Hybrid Tail Risk and Expected Stock Returns: When Does the Tail Wag the Dog?

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1 Hybrd Tal Rsk and Expected Stock Returns: When Does the Tal Wag the Dog? Turan G. Bal, a Nusret Cakc, b and Robert F. Whtelaw c* ABSTRACT Ths paper ntroduces a new, hybrd measure of covarance rsk n the lower tal of the stock return dstrbuton, motvated by the under-dversfed portfolo holdngs of ndvdual nvestors, and nvestgates ts performance n predctng the cross-sectonal varaton n stock returns over the sample perod July 1963-December Our key nnovaton s that the covarance s measured across the states of the world n whch the ndvdual stock return s n ts left tal, not across the correspondng tal states for the market return as n standard systematc rsk measures. The results ndcate a postve and sgnfcant relaton between what we label hybrd tal covarance rsk (H- TCR) and expected stock returns, n contrast to the nsgnfcant or negatve results for purely stock-specfc or standard systematc tal rsk measures. A tradng strategy that goes long stocks n the hghest H-TCR decle and shorts stocks n the lowest H-TCR decle produces average raw and rsk-adjusted returns of 6% to 8% per annum, consstent wth results from a cross-sectonal regresson analyss that controls for a battery of known predctors. Key words: tal rsk, downsde rsk, under-dversfcaton, expected stock returns JEL classfcaton: G10, G11, C13 Frst draft: March 2007 Ths draft: September 9, 2011 a McDonough School of Busness, Georgetown Unversty, Washngton, D.C Phone: (202) , Fax: (202) , E-mal: tgb27@georgetown.edu. b School of Busness, Fordham Unversty, 113 West 60th Street, New York, NY 10023, Phone: (212) , Fax: (212) , E-mal: cakc@fordham.edu. c Correspondng author. Stern School of Busness, New York Unversty, 44 W. 4 th Street, Sute 9-190, New York, NY 10012, and NBER. Phone: (212) , E-mal: rwhtela@stern.nyu.edu. * We would lke to thank semnar partcpants at the Unversty of Utah for helpful comments.

2 In spte of the domnance of the CAPM paradgm, there has been a longstandng nterest n the lterature on the queston of whether downsde or tal rsk plays a specal role n determnng expected returns. Such a role could come about, for example, due to preferences that treat losses and gans asymmetrcally, 1 return dstrbutons that are asymmetrc, or some combnaton of the two. Whle systematc downsde or tal rsk s a natural startng pont, there s ncreasng evdence that non-market rsk may play an mportant role n determnng the cross-secton of expected returns. 2 Thus, we consder a settng where nvestors hold concentrated stock holdngs n addton to a fracton of ther wealth n a well-dversfed portfolo, e.g., a mutual fund wthn a retrement account, consstent wth exstng emprcal evdence on the holdngs of ndvdual nvestors. 3 In ths settng the contrbuton of an ndvdual stock to the tal rsk of the portfolo can be decomposed nto three components a systematc component, a stock-specfc component, and a thrd, hybrd component that depends on co-tal rsk of the stock and the market portfolo. Based on ths decomposton, we conduct a thorough re-examnaton of the role of downsde rsk n determnng the cross-secton of expected returns. Specfcally, controllng for the usual determnants of expected returns, we nvestgate the predctve power of varous downsde rsk measures that vary across two dmensons: () the fracton of the lower half of the return dstrbuton that they measure and on whch they are calculated,.e., the extent to whch they are tal rsk measures, and () the extent to whch they capture systematc versus dosyncratc or total rsk. Our rsk measures buld on the noton of sem-varance or the lower partal moment (LPM) of Markowtz (1959). The LPM of an asset or portfolo s defned as: LPM = h ( R h) 2 f p ( R) dr, (1) where h s the target level of returns and f p (R) represents the probablty densty functon of returns for portfolo p. 4 That s, the sem-varance s the expected value of the squared negatve devatons of the possble outcomes from the mean, whle the more general LPM uses a chosen pont of reference (h). The man heurstc motvaton for the use of the LPM n place of varance 1 See, for example, Kahneman et al. (1990) for some of the extensve expermental evdence on loss averson. 2 For recent examples, see Ang et al. (2006, 2009) and Bal, Cakc and Whtelaw (2011). 3 Polkovnchenko (2005), Van Neuwerburgh and Veldkamp (2010). 4 For expected utlty maxmzng nvestors, Bawa (1975) provdes a theoretcal ratonale for usng sem-varance or the lower partal moment as the measure of portfolo rsk. 1

3 as a measure of rsk s that the LPM measures losses (relatve to some reference pont), whereas varance depends on gans as well as losses. Of course, for symmetrc dstrbutons and a reference pont equal to the mean of the dstrbuton, ths dstncton s meanngless. In our smplfed settng n whch nvestors hold an under-dversfed portfolo consstng of a poston n an ndvdual stock and a dversfed fund, systematc tal rsk matters n that the tal rsk of the market portfolo contrbutes to the tal rsk of the overall portfolo. We take our systematc rsk measure from the mean-lower partal moment captal asset prcng model (EL- CAPM) of Bawa and Lndenberg (1977): E([ R h][ Rm h] Rm < h) β LPM =, (2) 2 E([ R h] R < h) m where E R ) s the expected return on asset, E R ) s the expected return on the market ( portfolo, r f s the rsk-free nterest rate, and m ( m β LPM s a measure of downsde systematc rsk for a target level of returns h. That s, the relevant beta n the model s the co-lower partal moment (CLPM) of the asset return wth the market return dvded by the LPM of the market return, where the moments are condtonal on the market return beng below a specfed threshold. Earler studes on the EL-CAPM use alternatve measures of downsde market rsk based on dfferent return thresholds, such as the mean excess market return, the rsk-free rate, or zero. 5 More recently, Ang, Chen, and Xng (2006) re-examne these downsde betas. Motvated by the possblty that t s more extreme negatve realzatons about whch nvestors care or that t s asymmetres n the tal of return dstrbutons that are mportant, we examne alternatve measures of downsde beta based on the observatons n the lower tals of the market return dstrbuton. There s recent evdence that systematc crash rsk s prced n the cross-secton of returns (Kelly (2010) and Sptzer (2006)), but these studes consder nfrequent events of extreme magntude, n the sprt of the rare dsaster models of Retz (1988) and Barro (2006), usng emprcal technques from extreme value theory. In contrast, we consder the more frequent but less extreme tal events that occur on a regular bass, usng more tradtonal rsk measures. Clearly, the tal rsk of the ndvdual stock wll also matter for the tal rsk of the underdversfed portfolo. For stock-specfc tal rsk we use the LPM of ndvdual stock returns. Fnally, we propose a new, hybrd measure of tal rsk. Gven that ndvdual stocks generally have substantally hgher volatltes than the market portfolo and assumng a 5 See, for example, Jahankhan (1976), Prce, Prce, and Nantell (1982), and Harlow and Rao (1989). 2

4 suffcent weght n the stock n the portfolo, the tal events for such an under-dversfed portfolo wll concde more wth the tal events of the ndvdual stocks than wth the tal events of the dversfed holdngs. Thus, we construct a measure called hybrd tal covarance rsk (H- TCR) defned as: 6 H TCR = E([ R h ][ R h ] R < h ), (3) - m m where h denotes the return threshold, e.g., the 10 th percentle of the return dstrbuton of the stock or market. H-TCR s the CLPM between extreme daly returns on stock and the correspondng daly returns on the market portfolo, condtonal on the stock return beng below the specfed threshold. H-TCR s analogous to the numerator of the EL-CAPM beta defned above except that the moment s condtonal on the return on the ndvdual stock rather than on the return on the market. In our emprcal analyss, we compute the above measures of tal rsk (systematc, hybrd, and stock-specfc) for ndvdual stocks usng one year of daly data. We then ask whch, f any, of these measures have predctve power for returns over the subsequent month usng NYSE/AMEX/NASDAQ stocks over the July 1963-December 2009 sample perod. In addton to the standard controls n cross-sectonal tests, we are also careful to control for volatlty (Ang et al. (2006, 2009)) and extreme returns (Bal, Cakc and Whtelaw (2011)) snce these stockspecfc dstrbutonal characterstcs are lkely to be correlated wth both the LPM of stock returns and our hybrd measure of tal rsk. The results are strkng. Frst, systematc rsk has lttle or no explanatory power for future returns, whether measured relatve to the center or the tal of the dstrbuton. Second, stockspecfc rsk s, f anythng, prced negatvely,.e., n the opposte drecton of that mpled by theory. However, these results should be nterpreted wth cauton due to the dffculty of dstngushng any tal rsk effect from the prcng of other dstrbutonal characterstcs. Thrd, and most mportant, n marked contrast to these results, H-TCR has sgnfcant and robust postve predctve power for future returns. Unvarate portfolo level analyses ndcate that a tradng strategy that goes long stocks n the hghest H-TCR decle and shorts stocks n the lowest H-TCR decle yelds average raw and rsk-adjusted returns of up to 8% per annum. Frm-level, cross-sectonal regressons that control for well known prcng effects, ncludng sze, book-to-market (Fama and French (1992, 1993)), 6 We motvate H-TCR more formally n the context of a stylzed model n the next secton. 3

5 momentum (Jegadeesh and Ttman (1993)), short-term reversals (Jegadeesh (1990)), lqudty (Amhud (2002)), co-skewness (Harvey and Sddque (1999, 2000)), volatlty (Ang et al. (2006, 2009)), and preference for lottery-lke assets (Bal, Cakc, and Whtelaw (2011)) generate smlar results. Moreover, there s strong evdence that the prcng of H-TCR s a tal rsk, rather than a more general downsde rsk, phenomenon as the effect attenuates sgnfcantly as the fracton of observatons used to calculate the measure ncreases. As robustness checks, we test whether the postve relaton between tal covarance rsk and the cross-secton of expected returns holds n bvarate dependent sorts, usng sze and bookto-market matched benchmark portfolos smlar to Danel and Ttman (1997), and once we screen for extreme stocks across numerous dmensons. Throughout our emprcal analyss, the evdence s consstent wth sgnfcant prcng effects generated by ndvdual nvestors who care about how the tal rsk of ther concentrated postons nteracts wth ther dversfed holdngs. The paper s organzed as follows. Secton 1 motvates our tal covarance rsk measure n the context of a stylzed model and also presents motvatng emprcal evdence. Secton 2 contans the data and varable defntons. Secton 3 presents evdence of cross-sectonal predctablty n the context of unvarate portfolo sorts and also looks more closely at the characterstcs of the stocks wthn these portfolos. Secton 4 examnes the sgnfcance of a cross-sectonal relaton between tal covarance rsk and expected stock returns usng frm-level regressons. Secton 5 provdes bvarate portfolo level analyses, controllng for sze, book-tomarket, momentum, short-term reversals, lqudty, co-skewness, volatlty, and extreme returns, whle also examnng the predctve power of tal covarance rsk usng the characterstc matched benchmark portfolos. Secton 6 provdes addtonal robustness checks, and Secton 7 concludes. 1. Motvatng Theory and Emprcal Evdence In order to motvate our three measures of tal rsk, we develop a relatvely stylzed, 1-perod, dscrete state space model n whch systematc, stock-specfc and hybrd tal rsk arse as approprate measures of rsk for an ndvdual stock. The assumptons of ths model are, n turn, motvated by exstng emprcal evdence on the stock holdngs of ndvdual nvestors. Although dversfcaton s crtcal n elmnatng dosyncratc rsk, a closer examnaton of the portfolos of ndvdual nvestors suggests that these nvestors are, n general, not well- 4

6 dversfed. For example, Polkovnchenko (2005) examnes a survey of 14 mllon households and shows that the medan number of stocks n household portfolos s two n 1989, 1992, 1995, and The medan ncreases to three stocks n Based on 40,000 stock accounts at a brokerage frm, Goetzmann and Kumar (2008) fnd that the medan number of stocks n a portfolo of ndvdual nvestors s three n the perod. These results are smlar to the fndngs of earler studes. For example, Blume and Frend (1975) and Blume, Crockett, and Frend (1974) provde evdence that the average number of stocks n household portfolos s about 3.41 n Odean (1999) and Barber and Odean (2001) also report the medan number of stocks n ndvdual nvestors portfolos as two to three. In recent work, Dorn and Huberman (2005, 2010) use tradng records between 1995 and 2000 of over 20,000 customers of a German dscount brokerage and fnd that the typcal portfolo conssts of lttle more than three stocks. However, these ndvdual stock holdngs often do not consttute the full fnancal asset portfolos of these nvestors. Polkovnchenko (2005) reports the fracton of ndvdual equtes relatve to total fnancal assets as 33% n 1989, 39% n 1992, 49% n 1995, 53% n 1998, and 57% n That s, nvestors have a sgnfcant fracton of ther wealth n concentrated holdngs, but they also hold wealth n other nvestments that may take the form, for example, of dversfed mutual funds n retrement accounts. Based on ths evdence, consder an nvestor that holds a portfolo consstng of postons n two assets equty n an ndvdual frm and the market portfolo. Assume that over the next perod the returns on the frm (R ) and on the market (R m ) can take on J and K dscrete values, respectvely, ndexed by j=1,, J and k=1,, K and n order of ncreasng returns. That s, the return n state j on frm s greater that the return n state j-1 (R,j >R,j-1 ) and smlarly for the market (R m,k >R m,k-1 ). There are J x K possble states of the world, each occurrng wth probablty p jk, where the probablty of a gven state can be zero. Denote the nvestor s non-negatve portfolo weghts n the two assets as w and w m, where w w = 1. The portfolo return n each state s + m R P, jk w R, j + ( 1 w ) Rm, k =. (4) Now assume further that the relevant measure of rsk s the LPM of the portfolo, as defned n equaton (1), for a specfed threshold h. Ths calculaton requres an orderng of the J x K states n terms of the assocated portfolo return n order to compute the probablty-weghted sum of the states wth returns less than h, but the portfolo return and therefore the orderng 5

7 depends on both the magntudes of the returns on the two assets n each state and the portfolo weghts. A smple numercal example s suffcent to llustrate ths pont. Consder two states, one wth a frm return of -20% and a market return of -10%, the second wth frm and market returns of -15%. For relatvely larger (smaller) fractons nvested n the frm, the former wll have a lower (hgher) portfolo return than the latter as llustrated n the table below. State w w m R R m R P % -10% -16% % -15% -15% % -10% -14% % -15% -15% In spte of ths dependence on the model parameters, there are some thngs that can be sad about the relevant measures of tal rsk n ths settng. Frst, holdng all else fxed, the more extreme the negatve returns on the frm, the larger s the LPM of the portfolo,.e., the greater the tal rsk. Due to the under-dversfed nature of the portfolo, stock-specfc rsk matters. In the context of tal rsk, a natural measure of ths stock-specfc rsk s the LPM of the stock return: 2 LPM ( R ) = ( R h ). (5) Second, agan holdng all else fxed, the more extreme the negatve returns on the market, the larger s the LPM of the portfolo. Therefore, n addton to the stock-specfc rsk, a stock s rsky to the extent that t contrbutes to the tal rsk of the market portfolo. The natural measure of ths component of rsk s the beta n the EL-CAPM settng: R < h R < h ( R h h m m β, LPM =. (6) 2 ( R h ) R < h Thrd, and perhaps most nterestng, s the rsk component assocated wth co-movement of the frm and the market n the tal of the dstrbuton of the portfolo return. If the tal events for the frm and the market concde, then these states wll also be the tal states for the portfolo return, and the LPM of the portfolo wll be hgh. On the other hand, f they do not concde, then m m m m )( R m m m ) 6

8 we need to develop an easly mplementable emprcal proxy for ths co-movement. As noted above, the dentty of the tal states for the portfolo depends on the model parameters that determne the orderng of the returns across states. State (j=1, k=1) s obvously the state wth the lowest portfolo return ndependent of portfolo weghts. The next lowest return state s ether (j=1, k=2) or (j=2, k=1) wth returns R R P,12 P,21 = w R = w R The former wll have the lower return as long as w,1,2 + (1 w ) R + (1 w ) R m,2 m,1 ( R, 2,1 m,2 m, 1. (7) R ) > (1 w )( R R ). (8) Ths smple nequalty generates some nsght. Specfcally, condtonng on states wth low frm returns (as opposed to low market returns),.e., selectng (j=1, k=2) versus (j=2, k=1), s the ntutvely correct thng to do as long as the frm s more volatle than the market. The dfference between returns across dscrete states s the analogue to volatlty, and, as long as the weght n the frm s suffcently hgh, the set of low portfolo return states wll be those wth low frm returns and varyng market returns rather than low market returns and varyng frm returns. Ths ntuton motvates the constructon of our hybrd measure of tal rsk, whch we call hybrd tal covarance rsk (H-TCR). Specfcally, we defne H - TCR = ( R h )( R h ). (9) R < h The key dstncton between ths measure and the LPM beta n equaton (6) s that H-TCR condtons on the states of the world wth low stock returns, not low market returns. Note that for the purposes of cross-sectonal analyses, the denomnator n equaton (6) s rrelevant snce t s equal across all stocks; t smply serves to normalze the systematc rsk measure. As a check on the possble economc mpact of ths dstncton, we perform a smple emprcal exercse. For each month t, one year of daly returns from month t to t 12 (approxmately 250 daly observatons) are used to determne the tal observatons for the market portfolo and also for ndvdual stocks at the 10% level,.e., we dentfy the 25 days on whch the market fell the most, and we also dentfy separately the 25 days on whch each ndvdual stock fell the most. We then count the number of days these two sets have n common for each ndvdual stock. The table below shows percentles for the number of common days over the sample perod. The medan number of common days s about 5.56, ndcatng that m m 7

9 there s only an approxmately 22% overlap between the tals of the market return dstrbuton and that of a typcal stock. Even the 99 th percentle for the number of common days s only (a 50% overlap). Clearly the tal events for the market and ndvdual stocks do not concde. In other words, tal events for ndvdual stocks are prmarly dosyncratc. Thus, there s a realstc possblty that H-TCR wll dffer sgnfcantly from downsde beta and, moreover, that ths rsk measure wll better capture tal rsk for nvestors wth meanngful fractons of ther wealth n concentrated postons. Percentles for the Number of Common Days n the 10% Tal 1% 5% 10% 30% 50% 70% 90% 95% 99% We also look drectly at the emprcal, cross-sectonal determnants of the LPM of a concentrated portfolo of the type descrbed n equaton (4). Usng daly returns over a one year perod, the LPM of the portfolo s calculated as 2 LPM ( R ) = ( R h ), (10) p R p < h p where the sum s taken over the days for whch the portfolo return s less than the specfed threshold. Intutvely, ths portfolo LPM wll depend on the three components of tal rsk dscussed above systematc, stock-specfc, and hybrd. We consder two sets of portfolos weghts 50% n the stock and 50% n the market, and 30% n the stock and 70% n the market and thresholds at the 10 th percentle of the relevant return dstrbutons. Each month, we look back over the precedng 12 months and calculate the four quanttes n equatons (5), (6), (9), and (10). We then run frm-level Fama-MacBeth crosssectonal regressons of LPM ( R p ) on LPM ( R ), β, LPM, and H-TCR for each month from July 1963 to December 2009: p LPM, t λ 0, t + λ1, t LPM, t + λ2, t β, LPM, t + λ3, t H - TCR, t + ε, t p p =. (11) For the 50/50 weghts, the average slope coeffcent on LPM s estmated to be 0.26 wth a Newey-West t-statstc of 480.0, the average coeffcent on H-TCR s 0.36 wth a t-statstc of 76.7, and the average coeffcent on β, LPM s wth a t-statstc of 3.8. All the coeffcents are statstcally sgnfcant, but the magntudes of the t-statstcs ndcate ther margnal explanatory power. The average R-squared of the monthly cross-sectonal regressons n equaton 8

10 (11) s 98.5%,.e., the three tal rsk measures capture almost all the cross-sectonal varaton n portfolo LPM. For the 30/70 weghts, the average estmated slope coeffcent on LPM s lower, 0.09, wth a t-statstc of 314.4, because the nvestor allocates a smaller amount to the ndvdual stock. The average coeffcent on H-TCR s also lower at 0.25 wth a t-statstc of 65.0, whle that on β, LPM s hgher at , wth a correspondng t-statstc of 4.0. The average R-squared of the monthly cross-sectonal regressons s 95.2%. In both cases, the sgnfcance of the coeffcent on H-TCR and the hgh explanatory power of the cross-sectonal regressons valdate our choce of the hybrd tal rsk measure. Moreover, whle varatons of the H-TCR measure gve smlar results, these varatons generally yeld lower cross-sectonal R-squareds than those for H-TCR. Overall, these emprcal results ndcate that the H-TCR s an approprate measure of rsk n our framework. Theoretcally, the dependence of H-TCR on the CLPM of frm and market returns follows from the assumpton that the LPM of the portfolo return s the correct measure of rsk at the portfolo level. Fnally, as s true wth any model that assumes concentrated holdngs, the total rsk of these ndvdual assets wll also contrbute to portfolo rsk and therefore may requre compensaton n equlbrum. 2. Data and Varable Defntons The frst dataset ncludes all New York Stock Exchange (NYSE), Amercan Stock Exchange (AMEX), and NASDAQ fnancal and nonfnancal frms from the Center for Research n Securty Prces (CRSP) for the perod from July 1962 through December We use daly stock returns to estmate alternatve measures of rsk. The second data set s COMPUSTAT, whch s prmarly used to obtan the book values for ndvdual stocks. For each month from July 1963 to December 2009, we compute the three tal rsk measures for each frm n the sample (1) the LPM of the return on the stock, (2) the LPM beta of the stock wth respect to the market, and (3) hybrd tal covarance rsk, as defned n equatons (5), (6), and (9). In all cases we use daly returns over the past year, except for certan 7 Followng Harrs (1994), Jegadeesh and Ttman (2001), and Ang, Chen, and Xng (2006), we remove small, llqud, and low-prced stocks from the sample. Specfcally, for each month, all NYSE stocks on CRSP are sorted by frm sze to determne the NYSE decle breakponts for market captalzaton. Then, we exclude all NYSE/AMEX/NASDAQ stocks wth market captalzatons that would place them n the smallest NYSE sze decle. We also exclude stocks whose prce s less than $5 per share. 9

11 extensons n Secton 5, and the return thresholds for the stock and market return are determned by the relevant emprcal percentles over the same sample. For much of the analyss we employ the 10 th percentle as our measure of the tal of the dstrbuton, but we also report results for thresholds rangng from the 5 th percentle to the 50 th percentle. We also employ an extensve set of control varables. As Subrahmanyam (2010) ponts out, over 50 varables have been shown to have predctve power for stock returns n the crosssecton. It s nfeasble to control for all of these varables, but we select both the most popular varables n the lterature and those that, ntutvely, are most lkely to be correlated wth our tal rsk measures. The frst four varables are those from the Fama-French-Carhart 4-factor model market beta, sze, book-to-market, and momentum. We also control for mcrostructure-related phenomena n the form of short-term reversals and lqudty. Fnally we nclude three varables co-skewness, total volatlty, and extreme postve returns that are drectly related to the dstrbuton of returns, and thus possbly tal rsk as well, and that have been shown to have sgnfcant predctve power. The detaled defntons of all these varables are provded n the Appendx. 3. Prelmnary Evdence Gven the number of potental control varables,.e., other stock characterstcs that may nfluence returns, the Fama-MacBeth cross-sectonal regresson approach may be the natural way to examne the predctve power of measures of tal rsk. We turn to these regressons n Secton 4; however, n order to get an ntal feel for the data, we frst look at unvarate sorts on the bass of our three tal rsk measures and the assocated characterstcs of the portfolos Average Returns for Unvarate Portfolo Sorts Table 1 presents the average monthly returns for the equal-weghted, decle portfolos that are formed by sortng the NYSE, AMEX, and NASDAQ stocks based on our three tal rsk measures H-TCR, LPM(R ), and β LPM. We also report the average across months of the medan tal rsk measure wthn each portfolo. The returns are reported for the sample perod July 1963 to December 2009, whle the measures of tal rsk are computed over the precedng year. Portfolo 1 (Low) contans stocks wth the lowest tal rsk n the prevous year and Portfolo 10 (Hgh) ncludes stocks wth the hghest tal rsk n the prevous year. 10

12 We turn frst to our new hybrd measure of tal covarance rsk. By constructon, the average H-TCR of ndvdual stocks n the unvarate sort ncreases monotoncally across the decles, from for Portfolo 1 to for Portfolo 10. Snce we are condtonng on states n whch the ndvdual stock return s less than the specfed threshold, the stock specfc term n equaton (9) s always negatve. Thus, a negatve (postve) H-TCR ndcates that the market term s postve (negatve), on average, n these same states. Postve and large H-TCRs correspond to stocks whose low returns concde wth those of the market as a whole. In other words, they have substantal tal rsk because a portfolo wth sgnfcant weghts n both the stock and the market wll tend to have returns n the left tal due to the concdence of tal events for both assets. As shown n the frst column, the average return of ndvdual stocks s about 0.65% per month for the low H-TCR decle (Portfolo 1) and 1.13% per month for the hgh H-TCR decle (Portfolo 10). The raw average return dfference between decle 10 and 1 s 0.48% per month wth a Newey-West (1987) t-statstc of In other words, there s evdence that our hybrd measure of tal rsk s prced n the cross-secton consstent wth the model n Secton 1. However, there s also some evdence of non-monotoncty n the average portfolo returns, and we have not yet made an effort to control for other prced rsks that may vary across these portfolos. The results for the other two tal rsk measures are n sharp contrast to those for H-TCR. When stocks are sorted on the LPM of ther daly returns over the past year, ths measure of stock-specfc tal rsk s negatvely assocated wth raw portfolo returns. That s, the average returns on stocks wth hgh LPMs are lower than those wth less rsk, wth a return dfference of % that s economcally large n magntude and margnally statstcally sgnfcant (wth a t- statstc of -1.77). Whle ths result s somewhat dsappontng from the perspectve of uncoverng prced tal rsk n our framework of under-dversfed holdngs, t s perhaps not totally surprsng. As we analyze n more detal below, LPM s correlated wth other measures of stock-specfc rsk, specfcally volatlty (Ang et al. (2006, 2009)) and extreme returns (Bal et al. (2011)), that have been shown to have a strong relaton to returns n the cross-secton. Thus, solatng the effect of stock-specfc tal rsk may be extremely dffcult. Fnally, our measure of systematc tal rsk, β LPM, s also negatvely assocated wth portfolo returns n the cross-secton n the sense that the returns on Portfolo 1 are larger than those on Portfolo 10. However, the dfference s statstcally nsgnfcant, and the portfolo 11

13 returns are clearly non-monotonc. In lght of the volumnous lterature attemptng, and n many cases falng, to fnd sgnfcant prcng of systematc rsk measures n the cross-secton, ths result s not totally unexpected Descrptve Statstcs for Tal Rsk Portfolos Whle the raw return dfferences between the hgh and low H-TCR decles are economcally and statstcally sgnfcant, the pattern across decles n raw returns s not qute monotonc. Moreover, stock-specfc tal rsk, as measured by LPM, appears to be negatvely prced n raw returns. These patterns n the data could be the result of addtonal prced rsk factors, and these factors mght also nfluence the rsk-adjusted return dfferences across portfolos. To hghlght the frm characterstcs and rsk attrbutes of stocks n the portfolos of Table 1, Table 2 presents descrptve statstcs for the stocks n the varous decles. Specfcally, Panels A through C report the average across the months n the sample of the medan values wthn each month of varous characterstcs for the stocks n each decle sorted by H-TCR, LPM(R ), and β LPM, respectvely. In each case, we report values for the three tal rsk measures, the prce (n dollars), the market beta, the log market captalzaton (n mllons of dollars), the book-to-market (BM) rato, the return over the 6 months pror to portfolo formaton (MOM), the return n the portfolo formaton month (REV), a measure of llqudty (scaled by 10 5 ), the co-skewness, the total volatlty, and the maxmum daly return n the portfolo formaton month (MAX). Defntons of these varables are gven n the Appendx. Table 2, Panel A reports the characterstcs for the portfolos sorted on H-TCR. 8 Our hybrd measure of tal rsk s postvely related to systematc tal rsk, as measured by β LPM, but non-monotoncally related to stock-specfc tal rsk, as measured by LPM(R ). Ths latter result s a manfestaton of the fact that many tal events for ndvdual stocks are dosyncratc. Stocks wth large dosyncratc negatve returns have hgh values of LPM but low values of H-TCR, whereas stocks wth large systematc negatve returns have hgh values of both LPM and H- TCR. Interestngly, stocks wth hgh H-TCR are larger, hgher prced, and more lqud stocks, on average. The ntuton behnd ths result s that whle smaller stocks tend to have more 8 The average across months of the medan H-TCR for each portfolo dffers slghtly from that reported n Table 1 because the sample s slghtly smaller due to the data requrements necessary to calculate some of the other varables. 12

14 extreme negatve returns, these tal events are also more lkely to be dosyncratc. Thus, n the context of our hypotheszed portfolos of concentrated postons n ndvdual stocks plus addtonal wealth n a well-dversfed fund, t s the larger stocks that generate more portfolo tal rsk after controllng for the stock-specfc component. Ths sze and lqudty dscrepancy suggests that the raw return dfference wll hold up to rsk adjustment on these dmensons. Large stocks and lqud stocks, on average, have low returns, whereas stocks wth low systematc rsk n the left tal (low H-TCR) are small, llqud stocks that should have hgh returns, all else equal. Apparently, n the raw returns, the effect of hybrd tal rsk domnates the effect of sze or lqudty on future returns. Market beta also ncreases as H-TCR ncreases, mplyng that stocks wth hgh hybrd covarance rsk n the lower tal of the stock return dstrbuton are more exposed to market rsk. Of course, ths systematc rsk lkely does not explan the raw return dfferences across portfolos n Table 1 snce market beta s weakly prced, at best, n the cross-secton of returns. In contrast, t wll be mportant to control for momentum when rsk-adjustng returns. Stocks wth hgh H-TCR (low H-TCR) are generally past wnners (losers) over an horzon of 6 months, and thus H-TCR-sorted portfolos should exhbt the well-documented ntermedate-term momentum phenomenon. On the other hand, both medan book-to-market ratos (BM) and average returns n the portfolo formaton month (REV) are smlar across the H-TCR portfolos, ndcatng no assocaton between H-TCR and the value premum or short-term reversals. COSKEW (Harvey and Sddque (2000)) measures the drecton and strength of the relaton between ndvdual stock returns and the squared market returns, mplyng that stocks wth hgh co-skewness have hgh (low) returns when market volatlty s hgh (low). A preference for postve skewness suggests a negatve prce for co-skewness rsk. Panel A ndcates that stocks wth hgh H-TCR also have hgh co-skewness, ndcatng that ths phenomenon s unlkely to be an explanaton for our results. The fnal 2 columns of Panel A examne two propertes of the stock return dstrbuton total volatlty and the prevalence of extreme postve returns both of whch have been lnked to expected returns n the lterature. Stocks wth hgh H-TCR seem to have somewhat lower volatlty and lower maxmum daly returns n the portfolo formaton month. Interestngly, the patterns across portfolos n both TVOL and MAX do superfcally resemble those n the raw returns n Table 1. 13

15 Panel B reports the same characterstcs as Panel A, for portfolos sorted on LPM(R ) rather than H-TCR. These characterstcs may suggest a potental explanaton for the anomalous negatve relaton between raw returns and stock-specfc tal rsk n Table 1. Clearly, such an explanaton cannot rely on market beta, sze, or llqudty, snce these effects go n the opposte drecton to the raw returns across the decles. More lkely canddates are the stock-specfc return dstrbuton measures, total volatlty (TVOL) and extreme postve returns (MAX). Both of these varables have a strong negatve relaton to returns n the cross-secton and ncrease monotoncally across the LPM(R )-sorted portfolos. Ths assocaton between LPM, volatlty, and extreme returns s both expected and probably dffcult to resolve emprcally. In Panel C, we report the characterstcs of portfolos sorted on our fnal tal rsk measure, systematc tal rsk as measured by β LPM. There s lttle or nothng surprsng n the results. Tal beta s postvely assocated market beta, co-skewness, volatlty, and extreme postve returns. To further understand the nteracton of tal rsk wth frm characterstcs and rsk attrbutes, we compute the frm-level cross-sectonal correlatons of all the varables for each month from July 1963 to December Table 2, Panel D reports the tme-seres averages of the cross-sectonal correlatons. Confrmng our earler fndngs at the portfolo level, hybrd tal rsk, H-TCR, s postvely correlated wth market beta, sze, momentum, and co-skewness and negatvely correlated wth llqudty, volatlty, and MAX. The correlatons of H-TCR wth book-to-market and reversals are very small, wth magntudes less than Stock-specfc tal rsk, LPM(R ), s strongly postvely correlated wth both volatlty and MAX, as s systematc tal rsk, β LPM. 4. Frm-Level Cross-Sectonal Regressons The unvarate-sort portfolo results n Table 1 are certanly consstent wth H-TCR beng prced n the cross-secton, but Table 2 dentfes a number of rsk factors that may play a role n the results. Therefore, we now examne the cross-sectonal relaton between H-TCR and expected returns at the frm level usng the Fama and MacBeth (1973) methodology. Specfcally, we run the followng multvarate specfcaton and nested versons thereof: R, t+ 1 = λ0, t + λ1, t X, t + λ2, tbeta, t + λ3, tsize, t + λ4, tbm, t + λ5, tmom, t + λ6, tz, t + ε, t+ 1, (12) 14

16 where X,t s one of the the three tal rsk measures H-TCR, LPM(R ), and β LPM ; BETA, SIZE, BM, and MOM are the four Fama-French-Carhart factors; and Z,t represents the possble ncluson of other control varables. Table 3, Panel A reports the tme seres averages of the slope coeffcents over the sample perod July 1963-December 2009 (558 monthly observatons) from the unvarate regressons of one-month ahead stock returns on our three tal rsk measures and multvarate regressons on each tal rsk measure wth the four Fama-French-Carhart factors. The average slopes provde standard Fama-MacBeth tests for determnng whch explanatory varables on average have nonzero premums, and the Newey-West adjusted t-statstcs are gven n parentheses. Not surprsngly, the unvarate cross-sectonal regresson results are consstent wth the raw return dfferences across portfolos from the unvarate portfolo sorts n Table 1. The average slope on H-TCR s 1.39 wth a t-statstc of Gven a dfference n medan H-TCR of approxmately 0.7 between the hgh and low H-TCR decles, ths coeffcent estmate translates nto a monthly return dfference of almost 1%, about twce the sze of the effect seen n Table 1. For both LPM(R ), and β LPM, the coeffcents are negatve, albet statstcally nsgnfcant. In the multvarate regressons, the coeffcents on the Fama-French-Carhart factors are also as expected. The average slope on BETA s negatve and statstcally nsgnfcant, whch s consstent wth pror emprcal evdence. The average slope on SIZE s negatve, but only sporadcally sgnfcant at the 10% level. Ths lack of statstcal sgnfcance s due to the excluson of small and low-prced stocks from our sample, the sector of the market n whch the tradtonal sze effect s concentrated. There s a sgnfcantly postve value premum as the average slope on BM s postve and sgnfcant wth t-statstcs between 2.6 and 2.8. Fnally, stocks exhbt strong ntermedate-term momentum. Of greater nterest are the average slope coeffcents on the tal rsk measures. In the multvarate regresson, the coeffcent on H-TCR s 1.01 wth a t-statstc of The mpled economc magntude of the effect s slghtly lower than n the unvarate regresson, but at a dfference of approxmately 70 bass ponts per month between medan stocks n the hgh and low H-TCR decles, t s stll large. The slght attenuaton s due partly to the ncluson of the momentum factor, wth whch H-TCR s postvely correlated. The coeffcent n a smlar regresson wth only the three Fama-French factors s larger. 15

17 Controllng for the addtonal rsk factors has a large effect on the LPM(R ) coeffcent. The magntude almost doubles to -0.12, and t becomes statstcally sgnfcant at all standard levels wth a t-statstc of However, the sgn on the coeffcent s nconsstent wth LPM(R ) beng a measure of prced tal rsk hgh rsk stocks apparently have lower expected returns. Ths anomalous result strongly suggests that ths varable s proxyng for an omtted factor. For systematc tal rsk, the ncluson of the addtonal factors has lttle effect. The coeffcent remans small n magntude and of the wrong sgn, albet statstcally nsgnfcant. Table 3, Panel B reports results for multvarate regressons that nclude each of the addtonal control varables from Table 2 n turn. The coeffcents on these varables are generally n lne wth the exstng lterature. The average slope on REV s negatve and hghly sgnfcant, mplyng that stocks exhbt strong short-term reversals. There s evdence of statstcally sgnfcant coeffcents on llqudty and co-skewness, but n both cases the sgn s the opposte of that mpled by the theory. For llqudty, ths result s lkely due to the excluson of small and low-prced stocks from the sample. Fnally, consstent wth the fndngs of Ang et al. (2006) and Bal, Cakc, and Whtelaw (2011), the results ndcate a negatve and sgnfcant relaton between expected returns and the TVOL and MAX varables. Agan, t s the tal rsk varables that are of prmary nterest. For H-TCR, the ncluson of controls for return reversals, llqudty, or co-skewness has lttle effect. However, both TVOL and MAX reduce the coeffcent on H-TCR by approxmately 50%. The effects of these control varables are extremely strong n the data wth a sgn opposte to that of H-TCR, and they are negatvely correlated wth H-TCR. Nevertheless, H-TCR remans economcally and statstcally sgnfcant n all the specfcatons. Interestngly, the pattern of coeffcents across regressons s smlar for LPM(R ). Reversals, llqudty and co-skewness have lttle effect, but the ncluson of ether TVOL or MAX reduces the coeffcent by about 50%. In ths case, the sgn of the coeffcent on LPM(R ) and the control varables s the same, but the varables are postvely correlated. Regardless of the specfcaton, the anomalous sgn on stock-specfc tal rsk s preserved. For systematc tal rsk, t s the control for co-skewness that has the largest effect on the results. When ths control varable s added to the regresson, the magntude of the coeffcent on β LPM ncreases and becomes statstcally sgnfcant. However, gven that the sgns of the 16

18 coeffcents on both systematc tal rsk and co-skewness are the opposte of those mpled by theory, ths partcular result should be nterpreted wth cauton. The tal rsk measures n Tables 1, 2, and 3 are constructed usng the 10% tals of the relevant return dstrbutons over the precedng year. The choce of 10% s essentally arbtrary, although t ntutvely provdes a reasonable tradeoff between a suffcent number of observatons to lmt estmaton error and the desre to get a measure of tal rsk rather than more general downsde rsk. Moreover, there s the more fundamental ssue of the nature of nvestor preferences or the asymmetry propertes of jont return dstrbutons that generate a role for tal rsk. To partally address both these ssues, we re-run the multvarate cross-sectonal regressons above wth our three tal rsk measures and the Fama-French-Carhart rsk factors, varyng the fracton of the lower half of the return dstrbuton over whch we calculate tal rsk. The results are reported n Table 4, whch contans the average slope coeffcents and assocated t-statstcs for the tal rsk measures but omts the coeffcents for the other rsk factors n the nterest of brevty. The results n the second column (labeled 10% ) are the same as those reported n Table 3, Panel A. For H-TCR, the pattern n coeffcents as the defnton of the tal of the dstrbuton changes s consstent wth the theoretcal ntuton. Both the coeffcent and statstcal sgnfcance peak when H-TCR s calculated usng the 10% tal. When H-TCR s measured over the full lower half of the dstrbuton,.e., when t becomes a downsde rsk rather than tal rsk measure, the magntude of coeffcent s small, the sgn s reversed, and t s statstcally nsgnfcant. The pattern for stock-specfc rsk, as measured by LPM(R ), s markedly dfferent. The coeffcent remans large n magntude and ts statstcal sgnfcance s preserved for all tal values from 5% to 50%. Ths evdence confrms the concluson above that the sgnfcance of LPM(R ) s not due to ts ablty to pck up tal rsk at all. Instead, t s proxyng for the more general features of the return dstrbuton assocated wth volatlty and extreme returns. One should not conclude from these results that stock-specfc tal rsk s unprced, rather that dsentanglng the prcng of ths rsk from the prcng of related dstrbutonal rsks s an extremely challengng emprcal exercse, whch s beyond the scope of ths paper. Fnally, the negatve but statstcally nsgnfcant coeffcents for systematc tal rsk are evdent across the varous defntons of the tal. Lke many exstng papers, we are unable to 17

19 document that systematc rsk, n our case tal rsk, s prced n the cross-secton. However, the results for the 50% tal appear to contradct those of Ang, Chen, and Xng (2006), who report sgnfcant compensaton n the form of hgher expected returns for those stocks wth hgher systematc downsde rsk. Our sample perod and methodology dffer somewhat, but the prmary reason for the dscrepancy appears to be sample selecton. Ang, Chen, and Xng (2006) restrct ther sample to NYSE stocks and elmnate the 20% of these stocks wth the hghest volatlty, whereas we use the broader NYSE/AMEX/NASDAQ sample and elmnate small and low prced stocks due to concerns about lqudty and the assocated mcrostructure ssues. The clear concluson from our emprcal analyss s that cross-sectonal regressons provde strong evdence for an economcally and statstcally sgnfcant postve relaton between hybrd tal covarance rsk and future returns, consstent wth models that suggest that rsk n the left tal of portfolo returns s prced and that prces are nfluenced by nvestors wth concentrated holdngs n ndvdual securtes and postons n more dversfed portfolos. The evdence for both stock-specfc and systematc tal rsk measures s more mxed. In the former case, the results strongly suggest an nablty to dstngush the desred effect from other effects assocated wth the stock-specfc dstrbuton of returns. In the latter case, the effect may be too small to detect wth any degree of precson gven measurement ssues. 5. Further Evdence on Hybrd Tal Covarance Rsk Gven the evdence above of statstcally and economcally sgnfcant prcng of hybrd tal covarance rsk n the cross-secton of stocks, we now proceed to examne ths phenomenon n more detal, examnng frst the tme seres persstence of H-TCR and then turnng to alternatve measures of tal covarance rsk and tal beta Persstence As s approprate for a study of cross-sectonal expected returns, we calculate tal rsk over a specfc tme wndow (one year n the prevous analyses) and examne returns over the subsequent month. However, n a ratonal settng, nvestors should only care about hstorcal hybrd tal covarance rsk to the extent that t predcts future rsk. Alternatvely, f one vews the phenomenon we document as msprcng, then the portfolo turnover assocated wth a strategy that explots ths msprcng s of nterest. 18

20 To examne ths ssue we compute the transton matrx for decle portfolos formed by sortng on H-TCR. Table 5, Panel A reports the average of the month-to-month transton matrces for the stocks n these portfolos,.e., the average probablty (n percent) that a stock n decle (as gven by the rows of the matrx) n one month wll be n decle j (as gven by the columns of the matrx) n the subsequent month. Because we calculate H-TCR over one year, there s an 11-month overlap between H-TCR calculated n two adjacent months, whch generates persstence n portfolo membershp by constructon. Thus, the dagonals of the transton matrx reflect both the overlap and the persstence n tal rsk. For the extreme decles, portfolos 1 (Low H-TCR) and 10 (Hgh H-TCR), more than 80% of the stocks reman n the extreme decle n the followng month. To elmnate the persstence caused by the overlap, Panel B reports the average of the 12- month lag transton matrces for the stocks n these portfolos,.e., the average probablty (n percent) that a stock n decle (as gven by the rows of the matrx) n one month wll be n decle j (as gven by the columns of the matrx) 12 months later. Despte the hgh hurdle presented by the one year lag, ths transton matrx also shows substantal evdence of persstence. Approxmately 25% of stocks n the extreme portfolos are stll n the same portfolos a year later, and more than 40% of the stocks are n the top or bottom two decles. In other words, H-TCR predcts both returns and future tal rsk n the cross-secton Alternatve Measures of Hybrd Tal Covarance Rsk As presented n equaton (9), we have so far used one year of daly extreme returns from month t to t 11 to estmate hybrd tal covarance rsk (H-TCR) for predctng returns n month t+1. Extreme daly returns are obtaned from the 10% lower tal of the daly stock return dstrbuton over the past 12 months. In order to get addtonal nsght nto the relaton between tal rsk and returns, we generate alternatve measures of tal covarance rsk and test ther predctve power for the cross-secton of expected returns. Specfcally, alternatve measures of H-TCR are computed based on the daly returns from the () 5% lower tal of the daly return dstrbuton over the past 12 months; () 10% lower tal of the daly return dstrbuton over the past 6 months; () 20% lower tal of the daly return dstrbuton over the past 6 months; and (v) 20% lower tal of the daly return dstrbuton over 19

21 the past 3 months. As the calculaton wndow s reduced, we ncrease the tal sze n order to keep the number of daly observatons used n the computng H-TCR suffcently large. Table 6, Panel A shows that for all measures of H-TCR the average raw return ncreases when movng from the low H-TCR to the hgh H-TCR portfolo. The average raw return dfferences are n the range of 0.48% to 0.70% per month wth Newey-West t-statstcs rangng from 2.74 to The results, both n terms of the magntude of the return dfference and ts statstcal sgnfcance, seem to mprove as the wndow length s reduced,.e., we use more recent nformaton, and as we focus on a smaller fracton of the more extreme tal observatons. In addton to the average raw returns, Panel A also presents the magntude and statstcal sgnfcance of the dfference n ntercepts from the regresson of the equal-weghted H-TCR portfolo returns on a constant, the excess market return, a sze factor (SMB), a book-to-market factor (HML), followng Fama and French (1993), and Carhart s (1997) momentum (MOM) factor. 9 The four-factor alpha dfferences between the low H-TCR and hgh H-TCR portfolos are n the range of 0.46% to 0.80% per month and are hghly sgnfcant, wth the t-statstcs rangng from 3.01 to These results ndcate an economcally and statstcally sgnfcant, postve relaton between hybrd tal covarance rsk and the cross-secton of expected returns. An nvestment strategy that goes long stocks n the hghest H-TCR decle and shorts stocks n the lowest H-TCR decle produces average raw and rsk-adjusted returns n the range of 5.8% to 8.4% on an annualzed bass. A fnal pont from Table 6, Panel A s that, n terms of economc and statstcal sgnfcance, the 10% H-TCR measure over the past one year, the bass for all our prevous analyses, s domnated by both the 10% H-TCR over 6 months and the 20% H-TCR over 3 months. Hence, the earler return dfferences n long-short portfolos and the rsk premums n frm-level cross-sectonal regressons should be vewed as conservatve. 9 SMB (small mnus bg), HML (hgh mnus low), and MOM (wnner mnus loser) are descrbed n and obtaned from Kenneth French s data lbrary: 10 The three-factor Fama-French alpha dfferences are even larger and more statstcally sgnfcant than those that control for momentum; however, we report only the latter for the sake of brevty. 20

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