Does Stock Return Predictability Imply Improved Asset Allocation and Performance? Evidence from the U.S. Stock Market ( )

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1 Does Stock Return Predctablty Imply Improved Asset Allocaton and Performance? Evdence from the U.S. Stock Market ( ) Puneet Handa * Ashsh war ** Current Draft: November, 004 Key words: Predctablty, parameter uncertanty, asset allocaton, predctve dstrbuton, condtonng nformaton, performance evaluaton, market tmng. * John Hawknson Research Fellow n Fnance and Assocate Professor, Henry B. ppe College of Busness, Unversty of Iowa, Iowa Cty 54, Ph.: , E-mal: puneet-handa@uowa.edu. ** Matthew Bucksbaum Research Fellow n Fnance and Assstant Professor, Henry B. ppe College of Busness, Unversty of Iowa, Iowa Cty 54, Ph.: , E-mal: ashsh-twar@uowa.edu. We are grateful to the referee for numerous suggestons that have helped us mprove the paper substantally. We would also lke to thank Mchael Cooper, om George, Campbell Harvey, Luboš Pástor, Mke Stutzer, and the semnar partcpants at the Unversty of Iowa and the 00 Amercan Fnance Assocaton meetngs, for ther helpful comments.

2 Abstract hs paper provdes evdence on the economc sgnfcance of U.S. stock return predctablty wthn an asset allocaton framework n a real-tme context. We examne the performance of a Bayesan, rsk averse nvestor (the mutual fund nvestor) who reles on condtonng nformaton (e.g., dvdend yeld, -bll yeld, default spread and term spread) to forecast future returns, and contrast t wth that of an otherwse dentcal nvestor who beleves that the returns are..d. (the..d. nvestor). Our major fndng s that the relatve performance of the mutual fund strategy s unstable over tme, beng notceably poor durng the most recent sub-perod ( ). In marked contrast, the performance of the mutual fund strategy s sgnfcantly better when t reles on a model-based approach, characterzed by varyng degrees of pror confdence n the CAPM. 1

3 Does Stock Return Predctablty Imply Improved Asset Allocaton and Performance? Evdence from the US Stock Market ( ) I. Introducton he predctablty of excess stock returns s by now a well-establshed phenomenon n the asset prcng lterature. As Cochrane (1999) notes n hs revew artcle, durng the past two decades several studes have demonstrated that a substantal amount of stock return varaton can be explaned by varables such as the dvdend yeld and -bll yeld, among others. here are legtmate concerns expressed n the lterature about the statstcal sgnfcance of the evdence and ts out-of-sample valdty. 1 However, as Cochrane notes, the consensus vew now s that stock returns are predctable. In contrast, there has been less dscusson and agreement on the economc sgnfcance of stock return predctablty,.e., whether the documented predctablty leads to mproved performance. Breen, Glosten and Jagannathan (1989) fnd that the predctve ablty of the onemonth reasury bll rates s economcally sgnfcant. In a smlar ven, Solnk (1993) shows that the use of condtonng nformaton leads to economcally sgnfcant mprovement n the performance of nternatonal asset allocaton strateges. On the other hand, Pesaran and mmermann (1995) show that the predctve ablty of a number of economc factors vares over tme although t s economcally sgnfcant durng the volatle markets of the 1970s. More recently, Cooper, Guterrez, and Marcum (001), and Cooper and Gulen (00) fnd that predctablty s not evdent n a real-tme context. Except for Solnk (1993), the above studes adopt a framework that nvolves shftng one s portfolo entrely to stocks or to -blls, nstead of a utlty-based asset allocaton framework whch would be more reflectve of the mpact of predctablty on nvestor decsons. Most of these studes also gnore the uncertanty nherent n the parameters of the forecastng 1 For a crtque of the evdence on grounds of ms -specfed test statstcs and/or fnte sample bases, see Rchardson and Stock (1989), Rchardson and Smth (1991), Km, Nelson and Startz (1991), Nelson and Km (1993), and Goetzmann and Joron (1993), among others. More recently, Bossaerts and Hllon (1999) and Goyal and Welch (1999) have used statstcal measures of model performance to queston the out-of-sample predctve ablty of the return forecastng models. he mpact of data snoopng bases has been examned by Lo and Macknlay (1990), and Foster, Smth and Whaley (1997).

4 model used to form expectatons of future asset returns. From the standpont of practcal advce to portfolo nvestors, a natural queston to ask s whether the documented stock return predctablty translates nto mproved out-of-sample asset allocaton and performance when nvestors account for parameter uncertanty. hs queston s mportant because as Kandel and Stambaugh (1996) argue, the ssue of statstcal sgnfcance of a predctve model s qute dstnct from ts mpact on portfolo allocaton. In partcular, a predctve varable may sgnfcantly mpact portfolo allocaton despte havng low statstcal sgnfcance. Hence, an asset allocaton framework provdes us wth an ntutvely appealng metrc by whch to judge the value of nformaton contaned n a predctor varable, namely the qualty of portfolo allocaton resultng from ts use. here s relatvely lttle evdence n the lterature on the above ssue,.e., whether predctablty of stock returns leads to mproved asset allocaton and performance, n a real-tme context. In ths paper we provde evdence on ths ssue n the context of a Bayesan nvestor who accounts for parameter uncertanty and faces constrants such as the presence of transacton costs, lmted short sellng, and real-tme decson makng. 3 Prevous studes such as Kandel and Stambaugh (1996) and Barbers (000) have provded valuable nsghts n regard to the mpact of return predctablty on portfolo allocaton n a Bayesan settng. However, these studes make use of the full hstorcal sample nformaton and the end-of-sample values of the predctor varables. Our analyss complements and adds to the prevous lterature by sheddng lght on the portfolo allocaton mplcatons of return predctablty when nvestors are faced wth parameter uncertanty and market frctons n a real-tme context. 4 At the start of each month, our Bayesan nvestor makes portfolo allocatons based on the belef that stock returns are predctable. he portfolos allocatons are updated every month. We recently became aware of a study by We and Zhang (000) that examnes return predctablty n an asset allocaton framework. hs study dffers from our work as t adopts a non-bayesan framework. he benefts of nternatonal dversfcaton wth regme shfts have been examned by Ang and Bekaert (00) who fnd that such benefts persst despte the presence of bear market regmes. hey do not consder transacton costs or parameter uncertanty n ther analyss. In a dfferent context, Flemng, Krby, and Ostdek (001) study volatlty tmng n an asset allocaton framework. 3 In our context, the real-tme envronment allows the nvestor access to just the hstorcal data whle makng portfolo allocaton decsons. 4 As Kandel and Stambaugh (1996) note, ther analyss ntentonally omts any consderaton of the typcal performance of the optmal condtonal asset allocatons. Our analyss may be vewed as provdng some nsghts nto the typcal performance of the strategy that reles on predctablty. 3

5 We contrast hs performance to that of a Bayesan nvestor who beleves that stock returns are ndependent and dentcally dstrbuted (..d.) and gnores any evdence of stock return predctablty n makng hs monthly portfolos allocaton choces. Both nvestors are assumed to be mean-varance optmzers wth a sngle-perod horzon. We refer to the former strategy as the mutual fund strategy and the latter as the..d. strategy. Our study spans the perod Our research desgn allows for both a data-based approach as well as an alternatve model-based approach to portfolo allocaton. Under the data-based approach the mutual fund nvestor uses condtonng varables such as lagged values of the dvdend yeld, one month - bll yeld, default spread, and term spread, to form expectatons of next month s stock returns. A key feature of our research desgn s that at the start of each month the mutual fund strategy s allowed to select an optmal forecastng model based on a recursve utlty maxmzaton crteron. 5 At the start of each month, the mutual fund strategy selects an optmal model from the full set of 16 (= 4 ) possble models, ncludng the..d. model. he model chosen by the mutual fund strategy s the one that performs the best n terms of the realzed utlty, to date. he mutual fund strategy and the..d. strategy are allowed to form portfolos by nvestng n a rsk-free asset and (a) a sngle stock portfolo, or (b) three stock portfolos. We also adopt the alternatve model-based approach of Pástor and Stambaugh (1999, 000) and Pástor (000) under whch the mutual fund strategy s allowed to form expectatons of next month s stock returns based on varyng degrees of pror confdence n the Captal Asset Prcng Model. Under ths framework we allow both strateges to nvest n ten ndustry-based portfolos n addton to the rsk free asset. As an extenson of ths framework we allow the mutual fund strategy to account for tme-varaton n alphas, betas, and the market rsk premum. Smlar to Shanken (1990), Ferson and Harvey (1999), Avramov (004), and Avramov and Chorda (004), we model the alphas and betas as well as the market rsk premum as lnear functons of the condtonng varables. 5 wo recent studes, by Avramov (00) and Cremers (00), use Bayesan model averagng technques to examne predctablty n the presence of model uncertanty. hey use statstcal measures to conclude that the out-of-sample performance of the Bayesan approach s superor to that of the classcal statstcal model selecton methods. However, nether study addresses the queston of economc sgnfcance of return predctablty n a real-tme context. 4

6 We calculate the certanty equvalent rates of return (CER) for each strategy to judge ts relatve performance. he CER for each strategy represents the rate of return earned wth certanty that would provde the nvestor wth the same utlty as the utlty realzed from the portfolo allocatons resultng from the strategy. We also report the adjusted Sharpe rato of Graham and Harvey (1997) for the mutual fund strategy and assess ts market tmng performance. In our analyss we ncorporate several features of the economc envronment ncludng out-of-sample forecastng, parameter uncertanty, portfolo allocaton across both sngle and multple rsky assets, transactons costs and lmted short sellng and margn purchases. 6 Our study s desgn allows nvestors to contnuously update the parameters of the predctve model utlzed by them. hs s potentally mportant f the dynamcs of the stock return process or the predctve ablty of condtonng varables changes over tme. Our constructon of the two hypothetcal strateges can be thought of as a devce to explore the value of nformaton contaned n (a) the predctor varables n the data-based approach, and (b) the asset prcng model utlzed n the model-based approach. If, for example, the predctor varables contan useful nformaton about future returns, n the context of our repeated experments, we expect the mutual fund strategy to outperform the..d. strategy. Our approach s smlar n sprt to the utlty-based metrc proposed by McCulloch and Ross (1990) to judge the economc sgnfcance of departures from the arbtrage prcng theory. As McCulloch and Ross note, such an approach may result n conclusons that dffer markedly from tradtonal sgnfcance testng. We examne the relatve performance of the mutual fund and..d. strateges n real-tme over the perod , wth the frst fve years used as an ntal learnng perod. Specfcally, we allow each strategy to use only hstorcal data from 1954 to date, to make portfolo allocatons for the next perod. he mutual fund strategy that reles on a data-based approach has unstable out-of-sample performance that s perod specfc. It s unable to outperform the..d. strategy on a consstent bass. he mutual fund strategy s performance s generally favorable durng the sub-perods and For example, n the case where portfolo allocatons nvolve the rsk free asset and a sngle stock portfolo, the mutual 6 In partcular, the multple rsky asset scenaro allows us to capture sectoral dfferences that may exst over economc cycles (see for example, Perez-Quros and mmermann (000)). 5

7 fund strategy outperforms the..d. strategy by 37 bass ponts per month durng the two subperods n terms of CER, when averaged across the three levels of rsk averson consdered by us. However, the strategy s performance s notceably poor durng the most recent sub-perod, , where t underperforms the..d. strategy wth an average CER dfference of 38 bass ponts per month. Moreover, the portfolo allocatons of the mutual fund strategy are consderably more volatle than the allocatons of the..d. strategy. We also fnd that the mutual fund strategy performs only margnally better than a random con toss n terms of ts market tmng ablty. When judged n terms of statstcal sgnfcance, for the full sample perod, the performance of the mutual fund strategy s not sgnfcantly dfferent than that of the..d. strategy. o get a better understandng of the performance of the mutual fund strategy, we focus on the model choces made by the strategy over tme. We fnd that the mutual fund strategy makes use of the..d. model nfrequently, suggestng that there s support for predctablty n the data. However, we fnd that the shfts n the economc envronment over tme make t dffcult for the mutual fund strategy to uncover the approprate model for a partcular tme perod leadng to unstable performance across sub-perods. In our base set of results, the mutual fund strategy chooses the optmal forecastng model each month based on the canddate models performance to date. Shortenng the length of the hstorcal wndow used for model evaluaton and selecton does not qualtatvely alter our results. In contrast to the data-based approach, the results from the model-based approach, under whch the mutual fund strategy reles on varyng degrees of pror belef n the CAPM to form expectatons of future stock returns, are much more favorable for the strategy. We fnd that under the standard CAPM approach, the mutual fund strategy sgnfcantly outperforms the..d. strategy for moderate to hgh levels of rsk averson. he performance advantage s sgnfcant n both statstcal and economc terms, especally when the nvestor has a hgh degree of pror confdence n the CAPM. Allowng for tme varaton n the alphas, betas, and the market rsk premum does not lead to sgnfcant mprovement relatve to the standard CAPM framework. Our results suggest that varables such as the -bll yeld, dvdend yeld, default spread, and term spread are valuable for asset allocaton. hs s borne out by the fact that the Bayesan mutual fund nvestor contnues to use these varables nstead of optng for the..d. model n most cases. Also, n economc terms, there are mpressve gans from usng these varables n 6

8 some sub-perods. However, the gans are not consstent over tme. Shfts n the economc envronment often make t dffcult to dentfy the model that s most approprate n a partcular sub-perod. hus, our results provde support for predctablty n the data. At the same tme they suggest that translatng the predctablty nto consstently superor performance s a challengng task and therefore allocaton strateges relyng on data-based predctor varables should be tempered wth cauton. Our analyss helps explan the gap between the fndngs of stock return predctablty n the academc lterature on the one hand, and the documented lack of superor performance of the average professonal money manager, on the other. 7 Interestngly, our results also serve to hghlght the value of a model-based approach motvated by theoretcal consderatons, for achevng optmal asset allocatons. Recently, studes such as Balduzz and Lynch (1999) and Campbell and Vcera (1999) have argued that the utlty costs of gnorng the sample evdence of stock return predctablty are substantal. hese studes typcally nvolve a calbraton of a data generatng process for excess stock returns based upon the U.S. sample hstory. In ths context, our analyss can be vewed as a real-tme experment desgned to capture the actual experence of a strategy that reles on predctablty and s subject to realstc constrants such as transactons costs, and lmted short sellng and margn purchases. As dscussed below, our results suggest that the utlty gans from relyng on predctablty that have been antcpated by prevous studes, may represent an deal that s dffcult to attan for nvestors makng decsons n real-tme. In ths respect our results complement those of Carlson, Chapman, Kanel, and Yan (001) who examne the mpact of return predctablty on the optmal portfolo and consumpton choces of an nvestor n a general equlbrum settng. hey fnd that allocatons based on the observed sample estmates can be severely based and have large varance n repeated smulatons of the model economy. hey conclude that calbraton exercses based on pont estmates from a smple sample path can be qute msleadng. 8 A smlar note of cauton s sounded by Aït-Sahala and 7 For example, begnnng wth Jensen (1968), a large number of studes have shown that the average actvely managed mutual fund fals to outperform the relevant benchmark(s), net of the management fees (see, also, Gruber (1996), and Carhart (1997), among others). 8 In related work Johannes, Polson, and Stroud (00) examne the value of market tmng and volatlty tmng whle allowng for estmaton rsk. hey fnd that market tmng strateges perform poorly n contrast to the performance of volatlty tmng strateges. Smlarly, Gomes (003) fnds that utlty gans from strateges based on explotng short-run stock return autocorrelaton are qute small and not robust to parameter uncertanty, n contrast to the performance of strateges based on volatlty tmng that yeld sgnfcant gans. Both studes analyze strateges 7

9 Brandt (001) who solve the sample analogues of the condtonal Euler equatons that characterze nvestors optmal portfolo choce, to determne whch economc varables are mportant for such a choce. hey conclude that the magntude of predctablty n returns s small and subject to a tremendous amount of nose, especally at short horzon. (p. 1349). he paper s organzed as follows. In Secton II, we outlne how we mplement the portfolo allocaton problem n a Bayesan context. Secton III descrbes our research desgn whle Secton IV presents descrptve statstcs and the results of prelmnary regresson analyss. In Secton V we present the results of the data-based approach where the mutual fund nvestor reles on the predctve ablty of the condtonng varables to make asset allocatons. Secton VI presents results for the alternatve model-based approach whle Secton VII concludes. II. Asset allocaton A. Investor preferences We assume that nvestors expected utlty s defned over the expected return, E r ), ( p and varance, σ p, of ther portfolo: E 1 ( U ) = E( r p ) λσ (1) p where λ represents the nvestors degree of relatve rsk averson. 9 We consder two nvestors. In our framework the mutual fund nvestor, beleves that excess stock returns are predctable. Under the data-based approach to asset allocaton, each month the nvestor estmates the parameters of the followng model usng hstorcal data, and uses the estmates as n nput n forecastng the next month s return: ~ r, = X B + ε ~ t t 1, t () nvolvng a sngle rsky asset and nether study examnes the performance of the strateges n the context of repeated samples. 9 Samuelson (1970) provdes a theoretcal justfcaton for the mean-varance crteron. For recent applcatons of ths utlty specfcaton see Solnk (1993) and Pàstor and Stambaugh (000), among others. 8

10 where X (, x, x,...., x ) t 1 = 1 1, t 1, t 1 k, t 1 s a k-dmensonal vector of explanatory (predctor) varables observed at the end of month t-1, B s a k x 1 vector of regresson coeffcents, and ε ~ ~... (0,, t d N σ ). he excess return on a stock portfolo, r,t, s defned as the portfolo s monthly return n excess of the return on the one-month treasury bll. he vector X t-1 conssts of varables such as the dvdend yeld, one month -bll yeld, default spread and the term spread n addton to a constant term. 10 At the start of each month the nvestor chooses the optmal combnaton of predctor varables from the full set. We descrbe the predctor varables and the process by whch the nvestor chooses the approprate varables to use n the forecastng model, n more detal n Secton III. he..d. nvestor, beleves that stock returns are ndependently and dentcally dstrbuted. In partcular, the nvestor beleves that the excess return on a portfolo has the followng dstrbuton: ~ = ε~ (3) r, t µ +, t ~, t where ε ~.. d. N(0, σ ). Hence, the expected next perod excess return for the..d. nvestor s gven by equaton () where the set of predctor varables X t-1 ncludes just the constant 1, and the estmated regresson coeffcent represents the sample mean excess return. hs yelds a set of optmal portfolo allocatons for the..d. nvestor for that month. he portfolo allocaton s updated every month. B. Portfolo Optmzaton Problem Consder an nvestor who forms a portfolo P at the start of month t by choosng the optmal proportons to nvest n multple rsky assets and a rsk-free asset. Wth multple rsky assets, an nvestor chooses the optmal allocaton to the rsky assets by solvng the followng problem: 10 Our framework s smlar to Campbell (1991), Hodrck (199), Kandel and Stambaugh (1996) and Barbers (000), among others. 9

11 Max w Er w s. t. n = 1 w 1 λ w Sw n = 1 κ ( w w ), t, t 1 (4) where Er denotes a nx1 vector of expected excess returns on the n rsky assets, S denotes the nxn covarance matrx of returns and w denotes the nx1 vector of weghts on the rsky assets chosen by the nvestor at the start of month t. Note that the proporton of the portfolo nvested n the rsk-free asset s gven by = n 1 w, t. In the above equaton κ s the proportonal transacton = 1 cost ncurred by the nvestor. We assume that the value of κ s 0.005; ths s equvalent to a proportonal transacton cost of 50 bass ponts for a round trp trade n stocks. 11 We further assume that nvestors can costlessly alter ther postons n the rsk-free asset. In addton, we allow nvestors to take short or long postons consstent wth a 50% margn requrement. 1 In equaton (4) an nvestor s expected portfolo return, E(r p,t ) and varance of return, by: s p,t, are gven E( r ) = r + w Er σ = w Σw (5) p, t f, t ; p, t where r f,t s the rsk-free rate of return for month t observed at the start of the month. Both the..d. nvestor and the mutual fund nvestor form an estmate of the expected excess return matrx, Er and the covarance matrx, S. In our analyss we consder Bayesan nvestors who account for parameter uncertanty. Hence, n order to solve for the optmal portfolo allocaton they use the moments of the predctve dstrbuton of returns based on ther respectve belefs. 13 hs process s outlned n more detal below. 11 hs s consstent wth estmates used n the lterature (for example, Balduzz and Lynch, 1999). 1 As an example, under a 50% margn requrement, a $100 stock nvestment may be fnanced wth $50 n captal, whch corresponds to a portfolo weghtng of 00% n the stock. A smlar calculaton apples to short-sales subject to a 50% margn requrement. We also allow for margn purchases by borrowng at the rsk-free rate. 13 he mpact of parameter uncertanty on portfolo choce has been studed by Klen and Bawa (1976), Bawa, Brown and Klen (1979), and more recently, n our context by Kandel and Stambaugh (1996) and Barbers (000). In related work Lewellen and Shanken (00) show that Bayesan learnng n the presence of parameter uncertanty can lead to stock return predctablty, although nvestors can nether perceve t nor explot t. 10

12 B.1. Predctve Dstrbuton for the Mutual Fund Investor As prevously dscussed, the mutual fund nvestor reles on a set of predctve varables n formng expectatons of future stock returns at the start of each month t. Consder the case where the nvestor forms a portfolo by nvestng n a sngle rsky asset, n addton to the rsk free asset. 14 We can express the predctve model descrbed n equaton () as: 1 ( 0, h ) R = X β + ε ; ε X ~ N (6) where n ths context R s a x 1 matrx of excess returns, X s a x k matrx of k predctor varables, and B s a k x 1 matrx of regresson coeffcents. he parameter h s the precson of the..d. dsturbances and equals the nverse of ( ε ) = σ. I Var he mutual fund nvestor faces uncertanty about hs estmate of the return varance and also hs estmate of the regresson coeffcent vector, B. We assume that the nvestor has the standard normal-gamma conjugate pror: t s h β h ~ χ ~ N ( ν ) 1 1 ( β, h ) H (7) Smlar to Pástor and Stambaugh (000), we specfy the degrees of freedom ν to be 15 so that the pror contans only as much nformaton as 15 observatons. he value of β s set equal to the OLS estmate of β usng data for the perod Smlarly, we set s equal to the average of the sample estmate of the varance over the perod We further specfy H = s? ( ). he posteror dstrbuton for the parameters of nterest s then gven by: ( s + Q) h R ~ χ ( + ν ) β R, h 1 ~ N 1 1 ( β, h H ) (8) where 14 he multple rsky assets case s smlar n sprt and we omt the detals for the sake of brevty. 11

13 H = Q = 1 H + X X ; β = H ( H β + X Xβˆ ), and ( βˆ β ) X X ( βˆ β ) + ( β β ) H( βˆ β ) (9) he term βˆ s the k x 1 vector of estmated OLS coeffcents usng the -perod sample of data avalable to the nvestor. he parameters of the posteror dstrbuton gven n equaton (9) have an ntutve form. For example, note that the precson of the posteror dstrbuton, H, s the sum of the pror precson and the term X X whch would be the precson f the pror precson were 0. In smlar fashon the mean of the posteror dstrbuton s a weghted average of the pror β and the sample estmate, βˆ. In other words, the sample estmate s shrunk towards the pror. he predctve return dstrbuton for ~ = (,...., ) multvarate t-dstrbuton.e., r r + r + f ~ ~ ~ ( R, X ) ~ t [ Xβ,( s + Q)( XH X + I );ν ] 1 at tme s n the form of a ~ (10) r 1 f + At the start of each month, the mutual fund nvestor uses the mean and the varance of the one-step ahead predctve dstrbuton as hs expectaton for the expected excess returns, and the expected varance, of the rsky assets. Usng these nputs, the nvestor solves the problem n equaton (4) and chooses the nvestment proportons n the rsky assets and the rsk-free asset, respectvely. B.. Predctve Dstrbuton for the..d. Investor he..d. nvestor s also assumed to have a conjugate pror smlar to the mutual fund nvestor. However, the matrx of explanatory varables X used by ths nvestor conssts solely of a vector of ones. he predctve return dstrbuton s agan of the Student t-form. he..d. nvestor uses the moments of the predctve return dstrbuton as nputs n solvng for the optmal allocaton to the equty portfolo, w, n equaton (4). C. Alternatve Predctve Models As a robustness check we allow the mutual fund nvestor to employ two alternatve forecastng models n formng hs expectatons. We dscuss each of these cases below. 1

14 C.1. A VAR specfcaton Followng Kandel and Stambaugh (1996), we allow the mutual fund nvestor to model the stock returns and the predctor varables as a frst-order vector autoregresson. hat s, the mutual fund nvestor beleves that Y = XB + U where r t s the frst element of Y t and X t s a 1 x k vector of predctor varables, and U s a vector of ndependent mean-zero error terms. C.. Expectatons based on an asset prcng model We extend our analyss to allow our nvestors to nvest n ten rsky assets along wth the rsk-free assets. Addtonally, we allow the mutual fund nvestor to use the Captal Asset Prcng Model (CAPM) to form estmates of expected returns rather than relyng on the predctor varables consdered earler. We adopt the framework suggested by Pástor and Stambaugh (000) who study the optmal portfolo allocatons resultng from pror belefs n dfferent asset prcng models as a way of comparng the models. he framework s descrbed n more detal n Secton VI. As a further extenson of the standard CAPM based framework, we consder a settng where the alphas, betas and the market rsk premum are allowed to potentally vary wth the condtonng varables consdered by us. III. Research Desgn A. Data Data for the study from the perod January 1954 to December 00, are obtaned from the CRSP database. As a proxy for the rsk-free asset, we use the 1-month US -bll yeld from the CRSP fles. For the case nvolvng nvestment n a rsk-free asset and one rsky asset, we use the CRSP value-weghted ndex of Nyse/Amex/Nasdaq stocks as the rsky asset. For the case nvolvng a rsk-free asset and three rsky assets, we use three CRSP cap-based portfolos. We form a large cap portfolo by value-weghtng the returns on the CRSP decle portfolos 1 through 3. Smlarly, we form a medum cap portfolo by value-weghtng the returns on the CRSP decle portfolos 4 though 7, and we form a small cap portfolo by value-weghtng the returns on the CRSP decle portfolos 8 though 10. We construct monthly excess returns by 13

15 subtractng the one-month -bll return from the monthly returns on the respectve stock portfolos. In subsequent analyss (reported n Secton VI) we utlze returns on ten ndustrybased portfolos provded by Ken French. B. Predctor Varables At the start of each month we allow the mutual fund nvestor to choose from a base set of four predctor varables. hese nclude the dvdend yeld, the one month -bll yeld, a measure of the default spread and a measure of the term spread. 15 Our prmary motvaton for focusng on these varables s that they are busness-cycle related varables (see, for example, the dscusson n Fama and French (1989)) and have therefore attracted a lot of attenton n the lterature. he dvdend yeld seres s constructed as the natural log of the cumulated dvdends on the CRSP value-weghted market ndex (coverng NYSE, AMEX and NASDAQ stocks) over the prevous 1 months dvded by the current level of the ndex. he -bll yeld s the one-month -bll yeld mnus ts 1-month backward movng average. hs stochastc detrendng method for the short rate has been used by Campbell (1991) and Hodrck (199), among others. he default spread s the yeld dfference between Moody s BAA rated corporate bonds and AAA rated corporate bonds. he term spread s the dfference between the 10-year and the 3-month treasury yelds. Data for the default spread and the term spread are obtaned from the Federal Reserve Bulletn whle the relatve -bll yeld s constructed from CRSP data. All condtonng varables are lagged by one month. C. Methodology he study s conducted on a monthly bass and spans the perod January 1954 to December 00. In our tests we contrast the performance of the mutual fund nvestor wth that of the..d. nvestor n two cases: () the sngle rsky asset case when nvestors have the choce of nvestng n one rsky asset and a rsk-free asset, and () the multple rsky asset case where nvestors have the choce of nvestng n three rsky assets and a rsk-free asset. Specfcally, we 15 A partal lst of predctor varables examned n the lterature ncludes lagged returns [Fama and French (1988a)], dvdend yeld [(Rozeff (1984), Shller (1984), Campbell and Shller (1988), Fama and French (1988b, 1989), Hodrck (199)], book-to-market rato [Kothar and Shanken (1997), Pontff and Schall (1998), Lewellen (1999)], the short-term -bll rate [Fama and Schwert (1977), Campbell (1991)], default spread [Kem and Stambaugh (1986), Fama and French (1989)], term spread [Campbell (1987), Fama and French (1989)], and the aggregate consumpton-wealth rato (Lettau and Ludvgson (001)). 14

16 track the certanty equvalent rates of return (CER) and the adjusted Sharpe ratos for the two nvestors. Both nvestors make ther portfolo choces at the start of a month and are assumed to ncur a proportonal transacton cost on the change n equty holdngs equvalent to 50 bass ponts for a round trp trade. We conduct tests usng three levels of nvestor rsk-averson,.e., low rsk-averson (when the coeffcent of relatve rsk averson s set at ), moderate rskaverson (when the coeffcent of relatve rsk averson s set at 5), and hgh rsk-averson (when the coeffcent of relatve rsk averson s set at 10). Henceforth, we use the terms low, moderate and hgh to refer to rsk averson values of, 5 and 10, respectvely. As the mutual fund strategy and the..d. strategy both requre an ntal learnng phase, we reserve the perod January 1954 to December 1958 (60 months) for ths purpose. Hence, the actual estmaton starts from January 1959 to December 00 (58 months). As we move forward from January 1959 to December 00, the nvestors learn from hstorcal market performance. Hence, an mportant feature of the emprcal methodology s that we allow the nvestors to use all hstorcal data from an expandng wndow from January 1954 to the month precedng the month n whch estmaton s carred out. As a robustness check we also allow nvestors to consder non-expandng, movng estmaton wndows. We dvde the performance evaluaton perod nto three sub-perods, where the frst sub-perod extends from January 1959 to December 1973 (180 months), the second from January 1974 to December 1988 (180 months) and the last from January 1989 to December 00 (168 months). An mportant research desgn ssue s the choce of the forecastng model employed by the mutual fund nvestor. Clearly, the nvestor s faced wth the problem of model uncertanty n real tme. Whle a full Bayesan treatment of model uncertanty s beyond the scope of ths paper, we address ths ssue usng the recursve modelng framework advocated by Pesaran and mmermann (1995). 16 Specfcally, at the start of each month the mutual fund nvestor s allowed to search over the base set of the four predctor varables and use only hstorcal data to choose an optmal forecastng model based on a predefned model selecton crteron. he nvestor chooses one model out of the set of 16 ( 4 ) possble models, that ncludes the..d. model, and uses t to predct the excess stock returns. Whle a number of statstcal model selecton crtera may be used to select the approprate forecastng model, we present results 15

17 based on a recursve utlty maxmzaton crteron that s consstent wth the nvestor s objectve (expressed n equaton (4)) and that takes nto account constrants such as transacton costs and short sales lmts. 17 o mplement ths model selecton rule, the mutual fund nvestor tracks the realzed performance to date of the portfolo allocatons resultng from the use of each of the 16 canddate models, and chooses the model that provdes the maxmum realzed utlty. D. Performance Measures he prmary measure of performance that we use n our analyss s the certanty equvalent rate (CER) of return for a portfolo defned as: where 1 CER = r p λσ p (17) r p denotes the mean realzed return on a portfolo net of transacton costs. he CER of a rsky portfolo represents the rate of return earned wth certanty that would provde the nvestor wth the same utlty as the expected utlty derved from the rsky portfolo. he CER s a wdely used measure of performance n the lterature [see, for example, Kandel and Stambaugh (1996)]. We also report the adjusted Sharpe rato measure proposed by Graham and Harvey (1997). he adjusted Sharpe rato s computed as the dfference between the mutual fund portfolo s return and the return on a volatlty-matched portfolo obtaned by combnng the..d. portfolo wth - blls, over the same evaluaton perod. Specfcally, we frst compute the standard devaton of the mutual fund portfolo over a partcular evaluaton perod. We next lever or unlever the..d. portfolo by combnng t wth -blls n order to match the standard devaton of the mutual fund portfolo. We report the dfference n the two portfolo s returns as the adjusted Sharpe rato. he measure reported here s analogous to the GH1 measure of Graham and Harvey (1997). Intutvely, the dfference represents the gan (or loss) n return from nvestng n the mutual fund portfolo relatve to a passve strategy, for an nvestor wth a target level of volatlty equal to the mutual fund portfolo s volatlty. We note that n our context the CER s the most relevant performance measure as t s consstent wth the nvestor s utlty maxmzaton objectve presented n Equaton (4). 16 We thank the referee for hs or her suggestons n ths regard. 17 We confrm that our results are robust to alternatve model selecton crtera such as the adjusted R, among others. 16

18 We report the dfference n the CER of the mutual fund and the..d. portfolo that we refer to as the CER dfferental, along wth the adjusted Sharpe rato. We assess the sgnfcance of the CER dfferental and the adjusted Sharpe rato usng the statonary bootstrap technque proposed by Polts and Romano (1994). 18 he technque s partcularly suted to weakly dependent statonary tme seres. he statonary bootstrap nvolves re-samplng from the orgnal tme seres of returns usng a block-samplng scheme. Brefly, the steps nvolved n the bootstrap procedure are: (1) randomly select an observaton, say, R t, from the orgnal tme seres, () wth a fxed probablty q, select the next observaton randomly from the orgnal tme seres, and wth probablty (1-q) select t as the next observaton to R t (.e., select R t+1 ) from the orgnal tme seres, (3) repeat ths process to generate a pseudo tme seres of desred length. hs procedure ensures that the data are re-sampled n blocks where the block length has a geometrc dstrbuton wth a mean of 1/q. In our procedure we use a value of q that corresponds to a mean block length of 60 months (5 years). 19 We replcate the above process 1000 tmes. For each such replcaton, we compute the optmal allocatons for each nvestor through 58 months. At every month, the nvestors are allowed to utlze just the nformaton avalable up to that pont n tme. We calculate the dfference n certanty equvalent between the two strateges and the adjusted Sharpe rato for each replcaton. We count the proporton of tmes n 1000 replcatons that these dfferences exceed the certanty equvalent and adjusted Sharpe rato based on the orgnal data for a gven set of results. he sgnfcance of the orgnal dfferences n these statstcs s nferred based upon these emprcal p-values. he bootstrap p-values provde us wth a measure of the statstcal sgnfcance of the relatve performance of the two strateges n the context of repeated samples. We beleve ths s mportant to consder, both for a potental nvestor as well as for the academc communty, snce we have access to only a sngle sample path of the U.S. economy. As emphaszed by Carlson, et al. (001), t s hazardous to base nference on estmates from a sngle sample path. Of course, t s also mportant to pay attenton to the economc magntude of the performance dfferences when makng 18 Sullvan, mmerman and Whte (1999) employ a smlar bootstrap technque to assess the performance of techncal tradng strateges. 19 Our choce of q s consstent wth slow mean-reverson n returns over long horzons documented n Fama and French (1988a), for example. We tested for robustness by varyng the value of q and found that our results are not senstve to our choce of q. 17

19 an nference about the value of the predctor varables beng examned. In our dscusson of the results we pay attenton to both of these aspects of performance evaluaton. 0 We examne the relatve performance of the mutual fund strategy for three sub-perods and n each sub-perod we consder three levels of rsk averson. hs gves us a total of nne subperod/rsk averson permutatons and correspondngly nne observatons on the CER dfferentals. A postve CER dfferental ndcates that the mutual fund strategy outperforms the..d. strategy and vce versa. Instead of basng nference on bootstrap p-values, an alternate way to conduct nference about the performance of the mutual fund strategy s to ask the queston whether postve CER dfferentals are equally lkely as negatve dfferentals. In order to assess ths ssue, we perform a non-parametrc sgn test wth the null hypothess that postve performance s equally lkely as negatve performance. he sgn test s a robust, non-parametrc test, well suted for ths task. Rejecton of the null hypothess would ndcate that the relatve performance of the mutual fund strategy s superor. Yet another way to judge the performance of the mutual fund strategy s to examne the qualty of market tmng t acheves. Superor market tmng ablty s a natural outcome of the mutual fund strategy f the forecastng model and the predctors t reles on have value, and s a necessary condton that t should satsfy. o assess market tmng ablty (mpled by the predctor varables) we allow the mutual fund nvestor to follow a pure swtchng strategy. 1 At the start of each month, he shfts hs portfolo entrely nto stocks or nto the rsk-free asset, based on hs expected return on these assets for the next month. hs s equvalent to assumng rsk neutralty on the part of the nvestor. We restrct ths analyss to the case where the mutual fund nvestor s allowed to nvest n a sngle rsky asset and a rsk-free asset, snce ths settng allows for a more natural market tmng nterpretaton. Perfect market tmng would mean that the nvestor s fully nvested n stocks n up-markets (when the value-weghted stock ndex outperforms the -blls) and fully nvested n -blls n down-markets. Hence, perfect market tmng would requre an nvestor to correctly predct market drecton (up versus down market) and then fully shft hs portfolo from stocks to bonds or vce versa. We measure the percent of 0 We are grateful to the referee for pontng out the need for nterpretng the p-values n the context of the economc sgnfcance of the observed performance dfferences. 1 ypcally, the lterature has tested market tmng ablty usng such a swtchng strategy. See, for example, Pesaran and mmerman (1995). 18

20 months the mutual fund nvestor correctly predcts an up-market and smlarly, the percent of tmes he correctly predcts a down-market. he weghted average gves hs overall market tmng ablty. IV. Prelmnary Analyss A. Descrptve Statstcs able 1 presents descrptve statstcs for monthly stock portfolo excess returns, 1-month -bll yeld, dvdend yeld on a value-weghted market portfolo, 1-month detrended -bll yeld, default spread and the term spread for the perod he mean value-weghted portfolo excess return was 0.55% per month wth a standard devaton of 4.38%. he small cap, medum cap and the large cap portfolo excess returns averaged 0.77%, 0.70% and 0.53%, respectvely. he annualzed log dvdend yeld on the value-weghted market portfolo averaged wth a standard devaton of he summary statstcs reported n the table also nclude the frst order autocorrelatons for each seres. As expected, the dvdend yeld seres s qute persstent wth an autocorrelaton coeffcent of Let us consder the overall sample perod used n the study as well as the three sub-perods we examned.e., , and he frst sub-perod from 1959 to 1973 was characterzed by three economc expansons and two economc contractons, accordng to the Natonal Bureau of Economc Research (NBER). he average monthly equty rsk premum over ths sub-perod was 0.30%. he second sub-perod from 1974 to 1988 was more volatle, consstng of three expansons and three contractons. It ncluded two severe downturns from December 1973/January 1974 to March 1975 (16 months), and August 1981 to November 198 (16 months). he average monthly equty rsk premum over the sub-perod was 0.45%. he fnal sub-perod from 1989 to 00 conssted of a long expanson broken only by two eght-month downturns from August 1990 to March 1991 and from Aprl 001 to November 001. It ncluded the longest post-war expanson n our sample, spannng Aprl 1991 to March 001 (10 months). he average monthly equty rsk premum over the perod was 0.56%. In summary, the frst sub-perod ( ) was a perod of low to moderate expanson coupled wth low volatlty, the second perod ( ) was a perod of hgh volatlty and moderate 19

21 economc actvty, and the fnal perod ( ) was characterzed by strong expanson and hgh equty premum. B. In-sample Regresson Evdence Our framework allows for four predctor varables: dvdend yeld, -Bll yeld, default spread and term spread. Our mutual fund nvestor chooses the model for predctng the next perod returns from the 16 ( 4 ) canddate models gven by these four predctor varables. hese four busness cycle related predctor varables have been proposed and tested n the lterature by varous researchers who have found them to be of hgh predctve value for stock returns. In order to allow for comparsons to prevous results n the lterature and for the sake of completeness, we present evdence on predctve regressons usng these varables. We estmate each of the 16 regresson models, labeled A through P, over the sample perod for the value-weghted market portfolo of all NYSE, AMEX and NASDAQ stocks. hese results appear n able. Note that model P s smply the..d. model. he table presents the estmated coeffcents, the correspondng t-statstcs based on the Newey-West covarance matrx, and the adjusted R values for each regresson. Dvdend yeld appears n 8 of the 16 regressons ndvdually and n combnaton. Its coeffcent s sgnfcant at the 5% level n 6 of the 8 regressons and at the 10% level for the unvarate regresson wth an R of 0.58%. he coeffcent for -bll yeld s negatve and sgnfcant at the 1% level for all 8 regressons n whch t appears. he coeffcent s wth a t-statstc of n the unvarate regresson wth an R of.99%. In the multvarate regresson wth all varables (Model O), the coeffcent s wth a t-statstc of Interestngly, default spread s nsgnfcant over the full sample perod n all the 8 regressons t appears n, ncludng the unvarate regresson. Of course, there s every reason to beleve default spread s sgnfcant n sub-samples across tme and across specfc portfolos. he term spread s sgnfcant n 4 of the 8 regressons. It s sgnfcant n the unvarate regresson wth a coeffcent of and a t-statstc of.48. hs s not true n the multvarate Bases n predctve regressons nvolvng stock returns have been analyzed by Stambuagh (1999), and Ferson, Sarksson, and Smn (00), among others. 0

22 regresson wth all varables (Model O) where t s nsgnfcant wth a coeffcent of and a t-statstc of 0.5. In the unvarate regressons, three varables out of four are sgnfcant at conventonal levels, the only excepton beng default spread. In the multvarate regresson wth all varables,.e., Model O, only dvdend yeld and -Bll yeld are sgnfcant. V. Out-of-sample results In ths secton we present our man results relatng to the out-of-sample performance of the mutual fund and the..d. strateges. A. Portfolo Performance Panel A of able 3 presents the performance of the..d. and mutual fund strateges when both strateges adjust allocatons every month between the rskfree asset and an equty portfolo. Performance results are presented for the three sub-perods January 1959 December 73, January 1974 December 88, January 1989 December 00, and for the overall sample perod January 1959 December 00, for three coeffcents of relatve rsk averson (, 5 and 10). We report the dfferences n the average return and certanty equvalent rate of return between the mutual fund strategy and the..d. strategy. We also report the adjusted Sharpe rato for the mutual fund strategy. For both the certanty equvalent rate of return and the adjusted Sharpe rato, we report the bootstrapped p-values. It s clear from the results n Panel A that the relatve performance of the mutual fund strategy s unstable across sub-perods. he results are generally postve for the and sub-perods, and generally negatve for the last sub-perod, Also, the results n the thrd sub-perod suggest that an aggressve mutual fund strategy s more lkely to fal, compared to a conservatve strategy. For the three sub-perods and for the three rsk-averson cases (.e., for the nne possble sub-perod/rsk averson permutatons), judgng by the bootstrap p-values, the dfferences n performance of the two strateges are not sgnfcant at the conventonal levels n each case. Notwthstandng the p-values, the mutual fund strategy tends to outperform the..d. strategy n the frst two sub-perods and conversely, the..d. strategy generally outperforms the mutual fund 1

23 strategy n the last sub-perod. For the full sample perod , the mutual fund strategy does better than the..d. strategy for the low and hgh rsk-averson cases. he CER dfferences and the adjusted Sharpe ratos are postve n sx of the nne cases. We subject ths hypothess to a non-parametrc sgn test. he test s that the fndng of sx out of nne postve performances s consstent wth the null hypothess that postve performance s equally lkely as negatve performance. 3 he sgn test yelds a p-value of 0.539, falng to reject the null hypothess at conventonal levels of sgnfcance. he bootstrap p-values provde us wth a measure of the statstcal sgnfcance of the dfference n performance between the mutual fund and the..d. strateges n the context of repeated samples. However, as we stated earler, we are also nterested n the economc sgnfcance of the performance dfference. In the low rsk averson case, for the full sample perod of 44 years, the monthly dfference n CER between the mutual fund and the..d. strategy s 37 bps per month that s clearly sgnfcant n economc terms. However, consder the hgh rsk averson case where the monthly CER dfferental of 4 bass ponts has to be vewed n context of the correspondng bootstrap p-value suggestng that a smlar mprovement n performance could have been obtaned wth a 74.4% probablty even f the data had no predctablty but had a smlar correlaton structure as the observed returns n the economy. It would be safe to say that the mutual fund nvestor would balk at gvng up much utlty to get access to the predctor varables f there was only a 5.6% level of confdence n outperformng the..d. nvestor. Panel B reports results for the..d. and mutual fund strateges when the strateges adjust allocatons every month between the rskfree asset and three equty portfolos. he results are qualtatvely smlar to the sngle rsky asset case. here contnues to be nstablty across the three sub-perods. In the frst sub-perod the mutual fund strategy tends to outperform the..d. strategy whle n the thrd sub-perod t tends to underperform. For the full sample perod, as before, n the low rsk averson case the mutual fund strategy enjoys an monthly CER performance advantage of 38 bass ponts whle n the other two rsk averson cases the dfference n performance s neglgble. 3 he mplct assumpton that we make s that there are nne ndependent draws of whch sx are postve.