Marke Timing & Trading Sraegies using Asse Roaion Panagiois Schizas * and Dimirios D. Thomakos Deparmen of Economics Universiy of Peloponnese 22 00 Greece 2/6/200 Absrac We presen empirical resuls on he saisical and economic viabiliy of a marke iming rading sraegy ha is based on roaion beween wo risky asses. Using daa on Exchange Traded Funds (ETFs), and models for boh he reurns and he volailiy of he underlying asses, we compare he performance of he suggesed models wih he sandard benchmarks of a buy-andhold sraegy and an equally weighed porfolio. The underlying inuiion for he use of such a sraegy ress wih lieraure on sign and volailiy predicabiliy. The roaion sraegy, as we apply i in his paper, is no risk-neural and assumes he presence of arbirage opporuniies in he markes. Furhermore, he model specificaion uses he inerplay beween relaive reurns and relaive volailiies in picking-up he asse wih he highes reurn. Our resuls show ha even a naive model ha is based on a moving average of relaive reurns can ouperform boh benchmarks and ha more elaborae specificaions for he roaion model may yield addiional performance gains. We also find ha, in many cases, he roaion sraegy yields saisically significan sign predicions of he relaive reurns and volailiy. While our resuls are condiional o he daa ha we have used in our analysis hey, neverheless, suppor he marke iming lieraure and show ha an acive rading sraegy can be based on he concep of roaion. JEL Classificaion: C5, C53, G0 Keywords: Exchange Traded Funds, forecasing; marke iming; sign predicions; saisical arbirage; echnical rading, volailiy iming. * Corresponding auhor. Deparmen of Economics, Universiy of Peloponnese, 22 00, Greece. Email: pschizas@uop.gr, panagiois.schizas@gmail.com. Corresponding auhor. Deparmen of Economics, Universiy of Peloponnese, 22 00, Greece. Email: homakos@uop.gr, homakos@gmail.com. Tel.: +30-270-23028, Fax: +30-270-23039. Elecronic copy available a: hp://ssrn.com/absrac=53794
. Inroducion Acive rading based on asse roaion is no a new idea. According o Skidelsky (992), Keynes a he urn of he 20 h cenury examined he variaion on sock reurns according o he business cycle and suggesed a rading sraegy under he name of Acive Invesmen Policy. His sraegy was based on a consan swiching beween shor and long mauriy asses under forecas esimaes following changes in he ineres rae. This kind of sraegy is no far from wha modern marke iming sraegies ofen do and here is some empirical evidence ha asse roaion does work. However, his comes conrary o several heoreical and empirical hypoheses such as he efficien markes hypohesis, he random walk hypohesis, he no arbirage hypohesis and ohers. The empirical evidence is conflicing and, moreover, marke raders do use asse roaion in heir day-o-day operaion as he source of poenial profis. In paricular, and in addiion o he srand of he lieraure ha deals wih marke iming, here is evidence ha he sign and volailiy of a risky asse are boh predicable. In his case one can envision a sraegy ha explois sign and volailiy forecasabiliy and roaes beween asses based on heir relaive reurn or relaive volailiy. I is herefore of heoreical and pracical ineres o presen resuls on he viabiliy of a roaion sraegy ha ries o use a saisical sylized fac for arbirage rading in he hope of generaing economic gain. In his paper we presen a marke iming sraegy ha is based on a modificaion of he marke iming sraegy beween a risky and a risk-free asse. Here we use pairs of risky asses and generae predicions for heir relaive reurns or relaive volailiies which are hen ransformed ino rading signals via a predefined rule. Our approach is hus par of he marke iming and volailiy iming lieraure and ress on he assumpion of arbirage opporuniies ha can be exploied based on successful model forecass. Our empirical mehodology is sraighforward o implemen bu we have no found a similar implemenaion in he res of he lieraure. 2 To preview our resuls, we find ha he proposed mehodology appears o work well. In paricular, we find ha various specificaions for he roaion mehodology we propose do work in he sense of ouperforming a buy-and-hold sraegy and an equally weighed porfolio formed beween he wo asses in he pair. Since his is a sraegy ha has he invesor always exposed in he marke i is riskier han an equally weighed porfolio bu is a priori less risky for someone considering a buy-and-hold sraegy for a single asse. An ineresing by-produc of our analysis is ha we find ha he relaive sign of he wo asses can be prediced more han 50% of he A brief review of he marke iming and relaed lieraure is given in he nex secion. 2 The proposed sraegy is similar o wha is called a quaniaive direcional equiy rading in he indusry. There is a growing ineres for his kind of sraegies. 2 Elecronic copy available a: hp://ssrn.com/absrac=53794
ime, wih he predicion being saisically significan. This is useful as one can devise more elaborae rading sraegies based on he sign forecass han he ones we implemened in his paper (such as hedging o make he roaion a risk-neural sraegy). The res of our paper is srucured as follows. In secion 2 we presen a brief lieraure review on marke iming and roaion sraegies. In secion 3 we presen he daa we used in our analysis. In secion 4 we presen he economeric and rading mehodology for he roaion sraegies we implemen. In secion 5 we presen and discuss our resuls. Finally, in secion 6 we offer some concluding remarks and exensions o he curren work. All ables and figures are given in he appendix. 2. Lieraure review on marke & volailiy iming and asse roaion Marke iming requires a model selecion mehodology for generaing predicions, a rading sraegy (he rading rule) and a rading cos esimae o be complee. Trading coss are of course a crucial aspec in he consrucion of rading rules and sraegies as high ransacion coss usually erode he profiabiliy of sraegies. Our analysis draws from wo srands of he lieraure: he srand on marke and volailiy iming and he srand on sign and volailiy predicabiliy. We are no going o be exhausive in our review and concenrae on papers ha relae o he presen work. We sar wih he marke and volailiy iming lieraure. Vandell and Sevens (989) is an early reference presening evidence on he superior performance of a marke iming approach. Breen, Glosen and Jagannahan (989) is also an early and influenial paper on marke iming; hey examined he marke iming abiliy of a model ha uses a risk-free asse and a risky asse and roaes beween he risky asse and cash and found evidence of explanaory power for he shor-erm ineres rae. Sy (990) quesions he viabiliy of marke iming. Beebower and Varikooy (99) discuss ways of measuring he success of marke iming sraegies and Shilling (992) argues ha marke iming can bea he buy-and-hold sraegy. Pesaran and Timmermann (994, 995) examine he predicabiliy and profiabiliy of a similar marke iming approach across differen frequencies and ake ino consideraion he effecs of rading coss. Larsen and Wozniak (995) argue ha marke iming is a viable sraegy for he real world while Levis and Liodakis (996) examine syle roaion sraegies for he U.S. Benning (997) discusses he predicion skills of raders ha apply iming mehodologies in he real world. Lee (997) connecs marke iming wih shor-erm ineres raes, much in he syle of Breen, Glosen and Jagannahan (989) and Wanger (997) offers some explanaions as o why marke iming works. Whielaw (997) used a Sharpe raio-based 3
approach o consruc marke iming sraegies and found evidence ha such sraegies can ouperform a buy-and-hold sraegy. Johannes e al (2002) defined marke iming as ha behaviour when invesors increase heir allocaion in risky asses in periods of bull markes while volailiy iming as ha behaviour when invesors are decreasing heir allocaion in risky asses in periods of high volailiy. They conclude ha a sraegy based exclusively on volailiy iming can ouperform marke iming sraegies. A similar approach, based again on volailiy iming, is aken by Fleming, Kirby and Osdiek (200, 2003). Li e al (2002) considered opimal marke iming sraegies under ransacion coss and argued ha as ime elapses he opimal sraegy confirms he momenum index rading rule. Lam e al (2004) examine again he opimal marke iming sraegies and he relaionship of is performance wih he percenage of correc sign predicions and he magniude of ransacion coss. Jiang (2003) applied a nonparameric es in order o examine he marke iming abiliy in a large daa sample of muual funds and showed a superior iming abiliy among acively managed equiy funds. Wang (2005) argued ha roaing sraegies over equiy syles could generae significan reurns. Brooks e al (2006) compare and evaluae a number of differen marke iming sraegies. Thomakos, Wang, and Wu (2007) employed marke iming swiching sraegies similar o Breen, Glosen and Jagannahan (989) and Pesaran and Timmermann (994, 995) bu heir roaion is based no only on a risk-free and risky asse bu also beween pairs of risky asses. They also inroduced asymmeric response erms for he relaive reurns on he pair of asses ha is being roaed. In his paper we apply he mehodology of Thomakos, Wang and Wu (2007) wih cerain modificaions and exensions. We nex urn o he lieraure on sign forecasing. 3 The work of Chrisoffersen and Diebold (2006) and Chrisoffersen e al (2007) examines in deail he predicabiliy of he direcion of he reurns and connecs i o he predicabiliy of asse reurn volailiy, claiming ha sign and volailiy predicabiliy does no violae he efficien marke hypohesis. Thomakos and Wang (200) generalize and exend he work of Chrisoffersen and Dievold (2006) by showing ha zero is no necessarily an opimal hreshold for direcional predicions and ha a ime-varying hreshold based on volailiy is more suiable when making direcional predicions. Hong and Chung (2003) and Chung and Hong (2005) propose ways of esing and assessing sign predicabiliy for asse reurns. All he above references offer evidence ha signs, and he direcion in general, of asse reurns is predicable if one accouns for he volailiy of he reurns. 3 Here i s imporan o noe ha sign and direcional predicions have been found o heavily depend on volailiy forecasabiliy for which here is a raher large lieraure which we will no review here. 4
3. Daa A feaure of our analysis is ha we use he class of Exchange Traded Funds (ETFs) o apply our roaion mehodology. We focus on 4 broadly defined passive ETFs which we have seleced based on crieria like marke capializaion, long(er) hisorical racking record, and high rading volume. These series are: S&P500 (icker: SPY) is he firs ETF in he US and was launched on 29 January 993 (second globally afer he TIPS), on he American Sock Exchange, under he name SPDRs - Sandard & Poor s Deposiory Receips or Spiders. I racks he S&P 500 index and is he larges ETFs in he world wih 6.4 billion worh of asses under managemen. Financial Selec Secor SPDR ETF (icker: XLF) was launched on 6 February 998. I belongs o he group of Sandard & Poor s Deposiory Receips or Spiders and i is raded on AMEX. I has 3.98 billion worh of asses under managemen. PowerShares QQQ Cubes (icker: QQQQ) is designed o rack he NASQAD 00 Sock Index and has been launched on 0 March 999, being raded on AMEX. Due o he underlying index i belongs among he mos popular ETFs wih 0.26 billion worh of asses under managemen. Oil Services HOLDRs rus (icker: OIH) has been designed as a baske of specified companies wih exposure o oil service indusry. They are currenly 20 companies which are among he larges and mos liquid wih U.S. The respecive ETF launched on 6 of February 200 and has.55 billion worh of asses under managemen. I is raded on NYSE. While a review on ETFs is beyond he scope of our paper i is useful o presen some perinen facs abou his asse class. By he end of he firs quarer of 2008 here were.280 ETFs wih 2.65 lisings wih $760 billion worh of asses under managemen, managed by 79 managers and lised on 42 exchanges hroughou he world. 4 An ETF can be defined as a fund ha duplicaes a sock index or a baske of socks, from one or more secors and indusries hese socks are embedded in he ETF and are bough and sold as a uni. In oher words, whaever racks a specific index or a specific baske of socks and i is raded as a uni could be called ETF. Wihin he universe of risky asses ETFs cover a broad specrum of invesmen soluions, including marke capializaion, invesmen syles and secors, counries, fuures and opion conracs as well as he opporuniy of shoring major indices on he spo marke. ETFs have an exposure o fuures markes wih more han 300 opions and 3 fuures lised on he markes on US, Canada and Europe. 5 The ETF srucure combines he dynamics of index-racking uni russ wih he meris and radabiliy of lised invesmen companies. ETFs have lower operaing 4 According o he 2008 Morgan Sanley ETF Global Indusry Review. 5 According o Morgan Sanley in he US here are 288 opions, which means ha 47% of US lised ETFs have opions. 5
expenses, more rading liquidiy, and more efficien ax srucures han he convenional indexracking muual funds. Finally, ETFs have lower ransacion coss and are offered as an affordable soluion o low-budge invesors. For he four ETF series noed above we use weekly OHLC observaions. From he daily observaions we calculae he daily range-based volailiy esimaor (Parkinson, 980) and hen we consruc he realized weekly range-based esimaor by summing he daily range-based esimaes. This is a good compromise beween he uses of a non-parameric volailiy esimaor versus a parameric approach based solely on he weekly observaions. In our analysis we experimened wih differen days of he week, Monday, Wednesday and Friday. Afer he consrucion of he corresponding volailiy series we mach he daa according o he hree pairs ha we form, namely: SPY-OIH, SPY-XLF and SPY-QQQ. We deliberaely used only he pairs wih S&P500 ETF since he sraegy should perform a leas on par wih he marke. The full samples n=n 0 +n for he hree pairs were obained and he relevan series are presened in Figures o 3. In our applicaion we used a rolling esimaion window of n 0 =04 weeks and an evaluaion window ha differed in lengh according o he day and he pair being examined; for each of he pairs SPY-OIH, SPY-XLF and SPY-QQQ we have: for Monday n =232, 33, 322; for Wednesday n =264, 374, 363, and for Friday n =256, 33, 353. 4. Mehodology 4.. Model specificaion In his secion we presen and discuss our roaion mehodology and he models and approach we use o implemen i. As in mos forecasing exercises we leave ou par of our sample for esing and use a rolling window of observaions o forecas and rade in hisorical real ime. Our resuls are for a rolling window of 04 weeks and oher resuls for differen esimaion lenghs are available on reques. Le us define by R i he weekly reurn, defined as he logarihmic difference of he weekly ETF closing price, of he i h asse and by V i is corresponding volailiy. For he measuremen of volailiy we use he realized weekly range-based volailiy esimaor given by: 5 Vi = s = σ s, i () where formula as: σ s, i is he daily range-based volailiy esimaed from he high-low Parkinson (980) [ log( 2) ] [ log H L ] 2 σ, = log (2) s i 4 s s 6
wih H, L being he s h day s high and low prices for he i h asse. s s In our roaion-based models he dependen variable is eiher he relaive reurn (difference in reurns) or he relaive volailiy beween wo asses, The relaive reurn is defined as: y = R R (3) i j which is equivalen o he reurn of relaive prices, i.e. o log( C / C ) log( C C ) i j y i j, i /, j =, wih C i being he closing price. This is an aracive feaure of roaion modelling, i.e. ha deals wih he economically inerpreable noion of relaive prices. The relaive volailiy is defined using levels and logarihms as: V = V V and v = log( V ) log( V ) (4) i j and we experimen wih he direc modeling of v bu also wih he modeling of he individual log-volailiies as well. All roaion models we consider are using eiher y or v as heir dependen and decision variables and follow a sandard regression specificaion: y = x β + u (6) T where x is he regressor vecor, whose dimension and included variables differs across model specificaions, and u is he regression error. The simples roaion model we consider is he naïve model ha does no include any explanaory variables oher han he mean of relaive reurns and ignores dynamics poenially presen, i.e. is given as: y β + (7) = 0 u Anoher simple model comes if we include any dynamics ha are presen in he regression error erm using, as we do, a moving average such as: q = β ( ε ; θ ), ( θ ) = ε + = ε y + 0 u u ε ; θ (8) k k k A more plausible alernaive o hese benchmarks is a model ha includes some explanaory variables in he righ-hand side. We experimened wih he inclusion of a lagged dependen variable, lagged values of he relaive volailiy and asymmeric response erms for boh of hem, as well as cross-erms. For a model wih a single lag his specificaion is given by: y β I + u (9) y V y V V = 0 + β y + β 2V + β 3I + β 4I + β 5 y I + β 6V I + β 7 y I + β8v y where = I( y < 0) I is a dummy variable capuring he asymmeric response of relaive V reurns, and similarly for I I( V c) = of relaive volailiies c being a fixed hreshold which < we discuss laer. This approximaion is piecewise linear and can capure he poenially differen 7 y
behaviour of relaive reurns in periods when one of he wo asses ouperforms he oher depending on boh he asymmeric response of relaive reurns and relaive volailiies. Finally, we consider he following auoregressive models for he individual log-volailiies and for he relaive volailiy of he form: paic ( V i ) = φ 0 + φ log( V k i ) + k= k, log η, v paic 0 + α k k = α = v + w (0) k where he orders of boh models are seleced by he AIC crierion, which is known o overfi and is suiable for he presence of long-memory in he volailiy series. 4.2. Trading rules The roaion rading sraegy we implemen is based on he forecass generaed by he above models and is sraighforward since i involves a binary decision for he asse ha is o be bough. Noe ha in he conex of his sraegy all available capial roaes when a signal for a swich from one asse o he oher is given. Given a sample of n observaions suppose ha a rolling window of n 0 observaions 6 is o be used for he hisorical evaluaion of he sraegy. The seps involved in he compuaions are as follows: - A ime esimae he models m =,2,..., M and compue he one-week ahead forecass ˆ,log( V, i ) vˆ +, y + + ˆ. ) ˆ > + ( m - Based on he forecass ener ino a posiions as follows: if y 0 hen ener a long posiion for asse i, else ener a long posiion for asse j. Noe ha a swich occurs a ime only if he posiion was in a differen asse han he curren signal a ime -. ˆ + > Similarly for 0 v and for ( ˆ ) log( Vˆ ) V, i, j log + > +. - A ime + evaluae he realized reurn of he sraegy R + and rack he correc sign R I yˆ r I yˆ + = > 0 + < 0 r +, for he sraegy reurn 7 predicions as follows: ( + ) +, i ( + ) j and I( y > 0) I( y > 0) + I( yˆ < 0) I( y 0) ˆ + = + + + + < S for he couner of correc sign predicions. Noe ha in he conex of he naïve model he roaion sraegy coincides wih a momenum sraegy based on local smoohing: he comparison is beween wo moving averages of he same size since yˆ ˆ + = β 0 = yn = Ri, n Rj n. A similar commen can be made for he oher models 0 0, 0 6 The window size is n 0 =04 weeks for he resuls presened in he nex secion. 7 Noe ha he sraegy s reurn depends on he realized values of he individual asse reurns, denoed wih small case leers, and is uncerainy comes only from he uncerainy of he forecas. 8
we use alhough he momenum naure of he sraegy is no as clear in hem. However, we sill need o emphasize ha is no jus momenum ha makes he sraegy works: i is also he percenage of correc sign predicions, somehing ha we furher discuss in he resuls. Also noe ha he, ime, condiional expeced reurn and volailiy for he roaion sraegy are given by: ( R ) r j P( yˆ + = +, + y+ > 0)( r +, i r + j ), ( R ) P( yˆ > 0) [ P( yˆ > )]( r r ) 2 E, Var + y+ y+ 0 +, i +, j = () From he above we can easily see ha he expeced reurn is posiive if ( r / r ) > ( P ) +, i +, j, i.e. when he relaive realized reurn is greaer han a negaive hreshold ha depends on he probabiliy of making a posiive predicion. Therefore when boh reurns are posiive he sraegy s expeced reurn is also posiive. We can also see ha he volailiy of he roaion sraegy is maximized when he probabiliy of making a posiive predicion is close o one-half or when he difference beween he wo realized reurns is increasing or boh. Essenially boh he expeced reurn and volailiy of he sraegy depend on he disribuion of he models forecass. The above observaions show ha he pair selecion of he wo asses is crucial for he roaion sraegy o work. For example, pairs of asses ha move ogeher, in he sense ha r, i r +, j +, are no suiable for roaion rading; i s obviously sufficien o say wih a single asse or possibly an equally weighed porfolio. Noe ha asses ha consisenly exhibi similar reurn pahs will also have similar volailiy pahs his observaion will be imporan in wha follows. 5. Resuls and discussion 5.. Model & wealh-based performance Our main performance resuls are given in Tables o 3. In hese ables we presen some performance saisics over he 04 weeks of he evaluaion period. These saisics include he cumulaive reurn (erminal wealh of $ invesed a he beginning of he evaluaion period), he average weekly reurn, weekly sandard deviaion, weekly Sharpe raio and he maximum loss of each sraegy. Resuls are given for hree weekdays, Monday, Wednesday and Friday in Tables, 2 and 3 respecively. In each able here are hree panels, one for each pair: Panel A for he SPY and OIH pair, Panel B for he SPY and XLF pair and Panel C for he SPY-QQQ pair. The easies way o summarize he relaive performance of he proposed roaion mehodology is o rank he models wih respec o one or more performance measures and o see how many imes he roaion models (and which one in paricular) ouperform he buy-and-hold sraegy 9
and he equally weighed porfolio. We discuss our resuls based on erminal wealh and weekly Sharpe raio. A careful look a he ables shows us ha he roaion models are he bes performers for 8 ou of 9 pairs wih he models based on he differences in reurns being op performers in 6 ou of 9 pairs. In paricular, he roaion based on he comprehensive model of equaion (9) is he op performer in 3 pairs (SPY-XLF and SPY-QQQ on Monday and SPY- QQQ on Friday); he roaion based on he moving average model of equaion (8) is he op performer in 2 pairs (SPY-OIH on Wednesday and Friday); he roaion based on he differences in volailiy model using (he firs of) equaion (0) is he op performer also in 2 pairs (SPY- OIH on Monday and SPY-XLF on Wednesday) and he roaion based on he naïve model o equaion (7) is he op performer in pair (SPY-XLF on Wednesday), along wih SPY (SPY- QQQ on Wednesday) and OIH (SPY-OIH on Monday ha ies wih he differences in volailiy). Noe ha he equally weighed porfolio does no ener ino he lis of op performers nor does he XLE (financials) ETF. Comparing he erminal wealh performance of he bes roaion models wih he erminal wealh of he bes performing asse in each pair and wih ha of he equally weighed porfolio we find he following 8. Wih respec o he bes performing asse he minimum difference is - 5 and he maximum difference is 33. The average difference is 4 wih a sandard deviaion of 2 (calculaed across he 9 pairs ha we used). Wih respec o he equally weighed porfolio he minimum difference is 0.7 and he maximum difference is 7. The average difference is 39 wih a sandard deviaion of 24. The average difference across boh he bes performing asse and he equally weighed porfolio is 27 wih a corresponding sandard deviaion of 23. Overall hese resuls sugges ha, on average, we are beer off using a marke iming sraegy han using he bes asse buy-and-hold sraegy or he equally weighed porfolio. I is ineresing o noe ha our resuls also show a day-of-he-week effec: for he SPY-OIH pair he bes days are Wednesday and Friday, for he SPY-XLF pair he bes day is Monday and for he SPY-QQQ pair he bes day is Friday. 9 The ranking of he models based on heir average weekly reurn and on heir sandard deviaions is no very meaningful since hey are frequenly he same across models. We also know ex ane ha he roaion sraegy is going o be riskier han he equally weighed porfolio and his shows up clearly in he resuls along wih he fac ha he maximum loss of he roaion sraegies always corresponds o he maximum loss of he wo asses in he pair. The laer is based on he 8 Resuls are given in for an invesmen of $; a posiive difference shows ha he roaion sraegy is beer; compare across all hree rading days. 9 See, among ohers, Gibbons and Hess (979), French (980), Conrad and Kaul (988), Rogalski (984) and Chordia e al (200) for a discussion on he day-of-he-week effec. 0
naure of he roaion (you are expeced o make sign errors frequenly and his invariably happens a he mos difficul swiching poin). Making a comparison beween models based on heir volailiy characerisics we see ha for he SPY-OIH pair he difference in volailiy is he highes and for he pair SPY-QQQ he difference in volailiy is he lowes. Noe now ha for he SPY-OIH pair he piecewise linear model fails o ouperform he volailiy iming models. On he conrary, for he SPY-QQQ pair, even if our resuls are condiional on he weekday, wo ou of he hree days he model of equaion (9) ouperforms he alernaive specificaions. This can possible indicae ha he relaive performance of he piecewise linear specificaion works beer as he volailiy of he one asse of he pair is no dominaing he oher asse. A nice visual summary on he reurns of he roaion sraegies is given in Figures 4 o 6. There we plo he kernel densiy esimaes of he reurns from he sraegies in overlap for each rading day. We can see several ineresing characerisics in hese plos. For example, we can see ha he densiy of he moving average model is consisenly more peaked around zero han ha of he oher models. We can also see ha he piecewise linear model is more peaked around zero, compared o oher models, bu also has lower probabiliy on he negaive side of he reurns. The naïve and volailiy models are consisenly less peaked around zero, and hus has heir probabiliy spread on eiher side of zero bu hey are also more risky: hey have higher probabiliy on ending up wih negaive reurns. While here are differences across he rading days we can possibly conclude ha he piecewise linear model appears, based on is densiy characerisics, a safer model o follow. 5.2 Trading behavior of he roaion To furher examine he rading behavior of he roaion models we compue saisics wih respec o he number of rades, he rading ime, and he rade duraion and ransiion probabiliies. 0 We have calculaed hese quaniies for all pair combinaions ha appear in Tables o 3 bu repor only hree represenaive ables, Tables 4, 5 and 6 based on he Wednesday resuls and he roaion models of relaive reurns he res are available on reques. For he SPY-OIH pair we see ha for he naïve model he number of rades is 233 wih a mean rading ime of 88.6%, compared o only 44 (55%) and 49 (57%) rades (mean rading imes) for he moving average and complee model (of equaions (8) and (9)). Noe ha for his pair and day he bes performance was by he moving average model followed by he naïve model. 0 Manganelli (2006) proposed a framework o model duraion, volume and reurns simulaneously, obaining an economeric modelling which incorporaes he ineracion among hese variables.
Therefore he moving average model would be way beer in he presence of any ransacion coss. The ransiion probabiliies for he moving average model show more persisence in he case of rading raher han no rading (here a roaion couns as a rade) and are close o 55% - he corresponding probabiliies for he naïve model are much higher, in excess of 80%. The maximum duraion wih no rading is 9 weeks for he naïve model, 6 weeks for he moving average model and 0 weeks for he complee model. The maximum duraion wih rading (i.e. coninuous swiching) is 20, 8 and 4 weeks respecively. As we will noe laer on, he moving average model has a higher sign success raio han he naïve model and herefore achieves beer performance wih far fewer rades. For he SPY-XLF pair we see a differen paern. Here for he naïve model he number of rades is only 37 wih a mean rading ime of 0%, compared o 63 (44%) and 4 (38%) rades (mean rading imes) for he moving average and complee model. The naïve model is now he bes performer, alhough only marginally wih respec o he bes performing asse. I is now he naïve model ha would be beer in he presence of any ransacion coss. The maximum duraion wih no rading is 98 weeks for he naïve model, 0 weeks for he moving average model and 34 weeks for he complee model. The maximum duraion wih rading (i.e. coninuous swiching) is 8, 6 and 5 weeks respecively. Noe ha hese resuls are differen han he previous pair, especially for he no rading ime. Now he naïve model has a higher sign success raio han he res and herefore achieves beer performance wih far fewer rades; his is imporan and documened in he ables ha follow. Finally, for he SPY-QQQ pair and Wednesday no model is beer han he bes performing asse bu he rading characerisics are similar o hose of he SPY-OIH pair. However, he sign success raio is no as good as before and herefore he rading performance is correspondingly lower. 5.2. Sign success raio and volailiy levels In Table 7 we presen he sign success raio ( m S ) + / n, i.e. he average number of imes ha a roaion was correc. A boosrap-based sandard error is also given. We can immediaely see ha, as noed jus above, for he cases where he roaion models ouperform he benchmarks we have ha he sign success raio is significanly higher han 50%. For example, for he case of he SPY-OIH pair and he moving average model i is equal o 54% wih a sandard error of 2.6% and for he SPY-XLE pair is equal o 57.5% wih a sandard error of 2.6%. On he oher Leich and Tanner (99) argued ha he ranking of forecass based on sign ess is closely relaed o heir ranking of correc predicions in simple rading sraegies. Pesaran and Timmerman (2005) also argued abou he imporance of having correc sign predicions in he conex of a rading mehodology. Lam and Li (2004) suggesed ha a correc predicion probabiliy should be around 60% in order for a rading sraegy o be economically significan wih a 0.% ransacion cos. 2
hand for he SPY-QQQ pair he sign success raio is no significanly differen from 50% as i is compued for all models a or below 50% wih sandard errors of almos 2.5%. An ineresing quesion, ha relaes o he success (or no!), of he proposed models and sraegy has o do wih he role ha volailiy plays in generaing correc sign predicions and successful rading signals. To examine his dependence of sign forecasing, rading signals and volailiy we compue a chi-square ype es on binary variables ha are defined as he following combinaions 2 : le ( m ) ( ( m Y ˆ ) + I y+ 0) = > be he variable ha couns he posiive predicions (signals o rade he firs asse), le ( m ) ( ( m Z I yˆ ) 0) I( y 0) = > > be he variable ha couns + + + he correc posiive predicions (i.e. signals o correcly rade he firs asse) and le X ( ) = I V > be he variable ha couns wheher he volailiy of he firs asse was higher + i 0 han ha of he second asse. For he pairs Y Z and + + + + we compue chi-square ype ess o examine heir dependence. 3 The resuls from hese ess are given in Table 8 and we find ha some dependence exiss for he following pair-day combinaions: for he SPY-OIH pair and Monday we find dependence beween Y + + for he piecewise linear model (which is he relaive reurn rading model wih bes Sharpe raio for Monday); for he SPY-OIH pair and Wednesday we find dependence beween boh Y Z and + + + + for he moving average and he piecewise linear model (moving average is he op model here); for he SPY-OIH pair and Friday we find dependence beween boh ( m Y ) Z and ( m ) + + + + for he naïve model (second bes model here). For he SPY-XLE and SPY-QQQ pairs and all hree days we find dependence beween boh Y Z and + + + + for he naïve model only. These resuls are mixed, offering no conclusive evidence on he poenial relaionship beween higher volailiy and higher sign success raios, excep for he SPY-OIH pair ha has exhibied a srong upward rend during he evaluaion period. On he oher hand, for he oher wo pairs i is only he naïve model ha appears o be drive by volailiy consideraions (and is no he bes performer across days and he wo pairs). This resul migh no be surprising since he naïve model does no incorporae volailiy dynamics direcly bu only hrough heir effec in reurns, while he piecewise linear model does include volailiy dynamics in is specificaion. Finally, noe ha again we see ha he day of he week can be a crucial facor in erms of boh rading performance and he explanaion ha one can aribue o such performance. 2 Kavajecz and Odders-Whie (200) examined volailiy wihin hree relaed inra-day series, ransacion reurns, quoe midpoin reurns, and limi order book midpoin reurns using as daa span NYSE lised socks using GARCH mehodology. 3 Boosrap-based p-values are being repored in Table 8. 3
6. Concluding remarks In his paper we invesigaed he saisical and economic viabiliy and rading performance of marke iming models based on asse roaion. Using daa on hree of he mos widely raded Exchange Traded Funds (ETF) we find supporive evidence in favour of he marke iming lieraure: he performance of a roaion sraegy is op in 8 ou of he 9 examined model combinaions, across hree asse pairs and hree weekdays (Monday, Wednesday and Friday). Our resuls show ha even a naive model wih no dynamics can be useful, or a leas should be used as a benchmark, in roaion-based on oher marke iming sraegies. Furhermore, we find srong evidence ha sign success is imporan for good rading performance and link good performance in sign predicion wih fewer rading signals. Our analysis suggess he ime-varying relaive reurns and volailiies can lead o poenially profiable rading sraegies. Overall, our sudy propose ha marke iming sraegies can be profiable even when execued from simple models and ha here is room for much improvemen in he srucure of he models used for implemening he roaion mehodology. 4
References Benning C., J., 997, Predicion skills of real-world marke imers, The Journal of Porfolio Managemen, 23, pp55 65. Breen W., Glosen L.,R., Jagannahan R., 990, Economic Significance of predicable variaions in sock index reurns, Journal of Finance, 44 pp77 89. Beebower G.,L., Varikooy A.,P., 99, Measuring marke iming sraegies, Financial Analyss Journal; 47 pp78 84. Brooks, C., 996, Linear and Non-linear (Non-) Forecasabiliy of High-frequency Exchange Raes, Journal of Forecasing, 6, pp 25-45. Fleming, J., Kirby, C., Osdiek, B., 200, Economic value of Volailiy Timing, Journal of Finance,, pp. 329-352. Fleming, J., Kirby, C., Osdiek, B., 2003, The economic value of volailiy iming using realized volailiy, Journal of Financial Economics 67, pp 473 509 Johannes M.,S., Polson, N., Sroud, J.,R., 2002, Sequenial Opimal Porfolio Performance: Marke and Volailiy Timing, working paper Jiang, W., 2003, A nonparameric es of marke iming Journal of Empirical Finance, 4, pp 399-425 Kavajecz, K., Odders-Whie, R.,E., 200, Volailiy and marke srucure, Journal of Financial markes, 4 pp359-384. Lam, K., Lee, L., 2004, Is he Perfec Timing Sraegy Truly Perfec?, Review of Quaniaive Finance and Accouning 22, pp 39-5 W., Li, W., Lam, K., 2002, Opimal marke iming sraegies under ransacion coss, The Inernaional Journal of Managemen Science 30, pp 97-08. Lee W., 997, Marke iming and shor-erm ineres raes, The Journal of Porfolio Managemen 23, pp 35 46. Levis M., Liodakis M.,. 996, The profiabiliy of syle roaion sraegies in he Unied Kingdom, Journal of Porfolio Managemen 26, pp 73 86. Larsen G.,A., Wozniak G.,D., 995, Marke iming can work in he real world, The Journal of Porfolio Managemen, 2 pp74 8. Manganelli, S., 2005, Duraion volume and volailiy impac of rades, Journal of Financial markes, 4 pp377-399. Pesaran M.,H, Timmermann A., 994, Forecasing sock reurns: and examinaion of sock marke rading in he presence of ransacion coss, Journal of Forecasing; 3 pp 335 367. Pesaran M.,H, Timmermann, A., 995 A. Predicabiliy of sock reurns: robusness and economic significance, The Journal of Finance, 50, pp20-228. Sy M., 990, Marke iming: is i a folly?, Journal of Porfolio Managemen, 6, pp 6. 5
Shilling, A.,G., 992 Marke iming: beer han a buy-and-hold sraegy, Financial Analyss Journal, 48, pp 46 50. Taylor S.,J., 986, Modelling financial ime series. Chicheser: Wiley. Thomakos, D.,D., Wang, T., Wu J., 2007, Marke cap and Roaion Sraegies Mahemaical and Compuer Modelling, 46, pp 278-29 Vandell R.,F., Sevens J.,L., 989, Evidence of superior performance form iming, The Journal of Porfolio Managemen, 5, pp38 43. Wang, K.,Q., 2005, Mulifacor Evaluaion of Syle Roaion, Journal of Financial and Quaniaive Analysis, 40, pp 349-372 Wagner J.,C., 997, Why marke iming works, The Journal of Invesing, 6, pp78 8. 6
Table Trading Performance - Monday The resuls correspond o he rading performance of he models of equaions (7), (8), (9) and (0):naïve, moving average, piecewise linear, differences in volailiy and volailiy raio. The sample periods are: for he SPY-OIH pair from 7 February 200 o April 2008; for he SPY-XLF pair from 22 December o 4 of April 2008; and for he SPY-QQQ pair from he 0 March 999 o 4 April 2008. The rolling window is 04 weeks Panel A: S&P500-OIH pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 OIL Equally Weighed End wealh 2.069.589 2.47 2.47.996.387 2.47.767 Average Reurn 0.46 0.254 0.494 0.494 0.429 0.66 0.494 0.33 Sandard Deviaion 3.87 2.830 3.932 3.932 3.55.827 3.932 2.477 Sharpe Raio 0.9 0.090 0.26 0.26 0.22 0.09 0.26 0.33 Minimum Realized Reurn -0.77-0.09-0.77-0.77-0.77-5.537-0.77-7.570 Panel B: S&P500-XLF pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 Financial Secor Equally Weighed End wealh.888.702 2.029 2.75 2.228.45.899.522 Average Reurn 0.268 0.22 0.3 0.355 0.37 0.044 0.272 0.58 Sandard Deviaion 3.484 3.060 3.48 3.454 3.22 2.45 3.498 2.670 Sharpe Raio 7.702 6.932 8.932 0.282.58.782 7.769 5.905 Minimum Realized Reurn -4.675-3.744-4.675-4.675-3.744 -.632-4.675-2.688 Panel C: S&P500-QQQ pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 Nasdaq 00 Equally Weighed End wealh.032 0.856 0.92 0.967.282.045 0.948 0.997 Average Reurn 0.00-0.0446-0.0273-0.003 0.0877 0.04-0.06-0.000 Sandard Deviaion 2.75 3.50 3.722 3.729 2.980 2.445 3.73 2.982 Sharpe Raio 0.369 -.46-0.732-0.277 2.942 0.578-0.432-0.033 Minimum Realized Reurn -.632 -.632-4.320-4.320 -.632 -.632-4.320-0.779 7
Table 2 Trading Performance - Wednesday The resuls correspond o he rading performance of he models of equaions (7), (8), (9) and (0):naïve, moving average, piecewise linear, differences in volailiy and volailiy raio. The sample periods are: for he SPY-OIH pair from 7 February 200 o April 2008; for he SPY-XLF pair from 22 December o 4 of April 2008; and for he SPY-QQQ pair from he 0 March 999 o 4 April 2008. The rolling window is 04 weeks Panel A: S&P500-OIH pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 OIL Equally Weighed End wealh 2.323 2.464 2.234 2.234.829.523 2.234.878 Average Reurn 0.005 0.006 0.005 0.005 0.003 0.002 0.005 0.003 Sandard Deviaion 0.035 0.028 0.036 0.036 0.03 0.07 0.036 0.023 Sharpe Raio 0.44 0.97 0.29 0.29 0.02 0.8 0.29 0.45 Minimum Realized Reurn -0.7-0.086-0.7-0.7-0.7-0.060-0.7-0.080 Panel B: S&P500-XLF pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 Financial Secor Equally Weighed End wealh.87.638.822.6.709.025.83.428 Average Reurn 0.233 0.7 0.220 0.63 0.90 0.007 0.222 0.4 Sandard Deviaion 3.23 2.738 3.056 3.4 2.875 2.250 3.7 2.404 Sharpe Raio 7.456 6.234 7.93 5.243 6.598 0.298 7.33 4.763 Minimum Realized Reurn -.5-0.908 -.5 -.5 -.5-0.908 -.5 -.030 Panel C: S&P500-QQQ pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 Nasdaq 00 Equally Weighed End wealh.02 0.989.03.04 0.973.50.039.095 Average Reurn 0.028-0.00293 0.0087 0.02-0.0073 0.043 0.007 0.0260 Sandard Deviaion 2.438 3.092 3.66 3.675 2.903 2.27 3.676 2.837 Sharpe Raio.52-0.095 0.237 0.305-0.252.865 0.292 0.98 Minimum Realized Reurn -0.908-6.454-6.454-6.454-6.454-0.908-6.454-3.68 8
Table 3 Trading Performance - Friday The resuls correspond o he rading performance of he models of equaions (7), (8), (9) and (0):naïve, moving average, piecewise linear, differences in volailiy and volailiy raio. The sample periods are: for he SPY-OIH pair from 7 February 200 o April 2008; for he SPY-XLF pair from 22 December o 4 of April 2008; and for he SPY-QQQ pair from he 0 March 999 o 4 April 2008. The rolling window is 04 weeks Panel A: S&P500-OIH pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 OIL Equally Weighed End wealh 2.67 2.400 2.53 2.88.957.496 2.88.842 Average Reurn 0.005 0.005 0.005 0.005 0.004 0.002 0.005 0.003 Sandard Deviaion 0.036 0.028 0.037 0.037 0.032 0.08 0.037 0.023 Sharpe Raio 0.28 0.97 0.22 0.25 0.6 0.0 0.25 0.40 Minimum Realized Reurn -0.22-0.06-0.22-0.22-0.22-0.059-0.22-0.083 Panel B: S&P500-XLF pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 Financial Secor Equally Weighed End wealh.829.425 2.049.996.49.05.880.448 Average Reurn 0.229 0.8 0.290 0.275 0.36 0.004 0.243 0.24 Sandard Deviaion 3.45 2.678 3.03 3.4 2.92 2.239 3.39 2.40 Sharpe Raio 7.282 4.389 9.338 8.834 4.66 0.86 7.747 5.32 Minimum Realized Reurn -3.76 -.8 -.8 -.8 -.8 -.8-3.76-2.440 Panel C: S&P500-QQQ pair Naïve ARIMA Differ. of Volailiy Volailiy Raio Piecewise Linear S&P500 Nasdaq 00 Equally Weighed End wealh.092.206.66.93.340.89.234.2 Average Reurn 0.026 0.0584 0.047 0.0548 0.0964 0.0535 0.0662 0.0598 Sandard Deviaion 2.434 3.268 3.789 3.784 2.556 2.2 3.788 2.876 Sharpe Raio.072.787.242.448 3.772 2.49.747 2.080 Minimum Realized Reurn -.8-7.853-7.853-7.853 -.8 -.8-7.853-4.485 9
Table 4 Trading Time & Duraion Saisics - Wednesday The able presens saisics based on he rading ime and duraion characerisics of he roaion sraegy. A rade is couned as a binary variable ha indicaes a swich from one asse o he oher. Sample period evaluaion is as in Table. Saisic SPY-OIH pair Naïve Mov. Avg. Piecewise Linear Number of rades 233 44 49 Mean rading ime 0.886 0.548 0.567 No Trade To No Trade 0.833 0.462 0.434 No Trade To Trade 0.67 0.538 0.566 Trade To No Trade 0.022 0.448 0.436 Trade To Trade 0.978 0.552 0.564 Mean Duraion No Trade 6.86.75 Max Duraion No Trade 9 6.00 0.00 Mean Duraion Trade 8 2.23 2.29 Max Duraion Trade 20 8.00 4.00 Table 5 Trading Time & Duraion Saisics - Wednesday The able presens saisics on he rading ime and duraion characerisics of he roaion sraegy. A rade is couned as a binary variable ha indicaes a swich from one asse o he oher. Sample evaluaion is as in Table 2. SPY-XLF pair Naïve Mov. Avg. PieceWise Linear Number of rades 37 63 4 Mean rading ime 0.099 0.436 0.377 No Trade To No Trade 0.982 0.586 0.685 No Trade To Trade 0.08 0.44 0.35 Trade To No Trade 0.89 0.540 0.525 Trade To Trade 0.8 0.460 0.475 Mean Duraion No Trade 8.500 2.402 3.0 Max Duraion No Trade 98.000 0.000 34.000 Mean Duraion Trade 5.286.852.905 Max Duraion Trade 8.000 6.000 5.000 20
Table 6 Trading Time & Duraion Saisics - Wednesday The able presens saisics based on he rading ime and duraion characerisics of he roaion sraegy. A rade is couned as a binary variable ha indicaes a swich from one asse o he oher. Sample period evaluaion is as in Table 3. Saisic SPY-QQQ pair Naïve Mov. Avg. Piecewise Linear Number of rades 43 75 99 Mean rading ime 0.394 0.482 0.548 No Trade To No Trade 0.932 0.553 0.59 No Trade To Trade 0.068 0.447 0.409 Trade To No Trade 0.06 0.483 0.338 Trade To Trade 0.894 0.57 0.662 Mean Duraion No Trade 4.667 2.238 2.448 Max Duraion No Trade 25 7 2 Mean Duraion Trade 9.067 2.07 2.940 Max Duraion Trade 65 7 3 p-value for Independen Tes 0.000 0.80 0.000 Table 7 Sign Success Raio - Wednesday The able represens he esimaed sign success raio (average number of correcly prediced signs over he evaluaion period) along wih is sandard error. The sandard error is calculaed using he saionary boosrap and 400 ieraions. Sample period evaluaion is as in Table 2. Panel A: SPY-OIH pair Model --> Naïve Mov.Avg. Differ. in Volailiy Volailiy Raio Piecewise Linear Sign success raio 0.532 0.540 0.535 0.535 0.523 Sandard Error 0.027 0.026 0.025 0.026 0.02 Panel B: SPY-XLF pair Sign success raio 0.575 0.484 0.567 0.550 0.538 Sandard Error 0.026 0.023 0.023 0.028 0.032 Panel C: SPY-QQQ pair Sign success raio 0.506 0.486 0.480 0.479 0.478 Sandard Error 0.024 0.025 0.024 0.024 0.026 2
Table 8 Tes for Independence beween Sign Success Raio and Volailiy The able presens boosrap based p values for a chi squared es of ( m ) ˆ ( m Y = I y ) > 0, independence beween he binary variables ( ) + + ( m ) ( ( m Z ˆ ) + = I y+ > 0) I( y+ > 0) and X = I( V > + 0) Y Z + + + + Panel A: SPY OIL pair Y Z Y + + + + + + Z + + Monday Wednesday Friday Naïve 0.2 0.35 0.625 0.25 0.00 0.002 MA 0.693.0 0.076 0.073 0.75 0.742 Piecewise 0.079 0.29 0.065 0.062 0.756 0.45 Y Z + + + + Panel B: SPY XLF pair Y Z Y + + + + + + Z + + Monday Wednesday Friday Naïve 0.000 0.000 0.002 0.00 0.005 0.005 MA 0.29 0.278 0.898.0 0.089 0.893 Piecewise 0.005 0.284 0.378.0 0.2 0.67 Y Z + + + + Panel B: SPY QQQ pair Y Z Y + + + + + + Z + + Monday Wednesday Friday Naïve 0.089 0.273 0.0 0.062 0.005 0.097 MA 0.535.0.0 0.826 0.345 0.633 Piecewise 0.829 0.83 0.434 0.667 0.420 0.53 22
re -0.0 0.00 0.0 0.0000 0.005 0.0030 0 00 200 300 400 0 00 200 300 400 Index Index re2-0.0 0.00 0.0 vol2 0.0000 0.005 0.0030 vol 0 00 200 300 400 0 00 200 300 400 Index Index Figure : Figure : Log weekly reurns and weekly realized range-based volailiy for Wednesday and for he SPY-OIH pair. The sample is from March 200 o April 2008. 23
re -0.0 0.00 0.0 0.0000 0.005 0.0030 0 00 200 300 0 00 200 300 Index Index re2-0.5 0.00 0.0 vol2 0.000 0.002 0.004 vol 0 00 200 300 0 00 200 300 Index Index Figure 2: Log weekly reurns and weekly realized range-based volailiy for Wednesday and for he SPY-XLF pair. The sample is from December, 998 o April, 2008. 24
re -0.0 0.00 0.0 0.0000 0.005 0.0030 0 00 200 300 400 0 00 200 300 400 Index Index re2-0.2 0.0 0. 0.2 vol2 0.000 0.00 vol 0 00 200 300 400 0 00 200 300 400 Index Index Figure 3: Log weekly reurns and weekly realized range-based volailiy for Wednesday and for he SPY-QQQ pair. The sample is from March, 999 o April, 2008. 25
Densiy Esimaes of Sraegies' Reurns - Wednesday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn 26
Densiy Esimaes of Sraegies' Reurns - Friday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn Figure 4: Densiy disribuions for he hree days of he week (Monday-Wednesday-Friday) and for he SPY-OIH pair. The sample is from March 200 o April 2008. 27
Densiy Esimaes of Sraegies' Reurns - Monday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn Densiy Esimaes of Sraegies' Reurns - Wednesday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn 28
Densiy Esimaes of Sraegies' Reurns - Friday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn Figure 5: Densiy disribuions for he hree days of he week (Monday-Wednesday-Friday) and for he SPY-XLF pair.. The sample is from December, 998 o April, 2008. 29
Densiy Esimaes of Sraegies' Reurns - Monday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn Densiy Esimaes of Sraegies' Reurns - Wednesday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn 30
Densiy Esimaes of Sraegies' Reurns - Friday 0 5 0 5 20 Naive Mov.Avg. Differ. in Vol. Vol. Raio PieceWise Linear -0.0-0.05 0.00 0.05 0.0 Reurn Figure 6: Densiy disribuions for he hree days of he week (Monday-Wednesday-Friday) and for he SPY-QQQ pair.. The sample is from March, 999 o April, 2008. 3