How dversfable s frm-specfc rsk? James Bennett and Rchard W. Sas * October 0, 006 JEL: G0, G, G, G4 Keywords: dversfcaton, dosyncratc rsk * Bennett s from the Department of Accountng and Fnance, Unversty of Southern Mane, Portland, Mane 0404-9300, (07) 780-4080, james.bennett@mane.edu. Sas s from the Department of Fnance, Insurance, and Real Estate, College of Busness and Economcs, Washngton State Unversty, Pullman, Washngton, 9964-4746, (509) 335-347, sas@wsu.edu. The authors thank Jack Bogle, Mark Hulbert, Harry Markowtz, Mer Statman, Harry Turtle, Davd Whdbee, and semnar partcpants at the Inqure UK Fall 006 conference, Washngton State Unversty, and Unversty of Southern Mane for ther helpful comments. Copyrght 006 by the authors.
How dversfable s frm-specfc rsk? Abstract Contrary to conventonal wsdom, we demonstrate that: () there s no evdence nvestors can, or have ever been able to, easly form a well-dversfed portfolo, and () there are, and always have been, substantal dversfcaton gans avalable beyond a relatvely small portfolo (e.g., 0-50 stocks). Investors ablty to elmnate frm-specfc rsk s, n fact, much more lmted than prevously recognzed. In recent years, for example, one-ffth of randomly-chosen 50-stock portfolos experenced, on average, an annual frm-specfc shock of 6%.
How dversfable s frm-specfc rsk?. Introducton Ths study re-examnes two fundamental questons how large must a portfolo be to ensure neglgble frm-specfc rsk? And, at what pont do gans from addtonal dversfcaton become meanngless? Our analyss yelds surprsng answers. Contrary to conventonal wsdom, there s no evdence nvestors can, or have ever been able to, form well-dversfed portfolos and there are, and always have been, substantal dversfcaton gans from holdng much larger portfolos than tradtonally recommended. A portfolo s well-dversfed when an nvestor s assured that the portfolo s frm-specfc return wll dffer neglgbly from zero (Ross, 976). Determnng the number of securtes requred to form such portfolos s fundamental to fnancal economcs because of the central role well-dversfed portfolos play n market effcency, portfolo management, and asset prcng theores. These theores rely, at least n part, on arbtrageurs to rsklessly correct msprcngs and ensure that only systematc exposures are prced. The dea that nvestors can easly elmnate frm-specfc rsk s ngraned and pervasve. Alexander, Sharpe, and Baley (00, pages 63 and 6), for example, note, Roughly speakng, a portfolo that has equal proportons of 30 or more randomly selected securtes n t wll have a relatvely small amount of unque rsk. Its total rsk wll be only slghtly greater than the amount of market rsk that s present. Such portfolos are well dversfed and For a well-dversfed portfolo, nonfactor rsk wll be nsgnfcant. Smlarly, Francs and Ibbotson (00, page 399) mantan, Dversfable rsk may be easly dversfed away to zero n a portfolo that contans more than about 36 random stocks because the unsystematc peces of good luck and bad luck from randomly selected assets tend to average out to zero. The Securtes Exchange Commsson (005) tells nvestors that, You ll need at least a dozen carefully selected ndvdual stocks to be truly dversfed. Campbell, Lettau, Malkel, and Xu (00) suggest, however, that the declne n the average correlaton between stocks and the rse n frm-specfc rsk over tme has ncreased the number of securtes needed for a well-dversfed portfolo. The 003 edton of Malkel s classc A Random Walk Down Wall Street, for nstance, mantans the golden number has ncreased to 50.
The wdely-accepted belef that nvestors can easly elmnate frm-specfc rsk and there are effectvely no gans to dversfcaton beyond a relatvely small portfolo appears to be largely based on the erroneous nterpretaton of the relaton between the number of securtes n a portfolo, the expected standard devaton of the portfolo, and the return standard devaton of the market portfolo. For example, the most wdely used textbook (Brealey, Myers, and Allen, 006) n the top MBA programs (see Womack, 00) shows a graph analogous to Fg., and reports, In Fgure 7.9 we have dvded rsk nto two parts unque rsk and market rsk. If you have only a sngle stock, unque rsk s very mportant; but once you have a portfolo of 0 or more stocks, dversfcaton has done the bulk of ts work. For a reasonably well-dversfed portfolo, only market rsk matters. Most fnance textbooks contan a smlar graph and nterpretaton. [Insert Fgure about here] The problem wth Fg. s that t does not, and cannot, dvde total rsk nto systematc and frmspecfc components. A portfolo s total varance s the sum of ts systematc and frm-specfc varances. A portfolo s standard devaton, however, s not the sum of ts systematc and frm-specfc standard devatons. Consder a smple numercal example of what ths mstake means. Assume that the return standard devatons for the market portfolo and a gven -asset portfolo are 4% and 5%, respectvely. Fg., therefore, erroneously suggests frm-specfc rsk for the gven -asset portfolo s % when, n fact, the portfolo s frm-specfc rsk s 3%,.e., although 5 =4 +3, 5 4+3. Fg. demonstrates how ths error mpacts estmates of frm-specfc rsk n practce. Specfcally, Fg. plots the relaton between the number of securtes n a portfolo and expected annual total rsk, market rsk, and frm-specfc rsk (estmated over the 000-004) for portfolos of 0 to 500 securtes. As shown n Fg., total standard devaton s not the sum of market and frm-specfc standard devatons. Denotng the dfference between total rsk (the top lne) and market rsk (the broken lne) as frm-specfc rsk s analogous to denotng the dfference between total rsk (the top lne) and frm-specfc rsk (the bottom lne) as market rsk. A portfolo s well-dversfed when the bottom lne (expected frm-specfc rsk) s neglgbly dfferent from the horzontal axs. The fact that total rsk (the top lne) and market rsk (the broken lne) are close, does not mply that a portfolo s well-dversfed (.e., that the bottom lne s neglgbly dfferent from the
horzontal axs). Because an arbtrageur, by defnton, takes long and short postons n portfolos wth equal systematc exposures (and therefore perfectly hedges the systematc rsk), the dfference between the total rsk of an -stock portfolo and that of the equal-weghted market portfolo s meanngless all that matters to an arbtrageur s the uncertanty regardng the frm-specfc shocks. [Insert Fgure about here] Consder, for example, the rsk faced by an nvestor attemptng to explot an arbtrage opportunty wth 50-stock portfolos. In recent years (000-004) the expected annual return varance for a 50-stock portfolo was 4.36% (consstng of, approxmately, an equal-weghted market varance of 3.58% plus frmspecfc varance of 0.78%). Because frm-specfc and systematc shocks are ndependent, an nvestor holdng the typcal 50-stock portfolo had only moderately greater total return uncertanty than an nvestor holdng the market portfolo. As shown n Fg., the average 50 stock portfolo had a 0.9% annual return standard devaton ( (0.0358+0.0078)) versus 8.9% ( 0.0358) for the market portfolo. The relatvely small dfference n total return uncertanty, however, does not mean frm-specfc uncertanty s small the fact the two dstrbutons have smlar parameters (.e., same mean, smlar standard devatons) and are nherently related (.e., the expected systematc returns on a randomly-selected 50-stock portfolo and on the market portfolo are equal) does not mply that a draw from one dstrbuton wll dffer only slghtly from a draw from the other dstrbuton. Specfcally, the expected annual frm-specfc varance of 0.78% for a 50-stock portfolo means the average 50-stock portfolo has a frm-specfc standard devaton of 8.8% ( 0.0078). Thus, the arbtrageur actually faces substantal frm-specfc rsk most of the tme, at least one of the arbtrageur s 50- stock portfolos would experence a frm-specfc shock of at least 8.8%. Moreover, for most other (.e., non-arbtrageurs) nvestors, dosyncratc rsk s lkely to be more mportant than the dfference between total return uncertanty for a portfolo and return uncertanty for the market. Professonal managers who account for most tradng (e.g., Schwartz and Shapro, 99), for example, are nearly always judged on ther performance relatve to market ndces. Thus, most professonal nvestors lkely care more about the uncertanty regardng devaton from systematc returns than the total return uncertanty [see Warng and Segel (006) for further dscusson]. Smlarly, t s reasonable to hypothesze that many ndvdual nvestors also focus on devatons from market returns gven evdence that ndvdual nvestors chase mutual funds that experence postve frm-specfc shocks (e.g., Gruber, 996). By defnton, the arbtrageur has net systematc exposures equal to zero. Thus, the expected standard devaton of the un-hedged porton of each portfolo s return s 8.8%. Because the frm-specfc shock of each portfolo s ndependent of the other, the lkelhood that at least one portfolo has a frm-specfc shock of at least 8.8% s 54%.e., -0.68 (assumng the dstrbuton s normal). 3
Smlarly, the fact that most gans from dversfcaton occur wth the frst few securtes does not mply that the gans to further dversfcaton are effectvely meanngless. For example, under the same set of assumptons, an nvestor holdng arbtrage portfolos each consstng of 00 stocks faces only one-half the frm-specfc rsk as the 50-stock arbtrageur,.e., the expected frm-specfc standard devaton of a 00-stock portfolo s 4.4% versus 8.8% for a 50-stock portfolo. Although the gans from addtonal dversfcaton are small relatve to the gans assocated wth the frst few securtes, they can be substantal n absolute terms. That s, the bottom lne n Fgure reveals that there s room for substantal dversfcaton gans beyond a relatvely small portfolo. A second problem wth nterpretng a 50-stock portfolo as well-dversfed s that the relaton between the number of securtes n a portfolo and expected portfolo rsk s msleadng because t fals to account for the dstrbuton of possble volatltes around the expected value. For example, the frm-specfc rsk for (approxmately) half of all 50-stock portfolos wll be greater than the expected frm-specfc rsk. Thus, as Elton and Gruber (977) pont out, dversfcaton gans assocated wth larger portfolos arse from both a declne n the expected frm-specfc rsk and from greater confdence n that estmate. Gven one cannot evaluate the dfference between the expected standard devaton of an -asset portfolo and the market portfolo to determne when a portfolo s well-dversfed, one must ether: () focus on varance, because unlke standard devaton, expected varance s the sum of frm-specfc and systematc varance, or () focus on the role of frm-specfc and systematc rsk n explanng cross-sectonal varaton n portfolo returns. We take both approaches, but demonstrate key benefts to focusng on the latter. Frst, unlke varance, the relatve role of frm-specfc rsk n explanng cross-sectonal varaton n portfolo returns s large and constant,.e., t does not declne wth the number of securtes added to the portfolo. If frmspecfc rsk, for example, accounts for 90% of the cross-sectonal varaton n ndvdual securty returns, then frm-specfc rsk wll also account for 90% of the expected cross-sectonal varaton n the returns of randomly-chosen portfolos of any gven sze. Because the relatve role of frm-specfc rsk n explanng crosssectonal return varaton s constant, one can easly determne f assets are suffcent for a well-dversfed portfolo by smply computng the return varaton across randomly-chosen -securty portfolos f the 4
cross-sectonal portfolo return varaton s non-neglgble, then frm-specfc rsk s also non-neglgble and assets are nsuffcent for a well-dversfed portfolo. Second, because the relatve role of frm-specfc rsk n explanng cross-sectonal varance n portfolo returns s ndependent of portfolo sze, conclusons regardng nvestors ablty to form well-dversfed portfolos are unlkely to be explaned by errors n decomposng returns nto systematc and frm-specfc components. As long as frm-specfc shocks account for most of the cross-sectonal varaton n ndvdual securty returns, then non-neglgble return varaton across random - asset portfolos ndcates that assets are nsuffcent to ensure a well-dversfed portfolo. Our emprcal results demonstrate that even very large portfolos have substantal frm-specfc rsk. In recent years, one n fve nvestors who held an equally-weghted random 00-stock (500-stock) portfolo, for example, averaged an annual frm-specfc shock of 7.8% (4.7%). Although we focus our analyss on a recent fve-year perod (000-004), we also show that both small (e.g., 0 stocks) and large portfolos (e.g., 500 stocks) have never been well-dversfed (at least as far back as 966). In sum, contrary to conventonal wsdom, we fnd no evdence that nvestors can form portfolos wth neglgble frm-specfc rsk or that there s ever a pont where the dversfcaton gans from holdng a larger portfolo are not statstcally sgnfcant. Our concluson s, of course, predcated on one s defnton of neglgble. As noted above, for example, one n fve randomly-selected 500-stock portfolos averaged an annual frm-specfc shock of 4.7%. We mantan that f one assumes the expected rsk premum on a randomly-selected portfolo s approxmately 6% per year, a one-n-fve chance of averagng a 4.7% frm-specfc shock ndcates that the nvestor s not assured of a frm-specfc return that s neglgbly dfferent from zero. Because nvestors ablty to elmnate frm-specfc rsk through dversfcaton s a central tenet of asset prcng, market effcency, and portfolo management theores, our results have a number of mplcatons. Frst, nvestors nablty to explot msprcng wthout sgnfcant exposure to frm-specfc rsk (.e., ther nablty to form arbtrage portfolos) suggests that observed returns may dffer from those predcted by asset prcng theores by more than prevously antcpated. Ths may help explan the persstence of some anomales and the possblty of bubbles n asset prces. Second, f frm-specfc rsk s not easly dversfable, then t may be prced (see Mayers, 976; Levy, 978; Merton, 987; Barbers and Huang, 00; 5
Malkel and Xu, 00; Goyal and Santa Clara, 003). Thrd, managers holdng relatvely large portfolos may not be closet ndexers even large portfolos wth total rsk only slghtly greater than market ndces may have substantal frm-specfc exposures. Fourth, nvestors portfolos are not nearly as well-dversfed as they have been led to beleve. There are, and always have been, substantal and statstcally sgnfcant dversfcaton gans from holdng much larger portfolos than tradtonally recommended.. The role of frm-specfc rsk n portfolo return and varance Defne the excess return on securty n perod t as the securty s return less the rsk-free rate (.e., R t = r t - r ft ) and assume that securty returns are generated by a lnear K-factor model wth the followng propertes: frm-specfc shocks have an expected value of zero (E(ε )=0), fnte varances = (( E( ε ) σ ( ε ) > 0 ), are ndependent across securtes (E(ε,ε j )=0), are ndependent from systematc factors (E(ε,γ k )=0,k), and are ndependent over tme (E(ε t,ε t- )=0): Rt K = β kγ kt + ε t. () k= The return on a portfolo s the weghted average return of the securtes n the portfolo (where w s the fracton of the portfolo nvested n securty ): R pt = = w R t = K k= β γ + ε, where β = w β k, and ε = w ε. () pk kt pt pk = k pt = t The expected portfolo return varance n ths framework s gven by: E K ( σ ( R ) Eσ β γ + E σ ( ε ) p = k= pk k ( ). p (3) For an equal-weghted portfolo, ε pt = = ε, and the expected varance of the portfolo s frm-specfc t shocks s: E( σ ( ε p ) = E( σ ( ε )). (4) 6
Eq. (4) demonstrates that, as s well-known, as approaches nfnty, the varance of frm-specfc shocks approaches zero, and thus, gven the expected frm-specfc shock s zero, an nvestor can be certan that the frm-specfc return wll be zero (.e., the portfolo s well-dversfed). Although examnaton of dversfcaton as the expected fracton of frm-specfc rsk elmnated s ndependent of the level of frm-specfc rsk (.e., the expected frm-specfc varance of a randomly-selected -asset portfolo s / th that of holdng a sngle randomly-selected securty), estmaton va Eq. (4) of the level of frm-specfc return uncertanty contaned n an -securty portfolo requres an estmate of the expected varance of frm-specfc returns at the ndvdual securty level,.e., E( σ ( ε ))... Estmates of expected frm-specfc rsk of a sngle securty The expected varance of frm-specfc shocks for a sngle (randomly-selected) securty (.e., E( σ ( ε )) n Eq. (4)) can be estmated by averagng the ndvdual estmated frm-specfc varances for each securty across all M securtes n the market (see Appendx A for proof). Lettng E( σ ( ε )) ndcate the ˆ TS ε estmated expected varance of a sngle randomly-selected securty and σ ( ) denote the estmated tmeseres frm-specfc varance for securty : M ( ( ε )) = ˆ σ ( ε ). Eˆ σ TS (5) M = Alternatvely, because the expected varance of a sngle randomly-selected securty s smply the varance of the dstrbuton of frm-specfc shocks pooled over all securtes, the cross-sectonal varance of observed frm-specfc shocks s also an unbased estmate of the expected varance of frm-specfc shocks ˆ for a randomly-selected securty,.e., E( σ ( ε )) n Eq. (4) (see Appendx A for proof). Lettng ˆ XS ( ε σ ) denote the estmated cross-sectonal frm-specfc varance n securty returns: ( ( ε )) ˆ σ ( ε ). ˆ XS E σ = (6).. Tme-seres and cross-sectonal varaton n random portfolos 7
The expected tme-seres varance of a randomly-selected -asset portfolo can be wrtten as (see Appendx A for proof): E K K K ( σ TS ( R p ) = Eσ TS βkγ k + Ecov TS βkγ k, β jkγ k + E σ TS ( ε ) k= k= k= ( ), (7) where the TS subscrpt ndcates the tme-seres varance and covarance. The sum of the frst two terms n Eq. (7) s the expected systematc rsk of a randomly-selected -asset portfolo (.e., the frst term on the rght-hand sde of Eq. (3)) whle the last term n Eq. (7) corresponds to the expected frm-specfc rsk for a random -securty portfolo (.e., Eq. (4) or the last term n Eq. (3)). As noted n the ntroducton, because frm-specfc rsk s manfested as devatons from systematc returns, one may examne cross-sectonal return varaton on randomly-selected -asset portfolos to evaluate the relaton between the number of securtes n a portfolo and frm-specfc rsk. 3 The ntuton s straghtforward portfolo (or securty) returns dffer only because the portfolos exhbt dfferent systematc exposures or because the portfolos receve dfferent frm-specfc shocks. Because the number of assets s fnte, however, the expected cross-sectonal return varance across random portfolos wll declne wth portfolo sze (regardless of the level of frm-specfc rsk) due to overlap n consttuent securtes across portfolos. For example, once portfolo sze exceeds half the number of securtes n the market, any two portfolos must hold common securtes. In the case of equal-weghted portfolos formed from a fnte pool of securtes, the expected cross-sectonal return varance can be wrtten as a functon of the portfolo sze 3 Several prevous studes suggest usng cross-sectonal return varaton to examne dversfcaton. Solnk and Roulet (000) develop a drect lnk between tme-seres correlaton between two assets (n ther case, nternatonal markets) and the cross-sectonal varaton n asset returns. Smlarly, de Slva, Sapra, and Thorley (00) argue that cross-sectonal varaton n mutual fund returns ncreases over tme because the cross-sectonal varaton n stock returns ncreases over tme. In fact, Eq. (5) n de Slva, Sapra, and Thorley (00) s smlar to the last term n Elton and Gruber s Eq. (B8) and our Eq. (8). In a study of mutual fund dversfcaton, O eal (997) examnes the dsperson of termnal wealth values for portfolos of mutual funds (see also Fsher and Lore, 970). In a concurrent workng paper, Statman and Sched (004) argue that the cross-sectonal standard devaton of total returns s a more ntutve measure of dversfcaton than tme-seres correlaton. Although these studes usually defne the cross-sectonal standard devaton of returns as dsperson, we avod that term because, as we demonstrate (n Appendx A), both the cross-sectonal varance of frmspecfc shocks and the cross-sectonal average of the tme-seres varance of frm-specfc shocks generate estmates of the expected varance of frm-specfc shocks for a randomly-selected securty,.e., both are estmates of the same quantty. In addton, although frm-specfc shock dsperson s an unbased estmate of frm-specfc rsk, total return dsperson s drven by both systematc and frm-specfc rsk. 8
(), the number of securtes n the market (M), and the cross-sectonal varance across systematc and frmspecfc returns on ndvdual securtes (see Appendx A for proof): 4 E = XS M k= K ( σ ( R ) Eσ β γ + E σ ( ε ) XS p k k M ( ). XS (8) The fracton of cross-sectonal return varance attrbuted to frm-specfc rsk s gven by the rato of the last term n Eq. (8) to the total: K % Frm Specfc = E( σ ( )) XS ε E σ. XS β kγ k + ε (9) k= Equatons (8) and (9) reveal two key benefts of examnng return varaton across randomly-formed portfolos to evaluate the relaton between the number of securtes n a portfolo and frm-specfc rsk. Frst, gven frm-specfc shocks account for most of the cross-sectonal varaton n securty returns, frm-specfc shocks wll also account for most of the cross-sectonal varaton n portfolo returns,.e., frm-specfc rsk plays a much larger role than systematc rsk n explanng cross-sectonal return varaton regardless of the number of securtes n the portfolo. 5 That s, the key dfference between the expected cross-sectonal portfolo varance and the expected tme-seres portfolo varance s that the tme-seres varance ncorporates how securtes move together over tme (.e., the second term n Eq. (7) dfferentates t from Eq. (8)). Because the week-to-week tme-seres varaton n portfolo returns s prmarly drven by systematc factor realzatons, even large frm-specfc shocks wll have a relatvely small mpact on total tme-seres volatlty,.e., the dfference between total rsk for an -asset portfolo and the market portfolo may be small even though frm-specfc rsk s large. Second, Eq. (9) llustrates that, unlke tme-seres varance, the relatve role of frmspecfc rsk n explanng cross-sectonal portfolo return varance s ndependent of portfolo sze (.e., all terms contanng cancel out). In sum, the relatve role of frm-specfc rsk n explanng cross-sectonal return varaton s both large and constant. 4 Our proof s a specal case of the general proof derved by Elton and Gruber (977). 5 Frm-specfc rsk accounts for, on average, approxmately 9% of the weekly cross-sectonal standard devaton n securty returns over the 966-004 perod based on a fve-factor (market, sze, value, momentum, and ndustry) model. 9
As noted above, as the number of securtes ncreases, portfolos wll have overlappng consttuent securtes and cross-sectonal varaton n portfolo returns wll fall regardless of frm-specfc rsk. onetheless, gven that the last term n Eq. (8) s the expected cross-sectonal varance n frm-specfc shocks for -asset portfolos, one can estmate the expected varance of frm-specfc shocks for -asset portfolos (.e., the last term n Eq. (3)) from the estmated cross-sectonal varance of frm-specfc shocks n randomlyselected -asset portfolos (see Appendx A for proof). Denotng E( σ ( ε P )) as the estmated expected frm- ˆ XS ε p specfc varance for an -asset portfolo and σ ( ) as the estmated cross-sectonal frm-specfc varance across random -asset portfolos) yelds: ( ε ) ˆ σ ˆ XS p E( σ ( ε p ) =. (0) M In sum, we can generate three estmates of expected frm-specfc varance for -asset portfolos: (/) tmes the average tme-seres varance of frm-specfc returns (.e., Eq. (5)), (/) tmes the cross-sectonal varance of frm-specfc returns across ndvdual securtes (.e., Eq. (6)), and the cross-sectonal varance n frm-specfc returns across randomly-formed portfolos dvded by (-(-)/(M-)), (.e., Eq. (0)). Emprcal tests reported n the next secton reveal that all three estmates are essentally dentcal. ˆ.3. Cross-sectonal varance n tme-seres varance by: E The expected cross-sectonal varance n tme-seres varances for random -asset portfolos s gven K K ( ( ( ) = + K σ XS σ TS R p E σ TS βkγ k cov TS βkγ k, β jkγ k + σ TS ( ε ) k= k= k= K K K σ cov, ( ) TS βkγ k TS βkγ k β jkγ k σ TS ε, () k= k= k= 0
where the sngle bars ndcate expectatons over securtes n a gven portfolo and the double bars ndcate expectatons over all securtes n the market. 6 Eq. () demonstrates that the cross-sectonal varance n tmeseres varance due to dfferences n frm-specfc rsk and that due to dfferences n systematc rsk wll declne as the number of securtes n an equal-weghted portfolo ncreases. Thus, as ponted out by Elton and Gruber (977), dversfcaton gans arse from both lower expected rsk and greater confdence n that expected value (.e., a tghter cross-sectonal dstrbuton about that expected value). 7 3. Emprcal results Most of our emprcal tests are based on annual results averaged over the 000-004 perod. We nclude each year all ordnary (.e., Center for Research n Securty Prces share code of 0 or ) ew York Stock Exchange, Amercan Stock Exchange, and asdaq securtes that have at least one hundred weeks of return data over the prevous 04 weeks and have complete weekly return data over that year. 8 The annual sample szes range from 4,835 securtes n 000 to 4,77 securtes n 004. Followng the lterature (e.g., Xu and Malkel, 003; Spegel and Wang, 005), we estmate the frmspecfc porton of a securty s return, each week, as the resdual from weekly rollng tme-seres regressons over the prevous two years (.e., the current and prevous 03 weeks) of each securty s excess return on a 6 Eq. () s derved as the expected squared dfference between Eq. (7) for a gven -asset portfolo and Eq. (7) for all -asset portfolos. Eq. () represents the expected cross-sectonal varance n tme-seres portfolo varance. The realzed cross-sectonal varance n tme-seres varance ncludes addtonal terms for the covarance between frm-specfc returns for securty and securty j, the covarance between frm-specfc returns of securty and the systematc returns of securty j, and the covarance between the systematc and frm-specfc returns of securty. Although each of these covarances has an expected value of zero, they are not restrcted to zero n any fnte sample. In fact, restrctng these covarances to zero volates the ndependence assumptons of the model. Robustness tests reported n Appendx B, however, reveal that our conclusons are not senstve to such restrctons (e.g., forcng the average resdual to zero by estmatng frm-specfc shocks through cross-sectonal regressons). 7 Elton and Gruber (977) develop an analytcal expresson for the expected cross-sectonal varance of equal-weghted portfolo tme-seres varance as a functon of the number of securtes n a portfolo and the number of securtes n the market. Unfortunately, due to computer lmtatons at the tme, the authors were unable to compute the necessary parameters based on all securtes n the market. (The authors, however, were able to compute the parameters for a random sample of 50 stocks.) Although current computer capablty allows us to compute the Elton and Gruber varance n varance formula for our sample, we do not use ther formula because the varance s a suffcent statstc only f the dstrbuton of tme-seres varances across random portfolos of a gven sze s normal and our emprcal tests reveal ths s not the case, especally for smaller portfolos. Instead, we report the average varances for the extreme varance decles across randomly-selected portfolos. We do fnd, however, that ther formula works well for larger (e.g., greater than one hundred stocks) portfolos. In addton, our prmary focus s on cross-sectonal varaton n frm-specfc rsk rather than total rsk. 8 The survvorshp bas we mpose by requrng securtes to have weekly returns for the entre year suggests our emprcal estmates of frm-specfc rsk are conservatve.
fve-factor model that ncludes the market return n excess of the rsk-free rate (denoted R mt ), a sze factor (denoted R SMBt ), a book-to-market equty factor (denoted R HMLt ), a momentum factor (denoted R UMDt ), and an ndustry factor (the value-weghted ndustry return n excess of the rsk-free rate, denoted R jt where securty s n ndustry j): 9 R t = β R + β R + β R + β R + β R + ε. () m mt HML HMLt SMB SMBt UMD UMDt j jt t We next randomly select, at the begnnng of each calendar year, portfolos of sze (=0 to 4,000 securtes) and compute each portfolo s annual return and realzed varance (based on weekly returns) as well as the systematc and frm-specfc components of each. 0 Because the cross-sectonal varance of realzed portfolo volatlty and returns declnes (and computer requrements ncrease) as the portfolos grow n sze, the number of randomly-selected portfolos we evaluate declnes wth. Specfcally, we form 5,000 random portfolos for szes =0 to =00, n 0-stock ncrements (e.g., 5,000 portfolos of sze 0, 0, 30,, 00),,000 random portfolos for szes =00 to 900 by 00 stock ncrements, and,000 random portfolos for szes =,000 to 4,000 by,000 stock ncrements. Ths procedure s repeated for each year n the sample perod, and most of the results presented n ths secton are based on averages across the fve calendar years (000-004) under evaluaton (we examne dversfcaton over earler perods n Secton 3.4). 3.. Equal-weghted portfolo return varance We begn by evaluatng the relaton between the number of securtes n a portfolo and the expected tme-seres varance of portfolo returns. The bold lne n Fg. 3A plots the average annual realzed equalweghted portfolo tme-seres varance as a functon of portfolo sze for portfolos from 0 to 4,000 9 Weekly market, sze, value, and momentum factors are generated by compoundng the daly values provded by Ken French. The 48 ndustres are as defned n Fama and French (993). Securtes wth SIC codes not used by Fama and French comprse the 49 th ndustry. Followng Durnev, Morck, and Yeung (004), the ndustry returns are value-weghted and computed excludng securty. 0 As dscussed n, footnote 6 realzed tme-seres varances also nclude realzed covarances between frm-specfc shocks across securtes n the portfolo, realzed covarances between the frm-specfc shocks of a securty and the systematc shocks of other securtes, and realzed covarances between the frm-specfc shocks of a securty and ts own systematc shocks. Smlarly, realzed cross-sectonal varances nclude realzed cross-sectonal covarances between frmspecfc and systematc shocks. We defne all such covarances as part of frm-specfc rsk. Smlarly, we defne the systematc porton of return as the compound weekly systematc return and the remanng porton of the annual return as frm-specfc. As noted n Appendx B, however, we fnd qualtatvely dentcal results when we restrct the crosssectonal covarance between systematc and frm-specfc shocks to average zero by estmatng frm-specfc shocks wth a cross-sectonal regresson.
securtes, based on weekly rebalancng to equal weghts. Fg. 3B through 3D partton Fg. 3A nto more nformatve unts of analyss. The bold lne reveals the well-documented relaton between the number of securtes n a portfolo and the average total rsk of the portfolo. The frst few securtes added to the portfolo sze greatly reduce the average portfolo varance. As more securtes are added, however, the margnal reducton n average varance declnes. [Insert Fgure 3 about here] There are two problems wth nterpretng the bold lne n Fg. 3 as evdence that nvestors can easly form well-dversfed portfolos: () the fact that a randomly-selected 50-stock portfolo s expected to exhbt only % (/) of the average securty s frm-specfc varance does not mean that every 50-stock portfolo does so, and () the fact that the average 50-stock portfolo exhbts only margnally greater total return volatlty than the market portfolo does not mean the remanng frm-specfc rsk s neglgble. We begn by focusng on the frst ssue. The realzed tme-seres varance of a specfc -securty portfolo may dffer from the average varance of -securty portfolos because the portfolo has dfferent systematc exposures (e.g., beta unequal to the average beta), or because the portfolo experences above- or below-average frm-specfc rsk (see Eq. ()). (Although our focus s on frm-specfc rsk, we evaluate cross-sectonal varaton n total, systematc, and frm-specfc rsk to provde a complete pcture of the relaton between portfolo sze and rsk.) The sold outsde lnes n Fg. 3 report the average annual realzed varance for the most (top lne) and least volatle (bottom lne) decles of our random samples. The nsde broken lnes n Fg. 3 partton the dfferences between the average realzed tme-seres varances on portfolos contaned n these decles and the average realzed tme-seres varance for all portfolos (.e., the bold lne) nto systematc and frm-specfc components the dstance between the average (bold lne) and the broken lne represents the porton due to dfferences n systematc rsk, whle the dstance between the broken lne and the outsde sold lne corresponds to the porton attrbuted to dfferences n frm-specfc rsk. We focus on the average varance of portfolos n the extreme decles rather than the cross-sectonal varance n tmeseres varance because the cross-sectonal varance does not provde a suffcent descrpton of the dstrbuton of varances for small portfolo szes, whch are hghly skewed. The cross-sectonal average varances are computed for stocks n the extreme decles each calendar year and then averaged over the 000-004 perod. 3
Examnaton of Fg. 3 demonstrates that realzed portfolo varance may dffer substantally from ts expected (average) value,.e., the cross-sectonal varaton n realzed tme-seres varance s relatvely large even when the number of securtes s large. For example, for 0% of the 00-stock portfolos, the realzed tme-seres varance averages over 33% greater than the tme-seres varance for the average 00-stock portfolo (the standard devaton averages about 5% greater (.e.,.33 0.5 -)). To formally test for the presence of dversfcaton gans n the form of an ncrease n varance certanty as portfolo sze grows, we estmate F- statstcs assocated wth the null hypothess that the cross-sectonal varance n tme-seres total return varances for each portfolo sze equals the cross-sectonal varance n varance for the next larger sze portfolo, e.g., each year we compare the cross-sectonal varance n random 0-stock portfolo varances to the cross-sectonal varance n random 0-stock portfolo varances. In addton, we run the tests for both total rsk and for frm-specfc rsk. We reject the null of equal cross-sectonal varance n total varance at the 5% level or better n 0 of the 05 tests (fve years of data and portfolo szes yeld 05 comparsons, (- )*5). Smlarly, we reject the null of equal cross-sectonal varance n frm-specfc varance at the 5% level or better n 00 of the 05 tests. Thus, the results demonstrate statstcally sgnfcant dversfcaton gans n the form of a reducton n varance n varance for portfolos up to 4,000 stocks. Ths can be seen n Fg. 3 as the dfference between the outsde bands, as well as the dfference between the outsde bands and the nsde broken lnes (cross-sectonal varaton n frm-specfc rsk), contracts as portfolo sze grows. We next consder the second ssue that elmnaton of most of the expected frm-specfc rsk of holdng a sngle securty and a relatvely small dfference between the expected total return uncertanty for an -asset portfolo and the market portfolo does not mean the remanng frm-specfc rsk s neglgble. The second column n Table reports the fracton of frm-specfc varance of holdng a sngle randomly-selected securty that s expected to be elmnated for varous sze portfolos (.e., -(/)) and ndcates that relatvely small portfolos greatly reduce expected frm-specfc varance. The thrd column n Table reports the dfference between the expected annual standard devaton of total returns for an -stock portfolo and the Because the tests are overwhelmngly sgnfcant, we do not report specfc results to conserve space. Tests for equalty of varances are one-taled and based on the folded F-statstc. 4
annual standard devaton of the equal-weghted market return. 3 The results n the second and thrd columns, consstent wth Fg., reveal that the expected frm-specfc rsk declnes quckly wth the frst few securtes added to a portfolo and the expected total return standard devaton for relatvely small portfolos s only margnally greater than that for the equal-weghted market portfolo. [Insert Table about here] Columns four, fve, and sx report the three estmates of the annualzed frm-specfc varance remanng n -securty portfolos: (/) tmes the realzed tme-seres varance of frm-specfc returns averaged across all ndvdual securtes (.e., Eq. (5)), (/) tmes the average weekly realzed cross-sectonal varance of frm-specfc returns (.e., Eq. (6) averaged over tme), and the realzed cross-sectonal varance n frm-specfc returns across the random portfolos dvded by (-(-)/(M-)), (.e., Eq. (0) averaged over tme). 4 The last three columns n Table report the square root of the respectve estmated expected frm-specfc varance as an estmate of the expected standard devaton of frm-specfc shocks for a portfolo of securtes. There are four mportant ponts to take from Table. Frst, the estmate of -asset portfolos expected frm-specfc rsk calculated from ndvdual securtes tme-seres varance (column four), the estmate calculated from ndvdual securtes cross-sectonal varance (column fve), and the estmate calculated from the cross-sectonal varance of random -asset portfolo returns dvded by (-(-)/(-M)) (column sx) are essentally dentcal. 5 Second, although relatvely small portfolos elmnate, on average, most of the frm-specfc rsk (column ) and exhbt only margnally greater total return standard devaton than the market portfolo (column 3), the remanng frm-specfc rsk s not neglgble. For example, although the average 50-stock portfolo elmnates 98% of the frm-specfc varance of the average securty (second 3 Computed as the dfference between the expected total standard devaton for an -asset portfolo and the total standard devaton of the equal-weghted market portfolo,.e., [(/)E(σ (ε ))+ E(σ (R M ))] [ E(σ (R M ))], where E(σ (ε )) s calculated, each year, as the cross-sectonal average of each securty s tme-seres estmated frm-specfc rsk and then averaged over the 000-004 perod. E(σ (R M )) s estmated as the tme-seres average annual varance of equal-weghted market portfolo over 000-004. 4 Tme-seres estmates are computed each year and averaged across securtes. Cross-sectonal estmates for both ndvdual securtes (.e., Eq. (6)) and random portfolos (.e., Eq. (0)) are computed each week. We then average the annualzed cross-sectonal estmates over tme each year (to allow drect comparson wth the tme-seres estmates; see Appendx A for addtonal dscusson). All three sets of estmates are then averaged over the fve years n the prmary sample perod (000-004). 5 Because frm-specfc resduals exhbt slghtly postve covarances across securtes (e.g., Hammer and Phllps, 99) the standard devatons for the random portfolos (last column) are slghtly greater than the estmated standard devatons assumng ndependence (prevous two columns). 5
column), and exhbts only % greater annual standard devaton than the equal-weghted market portfolo (thrd column), the expected standard devaton of the remanng frm-specfc rsk s approxmately 9% per year (last three columns). Thrd, the results n Table reveal that the standard devaton of frm-specfc returns are not neglgble for even very large portfolos. The standard devaton of annual frm-specfc shocks for portfolos of 500 securtes, for nstance, averages nearly 3%. Fourth, there are large dversfcaton gans beyond 0, 50, 00, or even 500 stocks. Specfcally, usng the sample of randomly-generated portfolos, we compute both non-parametrc and parametrc tests of the null hypothess that there are no dversfcaton gans assocated wth movng to the next largest portfolo (e.g., the average frm-specfc varance of a 0- stock portfolo equals the average frm-specfc varance of a 0-stock portfolo). Both sets of tests reject the hypothess n every case at the 5% level or better. 6 In fact, to elmnate (approxmately) half of the remanng standard devaton of a portfolo s frm-specfc shocks, an nvestor must quadruple the number of securtes n the portfolo. 7 The average 00-stock portfolo, for example, exhbts half the frm-specfc standard devaton as the average 50-stock portfolo (approxmately 4.4% per year versus 8.8%) and the average 800- stock portfolo stll contans one-quarter of the frm-specfc rsk of the average 50-stock portfolo (approxmately.% versus 8.8%). 3.. Frm-specfc returns on equal-weghted portfolos As dscussed n the ntroducton, because frm-specfc rsk s manfested as devatons from systematc returns, one can evaluate the mpact of frm-specfc rsk for varous sze portfolos by examnng the realzed return varaton across the randomly-selected portfolos. Thus, each calendar year, we compute the annual devaton from the equal-weghted market return for the random portfolos n the top and bottom 6 Specfc results are not reported to conserve space. We compute the cross-sectonal varance n frm-specfc returns each week (over the fve-year perod) for the random portfolos of each sze. We then nflate the estmates by (-(- )/(-M)) (see Eq. (0) and Appendx A; nflatng the estmates works aganst our ablty to reject the null). We then estmate Wlcoxon rank-sum tests of the null hypothess that the two sets of weekly varances have the same locaton parameter. In every case, we can reject the null at the 5% level or better. In addton, we compute parametrc t-tests for dfference n mean varances. Agan, for every case we reject the null (at the 5% level or better) that adjacent portfolo szes have equal mean frm-specfc rsk. 7 From Eq. (4), the rato of the expected varances of frm-specfc shocks for portfolos of sze k and s /k. The standard devaton of a portfolo of k securtes s thus expected to be /k.5 that of a portfolo of securtes. As a result, quadruplng the number of securtes (k=4), wll cut the portfolo s expected standard devaton of frm-specfc shocks n half. 6
performance decles. Analogous to Fg. 3, the sold lnes n Fg. 4A plot these annual devatons (averaged over the fve years n the sample perod) as a functon of portfolo sze. 8 The broken lnes n Fg. 4A partton the dfferences between the top and bottom decles average annual returns and the market return nto frmspecfc and systematc components (e.g., the dstance between the top sold lne and the top broken lne represents frm-specfc return). 9 As before, Fg. 4B through 4D partton the results nto more nformatve unts of analyss. [Insert Fgure 4 about here] Consstent wth Eq. (9), the relatve role of frm-specfc rsk n explanng cross-sectonal return varaton s approxmately constant across dfferent portfolo szes frm-specfc shocks (.e., the dfference between the broken lnes and the outsde lnes) account for an average of 86.8% of the total return dfference between the top and bottom decles (.e., the dfference between the outsde lnes), rangng from a low of 85.% (for,000-stock portfolos) to a hgh of 87.% (for 80-stock portfolos). Moreover, gven frm-specfc rsk s neglgble when the outsde lnes are neglgbly dfferent from the horzontal axs, the results n Fg. 4, consstent wth Table, reveal no evdence nvestors can elmnate frm-specfc rsk regardless of the number of securtes they hold. For example, the average return dfference between 50-stock portfolos n the top and bottom decles s 37% per year and nearly all of ths dfference (87%) s due to dfferences n frm-specfc shocks. In other words, over the sample perod, an nvestor holdng a randomly-selected 50-stock portfolo had a one n fve chance of averagng a return that dffered from the equal-weghted market return by 8% (.e., 37%/) over the next year. The results also confrm that even extremely large portfolos exhbt nonneglgble frm-specfc rsk (.e., are not well-dversfed) the annual return dfference between the top and bottom decles for,000-stock portfolos averages 7.3% and, agan, nearly all the dfference (85%) s attrbuted to frm-specfc shocks. In fact, as noted n the ntroducton, because frm-specfc rsk accounts for a large and constant fracton of cross-sectonal return varaton, -assets are suffcent for a well-dversfed 8 We focus on the extreme decles rather than the cross-sectonal standard devaton because standard devaton s a suffcent descrpton only f the cross-sectonal return dstrbuton s normal. In addton, we focus on the average return wthn each decle, rather than the decle breakpont, because although the relatve role of expected frm-specfc rsk s constant (Eq. (9)), the relatve role of frm-specfc rsk for a gven portfolo s not contant, e.g., the frm-specfc rsk n the one portfolo that serves as the decle breakpont for each portfolo sze, s not contant. 9 Although Fg. 3 and 4 are generated from the same set of random portfolos, the portfolos that form the top and bottom return decles n Fg. 4 are not necessarly the same that form the top and bottom varance decles n Fg. 3. 7
portfolo only when the return varaton across random -asset portfolos s neglgble. The results n Fg. 4 clearly ndcate non-neglgble cross-sectonal varaton n portfolo returns regardless of portfolo sze. Fg. 4 also ndcates substantal dversfcaton gans beyond 0, 50, 00, or even 500 securtes,.e., there are gans to dversfcaton as long as the dfferences between the outsde lnes and the nsde broken lnes n Fg. 4 contract. As a formal test of whether these dversfcaton gans are statstcally sgnfcant, each week we test the null hypothess that the cross-sectonal total return and frm-specfc return varances for portfolos of each sze are equal to the correspondng cross-sectonal varance for the next largest portfolo sze. Of the 5,48 test statstcs (6 weeks tmes comparsons each week), we reject the hypothess (at the 5% level or better) that adjacent portfolo szes exhbt equal cross-sectonal total or frm-specfc varance n more than 97% of the cases. 0 In short, the results n Fg. 4 demonstrate that: () there s no evdence nvestors can form portfolos wth neglgble frm-specfc rsk, and () there are substantal and statstcally sgnfcant dversfcaton gans possble beyond 0, 50, 00, or even 500 stocks. 3.3. Value-weghted portfolos Although t s most common to consder the role of frm-specfc rsk n equal-weghted portfolos, nvestors, n aggregate, hold a value-weghted portfolo. In addton, recent evdence shows that both ndvdual and professonal nvestors hold portfolos that devate substantally from equal weghts (e.g., Goetzmann and Kumar, 004; Cohen, Gompers, and Vuolteenaho, 00). In ths secton, we examne the relaton between frm-specfc rsk and portfolo sze for value-weghted portfolos that potentally provdes a better vew of dversfcaton n practce. Specfcally, we repeat the cross-sectonal varaton n returns analyss wth three changes we value-weght the portfolos, we set the probablty of ncluson proportonal to captalzaton rankng when formng random portfolos, and we only evaluate portfolo szes up to,000 securtes. We do not clam that these portfolos necessarly reflect nvestors actual portfolos any better 0 Test statstcs are not reported to conserve space. The tests are based on a one-tal test of the Folded-F statstc. The probablty of ncluson s set proportonal to captalzaton rankng, e.g., f there are 4,500 stocks n the sample, the largest stock s 4,500 tmes more lkely to be selected than the smallest stock. ote that, due to the extreme skewness n captalzaton, we cannot form larger portfolos (.e., beyond a few stocks) f we set the probablty of ncluson proportonal to market captalzaton. Although specfc results are not reported to conserve space, we fnd (not 8
than randomly-chosen equal-weghted portfolos. Rather, gven evdence that nvestors typcally do not hold equal-weghted portfolos, our focus s on understandng how value weghtng mpacts frm-specfc rsk. Fg. 5A (portfolo szes of 0 to 00 stocks) and 5B (portfolo szes of 00 to,000 stocks) report the average annual devaton from the value-weghted market portfolo return for the top and bottom decles of the value-weghted portfolos as a functon of portfolo sze. The results n Fg. 5 demonstrate that an nvestor holdng a value-weghted portfolo also faces very large frm-specfc rsk,.e., the outsde lnes are far apart and prmarly drven by frm-specfc shocks. For example, the average dfference n returns for 00-stock portfolos n the top and bottom decles s 34% per year (63% of the dfference s due to dfferences n frmspecfc shocks). Even,000-stock portfolos exhbt a tremendous amount of frm-specfc rsk the dfference n returns between the top and bottom decles of random,000-stock portfolos averages 9.7% wth 6% of the dfference attrbuted to frm-specfc shocks. The results also reveal contnual and nonneglgble gans to dversfcaton as the dfference between the outsde lnes and the adjacent broken lnes contnues to shrnk (and the outsde lnes contract toward the horzontal axs) as portfolo sze grows. As a formal test that these gans are statstcally meanngful, we test the null that there are no dversfcaton gans n movng to the next largest value-weghted portfolo by comparng the cross-sectonal frm-specfc varance for adjacent portfolo szes. Smlar to the results for the equal-weghted portfolos, we reject the null hypothess n over 96% of the 4,698 comparsons. [Insert Fgure 5 about here] 3.4. Frm-specfc rsk over tme Our results, focusng on a perod of hgh frm-specfc rsk relatve to hstorcal averages, may not hold over other perods although an nvestor holdng a 0-stock portfolo over the 000-004 perod bore very large levels of frm-specfc rsk, t s possble that 0-stock portfolos were well-dversfed n earler perods. Fg. 6 reports the year-by-year average return dfference (over 966-004) between portfolos n the surprsngly) even greater evdence of nvestors nablty to elmnate frm-specfc rsk n value-weghted portfolo when selecton s truly random (.e., not proportonal to captalzaton rankng). Specfc results are not reported to conserve space. The 4,698 tests are generated from 8 comparsons each week across 6 weeks. 9