween reurns and fuure volailiy. There is good reason o believe ha higher momens of reurns no jus second momens are imporan for asse pricing. A large heoreical lieraure, saring wih Kraus and Lizenberger (1976), and coninuing wih he macroeconomic disaser research of Riez (1988), Longsaff and Piazzesi (24), and Barro (26), hypohesises ha heavy-ailed shocks and lef-ail evens in paricular have an imporan role in explaining asse price behaviour. Barberis and Huang (27) and Mion and Vorkink (27) argue ha invesors look for idiosyncraic skewness, seeking asses wih loo-ype pay-offs. There is much empirical evidence suggesing ha marke skewness is ime varying, and ha i predics fuure reurns in boh he ime series (Kelly and Jiang, 214) and in he cross-secion (Harvey and Siddique, 2, and Ang, Hodrick, Xing and Zhang, 26). Boyer, Mion and Vorkink (21) and Conrad, Dimar and Ghysels (213) show ha high idiosyncraic skewness in individual socks oo is correlaed wih posiive reurns. Ghysels, Plazzi and Valkanov (216) show similar resuls for emerging marke indices. Bu here are wo serious problems in measuring hese momens a he long horizons (e.g. years) of ineres o asse pricing. Firs, he higher he momen, he more sensiive he esimae is o ouliers. Second, he longer he horizon, he smaller he number of independen observaions in any fixed daa sample. We show how hese problems can be miigaed by using informaion in shor horizon reurns o make esimaes of skewness and kurosis of long horizon reurns more precise. I is sandard pracice o use high frequency daa o esimae he second momen of long horizon reurns. Under he assumpion ha he price process is maringale, he annualized variance of reurns is independen of he sampling frequency and he realized variance compued from high frequency reurns is a good esimae of he variance of long horizon reurns. Bu his does no hold for higher momens - here is no necessary relaionship beween he higher momens of long and shor horizon reurns. If daily reurns are volaile, hen annual reurns are also volaile. Bu if daily reurns are highly skewed and i.i.d., hen annual reurns will show lile skew. Conversely, daily reurn disribuions can be symmeric, while annual reurns are skewed (e.g. in a Heson-ype model where volailiy is sochasic and shocks o volailiy are correlaed wih shocks o prices). Similar examples could be given for kurosis. The purpose of his paper is o demonsrae how o exploi he informaion in shor horizon reurns o esimae he skewness and kurosis of long horizon reurns. The only assumpion we make abou he price process is ha i is maringale, and ha he relevan momens exis. We - 2 -

3 prove ha he skewness of long horizon reurns can come from one of only wo sources: he skewness of shor horizon reurns; and he leverage affec, ha is he covariance beween lagged reurns and squared reurns. Similarly, he kurosis of long horizon reurns has jus hree sources: he kurosis of shor horizon reurns; he covariance beween cubed reurns and lagged reurns; and he covariance beween squared reurns and lagged squared reurns (which we refer o as he GARCH effec). When we ake hese heoreical resuls o he daa, we show ha he skewness of he US sock marke a long horizons is large and negaive and due almos enirely o he leverage effec. Kurosis in long horizon reurns is driven by he GARCH effec. Thus, he negaive skewness and he excess kurosis in annual sock marke reurns owe virually nohing o he skew and kurosis of daily reurns. To dae, he lieraure has used a variey of approaches o measure he higher momens of long horizon reurns. The mos sraighforward is o apply he sandard esimaors o hisoric reurns. Kim and Whie (24) show ha hese esimaors are subjec o large esimaion errors and advocae he use of robus esimaors such as hose developed by Bowley (192), which are based on he quaniles of he observed disribuion. 1 The aracion is ha quaniles can be esimaed wih much greaer precision han momens. This soluion is used in Conrad, Dimar and Ghysels (213) and he mehodology is furher developed in Ghysels, Plazzi and Valkanov (216). The weakness of he approach is ha i assumes ha he body of he disribuion, which is capured by he quaniles, is highly informaive abou he behaviour of he ails, which deermine he higher momens. Kelly and Jiang (214) follow an alernaive approach. They focus on he ails. They ge power no by aking a very long ime series, bu raher by exploiing he informaion in he crosssecion. They assume ha ail risk for individual socks is a combinaion of sable sock specific ail risk and ime-varying marke-wide ail risk. They can herefore exploi he exisence of a large number of socks o ge a much more precise esimae of marke-wide ail risk. The validiy of he inference depends no only on he assumed decomposiion of he ail componen, bu also on assumpions abou he dependence of reurns across socks. The opions marke is an aracive source of informaion abou momens. Whereas he underlying marke shows jus one realizaion of he price process, he opions marke reveals he enire implied densiy of reurns a any poin in ime. The echnology for exracing implied skewness and kurosis from opions prices is well-esablished (Bakshi, Kapadia and Madan, 23). The mehod can only be used on asses such as he major marke indices - ha suppor a liquid opions marke, and canno be used for managed porfolios. Bu here is a more fundamenal issue: implied measures reflec risk premia as well as objecive probabiliies. As demonsraed by Broadie, Chernov and Johannes (27), he wedge beween he objecive price process and he process as implied by opion prices (he so called risk neural process) can be very wide. 1 To esimae he skewness (kurosis) of a normally disribued random variable wih a sandard error of.1 requires a sample size of 6 (24). Even for monhly reurns, his would require 5 (2) years of reurns daa. If reurns are non-normal, he sandard errors are generally subsanially higher. Monhly reurns on he US marke over he las 5 years have a skew coefficien of -.98; he boosrapped sandard error is

4 We show by simulaion ha our measures of skewness and kurosis are indeed subsanially more powerful han sandard esimaors, reducing sandard errors on skewness by around 6% and on kurosis by around 3%. This is rue for all of he daa generaing processes ha we use in our simulaions. Focussing on skew esimaion, we show ha our mehod works prey much equally well regardless of how skewed reurns acually are and ha our esimaion echnique is subsanially more precise han a simple quanile-based skew esimaor. We apply our echnology o he US equiy marke using daa from he pas niney years. Our analysis suggess ha he skew coefficien of monhly reurns is around This skewness does no aenuae o any marked degree wih horizon. Our cenral esimae is ha he skewness of annual reurns is and of five year reurns is Thus, long-erm invesors should no hink ha he lef-ail evens ha are worrisome in daily or monhly reurns wash away when one aggregaes o an annual or longer horizon. To illusrae visually he meaning of skew coefficiens, Figure 1 shows he probabiliy densiy of annual year reurns on he assumpion ha log reurns are skew normal. The skew normal is a hree parameer family which includes he normal as a special case. I has been widely used in he lieraure o model skewed asse reurns (for example by Harvey e al, 21). The figure shows wo disribuions for reurns. In boh cases he mean reurn is zero, and he annualized volailiy is 18.5%. In he one disribuion, log reurns are assumed o be disribued normally (which gives zero skew in he way we define skew), and in he oher log reurns are skew normal, wih he coefficien of skewness se o -.7. Figure 1 Disribuion of annual reurns mean %, volailiy 18.5%, skewlognormal densiy skew = (lognormal) Skew % -5% -25% % 25% 5% 75% Log Reurn I would be nice o plo he graph o mach he skewness we observe in he daa, bu i is no possible o do so. The skew normal canno accommodae skewness coefficiens ha lie ouside he range (-1,+1). To illusrae annual reurns wih a skew of -1.32, we herefore use he binomial process. The binomial process can accommodae any level of skewness and has he added advanage ha i can be readily comprehended

5 The annual reurn akes he value u > 1 wih probabiliy p, and he value d < 1 wih probabiliy 1-p. We fix he mean reurn o be, and he annualized volailiy o be again 18.5%. In he absence of skew 2, u = exp(+18.5%) = 1.23, and d = exp(-18.5%) =.831, wih p = 45.4%. To keep he same firs and second momens wih a skew of -1.32, we need u = 1.99, and d =.75, wih p = 74.9%. To ge some sense of he economic imporance of hese levels of skew, consider he following quesion: how large does he equiy premium have o be for he represenaive agen o hold all his wealh in he marke porfolio? We follow he sandard approach o answering his quesion and assume he represenaive agen has power uiliy, wih consan relaive risk aversion coefficien g. We assume a horizon of 1 year. To persuade he invesor o inves 1% of heir wealh in he marke, he Euler condiion needs o be saisfied E é ( R ) - 1 R -g =, where R is he annual gross excess reurn. Expanding he expression o he hird order, his can be wrien as 1 3/2 E [ R]» gvar [ R] - g ( 3g -1) var [ R] skew [ R]. (1) 6 Wih volailiy of 18.5%, he equiy risk premium required o ge he invesor o inves fully in he marke depends on he coefficien of risk aversion of he invesor and he skew coefficien of he marke as shown below. Risk aversion Skew coefficien % 3.7% % 13.61% % 26.86% Wih low levels of risk aversion, he skew risk premium is small. Bu while he variance risk premium is proporional o g, he skew risk premium is proporional, roughly, o g 2 and, as he able shows, is significan a fairly moderae levels of risk aversion. The required variance risk premium, expressed as an annual rae, is independen of horizon since variance is linear wih horizon. By conras, equaion (1) shows ha he componen of he equiy premium aribuable o negaive skew aversion acually increases wih horizon unless he skew coefficien iself aenuaes wih he square roo of he horizon hence he 2 Our definiions of volailiy and skewness (discussed a lengh below) are non-sandard. If reurns are lognormal, our volailiy is equal o he sandard deviaion of log reurns, and our skewness is zero

6 imporance of undersanding he behaviour of skew wih horizon. We explore his relaionship in our empirical work. The res of he paper proceeds as follows. In Secion 1 we develop he heoreical relaionship beween low-frequency skewness and kurosis and heir high-frequency counerpars. In Secion 2 we demonsrae he power of he echnique hrough simulaion. Secion 3 provides an empirical applicaion o he US sock marke. Secion 4 concludes. 1. THE THEORY 1.1 Momens of price changes We work in a discree ime seing, ÎZ.. The asse has discouned price P ( he price ).We are concerned wih he disribuion of reurns from ime o +T. For breviy, we refer o he ime incremen as a day, and he long horizon as a monh, bu obviously nohing hangs on his. The erm kurosis is used specifically for excess kurosis. The problem we are ineresed in is [P]: Le : = { =...,,1,... } P P be a sricly posiive maringale process, whose associaed reurns process r, where r := P /P -1, is srongly saionary. The long horizon reurns process R is defined by R := P /P -T. How can one esimae higher momens of long horizon reurns R efficienly, assuming ha hese momens exis? Problem P is difficul because i deals wih reurns (raios) raher han wih price changes (differences). We herefore firs address a simpler problem, P*, and use he soluion as a guide o solving P. The simpler problem is: [P*]: Le P: { P...,,1,... } = = be a real-valued (no necessarily posiive) maringale process whose associaed difference process d, where d := P P -1, is srongly saionary. The long horizon difference process D is defined by D := P - P -T. How can one esimae higher momens of D efficienly, assuming ha hese momens exis? The soluion o P* is given by Proposiion 1 The volailiy, skewness and kurosis of monhly price changes is relaed o he disribuion of daily price changes in he following way - 6 -

7 [ D ] = [ d ] vol vol ; * 2 æ cov é 1, ö 1/2 skew [ ] skew [ ] 3 y - d - D = d + T ; 3/2 ç var[ ] è d ø * 3 * 2 æ cov é -1, cov é -1, ö -1 kur[ ] kur[ ] 4 y d = z d D d T ; 2 2 ç var[ ] var[ ] è d d ø (2) where T -1 * = å - -u u= y : P P T; and T -1 * = å - -u u= z : P P T. 2 (3) Proof: he full proof is in he Appendix. Proposiion 1 gives expressions for he volailiy (he square roo of he variance rae), he skewness and he excess kurosis of monhly price changes. The firs resul is familiar: he volailiy of price changes is he same wheher compued from monhly or daily daa. The second resul says ha skew a he monhly horizon has jus wo sources: daily skew and a erm we call leverage. Daily skew aenuaes wih horizon wih he square roo of ime. The leverage erm is proporional o he covariance beween squared price changes and he quaniy y*, which is equal o he difference beween he opening price on he day and he average price over he las monh. The final resul says ha he kurosis of monhly reurns has jus hree sources: daily kurosis aenuaing wih ime, he covariance beween cubed price changes and y*, and he covariance beween squared price changes and z*. z* is a measure of he average squared price change over he las monh. In order o demonsrae he logic underlying Proposiion 1 (and indeed he main resul in his paper, Proposiion 2) and also he role of he assumpions (maringale, sric saionariy), i is useful o review he proof of one par of he proposiion, ha concerning skewness. Sar wih an algebraic decomposiion of he hird power of he monhly price change T-1 T-1 T = u + u - T u + u - T u u= u= u= 2 å - 3å( ) - 3 å ( ) -. (4) D d P P d P P d Taking condiional expecaions of boh sides, he hird erm drops ou because of he maringale assumpion 3, so 3 If he price process were no maringale, here would be an addiional erm in he skew, he covariance beween price changes and pas volailiy. Bu here is reason o believe ha any such erm would be small, a leas in he - 7 -

8 T-1 T T D = å -T d-u + å -T ( P-u- 1 -P- T) d-u u= u= E é E é 3 E é. (5) Define T * -1 = -1- -u u= 1 å (6) y : P P T. y - is he difference beween oday s opening price and he T-day moving average 4. Using * 1 sric saionariy, he condiional expecaions can be replaced by uncondiional expecaions. Subsiuing * y in o (5) gives he following expression for he uncondiional hird momen 3 3 * 2 Eé D = TEé d + 3 TE é y-1d. (7) y - is mean zero, so he expecaion can be replaced by he covariance, giving * 1 A similar argumen shows ha ( ( -1 )) Eé D 3 3 3cov *, 2 = T E é d + y d. (8) 2 2 Eé D = TE é d. (9) The resul in proposiion 1 hen follows immediaely from he definiion of he skewness coefficien. 1.2 Momens of Reurns The objecive is o produce a resul akin o Proposiion 1, bu one ha applies o momens of reurns raher han o price changes. We now work wih daily reurns, r, = P P - 1 and monhly reurns, = ; R T P P we drop he argumen of R where i causes no confusion. -T The problem is inracable if we say wih he sandard definiions of momens. I is necessary o modify he definiion of momens. Define case of he equiy marke. As Bollerslev e al (213, p21) say: The mos sriking empirical regulariies o emerge from his burgeon lieraure are ha reurns are a bes weakly posiively relaed, and someimes even negaively relaed, o pas volailiies. 4 The aserisk is used o disinguish his variable from he corresponding variable in he problem P

9 ( L [ ] E é ) ( E [ ] E é ) ( 3 ) ( r) 3/2 [ r] ( 4 ) ( r) L 2 [ r] ( L ) L 2, 2, var r : = x r where x r : = 2 r-1-ln r ; ( E ) E 2, 2, var r : = x r where x r : = 2 rln r+ 1 -r ; E é ( 3 skew [ ]: x r = where x ) ( r) : = 6( ( r+ 1) ln r-2( r-1 )); L var E é ( 4 [ ] ) 2 kur : x r = - 3 where x ( r) : = 12( ( ln r) + 2( r+ 2) ln r-6( r-1 ));. var (1) x (2,L) approximaes he second power of log reurns, as does x (2,E). Similarly, x (3) and x (4) approximae he hird and fourh powers. This can be shown by doing a Taylor expansion, and is seen graphically in Figure 2. Modifying he definiions of momens in his way is no unprecedened. The Model Free Implied Variance is widely used by boh academics and praciioners. I is defined as Q ( 2, L [ ] = é ) MFIV R : =E x R, (11) where he Q superscrip denoes ha he expecaion is under he pricing measure. I also follows he definiion of realized variance in Bondarenko (214). A definiion of skewness similar o he above is seen in Neuberger (212). We also define volailiy of he reurn (vol[r]) as he square roo of he variance rae. Figure 2: approximaing he momens of reurns Wih hese definiions, we can now sae he main heoreical resul of his paper Proposiion 2 If P is a srongly saionary maringale process, he volailiy, skewness and kurosis of monhly reurns (as defined in equaion (1)) is relaed o he disribuion of daily reurns as follows - 9 -

10 L [ R ] = T [ r] L var var ; æ [ ] [ ] [ ] [ ] ( 2, E ) cov é y, x r -1-1/2 skew R = ç skew r + 3 T ; L 3/2 ç var [ r ] è æ cov éy, x r cov éz, x r ö kur R kur r 4 6 T ; ö ( L ) 3 2, = ç + + L 2 L 2 ç var [ r] var [ r] è ø ø (12) where T -1 å u= T -1 ( ) y : = R u -1 T and å ( ( )) z : = 2 R u -1-ln R u T. u= (13) Proof: he proof is similar o ha of Proposiion 1; deails in he Appendix. Proposiion 2 is very similar o Proposiion 1. I can be seen ha The volailiy of monhly reurns is idenical o he volailiy of daily reurns. The skew in daily reurns generaes a much smaller ( 1 T ) skew in monhly reurns. If monhly reurns have significan skew, i mus be hrough he leverage effec, he correlaion beween volailiy and pas reurns. Pas reurns are measured by y, which is he ne reurn relaive o he one monh moving average 5. Kurosis in daily reurns generaes a much smaller (1/T) kurosis in monhly reurns. If annual reurns are significanly lepokuric, i is for one of wo reasons: - because daily skew is correlaed wih pas reurns (as measured again by y); - or because of a GARCH effec whereby curren variance is correlaed wih pas variance. Pas variance is measured by z, which is a funcion of he average realized variance over horizons of up o one monh, again wih more recen experience having more weigh. The resuls are quie general; here is no presumpion abou any funcional form for he sochasic process driving he price. For example, in a Meron (1976) jump-diffusion model, he asymmeric jump creaes skewness and kurosis in high frequency reurns. The absence of any covariaion beween volailiy and lagged reurns and lagged squared reurns (volailiy is consan) ensures ha here is no leverage or GARCH effec, so skewness and kurosis aenuae rapidly wih he horizon. In a Heson (1993) model here is no skewness or condiional kurosis in shor horizon reurns, bu here is skewness in longer horizon reurns because correlaion beween innovaions in reurns and innovaions in volailiy, coupled wih he persisence of volailiy, creaes a correlaion beween volailiy and lagged reurns. The persisence of 5 The moving average in his case is he rolling harmonic mean

11 volailiy shocks also generaes kurosis. GARCH processes also generae kurosis in long horizon reurns hrough he persisence of volailiy shocks. To generae skewness in long horizon reurns in a model from he GARCH family, one mus addiionally have volailiy reacing asymmerically o posiive and negaive reurn shocks i.e. as wih Heson, a correlaion beween volailiy and lagged reurns. 1.3 Relaion o Opion-based realized skewness Proposiion 2 shows ha he skewness of long horizon reurns is relaed o he leverage effec he covariance beween insananeous realized variance and lagged reurns. Neuberger (212) relaes skewness o he covariance beween reurns and conemporaneous changes in opion implied variance. In his secion we show how hese wo resuls are relaed. The relaionship can be skeched ou informally. Skewness comes from leverage. Leverage a he monhly horizon is he covariance beween oday s realized variance and reurns over he las monh. By rearranging erms, i can be seen ha his is he same as he covariance beween oday s reurn and realized variance over he nex monh. If here exiss a sufficienly rich opions marke, we can observe he corresponding one monh implied variance. In he absence of risk premia in he opions marke, he implied one monh variance a any ime is he expecaion of he realized variance over he nex monh. We can hen obain a much more precise esimae of he leverage effec by looking no a he covariance beween daily reurns and realized variance over he nex monh, bu a he covariance beween daily reurns and conemporaneous daily changes in monhly implied variance. We can sae he argumen more precisely. Proposiion 2 shows he relaion beween he hird momen of long horizon reurns and high frequency reurns ( E { ( ) -1 )} Eé 3 é 3 x R = + 3cov, 2, T E x r y x r. (14) By reordering he erms, he leverage erm can be wrien as ( 2, E ( y ) - x ( r )) 1 = r - w+ T- 1 cov, cov 1, T -1 1 where w = R u-1 x r. å ( 2, E ) + u- T + u+ 1-T T u= 1 (15) (We ake advanage of he fac ha boh y and r-1 are mean zero.) The variable w is a measure of average fuure realized variance. In esimaing he skewness of long horizon reurns, i makes lile difference wheher one esimaes he righ-hand side of equaion (15) or he lef hand side. Bu suppose now ha we can observe he expecaion of fuure realized variance. Then here is poenial for considerable efficiency gains. r -1 is known a ime ; i is also mean zero. So ( r - w+ T- 1) = ( r - [ w+ T-1] -h- 1) cov 1, cov 1, E (16)

12 where h is any variable known a ime. Suppose we choose h -1 so ha i close o E [ w ], T-1 we can he esimae he righ hand side of equaion (16) much more precisely han he lef hand side since he change in he expecaion of fuure variance over he day is likely o have a far lower sandard deviaion han he realized variance iself. To ensure ha expecaions if fuure variance are observable, we need o make wo furher assumpions 1. ha he opions marke is complee, so ha in paricular we can replicae (and hence price) he so-called enropy conrac ha pays x (2,E) (P +T /P +1 ); 2. ha he price of opions, as well as of he underlying, are maringale (ie here is no volailiy or jump risk premium). The significance of he enropy conrac is wo-fold: he price of he enropy conrac (like he like conrac) a incepion is equal o is Black-Scholes implied variance. Second, he conrac, when dela-hedged, generaes he cash flow Tw +T. Denoe he price of he enropy conrac a ime +1 by q +1 hen he absence of risk premia means ha = E [ ] have he resul ha q Tw We herefore T. 1 cov( r - 1, w+ T-1) = cov( r -1, q - q- 1). (17) T Assuming complee markes and he absence of variance risk premia, he leverage effec can be esimaed from he covariance beween changes in he implied variance of he enropy conrac and conemporaneous reurns. 1.4 The erm srucure of momens in coninuous ime So far, we have worked in a discree ime seing. Given ha daa is discree, his makes i easy o implemen our resuls in pracice. Bu here are advanages in going o coninuous ime. The resuls are simpler, paricularly if he price process is coninuous. We can also derive a simple useful es for esimaing how reurn momens change wih horizon. We have assumed ha P is a posiive maringale, wih a sricly saionary reurns process wih well-defined momens. We now make he furher assumpion ha he process is a diffusion. We assume in paricular ha P can be represened by a sochasic differenial equaion dp P = v dz, (18) where v is predicable, and z is a sandard Brownian process. We reain he definiions of variance, skewness and kurosis ha we used in he discree ime seing. The counerpar o Proposiion 2 in a diffusion seing is hen Proposiion

13 If P is a srongly saionary maringale diffusion, he volailiy, skewness and kurosis of T-period reurns is relaed o he volailiy of insananeous reurns v as follows where = [ ] var L T TE v ; ( T) ( T) yt 3/2 [ ] zt 2 [ ] cov é, v Ev -1/2 skew = 3 T ; cov é, v Ev -1 kur = 6 T ; ( ) T y T : = R u -1 du T and Proof: he proof is in he Appendix. ò ò u= T u= ( 2, L) ( ) z T : = x R u du T. The mos significan difference beween Proposiions 2 and 3 is he dropping of he daily skewness from he period skewness, and he dropping of he daily kurosis and he cube effec from he period kurosis. Wih he diffusion assumpion, he higher order momens of high frequency reurns vanish. The disincion beween enropy variance and log variance vanishes in he limi. The definiions of y (he lagged reurn) and z (he lagged realized variance) are he naural limis of heir discree ime counerpars. We show in our empirical work ha, a leas so far as he equiy marke is concerned, jumps do no play any significan role in he momens of long horizon reurns. We now see ha, in he absence of jumps, all skewness in period reurns derives from he leverage effec, and all kurosis comes from he GARCH effec. The erm srucure of momens is a maer of considerable imporance; if skewness and kurosis end o zero a long horizons, hen hese higher momens are likely o be of limied significance for longer erm invesors. Proposiion 3 enables us o es his direcly. Corollary o Proposiion 3 Given wo horizons, T 1 and T 2 : éæ T ö 2 skew ( T2) > skew ( T1) if and only if cov ê ç y( T2) - y( T1), v ; T ú > êè 1 ø ú éæ T ö 2 kur ( T2) > kur ( T1) if and only if cov êç z( T2) - z( T1), vú >. è T1 ø (19)

14 We will use his corollary o es how he skewness of sock marke reurns changes wih horizon. 2. SIMULATION RESULTS 2.1 Resuls for variance, skewness and kurosis We now evaluae he performance of our esimaors of higher momens hrough a series of simulaion experimens. We compare our esimaors boh wih sandard mehods and wih a quanile-based approach. Reurns are simulaed from hree differen models; a geomeric Brownian moion (GBM), a Heson model and an EGARCH specificaion. For each model we simulae 1, pahs for daily reurns, each of lengh 5 (i.e. roughly 2 years). The parameers for each model are derived from fiing hem o spans of daily US sock marke reurns. For he GBM and he Heson model, he parameers are aken from Eraker (24). Those esimaions use daily S&P-5 reurns from January 2 nd 198 o December 31 s The EGARCH parameers are obained from our own fi of such a model o daily valueweighed CRSP US sock reurns covering he period from January 2 nd 198 o he end of December 215. Given he parameers for a paricular daa generaing process, he objecs ha we wish o measure are he sandard deviaion, skewness and kurosis of 25-day (i.e. roughly monhly) reurns, where hese are as defined in equaion (9). We use hree esimaion echniques for each momen. Firs, we consruc he sample momens of non-overlapping 25-day reurns (and we refer o hese subsequenly as Monhly esimaes). Second, we measure he sample momens using overlapping 25-day reurns (referred o laer as Overlapping esimaes). 6 Finally, we implemen he esimaors from Proposiion 2 (which we label NP ). Resuls from hese simulaions are given in Table 1. Panel A shows he simulaion resuls when reurns are generaed by a GBM, Panel B gives simulaion resuls for he Heson model and Panel C shows he EGARCH resuls. Each able gives saisics on he disribuion of esimaes from all hree esimaion echniques and for each of he hree momens from across he 1, sample pahs. In he discussion below, we focus on skewness and kurosis esimaes. Under he assumpion ha daily reurns follow a GBM, 25-day skewness and kurosis should boh be zero. Table 1 confirms ha, on average, his is rue for all hree esimaion echniques. More imporanly, he dispersion of he esimaes for he NP mehod are grealy reduced relaive o hose from monhly and overlapping esimaors. The sandard deviaions of esimaes from our mehod are beween 7% and 9% smaller han hose from he alernaives. The improvemen in esimaion accuracy for he NP mehod is mos sriking for skewness, bu only 6 So for each simulaed reurn pah of 5, daa poins, he Monhly esimaor uses 2 non-overlapping 25-day reurns and he Overlapping esimaor uses 4,976 overlapping 25-day reurns

15 slighly less impressive for kurosis. Overall, for boh skewness and kurosis, he use of daily daa o improve monhly momen esimaes provides a subsanial improvemen in accuracy. Table 1: simulaion resuls for NP and sandard esimaors Panel A: Geomeric Brownian Moion Sandard deviaion NP Monhly Overlap Mean STDEV Coefficien of Skewness NP Monhly Overlap Mean STDEV Excess Kurosis NP Monhly Overlap Mean STDEV Panel B: Heson model Sandard deviaion NP Monhly Overlap Mean STDEV Coefficien of Skewness NP Monhly Overlap Mean STDEV Excess Kurosis NP Monhly Overlap Mean STDEV Panel C: EGARCH model Sandard deviaion NP Monhly Overlap Mean STDEV Coefficien of Skewness NP Monhly Overlap Mean STDEV Excess Kurosis NP Monhly Overlap Mean STDEV

16 For he Heson model, we expec excess kurosis (as he variance of daily reurns is changing hrough ime) and negaive skew (as he innovaions o he variance and he reurn are negaively correlaed). All hree esimaion echniques pick hese feaures up, bu again use of he NP mehod resuls in a significan reducion in he spread of esimaion errors. For skewness, he sandard deviaion of esimaes for he new mehod is around 7% smaller han hose of he monhly or overlapping mehods, while for kurosis improvemens are beween 4% and 5%. Finally, he esimaed EGARCH model also generaes negaive skew and excess kurosis and hese appear in all esimaion mehods. Panel C shows ha he NP esimaion echnique dominaes in erms of accuracy under his model also bu ha he improvemens i delivers are less pronounced. The sandard deviaion of skewness esimaes is around 6% smaller for he new mehod bu he sandard deviaion of he kurosis esimaes drops by only 3 o 4%. Overall, regardless of which model we choose or which momen one focusses on, use of he esimaors described in Proposiion 2 leads o much more precise esimaes of monhly reurn momens. Improvemens are greaer for skewness esimaes han hey are for kurosis and are larger for he GBM and Heson models han hey are for he EGARCH specificaion. Bu in almos all cases, use of he NP momen esimaors leads o he dispersion of esimaed coefficiens being reduced by 5% or more. 2.2 Simulaions of he NP esimaor s performance using inra-day daa Given he improvemens in esimaion precision ha are available from using daily daa o esimae momens of monhly daa, i is naural o ask how he use of inra-day daa migh furher improve accuracy. From he resuls in papers such as Andersen, Bollerslev, Diebold and Labys (23) we know ha if we wish o esimae daily reurn variances, he use of finely sampled inra-day daa is valuable. Here we explore an analogous issue bu for esimaion of higher momens of lower frequency reurns. Thus we adjus our simulaions from he previous secion o generae daa sampled a N D equally spaced inervals across 1 day. We assume ha he daa generaing process is he same across he day (hus ignoring issues like overnigh periods). In our simulaions, we vary N D beween 1 (daily daa) and 16. We sar off wih a benchmark case where we assume ha reurns are generaed from a Geomeric Brownian Moion and hen move o a Heson model wih he same (daily) parameers as in he previous secion. The simulaion resuls for he NP esimaor only and for boh daa generaing processes are given in Table 2. The resuls in Table 2 demonsrae exacly wha one would expec in he GBM case. The use of inra-day daa increases he precision of he NP esimaors of monhly momens wih he raio of he sandard deviaion of he daily esimaor o ha of he inra-day esimaors equal o roughly N ". Thus, for example, sampling daa 16 imes a day reduces he sandard deviaion of he disribuion of momen esimaes by a facor of four relaive o he daily reurns case

17 Table 2: Inraday simulaion resuls Panel A: Geomeric Brownian Moion N D Sandard deviaion Mean STDEV Coefficien of Skewness Mean STDEV Excess Kurosis Mean STDEV Panel B: Heson model N D Sandard deviaion Mean STDEV Coefficien of Skewness Mean STDEV Excess Kurosis Mean STDEV Resuls for he Heson model are given in Panel B. Here, he inra-day daa deliver no gains in he precision wih which one can esimae monhly sandard deviaions. If daa is sampled 16 imes per day, hen he precision wih which skewness is esimaed improves by abou 2% and he corresponding figure for excess kurosis is 1%. These resuls are linked o he persisence in volailiy ha our Heson model displays. The (daily) mean reversion coefficien for he reurn variance is.17 and herefore volailiy is close o a random walk. Sampling such a persisen process more finely han daily does no help maerially in esimaing, for example, he covariance beween variance and lagged reurns ha is imporan in measuring skewness and so our esimaors derive smaller benefi from he use of inra-day daa in his case. 2.3 Performance of he NP skew esimaor across skew levels In order o invesigae how he performance of our skew esimaor changes wih he level of skew in reurns, we ake a Heson model and vary he correlaion beween reurn and variance innovaions beween -.9 and +.9 (wih he former giving large negaive skewness and he laer generaing large posiive skewness). All oher parameers are se a he values from Eraker

18 (24). For each parameer se, our simulaion conains 1, replicaions of 1, daily reurns and from hese we esimae 25-day skew. The resuls are summarised in Figure 3. The x-axis of his figure shows he correlaion parameer from he Heson model. Agains each correlaion parameer, we plo he average esimaed skewness from our 1, runs, as well as he 5 h and 95 h perceniles of he disribuion of skew esimaes. Also ploed on Figure 3 is he heoreical value of he coefficien ha one should obain from he Heson model a each parameer value. The Figure demonsraes ha he NP esimaor does an excellen job of racking skewness, on average, across he range of parameers. There is a sligh endency for he esimaor o be biased owards zero when he heoreical skew is large, hough, wih he larges bias around.1 when heoreical skewness is a a value of.9. The range beween 5 h and 95 h perceniles is fairly sable a a value of around.65. The bias in he esimaion of he coefficien of skewness arises due o he fac ha i is a raio of he esimaed hird momen o he cube of he esimaed sandard deviaion. Esimaes of boh of hese momen measures using he NP mehod are unbiased, bu esimaion errors in second and hird momens are correlaed and i is his ha causes he bias in he esimaed skewness. Figure 3: Theoreical and esimaed skew coefficien versus correlaion parameer: mean, 5 h and 95 h perceniles. 1.8 Theory NP Mean.6.4 Skew Coefficien Correlaion beween vol and reurn innovaions Obviously, as one increases he quaniy of high-frequency daa poins used o consruc low frequency skew, esimaion becomes more accurae. If one runs simulaions of ime-series of lengh 1,, raher han 1,, precision improves grealy, wih he bias dropping o close o zero and he range beween he 5 h and 95 h perceniles falling o around Comparison of NP and quanile-based skew measures Ghysels, Plazzi and Valkanov (216) (hereafer GPV) propose a skewness esimaor based on he quaniles of he reurn disribuion in heir recen work on inernaional asse allocaion. We

19 now compare he performance of our esimaor and heir preferred esimaor based on daa simulaed from a Heson model. We use exacly he same seup as in Secion 3.2.1, excep now we esimae NP skewness and GPV s quanile skewness for each simulaed se of daa. The GPV skew esimaor is as follows; 6,.. { q ( r * q,.. r * q,.. r * q /( r * }dα,,.., q ( r * q /( r * dα,.. q ( z dα,,.., q ( 5 z dα where r * are reurns measured a he frequency of ineres (e.g. monhly), q ( x is he ah quanile of he disribuion of x and he q ( z are he quaniles of he sandard Normal disribuion. In heir implemenaion, GPV approximae he inegrals in he firs raio by aggregaing across he following se of quaniles: [.99,.975,.95,.9,.85,.8,.75]. The GPV esimaor esimaes skew by looking direcly a he symmery (or lack of i) of a and 1-a quaniles wih respec o he median. This is capured by he numeraor of he firs raio in he equaion while he oher erms are jus scaling facors. For each simulaion run, we apply he GPV esimaor o overlapping 25-day reurns. I is worh re-ieraing ha he GPV esimaor and he esimaor proposed here are designed o arge slighly differen measures of skewness. GPV propose an esimaor of he radiional skewness coefficien whereas our esimaor is of he modified skew coefficien as defined in equaion (9). However, differences in hese arges are minor. Figure 4: esimaed skew coefficiens from he NP and quanile mehods: means and 5 h and 95 h perceniles. Noes: solid lines give he mean value of he esimaed coefficiens and he dashed lines give he 5 h and 95 h perceniles of he disribuion of esimaed coefficiens. Lines marked wih * symbols are for he NP esimaors and hose marked wih + symbols are he quanile-based esimaors

20 The resuls from our comparison are displayed in Figure 4. As before, he x-axis values are he Heson correlaion parameers and skewness is on he y-axis, and again we run 1 simulaions of 1 daily reurns from which we esimae 25 day skewness. The resuls are encouraging. The NP and he GPV mean esimaes are very close ogeher, bu he precision of he NP esimaor is much greaer. The 5 h -95 h percenile range of he GPV esimaes average around 1.3 i.e. around wice as large as ha of he NP esimaor. Thus, overall, our esimaion echnique works well. I is more precise han compeing esimaors and is precision shows lile variaion as he parameers of he chosen model change. 3. Applicaion o he US Equiy Marke In his secion, we apply our echnology o he US sock marke. Unless saed oherwise, he reurns used in his analysis run from and were rerieved from Ken French s daa library. Firs, we documen how momens of annual and monhly reurns have evolved over he las niney years, and he imporance of he componens of each monhly momen as described in Proposiion 2. We hen focus on skew, and characerize he erm srucure of skewness and he relaionship beween our skew measure and hose derived from opions markes Before proceeding, i is worh poining ou a couple of implemenaion issues. Firs, for he sake of simpliciy, our heory has focussed on uncondiional momens. We can, however, adap he heory o deal wih condiional momens wih lile difficuly. The second issue o address is he mehod of esimaing covariances. Our skew esimaor requires us o esimae erms such as cov[y, v ] over some period [, S]. The obvious esimaor is he sample covariance Bu his is biased. Specifically 1 S å å Q : = y - y v -v S- 1 where y: = y S. = 1 = 1 S S- j E [ Q1] = c - å cj + c- j where cj : = cov év, c + j. S S j= 1-1 In our conex, y is a muli-period reurn variable ha is persisen by consrucion and v is an insananeous variance which is also persisen. The cross correlaions beween he series are subsanial, so he bias is significan. The bias of Q 1 arises from he fac ha he means of y and v are esimaed in-sample. We can avoid he bias by esimaing he means ex ane. In our empirical work, we use he maringale assumpion o se he esimaed mean of y o, while we se he esimaed mean of v o is sample mean in he period prior o he sample period (empirically we use a 5-year period before he sar of he sample). Denoing his mean by v, he covariance esimaor we use is S - 2 -

21 3.1 Momens of annual reurns S å Q : = y v -v S. 2 = 1 We begin by looking a he ime-variaion in annual (by which we mean 25 day) reurn momens across our daa se. The annual momens are esimaed using a rolling window of 125 days of daa (and hus are auocorrelaed by consrucion). The op lef panel of Figure 5 shows he (log) marke level across our sample. Is esimaed volailiy is shown in he op righ panel (along wih a 95% confidence inerval). Over he 9 or so years ha our daa cover, US sock marke volailiy is iniially high (around he Grea Depression) and also high a he end (around he 28 Financial Crisis). In beween volailiy is smaller, puncuaed by infrequen upward spikes. There was, for example, subsanial marke volailiy around he oil price shocks of he early 197s and he sock marke crash of Figure 5: 25-day momens of US sock marke reurns The NP skew daa in he boom lef panel indicaes ha annual sock marke skew is almos always significanly negaive (wih a mean of -1.5) over our 9 years of daa. The only excepions o his are a couple of isolaed years in he mid-198s and lae 199s when skew is significanly posiive, alhough very small in magniude. Times of paricularly severe negaive skewness include he Grea Depression (wih skew below -6) and he mid-199s (wih skew around -4) and skew has also reached a level of close o -3.5 in he mos recen par of our daa. Ineresingly, he 28 Financial Crisis does no appear o be associaed wih remendously

22 large negaive skew. Overall, here is very clear evidence of consisenly large, significan and negaive skew in annual US sock marke reurns. The boom righ panel of Figure 5 shows esimaes of annual excess reurn kurosis esimaed from daily daa. As expeced, excess kurosis is posiive on average, wih a mean of 2.8, and is almos never negaive. Excess kurosis is larger in he firs half of he 2 h cenury han i is in he second half, bu in he second half of he cenury i is a is highes level in he mos recen par of he daa. Overall, our esimaes of higher momens sugges ha long-erm invesors (i.e. hose wih an annual ime horizon) should no assume ha he negaive skew and fa ails we see in daily reurns wash away as he reurn measuremen horizon is exended. 3.2 Momens of monhly reurns from non-overlapping years of daa While annual momens are ineresing from an invesmen risk perspecive, previous auhors have focussed on monhly measures of higher momens (e.g. ail risk and quanile-based skew). Thus in his secion we presen he same informaion as in Secion 3.1 bu for 25-day momens. We ake each year of he sample separaely and using daa from wihin ha year compue monhly volailiy, skew and excess kurosis. Time-series plos of he hree momens esimaed using he NP mehod plus he quanile based skew measure of GPV are presened in Figure 6. 9 While Figure 6 leads o broadly he same resuls as Figure 5 (i.e. he US sock marke reurn is on average very negaively skewed and displays excess kurosis), as one would expec monhly skew and monhly kurosis are much more volaile han heir annual counerpars. A negaive correlaion beween monhly skew and excess kurosis becomes clearer, however. When monhly skew is large and negaive, monhly kurosis ends o be large and posiive. The quanile-based skew measure, in he boom righ panel, is also negaive on average (wih a mean of -.25), bu i is less easy o see a paern in he monhly skews here han i is in he NP esimaes. The quanile skew and NP skew measures are posiively correlaed, wih a correlaion coefficien of The componens of 25-day skew and kurosis As Proposiion 2 makes clear, skew in long horizon reurns is driven by skew in high-frequency reurns and by he leverage effec. Long-horizon kurosis has hree possible sources: kurosis in high-frequency reurns, covariaion beween lagged reurns and curren cubed reurns (which we refer o as he Cube componen) and covariaion beween curren and lagged squared reurns (which we will call he GARCH effec). 9 The quanile based skew measure is esimaed from he se of overlapping 25 day reurns ha can be consruced from he seleced year of daily reurns

23 Figure 6: ime-variaion in monhly momen esimaes for US sock marke Figure 7 shows he ime-series variaion in he wo monhly skew componens for he years 1935 o 215, wih he wo plos having idenical verical scales and wih 95% confidence bands ploed around each esimae. Clearly he leverage effec generaes boh he level and he variaion in skewness. The influence of skew in daily reurns is negligible and almos never saisically differen from zero. Thus, boh he average level of 25-day skew and is variaion hrough ime are aribuable o covariaion beween lagged reurns and curren squared reurns. This covariance is almos always negaive, usually significanly so and is almos never significanly greaer han zero. Figure 8 shows a similar decomposiion of 25-day kurosis ino is hree componens (i.e. daily kurosis, he cube erm and he GARCH erm). As wih skew, he conribuion of he daily momen is close o zero and is ime-series variaion is small. The Cube erm is also close o zero on average and so almos all he significan posiive excess kurosis apparen in he daa, as well as he ime-variaion in ha excess kurosis, comes from he GARCH componen. Thus, we have shown here ha he higher momens of low frequency reurns and daily reurns bear lile relaion o one anoher. Low-frequency skewness and kurosis are driven by leverage and GARCH effecs respecively raher han by jumps in or he momens of daily daa. This observaion is of considerable pracical imporance as, for example, researchers ofen use momens of daily daa o proxy he ail risks faced by invesors who (presumably) have

24 relaively long run invesmen horizons. Our resuls show ha hese proxies are largely irrelevan o he long-run invesor. Figure 7: ime variaion in componens of monhly skew coefficien

25 Figure 8: ime variaion in componens of monhly excess kurosis

26 3.3 Esimaes of monhly skewness using inra-day daa We now focus our aenion on he esimaion of monhly reurn sandard deviaions and monhly skewness. We ask wheher he use of inra-day daa maerially changes he esimaes of monhly skew ha we have obained from daily daa. To his end we have colleced S&P- 5 ETF reurns wih a 1 minue sampling frequency covering he period beween he beginning of 24 and he end of 216. Every 25 days hrough his period, we esimae 25-day reurn sandard deviaions, hird momens and coefficiens of skewness using 25 days of daily reurns, half-hourly reurns and 1 minuely reurns, respecively. Table 3 gives he resuls. The key resul from his able is ha, in all cases, using daily daa leads o monhly momen esimaes ha are exremely highly correlaed wih hose from 1 minue or 3 minue daa. The esimaes of sandard deviaions and hird momens from inra-day daa always have a correlaion wih he esimaes using daily daa ha is larger han.99. The coefficiens of skewness measured using inra-day and daily daa are somewha less highly correlaed, bu he number is sill above.9, and he average coefficien of skewness esimaed from inra-day daa is more negaive han ha from daily daa. Table 3: esimaion US sock marke skewness wih inra-day daa Sandard Deviaion 1min 3min Daily Mean STDEV Corr(Daily) Third momen 1min 3min Daily Mean STDEV Corr(Daily) Coefficien of skewness 1min 3min Daily Mean STDEV Corr(Daily) Thus, while he simulaions show ha he use of inra-day daa can lead o improvemens in esimaion precision, in our empirical work i seems ha he addiional value from collecing high-frequency daa is small. Of course hese resuls are based on an examinaion of monhly reurn momens. I is possible ha inra-day daa could be much more informaive for a researcher/invesor ineresed in, for example, weekly momens

27 3.4 The erm srucure of skewness Our analysis hus far confirms he exisence of significan negaive skew in monhly and annual US sock marke reurns. I is reasonable o ask how measured skewness varies across a range of possible horizons, from monhly o muli-year reurn skew. Via such analysis one can ask, for example, wheher invesors wih a 5 year invesmen horizon need o worry abou skewness and begin o commen upon he compensaion hey migh demand for holding porfolios wih skewed reurn series. As Proposiion 3 and is corollary make clear, he variaion in skewness wih horizon is driven by cov y T, v where T is horizon. In Table 4 we presen esimaes of skewness for T ranging from 25 days (roughly 1 monh) o 125 days (roughly 5 years). We presen hese numbers for he full sample of daa and hen separaely for daa up o he end of 197 and daa afer 197. Alongside each esimae of skewness we give a -es of he null hypohesis ha he skew a ha horizon is significanly differen from 25-day (i.e. annual) skewness. This -es is buil using he resuls in he corollary o Proposiion 3. Figure 9 plos he skew erm srucures (for he full sample and he wo subsamples respecively). Table 4 Full sample Up o 197 Afer 197 Horizon Skew -es Skew -es Skew -es Looking firs a he erm srucure for he full sample, i is remarkably fla across horizons. Monhly skew is roughly -1.3 and 5-year skew is jus above Saisically, no esimae of skew a any horizon is differen from annual skew. Thus, worryingly for long erm invesors and risk managers, here is no sign ha negaive skewness disappears a long horizons. The index reurns from he early par of he sample show a differen paern. Skew ends o become more negaive wih horizon, and very significanly so. In his subsample, 1-year skew is close o he figure from he full sample (-1.12 versus -1.24) bu 5 year skew is more han 5% larger han is full sample counerpar (-1.8 versus -1.2)