New facts in finance. John H. Cochrane

Transcription

1 New facs in finance John H. Cochrane Inroducion and summary The las 15 years have seen a revoluion in he way financial economiss undersand he invesmen world. We once hough ha sock and bond reurns were essenially unpredicable. Now we recognize ha sock and bond reurns have a subsanial predicable componen a long horizons. We once hough ha he capial asse pricing model (CAPM) provided a good descripion of why average reurns on some socks, porfolios, funds, or sraegies were higher han ohers. Now we recognize ha he average reurns of many invesmen opporuniies canno be explained by he CAPM, and mulifacor models are used in is place. We once hough ha long-erm ineres raes refleced expecaions of fuure shor-erm raes and ha ineres rae differenials across counries refleced expecaions of exchange rae depreciaion. Now, we see ime-varying risk premiums in bond and foreign exchange markes as well as in sock markes. We once hough ha muual fund average reurns were well explained by he CAPM. Now, we see ha funds can earn average reurns no explained by he CAPM, ha is, unrelaed o marke risks, by following a variey of invesmen syles. In his aricle, I survey hese new facs, and I show how hey are variaions on a common heme. Each case uses price variables o infer marke expecaions of fuure reurns; each case noices ha an offseing adjusmen (o dividends, ineres raes, or exchange raes) seems o be absen or sluggish. Each case suggess ha financial markes offer rewards in he form of average reurns for holding risks relaed o recessions and financial disress, in addiion o he risks represened by overall marke movemens. In a companion aricle in his issue, Porfolio advice for a mulifacor world, I survey and inerpre recen advances in porfolio heory ha address he quesion, Wha should an invesor do abou all hese new facs? Firs, a slighly more deailed overview of he facs hen and now. Unil he mid-1980s, financial economiss view of he invesmen world was based on hree bedrocks: 1. The CAPM is a good measure of risk and hus a good explanaion of he fac ha some asses (socks, porfolios, sraegies, or muual funds) earn higher average reurns han ohers. The CAPM saes ha asses can only earn a high average reurn if hey have a high bea, which measures he endency of he individual asse o move up or down wih he marke as a whole. Bea drives average reurns because bea measures how much adding a bi of he asse o a diversified porfolio increases he volailiy of he porfolio. Invesors care abou porfolio reurns, no abou he behavior of specific asses. 2. Reurns are unpredicable, like a coin flip. This is he random walk heory of sock prices. Though here are bull and bear markes; long sequences of good and bad pas reurns; he expeced fuure reurn is always abou he same. Technical analysis ha ries o divine fuure reurns from paerns of pas reurns and prices is nearly useless. Any apparen predicabiliy is eiher a saisical arifac which will quickly vanish ou of sample or canno be exploied afer ransacion coss. Bond reurns are no predicable. This is he expecaions model of he erm srucure. If long-erm bond yields are higher han shor-erm yields if he yield curve is upward sloping his does no mean ha you expec a higher reurn by holding long-erm bonds raher han shor-erm bonds. Raher, i means John H. Cochrane is he Sigmund E. Edelsone Professor of Finance in he Graduae School of Business a he Universiy of Chicago, a consulan o he Federal Reserve Bank of Chicago, and a research associae a he Naional Bureau of Economic Research (NBER). The auhor hanks Andrea Eisfeld for research assisance and David Marshall, John Campbell, and Rober Shiller for commens. The auhor s research is suppored by he Graduae School of Business and by a gran from he Naional Science Foundaion, adminisered by he NBER. 36 Economic Perspecives

2 ha shor-erm ineres raes are expeced o rise in he fuure. Over one year, he rise in ineres raes will limi he capial gain on long-erm bonds, so hey earn he same as he shor-erm bonds over he year. Over many years, he rise in shor raes improves he rae of reurn from rolling over shor-erm bonds o equal ha of holding he long-erm bond. Thus, you expec o earn abou he same amoun on shor-erm or longerm bonds a any horizon. Foreign exchange bes are no predicable. If a counry has higher ineres raes han are available in he U.S. for bonds of a similar risk class, is exchange rae is expeced o depreciae. Then, afer you conver your invesmen back o dollars, you expec o make he same amoun of money holding foreign or domesic bonds. In addiion, sock marke volailiy does no change much hrough ime. No only are reurns close o unpredicable, hey are nearly idenically disribued as well. Each day, he sock marke reurn is like he resul of flipping he same coin, over and over again. 3. Professional managers do no reliably ouperform simple indexes and passive porfolios once one correcs for risk (bea). While some do beer han he marke in any given year, some do worse, and he oucomes look very much like luck. Funds ha do well in one year are no more likely o do beer han average he nex year. The average acively managed fund performs abou 1 percen worse han he marke index. The more acively a fund rades, he lower he reurns o invesors. Togeher, hese views reflec a guiding principle ha asse markes are, o a good approximaion, informaionally efficien (Fama, 1970, 1991). Marke prices already conain mos informaion abou fundamenal value and, because he business of discovering informaion abou he value of raded asses is exremely compeiive, here are no easy quick profis o be made, jus as here are no in any oher well-esablished and compeiive indusry. The only way o earn large reurns is by aking on addiional risk. These views are no ideological or docrinaire beliefs. Raher, hey summarize he findings of a quarer cenury of careful empirical work. However, every one of hem has now been exensively revised by a new generaion of empirical research. The new findings need no overurn he cherished view ha markes are reasonably compeiive and, herefore, reasonably efficien. However, hey do subsanially enlarge our view of wha aciviies provide rewards for holding risks, and hey challenge our undersanding of hose risk premiums. Now, we know ha: 1. There are asses whose average reurns can no be explained by heir bea. Mulifacor exensions of he CAPM dominae he descripion, performance aribuion, and explanaion of average reurns. Mulifacor models associae high average reurns wih a endency o move wih oher risk facors in addiion o movemens in he marke as a whole. (See box 1.) 2. Reurns are predicable. In paricular: Variables including he dividend/price (d/p) raio and erm premium can predic subsanial amouns of sock reurn variaion. This phenomenon occurs over business cycle and longer horizons. Daily, weekly, and monhly sock reurns are sill close o unpredicable, and echnical sysems for predicing such movemens are sill close o useless. Bond reurns are predicable. Though he expecaions model works well in he long run, a seeply upward sloping yield curve means ha expeced reurns on long-erm bonds are higher han on shorerm bonds for he nex year. These predicions are no guaranees here is sill subsanial risk bu he endency is discernible. Foreign exchange reurns are predicable. If you pu your money in a counry whose ineres raes are higher han usual relaive o he U.S., you expec o earn more money even afer convering back o dollars. Again, his predicion is no a guaranee exchange raes do vary, and a lo, so he sraegy is risky. Volailiy does change hrough ime. Times of pas volailiy indicae fuure volailiy. Volailiy also is higher afer large price drops. Bond marke volailiy is higher when ineres raes are higher, and possibly when ineres rae spreads are higher as well. 3. Some muual funds seem o ouperform simple indexes, even afer conrolling for risk hrough marke beas. Fund reurns are also slighly predicable: Pas winning funds seem o do beer han average in he fuure, and pas losing funds seem o do worse han average in he fuure. For a while, his seemed o indicae ha here is some persisen skill in acive managemen. However, mulifacor models explain mos fund persisence: Funds earn persisen reurns by following fairly mechanical syles, no by persisen skill a sock selecion. Again, hese saemens are no dogma, bu a cauious summary of a large body of careful empirical work. The srengh and usefulness of many resuls are holy debaed, as are he underlying reasons for many of hese new facs. Bu he old world is gone. Federal Reserve Bank of Chicago 37

3 BOX 1 The CAPM and mulifacor models The CAPM uses a ime-series regression o measure bea, β, which quanifies an asse s or porfolio s endency o move wih he marke as a whole, i f R - R = ai + bim( R - R ) + e; = 1, 2... T for each asse i. Then, he CAPM predics ha he expeced excess reurn should be proporional o bea, i f E( R - R ) =b l for each i. λ m gives he price of bea risk or marke risk premium he amoun by which expeced reurns mus rise o compensae invesors for higher bea. Since he model applies o he marke reurn as well, we can measure λ m via im m m m l m = ER - f ( R). f i Mulifacor models exend his heory in a sraighforward way. They use a ime-series muliple regression o quanify an asse s endency o move wih muliple risk facors F A, F B, ec. i f m 3) R - R = ai + bim( R - R ) + biaf + bibf i e ; = 12,... T for each asse i. Then, he mulifacor model predics ha he expeced excess reurn is proporional o he beas i f 4) ER ( - R ) = bl im m + bl ia A + bl ib B +... for each i. The residual or unexplained average reurn in eiher case is called an alpha, i f ai ER -R - bimlm + biala + biblb + ( ) (...). f A B The CAPM and mulifacor models The CAPM The CAPM proved sunningly successful in a quarer cenury of empirical work. Every sraegy ha seemed o give high average reurns urned ou o have a high bea, or a large endency o move wih he marke. Sraegies ha one migh have hough gave high average reurns (such as holding very volaile socks) urned ou no o have high average reurns when hey did no have high beas. Figure 1 presens a ypical evaluaion of he CAPM. I examine 10 porfolios of NYSE socks sored by size (oal marke capializaion), along wih a porfolio of corporae bonds and long-erm governmen bonds. As he verical axis shows, here is a sizable spread in average reurns beween large socks (lower average reurn) and small socks (higher average reurn) and a large spread beween socks and bonds. The figure plos hese average reurns agains marke beas. You can see how he CAPM predicion fis: Porfolios wih higher average reurns have higher beas. In fac, figure 1 capures one of he firs significan failures of he CAPM. The smalles firms (he far righ porfolio) seem o earn an average reurn a few percen oo high given heir beas. This is he celebraed small-firm effec, (Banz, 1981) and his deviaion is saisically significan. Would ha all failed economic heories worked so well! However, he plo shows ha his effec is wihin he range ha saisicians can argue abou. Esimaing he slope of he line by fiing a cross-secional regression (average reurn agains bea), shown in he colored line, raher han forcing he line o go hrough he marke and Treasury bill reurn, shown in he black line, halves FIGURE 1 CAPM Mean excess reurns vs. bea, version 1 mean excess reurns, percen 18 Means and beas 14 Fied marke premium Direc marke premium beas Noes: Average reurns versus beas on he NYSE value-weighed porfolio for en size-sored sock porfolios, governmen bonds, and corporae bonds. Sample period The black line draws he CAPM predicion by fiing he marke proxy and Treasury bill raes exacly (a ime-series es) and he colored line draws he CAPM predicion by fiing an OLS cross-secional regression o he displayed daa poins (a second-pass or crosssecional es). The small-firm porfolios are a he op righ. Moving down and o he lef, one sees increasingly large-firm porfolios and he marke index. The poins far down and o he lef are he governmen bond and Treasury bill reurns. 38 Economic Perspecives

4 FIGURE 2 CAPM Mean excess reurns vs. bea, version 2 mean excess reurns, percen 16 Means and beas Direc marke premium Fied marke premium beas Noes: CAPM using he equally weighed NYSE as he marke porfolio. Oherwise, his figure is idenical o figure 1. he small-firm effec. Figure 2 uses he equally weighed porfolio as marke proxy, and his change in specificaion eliminaes he small-firm effec, making he line of average reurns versus beas if anyhing oo shallow raher han oo seep. Why we expec muliple facors In rerospec, i is surprising ha he CAPM worked so well for so long. The assumpions on which i is buil are very sylized and simplified. Asse pricing heory recognized a leas since Meron (1973, 1971) he heoreical possibiliy, indeed probabiliy, ha we should need facors, sae variables or sources of priced risk, beyond movemens in he marke porfolio o explain why some average reurns are higher han ohers. (See box 1 for deails of he CAPM and mulifacor models.) Mos imporanly, he average invesor has a job. The CAPM (ogeher wih he use of he NYSE porfolio as he marke proxy) simplifies maers by assuming ha he average invesor only cares abou he performance of his invesmen porfolio. While here are invesors like ha, for mos of us evenual wealh comes boh from invesmen and from earning a living. Imporanly, evens like recessions hur he majoriy of invesors. Those who don acually lose jobs ge lower salaries or bonuses. A very limied number of people acually do beer in a recession. Wih his fac in mind, compare wo socks. They boh have he same sensiiviy o marke movemens. However, one of hem does well in recessions, while he oher does poorly. Clearly, mos invesors prefer he sock ha does well in recessions, since is performance will cushion he blows o heir oher income. If los of people feel ha way, hey bid up he price of ha sock, or, equivalenly, hey are willing o hold i a a lower average reurn. Conversely, he procyclical sock s price will fall or i mus offer a higher average reurn in order o ge invesors o hold i. In sum, we should expec ha procyclical socks ha do well in booms and worse in recessions will have o offer higher average reurns han counercyclical socks ha do well in recessions, even if he socks have he same marke bea. We expec ha anoher dimension of risk covariaion wih recessions will maer in deermining average reurns. 1 Wha kinds of addiional facors should we look for? Generally, asse pricing heory specifies ha asses will have o pay high average reurns if hey do poorly in bad imes imes in which invesors would paricularly like heir invesmens no o perform badly and are willing o sacrifice some expeced reurn in order o ensure ha his is so. Consumpion (or, more generally, marginal uiliy) should provide he pures measure of bad imes. Invesors consume less when heir income prospecs are low or if hey hink fuure reurns will be bad. Low consumpion hus reveals ha his is indeed a ime a which invesors would especially like porfolios no o do badly, and would be willing o pay o ensure ha wish. Alas, effors o relae asse reurns o consumpion daa are no (ye) a grea success. Therefore, empirically useful asse pricing models examine more direc measures of good imes or bad imes. Broad caegories of such indicaors are 1. The marke reurn. The CAPM is usually included and exended. People are unhappy if he marke crashes. 2. Evens, such as recessions, ha drive invesors noninvesmen sources of income. 3. Variables, such as he p/d raio or slope of he yield curve, ha forecas sock or bond reurns (called sae variables for changing invesmen opporuniy ses ). 4. Reurns on oher well-diversified porfolios. One formally jusifies he firs hree facors by saing assumpions under which each variable is relaed o average consumpion. For example, 1) if he marke as a whole declines, consumers lose wealh and will cu back on consumpion; 2) if a recession leads people o lose heir jobs, hen hey will cu back on consumpion; and, 3) if you are saving for reiremen, hen news ha ineres raes and average sock reurns have declined is bad news, which will cause you o lower consumpion. This las poin esablishes a connecion beween predicabiliy of reurns and he presence of addiional risk facors for undersanding Federal Reserve Bank of Chicago 39

5 he cross-secion of average reurns. As poined ou by Meron (1971), one would give up some average reurn o have a porfolio ha did well when here was bad news abou fuure marke reurns. The fourh kind of facor addiional porfolio reurns is mos easily defended as a proxy for any of he oher hree. The fied value of a regression of any pricing facor on he se of all asse reurns is a porfolio ha carries exacly he same pricing informaion as he original facor a facor-mimicking porfolio. I is vial ha he exra risk facors affec he average invesor. If an even makes invesor A worse off and invesor B beer off, hen invesor A buys asses ha do well when he even happens and invesor B sells hem. They ransfer he risk of he even, bu he price or expeced reurn of he asse is unaffeced. For a facor o affec prices or expeced reurns, i mus affec he average invesor, so invesors collecively bid up or down he price and expeced reurn of asses ha covary wih he even raher han jus ransferring he risk wihou affecing equilibrium prices. Inspired by his broad direcion, empirical researchers have found quie a number of specific facors ha seem o explain he variaion in average reurns across asses. In general, empirical success varies inversely wih heoreical puriy. Small and value/growh socks The size and book o marke facors advocaed by Fama and French (1996) are one of he mos popular addiional risk facors. Small-cap socks have small marke values (price imes shares ousanding). Value (or high book/marke) socks have marke values ha are small relaive o he value of asses on he company s books. Boh caegories of socks have quie high average reurns. Large and growh socks are he opposie of small and value and seem o have unusually low average reurns. (See Fama and French, 1993, for a review.) The idea ha low prices lead o high average reurns is naural. High average reurns are consisen wih he CAPM, if hese caegories of socks have high sensiiviy o he marke, high beas. However, small and especially value socks seem o have abnormally high reurns even afer accouning for marke bea. Conversely, growh socks seem o do sysemaically worse han heir CAPM beas sugges. Figure 3 shows his value size puzzle. I is jus like figure 1, excep ha he socks are sored ino porfolios based on size and book/marke raio 2 raher han size alone. The highes porfolios have hree imes he average excess reurn of he lowes porfolios, and his variaion has nohing a all o do wih marke beas. FIGURE 3 Mean excess reurns vs. marke bea, Fama French porfolios mean excess reurns marke bea Noes: Average monhly reurns versus marke bea for 25 sock porfolios sored on he basis of size and book/marke raio. In figure 4, I connec porfolios of differen sizes wihin he same book/marke caegory (panel A). Variaion in size produces a variaion in average reurns ha is posiively relaed o variaion in marke beas, as shown in figure 1. In panel B, I connec porfolios ha have differen book/marke raios wihin size caegories. Variaion in book/marke raio produces a variaion in average reurn ha is negaively relaed o marke bea. Because of his value effec, he CAPM is a disaser when confroned wih hese porfolios. To explain hese facs, Fama and French (1993, 1996) advocae a mulifacor model wih he marke reurn, he reurn of small less big socks (SMB), and he reurn of high book/marke less low book/marke socks (HML) as hree facors. They show ha variaion in average reurns of he 25 size and book/marke porfolios can be explained by varying loadings (beas) on he laer wo facors. Figure 5 illusraes Fama and French s resuls. As in figure 4, he verical axis is he average reurns of he 25 size and book/marke porfolios. Now, he horizonal axis is he prediced values from he Fama French hree-facor model. The poins should all lie on a 45 degree line if he model is correc. The poins lie much closer o his predicion in figure 5 han in figures 3 and 4. The wors fi is for he growh socks (lowes line, panel A), for which here is lile variaion in average reurn despie large variaion in size bea as one moves from small o large firms. Wha are he size and value facors? One would like o undersand he real, macroeconomic, aggregae, nondiversifiable risk ha is proxied by he reurns of he HML and SMB porfolios. Why 40 Economic Perspecives

6 FIGURE 4 Mean excess reurns vs. marke bea, varying size and book/marke raio A. Changing size wihin book/marke caegory mean excess reurn B. Changing book/marke wihin size caegory mean excess reurn marke bea marke bea Noes: Average reurns versus marke bea for 25 sock porfolios sored on he basis of size and book/marke raio. The poins are he same as figure 3. In panel A, lines connec porfolios as size varies wihin book/marke caegories; in panel B, lines connec porfolios as book/marke raio varies wihin size caegories. are invesors so concerned abou holding socks ha do badly a he imes ha he HML (value less growh) and SMB (small-cap less large-cap) porfolios do badly, even hough he marke does no fall? The answer o his quesion is no ye oally clear. Fama and French (1995) noe ha he ypical value sock has a price ha has been driven down due o financial disress. The socks of firms on he verge of bankrupcy have recovered more ofen han no, which generaes he high average reurns of his sraegy. 3 This observaion suggess a naural inerpreaion of he value premium: In he even of a credi crunch, liquidiy crunch, or fligh o qualiy, socks in financial disress will do very badly, and his is precisely when invesors leas wan o hear ha heir porfolio is losing money. (One canno coun he disress of he individual firm as a risk facor. Such disress is idiosyncraic and can be diversified away. Only aggregae evens ha average invesors care abou can resul in a risk premium.) FIGURE 5 Mean excess reurn vs. hree-facor model predicions A. Changing size wihin book/marke caegory acual mean excess reurn, E(R i R f ) 1.2 B. Changing book/marke wihin size caegory acual mean excess reurn, E(R i R f ) prediced, β i,m E(R m R f ) + β i,h E(HML) +β i,s E(SMB) prediced, β i,m E(R m R f ) + β i,h E(HML) +β i,s E(SMB) Noes: Average reurns versus marke bea for 25 sock porfolios sored on he basis of size and book/marke raio versus predicions of Fama French hree-facor model. The predicions are derived by regressing each of he 25 porfolio reurns, R i, on he marke porfolio, R m, and he wo Fama French facor porfolios, SMB (small minus big) and HML (high minus low book/marke). (See equaion 4 in box 1.) Federal Reserve Bank of Chicago 41

7 Heaon and Lucas s (1997) resuls add o his sory for he value effec. They noe ha he ypical sockholder is he proprieor of a small, privaely held business. Such an invesor s income is, of course, paricularly sensiive o he kinds of financial evens ha cause disress among small firms and disressed value firms. Therefore, his invesor would demand a subsanial premium o hold value socks and would hold growh socks despie a low premium. Liew and Vassalou (1999), among ohers, link value and small-firm reurns o macroeconomic evens. They find ha in many counries, counerpars o HML and SMB conain supplemenary informaion o ha conained in he marke reurn for forecasing gross domesic produc (GDP) growh. For example, hey repor a regression GDP +4 = a MKT HML 4 + ε +4, where GDP +4 denoes he following year s GDP growh and MKT 4 and HML 4 denoe he previous year s reurn on he marke index and HML porfolio. Thus, a 10 percen HML reurn raises he GDP forecas by 0.5 percenage poins. (Boh coefficiens are significan wih -saisics of 3.09 and 2.83, respecively.) The effecs are sill under invesigaion. Figure 6 plos he cumulaive reurn on he HML and SMB porfolios; a link beween hese reurns and obvious macroeconomic evens does no jump ou. Boh porfolios have essenially no correlaion wih he marke reurn, hough HML does seem o move inversely wih large marke declines. HML goes down more han he marke in some business cycles, bu less in ohers. On he oher hand, one can ignore Fama and French s moivaion and regard he model as an arbirage pricing heory (APT) following Ross (1976). If he reurns of he 25 size and book/marke porfolios could be perfecly replicaed by he reurns of he hree-facor porfolios if he R 2 values in he imeseries regressions of he 25 porfolio on he hree facors were 100 percen hen he mulifacor model would have o hold exacly, in order o preclude arbirage opporuniies. To see his, suppose ha one of he 25 porfolios call i porfolio A gives an average reurn 5 percen above he average reurn prediced by he Fama French model, and is R 2 is 100 percen. Then, one could shor a combinaion of he hreefacor porfolios, buy porfolio A, and earn a compleely riskless profi. This logic is ofen used o argue ha a high R 2 should imply an approximae mulifacor model. If he R 2 were only 95 percen, hen an average reurn 5 percen above he facor model predicion FIGURE 6 Cumulaive reurns on marke porfolios cumulaive reurn HML SMB Marke Noes: Cumulaive reurns on he marke RMRF, SMB, and HML porfolios. The SMB reurn is formed by R TB + asmb ; a = σ(rmrf)/σ(smb). In his way i is a reurn ha can be cumulaed raher han a zero-cos porfolio, and is sandard deviaion is equal o ha of he marke reurn. HML is adjused similarly. The verical axis is he log base 2 of he cumulaive reurn or value of $1 invesed a he beginning of he sample period. Thus, each ime a line increases by 1 uni, he value doubles. would imply ha he sraegy long porfolio A and shor a combinaion of he hree-facor porfolios would earn a very high average reurn wih very lile, hough no zero, risk a very high Sharpe raio. In fac, he R 2 values of Fama and French s (1993) ime-series regressions are all in he 90 percen o 95 percen range, so exremely high risk prices for he residuals would have o be invoked for he model no o fi well. Conversely, given he average reurns from HML and SMB and he failure of he CAPM o explain hose reurns, here would be near-arbirage opporuniies if value and small socks did no move ogeher in he way described by he Fama French model. One way o assess wheher he hree facors proxy for real macroeconomic risks is by checking wheher he mulifacor model prices addiional porfolios, especially porfolios whose ex-pos reurns are no well explained by he facors (porfolios ha do no have high R 2 values in ime-series regressions). Fama and French (1996) find ha he SMB and HML porfolios comforably explain sraegies based on alernaive price muliples (price/earnings, book/marke), five-year sales growh (his is he only sraegy ha does no form porfolios based on price variables), and he endency of five-year reurns o reverse. All of hese sraegies are no explained by CAPM beas. However, hey all also produce porfolios wih high R 2 values in a ime-series regression on he HML and SMB porfolios. This is good and bad news. I migh 42 Economic Perspecives

8 mean ha he model is a good APT, and ha he size and book/marke characerisics describe he major sources of priced variaion in all socks. On he oher hand, i migh mean ha hese exra ways of consrucing porfolios jus haven idenified oher sources of priced variaion in sock reurns. (Fama and French, 1996, also find ha HML and SMB do no explain momenum, despie high R 2 values. I discuss his anomaly below.) The porfolios of socks sored by indusry in Fama and French (1997) have lower R 2 values, and he model works less well. A final concern is ha he size and book/marke premiums seem o have diminished subsanially in recen years. The sharp decline in he SMB porfolio reurn around 1980 when he small-firm effec was firs popularized is obvious in figure 6. In Fama and French s (1993) iniial samples, , he HML cumulaive reurn sars abou one-half (0.62) below he marke and ends up abou one-half (0.77) above he marke. On he log scale of he figure, his corresponds o Fama and French s repor ha he HML average reurn is abou double (precisely, = 2.6 imes) ha of he marke. However, over he enire sample of he plo, he HML porfolio sars and ends a he same place and so earns almos exacly he same as he marke. From 1990 o now, he HML porfolio loses abou one-half relaive o he marke, meaning an invesor in he marke has increased his money one and a half imes as much as an HML invesor. (The acual number is 0.77 so he marke reurn is = 1.71 imes beer han he HML reurn.) Among oher worries, if he average reurns decline righ afer publicaion i suggess ha he anomalies may simply have been overlooked by a large fracion of invesors. As hey move in, prices go up furher, helping he apparen anomaly for a while. Bu once a large number of invesors have moved in o include small and value socks in heir porfolios, he anomalous high average reurns disappear. However, average reurns are hard o measure. There have been previous en- o 20-year periods in which small socks did very badly, for example he 1950s, and similar decade-long variaions in he HML premium. Also, since SMB and HML have a bea of essenially zero on he marke, any upward rend is a violaion of he CAPM and says ha invesors can improve heir overall mean variance radeoff by aking on some of he HML or SMB porfolio. Macroeconomic facors I focus on he size and value facors because hey provide he mos empirically successful mulifacor model and have araced much indusry as well as academic aenion. Several auhors have used macroeconomic variables as facors. This procedure examines direcly wheher sock performance during bad macroeconomic imes deermines average reurns. Jagannahan and Wang (1996) and Reyfman (1997) use labor income; Chen, Roll, and Ross (1986) look a indusrial producion and inflaion among oher variables; and Cochrane (1996) looks a invesmen growh. All hese auhors find ha average reurns line up wih beas calculaed using he macroeconomic indicaors. The facors are heoreically easier o moivae, bu none explains he value and size porfolios as well as he (heoreically less solid, so far) size and value facors. Meron s (1973, 1971) heory says ha variables which predic marke reurns should show up as facors ha explain cross-secional variaion in average reurns. Campbell (1996) is he lone es I know of o direcly address his quesion. Cochrane (1996) and Jagannahan and Wang (1996) perform relaed ess in ha hey include scaled reurn facors, for example, marke reurn a muliplied by d/p raio a 1; hey find ha hese facors are also imporan in undersanding cross-secional variaion in average reurns. The nex sep is o link hese more fundamenally deermined facors wih he empirically more successful value and small-firm facor porfolios. Because of measuremen difficulies and selecion biases, fundamenally deermined macroeconomic facors will never approach he empirical performance of porfolio-based facors. However, hey may help o explain which porfolio-based facors really work and why. Predicable reurns The view ha risky asse reurns are largely unpredicable, or ha prices follow random walks, remains immensely successful ( Malkiel, 1990, is a classic and readable inroducion). I is also widely ignored. Unpredicable reurns mean ha if socks wen up yeserday, here is no exploiable endency for hem o decline oday because of profi aking or o coninue o rise oday because of momenum. Technical signals, including analysis of pas price movemens rading volume, open ineres, and so on are close o useless for forecasing shor-erm gains and losses. As I wrie, value funds are reporedly suffering large ouflows because heir socks have done poorly in he las few monhs, leading fund invesors o move money ino blue-chip funds ha have performed beer (New York Times Company, 1999). Unpredicable reurns mean ha his sraegy will no do anyhing for invesors porfolios over he long run excep rack up rading coss. If funds are selling socks, hen Federal Reserve Bank of Chicago 43

9 conrarian invesors mus be buying hem, bu unpredicable reurns mean ha his sraegy can no improve performance eiher. If one can no sysemaically make money, one can no sysemaically lose money eiher. As discussed in he inroducion, researchers once believed ha sock reurns (more precisely, he excess reurns on socks over shor-erm ineres raes) were compleely unpredicable. I now urns ou ha average reurns on he marke and individual securiies do vary over ime and ha sock reurns are predicable. Alas for would-be echnical raders, much of ha predicabiliy comes a long horizons and seems o be associaed wih business cycles and financial disress. Marke reurns Table 1 presens a regression ha forecass reurns. Low prices relaive o dividends, book value, earnings, sales, or oher divisors predic higher subsequen reurns. As he R 2 values in able 1 show, hese are long-horizon effecs: Annual reurns are only slighly predicable and monh-o-monh reurns are sill srikingly unpredicable, bu reurns a five-year horizons seem very predicable. (Fama and French, 1989, is an excellen reference for his kind of regression). The resuls a differen horizons are reflecions of a single underlying phenomenon. If daily reurns are very slighly predicable by a slow-moving variable, ha predicabiliy adds up over long horizons. For example, you can predic ha he emperaure in Chicago will rise abou one-hird of a degree per day in spring. This forecas explains very lile of he day o day variaion in emperaure, bu racks almos all of he rise in emperaure from January o July. Thus, he R 2 rises wih horizon. Precisely, suppose ha we forecas reurns wih a forecasing variable x, according o 1) R R a bx TB = + + e + 1 2) x = c + rx + d Small values of b and R 2 in equaion 1 and a large coefficien ρ in equaion 2 imply mahemaically ha he long-horizon regression as in able 1 has a large regression coefficien b and large R 2. This regression has a powerful implicaion: Socks are in many ways like bonds. Any bond invesor undersands ha a sring of good pas reurns ha pushes he price up is bad news for subsequen reurns. Many sock invesors see a sring of good pas reurns and become elaed ha we seem o be in a bull marke, concluding fuure sock reurns will be good as well. The regression reveals he opposie: A sring of good pas reurns which drives up sock prices is bad news for subsequen sock reurns, as i is for bonds. Long-horizon reurn predicabiliy was firs documened in he volailiy ess of Shiller (1981) and LeRoy and Porer (1981). They found ha sock prices vary far oo much o be accouned for by changing expecaions of subsequen cash flows; hus changing discoun raes or expeced reurns mus accoun for variaion in sock prices. These volailiy ess urn ou o be almos idenical o regressions such as hose in able 1 (Cochrane, 1991). Momenum and reversal Since a sring of good reurns gives a high price, i is no surprising ha individual socks ha do well for a long ime (and reach a high price) subsequenly do poorly, and socks ha do poorly for a long ime (and reach a low price, marke value, or marke o book raio) subsequenly do well. Table 2, aken from Fama and French (1996) confirms his hunch. (Also, see DeBon and Thaler, 1985, and Jegadeesh and Timan, 1993.) The firs row in able 2 racks he average monhly reurn from he reversal sraegy. Each monh, allocae all socks o en porfolios based on performance from year 5 o year 1. Then, buy he bes-performing porfolio and shor he wors-performing porfolio. This sraegy earns a hefy 0.74 percen monhly reurn. 4 Pas long-erm losers come back and pas winners do badly. Fama and French (1996) verify ha hese porfolio reurns are explained by heir hreefacor model. Pas winners move wih value socks, TABLE 1 OLS regression of excess reurns on price/dividend raio Horizon k b Sandard error R 2 1 year years years years Noes: OLS regressions of excess reurns (value-weighed NYSE Treasury bill rae) on value-weighed price/dividend raio. VW TB R + k - R + k = a + b( P / D) + e+ k. R +k indicaes he k year reurn. Sandard errors use GMM o correc for heeroskedasiciy and serial correlaion. 44 Economic Perspecives

10 TABLE 2 Average monhly reurns, reversal and momenum sraegies Porfolio Average Sraegy Period formaion reurn, 10 1 (monhs) (monhly %) Reversal July 1963 Dec Momenum July 1963 Dec Reversal Jan Feb Momenum Jan Feb Noes: Each monh, allocae all NYSE firms o 10 porfolios based on heir performance during he porfolio formaion monhs inerval. For example, forms porfolios based on reurns from 5 years ago o 1 year, 1 monh ago. Then buy he bes-performing decile porfolio and shor he wors-performing decile porfolio. Source: Fama and French (1996, able 6). and so inheri he value sock premium. (To compare he sraegies, he able always buys he winners and shors he losers. In pracice, of course, you buy he losers and shor he winners o earn percen monhly average reurn.) The second row of able 2 racks he average monhly reurn from a momenum sraegy. Each monh, allocae all socks o en porfolios based on performance in he las year. Now, he winners coninue o win and he losers coninue o lose, so ha buying he winners and shoring he losers generaes a posiive 1.31 percen monhly reurn. Momenum is no explained by he Fama French (1996) hree-facor model. The pas losers have low prices and end o move wih value socks. Hence, he model predics ha hey should have high average reurns, no low average reurns. Momenum socks move ogeher, as do value and small socks, so a momenum facor works o explain momenum porfolio reurns (Carhar, 1997). This sep is so obviously ad hoc (ha is, an APT facor ha will only explain reurns of porfolios organized on he same characerisic as he facor raher han a proxy for macroeconomic risk) ha mos people are uncomforable adding i. We obviously do no wan o add a new facor for every anomaly. Is momenum really here, and if so, is i exploiable afer ransacion coss? One warning is ha i does no seem sable over subsamples. The hird and fourh lines in able 2 show ha he momenum effec essenially disappears in he earlier daa sample, while reversal is even sronger in ha sample. Momenum is really jus a new way of looking a an old phenomenon, he small apparen predicabiliy of monhly individual sock reurns. A iny regression R 2 for forecasing monhly reurns of (0.25 percen) is more han adequae o generae he momenum resuls of able 2. The key is he large sandard deviaion of individual sock reurns, ypically 40 percen or more on an annual basis. The average reurn of he bes performing decile of a normal disribuion is 1.76 sandard deviaions above he mean, 5 so he winning momenum porfolio wen up abou 80 percen in he previous year and he ypical losing porfolio wen down abou 60 percen. Only a small amoun of coninuaion will give a 1 percen monhly reurn when muliplied by such large pas reurns. To be precise, he monhly individual sock sandard deviaion is abou 40% / 12 12%. If he R 2 is , he sandard deviaion of he predicable par of reurns is œ 12% 0. 6%. Hence, he decile prediced o perform bes will earn 176. œ 0. 6% 1% above he mean. Since he sraegy buys he winners and shors he losers, an R 2 of implies ha one should earn a 2 percen monhly reurn by he momenum sraegy. We have known a leas since Fama (1965) ha monhly and higher frequency sock reurns have sligh, saisically significan predicabiliy wih R 2 abou Campbell, Lo, and MacKinlay (1997, able 2.4) provide an updaed summary of index auocorrelaions (he R 2 is he squared auocorrelaion), par of which I show in able 3. Noe he correlaion of he equally weighed porfolio, which emphasizes small socks. 6 However, such small, hough saisically significan, high-frequency predicabiliy has hus far failed o yield exploiable profis afer one akes ino accoun ransacion coss, hin rading of small socks, and high shor-sale coss. The momenum sraegy for exploiing his correlaion may no work in pracice for he same reasons. Momenum does require frequen rading. The porfolios in able 2 are re-formed every TABLE 3 Firs-order auocorrelaion, CRSP value- and equally weighed index reurns Frequency Porfolio Correlaion ρ 1 Daily Value-weighed 0.18 Equally weighed 0.35 Monhly Value-weighed Equally weighed 0.17 Noe: Sample Source: Campbell, Lo, and MacKinlay (1997). Federal Reserve Bank of Chicago 45

11 monh. Annual winners and losers will no change ha ofen, bu he winning and losing porfolio mus be urned over a leas once per year. In a quaniaive examinaion of his effec, Carhar (1997) concludes ha momenum is no exploiable afer ransacion coss are aken ino accoun. Moskowiz and Grinbla (1999) noe ha mos of he apparen gains from he momenum sraegy come from shor posiions in small illiquid socks. They also find ha a large par of momenum profis come from shor posiions aken in November. Many invesors sell losing socks oward he end of December o esablish ax losses. By shoring illiquid losing socks in November, an invesor can profi from he selling pressure in December. This is also an anomaly, bu i seems like a glich raher han a cenral principle of risk and reurn in asse markes. Even if momenum and reversal are real and as srong as indicaed by able 2, hey do no jusify much of he rading based on pas resuls ha many invesors seem o do. To ge he 1 percen per monh momenum reurn, one buys a porfolio ha has ypically gone up 80 percen in he las year, and shors a porfolio ha has ypically gone down 60 percen. Trading beween socks and fund caegories such as value and blue-chip wih smaller pas reurns yields a bes proporionally smaller resuls. Since much of he momenum reurn seems o come from shoring small illiquid socks, mild momenum sraegies may yield even less. And we have no quanified he subsanial risk of momenum sraegies. Bonds The venerable expecaions model of he erm srucure specifies ha long-erm bond yields are equal o he average of expeced fuure shor-erm bond yields (see box 2). For example, if long-erm bond yields are higher han shor-erm bond yields if he yield curve is upward sloping his means ha shor-erm raes are expeced o rise in he fuure. The rise in fuure shor-erm raes means ha invesors can expec ( N Le p ) denoe he log of he N year discoun bond price a ime. The N period coninuously ( N ) compounded yield is defined by y N p ( N =- 1 ). The coninuously compounded holding period reurn is he selling price less he buying price, ( N ) ( N -1) ( N hpr + 1 = p p ). The forward rae is he rae a which an invesor can conrac oday o borrow money N 1 years from now, and repay ha money N years from now. Since an invesor can synhesize a forward conrac from discoun bonds, he forward rae is deermined from discoun bond prices by ( N ) ( N ) 1 ( N ) f = p p. The spo rae refers, by conras wih a forward rae, o he yield on any bond for which he invesor ake immediae delivery. Forward raes are ypically higher han spo raes when he yield curve rises, since he yield is he average of inervening forward raes, ( N ) 1 ( 1) ( 2) ( 3) ( N ) y = ( N f + f + f f ). The expecaions hypohesis saes ha he expeced log or coninuously compounded reurn should be he same for any bond sraegy. This saemen has hree mahemaically equivalen expressions: BOX 2 Bond definiions and expecaions hypohesis 1. The forward rae should equal he expeced value of he fuure spo rae, ( N ) ( 1) = + N-1 f E ( y ). 2. The expeced holding period reurn should be he same on bonds of any mauriy ( E ( hpr ) ) ( E ( hpr ) ) ( ) y. = = N + 1 M The long-erm bond yield should equal he average of he expeced fuure shor raes, ( N 1 () 1 () 1 () 1 y = N E ( y + y... y N-1). The expecaions hypohesis is ofen amended o allow a consan risk premium of undeermined sign in hese equaions. Is violaion is hen ofen described as evidence for a ime-varying risk premium. The expecaions hypohesis is no quie he same hing as risk-neuraliy, because he expeced log reurn is no equal o he log expeced reurn. However, he issues here are larger han he difference beween he expecaions hypohesis and sric risk-neuraliy. 46 Economic Perspecives

12 TABLE 4 Zero-coupon bond reurns Mauriy Average holding Sandard Sandard N period reurn error deviaion Noe: Coninuously compounded one-year holding period reurns on zero-coupon bonds of varying mauriy. Annual daa from CRSP he same rae of reurn wheher hey hold a long-erm bond o mauriy or roll over shor-erm bonds wih iniially low reurns and subsequen higher reurns. As wih he CAPM and he view ha sock reurns are independen over ime, a new round of research has significanly modified his radiional view of bond markes. Table 4 calculaes he average reurn on bonds of differen mauriies. The expecaions hypohesis seems o do prey well. Average holding period reurns do no seem very differen across bond mauriies, despie he increasing sandard deviaion of longermauriy bond reurns. The small increase in average reurns for long-erm bonds, equivalen o a sligh average upward slope in he yield curve, is usually excused as a liquidiy premium. Table 4 is jus he ip of an iceberg of successes for he expecaions model. Especially in imes of significan inflaion and exchange rae insabiliy, he expecaions hypohesis has done a very good firs-order job of explaining he erm srucure of ineres raes. However, if here are imes when long-erm bonds are expeced o do beer and oher imes when shorerm bonds are expeced o do beer, he uncondiional averages in able 4 could sill show no paern. Similarly, one migh wan o check wheher a forward rae ha is unusually high forecass an unusual increase in spo raes. Table 5 updaes Fama and Bliss s (1987) classic regression ess of his idea. Panel A presens a regression of he change in yields on he forward-spo spread. (The forward-spo spread measures he slope of he yield curve.) The expecaions hypohesis predics a slope coefficien of 1.0, since he forward rae should equal he expeced fuure spo rae. If, for example, forward raes are lower han expeced fuure spo raes, raders can lock in a borrowing posiion wih a forward conrac and hen lend a he higher spo rae when he ime comes. Insead, a a one-year horizon we find slope coefficiens near zero and a negaive adjused R 2. Forward raes one year ou seem o have no predicive power whasoever for changes in he spo rae one year from now. On he oher hand, by four years ou, we see slope coefficiens wihin one sandard error of 1.0. Thus, he expecaions hypohesis seems o do poorly a shor (one-year) horizons, bu much beer a longer horizons. If he expecaions hypohesis does no work a one-year horizons, hen here is money o be made one mus be able o foresee years in which shor-erm bonds will reurn more han long-erm bonds and vice versa, a leas o some exen. To confirm his implicaion, panel B of able 5 runs regressions of he one-year excess reurn on long-erm bonds on he forwardspo spread. Here, he expecaions hypohesis predics a coefficien of zero: No signal (including he TABLE 5 Forecass based on forward-spo spread A. Change in yields B. Holding period reurns Sandard Sandard Sandard Sandard error, error, Adjused error, error, Adjused N Inercep inercep Slope slope R 2 Inercep inercep Slope slope R Noes: OLS regressions, annual daa. Panel A esimaes he regression y (1) y (N+1) (1) = a +b (f +n (1) y ) + ε +N and panel B esimaes he regression hpr (N) y (1) (N+1) = a + b (f +1 y (1)) + ε, where y (N) denoes he N-year bond yield a +1 (N) dae ; f denoes he N-period ahead forward rae; and hpr (N ) denoes he one-year holding period reurn a dae on an N-year bond. Yields and reurns in annual percenages. Federal Reserve Bank of Chicago 47

13 forward-spo spread) should be able o ell you ha his is a paricularly good ime for long bonds versus shor bonds, as he random walk view of sock prices says ha no signal should be able o ell you ha his is a paricularly good or bad day for socks versus bonds. However, he coefficiens in panel B are all abou 1.0. A high forward rae does no indicae ha ineres raes will be higher one year from now; i seems o indicae ha invesors will earn ha much more by holding long-erm bonds. 7 Of course, here is risk. The R 2 values are all , abou he same values as he R 2 from he d/p regression a a one-year horizon, so his sraegy will ofen go wrong. Sill, is no zero, so he sraegy does pay off more ofen han no, in violaion of he expecaions hypohesis. Furhermore, he forward-spo spread is a slow-moving variable, ypically reversing sign once per business cycle. Thus, he R 2 builds wih horizon as wih he d/p regression, peaking in he 30 percen range (Fama and French, 1989). Foreign exchange Suppose ineres raes are higher in Germany han in he U.S. Does his mean ha one can earn more money by invesing in German bonds? There are several reasons ha he answer migh be no. Firs, of course, is defaul risk. Governmens have defauled on bonds in he pas and may do so again. Second, and more imporan, is he risk of devaluaion. If German ineres raes are 10 percen and U.S. ineres raes are 5 percen, bu he euro falls 5 percen relaive o he dollar during he year, you make no more money holding he German bonds despie heir aracive ineres rae. Since los of invesors are making his calculaion, i is naural o conclude ha an ineres rae differenial across counries on bonds of similar credi risk should reveal an expecaion of currency devaluaion. The logic is exacly he same as ha of he expecaions hypohesis in he erm srucure. Iniially aracive yield or ineres rae differenials should be me by an offseing even so ha you make no more money on average in one mauriy or currency versus anoher. 8 As wih he expecaions hypohesis in he erm srucure, he expeced depreciaion view sill consiues an imporan firs-order undersanding of ineres rae differenials and exchange raes. For example, ineres raes in eas Asian currencies were very high on he eve of he recen currency umbles, and many banks were making idy sums borrowing a 5 percen in dollars o lend a 20 percen in local currencies. This suggess ha raders were anicipaing a 15 percen devaluaion, or a smaller chance of a larger devaluaion, which is exacly wha happened. Many observers aribue high nominal ineres raes in roubled economies o igh moneary policy aimed a defending he currency. In realiy, high nominal raes reflec a large probabiliy of inflaion and devaluaion loose moneary policy and correspond o much lower real raes. Sill, does a 5 percen ineres rae differenial correspond o a 5 percen expeced depreciaion, or does some of i represen a high expeced reurn from holding deb in ha counry s currency? Furhermore, while expeced depreciaion is clearly a large par of he ineres rae sory in high-inflaion economies, how does he sory play ou in economies like he U.S. and Germany, where inflaion raes diverge lile bu exchange raes sill flucuae a large amoun? The firs row of able 6 (from Hodrick, 2000, and Engel, 1996) shows he average appreciaion of he dollar agains he indicaed currency over he sample period. The dollar fell agains he deuschemark, yen, and Swiss franc, bu appreciaed agains he pound serling. The second row gives he average ineres rae differenial he amoun by which he foreign ineres rae exceeds he U.S. ineres rae. 9 According o he expecaions hypohesis, hese wo numbers should be equal ineres raes should be higher in counries whose currencies depreciae agains he dollar. TABLE 6 Forward discoun puzzle Deusche- Pound Swiss mark serling Yen franc Mean appreciaion Mean ineres differenial b, R b, b, 10-year horizon Noes: The firs row gives he average appreciaion of he dollar agains he indicaed currency, in percen per year. The second row gives he average ineres differenial foreign ineres rae less domesic ineres rae, measured as he forward premium he 30-day forward rae less he spo exchange rae. The hird hrough sixh rows give he coefficiens and R 2 in a regression of exchange rae changes on he ineres differenial, s +1 s = a + b (r f r d ) + ε +1, where s = log spo exchange rae, r f = foreign ineres rae, and r d = domesic ineres rae. Source: Hodrick (2000), Engel (1996), and Meredih and Chinn (1998). 48 Economic Perspecives

14 The second row shows roughly he expeced paern. Counries wih seady long-erm inflaion have seadily higher ineres raes and seady depreciaion. The numbers in he firs and second rows are no exacly he same, bu exchange raes are nooriously volaile so hese averages are no well measured. Hodrick (2000) shows ha he difference beween he firs and second rows is no saisically differen from zero. This fac is analogous o he evidence in able 4 ha he expecaions hypohesis works well on average for U.S. bonds. As in he case of bonds, however, we can ask wheher imes of emporarily higher or lower ineres rae differenials correspond o imes of above- and below-average depreciaion as hey should. The hird and fifh rows of able 6 updae Fama s (1984) regression ess. The number here should be +1.0 in each case 1 percenage poin exra ineres differenial should correspond o 1 percenage poin exra expeced depreciaion. On he conrary, as able 6 shows, a higher han usual ineres rae abroad seems o lead o furher appreciaion. This is he forward discoun puzzle. See Engel (1996) and Lewis (1995) for recen surveys of he avalanche of academic work invesigaing wheher his puzzle is really here and why. The R 2 values shown in able 6 are quie low. However, like d/p and he erm spread, he ineres differenial is a slow-moving forecasing variable, so he reurn forecas R 2 builds wih horizon. Bekaer and Hodrick (1992) repor ha he R 2 rises o he 30 percen o 40 percen range a six-monh horizons and hen declines. Tha s high, bu no 100 percen; aking advanage of any predicabiliy sraegy is quie risky. The puzzle does no say ha one earns more by holding bonds from counries wih higher ineres raes han ohers. Average inflaion, depreciaion, and ineres rae differenials line up as hey should. The puzzle does say ha one earns more by holding bonds from counries whose ineres raes are higher han usual relaive o U.S. ineres raes (and vice versa). The fac ha he usual rae of depreciaion and ineres differenial changes hrough ime will, of course, diminish he ou-of-sample performance of hese rading rules. One migh expec ha exchange rae depreciaion works beer for long-run exchange raes, as he expecaions hypohesis works beer for long-run ineres rae changes. The las row of able 6, aken from Meredih and Chinn (1998) verifies ha his is so. Ten-year exchange rae changes are correcly forecas by he ineres differenials of en-year bonds. Muual funds Sudying he reurns of funds ha follow a specific sraegy gives us a way o assess wheher ha sraegy works in pracice, afer ransacion coss and oher rading realiies are aken ino accoun. Sudying he reurns of acively managed funds ells us wheher he ime, alen, and effor pu ino picking securiies pays off. Mos of he lieraure on evaluaing fund performance is devoed o he laer quesion. A large body of empirical work, saring wih Jensen (1969), finds ha acively managed funds, on average, underperform he marke index. I use daa from Carhar (1997), whose measures of fund performance accoun for survivor bias. Survivor bias arises because funds ha do badly go ou of business. Therefore, he average fund ha is alive a any poin in ime has an arificially good rack record. As wih he sock porfolios in figure 1, he fund daa in figure 7 show a definie correlaion beween bea and average reurn: Funds ha did well ook on more marke risks. A cross-secional regression line is a bi flaer han he line drawn hrough he Treasury bill and marke reurn, bu his is a ypical resul of measuremen error in he beas. (The daa are annual, and many funds are only around for a few years, conribuing o bea measuremen error.) The average fund underperforms he line connecing Treasury bills and he marke index by 1.23 percen per year (ha is, he average alpha is 1.23 percen). The wide dispersion in fund average reurns in figure 7 is a bi surprising. Average reurns vary across funds almos as much as hey do across individual socks. This fac implies ha he majoriy of funds are no holding well-diversified porfolios ha would reduce reurn variaion, bu raher are loading up on specific bes. Iniially, he fac ha he average fund underperforms he marke seems beside he poin. Perhaps he average fund is bad, bu we wan o know wheher he good funds are any good. The rouble is, we mus somehow disinguish skill from luck. The only way o separae skill from luck is o group funds based on some ex-ane observable characerisic, and hen examine he average performance of he group. Of course, skillful funds should have done beer, on average, in he pas, and should coninue o do beer in he fuure. Thus, if here is skill in sock picking, we should see some persisence in fund performance. However, a generaion of empirical work found no persisence a all. Funds ha did well in he pas were no more likely o do well in he fuure. Federal Reserve Bank of Chicago 49

15 FIGURE 7 Average reurns of muual funds vs. marke beas average reurn = value = neural = growh bea Noes: Average reurns of muual funds over he Treasury bill rae versus heir marke beas. Sample consiss of all funds wih average oal ne asses greaer han $25 million and more han 25 percen of heir asses in socks, in he Carhar (1996) daabase. Daa sample The average excess reurn is compued as E(R i R f ) = a i + β i x 9%. a i and β i are compued from a ime-series regression of fund annual excess reurns on marke annual excess reurns over he life of he fund. The o, +, and x labels in he figure sor funds ino hirds based on heir regression coefficien h on he Fama French value (HML) porfolio. The breakpoins are h = 0.084, The dashed line gives he fi of a cross-secional ordinary leas squares regression of a i on β i ; The solid line connecs he Treasury bill (β and excess reurn = 0) and he marke reurn (β = 1, excess reurn = 9%). Since he average fund underperforms he marke, and fund reurns are no predicable, we conclude ha acive managemen does no generae superior performance, especially afer ransacion coss and fees. This fac is surprising. Professionals in almos any field do beer han amaeurs. One would expec ha a rained experienced professional who spends all day reading abou markes and socks should be able o ouperform simple indexing sraegies. Even if enry ino he indusry is so easy ha he average fund does no ouperform simple indexes one would expec a few sars o ouperform year afer year, as good eams win championship afer championship. Alas, he conrary fac is he resul of pracically every invesigaion, and even he anomalous resuls documen very small effecs. Funds and value Given he value, small-firm, and predicabiliy effecs, he idea ha funds cluser around he marke line is quie surprising. All of hese new facs imply inescapably ha here are simple, mechanical sraegies ha can give a risk/reward raio greaer han ha of buying and holding he marke index. Fama and French (1993) repor ha he HML porfolio alone gives nearly double he marke Sharpe raio he same average reurn a half he sandard deviaion. Why don funds cluser around a risk/reward line significanly above he marke s? Of course, we should no expec all funds o cluser around a higher risk/reward radeoff. The average invesor holds he marke, and if funds are large enough, so mus he average fund. Index funds, of course, will perform like he index. Sill, he ypical acively managed fund adverises high mean and, perhaps, low variance. No fund adverises cuing average reurns in half o spare invesors exposure o nonmarke sources of risk. Such funds, apparenly aimed a mean variance invesors, should cluser around he highes risk/reward radeoff available from mechanical sraegies (and more, if acive managemen does any good). Mos roubling, funds who say hey follow value sraegies don ouperform he marke eiher. For example, Lakonishok, Shleifer, and Vishny (1992, able 3) find ha he average value fund underperforms he S&P500 by 1 percen jus like all he ohers. We can resolve his conradicion if we hink ha fund managers were simply unaware of he possibiliies offered by our new facs, and so (despie he adverising) were no really following hem. Tha seems o be he implicaion of figure 7, which sors funds by heir HML bea. One would expec he high-hml bea funds o ouperform he marke line. Bu he cuoff for he op one-hird of funds is only a HML bea of 0.3, and even ha may be high (many funds don las long, so beas are poorly measured; he disribuion of measured beas is wider han he acual disribuion). Thus, he value funds were really no following he value sraegy ha earns he HML reurns; if hey were doing so hey would have HML beas of 1.0. Similarly, Lakonishok, Shleifer, and Vishny s (1992) documenaion of value funds underperformance reveals ha heir marke bea is close o 1.0. These resuls imply ha value funds are no really following a value sraegy, since heir reurns correlae wih he marke porfolio and no he value porfolio. Ineresingly, he number of value and small-cap funds (as revealed by heir beas, no heir markeing claims) is increasing quickly. Before 1990, 14 percen of funds had measured SMB beas greaer han 1.0, 50 Economic Perspecives

16 and 12 percen had HML beas greaer han 1.0. In he full sample, boh numbers have doubled o 22 percen and 23 percen. This rend suggess ha funds will, in he fuure, be much less well described by he marke index. The view ha funds were unaware of value sraegies, and are now moving quickly o exploi hem, can explain why mos funds sill earn near he marke reurn, raher han he higher value reurn. However, his view conradics he view ha he value premium is an equilibrium risk premium, ha is, ha everyone knew abou he value reurns bu chose no o inves all along because hey feared he risks of value sraegies. If i is no an equilibrium risk premium, i won las long. Persisence in fund reurns The fund counerpar o momenum in sock reurns has been more exensively invesigaed han he value and size effecs. Fund reurns have also been found o be persisen. Since such persisence can be inerpreed as evidence for persisen skill in picking socks, i is no surprising ha i has araced a grea deal of aenion, saring wih Hendricks, Pael, and Zeckhauser (1993). Table 7, aken from Carhar (1997), shows ha a porfolio of he bes-performing one-hirieh of funds las year ouperforms a porfolio of he wors-performing one-hirieh of funds by 1 percen per monh (column 2). This is abou he same size as he momenum effec in socks, and similarly resuls from a small auocorrelaion plus a large sandard deviaion in TABLE 7 Porfolios of muual funds formed on previous year s reurn Average 4-facor Las year rank reurn CAPM alpha alpha ( percen ) 1/ / / / / / Noes: Each year, muual funds are sored ino porfolios based on he previous year s reurn. The rank column gives he rank of he seleced porfolio. For example, 1/30 is he bes performing porfolio when funds are divided ino 30 caegories. Average reurn gives he average monhly reurn in excess of he T-bill rae of his porfolio of funds for he following year. Four-facor alpha gives he average reurn less he predicions of a mulifacor model ha uses he marke, he Fama French HML and SMB porfolios, and porfolio PR1YR which is long NYSE socks ha did well in he las year and shor NYSE socks ha did poorly in he las year. Source: Carhar (1997). individual fund reurns. This resul verifies ha muual fund performance is persisen. Perhaps he funds ha did well ook on more marke risks, raising heir beas and, hence, average reurns in he following year. The hird column in able 7 shows ha his is no he case. The cross-secional variaion in fund average reurns has nohing o do wih marke beas. Jus as in he case of individual sock reurns, we have o undersand fund reurns wih mulifacor models, if a all. The las column of able 7 presens alphas (inerceps, he par of average reurn no explained by he model) from a model wih four facors he marke, he Fama French HML and SMB facors, and a momenum facor, PR1YR, ha is long NYSE socks ha did well in he las year and shor NYSE socks ha did poorly in he las year. In general, one should objec o he inclusion of so many facors and such ad-hoc facors. However, his is a performance aribuion raher han an economic explanaion use of a mulifacor model. We wan o know wheher fund performance, and persisence in fund performance in paricular, is due o persisen sock-picking skill or o mechanical sraegies ha invesors could jus as easily follow on heir own, wihou paying he managemen coss associaed wih invesing hrough a fund. For his purpose, i does no maer wheher he facors represen rue, underlying sources of macroeconomic risks. The alphas in he las column of able 7 are almos all abou 1 percen o 2 percen per year negaive. Thus, Carhar s model explains ha he persisence in fund performance is due o persisence in he underlying socks, no persisen sockpicking skill. These resuls suppor he old conclusion ha acively managed funds underperform mechanical indexing sraegies. There is some remaining puzzling persisence, bu i is all in he large negaive alphas of he boom one-enh o boom one-hirieh of performers, which lose money year afer year. Carhar also shows ha he persisence of fund performance is due o momenum in he underlying socks, raher han momenum funds. If, by good luck, a fund happened o pick socks ha wen up las year, he porfolio will coninue o go up a bi his year. In sum, he new research does nohing o dispel he disappoining view of acive managemen. However, we discover ha passively managed syle Federal Reserve Bank of Chicago 51

17 porfolios can earn reurns ha are no explained by he CAPM. Caasrophe insurance A number of prominen funds have earned very good reurns (and ohers, specacular losses) by following sraegies such as convergence rades and implici pu opions. These sraegies may also reflec high average reurns as compensaion for nonmarke dimensions of risk. They have no been examined a he same level of deail as he value and small-cap sraegies, so I offer a possible inerpreaion raher han a documened one. Convergence rades ake srong posiions in very similar securiies ha have small price differences. For example, a 29.5-year Treasury bond ypically rades a a slighly higher yield (lower price) han a 30-year Treasury bond. (This was he mos famous be placed by LTCM. See Lewis, 1999.) A convergence rade pus a srong shor posiion on he expensive securiy and a srong long posiion on he cheap securiy. This sraegy is ofen mislabeled an arbirage. However, he securiies are similar, no idenical. The spread beween and 30-year Treasury bonds reflecs he lower liquidiy of he shorer mauriy and he associaed difficuly of selling i in a financial panic. I is possible for his spread o widen. Noneheless, panics are rare, and he average reurns in all he years when hey do no happen may more han make up for he specacular losses when hey do. Pu opions proec invesors from large price declines. The volailiy smile in pu opion prices reflecs he surprisingly high prices of such opions, compared wih he small probabiliy of large marke collapses (even when one calibraes he probabiliy direcly, raher han using he log-normal disribuion of he Black Scholes formula). Wriers of ou-of-hemoney pus collec a fee every monh; in a rare marke collapse hey will pay ou a huge sum, bu if he probabiliy of he collapse is small enough, he average reurns may be quie good. All of hese sraegies can be hough of as caasrophe insurance (Hsieh and Fung, 1999). Mos of he ime hey earn a small premium. Once in a grea while hey lose a lo, and hey lose a lo in imes of financial caasrophe, when mos invesors are really anxious ha he value of heir invesmens no evaporae. Therefore, i is economically plausible ha hese sraegies can earn posiive average reurns, even when we accoun for sock marke risk via he CAPM and we correcly measure he small probabiliies of large losses. The difficuly in empirically esimaing he rue average reurn of such sraegies, of course, is ha rare evens are rare. Many long samples will give a false sense of securiy because he big one ha jusifies he premium happened no o hi. The value, yield curve, and foreign exchange sraegies I survey above also exhibi feaures of caasrophe insurance. Value socks may earn high reurns because disressed socks will all go bankrup in a financial panic. Buying bonds of counries wih high ineres raes leaves one open o he small chance of a large devaluaion, and such devaluaion is especially likely o happen in a global financial panic. Similarly, buying long-erm bonds in he deph of a recession when he yield curve is upward sloping may expose one o a small risk of a large inflaion. If hese inerpreaions bear ou, hey also sugges ha he premiums he average reurns from holding socks sensiive o HML or from following he bond and foreign exchange sraegies may be oversaed in he daa. The markes have had an unusually good 50 years, and devasaing financial panics have no happened. Implicaions of he new facs While he lis of new facs appears long, similar paerns show up in every case. Prices reveal slowmoving marke expecaions of subsequen reurns, because poenial offseing evens seem sluggish or absen. The paerns sugges ha invesors can earn subsanial average reurns by aking on he risks of recession and financial sress. In addiion, here is a small posiive auocorrelaion of highfrequency reurns. The effecs are no compleely new. We have known since he 1960s ha high-frequency reurns are slighly predicable, wih R 2 of 0.01 o 0.1 in daily o monhly reurns. These effecs were dismissed because here didn seem o be much one could do abou hem. A 51/49 be is no very aracive, especially if here is any ransacion cos. Also, he increased Sharpe raio (mean excess reurn/sandard deviaion) from exploiing predicabiliy is direcly relaed o he forecas R 2, so a iny R 2, even if exploiable, did no seem imporan. Now, we have a greaer undersanding of he poenial imporance of hese effecs and heir economic inerpreaions. For price effecs, we now realize ha he R 2 rises wih horizon when he forecasing variables are slowmoving. Hence, a small R 2 a shor horizons can mean a really subsanial R 2 in he 30 percen o 50 percen range a longer horizons. Also, he naure of hese effecs suggess he kinds of addiional sources of priced risk ha heoriss had anicipaed for 20 years. For momenum effecs, he abiliy o sor socks and 52 Economic Perspecives

18 funds ino momenum-based porfolios means ha very small predicabiliy imes porfolios wih huge pas reurns gives imporan subsequen reurns, hough i is no oally clear ha his amplificaion of he small predicabiliy really does survive ransacion coss. Price-based forecass If expeced reurns rise, prices are driven down, since fuure dividends or oher cash flows are discouned a a higher rae. A low price, hen, can reveal a marke expecaion of a high expeced or required reurn. 10 Mos of our resuls come from his effec. Low price/dividend, price/earnings, or price/book values signal imes when he marke as a whole will have high average reurns. Low marke value (price imes shares) relaive o book value signals securiies or porfolios ha earn high average reurns. The small-firm effec derives from low prices oher measures of size such as number of employees or book value alone have no predicive power for reurns (Berk, 1997). The five-year reversal effec derives from he fac ha five years of poor reurns lead o a low price. A high long-erm bond yield means ha he price of longerm bonds is low, and his seems o signal a ime of good long-erm bond reurns. A high foreign ineres rae means a low price on foreign bonds, and his seems o indicae good reurns on he foreign bonds. The mos naural inerpreaion of all hese effecs is ha he expeced or required reurn he risk premium on individual securiies as well as he marke as a whole varies slowly over ime. Thus we can rack marke expecaions of reurns by waching price/dividend, price/earnings, or book/marke raios. Absen offseing evens In each case, an apparen difference in yield should give rise o an offseing movemen, bu does no seem o do so. Somehing should be predicable so ha reurns are no predicable, and i is no. Figure 8 provides a picure of he resuls in able 5. Suppose ha he yield curve is upward sloping as in panel A. Wha does his mean? If he expecaions model were rue, he forward raes ploed agains mauriy would ranslae one for one o he forecas of fuure spo raes in panel B, as ploed in he black line marked Expecaions model. A high long-erm bond yield relaive o shor-erm bond yields should no mean a higher expeced long-erm bond reurn. Subsequen shor raes should rise, cuing off he one-period advanage of long-erm bonds and raising he muliyear advanage of shor-erm bonds. In figure 8, panel b, he colored line marked Esimaes shows he acual forecas of fuure spo ineres raes from he resuls in able 5. The essence of he phenomenon is sluggish adjusmen of he shor raes. The shor raes do evenually rise o mee he forward rae forecass, bu no as quickly as he forward raes predic hey should. Shor-erm yields should be forecasable so ha reurns are no forecasable. In fac, yields are almos unforecasable, so, mechanically, bond reurns are. The roughly 1.0 coefficiens in panel B of able 5 mean ha a 1 percenage poin increase in he forward rae ranslaes ino a 1 percenage poin increase in expeced reurn. I seems ha old fallacy of confusing bond yields wih heir expeced reurns for he firs year conains a grain of ruh. FIGURE 8 Yield curve and forecas one-year ineres raes A. Curren yield curve percen 7.0 B. Forecas one-year ineres raes forecas ineres rae Forward rae 6.5 Expecaions model 6.0 Yield Esimaes mauriy, years ime in he fuure, years Noes: Assuming ha he curren yield curve is as shown in panel A, he black line in panel B gives he forecas from he expecaions hypohesis, in which case forward raes oday are he forecas of fuure spo raes. The colored line in panel B gives he acual forecas of fuure spo raes, consruced from he esimaes in able 5. Federal Reserve Bank of Chicago 53

19 In he same way, a high dividend yield on a sock or porfolio should mean ha dividends grow more slowly over ime, or, for individual socks, ha he firm has aken on more marke risk and will have a higher marke bea. These endencies seem o be compleely absen. Dividend/price raios do no seem o forecas dividend growh and, hence, (mechanically) hey forecas reurns. The one-year coefficien in able 1 is very close o 1.00, meaning ha a 1 percenage poin increase in he dividend yield ranslaes ino a 1 percenage poin increase in reurn. I seems ha he old fallacy of confusing increased dividend yield wih increased oal reurn does conain a grain of ruh. A high foreign ineres rae relaive o domesic ineres raes should no mean a higher expeced reurn. We should see, on average, an offseing depreciaion. Bu here, he coefficiens are even larger han 1.0. An ineres rae differenial seems o predic a furher appreciaion. I seems ha he old fallacy of confusing ineres rae differenials across counries wih expeced reurns, forgeing abou depreciaion, also conains a grain of ruh. Economic inerpreaion The price-based predicabiliy paerns sugges a premium for holding risks relaed o recession and economy-wide financial disress. Sock and bond predicabiliy are linked: The erm spread (forward-spo, or long yield shor yield) forecass sock reurns as well as bond reurns (Fama and French, 1989). Furhermore, he erm spread is one of he bes variables for forecasing business cycles. I rises seeply a he boom of recessions and is invered a he op of a boom. Reurn forecass are high a he boom of a business cycle and low a he op of a boom. Value and small-cap socks are ypically disressed. Empirically successful economic models of he recession and disress premiums are sill in heir infancy (Campbell and Cochrane, 1999, is a sar), bu he sory is a leas plausible and he effecs have been expeced by heoriss for a generaion. To make his poin come o life, hink concreely abou wha you have o do o ake advanage of he predicabiliy sraegies. You have o buy socks or long-erm bonds a he boom, when sock prices are low afer a long and depressing bear marke, in he boom of a recession or he peak of a financial panic. This is a ime when few people have he gus or he walle o buy risky socks or risky long-erm bonds. Looking across socks raher han over ime, you have o inves in value or small-cap companies, wih years of poor pas reurns, poor sales, or on he edge of bankrupcy. You have o buy socks ha everyone else hinks are dogs. Then, you have o sell socks and long-erm bonds in good imes, when sock prices are high relaive o dividends, earnings, and oher muliples and he yield curve is fla or invered so ha long-erm bond prices are high. You have o sell he popular growh socks, wih good pas reurns, good sales, and earnings growh. You have o sell now, and he socks ha you should sell are he blue-chips ha everyone else seems o be buying. In fac, he marke iming sraegies said o sell long ago; if you did so, you would have missed much of he runup in he Dow pas he 6,000 poin. Value socks oo have missed mos of he recen marke runup. However, his shouldn worry you a sraegy ha holds risks uncorrelaed wih he marke mus underperform he marke close o half of he ime. If his feels uncomforable, wha you re feeling is risk. If you re uncomforable waching he marke pass you by, perhaps you don really only care abou long-run mean and variance; you also care abou doing well when he marke is doing well. If you wan o say fully invesed in socks, perhaps you oo feel he ime-varying aversion o or exposure o risk ha drives he average invesor o say fully invesed despie low prospecive reurns. This line of explanaion for he foreign exchange puzzle is sill a bi farher off (see Engel, 1996, for a survey; Akeson, Alvarez, and Kehoe, 1999, offer a recen sab a an explanaion). The sraegy leads invesors o inves in counries wih high ineres raes. High ineres raes are ofen a sign of moneary insabiliy or oher economic rouble, and hus may mean ha he invesmens are more exposed o he risks of global financial sress or a global recession han are invesmens in he bonds of counries wih low ineres raes, which are ypically enjoying beer imes. Reurn correlaion Momenum and persisen fund performance explained by a momenum facor are differen from he price-based predicabiliy resuls. In boh cases, he underlying phenomenon is a small predicabiliy of high-frequency reurns. The price-based predicabiliy sraegies make his predicabiliy imporan by showing ha, wih a slow-moving forecasing variable, he R 2 builds over horizon. Momenum, however, is based on a fas-moving forecas variable he previous year s reurn. Therefore, he R 2 declines raher han building wih horizon. Momenum makes he small predicabiliy of high-frequency reurns significan in a differen way, by forming porfolios of exreme winners and losers. The large volailiy of reurns means ha he exreme porfolios will have exreme pas reurns, so only a small coninuaion of pas reurns gives a large curren reurn. 54 Economic Perspecives

20 I would be appealing o undersand momenum as a reflecion of slowly ime-varying average expeced reurns or risk premiums, like he price-based predicabiliy sraegies. If a sock s average reurn rises for a while, ha should make reurns higher boh oday and omorrow. Thus, a porfolio of pas winners will conain more han is share of socks ha performed well because heir average reurns were higher, along wih socks ha performed well due o luck. The average reurn of such a porfolio should be higher omorrow as well. Unforunaely, his sory has o posi a subsanially differen view of he underlying process for varying expeced reurns han is needed o explain everyhing else. The rouble is ha a surprise increase in expeced reurns means ha prices will fall, since dividends are now discouned a a greaer rae. This is he phenomenon we have relied on o explain why low price/dividend, price/earnings, book/marke, value, and size forecas higher subsequen reurns. Therefore, posiive correlaion of expeced reurns ypically yields a negaive correlaion of realized reurns. To ge a posiive correlaion of realized reurns ou of slow expeced reurn variaion, you have o imagine ha an increase in average reurns oday is eiher highly correlaed wih a decrease in expeced fuure dividend growh or wih a decrease in expeced reurns in he disan fuure (an impulse response ha sars posiive bu is negaive a long horizons). Campbell, Lo, and MacKinlay (1997) provide a quaniaive exposiion of hese effecs. Furhermore, momenum reurns have no ye been linked o business cycles or financial disress in even he informal way ha I suggesed for price-based sraegies. Thus, momenum sill lacks a plausible economic inerpreaion. To me, his adds weigh o he view ha i isn here, i isn exploiable, or i represens a small illiquidiy (ax-loss selling of small illiquid socks) ha will be quickly remedied once a few raders undersand i. Remaining doubs The size of all hese effecs is sill somewha in quesion. I is always hard o measure average reurns of risky sraegies. The sandard formula s / T for he sandard error of he mean, ogeher wih he high volailiy σ of any sraegy, means ha one needs 25 years of daa o even sar o measure average reurns. Wih σ = 16 percen (ypical of he index), even T = 25 years means ha one sandard error is 16/5 3 percen per year, and a wo-sandard error confidence inerval runs plus or minus 6 percenage poins. This is no much smaller han he average reurns we are rying o measure. In addiion, all of hese facs are highly influenced by he small probabiliy of rare evens, which makes measuring average reurns even harder. Finally, viewed he righ way, we have very few daa poins wih which o evaluae predicabiliy. The erm premium and ineres rae differenials only change sign wih he business cycle, and he dividend/price raio only crosses is mean once every generaion. The hisory of ineres raes and inflaion in he U.S. is dominaed by he increase, hrough wo recessions, o a peak in 1980 and hen a slow decline afer ha. Many of he anomalous risk premiums seem o be declining over ime. Figure 6 shows he decline in he HML and SMB premiums, and he same may be rue of he predicabiliy effecs. The las hree years of high marke reurns have cu he esimaed reurn predicabiliy from he dividend/price raio in half. This fac suggess ha a leas some of he premium he new sraegies yielded in he pas was due o he fac ha hey were simply overlooked. Was i really clear o average invesors in 1947 or 1963 (he beginning of he daa samples) ha socks would earn 9 percen over bonds, and ha he sraegy of buying disressed small socks would double even ha reurn for he same level of risk? Would average invesors have changed heir porfolios wih his knowledge? Or would hey have sayed pa, explaining ha hese reurns are earned as a reward for risk ha hey were no willing o ake? Was i clear ha buying socks a he boom in he mid-1970s would yield so much more han even ha high average reurn? If we inerpre he premiums measured in sample as rue risk premiums, he answer mus be yes. If he answer is no, hen a leas some par of he premium was luck and will disappear in he fuure. Since he premiums are hard o measure, one is emped o pu less emphasis on hem. However, hey are crucial o our inerpreaion of he facs. The CAPM is perfecly consisen wih he fac ha here are addiional sources of common variaion. For example, i was long undersood ha socks in he same indusry move ogeher; he fac ha value or small socks also move ogeher need no cause a ripple. The surprise is ha invesors seem o earn an average reurn premium for holding hese addiional sources of common movemen, whereas he CAPM predics ha (given bea) hey should have no effec on a porfolio s average reurns. The behavior of funds also suggess he overlooked sraegy inerpreaion. As explained earlier, fund reurns sill cluser around he marke line. I urns ou ha very few fund reurns acually followed he value or oher reurn-enhancing sraegies. However, he number of small, value, and relaed funds funds Federal Reserve Bank of Chicago 55