The Pitfall of Using Sharpe Ratio

Transcription

1 Finnce Lettes, 3, 1, he Pitfll of Using Shpe Rtio Mei-Chen Lin * nd Pin-Hung Chou b Ntionl United Univesity, iwn, ROC b Ntionl Centl Univesity, iwn, ROC Abstct We show tht when etuns e iid, the Shpe tio clculted ove -peiod holding hoizon will fist ise nd then fll s inceses, insted of monotonic function of if one ignoes the compounding effect in clculting long-tem etuns. Specificlly, we show tht ignoing the compounding tem will yield bised estimte of Shpe tio, nd the bis enlges when long investment hoizon is consideed. o clculte long-hoizon Shpe tios, we popose the use of block esmpling to etin the seil dependency in the dt. Bsed on smple of size potfolios, we find tht nkings bsed on Shpe tios of diffeent holding hoizons will diffe when the compounding effect nd the time-seies dependency in the dt e both consideed. Key Wods: Shpe tio; block esmpling; investment hoizon. JEL Clssifiction: G, G1, G 1. Intoduction he Shpe tio hs been extensively used to evlute potfolio pefomnce. While obin (1965) pioneeed the wok on the effect of heteogeneous investment hoizon on potfolio choices, Levy (197) ws the fist to show tht the Shpe tio tends to chnge with diffeent investment hoizons. He shows tht s long s the intended investment hoizon is diffeent fom the hoizon used to compute the tio, the Shpe tio exhibits systemtic bises nd ny sset-lloction decisions bsed on the Shpe tio will be misleding. heefte, sevel studies, theoeticl o empiicl, hve identified the hoizon s n impotnt fcto ffecting the pefomnce mesues (Chen nd Lee (1981), Levy (1981), Levy (1984), Chen nd Lee (1986), Levy nd Smuelson (199), nd Gunthope nd Levy (1994)). A potentil poblem with the pevious studies is tht most hve been done by ssuming tht the etuns of the undelying potfolios e independently nd identiclly distibuted (iid). A ecent wok by Lo () is pehps the only exception tht deives the smpling distibution fo Shpe tio of diffeent investment hoizons while llowing etuns to be non-iid. Fo deivtionl convenience, Lo () ppoximtes the long-hoizon etun s the ithmetic sum of single-peiod etuns nd ignoes the effects of compounding. Howeve, s it is well known tht the ppoximtion deteiotes s the etuns become voltile (see, e.g., Bodie, Kne nd Mcus (), p. 89), which ppes to be the cse fo longe investment hoizons, Lo s mesues should be used with cution. We show tht the Shpe tio, when expessed s the function of the investment hoizon, will exhibit n nti-u shpe, whees it will be monotoniclly incesing in the length of the hoizon if the compounding tem is ignoed. Hodges, ylo nd Yode (1997) point out tht the nking bsed on Shpe tios clculted ove shot-tem (monthly, qutely, nd nnul) etuns my not be vlid fo long-tem investos. he intuition is tht if the sset etuns e elly geneted fom n iid pocess, which implies tht the investment oppotunity set emins unchnged ove time, the length of the hoizon should not mtte. On the othe hnd, the investment hoizon mttes when the sset etuns e seilly coelted. As esul they popose the use of simultion to clculte long-hoizon Shpe tios. Hodges, ylo nd Yode (1997), howeve, genete the -holding-peiod etuns by ndomly smpling * Coesponding utho. E-mil: [email protected] ISSN Globl EcoFinnce All ights eseved. 84

2 Lin nd Chou 85 etuns out of histoicl etuns with eplcement (i.e., independently esmpling the individul etuns). he pocedue beks the time-seies dependency of the undelying seies nd genetes independent etuns. Since it is well documented tht sset pices do not follow ndom wlks nd sset etuns e to some extent pedictble, independent esmpling my not be ppopite. Specificlly, it will oveestimte the Shpe tio in the cse of positively seil coeltion, nd undeestimte the Shpe tio in the cse of negtively seil coeltion. o void the poblem, we popose the use of block smpling to compute long-hoizon Shpe tios tht llow fo cptuing the seil dependency in the dt. We pesent the evidence by using the following pocedue. Fis bsed on smple of thee size potfolios we show tht potfolio etuns e seilly coelted. his implies tht the nkings of the Shpe tio with diffeent coelted pttens will diffe ove diffeent holding peiods. While, the nkings of the Shpe tio will emin unchnged if independent esmpling wee used. hus, we dvocte using block esmpling to clculte the Shpe tio the thn using independent esmpling. Once the seil coeltion is tken into ccoun the optiml potfolio chnges fom lge-sized potfolio to medium-sized potfolios when the investment hoizon is lengthened. his ppe is ognized s follows. Section shows tht the Shpe tio is not independent of the investment hoizon even unde the iid ssumption. Section 3 intoduces dt nd compes the empiicl esults fom ndomly smpling individul obsevtions with those fom ndomly smpling block dt. he finl section mkes conclusion.. Investment hoizons nd pefomnce mesue In this section, we show tht when etuns e iid, the Shpe tio clculted ove -peiod holding hoizon will fist ise nd then fll s inceses, insted of monotonic function of, s suggested in Lo () which ignoes the compounding effect in clculting long-tem etuns. Specificlly, we show tht ignoing the compounding tem will yield bised estimte of Shpe tio, nd the bis will enlge when long investment hoizon is consideed. o begin, define the -peiod etun, t+), of secuity s follows: Pt+ - Pt t + ) =, (1) Pt whee P t is the pice of one secuity. It is the multiplicity of single peiod simple etuns. ht is: t + ) = Π = 1(1 + t + i -1, t i)) -1, () t + If P t follows geometic Bownin motion such tht R t hs n iid noml distibution with men μnd vince σ, t+) hs the following expected vlue nd vince (see Jobson nd Kotz (197)): µ + σ / E( t + )) = e -1, (3) e ( (, )) (µ +σ ) ( σ -1) V R t t + = e. (4) As esul the men nd vince of -peiod etun e not linely popotionl to. Define the Shpe mesue of ove -peiod investment hoizon s the following: E( t + )) - R f ( t + ) Shpe( ), V( t + )) whee R f (t+) is the -peiod isk-fee etun. Unde iid nomlity distibution, the -peiod Shpe mesue hs the following expession: µ + σ / e e Shpe( ) = (5) ( µ + σ / ) σ 1 e e Figue 1 shows tht the Shpe tio, expessed s function of the investment hoizon, will fist ise nd then fll s the length of the hoizon inceses. 1 f 1 he Shpe tio of -peiod simple etun computed by Hodges, ylo nd Yode (1997) with ndomly esmpling individul etun ns lso ises fist nd then deceses.

3 Lin nd Chou 86 Fo deivtionl convenience, Lo () ppoximtes the long-hoizon etun s the ithmetic sum of single-peiod simple etuns nd ignoes the effects of compounding. i.e. t + ) = Σ t = 1 t + i -1, t + i)) When etuns e iid nd one ignoes the compounding effec Lo () show tht the -peiod Shpe tio will be monotoniclly incesing in. Specificlly, the -peiod Shpe tio stisfies the simple eltionship (see, e.g., Lo (), eqution 17): Shpe()= Shpe(1). hus, ignoing the compounding tem my yield bised estimte of Shpe tio even unde iid etuns, nd the bis will enlge especilly when long hoizon is consideed. When etuns e not iid, the stoy will be even moe complicted. Lo () shows tht the vince of -peiod etun cn be expessed s: V( t + )) = σ ( + 1 k = 1 ( k) ρ ), whee ρ k = Cov(R t,r t-k )/V(R t ) is the kth-ode utocoeltion of t. Agin, the expession ignoes the compounding tem becuse tctble nlyticl expession does not exist if the compounding tem is consideed. his mens tht the vince of -peiod etun eflects the utocoeltion of the etuns up to the ode of -1. Vince tio hs been used to summize the time-seies ptten of the undelying seies: V( t + )) VR ( ) =. V( t + 1)) When plotted gin the holding hoizon, the vince tio will exhibit n upwd tend if etuns e positively seilly coelted, nd downwd tend if negtively seilly coelted. As it is well known tht stock etuns e not seilly uncoelted, the Shpe mesue will not be constnt s the holding peiods chnge. Specificlly, if sset etuns e positively seilly coelted, then longe investment hoizon coesponds to eltively highe isk level nd smll Shpe mesue. Since independent esmpling beks the time-seies ptten nd genetes independent etuns, we shll use block smpling in the next section to cptue the time-seies dependency in the dt nd etins the compounding effect in clculting long-tem etuns. he simultion detils e lso discussed in next section. 3. Methodology nd Results k o exploe the eltionship between Shpe pefomnce nd the investment hoizon, we clculte the tios fo thee size potfolios fom the CRSP dtbse fo investment hoizons nging fom one to twenty-five yes. he smll-, medium-, nd lge-sized potfolios of nnul etuns e downloded fom Kenneth Fench s web site. he constuction of the thee potfolios is s follows. All stocks listed on the NYSE, Amex nd Nsdq e fist divided into thee ctegoies. he stocks with mket equity within bottom 3% e ssigned to the smll-sized potfolio; the stocks with mket equity within middle 4% e ssigned to the medium-sized potfolio; the stocks with mket equity within top 3% e ssigned to the lge-sized potfolio. he potfolios e constucted t the end of ech June using the June mket equity. he potfolios fo July of ye t to June of t+1 include ll stocks fo which mket equity dt e vilble fo June of ye t. esuy bills te is used s poxy fo the isk-fee te of inteest. he smple coves the peiod fom 197 to 1. We fist use vince tios to exmine if the potfolio etuns exhibit ny seil dependency. We then evlute the vege pefomnce of these thee types of potfolios ove vious investment hoizons. If the etuns e iid, the vince of the -peiod etun should be equl to times the vince of the one-peiod etun, nd the vince tio should be equl to one fo ll hoizons. If etuns e positively seilly coelted, then the vince tio will gow t n incesing te s the etun intevl inceses, in which cse shote investment hoizon implies eltively lowe isk level nd highe Shpe mesue. On the conty, if etuns e negtively seilly coelted, then vince tio will decline s the etun intevl inceses, nd imply eltively lowe isk level fo longe investment hoizon. Figue shows th with the lengthening of the holding peiod, the vince tios ise fis then decline, he web site is:

4 Lin nd Chou 87 nd eventully incese fo ll potfolios. Specificlly, fo the smll-sized potfolio the vince tio eches the top on the thid ye, declines ftewds, nd evets to n upwd tend fo hoizons longe thn ten yes. his suggests tht etuns on the smll-sized potfolio e positively utocoelted in the shot tem, but the tend is evesed ftewds. Fo medium- nd lge-sized potfolios, the vince tios lso hve the sme ptten s the smll-size potfolios. he esult confims the litetue tht smlle-sized potfolios e iskie thn lge-sized potfolio in the shot tem. Supisingly, howeve, the esult in Figue indictes tht lge-sized potfolio my become iskie fo investment hoizons longe thn ten yes becuse of the stonge positive highe-ode utocoeltions. Since etuns e seil coelted, it is not supising tht the nking of the Shpe mesues of n sset will diffe when dt of diffeent intevls e used. he esults in Figue lso indicte tht when investment hoizons e lengthened long enough (e.g., seven yes fo lge potfolios nd ten yes fo medium nd smll potfolios), long-tem investment does be eltively lowe isk. his confims the conventionl wisdom tht the isk in the long-tem investment is eltively lowe, nd suggests tht the mkets oveect nd depict men-eveting phenomenon (e.g., Fm nd Fench, 1988; Poteb nd Summes, 1988). 3 o compute the Shpe tios fo vious lengths of hoizon, simultion is used hee to genete smple etun distibutions fo potfolios of smll, medium, nd lge stocks fo holding peiods fom one ye to twenty-five yes. Fo compison puposes, we clculte the Shpe tios bsed on block smpling nd independent smpling s well. Figue 3 plots the Shpe tios fo the thee potfolios bsed on independent smpling. Specificlly, fo given holding peiod, sy, we genete smple of -peiod etuns fo ech potfolio by ndomly selecting histoicl nnul etuns with eplcement nd compute the compounded etun. Fo exmple, conside the cse of the smll-stock potfolio with thee-ye investment hoizon. hee nnul etuns e selected t ndom fom the histoicl etuns ove the peiod, nd then thee-ye holding peiod etun is computed using eqution (3). his pocess is then epeted 5, times, yielding smple of 5, potfolio etuns fo ech holding peiod. hen, the -peiod Shpe tio cn be clculted bsed on the smple estimtes of men nd vince of the tificil smple. Since etuns e geneted independently, the time-seies ptten embedded in the oiginl seies is boken down, nd the pocedue genetes independent etuns. Indeed, fom Figue 3 it cn be seen tht the shpes of the vince tios e simil to the one plotted in Figue 1. his is not supising becuse independent smpling elimintes ll seil dependency in the dt. Anothe inteesting finding in Figue 3 is tht the nking of the potfolios emins the sme though ll hoizons. his is lso expected fo the sme eson. Clely, such esult is entiely diven by the independent ntue of the simultion. o void the bove poblem (independent esmpling beks the time-seies ptten nd genetes independent etuns), we edo the nlysis bsed on block smpling to cptue the dependency in the dt. Specificlly, fo k-peiod holding hoizon, we fist ndomly pick ye, sy q, between 197 nd (1-k+1) with ech ye being selected with equl pobbility, nd then pick q,q+k-1) s k-peiod holding etun. his etins the time-seies popety within the etun. hen we epet the pocedue 5, times nd compute the excess men nd stndd devition of the 5, etuns. Dividing the excess men by stndd devition, we get the Shpe tio. 4 Figue 4 pesents the esult bsed on block smpling. Unlike the finding fom independent esmpling, the esult in Figue 4 indictes tht with block esmpling the length of the investment hoizon becomes elevnt. Moeove, with block dt, the Shpe tio fo ech potfolio lso fist inceses nd then deceses s the holding peiod is extended. Fo exmple, the Shpe tio fo smll-sized potfolio (medium, lge) goes up the pek t the hoizon of bout ten (ten, nine) yes, nd then deceses gdully ftewds. Although the Shpe tios still etin n nti-u shpe s in Figue 3, the nking of potfolios' pefomnce chnges s the investment hoizon lengthens. Fo exmple, the lge-sized potfolio pefoms the best when investment hoizon is less thn fou yes, yet becomes dominted by the medium-sized potfolios when holding peiods e lengthened to five yes o moe. 5 his indictes tht the Shpe pefomnce mesue computed bsed on 3 One impliction is tht funds with diffeent time-seies popeties of investment sttegies (e.g., momentum o men evesion) cnnot be evluted bsed on the sme investment hoizon. 4 Hee we follow the litetue on vince tio (e.g., Lo nd McKinly (1988)) tht uses ovelpping dt to impove the pefomnce of sttistics in finite-smples. 5 Hodges, ylo nd Yode (1997) find tht lge common stocks consistently outpefom smll stocks when etuns e smpled independently.

5 Lin nd Chou 88 independent esmpling is inppopite when etuns e seilly coelted. If sset etuns e seilly coelted nd we estimted the etun voltility using independent smpling, we my obtin incoect nking in pefomnce, nd mke wong sset lloction decision. Besides, ecll tht the medium-sized potfolio hs the lowest vince tio nd the highest Shpe tio fo longe holding peiods. heefoe, the medium-sized potfolio ppes to be moe ttctive fo long-tem investment. Figue1:he Shpe Rtio of Simple Retun nd Investment Hoizon unde Noml Distibution (ke Men=.1 nd Stndd Devition=.5 fo Exmple).4 t io.3 R e. p h.1 S Investment hoizon Figue:Vinces Rtios fo Smll-, Medium-, nd Lge-sized Potfolios o ti R e p h S Yes Smll Medium Lge Shpe Rtio Figue 3: he Shpe Rtio of -peiod Simple Retun with Rndom Smpling Individul Obsevtions Yes Smll Medium Lge Figue4: he Shpe Rtio of -peiod Simple Retun with Rndom Smpling Ovelpping Obsevtions 1.5 o i t e p h S Yes Smll Medium Lge

6 Lin nd Chou Conclusions In this ppe, we popose the use of block esmpling to obtin pope estimtes of Shpe tio fo vious investment hoizons. Using block esmpling etins the compounding effect in clculting long-tem etuns nd the time-seies dependency in the dt. We find tht nkings bsed on the Shpe tio vy substntilly with the investment hoizon. In conts investment hoizons e ielevnt when the estimtion of Shpe tio is bsed on independent smpling. Becuse investos diffe in thei isk ttitudes nd in holding hoizons, it is unesonble to evlute potfolio pefomnce bsed on one single investment hoizon. Pcticl implementtion of the Shpe tio is esonble only if the intended investment hoizon equls to the holding peiod of the etuns used to compute the tio. Howeve, mny investment compnies epot Shpe tio only bsed on the etuns fo fixed investment hoizon (e.g., monthly o nnul etuns). A gph of Shpe tio ginst the investment hoizon my be moe ppopite fo investos with multiye investment hoizons. he Shpe pefomnce nkings bsed on shot etun will be vlid only fo shot-tem investos, but not fo long-tem investos. Refeences Bodie, Z., A. Kne, nd A.J. Mcus (), Investments, 5 th edition, Boston: McGw-Hill Inc. Chen, S., nd C. Lee (1981), he Smpling Reltionship Between Shpe's Pefomnce Mesue nd its Risk Poxy: Smple Size, Investment Hoizon nd Mket Conditions, Mngement Science, 7, Chen, S., nd C. Lee (1986), he Effects of the Smple Size, the Investment Hoizon nd Mket Conditions on the Vlidity of Composite Pefomnce Mesues: A Geneliztion, Mngement Science, 3, Fm, F. nd K.R. Fench (1988), Pemnent nd empoy Components of Stock Pices, Jounl of Politicl Economy, 96, Gunthope, Deboh nd H. Levy (1994), Potfolio Composition nd the Investment Hoizon, Finncil Anlyst Jounl, 5, Jn.-Feb Hodges, Chles W., Wlton R.L. ylo, nd Jmes A. Yode (1997), Stock, Bonds the Shpe Rtio, nd the Investment Hoizon, Finncil Anlysts Jounl, 53, Nov.-Dec Johnson, N. L. nd Kotz, S. (197), Distibutions in Sttistics: Continuous Multivite Distibutions, New Yok: John Wiley nd Sons. Levy H. (197), Potfolio Pefomnce nd the Investment Hoizon, Mngement Science, 18, Levy H. (1981), he CAPM nd the Investment Hoizon, Jounl of Potfolio Mngemen 7, 3-4. Levy H. (1984), Mesuing Risk nd Pefomnce Ove Altentive Investment Hoizons, Finncil Anlyst Jounl, 4, Levy H., nd Pul A. Smuelson (199), he Cpitl Asset Picing Model with Divese Holding Peiods, Mngement Science, 38, Lo, A.W. nd A.C. McKinly (1988), Stock Pices do not Follow Rndom Wlks: Evidence fom Simple Specifiction es Review of Finncil Studies, 1, Lo, Andew (), he Sttistics of Shpe Rtios, Finncil Anlysts Jounl, 58, July-August Poteb, J.M. nd L.H. Summes (1988), Men-Revesion in Stock Pices Evidence nd Implictions, Jounl of Finncil Economics,, Shpe, W.F. (1994), he Shpe Rtio, he Jounl of Potfolio Mngemen 49, obin, J. (1965), he heoy of Potfolio Selection, he heoy of Inteest Rtes, 3-51, edited by F. H. Hhn nd F. P. R. Bechling. London: Mcmilln.