TIME-VARYING SHARPE RATIOS AND MARKET TIMING

Transcription

1 TIME-VARYING SHARPE RATIOS AND MARKET TIMING Yi Tang a and Rober F. Whielaw b* Curren version: Augus 20 Absrac This paper documens predicable ime-variaion in sock marke Sharpe raios. Predeermined financial variables are used o esimae boh he condiional mean and volailiy of equiy reurns, and hese momens are combined o esimae he condiional Sharpe raio, or he Sharpe raio is esimaed direcly as a linear funcion of hese same variables. In sample, esimaed condiional Sharpe raios show subsanial ime-variaion ha coincides wih he phases of he business cycle. Generally, Sharpe raios are low a he peak of he cycle and high a he rough. In an ouof-sample analysis, using 0-year rolling regressions, relaively naive marke-iming sraegies ha exploi his predicabiliy can idenify periods wih Sharpe raios more han 45% larger han he full sample value. In spie of he well-known predicabiliy of volailiy and he more conroversial forecasabiliy of reurns, i is he laer facor ha accouns primarily for boh he in-sample and ou-of-sample resuls. * a Fordham Universiy, School of Business; b New York Universiy, Sern School of Business and NBER. Thanks o Kobi Boudoukh, Michael Brennan, Ravi Jagannahan, and Ma Richardson for helpful commens. Conac: Prof. R. Whielaw, NYU, Sern School of Business, 44 W. 4 h S., Suie 9-90, New York, NY 002; rwhiela@sern.nyu.edu.

2 . Inroducion The empirical lieraure conains a wealh of evidence on predicable variaion in he mean and volailiy of equiy reurns. Given he apparen join predicabiliy of he mean and volailiy, i is perhaps somewha surprising ha he lieraure has been relaively silen on predicable variaion in equiy marke Sharpe raios. 2 Of course, predicable variaion in he individual momens does no imply predicable variaion in he Sharpe raio. The key quesion is wheher hese momens move ogeher, leading o Sharpe raios which are more sable and poenially less predicable han he wo componens individually. The empirical evidence on his issue is somewha mixed. Earlier work (e.g., French, Schwer and Sambaugh (987)) suggess a weak posiive relaion beween expeced reurns and volailiy. However, oher sudies (e.g., Glosen, Jagannahan, and Runkle (993), Whielaw (994), and Boudoukh, Richardson, and Whielaw (997)) appear o uncover a more complex relaion. Specifically, on an uncondiional basis, several empirical specificaions, including a modified GARCH-M model and nonparameric kernel esimaion, sugges a negaive relaion beween he condiional mean and volailiy of reurns. This evidence would indicae he likelihood of subsanial predicable variaion in marke Sharpe raios. Time-variaion in sock marke Sharpe raios is of ineres for a number of reasons. Firs he Sharpe raio is popular for performance evaluaion in an asse managemen conex, and he uncondiional Sharpe raio of he marke is ofen used as a convenien benchmark. If his raio shows subsanial predicable variaion, hen his variaion needs o be accouned for when using he marke as a performance benchmark. Second, ime-variaion in he marke Sharpe raio migh provide clues o he fundamenal economics underlying he economy and asse pricing. For example, he Sharpe raio could indicae he iming and magniude of flucuaions of risk aversion in a represenaive agen framework. Third, mean-variance invesors would have an obvious ineres in predicable Sharpe raios as his variaion could poenially lead o opimal rading sraegies ha differ markedly from simple buy-andhold sraegies. For example, in a parial equilibrium seing, he Sharpe raio deermines he fracion of wealh ha an agen invess in he risky marke porfolio. 3 This paper provides an empirical invesigaion of ime-variaion in monhly, equiy marke Sharpe raios over he sample period May 953 o December 200. We employ wo differen, bu relaed, See, for example, Breen, Glosen, and Jagannahan (989), Fama and French (989), Kandel and Sambaugh (990), Keim and Sambaugh (986), and Schwer (989). 2 An excepion is Kandel and Sambaugh (990), who invesigae he implicaions of condiional momens of consumpion growh for he price of risk over one-quarer and five-year horizons. 3 See, for example, Kandel and Sambaugh (996), who focus on he predicabiliy of expeced reurns and is effec on asse allocaion, and Fleming, Kirby, and Osdiek (200), who focus on he value of informaion abou imevarying variances and covariances of reurns.

3 mehodologies o come up wih four differen esimaed condiional Sharpe raios. In he firs mehodology, he condiional mean and volailiy of equiy reurns are modeled as linear funcions of he four predeermined financial variables used in Whielaw (994) plus lagged realized volailiy. The raio of hese momens provides he esimae of he condiional Sharpe raio. In order o decompose he predicabiliy of he Sharpe raio, we also consider wo special cases of he above esimae in which eiher he mean or volailiy is fixed a i is uncondiional average, and he oher momen is allowed o vary over ime. The second mehodology esimaes he condiional Sharpe raio direcly by projecing he monhly realized Sharpe raio on o he same se of five variables. We examine he forecasing power of hese esimaes on boh an in-sample and ou-of-sample basis. In sample, esimaed condiional Sharpe raios exhibi subsanial ime-variaion, wih monhly values generally in he range of -0.2 o 0.9. Esimaes from he raio of he condiional momens exhibi similar paerns o hose from direc esimaion of he Sharpe raio, wih a correlaion beween he wo series of approximaely 0.9, bu he laer are subsanially higher due o he negaive correlaion beween reurns and volailiy. Ineresingly, he fixed-volailiy esimae racks he unconsrained esimaes closely, while he fixed-mean esimae is much less variable and less correlaed wih is counerpars ha allow expeced reurns o vary over ime. Thus, hese unconsrained esimaes are clearly driven primarily by variaion in he condiional expeced reurn. The unconsrained condiional Sharpe raios vary wih he business cycle, peaking a business cycle roughs and declining over he course of he expansionary phase of he cycle. For example, he average change in he condiional Sharpe raio, esimaed as he raio of he condiional momens of reurns, beween he peak of he cycle and he subsequen rough is 0.25 (more han 0.85 on an annualized basis); herefore, he esimaed risk-reurn radeoff is much more favorable afer recessions. Regression of realized Sharpe raios on he esimaes shows subsanial predicive power, bu he condiional unbiasedness of he forecass can generally be rejeced. On an ou-of-sample basis, using l0-year rolling regressions, esimaed condiional Sharpe raios again show saisically and economically significan predicive power for realized Sharpe raios in spie of he fac ha he rolling coefficiens exhibi subsanial insabiliy. However, he slope coefficiens in regressions of he realized raio on he forecass are much lower han for he in-sample exercise and are much lower han one, indicaing subsanial overfiing. Neverheless, he forecass provide valuable informaion in he conex of a relaively naive marke-iming sraegy, significanly ouperforming a buyand-hold sraegy. These acive rading sraegies involve swiching beween he marke and he risk-free asse depending on he level of he esimaed Sharpe raio relaive o a specific hreshold. The average realized Sharpe raios of he monhs in he marke are compared o ha of he buy-and-hold sraegy, and hey exhibi improvemens of as much as 45%. Using ex pos Sharpe raios compued using monhly 2

4 reurns, hese improvemens are even more impressive, ranging up o 50%. Of equal imporance, as he hreshold for he condiional Sharpe raio increases, so does he Sharpe raio of he acive sraegy. Ineresingly, forecass based on he raio of he condiional momens appear o perform he bes, and here is lile difference beween he forecas ha models volailiy wih he full se of condiioning variables and he one ha uses simply he average volailiy over he prior 0 years. This resul complemens resuls showing he economic value of volailiy iming in an asse allocaion conex (e.g., Fleming, Kirby, and Osdiek (200)). There are wo possible inerpreaions of hese resuls. Firs, hey could be a produc of mispricing, perhaps induced by flucuaions in consumer senimen associaed wih business cycle flucuaions. In oher words, a he peak of he cycle, over-opimisic invesors could be overpricing socks leading o poor fuure radeoffs beween risk and reurn, wih he reverse happening a he rough of he cycle. If so, his paper provides a framework for analyzing and exploiing his inefficiency in order o generae superior performance. Second, he empirical resuls could be due o raional, bu no perfecly correlaed, imevariaion in he condiional momens of reurns. For example, if aggregae risk aversion changes over he cycle, decreasing during expansions and increasing during recessions, as migh be implied by a habi model such as Campbell and Cochrane (999), ha would produce resuls generally in line wih hose repored in he paper. Alernaively, he world may resemble he ICAPM framework of Meron (973), where ime-varying invesmen opporuniies generae an addiional source of priced risk. For example, Whielaw (2000) develops an equilibrium model in which he mean and volailiy of marke reurns do no move ogeher, implying subsanial, raional, ime-variaion in sock marke Sharpe raios due o imevarying probabiliies of regime swiches. The remainder of he paper is organized as follows. Secion 2 provides a heoreical discussion and a seing in which o inerpre ime-varying Sharpe raios. Secion 3 inroduces he esimaion mehodology and documens he economic and saisical significance of in-sample ime-variaion in sock marke Sharpe raios. In Secion 4, he ou-of-sample analysis is performed, and he performance of sylized marke-iming sraegies is examined. Secion 5 concludes. 2. Theoreical Background Harrison and Kreps (979) show ha he absence of arbirage implies he exisence of a pricing kernel, or sochasic discoun facor (M ) ha prices all asses. Specifically, he expeced value of he produc of he pricing kernel and he gross asse reurn (R ) mus equal uniy, i.e., E [ M R ], () + + = where E is he expecaion based on informaion available a ime. Applying equaion (), he -period 3

5 risk-free ineres rae (R f ) can be can be wrien as he inverse of he expecaion of he pricing kernel R. (2) + = E[ M + ] Equaion () also implies ha he expeced risk premium on any asse is proporional o he condiional covariance of is reurn wih he pricing kernel, i.e., E R R ] = R cov [ M R ], (3) [ + f f + + where cov is he covariance condiional on informaion available a ime. Consequenly, he condiional Sharpe raio of any asse, defined as he raio of he condiional mean excess reurn o he condiional sandard deviaion of his reurn, can be wrien in erms of he volailiy of he pricing kernel and he correlaion beween he pricing kernel and he reurn: E[ R σ [ R + + R R f f ] R = ] f cov [ M σ [ R ] + + R + ] = R σ [ M f + ]corr [ M + R + ], (4) where σ and corr are he sandard deviaion and correlaion, condiional on informaion a ime, respecively. Denoing he condiional Sharpe raio of he sock marke a ime by S, equaion (4) shows ha his raio is proporional o he produc of he volailiy of he pricing kernel and he correlaion beween he pricing kernel and he reurn on he marke (R m ) S E[ R σ [ R m+ m+ R R f f ] = R ] σ [ M f + ]corr [ M + R m+ ], (5) Inuiively, if he Sharpe raio varies subsanially over ime, hen his variaion is aribuable o variaion in he condiional volailiy or condiional correlaion. Noe ha R f is he gross, risk-free rae, which has varied beween.00 and.02 for monhly, U.S. daa. Consider firs he condiional correlaion in equaion (5). The implicaions for ime-variaion in he Sharpe raio depend criically on he modeling of he pricing kernel. One approach is o specify M as a funcion of asse reurns. For example, modeling he pricing kernel as a linear funcion of he marke reurn produces he condiional CAPM. Risk aversion implies a negaive coefficien on he marke reurn; herefore, he correlaion is - and he marke Sharpe raio is approximaely consan over ime. Alernaively, modeling he discoun facor as a quadraic funcion of he marke reurn gives he condiional hree-momen CAPM, firs proposed by Kraus and Lizenberger (976). 4 This specificaion allows for some ime-variaion in marke Sharpe raios due o he pricing of skewness risk, bu he correlaion will sill be pushed owards -. Bansal and Viswanahan (993) esimae he pricing kernel as 4 Noe ha hese models can also be derived by imposing resricions on a represenaive agen's uiliy funcion. Harvey and Siddique (2000) provide an empirical invesigaion of hese specificaions and a more deailed discussion of he underlying heory. 4

6 a general, nonlinear funcion of he marke reurn, bu again ime-variaion in he correlaion is limied by a specificaion which relies on variaion in marke reurns o proxy for variaion in he discoun facor. A slighly differen approach o generalizing he one-facor, condiional CAPM, wihou abandoning a linear specificaion, is o model he pricing kernel as a linear funcion of muliple asse reurns. Based on explanaory and predicive power, a number of addiional facors, including small firm reurns and reurn spreads beween long-erm and shor-erm bonds, have been proposed and esed. 5 However, as above, correlaions beween he discoun facor and he marke reurn end o be relaively sable, implying sabiliy in he sock marke Sharpe raio. A second branch of he lieraure, using resuls from a represenaive agen, exchange economy (Lucas (978)), models he pricing kernel as he marginal rae of subsiuion over consumpion. The resuling consumpion CAPM has been analyzed and esed in numerous conexs. 6 While he consumpion CAPM lieraure is voluminous, here has been relaively lile aenion paid o he implicaions of he model for he risk/reurn relaion. 7 Inuiively, when he marginal rae of subsiuion depends on consumpion growh and he sock marke is modeled as a claim on aggregae consumpion, one migh expec he correlaion and he Sharpe raio o be relaively sable. In fac, Whielaw (2000) shows ha his resul holds when consumpion growh follows an auoregressive process. However, in a wo-regime model, where mean consumpion growh differs across he regimes and he probabiliies of regime shifs are ime-varying, his inuiion is overurned. In his seing, he mean and volailiy of marke reurns can be negaively correlaed. Alhough he magniude and variaion of he marke Sharpe raio are no invesigaed, i is clear ha he model implies economically significan ime-variaion. I is imporan o noe ha he regimes correspond loosely o he expansionary and conracionary phases of he business cycle. Moreover, volailiy and expeced reurns a any poin in ime depend criically on he probabiliy of a regime shif. Consequenly, his model predics business cycle relaed variaion in Sharpe raios and large movemens around ransiions beween he phases of he cycle. A differen approach is o modify he preferences of he represenaive agen. In he exernal habi model of Campbell and Cochrane (999), uiliy depends on he deviaion of consumpion from a reference level. As consumpion falls owards his reference level, for example during a recession, he effecive risk aversion of he agen can increase dramaically, hus increasing he volailiy of he pricing 5 Among he numerous papers on his opic are Campbell (987), Chen, Roll, and Ross (986), and Fama and French (993). 6 See, for example, Hansen and Singleon (983), Breeden, Gibbons, and Lizenberger (989), and Ferson and Harvey (992). 7 One excepion is Kandel and Sambaugh (990), who consruc a four-sae model of he mean and volailiy of consumpion growh. In heir framework, he price of risk shows business cycle variaion a long horizons due o variaion in invesmen opporuniies. 5

7 kernel in equaion (5). Through his ype of mechanism, ime-varying risk aversion can creae business cycle variaion in he Sharpe raio. 3. Time-Variaion in Marke Sharpe Raios In his secion we firs provide a brief descripion of he daa, and we hen urn o an explanaion of he mehodologies for esimaing he condiional Sharpe raio of marke reurns. The following subsecion presens he in-sample empirical resuls. 3.. The Daa For his analysis, boh he mean and volailiy of sock marke reurns are esimaed as funcions of predeermined financial variables. Four variables he Baa-Aaa spread, he commercial paper-treasury spread, he one-year Treasury yield, and he dividend yield are chosen based on heir proven predicive power in earlier sudies. The Baa-Aaa spread (DEF), he commercial paper-treasury spread (CP), and he one-year Treasury yield (YR) are obained from he Federal Reserve Saisical Release. Daa on he dividend-price raio (DIV) are available on Rober Shiller s websie. 8 All daa are monhly and cover he period April 953 o November 200. In addiion o he four financial variables, he analysis uses monhly and daily reurns on he value-weighed marke porfolio from he CRSP daa files from April 953 o December 200. Monhly excess reurns are calculaed by subracing he monhly yield on a 3-monh T-bill from he corresponding sock reurn. The 3-monh yield is used insead of he -monh yield because of he well-documened idiosyncrasies in his laer ime series (see Duffee (996)) Esimaing Sharpe Raios The firs sep in he analysis is o deermine if here is significan ime-variaion in monhly esimaed condiional and realized Sharpe raios, and, furher, o see if he variaion in hese wo series coincides. To compue he realized Sharpe raio, we firs calculae he realized volailiy on a monhly basis using he sum of squared daily reurns wihin he monh: N v ` = R n,, (6) n= where v is he realized volailiy for monh and R n, are daily reurns on he VW porfolio wihin he monh. Adjusing for he daily, wihin-monh mean reurn or subracing he daily risk-free rae has no 8 hp:// 6

8 meaningful effec on he resuls. The realized volailiy series is winsorized a he 99 h percenile o eliminae a few obvious ouliers. For example, he monhly realized volailiies in Oc. 987 and Oc exceed 23% (approximaely 80% on an annualized basis). Moreover, hese observaions are more han eigh sandard deviaions away from he mean of he series. The monhly realized Sharpe raio for monh is hen compued as S R R f,, (7) v where R is he monhly reurn on he VW porfolio and R f, is he corresponding monhly, risk-free rae a he beginning of he monh. We esimae he condiional Sharpe raio using wo differen mehodologies (i) we esimae he firs wo condiional momens of reurns separaely and hen ake he raio, and (ii) we esimae he condiional Sharpe raio direcly using a regression wih he realized Sharpe raio as he dependen variable. For he firs mehodology, expeced reurns are esimaed by regressing excess reurns on a vecor of predeermined variables, and he condiional volailiy is esimaed by projecing realized volailiy on o he same se of variables. Specifically, he condiional momens are modeled as follows: where E [ R + R = X β, (8) f ] SD [ R = β ] + X 2, (9) SD is he condiional sandard deviaion, and he corresponding regressions are specified as R v R X β ε. (0) + f = = X 2 + ε 2+ β. () The condiioning variables are chosen based on he resuls in Whielaw (994), wih he addiion of lagged realized volailiy as a condiioning variable in boh equaions. Specifically, we regress reurns and realized volailiy on he Baa-Aaa spread (denoed DEF for defaul spread), he dividend yield (DIV), he one-year yield (YR), he commercial paper-treasury spread (CP), and lagged realized volailiy ( v ). Fied values from equaions (0)-(), can be used o compue condiional Sharpe raios for each monh. Specifically, based on informaion available a ime and parameer esimaes from he esimaion, he esimaed condiional Sharpe raio is ˆ ˆ E[ R+ R f ] X β S, = =, (2) SD [ R ] X ˆ β where he subscrip on he Sharpe raio is o disinguish i from oher esimaes o be defined laer

9 Two alernaive esimaes of he condiional Sharpe raio in equaion (2) shu down variaion in one of he wo condiional momens of reurns, i.e., Sˆ R R + f 2, = X ˆ β 2 Sˆ 3, = X v ˆ + (3) β. (4) The condiional momen is replaced wih he uncondiional, in-sample mean of eiher excess reurns or realized volailiy. This esimae is equivalen o replacing he vecor of condiioning variables in he regressions in equaions (0)-() wih jus a consan. In an ou-of-sample conex, hese alernaives migh ouperform he more general specificaion due o overfiing or esimaion error. In sample, hey provide a decomposiion of he esimaed Sharpe raio ino is consiuens. Our second approach o esimaing he condiional Sharpe raio is o regress he realized Sharpe raio from equaion (7) on he same se of predeermined variables used o esimae he condiional momens of reurns above: S + = X 3 + ε 3+ The corresponding fied condiional Sharpe raio is 4, X β. (5) S ˆ = βˆ, (6) Thus, we now have four esimaes in oal (equaions (2)-(4) and (6)) Empirical Resuls Table, Panel A presens monhly resuls for he full sample period, May 953 o December 200 from he esimaion of he condiional momens of reurns based on he regressions in equaions (0)-(). The resuls are broadly consisen wih hose repored in he lieraure. Boh he dividend yield and he -year Treasury rae are significan predicors of he marke reurn a he % level. Lagged realized volailiy has a negaive, albei small and saisically insignifican, coefficien. To he exen ha his variable is a proxy for condiional volailiy, his resul coincides wih he inabiliy of many sudies o find a significanly posiive risk-reurn relaion a he marke level. 9 The R-squared of slighly less han 3% is lower han ha repored in some previous sudies, bu he sample, which includes he recen financial crisis, may accoun for his resul. (We examine his quesion in more deail below.) In he volailiy equaion, lagged realized volailiy, he defaul spread, he dividend yield, and he 9 See, for example, Guo and Whielaw (2006) for an exended discussion and possible resoluion of his puzzle. 8

10 commercial paper-treasury spread are all highly significan. The significan posiive serial correlaion in realized volailiy is a manifesaion of he auoregressive condiional heeroskedasiciy of monhly reurns. Realized volailiy is much more predicable han reurns, wih an R-squared exceeding 50%. To examine he influence of he financial crisis on he regression resuls, we re-esimae he same specificaion ending he sample in December 2007, wih he resuls repored in Table, Panel B. For his shorer sample, he coefficiens are somewha similar, bu he defaul spread is now significan in he mean equaion, and he -year Treasury rae is significan a he 0% level in he volailiy equaion. Of perhaps greaer ineres, he R-squared in he mean equaion increases o more han 4%. For he majoriy of he resuls ha follow, we use he full sample, bu i is imporan o keep in mind ha explanaory power is reduced by he exreme and unpredicable variaion of reurns associaed wih he crisis and is afermah. There is some concern in he lieraure abou srucural insabiliy of he mean reurn regression in Table (e.g., Welch and Goyal (2008)). In paricular, variables such as he dividend yield appear o exhibi nonsaionariy associaed wih srucural shifs during he sample period, which may, in urn, accoun for insabiliy in he regression coefficiens esimaed over shorer subsamples. Leau and Van Nieuwerburgh (2008) invesigae his issue and provide evidence of a shif in he mean of he dividend yield o a lower level afer 99. To incorporae his evidence, we consruc a new independen variable which is he dividend yield adjused for is mean wihin he wo subsamples up o an including 99 and he period hereafer. Table, Panel C repors resuls for he regression over he full sample using his adjused dividend yield (DIV-ADJ). The R-squared in he mean equaion is somewha higher han in he corresponding regression wih he unadjused dividend yield in Panel A, hus we chose his specificaion as our baseline, in-sample esimaion. The heoreical jusificaion for he adjusmen above would depend on he underlying mechanism behind he srucural shif in he dividend yield. For example, if dividend yields declined because of a decline in risk aversion and a corresponding decline in he required risk premium (in a represenaive agen seing), hen such an adjusmen would be inappropriae. The declining dividend yield would reflec a decline in expeced reurns. However, in his case, he adjusmen should no increase he explanaory power of he regression. Alernaively, he decline in he dividend yield could reflec a swich in corporae payou policy in favor of sock repurchases. Boudoukh e al. (2007) examine his issue in grea deail and presen evidence ha adjusing for sock repurchases does increase he explanaory power of predicive regressions. As confirmaion of his evidence Table, Panel D presens resuls for regressions wih he dividend plus repurchase yield (DIV+REP) as he dependen variable. Specifically, 9

11 we use he annual dividend plus repurchase yield from Michael Robers websie 0 o compue jus he repurchase yield, i.e., he amoun of sock repurchases divided by he marke capializaion, a he marke level. For each monh in our sample, we ake a weighed average of he relevan annual repurchase yields and add he number o our dividend yield series. For example, for he 2-monh dividend yield ha goes hrough March 997, we add in ¾ of he 996 repurchase yield and ¼ of he 997 repurchase yield. In erms of R-squared, he resuls are a sligh improvemen over hose in Panel C, which used he saisically adjused dividend yield. Of greaer imporance, he essenial implicaions of his analysis and he in-sample Sharpe raio analysis ha follows are insensiive o he precise specificaion. Of course, all hese resuls mus be considered in he conex of he well known problem wih daa snooping (see, for example, Foser, Smih, and Whaley (997)). These concerns are miigaed somewha by he fac ha he predicor variables have been used in he lieraure for wo decades, or more, and by he ou-of-sample exercise conduced laer in he paper. Table 2 presens resuls of our second esimaion mehodology, i.e., direc esimaion of he condiional Sharpe raio, using he adjused dividend yield described above for boh he full sample and pre-crisis sample, alhough resuls for he original dividend yield series and he dividend plus repurchase yield are similar. For boh samples, he dividend yield and one-year Treasury rae are highly significan. The posiive coefficien on he dividend yield is consisen wih he fac ha his variable posiively predics he reurn and negaively predics volailiy (see Table ). The negaive coefficien on he oneyear ineres rae reflecs is negaive relaion o reurns. The posiive, albei saisically insignifican, signs of he coefficiens on he defaul spread and commercial paper-treasury spread are more difficul o reconcile wih he earlier resuls given he posiive relaion beween hese variables and volailiy. Finally, for he full sample, lagged realized volailiy is a significan negaive predicor of he Sharpe raio a he 5% level, which is consisen wih he posiive serial correlaion in realized volailiy documened above. Table 3 provides descripive saisics for he four esimaed condiional Sharpe raio series, for he realized Sharpe raio, and for he firs wo condiional momens of reurns. There are several noable resuls. Firs, variaion in he esimaed condiional mean appears o dominae he variaion in he esimaed condiional Sharpe raios. The condiional mean is highly correlaed wih he condiional Sharpe raio esimaes in which boh momens are allowed o vary, eiher explicily or implicily (#, he raio of he condiional momens, and #4, he direc esimaion), bu he magniudes of he correlaions beween hese series and he condiional volailiy are much lower. Moreover, boh he esimae using he raio of he condiional momens (#) and he direcly esimaed condiional Sharpe raio (#4) are highly 0 hp://finance.wharon.upenn.edu/~mrrober/ The annual dividend plus repurchase yield series ends in 2003, so we assume a consan repurchase yield hereafer. 0

12 correlaed wih he esimae in which he volailiy is fixed a is sample average (#3), wih correlaions of approximaely 0.9. In conras, hese same series have correlaions of less han 0.5 wih he esimae in which he mean is fixed a is sample average (#2). This resul is no an arifac of he relaive variaion of he wo condiional momens, since he condiional volailiy has a subsanially higher sandard deviaion. Of course, hese are in-sample resuls. From an ou-of-sample perspecive, his phenomenon migh be bad news if he in-sample explanaory power is due o overfiing of he condiional mean. Second, all four esimaed condiional Sharpe raio series are posiively correlaed wih he realized Sharpe raio. For he hree esimaes ha allow for variaion in he condiional mean, hese correlaions are close o 0.2, somewha more han wice he magniude of he correlaion wih he esimae ha fixes he condiional mean (#2). Perhaps surprisingly, he correlaion for he esimae ha fixes he condiional volailiy (#3) is acually marginally higher han ha for he esimae ha allows boh momens o vary over ime (#). These correlaions are quie high given he sampling variaion in he realized Sharpe raio. This variaion, due in large par o he unexpeced componens of reurns and volailiy, is apparen in he high sandard deviaion of he realized Sharpe raio and is low firs order auocorrelaion. Third, he means of he Sharpe raio esimaes calculaed from he raio of he individual momens (#, #2, and #3) are approximaely half as larger as he mean of eiher he realized Sharpe raio or he direc esimae. 2 There is a Jensen s inequaliy effec, bu he primary reason for he difference lies in he ime series properies of reurns and volailiy. Specifically, he reurn and realized volailiy series are relaively srongly negaively correlaed, wih a correlaion of -0.27, a phenomenon ha has been called he leverage effec. 3 These series hemselves are made up of wo componens an expeced componen, refleced in he condiional momens series, and shocks. From Table 3, he correlaion beween he condiional momens is This phenomenon is he well known, and anomalous, negaive risk-reurn relaion a he aggregae level. 4 I is his negaive correlaion ha causes he mean of he raio of hese momens (#) o exceed he raio of he means of he wo momens, i.e., 0.8 > 0.55/3.60 = 0.5. However, his effec is even sronger for he realized Sharpe raio because here is subsanially more variaion in realized reurns and volailiy han in expeced reurns and condiional volailiy. In addiion, he shocks o he wo momens are more negaively correlaed han he momens, wih a correlaion beween he residuals from equaions (0)-() of This negaive correlaion beween unexpeced 2 The means of hese laer wo series coincide by consrucion, i.e., he average fied value equals he mean of he dependen variable in a regression conex. 3 If equiy is a levered claim on underlying asses, a negaive shock o he value of he asses, and herefore he equiy, causes an increase in he volailiy of he equiy as i becomes more levered, for a fixed deb claim. 4 See, for example, Whielaw (994) for a deailed discussion of his issue in a similar framework o ha used in his paper.

13 reurns and shocks o volailiy is consisen wih a volailiy feedback effec, which, in urn, is consisen wih a posiive risk-reurn radeoff. 5 A more formal way o address he relaion beween he esimaed condiional Sharpe raios and he realized Sharpe raio is o run a regression of he laer on he former: S α + β ˆ η, (7) + = S i, + i, + where i indexes he esimaed condiional Sharpe raio series. Running his regression for he direc esimaion (#4) is meaningless, since he esimaed condiional Sharpe raio comes from a regression of he realized raio on a se of predeermined variables. Therefore, such a regression will generae an inercep of zero, a slope coefficien of, and an R-squared equal o ha of he original regression. More generally, α = 0 and β = are he condiions for he forecas o be condiionally unbiased. Table 4 presens esimaion resuls for he regression in equaion (7) for he hree series consruced as he raio of he momens. The regressions are esimaed over he full sample, including he financial crisis, and he forecass are consruced using he adjused dividend yield, alhough, as before, using an alernaive dividend yield series generaes qualiaively similar inferences. The key resuls are hreefold. Firs, all he esimaes have saisically significan forecasing power for he realized Sharpe raio in ha he hypohesis ha he slope is zero can be rejeced in all cases. Moreover, i is no possible o rejec he hypohesis ha he slope is equal o one in each case. Second, condiional unbiasedness can be rejeced for series # and #3 due o he saisically significan inercep in he regression. This resul is hardly surprising given he differences in means beween he realized Sharpe raio and he forecass as documened in Table 3. Finally he R-squareds are relaively small, bu, wih he excepion of he fixed mean esimae, hey are comparable o he R-squared from he direc esimaion (Table 2). In erms of insample explanaory power, fixing he mean has serious negaive consequences, whereas fixing he variance has, if anyhing, a small posiive effec on he forecasing. Given he significan and predicable ime-variaion in sock marke Sharpe raios documened above, a naural quesion is how movemens in esimaed Sharpe raios correspond o flucuaions in economic aciviy. 6 Figure shows he esimaed condiional Sharpe raios and he NBER business cycle peaks and roughs (recessions, i.e., he peak-o-rough phases of he cycle, are marked by he shaded bars). There are only nine complee business cycles wihin he sample period; herefore, conclusions should be drawn wih cauion. Neverheless, here appears o be sriking cyclical variaion in Sharpe raios. Almos wihou excepion, business cycle peaks correspond o low Sharpe raios and business 5 See, for example, Guo and Whielaw (2006) and Smih (2007) for illusraions of idenifying he risk-reurn radeoff using he volailiy feedback effec. 6 Time-variaions in boh expeced reurns and volailiy have been previously linked o he business cycle. See, for example, Fama and French (989) and Schwer (989). 2

14 cycle roughs o high Sharpe raios. The las wo columns in he op panel of Table 3 provide one quanificaion of his phenomenon. We calculae he difference beween he condiional Sharpe raio a he peak of he cycle and he subsequen rough, and hen average hese differences across he cycles in he sample. This average is a measure of how much he Sharpe raio changes during he course of a recession. We also perform he same calculaion from rough o peak, bu, by consrucion, his average difference is approximaely he negaive of he change from peak o rough. The ime from peak o rough (i.e., he conracionary phase of he cycle) is shor, averaging less han 2 monhs, bu he average increase in he monhly esimaed rolling Sharpe raio is large, on he order of 0.25 for our unconsrained esimaors (# and #4), which would be greaer han 0.8 on an annualized basis. Ineresingly, boh he condiional mean and volailiy increases from peak o rough, bu he former effec dominaes. The daa indicae ha he reurn/volailiy radeoff is significanly beer enering expansions han i is leaving expansions. To some exen, his resul is consisen wih he heoreical resuls in Whielaw (2000). A he end of expansions, when he probabiliy of shifing o a conracion is high, he condiional volailiy is also high. However, equiy reurns depend on wheher a regime swich occurs, an even ha is independen of he marginal rae of subsiuion. Consequenly, he correlaion in equaion (5) is low and so is he expeced reurn. Clearly, his combinaion will yield he low Sharpe raios shown in Figure. The regime-shif model also generaes a similar predicion for ransiions from conracions o expansions, bu a decrease in Sharpe raios a his poin in he cycle is no eviden in he daa. The one miigaing facor in he model in Whielaw (2000) is ha regime swich probabiliies are much more sable wihin conracions. Consequenly, he volailiy and expeced reurn effecs are smaller. Figure has several oher noable feaures. Wih he excepion of he fixed-mean esimae, he condiional Sharpe raios appear o coincide, consisen wih heir high correlaions in Table 3. The fixedmean esimae shows subsanially less variaion, consisen wih is low sandard deviaion in Table 3, and here is no srong business cycle paern. The condiional Sharpe raios also go negaive a various poins in he sample, again wih he excepion of he fixed mean series, i.e., he expeced reurn on he marke is less han he risk-free rae. This resul is somewha puzzling, alhough negaive risk premiums are no heoreically precluded in he framework of Secion 2 (see Boudoukh, Richardson, and Whielaw (997)). Moreover, hese resuls are consisen wih he resuls in Kairys (993), who uses commercial paper raes o predic negaive risk premiums. 4. Exploiing Predicable Variaion We now urn o an ou-of-sample analysis of predicable variaion in sock marke Sharpe raios, using 3

15 boh rolling and expanding window regressions. Afer examining he properies of he esimaed condiional Sharpe raios and evaluaing heir forecasing power, we employ hese forecass o consruc simple, marke-iming, rading sraegies. 4. Ou-of-Sample Forecasing The previous secion documens saisically significan ime-variaion in condiional Sharpe raios, and saisically and economically significan predicive power for esimaes based on a simple linear model. From a pracical perspecive, however, he key issue is wheher he empirical model has economically significan ou-of-sample predicive power. Unforunaely, i is difficul o conduc a rue ou-of-sample es. The condiioning variables are chosen based on heir correlaion wih reurns and volailiy in a sample ha runs hrough April 989, leaving over 20 years of new daa, bu here is sill he issue of how choices abou which papers o wrie and publish creae heir own daa snooping problems. Neverheless, i is worhwhile o consider he predicive power of ou-of-sample regressions. There are wo possible ways o specify he ou-of-sample regressions for esimaing he condiional Sharpe raios. The firs mehod is o choose a fixed sample size and o run rolling regressions. Tha is, a fixed number of observaions are used o esimae each se of coefficiens, and he esimaion window is moved forward by one monh a a ime. The advanage of his approach is ha if he coefficiens vary over ime, eiher because of misspecificaion of he empirical model or srucural shifs, hen he coefficiens from he rolling regressions will adap o hese changes. The second mehod is o add he new monhly observaion o he esimaion daase as we move hrough ime. As a resul, he number of observaions increases hrough ime, and he laer coefficiens of hese cumulaive regressions will be less subjec o esimaion error if he empirical specificaion is correc. An addiional issue is he choice of dividend yield series. The adjused dividend yield series used in he in-sample analysis relies on an in-sample esimaion of he break poin in he original series (Leau and Van Nieuwerburgh (2008)). Insead, for he ou-of-sample analysis, we use he dividend plus repurchase yield series. The mos naural way o evaluae hese alernaives is o examine heir ou-of-sample performance. For boh he rolling and cumulaive regressions, he iniial esimaion period is chosen o be 0 years, i.e., 20 monhly observaions from May 953 o April 963 are used o esimae he firs se of coefficiens. These coefficiens and he daa on he explanaory variables from April 963 are hen used o esimae he condiional momens of sock marke reurns for May 963. The esimaion is hen rolled forward one monh, adding he mos recen observaion for boh he rolling and cumulaive regressions, bu also dropping he oldes observaion from he rolling regression. Boh echniques generae a series of 4

16 572 ou-of-sample condiional Sharpe raios. Before urning o he ou-of-sample forecasing power, one poenially imporan issue ha can be addressed using he rolling regressions is he insabiliy of he coefficien esimaes. Unsable coefficiens indicae eiher srucural shifs in he daa, a misspecified model, or significan esimaion error. In he case of srucural shifs, predicive power migh be gained from shorening he esimaion period furher, alhough here is a clear radeoff wih esimaion error as he number of observaions decreases. In he case of model misspecificaion, alernaive specificaions, specifically more flexible funcional forms, migh prove useful, bu hese formulaions will also likely increase esimaion error. Figure 3 shows he rolling coefficien esimaes for he condiional mean (Panel A), he condiional volailiy (Panel B), and he direc esimaion of he Sharpe raio (Panel C). Noe ha he dae on he x-axis refers o he las dae in he esimaion period, e.g., he coefficien for December 978 is based on he en years of daa from January 969 o December 978. The graphs show a good deal of insabiliy. In he mean equaion, consider he wo variables ha are significan in he in-sample regression he dividend plus repurchase yield and he one-year Treasury rae (Table, Panel D). The coefficien on he dividend yield ranges up o 7, wih a small period of negaive coefficiens associaed wih rise of he echnology bubble in he mid o lae 990s. The coefficien on he ineres rae is predominanly negaive, bu here are periods in he lae 990s and during he financial crisis when i is posiive. From an economic sandpoin, he swiching of he sign of he coefficien is especially worrisome, bu one mus keep in mind he subsanial sampling error associaed wih a 0-year esimaion window. In he volailiy equaion, all he variables bu he Treasury rae are significan in he in-sample regression (Table, Panel D). The coefficien on lagged realized volailiy is clearly he mos sable, illusraing he robusness of he persisence of volailiy. In conras, he coefficiens on he defaul spread, he dividend yield, and he commercial paper-treasury spread swich signs a leas wice, a differen poins in he sample. Again, however, hese resuls should be considered in he conex of a regression over a shor sample period wih subsanial esimaion error. Direc esimaion of he Sharpe raio does no appear o improve he sabiliy of he rolling 0- year coefficiens. The wo significan variables from he in-sample regression in Table 2, he dividend yield and he Treasury rae, show paerns ha are remarkably similar o hose from he esimaion of he condiional mean. Given hese variaions, any ou-of-sample forecasing power of he rolling regression migh perhaps be even more surprising. Table 5 addresses he issue of ou-of-sample forecasing power direcly. Panel A presens resuls from he regression of he realized Sharpe raio on he esimaed condiional Sharpe raios for he period 5

17 May 963 o December 200, using boh he rolling window and expanding window regressions described above. These regressions are comparable o he in-sample regressions in Table 4, excep ha hey omi he firs 0 years of he sample due o he necessiy of using his window o esimae he firs se of coefficiens. There is definie evidence of forecasing power. All he slope coefficiens are posiive, i.e., he realized Sharpe raio is posiively relaed o he esimaed condiional Sharpe raio, and hree of he four are saisically significan for he rolling regressions. While he magniudes of he slope coefficiens are comparable for he expanding window regressions, only wo of he four coefficiens are significan a he 0% level. The R-squareds for he rolling regressions reach jus over %, which is only slighly greaer han one hird of he magniude in he comparable in-sample regressions. Of he four esimaed Sharpe raios, saisical evidence of forecasing power is weakes for he esimae ha fixes he mean reurn a is value in he esimaion window. While he slope coefficien is larges for his series, he magniude reflecs he lack of variaion in he series. For neiher he rolling nor expanding window regressions is he coefficien saisically significan, and he R-squareds are less han 0.5%. This resul coincides wih he in-sample evidence, i.e., ime-variaion in he condiional expeced reurn appears o be he dominan componen of variaion in he Sharpe raio. Finally, here is also subsanial evidence of overfiing across all he series. The hypohesis ha he forecas is condiionally unbiased, i.e., an inercep of zero and a slope of one, can be rejeced for a significan majoriy of he series, and he failure o rejec for he ohers is due o lack of power, i.e., large sandard errors. Moreover, all he slope coefficiens are less han one. This overfiing is illusraed clearly in Table 5, Panel B. This panel repors wha we call he predicive R-squared for each forecas series. Specifically, we calculae an R-squared measure imposing condiional unbiasedness: 2 ( S + ˆ 2 Si, ) R =, (8) 2 ( S µ ) where + µ S is he mean realized Sharpe raio. The predicive R-squareds in Panel B are uniformly negaive. In oher words, while he condiional Sharpe raios have forecasing power for he realized Sharpe raio, he uncondiional mean provides a beer fi han does he condiional Sharpe raio. S 4.2. Trading Sraegies While he above resuls clearly demonsrae economically and saisically significan ou-of-sample forecasing power for he realized Sharpe raio, hey are no linked direcly o a feasible rading sraegy. The possible se of rading sraegies is huge; neverheless, i is worhwhile o look a he performance of a few sylized sraegies. Consider he sraegy of esimaing he condiional Sharpe raio using he 0-6

18 year rolling regression and comparing his number o a fixed hreshold. If he esimaed condiional Sharpe raio is larger han he hreshold, hen inves in he sock marke; if i is smaller, hen inves in he risk-free asse. We hen consider he Sharpe raio of he monhs when he sraegy is invesed in equiies. This raio can easily be compared o a buy-and-hold sraegy ha always holds he marke. 7 Table 6 repors he resuls from execuing four sraegies: a buy-and-hold sraegy and hree marke-iming sraegies where he hresholds for invesing in he marke are hree differen pre-specified condiional Sharpe raio levels 0.0, 0., and 0.2. Again, we consider all four of our condiional Sharpe raio series and we use he dividend plus repurchase yield hroughou. The hird column of he able gives he number of monhs, ou of a possible 572, in which he sraegy is invesed in he marke. The able also shows he mean and average realized volailiy of monhly sock marke reurns for he monhs in which he marke is held. For he buy-and-hold sraegy, hese are he sample averages for he full ime period. The las wo columns presen saisics calculaed from monhly reurns (raher han he daily reurns ha are used o compue realized volailiy). The ex pos Sharpe raio is he mean excess monhly reurn over he volailiy of he reurn when he sraegy is in he marke, while ex pos volailiy is jus he denominaor of his raio. There are several noable resuls. Firs, here is consisen evidence of forecasing power. Of he 2 sraegies (four predicors x hree hresholds), eleven have Sharpe raios ha exceed ha of he buyand-hold. The one excepion is he fixed mean predicor wih a hreshold of 0.0. Moreover, across he four signals, hree produce monoonically increasing Sharpe raios as he hreshold increases. For he fourh, he direc esimaion of he Sharpe raio, here is only a very small violaion of monooniciy. Perhaps of greaes imporance, he increase in Sharpe raio associaed wih he sraegies is economically large. For example, for he esimae consruced as he raio of he condiional momens, where boh momens vary over ime, he Sharpe raios of he sraegies exceed ha of he buy-and-hold by 30-45%. These sraegies are invesed in he sock marke in 36-57% percen of he monhs in he sample. Second, for he sraegies ha allow he expeced reurn o vary over ime (# and #3), he forecasing power is coming primarily from his variaion. Mean reurns increase monoonically wih he hreshold, and he average realized volailiy varies lile, increasing as he Sharpe raio increases if anyhing. Ineresingly, he fixed mean sraegy is able o idenify periods wih low realized volailiies and hence high realized Sharpe raios. However, his predicive power is much less marked for volailiies 7 These marke-iming sraegies ignore boh ransacion coss and informaion in he magniude of he condiional Sharpe raios relaive o he hreshold. A more sophisicaed, and poenially beer performing, sraegy migh involve ime-varying marke weighs ha depend on he prior posiion in he marke and he relaive magniude of he condiional Sharpe raio. Neverheless, he sylized sraegy is sufficien o illusrae he exen of predicable variaion. 7

19 and Sharpe raios calculaed using monhly reurns. For he direc esimaion of he Sharpe raio, he hreshold appears o be he leas imporan. The number of monhs invesed changes less as he hreshold varies and he performance saisics also vary lile. Overall, he resuls confirm he earlier conclusions from boh he in-sample and ou-of-sample analyses, i.e., ha here is economically significan, predicable variaion in sock marke Sharpe raios. 5. Conclusion This paper demonsraes he abiliy of a relaively sraighforward linear specificaions, of eiher he condiional mean and volailiy of equiy reurns or of he Sharpe raio direcly, o predic dramaic imevariaion in monhly, sock marke Sharpe raios. This predicabiliy is eviden boh in-sample and ou-ofsample, where marke-iming sraegies ouperform a buy-and-hold sraegy in erms of ex pos Sharpe raios. This evidence provides furher suppor for he conenion ha he mean and volailiy of sock marke reurns do no move ogeher. Variaions of he magniude documened are inconsisen wih he condiional CAPM and relaed models ha imply a close o consan marke Sharpe raio. One possible explanaion is ha he resuls are due o marke irraionaliy or inefficiency. However, he apparen relaion beween variaion in Sharpe raios and he business cycle suggess he possibiliy of an economic inerpreaion. Whielaw (2000) provides a raional expecaions, general equilibrium model ha is broadly consisen wih he empirical evidence. In his model, discree shifs beween expansions and conracions overurn he sandard reurn/volailiy relaion. Alernaively, large flucuaions in risk aversion, as in Campbell and Cochrane (999), could also accoun for significan ime-variaions in Sharpe raios. Furher research in his area is warraned. While he empirical evidence provides insighs ino he ime series properies of equiy reurns and heir underlying economics, i also has implicaions in oher areas. In paricular, subsanial, predicable ime-variaion in marke Sharpe raios cass doub on he abiliy of he volailiy of even broadly diversified porfolios o proxy for priced risk. Consequenly, sandard measures of invesmen performance and radiional porfolio asse allocaion rules may have o be re-hough. 8