The Sources of Portfolio Returns: Underlying Stock Returns and the Excess Growth Rate * Jason T. Greene Southern Illinois University Carbondale

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1 The Sources of Portfolo Returns: Underlyng Stock Returns and the Excess Growth Rate * Jason T. Greene Southern Illnos Unversty Carbondale Davd Rakowsk Southern Illnos Unversty Carbondale Abstract Ths aer decomoses ortfolo returns nto the underlyng sources arsng from the consttuent stocks growth rates, as well as ther varances and covarances. We emloy ths method to show that the dfference between large and small stock ortfolo returns s drven by a ortfolo excess growth rate that s nduced by the hgher volatlty of small stocks returns and not by the average growth rate of small stocks. Therefore, the sze effect s not a small frm effect, but a small frm ortfolo effect drven by the excess growth rate of the ortfolos. In contrast, ortfolos of hgh book-to-market stocks outerform due to hgher average levels of growth by the consttuent stocks and not due to ther varance-covarance structure. Our results demonstrate the mortance of consderng the sources of ortfolo erformance as ossbly dstnct from the erformance of the ortfolo s underlyng stocks when desgnng and nterretng studes of ortfolo erformance, cororate events, or the cross-secton of stock returns. Frst Draft: August 5, 010 Ths Draft: August, 011 JEL classfcaton: G11; G1; G14 Keywords: Portfolo Returns, Portfolo Growth Rates, Sze Effect, Long-Term Returns * The authors wsh to thank Anna Agaova, Adran Banner, Conrad Cccotello, Bob Ferguson, Bob Fernholz, Scott Glbert, Vnce Intntol, Robert Jennngs, Charles Hodges, Tomasz Kasowc, Jm Musumec, Vasleos Paathanakos, Mark Peterson, Jason Trow, Phll Whtman and semnar artcants at Southern Illnos Unversty Carbondale and Unversty of Dayton for ther helful comments. All errors reman the sole resonsblty of the authors. Corresondng author. Deartment of Fnance, College of Busness, Southern Illnos Unversty Carbondale, Carbondale, Illnos, 6901, , jgreene@busness.suc.edu. Electronc coy avalable at: htt://ssrn.com/abstract=180591

2 The Sources of Portfolo Returns: Underlyng Stock Returns and the Excess Growth Rate Abstract Ths aer decomoses ortfolo returns nto the underlyng sources arsng from the consttuent stocks growth rates, as well as ther varances and covarances. We emloy ths method to show that the dfference between large and small stock ortfolo returns s drven by a ortfolo excess growth rate that s nduced by the hgher volatlty of small stocks returns and not by the average growth rate of small stocks. Therefore, the sze effect s not a small frm effect, but a small frm ortfolo effect drven by the excess growth rate of the ortfolos. In contrast, ortfolos of hgh book-to-market stocks outerform due to hgher average levels of growth by the consttuent stocks and not due to ther varance-covarance structure. Our results demonstrate the mortance of consderng the sources of ortfolo erformance as ossbly dstnct from the erformance of the ortfolo s underlyng stocks when desgnng and nterretng studes of ortfolo erformance, cororate events, or the cross-secton of stock returns. 1 Electronc coy avalable at: htt://ssrn.com/abstract=180591

3 1. Introducton Research across many areas of fnance mlctly assumes that the erformance of a ortfolo s equvalent to the erformance of the underlyng stocks. Unfortunately, ths assumton can lead to ncorrect nferences regardng the ortfolo s consttuents and can obscure rcher observatons about a ortfolo s erformance and that of ts underlyng stocks. Ths aer focuses on two dstnct sources of ortfolo erformance. The frst source arses from the famlar frst moment of the ortfolo s underlyng stocks returns. The second source, though not a second order effect, arses from the varances and covarances of the ortfolo s underlyng stocks. We emrcally estmate these sources based on the Fernholz and Shay (198) mathematcal model of ortfolo returns. Ths arsng reveals nterestng new nsghts nto the sources of ortfolo returns n general, and the sources of returns to oft-studed characterstc-based ortfolos, such as sze and market-tobook ortfolos, n artcular. The outerformance of small frm ortfolos can be attrbuted to the nfluence of the consttuent stocks varances on the ortfolo s growth rate, not to the growth rate of the underlyng stocks themselves. We fnd the seemngly aradoxcal result that ortfolos of small stocks outerform, but small stocks undererform. The aarent aradox of havng outerformng ortfolos comrsed of undererformng stocks s easly resolved n the mathematcal arsng of ortfolo returns. Fernholz and Shay (198), hereafter FS, offer what we beleve to be the frst rgorous mathematcal analyss to dentfy the sources of the long-term erformance, or comound growth, of a ortfolo. Secfcally, FS show that a ortfolo that s rebalanced to the same constant weghts has a comound growth rate that s a functon of the underlyng stocks comound returns (.e., growth rates) and an excess growth rate that s due to the dfference between stocks varances and

4 the (dversfed) ortfolo s varance. 1 Ths dchotomy mles that a ortfolo s comound return can be ncreased by 1) choosng stocks that have hgher comound returns than ther eers; and/or ) choosng a more favorable mx of stocks based on ther varances and covarances. The former comonent arses from dfferences among stocks cross-sectonal returns, whle the latter comonent arses from dfferences among stocks volatltes over tme and ther volatltes relatve to one another over tme. The followng smle examle llustrates the underlyng economc ssue that s the focus of ths aer. Suose there are four stocks wth returns over two erods as gven n Exhbt 1. Frst, consder stocks A 1 and A. Each stock doubles n value one erod and loses half ts value n the other erod. Both stocks have a 5% arthmetc average return, but a zero geometrc average return, reflected by the average log return. Note that the average arthmetc return s aroxmated by the sum of the log return and half the stock s varance of log returns. Because the stocks are erfectly negatvely correlated, an equal weghted ortfolo ( EW ) of the two stocks earns 5% er erod wth no rsk. Furthermore, because the ortfolo has no rsk, ts average arthmetc and geometrc returns are both 5% er erod. Now consder stocks B 1 and B, whch both return 5% each erod. Any ortfolo of these two stocks also returns 5% each erod. Thus, ortfolo ortfolo A EW B erforms dentcally to EW A. Whle extreme, ths examle llustrates how a attern of ortfolo returns can conceal the source of the ortfolo returns. In ortfolo of the underlyng stocks. In ortfolo EW A, the returns come from the varance EW B, the returns come from the growth of the ndvdual 1 Brennan and Schwartz (1985) ndeendently offer a smlar dervaton of what they term a ostve bas n the comound return of an equally weghted ortfolo (a secal case of the FS analyss) over and above the comound return of an equally weghted geometrc ndex. Booth and Fama (199) searately derve a smlar decomoston of a ortfolo s comound return and attrbute a dversfcaton return to an asset s contrbuton to the ortfolo return that s n excess of the asset s comound return. Booth and Fama (199) rovde some analyss at the asset allocaton level across ndexes, but do not examne the otental effects that occur wthn ortfolos across stocks. 3

5 stocks. Portfolo comarsons can obscure otentally mortant economc henomena at work n the sources of ortfolo erformance at the level of the underlyng samle frms. Exhbt 1 Returns Log Returns Perod 1 Perod Average Varance Perod 1 Perod Average Varance A % -50% +5% A -50% +100% +5% EW A +5% +5% +5% B 1 +5% +5% +5% B +5% +5% +5% EW B +5% +5% +5% Ths examle also hels to hghlght how the sources of ortfolo returns are relevant when desgnng and nterretng the analyss of fnancal erformance. For examle, f these stocks are the subject of a cross-sectonal study n whch characterstc A and characterstc B are comared, what would we conclude about how these characterstcs are assocated wth the cross-secton of stock returns? Because they yeld dentcally-erformng ortfolos, should nvestors be ndfferent to holdngs stocks A 1 and A comared B 1 and B? Should these stocks have the same rsk remum? If these stocks are the subject of an event study n whch one ortfolo reresents a ortfolo of control stocks, what would we conclude about the event? Based on the ortfolo results, whch stocks generate more shareholder wealth? Whch comany s executves would realze more ncentve comensaton? Do managers have an ncentve to generate the returns of stocks A 1 and A or B 1 and B? If these ortfolos reresent actvely managed ortfolos, whch ortfolo has the better ortfolo manager? Whch manager s a better stock cker? Whch ortfolo s better Whle a value weghted ortfolo mght hel reveal some of the dfferences among these stocks, we focus on equal weghted ortfolos because of ther common use n emrcal studes, and ther mlct use n emrcal methods, such as the Fama-Macbeth (1973) regressons. We also use our method to examne the sources of return dfferences between value- and equal-weghted ortfolos. 4

6 dversfed? What f dfferent? EW A has a hgher ortfolo return than 5 EW B, but the sources of returns are These questons hghlght the challenge facng researchers n tryng to summarze the affect of a characterstc, event, or olcy on a samle of stocks or n tryng to draw nferences about ndvdual stocks based on the erformance of ortfolos (and average holdng erod returns) of those stocks. Ths challenge also justfes the careful attenton to methodology that s gven n many studes. Research methods, whether n studes of the cross-secton of stock returns, ortfolo erformance, or events, tycally analyze the erformance of ortfolos of samle frms. However, n many cases, t s an understandng of the sources of ortfolo erformance due to the underlyng frms erformance and characterstcs that s the ultmate goal of the study. The method examned n ths artcle contrbutes to the arsenal avalable to researchers by further decomosng ortfolo returns nto sources of erformance due stock-level returns or growth rates, stock varances, and ortfolo varance. We llustrate the mortance of ths method by emrcally estmatng the sources of ortfolos comound returns for several characterstc-based ortfolos that aear frequently n the lterature. Adotng the FS termnology, we show that small stock ortfolos have hgher excess growth rates than large stock ortfolos even though the underlyng small stocks have lower average comound returns than large stocks. In short, the sze effect s not a small frm effect, but a small frm ortfolo effect drven by the excess growth rate of the ortfolos. The hgher excess growth rates more than offset the lower average comound returns to small stocks, resultng n hgher returns for ortfolos of small stocks. Ths arsng shows that small stock ortfolos outerform due to the hgher varance of small stocks. Smlarly, equally weghted ortfolos outerform valueweghted ortfolos because of the equally weghted ortfolo s relatve overweghtng of stocks wth hgher varances, not by lacng greater weght on stocks wth hgher comound returns. We

7 confrm that cross-sectonal dfferences n ortfolos holdng erod returns related to sze are fully exlaned by stocks varances. Furthermore, we fnd no sgnfcant cross-sectonal relatonsh between a frm s growth rate and frm sze or varance. Unlke the results for sze-sorted ortfolos, the varance of the underlyng stocks (va the ortfolo excess growth rate) does not exlan the outerformance of hgh book-to-market ortfolos relatve to low book-to-market ortfolos. Rather, the average comound growth rates of the underlyng hgh book-to-market stocks account for ther ortfolos sueror erformance. These emrcal results hghlght the mortance of utlzng the attrbuton from the FS mathematcal descrton of ortfolo returns. Notably, the sze effect aears to be exlaned by the nfluence of stocks varances and covarances on ortfolo returns and not due to cross-sectonal dfferences among stocks comound returns. In contrast, the book-to-market effect n ortfolo returns s rmarly determned by the cross-sectonal dfferences among stocks based on book-to-market characterstcs. The remander of ths aer s organzed as follows. Secton rovdes the background for our analyss, revews the related lterature, and develos the emrcal model that arses the longterm return nto comonents ncludng the excess growth rate. Secton 3 examnes the crtcal role of the excess growth rate n determnng a ortfolo s erformance usng the erformance dfferences between equally weghted and value-weghted ortfolos as an llustraton. Secton 4 focuses on ortfolo growth rates for ortfolos formed usng sze and book-to-market decles. Ths secton shows that stocks varances and ther contrbuton to the ortfolo excess growth rate exlan the well-known result that ortfolos of small stocks outerform ortfolos of large stocks, but that the excess growth rate s not the rmary determnant of a book-to-market effect. Secton 5 examnes the cross-secton of stock returns and stock growth rates and corroborates the fndng that stock 6

8 varances fully exlan the sze effect. The summary and concluson n Secton 6 relates these fndngs to exstng research and suggests aths for further research.. Background Numerous academc studes have demonstrated how cross-sectonal varaton n the characterstcs of stocks s related to ortfolo erformance over tme (Brown and Warner, 1980; Barber and Lyon, 1997; Kothar and Warner, 1997; Lyon, Barber and Tsa, 1999; Boyton and Oenhemer, 006). However, ths generalzaton from ndvdual stocks to ortfolos and vce versa s subject to nherent lmtatons as the tme-seres atterns of ndvdual stock returns do not drectly aggregate to ortfolo return atterns. We therefore aroach the ssue from the ooste drecton. By mosng recse and consstent measurement methods on stock and ortfolo returns by usng log returns and fxed rebalancng frequences we are able to arse ortfolo returns nto the underlyng sources of contrbutons made by the consttuent stocks. Ths allows us to undertake an analyss of the cross-sectonal characterstcs of ndvdual stocks and ther contrbuton to a ortfolo s return that was not ossble n ast studes. Frm sze s an examle of a cross-sectonal stock-level characterstc that can be used to show how economc exlanatons can be greatly refned through our decomoston of ortfolo returns. An nformal survey of fnance academcs, racttoners, ublshed artcles and leadng textbooks shows that dscussons of the sze effect fall nto two categores: 1) ortfolo comarsons (e.g., Portfolos of small frms outerform ortfolos of large frms. ); and ) stock comarsons (e.g., Small frms earn hgher returns than large frms. ). When dscussng comound returns, one category does not mly the other. Indeed, t s ossble to have a ortfolo of stocks that outerform over the long term, even though the underlyng stocks are below-average erformers. Alternatvely, a ortfolo formed by selectng stocks that generally outerform ther eers mght fal 7

9 to outerform a ortfolo of eer stocks n the resence of a secfc covarance structure among the stocks. By makng the dstncton between the average comound return of a set of stocks and the average comound return of a ortfolo of the same set of stocks, we are not tryng to dg u decades-old debates as to the absolute suerorty of one measure over the other. Rather, ths aer offers the arsng of a ortfolo s comound return as a method that allows researchers more recson n analyzng and understandng the erformance of stocks and ortfolos. For examle, modern ortfolo theory suggests that the rce of a securty s determned by the contrbuton of that securty to a ortfolo s return. Our analyss demonstrates a new aroach that allows researchers to move beyond the frst moment of securty returns when analyzng the contrbuton of a consttuent securty to ts ortfolo s return. Addtonally, the tycal urose of cross-sectonal analyss s to make a drect comarson of one stock to another (or one stock to all others) wth the same characterstcs, as n many event studes. Our analyss ndcates why the use of ortfolo returns can lead to msnterretaton n such stuatons. In other cases, such as the evaluaton of a fund manager s stock selecton ablty, the urose of the cross-sectonal analyss mght roerly be consdered to be analyss of a stock s contrbuton to a ortfolo. Our decomoston allows sharened nferences about a fund manager s erformance and skll n ths context. Let ( ) be the comound growth rate, or log return, for stock at tme t over a holdng, t h erod of length h. 3 If ( ) s normally dstrbuted (.e., stock returns are log-normally dstrbuted), t h wth mean (h) and varance ( ), then the average growth rate and the average holdng erod h return, r (h), are related to one another as 3 Throughout the aer, we use the terms comound growth rates, growth rates, and log returns nterchangeably to dstngush these from the holdng erod return. Other common names for growth rates are contnuous returns, comound returns, and contnuously comounded returns. Whle the use of log returns s not essental to llustrate the excess growth rate of a ortfolo, t greatly smlfes the analyss by allowng us to use the tractable soluton rovded by FS. 8

10 9 ) ( ) ( ) ( h h h r. 4 (1) If stock holdng erod returns are not log normally then t s an emrcal whether (1) s a useful aroxmaton. If stock returns are lognormally dstrbuted, then ortfolo returns are not lognormally dstrbuted. Smlarly, f ortfolo returns are log normally dstrbuted, the average holdng erod return of ortfolo s can be aroxmated by ) ( ) ( ) ( h h h r. () Usng the defnton of the arthmetc average return on a ortfolo and Eq. (1), the arthmetc average return on a ortfolo that mantans a constant weght n each stock s gven by ), ( 1 ) ( ) ( ) ( ) ( ) ( N N N N h w h w h h w h w r h r (3) where w s the ortfolo weght n stock. Note that ths ortfolo s mlctly rebalanced every holdng erod h to mantan the constant weghts. Furthermore, the holdng erod h s the horzon over whch returns are measured to estmate the means and varances. Combnng (3) and (), we can solve for the comound growth rate on a ortfolo as ) ( ) ( 1 ) ( ) ( 1 1 h h w h w h N N. (4) 4 The exact relatonsh s ) ( ) ( ) ( ln 1 h h h r. We use the aroxmaton formula for ease of exoston and llustraton, drong the aroxmaton beyond Eq. (). The ultmate utlty of ths dervaton and aroxmaton s examned emrcally later n the aer.

11 Usng contnuous growth rates (.e., contnuously comounded returns) and contnuous rebalancng, Fernholz and Shay (198) derve a contnuous-tme verson of (4) and dentfy the second term as a ortfolo s excess growth. We denote the (dscrete tme) ortfolo excess growth rate as N * 1 ( h) w ( h) ( h). (5) 1 The urose n dentfyng the holdng erod length h n the dscrete tme verson s to emhasze the mortance of matchng the measurement nterval of the varances and covarances wth the frequency of rebalancng to the constant weghts. For examle, when usng monthly returns to estmate Eq. (4), monthly rebalancng would be mled. The rebalancng to constant weghts s not necessary for a ortfolo to dslay an excess growth rate, but t allows us to use Eq. (5) n order to estmate a ortfolo s excess growth rate by estmatng ts comonents ndeendently. Equaton (4) shows that a ortfolo s comound growth rate comes from two rmary sources: 1) the weghted average stock growth rates; and ) the excess growth rate. Not surrsngly, weghtng stocks wth hgh comound growth rates benefts a ortfolo s comound growth rate. Less obvous s the mact to a ortfolo s comound growth rate that arses from the excess growth rate. Secfcally, any ortfolo that heavly weghts stocks wth relatvely hgh varances should beneft, assumng that the stocks are not erfectly correlated so that the ortfolo s varance s not ncreased roortonally. 5 Whether secfc ortfolos ut more or less weght on hgh varance stocks relatve to a secfc benchmark ortfolo, such as a value-weghted ortfolo, s an emrcal queston. Potental canddates for such ortfolos are equally weghted ortfolos, momentum ortfolos, contraran ortfolos, or small stock ortfolos. To the extent that the stocks n these ortfolos have hgher varances, they should have hgher excess growth rates. 5 Booth and Fama (199) derve an analogous exresson for a ortfolo s average log return from the Taylor seres exanson of the natural logarthm. However, they focus on the asset s comound growth rate and ts contrbuton to the ortfolo s growth rate, referrng to the effect as the dversfcaton return. Ther analyss focuses on the dversfcaton return to varous asset-class-level ndexes and does not consder the effect wthn the ndexes and asset classes that they consder. 10

12 Equaton (4) s useful beyond ts mlcatons for ortfolo analyss. Secfcally, many studes utlze, ether exlctly or mlctly, constant weght ortfolos n ther analyss. Indeed, any study emloyng anel data that averages the returns across stocks and then consders the average of ths measure over tme s subject to the multle effects that act on a ortfolo s growth rate. Secfcally, the comound growth rate of Portfolo A could exceed the comound growth rate of Portfolo B f: 1) the stocks n Portfolo A have hgher comound growth rates than the stocks n ortfolo B; or ) the excess growth rate of Portfolo A exceeds that of Portfolo B; or both. It would be nadvsable to nfer from a hgher growth rate of Portfolo A comared to Portfolo B that the stocks n Portfolo A have hgher comound growth rates comared to stocks n Portfolo B just by consderng the ortfolos growth rates. Equaton (4) allows researchers to arse the effects and dstngush between an nterretaton that mles somethng about the underlyng frms and another nterretaton that mles somethng only about ortfolos of such frms. In some crcumstances, ths subtlety mght be consdered neglgble semantcs, whle n other crcumstances t mght be crtcal to the nterretaton and mlcatons of the study. After consderng the nfluence of excess growth rates, t should come as no surrse that researchers have long found that ortfolos comosed of hgh-volatlty stocks dslay hgher returns. Because small stocks are more volatle than large stocks we should exect ortfolos of small stocks to dslay hgher excess growth rates and, thus, hgher growth rates even f the comound growth rates of small frms and large frms are equal. When small stocks are sorted nto ortfolos and comared aganst ortfolos of larger stocks, ths s exactly what researchers have found (Banz, 1981; Brown, Kledon, and Marsh, 1983; Kem, 1983; Schwert, 1983; Chan, Chen, and Hseh, 1985; Fama and French, 199; Fama and French, 1996; Danel and Ttman, 1997; Conrad and Kaul, 1998; Fama and French, 008). Most studes of the sze effect have examned ortfolo returns and not the comound growth rates of ndvdual stocks. Ths subtlety s rarely recognzed as 11

13 many nterret the results of these studes to mly that small frms outerform large frms, when the more recse result of many of these studes s that small frm ortfolos outerform large frm ortfolos. A varety of artal exlanatons have been offered for the sze effect, rangng from mcrostructure ssues (Blume and Stambaugh, 1983; Roll, 1981), seasonalty (Kem, 1983; Schwert, 1983), tax effects (Roll, 1983, Renganum, 1983) and so on. Studes of ortfolo returns and eventstudy methods (Barber and Lyon, 1997; Kothar and Warner, 1997; Lyon, Barber, and Tsa, 1999; Boyton and Oenhemer, 006) have often nvoked these effects n order to account for atterns that we show are exlaned by the varance-drven excess growth rates of ortfolos. Our results do not necessarly nvaldate exlanatons of the sze effect, n that our analyss s desgned to cature the effects of alternatve exlanatons for the sze effect to the extent that they nfluence a ortfolo s excess growth rate. For examle, mcrostructure effects such as bd-ask bounce may ncrease the varance of small stock returns and thus ndrectly ncrease a ortfolo s measured excess growth rate, regardless of the effect on the comound returns of the underlyng stocks. Varables that drectly affect the comound returns to the underlyng stock, such as an unexlaned rsk factor, wll be held constant n our analyss. 3. The Excess Growth Rate of Equally Weghted and Value-Weghted Portfolos The revous secton shows the source of a ortfolo s growth rate as orgnatng from two comonents: 1) the weghted average of the ortfolo s stocks growth rates; and ) the ortfolo s excess growth rate. The excess growth rate s comrsed of the weghted average of the varance of the ortfolo s underlyng stocks varances and the ortfolo s varance. Below, we estmate each comonent of varous ortfolos growth rates accordng to Eq. (4) and comare the growth rate 1

14 estmates wth the ortfolos actual growth rates. We comare the growth rates of constant-weght (rebalanced) ortfolos and the sources of dfferences among the ortfolos growth rates. We collect monthly return and begnnng-of-month catalzaton data from the CRSP Database for the erod January 1960 through December 009. We form two ortfolos n January of each year: an equally weghted (EW) ortfolo and a seudo-value-weghted (VW) ortfolo. We rebalance each ortfolo to ts begnnng-of-year weghts and create a monthly return seres for each ortfolo. We have dentfed the value-weghted ortfolo as seudo-value-weghted because the ortfolo s rebalanced each month to ts begnnng-of-year value weghts to conform to the constant weght assumton of Eq. (4). 6 Stocks that do not have vald catalzaton data n January are not ncluded n ether of that year s ortfolos. However, stocks that dro out of the database durng a samle year are ncluded u to the ont at whch they dro out. Beyond that ont, the ortfolo s reconsttuted to nclude only the remanng stocks. By mantanng stocks at ther begnnng-of-year weghts, we can estmate each term n Eq. (4) and establsh whether Eq. (4) s a reasonable aroxmaton of ortfolo comound growth rates. We use each stock s monthly log returns to estmate ts average comound growth rate and growth rate varance durng that year. 7 Smlarly, we comute a ortfolo s actual comound growth rate by takng the natural log of ts gross return; and comute the ortfolo s varance of the monthly log returns for each year. To examne whether the ortfolo growth rate effects are lmted to or drven by stocks at the extreme low end of the catalzaton range, we erform our analyss on three unverses of securtes: all securtes n the CRSP database ( All CRSP ); the to 1,000 securtes by 6 Note that the rebalancng frequency (the horzon durng whch constant weghts are aled) matches the horzon over whch stock and ortfolo returns are measured to estmate means and varances. 7 Stocks that dro out of CRSP durng the year are ncluded n these estmates as long as the stock has seven or more monthly observatons. Note that stocks that dro out ror to July of each year are stll ncluded n the actual ortfolo returns, but are not ncluded n the estmated comonents of the ortfolo s growth rate. We exect ths requrement to nduce nose nto our estmates, but not to bas the estmates systematcally. 13

15 catalzaton rank n January of each year ( To 1000 ); and the to 500 securtes by catalzaton rank n January of each year ( To 500 ). Table 1 reorts the average and standard devaton of annualzed estmates of the terms n Eq. (4) across years from 1960 through 009. As s well known, the equally weghted ortfolos have hgher average actual comound returns than ther value-weghted counterarts. The largest dfference occurs n the samle that ncludes all CRSP stocks. The EW All CRSP ortfolo has an average annual comound return of 11.68% comared to the VW All CRSP ortfolo s 8.97%. Of artcular mortance to ths aer are the sources of the average erformance n the equally weghted and value-weghted ortfolos. The equally weghted average stock comound growth rate s only 0.33% er year, comared to the value-weghted average stock comound growth rate of 4.89% er year. In other words, the value-weghted ortfolo has more exosure or weght n stocks that have hgher comound growth rates. Ths suggests that smaller catalzaton stocks have lower average comound returns comared to larger catalzaton stocks, on average. The results for the ortfolos of the To 1000 and To 500 stocks are qualtatvely smlar, though the dfference n weghted average stock growth rates s not as large. For examle, the weghted average stock growth rate for the equally weghted To 1000 ortfolo averages 5.03% er year comared to 5.50% er year for the value-weghted To 1000 ortfolo. Smlarly, the weghted average stock growth rates for the equally weghted To 500 ortfolo averages 5.49% er year comared to 5.67% er year for the value-weghted To 500 ortfolo. The ncrease n the average stock growth rate as the unverse of stocks moves from the broadest unverse ( All CRSP ) to the larger catalzaton unverses ( To 1000 and To 500 ) rovdes further evdence that comound growth rates ncrease, on average, wth stocks catalzatons. In contrast to the exosures to stock growth rates, the excess growth rates of the equally weghted ortfolos exceed that of the value-weghted ortfolos. Indeed, the hgher excess growth 14

16 rates n the equally weghted ortfolos are more than enough to comensate for the exosure to lower average growth rate stocks comared to the value-weghted ortfolo. In the samle from all CRSP stocks, the average estmated excess growth rate of the equally weghted ortfolo s 11.05% comared to only 4.08% for the value-weghted ortfolo. Combned wth the weghted average stock growth rates, the estmated growth rate for the equally weghted ortfolo of all CRSP stocks s 11.38%. The excess growth rate accounts for more than 95% of the equally weghted ortfolo s average estmated growth rate for the samle usng all CRSP securtes. The rmary contrbutor to the excess growth rate s the average stock varance. When equally weghted, CRSP stocks have an average annual varance of monthly returns of 0.575, but when value-weghted, the average annual varance of monthly returns s only Smlarly, the To 1000 and 500 stocks have (equally weghted) average annualzed varances of only and 0.109, resectvely. In short, small stocks have hgher return varances than larger stocks, leadng to a hgher excess growth rate for ortfolos that lace more weght n these stocks. Table 1 shows that the dfference n the actual growth rates between the equally weghted and value-weghted ortfolos s only sgnfcant at the 16% level (t-stat = 1.43) for the samle of all CRSP stocks and at less sgnfcant levels for the To 500 and To 1000 samles. As dscussed above, ths total growth rate s a functon of lower average stock growth rates combned wth hgher excess growth rates n the equally weghted ortfolos comared to the value-weghted ortfolos. For the All CRSP samle, the dfference n the average stock growth rates between the equally weghted and value-weghted ortfolos s sgnfcant at the 5% level (t-stat = -.41), whle n the other samles the dfference s not statstcally sgnfcant. In contrast, the dfferental excess growth rate s statstcally sgnfcant at the 1% level across all samles. Indeed, the excess growth rate for 15

17 the equally weghted ortfolos exceeds that of the value-weghted ortfolos n every year of the samle. Equaton (4) rovdes us wth a method to estmate what a ortfolo s growth rate should be based on the underlyng average stock returns and varances under the assumton of log-normally dstrbuted returns. Usng all CRSP stocks, the equally weghted ortfolo s actual average annual growth rate s 11.68% comared to an estmate of 11.38% usng the comonents of Eq. (4). Fgures 1 lots each samle year s estmated growth rates aganst the actual annual growth rates for the equally weghted and value-weghted ortfolos usng all CRSP stocks. Regressons (not reorted) of the actual annual ortfolo growth rates on the estmated growth rates for each of the sx ortfolos n Table 1 yeld sloe coeffcent estmates of aroxmately 1.0, statstcally nsgnfcant ntercet coeffcents, and R-squared values rangng from to Ths suggests that, whle stock and ortfolo returns mght not be truly log normally dstrbuted, the growth rate formula of Eq. (4) nonetheless descrbes ortfolos returns very well. The accuracy of the growth rate formula n estmatng actual growth rates suorts ts use to decomose the erformance of ortfolos nto the effects due to stock comound growth rates and stock varances. 4. The Excess Growth Rate of Sze and Book-to-Market Portfolos Fama and French (199), among others, show a sze effect n whch ortfolos of small (Low ME) stocks have hgher average returns comared to ortfolos of large (Hgh ME) stocks. The analyss below shows that ths result s drven entrely by the ortfolos relatve excess growth rates. Secfcally, small stock ortfolos have hgher growth rates because small stocks have hgher varances, leadng to hgher excess growth rates for ortfolos comrsed of those small stocks. Moreover, small stock ortfolos have hgher growth rates deste the fact that small stocks have 16

18 lower comound growth, on average. In other words, small stocks do not outerform, but small stock ortfolos do. Fama and French (199) (hereafter, FF) also show that ortfolos of hgh book-to-market (Hgh BE/ME) stocks have hgher average returns comared to ortfolos of low book-to-market (Low BE/ME) stocks. The attrbuton of the ortfolo growth rates nto stock growth rate and excess growth rate comonents shows that, unlke the sze effect, the returns to BE/ME sorted ortfolos are drven by the underlyng stocks growth rates and not ther varances. Thus, the conclusons regardng growth rates of ortfolos of BE/ME stocks can be generalzed to ther underlyng stocks Sze Decle Portfolos For the samle erod of January 1960 through December 009, we match monthly return data and catalzaton (ME) data for common stocks wth Comustat data on book value of equty (BE). We construct sze decle ortfolos n a smlar manner as n FF. 8 In January of each year, stocks are assgned to decle ortfolos based on the revous year s endng market value (ME). As n FF, the decle breakonts are determned usng NYSE stocks, but stocks from NYSE, AMEX, and Nasdaq are ncluded n the ortfolos. Equally weghted average ortfolo returns are calculated for each month usng monthly stock returns. As n the revous secton, the natural log of (one lus) the returns s used to calculate comound growth rates. Smlarly, a month s actual ortfolo comound growth rate s the natural log of (one lus) the ortfolo s return n that month. Stocks comound growth rates are calculated n a smlar manner and annual varances are calculated usng monthly growth rates. 8 Because we are not focused on market betas, we do not estmate betas for stocks or ortfolos. Therefore, we do not requre estmaton erod data for stocks to be ncluded. In ths way, our samles nclude more stocks than aear n the FF samles. We have conducted the analyss wth the estmaton erod requrement and found nearly dentcal results. 17

19 Table shows the growth rates and estmated comonents of the growth rates from Eq. (4) for the sze decle ortfolos. Panel A reorts the results for the entre 50-year erod, whle Panel B reorts the results for 1963 through 1990 a erod substantally smlar to the FF samle erod. The last column, (act.), shows that the actual average ortfolo growth rates are smlar to those of Fama-French (199), although FF use ortfolo returns rather than growth rates. The growth rates of ortfolos of small stocks generally are hgher than the large stock ortfolo growth rates. By decomosng the ortfolo growth rate nto ts comonents usng Eq. (4), we see that the outerformance of the lower decle ortfolos (.e., ortfolos of smaller stocks) s drven by the excess growth rate, * ( est.), of those ortfolos. The average ortfolo excess growth rates decrease monotoncally as the sze decle ncreases from the smallest to the largest stocks. Ths decrease occurs deste the fact that the ortfolo varance decreases as the sze decle ncreases. Thus, the drvng factor n the outerformance of small stock ortfolos s the varance of the underlyng stocks. For the entre samle erod, the smallest sze decle ortfolo holds stocks wth an average varance of er month, whle the largest sze decle ortfolo holds stocks wth an average varance of only er month. Consstent wth the results for the equally weghted and value-weghted ortfolos examned n the revous secton, the average comound growth rates of the smaller stocks are lower than those of larger stocks. The outerformance of small stock ortfolos occurs deste the average undererformance of ndvdual small stocks. The varance of the small stocks s enough to generate ortfolo excess growth rates that makes u for the small stocks lower comound growth rates. For the FF erod of 1963 through 1990, the average stock comound growth rates are very smlar at aroxmately 0.65% er month for all but the lowest two decles. The declne n the ortfolo growth rates as the sze of the underlyng stocks ncreases s drven by declne n the ortfolos excess growth rates due to the declne n the underlyng stocks varances. 18

20 Fgure lots the cumulatve average stock growth rate, cumulatve ortfolo growth rate, and cumulatve excess growth rate from 1960 to 009 for the lowest and hghest sze (ME) decles from Panel A of Table. The small stock ortfolo s larger average excess growth rate s reflected n a steeer sloe of ts cumulatve excess growth rate comared to the large stock ortfolo s. Moreover, the sloe of the small stock ortfolo s cumulatve excess growth rate s consstently greater than the large stock ortfolo s. Interestngly, the average small stock s growth rate aears to be hgher than the average large stock s growth rate untl the early 1980 s, after whch the average small stock s growth rate s generally negatve. Fnally, as also ndcated n Table, the grah shows that all of the growth n the small stock ortfolo has come from ts excess growth rate, whle the cumulatve growth n the large stock ortfolo can be attrbuted almost equally to the average underlyng stock growth rate and the excess growth rate. 4.. Bd-Ask Sread Effects Extant research has noted the otental nfluence of bd-ask bounce n nflatng rebalanced ortfolo returns (Conrad and Kaul, 1993; Blume and Stambaugh, 1983; Roll, 1981). In general, f bd-ask sreads nduce negatve seral correlaton n stock returns, then a constant-weght (.e., rebalanced) ortfolo s returns reflect the effects of sellng stocks hgh and buyng stocks low, on average. Average ndvdual stock growth rates are mmune to ths effect, snce growth rates reflect buy-and-hold comound returns. Therefore, the frst comonent of the ortfolo growth rate, reflectng average stock growth rates, does not reflect ths otental asect of a rebalanced ortfolo s erformance. Rather, any bd-ask bounce should be embedded n the excess growth rate comonent, rmarly va the nflaton of stocks varances due to the bd-ask sread. Our use of monthly returns should mnmze the mact of these effects, but we examne the robustness of our results to otental effects of ths tye. 19

21 The ortfolo growth rate attrbuton s easly adated to examne addtonal nfluences on stocks growth rates or varances. We examne the otental nfluence of the bd-ask sread, or any other factor that may nduce autocorrelaton n returns, on the excess growth rate by consderng stocks autocorrelaton-adjusted varances n the estmaton of a ortfolo s excess growth rate. In ths alcaton, we are not concerned wth drectly estmatng the bd-ask sread. Rather, we are nterested n the varance of nnovatons n stock returns after controllng for autocorrelatonnducng effects such as bd-ask bounce. Therefore, we adot a model sutable for nferrng mcrostructure effects from low-frequency return data consstent wth Roll (1984) and dscussed n Holden (009). For each stock n each year, we estmate a model of a stock s monthly growth rate,,t, as a functon of ts lagged monthly growth rate, as gven by, t, t 1, t. (6) Usng the resduals from ths regresson, we estmate the autocorrelaton-adjusted return varance,, whch we use to estmate an autocorrelaton-adjusted excess growth rate. 9 We reort these results n Panel C of Table. The autocorrelaton-adjusted ortfolo estmated growth rates are smaller than the actual (and unadjusted estmated) ortfolo growth rates across all sze decles. The decrease n the growth rates s due to autocorrelaton-adjusted stock varances beng lower than stocks unadjusted varances. The excess growth rate decreases aroxmately 9% across all sze decles. Secfcally, the estmated autocorrelaton-adjusted average monthly excess growth rate n the smallest sze decle ortfolo s 1.5% comared to the estmated unadjusted excess growth rate (from Panel A) of 1.39%. The mact to the largest sze decle ortfolo excess s smaller, wth the estmated autocorrelaton-adjusted average monthly excess growth rate at 0.4% comared to 0.7% for the 9 We urosefully do not estmate an autocorrelaton-adjusted ortfolo varance, leavng ths varance as estmated n Panel A of Table. Though unlkely to be large, ths leaves any otental ostve bas to our ortfolo varance, leadng to a negatve bas n our estmated autocorrelaton-adjusted excess growth rate. 0

22 estmated unadjusted excess growth rate. Deste the larger declnes n the smaller stock decles, we contnue to observe the same attern for the estmated autocorrelaton-adjusted ortfolo growth rates as for the actual ortfolo growth rates, ndcatng that bd-ask sread effects do not exlan the attern of excess growth rates across sze-sorted ortfolos Book-to-Market Portfolos Table 3 shows the average growth rates and the estmated comonents of the average growth rates from Eq. (4) for ortfolos formed usng book-to-market rato (BE/ME) decles. Agan, these results are nearly dentcal to those of Fama-French (199), n that the lowest decles have the lowest average erformance. In contrast to the results for the sze decle ortfolos, the attern of average ortfolo growth rates across book-to-market decles aears to be drven rmarly by the underlyng stocks average growth rates. That s, the average comound growth rates of low BE/ME stocks are lower than the comound growth rates of hgh BE/ME stocks. The average stock growth rates ncrease almost monotoncally from the lowest BE/ME decle to the hghest BE/ME decle. The stocks n the lowest BE/ME decle have an average comound growth rate of -1.16% er month, whle the stocks n the hghest BE/ME decle have an average comound growth rate of 0.55% er month. The sread between the hghest and lowest BE/ME decles average stock growth rates s 1.71%, whle the sread between the hghest and lowest sze decles s only 0.55%. As n the sze decle results, the average ortfolo excess growth rates decrease monotoncally as the BE/ME decle ncreases because the varance of the ortfolos underlyng stocks decrease monotoncally excet for the hghest BE/ME decle. Across the entre samle erod, the lowest BE/ME decle ortfolo s excess growth rate s 1.5% comared to the hghest BE/ME decle ortfolo s excess growth rate of only 0.91%. However, ths decrease s smaller n magntude than n 1

23 the sze decles and s more than offset by the ncrease n the underlyng stocks average growth rates. The results for the erod corresondng to the FF samle erod are qualtatvely the same, wth average stock growth rates ncreasng and excess growth rates decreasng as BE/ME ncreases. As wth the entre samle erod, the FF erod s results ndcate that the attern across BE/ME decles s due rmarly to the attern of average stock growth rates. Unlke sze decle ortfolos, the drvng factor n the outerformance of low BE/ME stock ortfolos s the average comound growth rates of the underlyng stocks. 5. The Cross-Secton of Stock Growth Rates Begnnng wth Fama and Macbeth (1973), cross-sectonal regressons have been used to examne the assocaton between stock returns and stock characterstcs. Cross-sectonal analyss aears to have the desrable roerty of ease of nterretaton n that the sloe coeffcents from cross-sectonal regressons reflect a drect assocaton between a frm s return (deendent varable) and the frm s characterstcs (ndeendent varables). The attrbuton of the ortfolo s growth rate n the revous sectons shows that a dfference can exst between the comound growth rate to a ortfolo that s formed on a secfc stock characterstc and the comound return on the underlyng stocks wth that characterstc. To further examne the role of stocks varances n average returns and the assocaton between a stock s growth rate and ts other characterstcs, such as sze and market-to-book rato, we estmate cross-sectonal regressons over our samle erod. The results n the revous secton rase questons about the commonly acceted vew that smaller frms have hgher returns. Secfcally, the hgher ortfolo growth rates for small frm ortfolos comared to large frm ortfolos aears to be rmarly due to the varance of small stocks. On the other hand, the relatonsh between the book-to-market rato and returns that s mled by ortfolo returns aears to be due to underlyng stocks and not smly a functon of

24 stocks varances. To further examne ths, we nclude the same-year varance of a stock s monthly growth rates (Varance) as an ndeendent varable n the cross-sectonal regressons, n addton to usng ln(me) and ln(be/me) to measure a frm s sze and market-to-book rato, resectvely. If the sze effect n ortfolo returns s due to stock varances, then ln(me) should not be negatvely assocated wth returns when Varance s ncluded n the regresson. More mortantly, based on the result n the revous secton that average stock growth rates are hgher for frms n the larger sze decles, we exect the average sloe on ln(me) to be ostve when usng comound growth rates as the deendent varable. We reort the tme-seres average sloe coeffcents from the monthly cross-sectonal regressons n Table 4. As shown n Panel A, the average sloe on ln(me) n the monthly return regressons s negatve over the erod from January 1960 through December 009. Smlarly, Panel B shows the results from the FF samle erod, wth a negatve sloe on ln(me) that s nearly dentcal to that n FF. The average coeffcent on ln(be/me) s statstcally sgnfcant and ostve n both Panel A and Panel B. Includng both ln(me) and ln(be/me) n the monthly return regressons results n average coeffcents substantally the same as those reorted n FF. However, when stocks return varances are ncluded n the regressons, the average coeffcent on ln(me) s ostve, though statstcally nsgnfcant at tradtonal levels, whle the average coeffcent on Varance s ostve and statstcally sgnfcant. It s commonly acceted that smaller stocks are more volatle. As suggested by the growth rate decomoston and the cross-sectonal regressons, the volatlty of small stocks s the rmary contrbutor to small stock ortfolo s growth rates and arthmetc average returns, suggestng that cauton should be exercsed when nferrng any return remum due to ln(me). Wth the average cross-sectonal correlaton between ln(me) and Varance of log returns beng -0.46, the effect of multcollnearty n the cross-sectonal regressons must be consdered. We note that the average 3

25 sloe coeffcent on Varance remans qute stable when ln(me) s added to the model. Moreover, the results usng stocks growth rates (.e., log returns) as the deendent varable n regressons casts more doubt on a sze effect n ndvdual stock returns. Over the entre samle erod, the coeffcent on ln(me) n monthly stock growth rate regressons averages a ostve , but s statstcally nsgnfcant at the 5% level. The average coeffcent remans ostve but statstcally nsgnfcant n multvarate regressons over the entre samle and statstcally sgnfcant n the FF samle erod. The average sloe on stocks own varances s statstcally nsgnfcant n both samle erods negatve n the erod and ostve n the erod. We conduct a J test (Davdson and MacKnnon, 1981) to determne whether sze or varance aears to be the true underlyng factor n the cross-secton of monthly stock returns. Panel C of Table 4 shows that we are able to reject ln(me) n favor of Varance as the true underlyng factor n exlanng the cross-secton of stock returns. The results from the J test are consstent wth the results from the ortfolo growth rate decomoston that attrbutes the source of outerformance n small-stock ortfolos relatve to large stock ortfolos orgnatng from the varance of the underlyng stocks. Fnally, we note that the average sloe coeffcent for own stock varance n return regressons s aroxmately one-half, as mled by Eq. (1). Secfcally, the sloe coeffcent on Varance averages over the entre samle erod n the unvarate regressons and n the multvarate regressons that nclude both ln(me) and ln(be/me). Over the FF samle erod, the average sloe coeffcent s , whch s not statstcally dfferent from 0.50 at tradtonal levels. In also consderng the result of no statstcally sgnfcant relatonsh between growth rates and ether ln(me) or Varance, we conclude that the relatonsh between arthmetc average stock returns and stock varances smly reflects the mathematcal descrton of average stock returns n Eq. (1). That s, the use of arthmetc averages and holdng erod stock returns rather than 4

26 comound growth rates results n aarent relatonshs that have a mathematcal rather than an economc exlanaton. When varables that are hghly correlated wth varance, such as sze, are used n common research methods that mlctly nvolve ortfolos through the use of arthmetc averages, results must be nterreted wth cauton and the mathematcal mact of varance must be consdered. The use of comound growth rates (log returns) or the decomoston of ortfolo growth rates can ad the researcher n dstngushng between the effects of stocks characterstcs and the effects of varance. 6. Summary and Concluson Whenever anel data sets are emloyed, the researcher must judge how to erform the aggregaton across observatons and across tme erods, and then how to generalze the results along one dmenson to the other dmenson. Ths aer resents one method for erformng ths (ds)aggregaton, allowng an mroved nterretaton of how cross-sectonal varaton n frm sze s related to returns over tme. We emloy a mathematcal arsng of ortfolo returns nto two key sources: 1) the average growth rate of the ortfolo s underlyng stocks; and ) the excess growth rate of the ortfolo that s due to the underlyng stocks varances and covarances. Whle the decomoston that we examne has been known to the lterature for at least 5 years, t has been rarely cted; and, to our knowledge, we are the frst to utlze ths mathematcal relatonsh n comarng ndvdual stock characterstcs to ortfolo growth rates. We demonstrate that usng the growth rate comonents can yeld new and meanngful nsghts nto the sources of erformance of commonly examned ortfolos n the lterature. We show that equally weghted ortfolos outerform value-weghted ortfolos deste the fact that equally weghted ortfolos lace more weght on stocks wth lower average ndvdual erformance as measured by average comound growth rates. By lacng more weght on stocks 5