MANAGEMENT SCIENCE Vol. 54, No. 6, June 2008, pp. 1052 1064 ssn 0025-1909 essn 1526-5501 08 5406 1052 nforms do 10.1287/mnsc.1070.0845 2008 INFORMS How Do Decson Frames Influence the Stock Investment Choces of Indvdual Investors? Alok Kumar McCombs School of Busness, Unversty of Texas at Austn, Austn, Texas 78712, akumar@mal.utexas.edu Sonya Seongyeon Lm Kellstadt Graduate School of Busness, DePaul Unversty, Chcago, Illnos 60604, slm1@depaul.edu Ths study examnes whether the framng mode (narrow versus broad) nfluences the stock nvestment decsons of ndvdual nvestors. Motvated by the expermental evdence, whch suggests that separate decsons are more lkely to be narrowly framed than smultaneous decsons, we propose trade clusterng as a proxy for narrow framng. Usng ths framng proxy, we show that nvestors who execute more clustered trades exhbt weaker dsposton effects and hold better-dversfed portfolos. We also fnd that the degree of trade clusterng s related to nvestors stock preferences and portfolo returns. Collectvely, the evdence ndcates that the choce of decson frames s lkely to be an mportant determnant of nvestment decsons. Key words: narrow framng; trade clusterng; dsposton effect; portfolo dversfcaton; prospect theory Hstory: Accepted by George Wu, decson analyss; receved September 14, 2006. Ths paper was wth the authors 5 months for 2 revsons. Publshed onlne n Artcles n Advance May 7, 2008. 1. Introducton Tradtonal portfolo choce models post that nvestors formulate ther tradng decsons by maxmzng expected utlty defned over ther total wealth. In these models, nvestors evaluate each nvestment choce accordng to ts mpact on the aggregate wealth. However, the extant evdence from psychologcal research suggests that people tend to consder each decson as unque, often solatng the current choce from ther other choces. In other words, people often engage n narrow framng (e.g., Kahneman and Lovallo 1993, Kahneman 2003), where the nteractons among multple decsons are often gnored. For example, when offered a monetary gamble, an ndvdual wth a narrow decson frame would evaluate the potental payoffs from the gamble n solaton wthout combnng them wth her exstng wealth. In contrast, an ndvdual wth a broad decson frame would ntegrate the potental payoffs from the gamble wth her exstng wealth and evaluate the combned outcomes before makng a choce. The noton of framng s applcable to decson makng n very general settngs, ncludng nvestment decsons. Tversky and Kahneman (1981, p. 453) defne a decson frame as the decson-maker s concepton of the acts, outcomes, and contngences assocated wth a partcular choce. For any gven decson problem, many dfferent decson frames can be potentally nduced. The frame that s eventually chosen s nfluenced by the formulaton of the problem and the personal characterstcs and habts of the person makng the decson. For nstance, when people use smple heurstcs and make decsons n an ntutve manner, they are lkely to adopt the most readly avalable frame, whch s often narrow and suboptmal (Kahneman 2003). People mght also adopt narrow decson frames due to ther lmted cogntve capactes and because ntegraton requres sgnfcant cogntve costs (e.g., Mller 1956; Smon 1957; Tversky and Kahneman 1981, 1986). Some ndvduals would ntentonally choose a decson frame that ncreases perceved utlty and makes the outcome appear more favorable to them (Thaler 1985). In partcular, hedonc optmzers would pck a narrow decson frame that segregates gans nstead of usng a broad decson frame that combnes all gans (e.g., Thaler and Johnson 1990, Lm 2006). Addtonally, people can engage n narrow framng because nonconsumpton utlty such as regret nfluences ther decsons (Barbers et al. 2006). Although the mportance of framng n the contexts of portfolo choce and asset prcng has been recognzed n the recent theoretcal lterature (e.g., Barbers and Huang 2001, 2007), emprcal tests of narrow framng outsde of laboratores have been almost nonexstent. In ths study, we use the portfolo holdngs and trades of a group of ndvdual nvestors at a large U.S. dscount brokerage house and examne the nfluence of narrow framng on ther stock nvestment decsons. We focus on narrow framng n a cross-sectonal context, where people make each tradng decson n solaton and are unable or unwllng to aggregate gans and losses of ndvdual stocks n ther portfolos. Ths type of framng effect 1052
Management Scence 54(6), pp. 1052 1064, 2008 INFORMS 1053 s related to, but dstnct from, ntertemporal framng effects, where people s perceptons of gans and losses are nfluenced by the evaluaton perods (e.g., Benartz and Thaler 1995, Gneezy et al. 2003). We examne two mportant stock nvestment decsons that are lkely to be nfluenced by the degree of narrow framng. Frst, we consder the dsposton effect, whch refers to the emprcal observaton that nvestors are reluctant to realze losses but more wllng to sell the wnners n ther portfolos (Shefrn and Statman 1985). We conjecture that some nvestors exhbt a stronger dsposton effect because they narrowly frame stock-level gans and losses nstead of adoptng a broader frame that consders the overall portfolo performance. Under the prospect-theoretc explanaton of the dsposton effect, the tendency to frame decsons narrowly at the ndvdual stock level nstead of framng them broadly at the portfolo level nfluences nvestors propensty to sell portfolo wnners and losers. The value functon n prospect theory s concave n the gan doman and convex n the loss doman. When the value functon s defned for each stock, ths S-shape mples that nvestors exhbt rskaverse behavor when contemplatng the sale of a wnner. They dsplay a greater propensty to sell the wnner to obtan a certan payoff. However, when they face the decson to sell a loser, nvestors exhbt rsk-seekng behavor. In ths scenaro, they prefer to hold on to the loser rather than sellng t to experence a certan loss. Thus, narrow framng nvestors who are senstve to the performance of ndvdual stocks nstead of the overall portfolo performance exhbt a greater propensty to sell the wnners, relatve to the losers, n ther portfolos. In contrast, nvestors who adopt a broader decson frame and evaluate the total performance of ther respectve portfolos are less lkely to exhbt ths asymmetry, even f they evaluate ther portfolo gans and losses usng a prospect-theoretc value functon. 1 Second, we examne whether the degree of narrow framng nfluences nvestment choces even when they are not nduced by prospect-theoretc preferences. Specfcally, we focus on nvestors portfolo dversfcaton decsons and examne whether some nvestors underdversfy because they narrowly examne the rsks of ndvdual stocks and do not use a broader decson frame to evaluate the aggregate portfolo rsk that takes nto account the correlaton structure of the portfolo. Certan stocks appear 1 Barbers and Xong (2008) pont out that prospect theory does not necessarly lead to the dsposton effect n a sngle stock settng. However, framng at the ndvdual stock level s essental to explan why nvestors exhbt dfferent sellng propenstes toward wnners and losers wthn the same portfolo. unattractve when consdered n solaton, but they can brng consderable dversfcaton benefts when added to an exstng portfolo. Investors who frame ther decsons narrowly are unlkely to perceve such benefts. In contrast, nvestors who frame ther decsons broadly would examne the ncremental effect of each stock on the rskness of ther respectve portfolos and, therefore, they would hold relatvely betterdversfed portfolos. In sum, our man hypothess posts that nvestors who frame ther decsons narrowly would exhbt stronger dsposton effects and weaker dversfcaton sklls. Because nvestors decson frames cannot be observed, to test ths hypothess we rely on the fndngs n the expermental psychology lterature and dentfy an approprate narrow framng proxy. The extant expermental evdence (e.g., Tversky and Kahneman 1981, 1986; Redelmeer and Tversky 1992) ndcates that the decson frames that people adopt are nfluenced by the manner n whch dfferent alternatves are presented to them. In partcular, the tme nterval between two consecutve decsons s lkely to nfluence the decson frame, where temporally separated decsons are more lkely to be framed narrowly than smultaneous decsons (e.g., Read and Loewensten 1995, Read et al. 1999). 2 Motvated by ths psychologcal evdence, we propose the degree of clusterng n trades as a proxy for the level of narrow framng n stock nvestment decsons. Specfcally, we post that, ceters parbus, nvestors who execute less-clustered trades are more lkely to use narrower decson frames n ther nvestment decsons than are nvestors who execute more clustered trades. Usng the narrow framng proxy, we fnd that nvestors who execute less-clustered trades exhbt a greater propensty to realze gans than losses, and thus, they exhbt greater dsposton effects. The negatve relaton between trade clusterng and the dsposton effect remans strong even when we control for the effects of demographc and portfolo characterstcs that have been dentfed as mportant determnants of the dsposton effect n prevous studes (e.g., Feng and Seasholes 2005, Dhar and Zhu 2006). Examnng the relaton between trade clusterng and portfolo dversfcaton, we fnd that nvestors who execute more clustered trades make relatvely better dversfcaton decsons and construct portfolos wth relatvely lower varances. The dversfcaton trade clusterng relaton remans strong when we control for other known determnants of portfolo dversfcaton (e.g., Goetzmann and Kumar 2008). Because we use a narrow framng proxy nstead of a drect framng 2 Also, see Thaler and Johnson (1990), Lnvlle and Fscher (1991), and Hrst et al. (1994).
1054 Management Scence 54(6), pp. 1052 1064, 2008 INFORMS measure, we conduct several robustness checks and show that the dsposton effect trade clusterng and the dversfcaton trade clusterng relatons are robust and are not mechancally nduced. Our emprcal evdence also ndcates that the degree of trade clusterng s related to nvestors style preferences and portfolo performance. Investors who execute less-clustered trades exhbt a preference for small-cap and value stocks, and they earn hgher raw returns but lower rsk-adjusted returns. Collectvely, the emprcal evdence supports our man hypothess and ndcates that the framng mode s an mportant determnant of ndvdual nvestors stock nvestment decsons. The rest of ths paper s organzed as follows: In 2, we brefly descrbe the data and defne our tradeclusterng measures. We present our man emprcal results n 3 5 and summarze the results from our robustness tests n 6. We conclude n 7 wth a bref summary of the paper. 2. Clustered Trades and Framng Decsons 2.1. Data Sources The prmary data for our study consst of a sx-year (1991 1996) panel of all executed trades and monthly portfolo postons of a group of ndvdual nvestors at a major U.S. dscount brokerage house. For a subset of households, demographc nformaton such as age, ncome, occupaton, martal status, gender, etc. s also avalable. 3 There are 77,995 households n the database, of whch 62,387 have traded n stocks. From ths group, we choose 41,039 nvestors who have executed a mnmum of fve trades durng the sx-year sample perod. The portfolo and demographc characterstcs of nvestors n our chosen sample are very smlar to those of the overall sample. A typcal nvestor n our sample holds a four-stock portfolo (medan s three) wth an average sze of $35,629 (medan s $13,869) and executes fewer than 10 trades per year, where the average trade sze s $8,779 (medan s $5,239). Fewer than 10% of the nvestors hold portfolos over $100,000 and less than 5% hold more than 10 stocks. These nvestors execute an average of 41 trades (medan s 19) durng the sx-year sample perod, where the average monthly portfolo turnover rate (the average of purchase and sales turnover rates) s 6.59% (medan s 2.53%). Further detals on the nvestor database are avalable n Barber and Odean (2000). 3 The demographc nformaton s ether self-reported at the tme the brokerage account s opened or gathered through a survey conducted at the end of the sample perod. See Barber and Odean (2001) for detals. In addton to the ndvdual nvestor database, for each stock n our sample, we obtan monthly prces and returns from the Center for Research on Securty Prces. We also obtan the monthly tme-seres of Fama-French factors and the momentum factor from Professor Ken French s data lbrary avalable at http:// mba.tuck.dartmouth.edu/pages/faculty/ken.french/. 2.2. Potental Narrow Framng Proxes We measure the degree of narrow framng usng the degree of clusterng n nvestor s trades. In partcular, we dentfy whether an nvestor executes trades separately (.e., one trade at a tme) or multple trades smultaneously. One potental measure that captures the degree of trade clusterng (TC) s TC = 1 NTDAYS NTRADES (1) where NTDAYS s the total number of days on whch nvestor trades stocks, and NTRADES s the total number of stock trades executed by nvestor durng the sample perod. On any gven day, multple nvestor trades n the same stock are aggregated nto a sngle trade. A low TC measure for an nvestor ndcates that her trades are temporally separated and, thus, the degree of narrow framng s lkely to be hgher. In partcular, the trade-clusterng measure s zero for nvestors who execute each trade on a separate day. These nvestors are more lkely to adopt narrower decson frames n ther nvestment choces. For robustness, we consder alternatve measures of trade clusterng. Specfcally, we borrow the ndex of cluster sze (ICS) measure from the spatal analyss lterature (e.g., Baley and Gatrell 1995), whch measures clusterng usng the tme nterval between two consecutve trades. The measure explots the property that completely random events wth no clusterng follow a Posson process. There s evdence of clusterng f the varance of the tme nterval between events s greater than the varance of the tme nterval under the Posson process. We also measure TC and ICS based only on stock purchases. These alternatve clusterng measures are strongly correlated wth TC (all correlatons are above 0.760) and yeld very smlar results. For brevty, we only report the results usng the TC measure defned n Equaton (1). 2.3. Trade Clusterng at the Aggregate Level To begn, we compute the trade-clusterng measures for all nvestors who execute at least fve trades durng the sample perod. The mean TC measure s 0.226 (medan s 0.200), whch ndcates that a typcal nvestor n our sample executes 10 trades over 7.74
Management Scence 54(6), pp. 1052 1064, 2008 INFORMS 1055 tradng days. 4 About 16% of all nvestors execute each trade on a dfferent tradng date and, thus, they have a zero TC measure. Although a sgnfcant number of nvestors n our sample have a zero TC score, there s consderable heterogenety n the extent to whch ther trades are clustered over tme. About 10% of nvestors have TC greater than 0.50, and these nvestors execute an average of two trades or more on the days they trade. 2.4. Trade Clusterng and Investor Characterstcs How do the personal characterstcs of nvestors who execute more clustered trades dffer from those who execute temporally separated trades? Table 1, Panel A reports the average characterstcs of nvestors n the 10 TC decles. The TC measure s computed for each nvestor, usng all her trades executed durng the entre sample perod. The sortng results ndcate that nvestors n the hghest TC decle hold larger portfolos than those n the bottom decle. The average portfolo sze s $55,770 n TC decle 10 and $21,690 n TC decle 1. Investors n the hghest TC decle also execute more trades per year, but wth a smaller sze, compared wth those n the lowest TC decle. Moreover, hgh-tc nvestors have hgher turnover rates, although the relaton between TC and portfolo turnover s not monotonc. Those nvestors also have hgher ncomes, are slghtly older, and allocate a larger share of ther portfolos to mutual funds. To examne the relatve nfluences of demographc and portfolo varables on trade clusterng, we estmate a cross-sectonal regresson model, where the trade-clusterng varable s the dependent varable and demographc and portfolo varables are employed as ndependent varables. All varables are standardzed so that we can compare the relatve nfluence of each ndependent varable on the dependent varable. The regresson estmates are reported n Table 1, Panel B. The coeffcent estmates are broadly consstent wth the unvarate results reported n Panel A. We fnd that TC ncreases wth ncome, age, portfolo sze, trades per year, and number of stocks n the portfolo. Moreover, TC s hgher for nvestors who trade foregn securtes and mutual funds, whch suggests that TC mght be assocated wth a stronger preference for dversfcaton. 3. Trade Clusterng and the Dsposton Effect In the frst set of formal tests, we examne the frst part of our man hypothess, whch posts that 4 These statstcs do not change n a sgnfcant manner f we use dfferent cutoffs for mnmum trades. For nstance, wth 10 mnmum trades, the mean TC measure s 0.236 (medan s 0.207), whch s only margnally dfferent from the mean of 0.226 (medan of 0.200). nvestors who adopt narrower decson frames are lkely to exhbt a stronger dsposton effect. 3.1. The Dsposton Effect Measure We use Odean s (1998) PGR-PLR methodology to measure the dsposton effect of each nvestor. Consderng the actual trades and potental trades of nvestor durng the sample perod, we compute the proportons of gans realzed (PGR) and proporton of losses realzed (PLR) as N gr PGR = Ngr + N PLR = N lr gp N lr + N (2) lp where Ngr (N lr ) s the number of trades by nvestor wth a realzed gan (loss), and Ngp (N lp ) s the number of potental or paper trades for nvestor wth a gan (loss). We compute the dsposton effect (DE) of nvestor as DE = PGR PLR (3) A postve value of DE ndcates that a smaller proporton of losers s sold compared wth the proporton of wnners sold and, thus, nvestor exhbts the dsposton effect. 3.2. Peer Group Adjusted Trade Clusterng and Dsposton Effect Measures Odean (1998) notes that the PLR and PGR measures are senstve to portfolo sze and tradng frequency. Both proportons are lkely to be smaller for nvestors who hold larger portfolos and trade frequently because those portfolos contan a larger number of stocks wth captal gans and captal losses. If the DE measure s employed n a cross-sectonal analyss n ts orgnal form, these dependences are lkely to nduce mechancal assocatons between the DE and the varables that are correlated wth portfolo sze and tradng frequency. Because the tradeclusterng measure s correlated wth portfolo sze, number of stocks, and tradng frequency (see Table 1), there mght be concerns about a potental mechancally nduced relaton between TC and the DE. To guard aganst the possblty of ths mechancal relaton, we mnmze the potental nfluences of portfolo sze, number of stocks, and tradng frequency on TC and DE and defne peer group adjusted measures of both trade clusterng and the dsposton effect. 5 We proceed as follows: Frst, we perform an ndependent double sort usng the portfolo 5 Followng Dhar and Zhu (2006), we also expermented wth other related measures of the ndvdual-level dsposton effect that are not senstve to portfolo sze and tradng frequency. Specfcally, we used the followng two DE measures: () N N /N + N and gr lr gr lr () N /N N /N gr lr gp lp. The results wth these alternatve DE measures are very smlar to the reported results.
1056 Management Scence 54(6), pp. 1052 1064, 2008 INFORMS Table 1 Trade Clusterng and Investor Characterstcs Panel A: Summary statstcs TC decle PSze TSze Income Age PTurn MFund TPY Low TC (TC = 0) 21 69 8 88 86 92 49 15 5 48 0 18 2 95 D2 (0 < TC 0 063) 25 01 9 69 86 68 49 30 6 35 0 18 3 95 D3 (0 063 < TC 0 111) 29 99 9 01 86 72 49 58 7 72 0 19 5 83 D4 (0 111 < TC 0 154) 30 01 8 90 89 69 49 80 8 02 0 20 6 15 D5 (0 154 < TC 0 200) 37 53 8 76 88 98 50 49 7 68 0 20 7 91 D6 (0 200 < TC 0 250) 33 67 8 66 88 04 50 60 8 56 0 20 8 16 D7 (0 250 < TC 0 300) 37 88 9 03 87 91 50 58 8 42 0 20 9 84 D8 (0 300 < TC 0 375) 38 70 8 67 92 60 50 99 8 28 0 20 11 01 D9 (0 375 < TC 0 490) 45 18 8 45 92 66 51 68 7 39 0 22 11 98 Hgh TC (TC > 0 490) 55 77 7 72 94 57 51 56 7 90 0 26 50 41 Hgh TC Low TC 34 07 1 16 7 65 2 41 2 42 0 08 47 46 Panel B: Regresson estmates Varable Estmate t-statstc Intercept 0 004 0 661 Income 0 057 4 509 Log age 0 033 2 490 Professonal dummy 0 005 0 293 Retred dummy 0 047 2 738 Trade sze 0 041 2 101 Portfolo turnover 0 038 1 920 Mutual fund ownershp 0 062 4 711 Short sell dummy 0 005 0 545 Opton dummy 0 040 0 588 Foregn dummy 0 056 4 712 Number of stocks 0 189 12 227 Log trades per year 0 235 10 611 Portfolo sze 0 072 3 062 Number of nvestors 10,755 Adjusted R 2 10 88% Notes. Usng the trade-clusterng measure for the entre sample perod, we rank households and form 10 nvestor groups. The tradeclusterng measure for household s defned as, TC = 1 (NTDAYS )/(NTRADES ), where NTDAYS s the total number of days on whch household traded stocks and NTRADES s the total number of stock trades by household. In Panel A, we report the mean values of several nvestor and portfolo attrbutes for the 10 TC decles. Portfolo sze (t PSze) s the average sze (n thousand dollars) of the household portfolo, trade sze (TSze) s the average trade sze (n thousand dollars), ncome s the total annual household ncome (n thousand dollars), Age s the age of the head of the household, portfolo turnover (PTurn) s the average of monthly buyand-sell turnover rates, trades per year (TPY) s the number of trades executed per year by a household, and mutual fund ownershp (MFund) s the mutual fund holdng of a household as a fracton of the total equty portfolo. In Panel B, we report the cross-sectonal regresson estmates, where TC s the dependent varable and a set of nvestor characterstcs and portfolo varables are employed as ndependent varables. The professonal and retred dummy varables represent the occupaton categores where the professonal job category ncludes nvestors who hold techncal and manageral postons. The remanng nvestors belong to the nonprofessonal category that conssts of blue-collar workers, sales and servce workers, and clercal workers. Short sell dummy s a dummy varable that s set to one f a household makes at least one short sale durng the sample perod, opton dummy s set to one f a household makes at least one opton trade durng the sample perod, foregn dummy s set to one f a household makes at least one trade n a foregn asset (ADR, foregn stock, or a closed-end country fund), and number of stocks s the average number of stocks n the nvestor portfolo durng the sample perod. In Panel A, we use the Kolmogorov-Smrnov test to examne the statstcal sgnfcance of the dfferences n nvestor characterstcs. In Panel B, all varables are standardzed (mean s set to zero and the standard devaton s one) and the standard errors are corrected for heteroskedastcty. Denotes sgnfcance at the 1% level. sze and the mean monthly tradng frequency measures, and defne a 10 10 grd. Wthn each portfolo sze-tradng frequency category, we further sort portfolos usng the average number of stocks measure, and defne portfolo quntles. 6 Altogether, there are 6 The results are qualtatvely smlar when a coarser (5 5) or a fner (20 20) grd s employed to dentfy the peer groups. 500 bns, where each bn represents a peer group. Each nvestor s mapped nto one of the 500 bns, and a peer group s assgned to her. The peer groups contan an average of 71 nvestors. The largest peer group conssts of 212 nvestors, and no peer group has fewer than 23 nvestors. To facltate cross-sectonal comparsons across peer groups, we standardze the TC measure usng the
Management Scence 54(6), pp. 1052 1064, 2008 INFORMS 1057 peer group means and standard devatons. Specfcally, we obtan a peer group adjusted TC measure for each nvestor as ATC = TC MNTC peer SIGTC peer (4) where MNTC peer s the mean TC of the peer group of nvestor, and SIGTC peer s the standard devaton of the TC of the peer group of nvestor. 7 A postve (negatve) ATC measure for an nvestor ndcates that her trades are more (less) clustered than other nvestors who exhbt smlar tradng frequency and hold portfolos wth smlar number of stocks and of smlar sze (.e., her peers). The magntude of the ATC measure for an nvestor ndcates the number of standard devatons that the nvestor s away from the mean of her respectve peer group. Usng the orgnal DE measure, a peer group adjusted dsposton effect measure s defned and nterpreted n an analogous manner. 3.3. Trade Clusterng and the Dsposton Effect: Sortng Results To set the stage, we conduct nonparametrc tests and examne the relaton between adjusted trade clusterng (ATC) and adjusted dsposton effect (ADE). Frst, we compute the ADE and the ATC measures for each nvestor n the sample, where for greater accuracy we only consder nvestors who execute at least fve sell trades. Both the ATC and the ADE measures are computed usng all nvestor trades executed durng the sample perod. Next, we rank nvestors on the bass of ther ATC measures and defne fve nvestor quntles that contan the same number of nvestors. 8 Last, we measure the average ADE for each of the fve nvestor groups. Table 2 reports the results from our sortng tests. To show the ATC varaton across the ATC quntles, n Panel A, we report the range of the ATC measure for the fve ATC quntles. The ATC measures are negatve n the frst two quntles, postve n the last two quntles, and have mxed sgns n the mddle quntle. In Panel B, we report the mean ADE for ATC quntles. Consstent wth our hypothess, we fnd that the mean ADE decreases wth ATC. The mean ADE s 0.102 n the lowest ATC quntle, and t decreases 7 For robustness, we also consdered a related measure of adjusted trade clusterng, where the deflator s MNTC peer nstead of SIGTC peer. Because the scaled ATC measures are senstve to the small value of the denomnator, we consdered an unscaled clusterng measure, defned as ATC = TC MNTC peer. Our results are smlar when we use these alternatve defntons of adjusted trade clusterng. 8 The results are very smlar when we form 10 nvestor groups nstead of 5. Table 2 Trade Clusterng and the Dsposton Effect: Sortng Results Panel A: ATC breakponts ATC quntles ATC range Low Q2 Q3 Q4 Hgh Low 0 459 0 144 0 073 0 016 0 134 Hgh 0 144 0 073 0 016 0 134 0 683 Panel B: Double sort on portfolo sze and ATC ATC quntles Portfolo sze Low Q2 Q3 Q4 Hgh Hgh Low Small 0 163 0 104 0 039 0 063 0 169 0 332 Q2 0 065 0 039 0 084 0 101 0 170 0 235 Q3 0 086 0 078 0 045 0 043 0 153 0 239 Q4 0 068 0 097 0 049 0 117 0 073 0 141 Large 0 075 0 105 0 029 0 006 0 057 0 132 All 0 102 0 064 0 027 0 099 0 156 0 258 Panel C: Double sort on annual number of trades and ATC ATC quntles Trades per year Low Q2 Q3 Q4 Hgh Hgh Low Low 0 124 0 031 0 065 0 107 0 130 0 254 Q2 0 161 0 116 0 021 0 211 0 222 0 383 Q3 0 090 0 046 0 010 0 252 0 323 0 413 Q4 0 106 0 042 0 057 0 102 0 235 0 341 Hgh 0 084 0 159 0 083 0 014 0 024 0 108 Notes. Ths table reports the mean adjusted dsposton effect (ADE) of nvestor groups formed on the bass of portfolo sze, adjusted trade clusterng (ATC), and trades per year measures. In Panel A, we report the lower and the upper lmts of the ATC quntles. In Panel B, ADE s reported for nvestor groups formed by performng an ndependent double sort on portfolo sze and the trade-clusterng measures. In Panel C, nvestor groups are formed by performng an ndependent, double sort on number of trades per year and trade-clusterng measures. The portfolo sze s the average sze of an nvestor s portfolo durng the sx-year sample perod. The adjusted ATC measure s defned as ATC = (TC MNTC peer )/SIGTC peer s the mean TC of the peer group of nvestor and SIGTC peer, where MNTC peer s the standard devaton of the TC of the peer group of nvestor. TC s defned n Table 1. The peer group of each nvestor s defned usng portfolo sze (measured n dollar terms), number of stocks, and tradng frequency. The ADE measure for each nvestor s defned n an analogous manner usng the orgnal dsposton effect (DE) measure. The DE for an nvestor s defned as the dfference between an nvestor s propensty to realze gans and the propensty to realze losses. We use the Kolmogorov-Smrnov test to examne the statstcal sgnfcance of the dfferences n the adjusted DE measures., Denote sgnfcance at the 5% and 1% levels, respectvely. monotoncally along the ATC quntle (see the last row of Panel B). In the hghest ATC quntle, the mean ADE s negatve ( 0 156), whch ndcates that nvestors n the group exhbt a lower dsposton effect than ther respectve peer group means. The dfference n the mean ADE of the top and the bottom ATC quntles s large (= 0 258) and statstcally sgnfcant at the 1% level. To better understand how the trade clusterng and the dsposton effect measures are related, we per-
1058 Management Scence 54(6), pp. 1052 1064, 2008 INFORMS form two double sorts and examne the varaton n ADE as ATC vares wthn portfolo sze and tradng-frequency-based nvestor groups. In Panel B, we present the average ADE for nvestor groups that are formed by sortng on the ATC and the portfolo sze varables. In Panel C, we report the average ADE for nvestor groups formed by sortng on ATC and trades per year (.e., tradng frequency) varables. The results from the double sorts ndcate that the ADE measure exhbts some senstvty to the portfolo sze measure wthn the ATC quntles, especally n the extreme ATC quntle portfolos. However, the aggregate effect s not very strong because the senstvty s postve n a few nstances and negatve n others. Overall, the peer group adjustment methodology appears to be reasonably effectve n mnmzng the mechancal effects of portfolo sze and tradng frequency on the DE measure. More nterestngly, we fnd that the mean ADE decreases wth ATC across all portfolo sze and tradng frequency quntles. For nstance, among large portfolos (hghest portfolo sze quntle), the mean ADE s largest (=0 075) n the lowest ATC quntle and sgnfcantly lower (= 0 057) n the hghest ATC quntle. The ADE dfferental of 0 132 s sgnfcant at the 1% level. Smlarly, wthn the subset of nvestors who trade most often (hghest trades per year quntle), the mean ADE s largest (=0 084) n the lowest ATC quntle and sgnfcantly lower (= 0 024) n the hghest ATC quntle. Agan, the mean ADE dfferental of 0 108 s sgnfcant at the 1% level. The other results n the table ndcate that the magntude of the ADE dfferental s even stronger n the remanng four portfolo sze and tradng frequency quntles. 3.4. Dsposton Effect Regresson Estmates To better quantfy the relaton between trade clusterng and the dsposton effect, we estmate a cross-sectonal regresson model. In the regresson specfcaton, ADE s the dependent varable, and the ATC measure along wth a set of demographc and portfolo-related varables are employed as ndependent varables. The choce of ndependent varables other than ATC s motvated by the fndngs n Feng and Seasholes (2005) and Dhar and Zhu (2006), who show that the level of dsposton effect s related to a varety of nvestor characterstcs. The varables that are known to nfluence the trade-clusterng measure (see 2.4) are also ncluded n the set of control varables. As before, all regresson varables are standardzed, and we ensure that our estmates are robust to concerns about multcollnearty. The dsposton effect regresson estmates are reported n Table 3. The explanatory varables used n the regresson model have been defned earler (see Table 3 Trade Clusterng and the Dsposton Effect: Regresson Estmates Varable Estmate t-statstc Estmate t-statstc Intercept 0 005 0 610 0 006 0 328 Adjusted trade clusterng 0 107 5 464 0 105 4 541 Income 0 021 1 209 Log age 0 107 5 198 Professonal dummy 0 012 0 557 Retred dummy 0 022 0 941 Gender dummy 0 016 0 842 Portfolo turnover 0 102 3 272 Mutual fund ownershp 0 018 2 085 Number of stocks 0 008 0 238 Log trades per year 0 099 6 487 Portfolo sze 0 014 1 963 Number of nvestors 13,683 9,756 Adjusted R 2 6 33% 12 96% Notes. Ths table reports the estmates from cross-sectonal regressons, where the adjusted dsposton effect (ADE) of a household s the dependent varable. The adjusted trade-clusterng (ATC) measure and a set of household characterstcs are employed as ndependent varables. Gender dummy s set to one (zero) f the head of the household s male (female). Other ndependent varables are defned n Table 1. Both ndependent and dependent varables have been standardzed (mean s set to zero and standard devaton s one). The standard errors are corrected for heteroskedastcty. Table 1), wth the excepton of the gender dummy. It s set to one f the head of the household s male. Although we use peer group adjusted trade clusterng and dsposton effect measures, to further ensure that the relaton between the dsposton effect and trade clusterng that we dentfy s not mechancally nduced, we employ the three varables that are used to defne the peer groups as addtonal control varables. The dsposton effect regresson estmates ndcate that when the ATC measure s the only ndependent varable, the ATC coeffcent estmate s negatve and strongly sgnfcant (ATC coeffcent = 0 107, t-stat = 5 464), and the adjusted R 2 of the cross-sectonal regresson model s 6.33%. Thus, the ATC measure alone can explan a consderable porton of the crosssectonal varaton n the dsposton effect. Even when we ntroduce the control varables n the regresson specfcaton, the ATC coeffcent estmate remans hghly sgnfcant. Specfcally, the ATC coeffcent estmate s 0 105 wth a t-statstc of 4 541. Ths evdence ndcates that trade clusterng has an ncremental explanatory power over the known determnants of the dsposton effect. Furthermore, comparng the ATC coeffcent estmate wth other coeffcent estmates, we fnd that ATC s one of the strongest determnants of ADE. For robustness, we consder an alternate trade clusterng measure, where we estmate a regresson model to remove the effects of portfolo sze, tradng frequency, and number of stocks on trade clusterng. We defne a resdual trade-clusterng (RTC) measure
Management Scence 54(6), pp. 1052 1064, 2008 INFORMS 1059 that s the resdual from a regresson model, where the raw trade-clusterng measure s the dependent varable and the three varables used to defne the peer groups are the ndependent varables. We reestmate the dsposton effect regresson wth RTC as the man ndependent varable nstead of ATC. The RTC coeffcent estmates are very smlar to the ATC coeffcent estmates. When RTC s the only ndependent varable, the RTC coeffcent estmate s 0 090 wth a t-statstc of 7 391. When other control varables are ncluded n the regresson specfcaton, the RTC coeffcent estmate s 0 087 wth a t-statstc of 5 467. The RTC estmates are comparable to the ATC coeffcent estmates and ndcate that the negatve dsposton effect trade clusterng relaton s qute robust. 3.5. Interpretaton of the Dsposton Effect Results The sortng results and the DE regresson estmates provde strong and robust evdence of a relaton between trade clusterng and nvestors decsons to sell losers. We fnd that nvestors who execute more clustered trades exhbt a weaker dsposton effect. Ths evdence s consstent wth the frst part of our man hypothess, whch posts that nvestors who adopt narrower decson frames exhbt a stronger dsposton effect because they are lkely to focus on the gans and losses of ndvdual stock postons nstead of examnng the overall portfolo performance. There are two potental nterpretatons of our fndngs, both consstent wth our frst hypothess. One nterpretaton s that decson frames nfluence dsposton effect, even though people do not actvely choose a narrow decson frame. For example, when nvestors make ntutve decsons, separate decsons almost subconscously nduce them to adopt a narrow decson frame. As a result, nvestors exhbt stronger dsposton effect when they execute less-clustered trades. An alternatve nterpretaton of our evdence s that certan nvestors ntentonally choose broader decson frames. The choce of those frames nfluences the relaton between trade clusterng and the dsposton effect. In partcular, hedonc optmzers would choose a decson frame and a level of mental accountng that maxmzes ther perceved utlty (Thaler 1985, Thaler and Johnson 1990). Due to loss averson and dmnshng senstvty of value functon n prospect theory, the mpact of a loss s larger than the mpact of a gan of the same magntude, and the senstvty to gans and losses declnes as ther magntudes ncrease. Therefore, the hedonc edtng hypothess predcts that people would prefer to combne a loss wth a larger gan or wth another loss because the combned outcome generates hgher utlty than the total utlty generated by segregated outcomes. In fact, Lm (2006) shows emprcally that nvestors choose the tmng of ther trades to perceve outcomes more favorably. If nvestors optmally choose ther decson frames by tmng ther trades, the degree of trade clusterng would reflect the level of framng n ther tradng decsons. Specfcally, because a loss s less panful when t s ntegrated wth another loss or wth a larger gan, nvestors mght sell a loser when they decde to sell a wnner to reduce the pan. As a result, hedonc optmzers would attempt to overcome ther reluctance to realze losses by executng clustered trades. If trade clusterng and the dsposton effect are related through the channel of hedonc optmzaton, t s consstent wth our man hypothess. The ntegraton of outcomes through smultaneous trades reflects a desre to engage n portfolo-level thnkng. Thus, trade clusterng resultng from hedonc optmzng behavor s lkely to reflect a broader decson frame, whch s exactly the effect we are tryng to capture usng the trade-clusterng measure. 4. Trade Clusterng and Portfolo Underdversfcaton In the second set of formal tests, we examne the second part of our man hypothess, whch posts that nvestors who frame ther decsons narrowly hold less-dversfed portfolos because they pay less attenton to stock correlatons. 4.1. The Dversfcaton Measure Followng Goetzmann and Kumar (2008), we use a normalzed verson (NV) of the portfolo varance as the dversfcaton measure. The NV for portfolo p s defned as NV p = p 2 (5) 2 where p 2 s the portfolo varance and 2 s the average varance of all stocks n the portfolo. We compute the dversfcaton measure for each nvestor n the sample, where the average varance of each nvestor portfolo s estmated usng the monthly returns data from the prevous fve years, and the portfolo varance measure s estmated usng the realzed portfolo returns. A peer group adjusted dversfcaton (ADIV) measure s defned usng the negatve of the normalzed varance measure and the peer group adjustment methodology descrbed n 3.2, so that ADIV ncreases as the level of dversfcaton ncreases. 4.2. Trade Clusterng and Dversfcaton: Sortng Results Table 4 reports the ADIV measures of fve nvestor groups (quntles) formed by sortng on the ATC mea-
1060 Management Scence 54(6), pp. 1052 1064, 2008 INFORMS Table 4 Trade Clusterng and Portfolo Dversfcaton: Sortng Results Panel A: Double sort on portfolo sze and ATC ATC quntles Portfolo sze Low Q2 Q3 Q4 Hgh Hgh Low Small 0 035 0 049 0 010 0 005 0 069 0 104 Q2 0 025 0 018 0 060 0 002 0 099 0 124 Q3 0 085 0 061 0 054 0 054 0 156 0 241 Q4 0 105 0 050 0 049 0 014 0 218 0 323 Large 0 116 0 095 0 056 0 046 0 215 0 331 All 0 074 0 056 0 046 0 019 0 154 0 228 Panel B: Double sort on annual number of trades and ATC ATC quntles Trades per year Low Q2 Q3 Q4 Hgh Hgh Low Low 0 080 0 011 0 021 0 044 0 057 0 136 Q2 0 049 0 036 0 019 0 012 0 123 0 172 Q3 0 040 0 040 0 031 0 024 0 092 0 132 Q4 0 107 0 060 0 106 0 034 0 216 0 324 Hgh 0 092 0 122 0 054 0 007 0 268 0 360 Notes. Ths table reports the mean adjusted dversfcaton (ADIV) levels of nvestor groups, formed on the bass of portfolo sze (PSze), adjusted trade clusterng (ATC), and number of trades per year (TPY) varables. Portfolo dversfcaton s measured usng normalzed varance. The normalzed varance of a portfolo s defned as NV = 2/ 2 p, where 2 p s the varance of the gven portfolo and 2 s the average varance of all stocks n the portfolo. The peer group ADIV measure for each nvestor s defned analogously to the ATC and ADE measures usng the negatve of the normalzed varance measure. In Panel A, the mean dversfcaton measures are reported for nvestor groups formed by performng an ndependent double sort on portfolo sze and the adjusted trade-clusterng measures. In Panel B, the nvestor groups are formed by performng an ndependent double sort on number of trades per year and adjusted trade-clusterng measures. We use the Kolmogorov-Smrnov test to examne the statstcal sgnfcance of the dfferences n dversfcaton measures., Denote sgnfcance at 5% and 1% levels, respectvely. sure. Consstent wth our man hypothess, we fnd that the level of portfolo dversfcaton ncreases wth ATC. The mean adjusted dversfcaton of nvestor portfolos n the hghest and the lowest ATC quntles are 0.154 and 0 074, respectvely, and the dfferental of 0.228 s statstcally sgnfcant at the 1% level. These unvarate results suggest that the degree of trade clusterng s postvely related to the level of portfolo dversfcaton. For robustness, we perform two double sorts and examne the varaton n the level of dversfcaton as ATC vares wthn the nvestor subgroups. In the frst case we use the ATC and the portfolo sze as the sortng varables, and n the second cases we employ the ATC and the trades per year as the two sortng varables. These results are also reported n Table 4. We fnd that the dversfcaton level ncreases wth ATC across all portfolo sze quntles (see Panel A) and across all trades per year quntles (see Panel B). For nstance, among large portfolos (hghest portfolo sze quntle), the mean ADIV s 0.215 n the hghest ATC quntle and 0 116 n the lowest ATC quntle, and the dfferental of 0.331 s sgnfcant at the 1% level. Smlarly, wthn the group of nvestors who trade most frequently (hghest trades per year quntle), nvestors n the hghest ATC quntle have a mean ADIV of 0.268, whereas those n the lowest ATC quntle have a mean adjusted dversfcaton of 0 092. Agan, the dfferental of 0.360 s sgnfcant at the 1% level. 4.3. Dversfcaton Regresson Estmates To examne the ncremental effect of ATC on nvestors dversfcaton decsons, we estmate a cross-sectonal regresson model, where the adjusted dversfcaton measure of an nvestor s the dependent varable, and the ATC measure along wth varous portfolo and demographc varables are the ndependent varables. Our choce of portfolo and demographc attrbutes as control varables s motvated by the results n Goetzmann and Kumar (2008), who have dentfed them as determnants of portfolo dversfcaton. Furthermore, the set of varables that are known to nfluence trade clusterng (see 2.4) are employed as addtonal control varables. As before, the varables n the regresson model are standardzed, and we ensure that the estmates are robust to concerns about multcollnearty. The regresson estmates are reported n Table 5. All explanatory varables used n the regressons have been defned earler (see Table 1), wth the excepton of the local bas varable. The local bas measure s the dfference between the weghted dstance of the actual nvestor portfolo from her locaton and the weghted dstance of the market portfolo from her locaton. 9 We nclude ths varable n our regresson specfcaton as a control varable because Goetzmann and Kumar (2008) fnd that nvestors wth stronger local bas exhbt greater underdversfcaton. When the ATC measure s the only ndependent varable, the coeffcent estmate of ATC s strongly postve and sgnfcant (ATC estmate = 0 116, t-stat = 6 246). When we ntroduce demographc and portfolo characterstcs as control varables n the regresson model, the ATC estmate decreases but remans postve and statstcally sgnfcant. The ATC coeffcent estmate s 0.086 wth a t-stat of 5.235. In addton, the 9 The dstance between an nvestor s locaton and her portfolo s defned as D act = N w =1 D, where N s the number of stocks n the portfolo, w s the weght of stock n the portfolo, and D s the dstance between an nvestor s zp code and the zp code of a frm s headquarter. The dstance between an nvestor s locaton and the market portfolo s defned n an analogous manner. See Coval and Moskowtz (2001) and Zhu (2002) for detals of ths measure. The results are qualtatvely smlar when we employ other related local bas measures n the regresson specfcatons.
Management Scence 54(6), pp. 1052 1064, 2008 INFORMS 1061 Table 5 Trade Clusterng and Portfolo Dversfcaton: Regresson Estmates Varable Estmate t-statstc Estmate t-statstc Intercept 0 004 0 194 0 019 0 465 Adjusted trade clusterng 0 116 6 246 0 086 5 235 Income 0 007 1 159 Log age 0 049 1 968 Professonal dummy 0 005 1 011 Retred dummy 0 057 0 990 Portfolo turnover 0 035 1 679 Portfolo performance 0 030 0 743 Mutual fund ownershp 0 041 2 019 Foregn dummy 0 053 3 134 Short sell dummy 0 017 1 743 Opton dummy 0 019 1 348 Local bas 0 093 2 171 Number of stocks 0 028 1 415 Log trades per year 0 018 2 186 Portfolo sze 0 028 1 247 Number of nvestors 21,679 7,569 Adjusted R 2 2 34% 13 70% Notes. Ths table reports the estmates of cross-sectonal regressons, where the average adjusted dversfcaton (ADIV) measure of a household s the dependent varable. The adjusted trade-clusterng (ATC) measure and a set of household characterstcs and portfolo varables are used as ndependent varables. Portfolo performance s the rsk-adjusted performance (Sharpe Rato) of the household, and local bas s the dfference between the weghted dstance of an actual nvestor portfolo from her locaton ( N w =1 D, where N s the number of stocks n the portfolo, w s the weght of stock n the portfolo, and D s the dstance between an nvestor s zp code and the zp code of a frm s headquarter), and the weghted dstance of the market portfolo from her locaton. Other ndependent varables have been defned n prevous tables. Both ndependent and dependent varables have been standardzed (mean s set to zero and standard devaton s one). The standard errors are corrected for heteroskedastcty. coeffcent estmates of the control varables are consstent wth the results reported n Goetzmann and Kumar (2008). Comparng the magntudes of the coeffcent estmates, we fnd that local bas and trade clusterng are the two strongest determnants of portfolo dversfcaton. For robustness, we use the RTC measure as the proxy for narrow framng and reestmate the dversfcaton regresson model. When RTC s the only ndependent varable n the dversfcaton regresson, the coeffcent estmate ncreases from 0.116 to 0.129 (t-stat = 5 629). When other control varables are ncluded n the regresson specfcaton, the RTC coeffcent estmate s 0.093 wth a t-statstc of 4.202. The RTC coeffcent estmates ndcate that the postve dversfcaton trade clusterng relaton s qute robust. Collectvely, the sortng results and the dversfcaton regresson estmates suggest that nvestors who execute more clustered trades and tend to make smultaneous tradng decsons hold better-dversfed portfolos than nvestors who execute separate trades and tend to make separate decsons. Ths evdence supports the second part of our man hypothess, whch posts that nvestors who adopt narrower decson frames hold relatvely less-dversfed stock portfolos. 5. Trade Clusterng, Style Preferences, and Portfolo Performance In ths secton, we examne the relaton between adjusted trade clusterng and portfolo performance. We consder both raw and rsk-adjusted performance measures because the performance trade clusterng relaton s lkely to depend upon the choce of the performance measure. Investors who execute more clustered trades exhbt weaker dsposton effects and hold relatvely better-dversfed portfolos. These nvestors who adopt broader decson frames would also be better postoned to examne the nteractons among multple portfolo decsons. In partcular, they mght be able to better assess the total rsk of ther portfolos. Consequently, trade clusterng could be postvely correlated wth rsk-adjusted portfolo performance. In contrast, nvestors who execute lessclustered trades and adopt narrower decson frames could earn hgher raw returns because they prefer to hold rsker stocks. In ths scenaro, f rsker stocks earn hgher average returns, trade clusterng would be negatvely related to raw portfolo performance. To examne the relaton between trade clusterng and portfolo performance, we compute the ATC measure for each nvestor usng all trades executed by the nvestor durng the entre sample perod and defne ATC quntles. We consder both raw and rsk-adjusted performance measures and examne the average portfolo performance of nvestors n those fve categores. 10 For each nvestor n our sample, we estmate several factor models, ncludng the four-factor model (Fama and French 1993, Jegadeesh and Ttman 1993, Carhart 1997), to measure her rsk-adjusted performance. Table 6 reports the performance estmates, where for greater accuracy nvestors wth fewer than 24 monthly observatons are excluded from the analyss. Consstent wth the evdence n Barber and Odean (2000), we fnd that, on average, our sample of nvestors underperforms the common performance benchmarks. The mean four-factor alpha s 0 375, whch translates nto an annual, rsk-adjusted underperformance of 4.50%. Examnng the average portfolo characterstcs across the ATC quntles, we fnd that nvestors who execute more clustered trades hold less rsky portfolos and earn lower raw returns. Consequently, the average Sharpe rato does not vary 10 The results are qualtatvely smlar when we consder ATC decles nstead of quntles.
1062 Management Scence 54(6), pp. 1052 1064, 2008 INFORMS Table 6 Trade Clusterng, Style Preferences, and Portfolo Performance ATC quntles Performance measure Low Q2 Q3 Q4 Hgh Hgh Low Mean monthly return 1 251 1 227 1 228 1 216 1 122 0 129 Portfolo standard 9 029 8 826 8 661 8 344 7 594 1 435 devaton Sharpe rato 0 101 0 101 0 103 0 105 0 104 0 003 Jensen s alpha 0 127 0 142 0 166 0 158 0 217 0 090 Two-factor alpha 0 433 0 430 0 441 0 418 0 432 0 001 Three-factor alpha 0 483 0 457 0 431 0 405 0 353 0 130 Four-factor alpha 0 407 0 382 0 379 0 362 0 271 0 136 RMRF exposure 1 221 1 197 1 204 1 185 1 116 0 105 SMB exposure 0 939 0 890 0 869 0 813 0 626 0 323 HML exposure 0 250 0 226 0 189 0 159 0 121 0 129 UMD exposure 0 376 0 351 0 320 0 307 0 228 0 148 Notes. Ths table reports the average portfolo performance and style preferences of nvestor groups formed by sortng along the adjusted tradeclusterng (ATC) dmenson. Usng the ATC measure for the entre sample perod, nvestors are ranked and fve nvestor groups are defned. The performance measures for each nvestor portfolo are computed separately and the group averages are reported n the table. The followng performance measures are reported: () mean monthly return, () portfolo standard devaton, () Sharpe rato, and (v) k-factor alpha measures (k = 1 2 3 4). The k-factor alphas are estmated for each household by fttng a k-factor model to the sample perod monthly returns tme-seres. The one-factor model contans only the market factor (RMRF); the two-factor model contans RMRF and SMB; the three-factor model contans RMRF, SMB, and HML; and the four-factor model contans RMRF, SMB, HML, and UMD. RMRF s the market return n excess of the rsk-free rate, SMB s the sze factor, HML s the value factor, and UMD s the momentum factor. The average factor exposures are also reported. They are estmated for each household by fttng a four-factor model to the sample perod monthly returns tme-seres. Due to the possblty of cross-sectonal dependence, we use bootstrappng to conduct the tests of statstcal sgnfcance., Denote sgnfcance at 5% and 1% levels, respectvely. consderably across the ATC quntles. For nstance, nvestors n the lowest ATC quntle earn a mean monthly return of 1.251% and have a mean portfolo standard devaton of 9.029%. In contrast, nvestors n the hghest ATC quntle earn a lower mean monthly return (=1 122%), but they also have a lower mean portfolo standard devaton (=7 594%). The mean Sharpe ratos for the two groups are 0.101 and 0.104, respectvely. Ths evdence s consstent wth our dversfcaton regresson estmates, where we fnd that nvestors who execute more clustered trades hold less rsky and better-dversfed portfolos. When we examne the mean rsk-adjusted performance levels across the ATC quntles, we fnd that the low-atc nvestors contnue to outperform hgh- ATC nvestors when we measure portfolo performance usng Jensen s alpha. However, the low-atc nvestors underperform hgh-atc nvestors when we apply other asset-prcng models to defne the performance benchmarks. For nstance, the mean four-factor alpha for the lowest (hghest) ATC quntle nvestors s 0 407 ( 0 271). The alpha dfferental of 0 136% per month translates nto an annual rsk-adjusted performance dfferental of 1.632%. 11 To dentfy the man source of the performance dfferences across the ATC quntles, we examne whether nvestors stock preferences vary across the ATC quntles. Usng the mean factor exposure estmates, we fnd that nvestors who execute lessclustered trades hold hgher beta, smaller, hgher book-to-market (B/M), and low-momentum stocks. Ths evdence mght at least partly explan why the performance dfferental between the hgh- and the low-atc nvestor groups swtches sgn when we ntroduce sze and book-to-market factors to defne the performance benchmarks. In untabulated results, we fnd that the stock preference estmates are smlar when we examne the actual portfolo holdngs of nvestors n the low- and the hgh-atc quntles. For nstance, compared to the weght n the aggregate market portfolo, the lowest ATC quntle nvestors overweght hgh beta (hghest beta quntle) stocks by 15.27%, whereas the hghest ATC quntle nvestors overweght them by 8.96%. Smlarly, the lowest ATC quntle nvestors overweght smaller (lowest market-cap quntle) stocks by 17.02%, whereas the hghest ATC quntle nvestors overweght them by 12.14%. The portfolo weghts along other stock characterstcs (B/M and 12-month momentum) also yeld estmates that are consstent wth nvestors stock preferences as reflected by the mean factor exposures. To better understand the relaton between trade clusterng and portfolo performance, we estmate performance regressons where the portfolo performance of an nvestor s the dependent varable. The ndependent varables are the adjusted dsposton effect, the adjusted dversfcaton, and the adjusted trade-clusterng measures, along wth the known determnants of portfolo performance. Ths set ncludes the nvestor s age, nvestment experence, the annual household ncome, a male dummy, a retred dummy, portfolo sze, monthly portfolo turnover rate, and the portfolo dvdend yeld (e.g., Barber and Odean 2001, Kornots and Kumar 2007). To account for potental cross-sectonal dependence, we use robust standard errors adjusted for heteroskedastcty and clusterng wthn zp codes. Consstent wth the sortng results, we fnd a negatve performance clusterng relaton when the mean portfolo return or the Jensen s alpha s the dependent varable (ATC coeffcent estmates are 0 038 and 0 030 wth t-statstcs of 3 622 and 2 725, respectvely) and a postve performance clusterng relaton 11 Due to the possblty of cross-sectonal dependence n portfolo performance, we use bootstrappng to conduct the tests of statstcal sgnfcance.
Management Scence 54(6), pp. 1052 1064, 2008 INFORMS 1063 when the four-factor alpha s the dependent varable (ATC coeffcent estmate = 0 032, t-statstc = 2 948). Furthermore, as expected, the ADE measure has a statstcally negatve coeffcent estmate and the ADIV measure has a statstcally postve estmate n both the raw and the rsk-adjusted performance regressons. Taken together, our performance results provde mxed evdence on the relaton between trade clusterng and portfolo performance. Ths relaton depends upon the choce of the performance measure. It s negatve when we consder the raw performance measure or the Jensen s alpha, but postve when we measure rsk-adjusted performance usng the threefactor alpha or the four-factor alpha. 6. Addtonal Robustness Checks We conduct several addtonal tests to ensure that the observed postve (negatve) relaton between trade clusterng and portfolo dversfcaton (dsposton effect) s robust. For brevty, the detals of the robustness tests and the results are descrbed n the onlne appendx (provded n the e-companon), 12 but a summary s provded below. In our frst robustness check, we perform Monte Carlo smulatons to further elmnate the possblty of a purely mechancal relaton between trade clusterng and the dsposton effect and portfolo dversfcaton. Next, we show that low levels of trade clusterng do not ndcate superor nformaton that mght be assocated wth lower portfolo dversfcaton (Ivkovć et al. 2008). We also consder the possblty that our trade-clusterng measure mght capture lqudty tradng, transacton costs, passve lmt orders, tax-motvated trades n December, or day tradng, because these factors are lkely to be assocated wth the dsposton effect or portfolo dversfcaton measures. We fnd smlar results even after we control for the effects of these alternatve sources of trade clusterng. To address the concern that we do not observe nvestors entre portfolos, we use a compensated measure of portfolo dversfcaton, where we assume that the unobserved part of an nvestor s portfolo s nvested n the market portfolo. We also reestmate the dversfcaton regresson usng a subsample of nvestors whose portfolos are large relatve to ther ncome levels, because for these nvestors we expect the unobserved portfolo components to be small. The results from these robustness tests are very smlar to the reported results. 12 An electronc companon to ths paper s avalable as part of the onlne verson that can be found at http://mansc.journal. nforms.org/. To examne whether our results are senstve to the wndow sze that defnes smultaneous trades, we recompute trade clusterng usng a weekly rather than a daly tme nterval, and fnd smlar results. The results are also very smlar when we estmate the dsposton effect and the dversfcaton regressons for the two subperods 1991 1993 and 1994 1996. In the last robustness test, we estmate change regressons and fnd that an ncrease n trade clusterng s assocated wth a decrease n the dsposton effect and an ncrease n portfolo dversfcaton. Collectvely, the robustness test results further ndcate that the dsposton effect trade clusterng and the dversfcaton trade clusterng relatons are robust and are unlkely to be mechancally nduced. 7. Summary and Conclusons Ths paper examnes whether the framng mode nfluences the stock nvestment decsons of U.S. ndvdual nvestors. Motvated by the extant expermental evdence, whch suggests that separate decsons are more lkely to be narrowly framed than smultaneous decsons, we propose trade clusterng as a proxy for narrow framng. Usng ths framng proxy, we show that nvestors who execute more clustered trades exhbt weaker dsposton effects and hold better-dversfed portfolos. Because we use a narrow framng proxy nstead of a drect framng measure, we conduct several robustness checks and show that the dsposton effect trade clusterng and the dversfcaton trade clusterng relatons are robust and are not mechancally nduced. We also show that nvestors who execute less-clustered trades exhbt a preference for small-cap and value stocks, and they earn hgher raw returns but lower rskadjusted returns. Taken together, our results ndcate that the framng mode s an mportant determnant of nvestors stock nvestment decsons. Ths evdence complements the theoretcal research on narrow framng (e.g., Barbers and Huang 2007) and contrbutes to an emergng lterature that attempts to dentfy the fundamental determnants of behavoral bases (e.g., Graham et al. 2006, Barbers and Xong 2008). Our evdence also suggests that narrow framng would have broader nfluence on nvestors portfolo choces and tradng decsons beyond ts effect on rsk atttudes (e.g., Benartz and Thaler 1995, Gneezy et al. 2003, Barbers et al. 2006). Although our study does not examne the relaton between narrow framng and stock returns, our emprcal results suggest that nvestors framng choces are lkely to have mplcatons for stock returns. For nstance, we fnd that nvestors stock preferences vary systematcally wth the degree of trade clusterng. Ths evdence suggests that the concentraton of nvestors
1064 Management Scence 54(6), pp. 1052 1064, 2008 INFORMS who are more lkely to frame ther decsons narrowly would vary wth stock characterstcs n a predctable manner. Thus, consstent wth the theoretcal predctons of Barbers and Huang (2001), those stocks mght exhbt greater volatlty and lower correlatons wth other stocks wthn the same category. We hope to examne these questons n our future research. 8. Electronc Companon An electronc companon to ths paper s avalable as part of the onlne verson that can be found at http:// mansc.journal.nforms.org/. Acknowledgments The authors thank two anonymous referees, the assocate edtor, Hal Arkes, Brad Barber, Nck Barbers, Robert Battalo, Shane Corwn, Alex Edmans, Anand Goel, Olesya Grshchenko, Jeff Hales, Davd Hrshlefer, Steve Huddart, Pam Losefsky, Tm Loughran, Terrance Odean, Amyatosh Purnanandam, Paul Schultz, Mark Seasholes, Juergen Symanzk, George Wu (the department edtor), Nng Zhu, and semnar partcpants at the 2005 European Fnance Assocaton Meetngs (Moscow), Korea Advanced Insttute for Scence and Technology, Korea Unversty, Penn State Unversty, and Seoul Natonal Unversty for helpful dscussons and valuable comments. They also thank Itamar Smonson for makng the nvestor data avalable to them and Terrance Odean for answerng numerous questons about the nvestor database. The authors are responsble for all remanng errors and omssons. References Baley, T., A. Gatrell. 1995. Interactve Spatal Data Analyss. Longman Scentfc & Techncal, Harlow, Essex, England. Barber, B., T. Odean. 2000. 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