Asymmetric Jump Beta Estimation with Implications for Portfolio Risk Management

Transcription

1 Asymmetrc Jump Beta Estmaton wth Implcatons for Portfolo Rsk Management Ths verson: Monday 15 th February, 2016 (PRELIMINARY VERSION. PLEASE DO NOT CIRCULATE WITHOUT PERMISSION) Vtal Alexeev a,b, Wenyng Yao a, Govann Urga c,d a Tasmanan School of Busness and Economcs, Unversty of Tasmana, Hobart, Tasmana 7001, Australa b Department of Economcs and Fnance, Unversty of Guelph, 50 Stone Rd E, Guelph, ON N1G 2W1, Canada c Centre for Econometrc Analyss, Faculty of Fnance, Cass Busness School, 106 Bunhll Row, EC1Y 8TZ, London, UK d Department of Management, Economcs and Quanttatve Methods, Unversty of Bergamo, Bergamo, 24127, Italy Abstract In ths paper we study jump dependence of two processes usng hgh-frequency data concentratng only on segments around a few outlyng observatons that are nformatve for the jump nference. Assumng that nvestors care dfferently about downsde losses as opposed to upsde gans, we estmate jump senstvtes for the negatve and postve market shfts. We nvestgate the mplcatons of the dfference n negatve and postve senstvtes to market jumps for portfolo rsk management by contrastng the results for ndvdual stocks wth the results for portfolos wth varyng number of holdngs. In the context of a portfolo, we nvestgate to what extend the downsde and upsde jump rsks can be dversfed away. Ths can have a drect mpact on the prcng of jump rsks and subsequently, nvestors decson-makng. Varyng the jump dentfcaton threshold, we show that the asymmetry s more promnent for more extreme events and that the number of holdngs requred to dversfy portfolos senstvtes to negatve jumps s hgher than that requred for postve jump dversfcaton. We found that gnorng the asymmetry n senstvtes to negatve versus postve market jumps may result n under-dversfcaton of portfolos and ncreased exposure to extreme negatve market shfts. Keywords: Systematc rsk, jumps, hgh frequency, downsde beta JEL: C58, G11, C61

2 1. Introducton In ths paper, we contrbute to the lterature on portfolo dversfcaton by evaluatng the mpact of extreme market shfts on equty portfolos. An mportant feature explored n our study s the asymmetry n portfolos behavour durng extreme negatve market downturns versus extreme postve uprses. In studyng jump dependence of two processes, we use hghfrequency observatons focusng on segments of data on the frnges of return dstrbutons. Thus, we only consder a few outlyng observatons that, at the tme, are nformatve for the jump nference. In partcular, we study the relatonshp between jumps of a process for a portfolo of assets and an aggregate market factor, and we analyse the co-movement of the jumps n these two processes. Gven the predomnance of factor models n asset prcng applcatons, we focus on a lnear relatonshp between the jumps and portfolo of assets and we assess ts senstvty to jumps n the market. We fnd that gnorng the asymmetry n senstvtes to negatve versus postve market jumps results n under-dversfcaton of portfolos and ncreases exposure to extreme negatve market shfts. We show that nvestors care dfferently about extreme downsde losses as opposed to extreme upsde gans demandng addtonal compensaton for holdng stocks wth hgh senstvtes to these movements. Snce Retz (1988), researchers have modeled the possblty of rare dsasters, such as economc depressons or wars, to resolve the equty premum puzzle and related puzzles (e.g. Barro, 2006; Gabax, 2012). Retz (1988) asserts that nablty of asset prcng models to explan hgh equty rsk prema s due to not capturng the effects of possble market crashes. It has been shown that large rsk prema can be obtaned n equlbrum when the representatve nvestor treats jump and dffusve rsks dfferently. Barro (2006) shows that rare economc dsasters have the potental to explan the hgh equty premum. Bates (2008) consders nvestors who are both rsk and crash averse. Jump and dffusve rsks are both prced even n the absence of crash averson, but ntroducng crash averson allows for greater dvergence between the two rsk prema. An mportant feature of the model n Bates (2008) s a representatve nvestor who treats jump and dffusve rsks dfferently, whch formalzes the ntuton that nvestors can treat extreme events dfferently than they treat more common and frequent ones. Could t be that the markets treat rare events somewhat dfferently from common, more frequent events? In practce, bearng non-dversfable jump rsk s sgnfcantly rewarded. Ths s evdent, for example, from the expensveness of short-maturty optons wrtten on the market ndex wth strkes that are far from ts current level. Bates (2008) proposes crash averson to explan the observed tendency of stock ndex optons to overpredct realzed volatlty. Guo et al. (2015) show theoretcally and emprcally that jump rsk s also an mportant determnant of condtonal equty premum even when controlled for commonly used stock market return predctors. Lu et al. (2005) study the asset prcng mplcaton of mprecse knowledge about rare 2

3 events. The equlbrum equty premum has three components: the dffusve- and jump-rsk prema, both drven by rsk averson; and the "rare-event premum", drven exclusvely by uncertanty averson. In fnancal markets, we see daly fluctuatons and rare events of extreme magntudes. In dealng wth the frst type of rsks, one mght have reasonable fath n the model bult by fnancal economsts. For the second type of rsks, however, one cannot help but feel a tremendous amount of uncertanty about the model. And f market partcpants are uncertanty averse, then the uncertanty about rare events wll eventually fnd ts way nto fnancal prces n the form of a premum. Lu et al. (2005) examne the equlbrum when stock market jumps can occur and nvestors are both rsk averse and averse to model uncertanty wth respect to jumps; they obtan smlar prcng mplcatons for jump and dffusve rsk. Whle Santa-Clara and Yan (2010) study the tme-seres relaton between systematc jump rsk and expected stock market returns allowng both the volatlty of the dffuson shocks, Cremers et al. (2015) examne the prcng of jump rsk n the cross-secton of stock returns. They fnd that stocks wth hgh senstvtes to jump rsk have low expected returns. The results are sgnfcant and economcally mportant. It has long been recognsed that nvestors care dfferently about downsde losses versus upsde gans. Ang et al. (2006a) show that the cross secton of stock returns reflects a downsde rsk premum of 6% per annum. The reward for bearng downsde rsk s not smply compensaton for regular market beta, nor t s explaned by common stock market return predctors. Ang et al. (2006b) fnd a premum for bearng volatlty and jump rsk. Lettau et al. (2014) follow Ang et al. (2006a) n allowng a dfferentaton n uncondtonal and downsde rsk. Ths captures the dea that assets that have a hgher beta wth market returns condtonal on low realzaton of the market return are partcularly rsky. The economc ntuton underlyng downsde rsk s smple: Agents requre a premum not only for securtes the more ther returns covary wth the market return, but also, and even more so, when securtes co-vary more wth market returns condtonal on low market returns. However, nether Lettau et al. (2014) nor Ang et al. (2006a) nvestgate the prcng of jump rsk n the cross-secton of stock returns. Thus, ther analyss does not separate jump rsk from dffuson rsk. Thus, the effects documented by these authors could be related to volatlty rsk, jump rsk, or a combnaton of both. Cremers et al. (2015) employ separate measures for jump and volatlty rsk to dsentangle the correspondng asset prcng effects. They fnd that stocks wth hgh senstvtes to volatlty and jump rsk have low expected returns, that s, volatlty and jump rsk both carry negatve market prces of rsk. Guo et al. (2015) document asymmetrc effects of physcal jump rsk measures on condtonal equty premum and show that sgned jump rsk measures have statstcally sgnfcant forecastng power for excess market return. Further, they conclude that negatve jump rsk has sgnfcant effect on condtonal equty premum but postve jump rsk does not. The fact 3

4 that the predctve power s neglgble when when total realzed jump volatlty s used further strengthens the mportance of ths asymmetry effect. The remander of the paper s organsed as follows. Secton 2 sets up the model framework. The data used n our emprcal nvestgaton are detaled n Secton 3. We dscuss our prcng mplcaton for cross secton of ndvdual assets n Secton 4 and nvestgate the behavours of systematc negatve and postve jump rsk factors n portfolos of assets n Secton 5. We draw our conclusons n Secton Model Setup We start wth a panel of N assets over a fxed tme nterval [0, T]. Followng the conventon n hgh frequency fnancal econometrcs lterature, we assume the log-prce p,t of the th asset follows a sem-martngale plus jumps process n contnuous tme. In turn, the log-return of any asset, r,t, has the followng representaton: r,t dp,t = b,t dt + σ,t dw,t + κ,t dµ,t, t [0, T], = 1, 2,..., N, (1) where b,t s a locally bounded drft term, σ,t denotes the non-zero spot volatlty, W,t s a standard Brownan moton for asset. The last part n (1) represents the jump component. The jump measure dµ,t s such that dµ,t = 1 f there s a jump n r,t at tme t, and dµ,t = 0 otherwse. The sze of jump at tme t s represented by κ,t. In fact, κ,t can be defned as κ,t = p,t p,t n general, where p,t = lm s t p,s. It follows mmedately that κ,t = 0 for t {t : dµ,t = 0} under ths defnton. Return on the market portfolo r 0,t can be decomposed n a way smlar to (1) as: r 0,t = b 0,t dt + σ 0,t dw 0,t + κ 0,t dµ 0,t. (2) In order to smplfy the analyss, we assume that the jumps n processes (1) and (2) have only fnte actvty Jump Regresson We focus on the dependence between the jump components of ndvdual assets (or portfolos) and that of the market return, utlsng the methodology proposed by L et al. (2015a) and L et al. (2015b). Jumps are rare events but have substantally hgher market mpacts than the dffusve prce movements (Patton and Verardo, 2012). The unpredctable nature of jump 1 Fnte actvty jumps, as opposed to the nfnte actvty, means that there s only a fnte number of jumps n the gven process over [0, T]. We make ths assumpton n the paper snce we only focus on bg jumps wth szes bounded away from zero. For a detaled dscusson on fnte versus nfnte actvty n jumps see Aït-Sahala and Jacod (2012) among others. 4

5 makes ts rsk harder to be dversfed away. Startng wth Retz (1988), researchers have modeled the possblty of rare dsasters, such as economc depressons or wars, to resolve the equty premum puzzle and related puzzles (e.g., Barro (2006) and Gabax (2012)). The dsasters n ths lterature are smlar to the jumps that we are nvestgatng, but there are some dfferences. Dsasters are extremely rare and they do not match well the short-dated optons often used n dsentanglng the jump and volatlty rsk. We nvestgate extreme events usng hgh frequency data. These events or jumps are rare spatally but they occur rather frequently n calendar tme when compared to dsasters defned n the lterature cted above. For nstance, we present n Fgure 1 the frequency of postve and negatve jump occurrence for each year from 2003 to For more lberal truncaton thresholds, majorty of the years have more than 100 jumps occurrng, whle dsasters wouldn t happen ths often. Let T be the collecton of jump tmes for the market portfolo r 0,t,.e. T = {τ : dµ 0,τ = 1, τ [0, T]}. The set T has fnte elements almost surely gven the assumpton of fnte actvty jumps. As an analog to the classcal one-factor market model, we set the lnear factor model for jumps n the followng form κ,τ = β d κ 0,τ + ɛ,τ, τ T, = 1, 2,..., N, (3) where the superscrpt d stands for dscontnuous (or jump) beta, and ɛ,τ s the resdual seres. We only consder the jump tmes of the market portfolo T because β d s not dentfed elsewhere. Therefore, β d only exsts f there s at least one jump n r 0,t n [0, T]. Model (3) mplctly assumes that β d s constant over the nterval [0, T]. The jump beta β d n (3) has a smlar nterpretaton as the market beta n the CAPM model. It allows us to assess the senstvty of an asset (or a portfolo of assets) to extreme market fluctuatons. Lower β d would sgnfy a resstance of an asset to move as much as a market durng extreme event (jump defensve assets), whle hgher β d values represent hgh senstvty of an asset exacerbatng the effect of the market moves durng the extreme event (jump cyclcal assets) Isolatng Jumps from the Brownan Component Under dscrete-tme samplng, nether the jump tmes T nor jump szes κ,τ are drectly observable from the data. Suppose the prce and return seres are observed every nterval,.e. we obtan return seres r,, r,2,..., r,m, where m = T/, for = 0, 1,..., N. Our frst step n constructng the jump regresson model (3) s to dentfy the dscrete-tme returns on the market portfolo r 0,j = p 0,j p 0,(j 1) that contan jumps, j = 1, 2,..., m. We use the truncaton threshold proposed by Mancn (2001) for ths purpose (see also Mancn, 2009; Mancn and Renò, 2011). The threshold, denoted by u 0,m, s a functon of the samplng 5

6 nterval, and hence the samplng frequency m. The most wdely used threshold s u 0,m = α ϖ, wth α > 0 and ϖ (0, 1/2). (4) Takng nto account the tme-varyng spot volatlty of the return seres, the constant α s usually dfferent for dfferent assets, and could vary over tme (see, for example, Jacod, 2008). One example s to set α to be dependent on the estmated contnuous volatlty of the gven asset. As 0 and m, the condton r 0,j > u 0,m = α ϖ elmnates the contnuous dffusve returns on the market portfolo asymptotcally, and hence only keeps returns that contan jumps. We collect the ndces of these dscrete-tme ntervals where the market return exceeds the truncaton level, and denote ths set as J m = {j : 1 j m, r 0,j > u 0,m }. (5) We denote the collecton of nterval returns for J m by {r 0,j } j Jm. Correspondngly, n the contnuous-tme data generatng process for market return (2), for each jump tme τ T, we also fnd the ndex j such that the jump κ 0,τ occurs n the nterval ((j 1), j ], J = {j : 1 j m, τ ((j 1), j ] for τ T }. (6) An mportant result from L et al. (2015a) s that, under some general regularty condtons, the probablty that the set J m concdes wth J converges to one as 0. Ths s formally stated n Proposton 1 n L et al. (2015a). 2 Proposton 1. Under certan regularty assumptons, as 0, we have (a) P(J m = J ) 1; (b) ((j 1), r 0,j ) j Jm (τ, κ 0,τ ) τ T. P Note that part (a) of Proposton 1 mples that the the number of elements n the set J m consstently estmates the number of jumps n the process r 0,t. Furthermore, part (b) of Proposton 1 states that as 0, the startng pont of the nterval (j 1), consstently estmates the jump tme τ, and the nterval return r 0,j consstently estmates the jump sze κ 0,τ. The asymptotc results n Proposton 1 provdes a powerful tool of lnkng the dscrete-tme return observatons to the unobservable jumps and jump tmes n the contnuous tme. We could use the dscrete-tme return observatons to estmate the jump regresson (3) and obtan consstent estmator of the jump beta β d. 2 Please refer to L et al. (2015a) for more detaled assumptons. 6

7 2.3. Estmatng Jump Beta In accordance wth model (3), the dscrete-tme jump regresson model has the form r,j = β d r 0,j + ɛ,τ, j J m, = 1, 2,..., N. (7) Hence a nave consstent estmator of β d s the analog of the OLS estmator, β d = j J m r,j r 0,j j Jm (r 0,j ) 2, = 1, 2,..., N. (8) L et al. (2015a) propose an effcent estmator for β d. It s an optmal weghted estmator n the sense that t mnmzes the condtonal asymptotc varance among all weghtng schemes. The optmal weght w j s a functon of the prelmnary consstent estmator β d, and the approxmated pre-jump and post-jump spot covarance matrces Ĉ j and Ĉ j+ : where w j = 2 ( β d, 1)(Ĉ j + Ĉ j+ )( β d, 1), for j J m, (9) J m = {j J m : k m + 1 j m k m }, and k m s an nteger such that k m and k m 0 as 0. The spot covarance matrces are estmated n the followng manner. We construct the truncaton threshold for the vector r j (r 0,j, r,j ) jontly n the same way as n (4), and denote t as u m (u 0,m, u,m ). For any j J m, we have Ĉ j = 1 k m k m r (j+l k m 1) r (j+l k m 1) 1 { r(j+l k m 1) u m}, l=1 (10) Ĉ j+ = 1 k m k m r (j+l) r (j+l) 1 { r(j+l) u m }, l=1 (11) as the approxmated pre-jump and post-jump spot covarance matrces, respectvely. Gven any weghtng functon w j, the class of weghted estmators ˆβ d j can be represented as ˆβ d = j J m w j r,j r 0,j, = 1, 2,..., N. (12) j J m w j (r 0,j ) 2 Theorem 2 n L et al. (2015a) show that the weghtng functon n (9) combned wth the estmator (12) produces the most effcent estmate of the jump beta β d. The standard errors and subsequently the confdence ntervals of the estmators can be constructed usng the bootstrap procedure outlned n L et al. (2015b). 7

8 2.4. Asymmetrc Jump Beta In developng Modern Portfolo Theory n 1959, Henry Markowtz recognzed that snce only downsde devaton s relevant to nvestors, usng downsde devaton to measure rsk would be more approprate than usng standard devaton (Markowtz, 1971). Ang et al. (2006a) explore the asset prcng mplcatons of the downsde rsk wthout, however, separately consderng extreme events or jumps. 3 In ths secton, we focus on jump rsk and separate the postve and negatve jumps. Instead of poolng jumps at both postve and negatve ends together, we examne the jump covaraton between ndvdual asset (or portfolo) and the equally weghted market ndex at tmes of postve market jumps and negatve market jumps separately. In ths way we could accommodate separate rsk prema for these two components. Although we focus on the negatve jump n the market portfolo and the negatve jump beta assocated wth t, our modellng approach naturally gves rse to a smlar defnton of the postve jump beta. The nave estmators of the two asymmetrc betas β d+ and β d are as follows: β d = j J m r,j r 0,j 1 {r0,j <0} j Jm (r 0,j ) 2, (13) 1 {r0,j <0} β d+ = j J m r,j r 0,j 1 {r0,j >0} j Jm (r 0,j ) 2, (14) 1 {r0,j >0} for = 1, 2,..., N. When calculatng the weghted estmators, the weghtng functon (9) would dffer for the postve and negatve jump betas: w j = w + j = 2 ( β d, 1)(Ĉ j + Ĉ j+ )( β d, 1), for j J m and r 0,j < 0, (15) 2 ( β d+, 1)(Ĉ j + Ĉ j+ )( β d+, 1), for j J m and r 0,j > 0. (16) Here we assume that before and after the jumps, the spot covarance matrces are the same for postve and negatve jumps. These lead to the formaton of the weghted estmators of the asymmetrc betas: ˆβ d j ˆβ d+ j = j J m w j r,j r 0,j 1 {r0,j <0} j J m w j (r 0,j ) 2, (17) 1 {r0,j <0} = j J m w + j r,j r 0,j 1 {r0,j >0} j J m w + j (r 0,j ) 2, (18) 1 {r0,j >0} 3 We defne the downsde market movements n a dfferent way from Ang et al. (2006a). They look at market returns that are lower than ts mean, but we smply look at negatve returns that contan jumps. It shouldn t affect the results as most jumps are n the tal dstrbuton, whch would be far from the mean of the return. 8

9 for = 1, 2,..., N. In what follows, we wll use an emprcal dataset to demonstrate the smlartes and dfferences between the postve and negatve jump betas and ts mplcatons for portfolo rsk management Data We nvestgate the behavor of the β d+ and β d estmates over a nne year sample perod, from January 2, 2003 to December 30, 2011, whch ncludes the perod of the fnancal crss assocated wth the bankruptcy of Lehman Brothers n September 2008 and the subsequent perod of turmol n US and nternatonal fnancal markets. The underlyng data are 5-mnute observatons on prces for 501 stocks drawn from the consttuent lst of the S&P500 ndex durng the sample perod, obtaned from SIRCA Thomson Reuters Tck Hstory. Ths dataset was constructed by Dungey et al. (2012) and does not purport to be all the stocks lsted on the S&P500 ndex, but ncludes those wth suffcent coverage and data avalablty for hgh frequency tme seres analyss of ths type Data Processng The orgnal dataset conssts of over 900 stocks taken from the 0#.SPX mnemonc code provded by SIRCA for the S&P500 Index hstorcal consttuents lst. Ths ncluded a number of stocks whch trade OTC and on alternatve exchanges, as well as some whch altered currency of trade durng the perod; these stocks were excluded. We adjusted the dataset for changes n Reuters Identfcaton Code (RIC) code durng the perod through mergers and acqustons, stock splts, and tradng halts. We also removed some stocks wth nsuffcent observatons durng the sample perod. The data handlng process s fully documented n the web-appendx to Dungey et al. (2012). The fnal data set contans 501 ndvdual stocks, hence N = 501. The ntra-day returns and prces data start at 9:30 am and end at 4:00 pm, observatons wth tme stamps outsde ths wndow and overnght returns are removed. Mssng 5-mnute prce observatons are flled wth the prevous observaton, correspondng to zero nter-nterval returns. In the case where the frst observatons of the day are mssng, we use the frst non-zero prce observaton on that day to fll backwards. Approxmately 20 prce observatons whch are orders of magntude away from ther neghbourng observatons are also removed. Thus, we have 77 ntra-day observatons for 2262 actve tradng days. The 5-mnute samplng frequency s chosen as relatvely conventonal n the hgh frequency lterature, especally for unvarate estmaton, see, for example, Andersen et al. (2007), Lahaye et al. (2011), and for some senstvty to alternatves, see Dungey et al. (2009). Optmal samplng frequency s an area of ongong research, and despte the unvarate work by Band and Russell (2006), ths ssue s outstandng for analyzng multple seres wth varyng degrees of lqudty. 4 L et al. (2015a) also consder the postve and negatve jump betas separately n ther emprcal applcaton. 9

10 The 5-mnute frequency s much fner than those employed n Todorov and Bollerslev (2010) and Bollerslev et al. (2008), both of whch use 22.5-mnute data. Lower samplng frequences are generally employed due to concerns over the Epps effect (Epps, 1979); however, as the qualty of hgh frequency data and market lqudty have mproved n many ways, fner samplng does not threaten the robustness of our results. Estmates of β d+ and β d are computed on an annual bass. Hgh frequency data permts the use of 1-year non-overlappng wndows to analyse the dynamcs of our systematc rsk estmates. L et al. (2015a) also fnds n ther emprcal applcaton usng US equty market data that the postve or negatve jump beta remans constant over a year most of the tme. For each year, only lqud stocks are consdered. Gven the 5-mnute samplng frequency, we defne lqud stocks as ones that have at least 75% of the entre 1-year wndow as non-zero return data, whch ndcates that these stocks are heavly traded most of the tme. We construct an equally weghted portfolo of all nvestble stocks n each estmaton wndow as the benchmark market portfolo. We use equally weghted portfolos rather than value weghted ones to avod stuatons where the weght on one stock s dsproportonately large relatve to other portfolo consttuents Parameter Values In our emprcal applcaton we normalze each tradng day to be one unt n tme. Gven the number of observatons n each day m = 77, the samplng frequency s = 1/77. Parameters n the truncaton threshold (4) are chosen as follows. The constant ϖ = Takng nto account the tme-varyng volatlty σ,t, we set α to be a functon of the estmated daly contnuous volatlty component for each ndvdual asset. In fnte samplng, the contnuous volatlty s consstently estmated by the bpower varaton (Barndorff-Nelsen and Shephard, 2004, 2006): ( π ) BV = 2 m 1 r,j r,(j+1) j=1 P T σ,t 2 0 We set α = 5 BV, whch leads to the threshold dt as 0, = 0, 1,..., N. (19) u,m = 5 BV (1/77) (20) The choce of usng a multple of the estmated contnuous volatlty s relatvely standard n the lterature for dsentanglng jumps from the contnuous prce movements. It could serve the purpose of controllng for the possbly tme-varyng spot volatlty automatcally n jump de- 5 See Fsher (1966) for the dscusson of Fsher s Arthmetc Index, an equally weghted average of the returns on all lsted stocks. 10

11 tecton. In our emprcal applcaton, BV s calculated on a daly bass and hence the threshold s dfferent for each tradng day. It keeps a good balance such that we could fnd both postve and negatve jumps n each estmaton wndow (year) n the market portfolo, and hence β d+ and β d can be estmated for each year. 4. Emprcal Analyss In ths secton we present some statstcal propertes of betas estmated based on the overall market returns, as well as based on the upsde and downsde market returns separately. In Secton 4.3 we use the two-stage regresson framework proposed by Fama and MacBeth (1973) to estmate the rsk prema on the rsk factors of nterest Market Volatlty and Jumps Fgure 1 plots the square root of the daly bpower varaton of the equally weghted market ndex towards the left axs, and the number of postve (blue) and negatve (red) jumps for each year from toward the rght axs. The subsample before md-2007 s much less volatle than the second half of the sample whch ncludes the global fnancal crss (GFC). Evdently, market volatlty has ncreased consderably snce md-2007, whch s usually regarded as the ntal emergence of the GFC, and peaked n late 2008 durng the few months after the bankruptcy of Lehman Brothers, the balout of AIG and the announcement of the TARP (Troubled Asset Relef Program). Other hghly volatle perods nclude md-2010 durng the Greek debt crss, and late 2011 durng the European soveregn debt crss wth the deteroraton of economc condtons n the Eurozone as a whole. The two peak values of market volatlty after the GFC correspond to the May 6, 2010 flash crash and August 9, On August 5, 2011 Standard & Poor s downgraded Amerca s credt ratng for the frst tme n hstory, followed wth short-sellng ban by Greece on August 8, 2011, and other 4 EU countres on August 11, We dentfy jumps usng three dfferent levels of threshold. Usng lower thresholds results n dentfcaton of too many jumps. For example, for threshold α = 3 BV, the number of negatve and postve jumps dentfed s often n excess of 100 jumps a year, meanng that the jumps occur almost every other day. Jumps are rare events and should not happen that often. The number of jumps dentfed usng thresholds α = 4 BV and α = 5 BV appear more realstc. 6 Notably, Fgure 1 shows that the volatle perod durng the GFC corresponds to fewer jumps n both drectons. In partcular, n 2008, usng the most conservatve threshold, we do not observe more than 5 jumps per year (postve or negatve). Ths result s expected as the market volatlty s generally hgher durng crss than durng calmer perod, the threshold of detectng jump observatons wll be elevated accordngly. Black et al. (2012) also observe that 6 When usng α = 6 BV as our threshold, we faled to dentfed any jumps n a number of years. 11

12 Fgure 1: Daly bpower varaton of equally weghted market ndex versus the dentfed number of postve (blue) and negatve (red) jumps. Jumps where dentfed usng the followng thresholds: α = 3 BV (top panel), α = 4 BV (mddle) and α = 5 BV (bottom panel). BV, daly BV Postve jumps Negatve jumps Tme Number of jumps n a year BV, daly BV Postve jumps Negatve jumps Number of jumps n a year Tme 0 BV, daly BV Postve jumps Negatve jumps Number of jumps n a year Tme 0 12

13 the stock market has fewer jumps durng crss perods. Snce only the jump observatons are taken to estmate the non-weghted asymmetrc beta n (13) and (14), low number of observatons could certanly affect the qualty of the estmates. Hence, t s necessary to use the weghted estmators, (17) and (18), n order to reduce the small sample sze effect Estmaton Results In addton to the weghted estmators of the overall jump beta (12) and the asymmetrc betas (17) and (18), we also calculate the hgh-frequency CAPM beta that s obtaned usng the OLS regresson n the sprt of Andersen et al. (2006) s realzed beta ˆβ OLS = m j=1 r,j r 0,j m j=1 (r 0,j ) 2, = 1, 2,..., N, (21) where all 5-mnute return observatons wthn a year are used to construct ˆβ OLS. The OLS beta would be able to ncorporate the mpact of co-movements n the contnuous component of ndvdual asset (or portfolo) and the market ndex. We calculate the descrptve statstcs and the correlatons between any pars of the beta estmates as well as the average monthly returns of each asset, and present them n Table 1. The means of all four estmated beta measures are very close to one, and they are all postvely skewed. The three jump beta estmates are statstcally nsgnfcant, possbly due to the hgh heterogenety between dfferent assets and n dfferent tme perods. The monthly return of all stocks are even more dspersed, and negatvely skewed. Followng the work of Ang et al. (2006a), we sort stocks n terms of the estmated jump betas, and dvde them nto quntle portfolos. Table 2 tabulates the annualzed realzed monthly return on the equally weghted quntle portfolo and all averaged beta estmates for these quntle portfolos. Although our result does not agree wth the fundng by Ang et al. (2006a) that hgher downsde beta s assocated wth hgher return, we stress that Ang et al. (2006a) do not separate jump and dffusve components, and that ther result could be attrbuted ether to dffusve component of rsk, jump component, or both. We separate postve and negatve jumps and nvestgate whether postve and negatve jump rsk are prced rsk factors. We do so through portfolo sorts. We consder contemporaneous relatonshp between returns and factor rsks. The general fndng of Table 2 s that stocks wth hgher senstvtes to aggregate (market) jumps earn lower returns. Ths s consstent wth the result n Cremers et al. (2015) for symmetrc jump rsk, and apples to our case for the negatve jump betas. In fact, our results, show a negatve market prce of negatve jump rsk. Ths mples that stocks wth hgh senstvtes to negatve market jumps should earn low returns. Cremers et al. (2015) argue that ths makes sense economcally, as such stocks provde useful hedgng opportuntes for rsk-averse nvestors, who dslke hgh systematc negatve jump rsk. On the contrary, wth a postve rsk prema for postve jump rsk, stocks earn hgher return. 13

14 Table 1: Summary Statstcs a ˆβ OLS ˆβ d ˆβ d+ ˆβ d r Panel A: Descrptve Statstcs Mean Std.Dev Skew Panel B: Correlaton Table ˆβ OLS ˆβ d ˆβ d+ ˆβ d r ˆβ OLS ˆβ d ˆβ d ˆβ d r a Ths table shows the descrptve statstcs (Panel A) and tme-seres means of parwse correlatons (Panel B) for ndvdual frm contnuous betas, β OLS, jump beta, β d, postve and negatve jump betas, β d+ and β d, respectvely, as well as the (annualzed) realzed monthly returns, r. All estmates are obtaned usng 5-mnute data wthn each calendar month (wthout overlap). The estmates for all ndvdual stocks and all calendar months are pooled together n calculatng the statstcs n ths table. b Sgnfcance levels: : 10%, : 5%, : 1% Rsk Prema of the Asymmetrc Jump Betas We use the two-stage regresson by Fama and MacBeth (1973) to estmate the rsk prema of the asymmetrc jump betas. The beta estmates obtaned above are taken as the explanatory varables n the cross-sectonal regresson r,s = γ s ˆβ,s 1 + ɛ,s, = 1, 2,..., N, (22) and the tme ndex s = 2, 3,..., 9 ndcates that we estmate (22) for each year separately. The vector of coeffcents γ s 1 represent the estmated rsk premum awarded to each rsk factor n ˆβ,s 1. Also note that the realzed annual returns of each ndvdual stock on the left hand-sde of (22) are from the year after the beta estmates. Hence equaton (22) s estmated from 2004 onwards. Table 3 dsplays the estmated rsk prema γ for each year from 2004 to 2011, as well as ther t-statstcs n parentheses. We also take the tme-seres average of γ s over the 8 years and present t n the bottom panel of Table 3. There are two dfferent model specfcatons under consderaton: model (1) decomposes the overall systematc rsk nto the contnuous and dscontnuous factors wthout takng nto account asymmetry, whle model (2) separate the postve and negatve dscontnuous betas. Contrastng these two models helps us detect the exstence and degree of asymmetry n the jump rsk. The t-statstcs n Table 3 underneath the 14

15 Table 2: Average returns of sorted quntle portfolos a Portfolo r 0 ˆβ OLS Panel A: Stocks sorted by ˆβ d ˆβ d ˆβ d+ ˆβ d 1 - Low Hgh Hgh-Low Panel B: Stocks sorted by ˆβ d+ 1 - Low Hgh Hgh-Low Panel C: Stocks sorted by ˆβ d 1 - Low Hgh Hgh-Low Panel D: Stocks sorted by ˆβ d+ ˆβ d 1 - Low Hgh Hgh-Low Panel E: Stocks sorted by ˆβ d ˆβ d 1 - Low Hgh Hgh-Low a We pool the beta estmates and annualzed realzed monthly returns for all nvestable stocks from all months together, and create equally weghted quntle portfolos by sortng them based on the correspondng realzed betas: ˆβ d (Panel A), ˆβ d+ (Panel B), ˆβ d (Panel C), ˆβ d+ ˆβ d (Panel D), and ˆβ d ˆβ d (Panel E). All reported portfolo characterstcs are contemporaneous wth the betas used to construct the portfolo sorts. b Sgnfcance levels: : 10%, : 5%, : 1%. 15

16 estmated rsk prema are calculated usng heteroskedastcty and auto-correlaton consstent standard errors. Table 3: Fama-MacBeth regressons and rsk prema (γ) assocated wth the estmated rsk factors (betas) a Model (1) (2) ˆβ c ˆβ d ˆβ c ˆβ d+ ˆβ d Average (6.6478) (0.1635) (6.6221) (1.7649) (0.8146) (5.7792) (0.0598) (5.5499) (0.4578) (1.3969) (6.4133) (5.2799) (6.2249) (2.0819) (4.3569) (0.3645) (1.5598) (0.3215) (2.0272) (0.1770) (3.1671) (1.5894) (3.0701) (0.4768) (1.8461) (0.2810) (1.0018) (0.4903) (0.8668) (1.5761) (6.2132) (1.1540) (5.3502) (1.6087) (0.8345) (9.5325) (0.2233) (9.3413) (0.2451) (0.0536) (5.8674) (3.0441) (6.1028) (1.9633) (1.6983) a Ths table nvestgates the cross-sectonal prcng of the contnuous and jump rsks (both symmetrc and asymmetrc). The sample perod s from January 2003 to December We run Fama-MacBeth regressons of annual returns on lagged one-perod realzed betas. The t-statstcs are gven n parentheses underneath the coeffcent estmates. Table 3 shows that n both model specfcatons, the contnuous beta receves a negatve rsk premum, whch s hghly sgnfcant n most years except 2007 and Ths result s consstent wth the fndng of Alexeev et al. (2015). On the other hand, the jump beta always carres a postve rsk premum. The average γ over the entre sample perod for all beta estmates are statstcally sgnfcant. In model (2) both the postve and negatve jump betas have postve rsk prema pontng to the mportance of dstngushng between these two rsk factors. As expected, the negatve jump beta receve slghtly hgher premum than the postve jump beta on average. In ths secton we showed that ndvdual stock betas can vary greatly. Experenced nvestors wll always consder allocatng wealth n a number of assets n favour of nvestng all wealth n a sngle securty. As the number of holdngs ncrease, the range of portfolos betas wll become more lmted compared to betas of ndvdual securtes, eventually convergng to unty for an equweghted market benchmark. Gven that nvestng n all securtes lsted on a abroad 16

17 market benchmark may not be nformatonally or cost effectve, fndng the optmal number of holdngs n a portfolo to mtgate most jump rsk (postve or negatve) s of sgnfcant mportance. We dscuss ths n our next secton. 5. Portfolo Smulaton Ever snce the development of Modern Portfolo Theory by Harry Markowtz 7 the quest for optmal portfolo sze that reduces dversfable rsk has been an ntegral part of portfolo choce lterature. The concept of dversfcaton s smple: the level of return varablty falls as the number of holdngs n a portfolo ncreases. 8 The avalablty of hgh-frequency data allowed new nsghts nto portfolo dversfcaton. For example, Slvapulle and Granger (2001) nvestgate asset correlatons at the tals of return dstrbutons and dscusses the mplcatons for portfolo dversfcaton. Bollerslev et al. (2013) examne the relatonshp between jumps n ndvdual stocks and jumps n a market ndex. The authors fnd that jumps occur more than three tmes as often at the ndvdual stock level compared to jump occurrence n an aggregate equweghted ndex constructed from the same stocks. Ths may pont to the fact that jumps are dversfable. In fact, Pukthuanthong and Roll (2014) do consder mplcatons of jumps for nternatonal dversfcaton. They fnd that cross-country dversfcaton s less effectve f jumps are frequent, unpredctable, and strongly correlated. Studes drectly nvestgatng optmal portfolo sze usng hgh-frequency data only recently started to emerge (e.g., Alexeev and Dungey (2014)). In ths secton, usng extensve portfolo smulaton technques, we evaluate the varablty reducton of portfolo jump betas as the number of holdngs n these portfolos ncrease. 9 We analyse the spread of estmated betas n equweghted randomly constructed portfolos of dfferent szes focusng on the dfference n convergence between negatve and postve jump betas as the number of holdngs n portfolos ncreases. For nvestors, the knowledge that ndvdual stocks respond dfferently to the postve and negatve extreme events s lkely to be a valuable addton to n developng portfolo rsk management strateges. However, nvestors who hold several S&P500 stocks may be rghtfully concerned wth the overall exposure of ther portfolos to systematc jump rsk. Moreover, nvestors exhbt dfferent atttudes towards extreme gans and extreme losses. We assert that f an asset tends to move downwards n a declnng market more than t moves upward n a rsng market, such asset s unattractve to hold, especally durng market downturns when wealth of nvestors s low. Usng a 12-month estmaton wndow, for each year from 2003 to 2011 we construct 1,000 random equally weghted portfolos wth a number of holdngs rangng from 1 to 200. For 7 See Markowtz (1952) and Markowtz (1959). 8 Ths s true for any coherent measure of rsk (e.g., Artzner et al. (1997) and Artzner et al. (1999)). 9 Detaled descrpton of the technques employed s provded n the appendx. See Algorthm 1 on page

18 each of these portfolos we estmate several systematc dscontnuous rsks. We estmate β d wthout takng nto account the asymmetry, as well as β d+ and β d. We assess the stablty of the systematc portfolo rsks by analyzng the nter-quartle ranges of the beta dstrbutons as the number of stocks n portfolo ncreases. Defned as the dfference between two percentles, 75% and 25%, the nter-quartle range (IQR) s a stable measure that s robust to outlers. That s, IQR (n) EDF 1 (.75) EDF 1(.25), (23) (n) (n) where EDF (n) s the emprcal dstrbuton functon of the estmated betas (β d, β d+ or β d ) for randomly drawn n-stock portfolos. Fgure 2: Dstrbuton of β d (top panel) and β d+ (bottom panel) across portfolo szes. Red ponts represent maxma and mnma, black lnes represent nterpercentale range from 2.5% to 97.5%, blue lnes denote nterguantle range and the black crcles are the mdans of the dstrbutons. We use 2008 data n estmatng the results n ths fgure. 2.5 Monthly estmates usng 5-mn S&P500 data 2.5 Monthly estmates usng 5-mn S&P500 data β d- β d Number of stocks n portfolo Number of stocks n portfolo Fgure 2 depcts the typcal dstrbutons of β d and β d+ for equally weghted randomly drawn portfolos of n = 1,..., 200 stocks. Snce these central ranges are dependent on the partcular tme perod analyzed, For each year, we normalze the IQR for the n-stock portfolos and represent t as a fracton of the IQR of the sngle-stock portfolo. The normalzed IQRs, or IQR (n) /IQR (1), of ˆβ d, β d+ and β d for year 2008 are depcted n Fgure 3 for n = 1,..., 200. Snce the market ndex s an equally weghted portfolo consstng of all nvestble stocks, and s thus unque, t has IQR (N) = 0. As a result, the normalzed IQRs n Fgure 3 are bounded between 0 and 1. We fnd that the dfference among the normalzed IQRs for the three dfferent betas are more pronounced durng perods of hgh volatlty 10 and for more extreme events (consder the top vs the bottom panel n Fgure 3). Fgure 3 shows that the IQR of portfolo jump betas decrease substantally as the num- 10 Results for years other than 2008 are omtted for brevty. 18

19 Fgure 3: Normalzed IQR of betas across portfolo szes. Both panels dsplay results based on year The top panel dsplays results based on a threshold α = 3 BV (n Equaton 20) and the bottom panel s based on α = 5 BV. As can be observed from the fgures below, the asymmetry n sgned betas s more pronounced when more extreme events are consdered. The optmal number of holdngs n a portfolo s determned at the ntersecton of the normalsed IQR curve (red, blue and black) wth the desred level of varablty reducton (n ths case 0.2 denoted by horzonal purple lne) Normalsed IQR for β d, β d, and β d+ β d β d- β d+ Normalsed range, IQR (n) IQR (1) Number of stocks n portfolo Normalsed IQR for β d, β d, and β d+ β d β d- β d+ Normalsed range, IQR (n) IQR (1) Number of stocks n portfolo 19

20 Table 4: Portfolo szes, n, requred to reduce normalsed IQR, IQR (n) /IQR (1) to 0.2. a Year(s) α = 3 BV α = 4 BV α = 5 BV β d β d β d+ β d β d β d+ β d β d β d a Ths table shows the number of holdng n a portfolo requred to stablze portfolo betas, that s β d, β d or β d+. We nclude the results for a number of severtes of extreme events (threshold α used to dentfy jumps). We consder level of IQR (n) /IQR (1) = 0.2 as approprate: the IQR of of ndvdual stock betas can be reduced fvefold f a portfolo s constructed wth at least a number of stocks outlned n the table. For example, n 2003, for α = 5 BV, 39 stocks are requred to reduce portfolo senstvty to negatve jumps by a factor of 5 compared to when a sngle stock s held. To get the same reducton n senstvty to postve shocks, 35 stocks wll suffce. Ths s n contrast wth year 2008, where as many as 54 stocks are requred to reduce senstvty to negatve events wth only 26 stocks needed to get the same reducton n senstvty of portfolo to postve events. Note that the asymmetry n results s more pronounced for more extreme events (e.g., larger threshold level α). 20

21 ber of stocks, n, n the portfolo ncreases. Usng the normalzed IQRs enables us to contrast the requred portfolo szes at dfferent perods of tme, n order to acheve the same proportonal reducton n IQRs of beta for these portfolos relatve to the beta spreads of ndvdual securtes. Table 4 outlnes the requred portfolo szes to reduce the normalzed IQR fve fold, respectvely. It s evdent that durng perods of market dstressed charactersed by hgh volatlty, the number of stocks requred to reduce IQR and stablse negatve jump beta s consderably hgher than n the less volatle perods. A reducton n a spread of postve jump rsk component n majorty of portfolos by a factor of 5 relatve to the postve jump beta spread of ndvdual securtes requres roughly 1/4 less stocks durng these volatle perods wth no substantal dfference durng normal market perods. Durng the perods of normal market actvty, the market seems to be ndfferent to the dstncton between negatve and postve jump rsk. The dfference n the number of stocks requred n order to acheve the same proportonate reducton n beta spreads for negatve vs postve jumps only dffers substantally durng 2008 and Ignorng the asymmetry n senstvtes to negatve vs postve market jumps may result n under-dversfcaton of portfolos and ncreased exposure to extreme negatve market shfts. For example, consder portfolo szes optmzed for the most extreme jumps dentfed wth α = 5 BV threshold (last three columns n Table 4). If we gnore the asymmetry, optmal portfolo szes are 35 and 34 n 2008 and 2011 respectvely. However, f nvestor s concerned wth extreme negatve shfts n the market, t would be advsable to hold 54 and 51 stocks nstead, to reduce the senstvty of portfolo returns to extreme negatve market shfts compared to a sngle stock portfolo. 6. Concluson In ths paper we studed jump dependence of two processes usng hgh-frequency observatons concentratng only on segments of data around a few outlyng observatons that are nformatve for the jump nference. In partcular, we studed the relatonshp between jumps of a process for an asset (or portfolo of assets) and an aggregate market factor and analysed the co-movement of the jumps n these two processes. Gven the predomnance of factor models n asset prcng applcatons, we focused on a lnear relatonshp between the jumps and assessed the senstvty of jumps n (portfolos of) assets to jumps n the market. We show that nvestors care dfferently about downsde losses as opposed to upsde gans and demand addtonal compensaton for holdng stocks wth hgh senstvtes to downsde market movements. We estmated jump betas for the negatve and postve market shfts and nvestgated the mplcatons for portfolo rsk management usng upsde and downsde jump betas. We assert that f an asset tends to move downwards n a declnng market more than t moves upward n a rsng market, such asset s unattractve to hold, especally durng market downturns when wealth of nvestors s low. 21

22 In the context of portfolo of assets, we nvestgated to what extend the downsde and upsde jump rsk can be dversfed. Ths has mportant mplcatons for prcng of jump rsk and can have a drect mpact on nvestors decson-makng. We found that gnorng the asymmetry n senstvtes to negatve versus postve market jumps results n under-dversfcaton of portfolos and ncreases exposure to extreme negatve market shfts. These results are more pronounced for the most extreme events and durng perods of very hgh market volatlty. References Aït-Sahala, Yacne and Jean Jacod, Analyzng the Spectrum of Asset Returns: Jump and Volatlty Components n Hgh Frequency Data, Journal of Economc Lterature, 2012, 50 (4), Alexeev, Vtal and Mard Dungey, Equty portfolo dversfcaton wth hgh frequency data, Quanttatve Fnance, nov 2014, 15 (7), ,, and Wenyng Yao, Tme-varyng contnuous and jump betas: The role of frm characterstcs and perods of stress, Techncal Report, Unversty of Tasmana Workng Paper seres Andersen, Torben G., Tm Bollerslev, and Francs X. Debold, Roughng It Up: Includng Jump Components n the Measurement, Modelng, and Forecastng of Return Volatlty, Revew of Economcs and Statstcs, nov 2007, 89 (4), ,,, and Gnger Wu, Realzed Beta: Persstence and Predctablty, Vol. 20 of Econometrc Analyss of Fnancal and Economc Tme Seres (Advances n Econometrcs), Emerald Group Publshng Ltd, Ang, Andrew, Joseph Chen, and Yuhang Xng, Downsde Rsk, Revew of Fnancal Studes, 2006, 19 (4), , Robert J. Hodrck, Yuhang Xng, and Xaoyan Zhang, The Cross-Secton of Volatlty and Expected Returns, The Journal of Fnance, jan 2006, 61 (1), Artzner, Phlppe, Freddy Delbaen, Jean-Marc Eber, and Davd Heath, Thnkng Coherently, Rsk, 1997, 10, ,,, and, Coherent Measures of Rsk, Mathematcal Fnance, 1999, 9 (3),

23 Band, R.M and J.R. Russell, Separatng Mcrostructure Nose from Volatlty, Journal of Fnancal Economcs, 2006, 79, Barndorff-Nelsen, Ole E. and Nel Shephard, Power and Bpower Varaton wth Stochastc Volatlty and Jumps, Journal of Fnancal Econometrcs, 2004, 2 (1), and, Econometrcs of Testng for Jumps n Fnancal Economcs Usng Bpower Varaton, Journal of Fnancal Econometrcs, 2006, 4 (1), Barro, Robert J., Rare Dsasters and Asset Markets n the Twenteth Century, The Quarterly Journal of Economcs, 2006, 121 (3), Bates, Davd S., The market for crash rsk, Journal of Economc Dynamcs and Control, jul 2008, 32 (7), Black, Angela, Jng Chen, Oleg Gustap, and Julan M Wllams, The Importance of Jumps n Modellng Volatlty durng the 2008 Fnancal Crss, Techncal Report July Bollerslev, Tm, Tzuo Hann Law, and George Tauchen, Rsk, Jumps, and Dversfcaton, Journal of Econometrcs, May 2008, 144 (1), , Vktor Todorov, and Sopha Zhengz L, Jump Tals, Extreme Dependences, and the Dstrbuton of Stock Returns, Journal of Econometrcs, 2013, 172 (2), <ce:ttle>latest Developments on Heavy-Taled Dstrbutons</ce:ttle>. Cremers, Martjn, Mchael Hallng, and Davd Wenbaum, Aggregate Jump and Volatlty Rsk n the Cross-Secton of Stock Returns, The Journal of Fnance, mar 2015, 70 (2), Dungey, Mard, Matteo Lucan, and Davd Veredas, Rankng Systemcally Important Fnancal Insttutons, Workng Papers ECARES 2013/130530, ULB Unverste Lbre de Bruxelles 2012., Mchael McKenze, and Vanessa Smth, Emprcal Evdence on Jumps n the Term Structure of the US Treasury Market, Journal of Emprcal Fnance, 2009, 16, Epps, Thomas W., Comovements n Stock Prces n the Very Short Run, Journal of the Amercan Statstcal Assocaton, 1979, 74 (366), Fama, Eugene F. and James D. MacBeth, Rsk, Return, and Equlbrum: Emprcal Tests, Journal of Poltcal Economy, 1973, 81 (3), Fsher, Lawrence, Some New Stock-Market Indexes, The Journal of Busness, 1966, 39 (1),