62 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 A Sraegy for Tradig he S&P 500 Fuures Marke Edward Olszewski * Absrac A sysem for radig he S&P 500 fuures marke is proposed. The sysem is applied o S&P 500 fuures daa durig he period from Sepember 14, 1987, o Sepember 27, 1999. The sysem uses a momeum oscillaor for geeraig ery or exi prices. I addiio, he sysem uses aoher idicaor for predicig he direcio of he red. Whe oly he oscillaor is used for selecig rades, he sysem is o, i geeral, as good as buy-ad-hold. However, whe he red idicaor is used as a filer, he radig sysem is, a leas, as good as buy-ad-hold. (JEL G14) Iroducio Resuls of a previous sudy (EAO) suggesed ha iefficiecy migh have exised i some fuures markes (Olszewski 1998). I ha sudy a red-followig radig sysem was applied o e fuures markes ad show o geerae large, hypoheical profis for a few diverse markes; however, for some of he markes cosidered, oably he S&P 500, he radig sysem was uable o geerae profis cosisely. I ha sudy he reaso speculaed for his failure i he case of he S&P 500 was ha a more appropriae radig sysem migh be oe which geeraes buy or sell sigals couer o he direcio of marke momeum, i.e., buy o high egaive momeum or sell o high posiive momeum. The purpose of his sudy is o evaluae he profiabiliy of radig he S&P 500 fuures idex usig such a radig sysem. Evaluaig he profiabiliy of a radig sysem requires ha some bechmark be seleced for compariso. For his purpose he sraegy buy-ad-hold (BAH) has bee chose. The reaso for his choice is ha BAH has bee ad coiues o be a relaively profiable sraegy sice he early 1980s. The radig sysem i his sudy is based o priciples similar o hose o which he radig sysem i he EAO sudy is based, uilizig such iformaio as momeum ad red characerisics of prices. However, he radig sysem i his sudy differs from ha i he EAO sudy because i icorporaes a addiioal filer for rade selecio based o he commime of raders repor released biweekly by he Commodiy Fuures Tradig Commissio (CFTC). The orgaizaio of he paper is as follows. I he secod secio, daa used i his sudy are preseed, ad he procedure for cosrucig a coiuous price series is explaied. Nex, a * Edward Olszewski, Deparme of Physics, Uiversiy of Norh Carolia a Wilmigo, Wilmigo, NC 28403, moof@pooie.phy.ucwil.edu.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 63 rigorous defiiio of marke momeum derivable from he pah iegral represeaio of he Lagevi equaios is iroduced. Log-rage memory processes are he briefly described, ad he modified rescaled rage (R/S) saisic of Lo is preseed (Lo 1991) Fially, his secio cocludes wih he iroducio ad ierpreaio of a descripive saisic calculaed from daa i he commime of raders repor. I he hird secio, he log-rage depedece of he S&P 500 fuures daa is assessed usig he R/S saisic. The a radig sysem based o marke momeum is proposed. The radig sysem is applied o he S&P 500 fuures daa ad is profiabiliy evaluaed. The radig sysem is modified usig he descripive saisic o filer rades, ad he profiabiliy of he radig sysem is re-evaluaed. Fially, here is discussio abou how performace of he modified radig sysem compares wih buy-ad-hold. I he fourh secio, resuls are summarized, ad a fuure direcio of sudy is oulied. Daa ad Mehodology Daa The ime frame wihi which he radig sysem is applied covers he period from 9/14/87 o 9/27/99. This ime frame coais 3,352 daa pois. The daa se is divided io wo subses: he firs from 9/14/87 o 6/9/93 (1,451 radig days) ad he secod from 6/10/93 o 9/27/99 (1,583 radig days). The radig sysem is applied o each subse separaely. There are wo reasos for dividig he daa se. The ime frame of he firs subse correspods o he ime frame used i he EAO sudy so ha comparisos ca be made wih ha sudy. Also, divisio of he daase io wo subses permis chages i he profiabiliy of he radig sysem o be observed over ime. The daa cosis of daily high, low, ad seleme prices of he S&P 500 fuures idex for he delivery mohs March, Jue, Sepember, ad December. Also, 1,316 addiioal daa pois, prior o 6/14/87, are prepeded o he daa for iiializig parameers i he radig sysem. Thus, he oal umber of daa pois is 4,352, begiig o 6/30/82 ad edig o 9/27/99. O ay give day muliple fuures coracs rade; however, a sigle ime series mus be cosruced for he subseque aalysis. The mehod seleced for cosrucig he ime series has he advaage of mimickig he daily perce chage i price of a rade, which is coiually rolled over from he ear corac o he firs forward corac. The rade is rolled over a he coclusio of radig o he las radig day of he moh prior o he expiraio moh of he ear corac. The procedure for cosrucig he bivariae series (S, r ) proceeds as i EAO. Firs, o a give day for all coracs, he quaiies s ad r are calculaed from ad s = l(p ) - l(p - 1 ), (1) r = max Ï Ô Ì Ô Ó l( H ) - l( L ) -1 l( H ) - l( P ) -1 l( L ) - l( P ) -1, (2) where (H, L, P ) are he high, low, ad seleme prices of a give corac. A sigle bivariae ime series is obaied by selecig cosecuive pairs (s, r ) from he corac begiig wih he earlies expiraio dae ad coiuig uil he las radig day of he moh prior o he expiraio moh of ha corac. O ha day pairs are he seleced from he firs forward
64 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 corac. The procedure is coiued i his maer uil a sigle pair (s, r ) is obaied for each radig day. This procedure provides a cosise way of selecig a sigle pair (s, r ) ou of all such pairs from he differe fuures coracs radig o a give day. The coiuous bivariae, ime series (S, r ) used i he aalysis is he defied by: S Ï 100 = 0 = Ì ÓS exp( s ) oherwise -1, (3) The series S is arbirarily defied so ha S 0 = 100, where = 0 correspods o Jue 13, 1986. The firs compoe of he ime series, S, is a syheic price series reflecig perce chage i he value of he fuures corac. This series offers he advaage of beig devoid of gaps i price, which ypically occur whe prices are rolled over from he ear corac o he firs forward corac. The secod compoe, r, is a measure of daily marke volailiy. FIGURE 1. CONTINUATION CONTRACT OF DAILY SETTLEMENT PRICES FOR THE S&P 500 INDEX FROM SEPTEMBER 14, 1987 TO JUNE 8, 1993 160 150 140 130 120 110 100 90 80 70 400 600 800 1000 1200 1400 1600 Noes: The value of he corac o Jue 13, 1986, has bee arbirarily se o 100. Figures 1 ad 2 show he syheic price series, S, for he ime-spa uder cosideraio. From Figure1, which covers abou six years, i ca be see ha he idex has appreciaed approximaely 15.0 perce, represeig 2.45 perce compouded, aual reur. The fac ha he plo begis abou oe moh prior o he crash of 1987 coribues appreciably o he modes rae of reur. Figure 2 also spas a ime of approximaely six years. I his case he rae of reur is greaer. Over his spa of ime he price icreases approximaely 255 perce, resulig i a compouded, aual reur of 17.0 perce.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 65 FIGURE 2. CONTINUATION CONTRACT OF DAILY SETTLEMENT PRICES FOR THE S&P 500 INDEX FROM JUNE 9, 1993, TO SEPTEMBER 27, 1999 400 350 300 250 200 150 100 1800 2000 2200 2400 2600 2800 3000 3200 Momeum The radig sysem o be esed uilizes a momeum idicaor for eerig ad exiig posiios. The momeum idicaor ca be derived rigorously from heoreical cosideraios (see EAO ad refereces herei). Is derivaio is ow oulied here. The derivaio begis wih a se of firs-order, sochasic differeial equaios, specifically Lagevi equaios, whose soluio is q(), beig a coiuous ime parameer. Aleraively, he Lagevi equaios ca be represeed as a pah iegral which is he codiioal probabiliy desiy of he q(). A Lagragia ca be obaied from he pah iegral, ad he momeum ca be rigorously defied as a appropriae derivaive of he Lagragia. The specific discree versio of he sochasic equaio releva for he subseque aalysis is q - m = se, (4) where is a ieger. Here s ad m are parameers characerizig q. I addiio, e is zero mea ad ui variace whie oise. I ca be show ha he momeum correspodig o q i Equaio 4 is give by p = q - m. (5) s 2
66 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 Log-Rage Memory Processes The hydrologis H. E. Hurs pioeered he sudy of log-rage memory processes i his ivesigaio of he sorage capaciy of reservoirs (Peers 1994; Hurs 1951). Exedig Hurs's work, Madelbro ad ohers iroduced a family of Gaussia radom fucios, desigaed fracioal Browia moios (fbm s), which exhibi log-erm persisece similar o hose sudied by Hurs (Madelbro ad Va Ness 1968; Madelbro ad Wallis 1969a-c; Bera 1994). As described i EAO, fbm s possess a umber of ieresig properies. The specral desiies of fbm s are proporioal o f 1-2H, f beig he frequecy ad H, 0 < H < 1, beig a parameer characerizig he fbm. 1 They exhibi he followig scalig behavior i he variace s 2 () = 2 H s 2 (1), where s() is he sadard deviaio of he fbm a ime. As deailed i Madelbro ad Wallis for H >.5, he fuure ad pas are posiively correlaed, wih correlaios approachig 1 as H approaches 1. These fbm s exhibi a log-rage memory characerized by a persisece i posiive correlaio bewee eves which are icreasigly separaed i ime. Cosequely, heir auocorrelaios decay much more slowly ha shor-rage memory processes like auoregressive or movig average. For H =.5 he fuure ad pas are ucorrelaed. For H <.5 he fuure ad pas are egaively correlaed, wih correlaios decreasig o -.5. These fbm s are characerized by ai-persisece, i.e., correlaios are all egaive. Such series are subjec o more reversals ha Browia moio. Deecig log-rage memory depedece i ime series, such as is foud i fbm s, ca be complicaed by he presece of shor-rage memory depedece. Ideed, i has bee poied ou by Lo ha he classical R/S aalysis used o ideify log-rage memory depedece ca falsely idicae is presece i a ime series whe, i fac, oly shor-rage memory depedece is prese. Lo has derived a es saisic for deecig log-rage memory depedece similar o he classical R/S saisic, R (Lo 1991). I is less proe o he misspecificaio of classical R/S aalysis, while also exhibiig power agais rejecig log-rage memory depedece whe i is prese. The es saisic akes accou of shor-rage memory depedece by adjusig he variace used i R. Specifically, give cosecuive pois from a discree ime series {x }, he es saisic Q is defied as 2 Rƒ Q ( qƒ), (6) s where Rƒ = max( x + º + x - kx) - mi( x + º + x - kx), (7) 1 1 k 1 k 1 k k x = 1  k = 1 x k, (8) qƒ 2 2 2 s ( qƒ) = s + w q x x x x  (ƒ) Ï j Ì Â ( - )( - ), i i - j j = 1 Ói = j + 1 (9) 1 The case H =.5 correspods o ordiary Browia moio. 2 The x correspods o he s give i Equaio 1.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 67 s = 2 Â x - x k = 1 ( ) k, (10) ad w j j ( qƒ) = 1 - qƒ + 1. (11) The value of he rucaio lag, q ~, depeds o he daa beig cosidered. Is value mus be large eough o accou for shor-rage memory depedece i he daa, bu o so large as o aler he fiie sample disribuio of Q radically. Adrews suggess he followig daa depede rule for selecig q ~ (Lo 1991; Adrews 1991): qƒ = k, k = 3 3 [ ] Ê Ë ˆ 2 1 Ê 2 p ˆ 2 Ë 1 - p 2 3, (12) where [k ] is he greaes ieger less ha or equal o k ad r^ is he esimaed firs-order correlaio of he daa. Also, he weighs w j (q ~ ) i Equaio 11 are replaced by w j j ( qƒ) = 1 -. (13) k Lo defies he saisic V (q ~ ), which is based o he es saisic Q, Q V ( qƒ). (14) This saisic is ideical o he saisic V j used by Hurs ad Peers, excep ha he variace has bee adjused o accou for shor-rage memory effecs (Peers 1994). Lo derives he limiig disribuio of V for which V (q ~ ) is a esimaor, uder he assumpio of o log-rage memory depedece ad a geeral se of codiios which iclude srog mixig (shor-rage memory) ad codiioal heeroskedasiciy. I Table 1 he fraciles of he limiig disribuio of he V saisic are preseed. Small values of V correspod o ai-persisece, ad large values correspod o persisece. If V (q ~ ) is less ha.861, he hypohesis of ai-persisece a he 5 perce cofidece level would be acceped. Similarly, if V (q ~ )is greaer ha 1.747, he hypohesis of persisece a he 5 perce cofidece level would be acceped. Commime of Traders Repor The fuures oly compoe of he commime of raders repor is released every wo weeks, usually o Friday, a 3:30 p.m. easer ime by he CFTC (he commime of raders repor). The repor icludes exesive iformaio abou he ope ieres of fuures markes i which five or more raders hold posiios equal o or i excess of he limis esablished by he CFTC. Each repor coais iformaio abou ope ieres of he previous wo Tuesdays. The ope ieres is classified as reporable ad o-reporable. No-reporable posiios are hose raders posiios
68 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 TABLE 1. FRACTILES OF THE LIMITING DISTRIBUTION OF THE V STATISTIC UNDER THE ASSUMPTION OF NO LONG-RANGE MEMORY Prob(V < v).005.0250.050.100.200.300.400.500 v 0.721 0.809 0.861 0.927 1.018 1.090 1.157 1.223 Prob(V < v).543.600.700.800.900.950.975.995 v p 2 1.294 1.374 1.473 1.620 1.747 1.862 2.098 ha are below he limis esablished by he CFTC. Of ieres i his sudy are he reporable posiios, which are posiios held by raders i excess of limis esablished by he CFTC. Tha limi for he S&P 500 fuures marke is 600 coracs. The reporable posiios are subdivided io commercial (raders who are classified as reporable ad egage i hedgig) ad o-commercial. For he purpose of his sudy he posiios held by commercials are impora. The specific iformaio eeded from he repor is he umber of log posiios, N lo, ad he umber of shor posiios, N sh, held by commercials. From hese quaiies he followig descripive saisic is defied I = N N lo lo - N + N sh sh. (15) The saisic I, (-1 I +1), measures he fracio of oal ope ieres held by hedgers, i.e., commercials. Wheever I > 0, hedgers are e log; for I < 0, hedgers are e shor. If I = 0, he umber of log ad shor hedged posiios is equal. Tradig Sysem A radig sysem cosise wih he followig paradigm of marke behavior is proposed. While redig eiher up or dow, he marke may experiece eiher a over-bough or over-sold codiio. Buy (sell) sigals are geeraed whe he marke becomes over-sold (over-bough). Typically, ormalized momeum idicaors (oscillaors) ca be used o deermie wheher a marke is over-bough or over-sold. Some examples of ormalized momeum idicaors are Wilder s RSI, Lae s Sochasics, or Lamber s CCI. Murphy discusses he use of such oscillaors i his coex (Murphy 1986, p. 279-284). Momeum, as defied by Equaio 5, is aoher example of a ormalized momeum idicaor ad cosiues oe compoe of he radig sysem used i his sudy. Accordig o Equaio 5, momeum is obaied by subracig he codiioal mea from he price ad dividig he resul by he codiioal variace. The defiiio is iuiively appealig because momeum is based o oly o how much he price deviaes from wha is expeced bu also o how much i deviaes i compariso o curre volailiy. The drawback of his defiiio is ha oe eeds esimaes of he codiioal mea, i.e., wha is expeced, ad also he codiioal variace, i.e., curre volailiy, o calculae is value. The radig sysem uses momeum i cojucio wih he prediced direcio of he red. Specifically, buy sigals are ake oly whe he prediced direcio of he red is up, ad sell sigals oly whe i is dow. I is assumed ha he direcio of he red ca be prediced from he e ope ieres of hedgers. If hedgers are e log (shor), he red is prediced o be up (dow).
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 69 The hedge saisic I, Equaio 15, is used for his purpose. 3 The assumpio is debaable sice i is corary o he idea ha speculaors, o average, ear profis by assumig he risk of hedgers. The obvious implicaio is ha speculaors would, o average, lose moey, which may imply a egaive risk premium. Wheher his assumpio ca be jusified ress, i par, o wheher he radig sysem is profiable. Persise vs. Ai-Persise Behavior ad he Modified R/S Tes I is ow show ha he S&P 500 fuures idex has he edecy o exhibi ai-persise behavior. Such behavior jusifies usig momeum o deermie over-bough or over-sold codiios. If prices are ai-persise, he large price moves i a give direcio are likely o be followed by price moves i he opposie direcio. Thus, he presece of ai-persisece i price daa is compaible wih he idea of usig momeum o deermie wheher a marke is over-bough or over-sold. The R/S saisic V (q) give i Equaio 14 ca be used for his purpose. To es wheher he S&P 500 daa is ai-persise, he R/S saisic has bee calculaed for 46 cosecuive, overlappig subsamples of daa, each subsample comprisig 1,315 pois. Each successive subsample eds 66 radig days i he fuure of he previous subsample. To perform his subdivisio a addiioal 1,000 daa pois have bee prepeded o he daa se, resulig i he followig subsamples [1, 1,315], [67, 1,381], [133, 1,447], [199, 1,513],..., [2,905, 4,219]. 4 The saisic V (q) has bee calculaed for each subsample usig Equaios 12, 13, ad14. The reaso for selecig subsamples of 1,315 pois, i.e., = 1,315, is based i par o Lo s sudy of he fiie sample properies of V (q) i which he has simulaed daa possessig various log-rage or shor-rage depedecies[6]. I hese sudies Lo has used sample sizes (=100, 250, 500, 750, 1,000). Lo has foud ha for = 1,000 asympoic properies of he V saisic are reasoably well approximaed. 5 I Table 2 he R/S saisic V 1315 (q) is displayed for he various subsamples. For coras ad compariso he R/S saisic of he Japaese ye is also displayed, sice i is geerally believed ha he currecies exhibi srog redig characerisics ad hus should be persise (see Olszewski 1998 sudy of EAO). Accordig o Table 1, large values of V, (V > 1.223) correspod o persisece ad small values, i.e., (V < 1.223), o ai-persisece. Thus, i is appare from Table 2 ha he S&P 500 fuures prices have cosisely exhibied ai-persise behavior. This provides some jusificaio for esig he couer-o-momeum (C2M) radig sysem described i he ex secio. O he oher had, he resuls of Table 2 sugges ha, i he case of he Japaese ye, a red-followig radig sysem may be more appropriae. Ideed, i he sudy of EAO a red-followig radig sysem was applied o he ye ad foud o be profiable durig he ime period covered by he firs subse. Momeum-Based Tradig Sysem To operaioalize he previously preseed paradigm of marke behavior, i is assumed ha prices flucuae abou some approximae equilibrium value. This simplifies calculaios ad reduces he compuaioal ime for performig hem. 6 This assumpio implies he followig 3 The daa for calculaig I ypically become available every wo weeks, o Friday aferoo. Icluded i he daa is iformaio abou ope ieres for each of he wo previous Tuesdays. The value of I calculaed is based o he daa of he previous Tuesday. Sice he daa are available so lae i he day, he value of I is o used for radig uil he followig Moday. 4 These subsamples have bee seleced because each oe eds oe day before he begiig of a 66-day period used i esig he radig sysem. See Momeum-Based Tradig Sysem. 5 Oher ha his moivaio he selecio of = 1,315 has bee a ad hoc decisio. 6 More complicaed assumpios have bee ried wih o improveme i he perfomace of he radig sysem.
70 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 TABLE 2. V N (Q) STATISTIC PRIOR TO EACH OUT-OF-SAMPLE TRADING PERIOD Subse Oe Subse Two Period S&P 500 Japaese Ye Period S&P 500 Japaese Ye 1,315 1.19 1.81 2,767 1.00 1.51 1,381 0.96 2.02 2,833 1.03 1.49 1,447 0.96 1.98 2,899 1.06 1.49 1,513 0.96 1.96 2,965 1.06 1.52 1,579 0.95 2.02 3,031 1.00 1.22 1,645 0.95 1.91 3,097 1.02 1.02 1,711 0.96 2.01 3,163 1.13 1.07 1,777 0.97 2.16 3,229 0.92 0.87 1,843 0.97 2.00 3,295 0.96 1.01 1,909 0.96 2.02 3,361 0.81 1.13 1,975 0.95 2.00 3,427 1.28 1.70 2,041 0.95 2.00 3,493 1.10 1.77 2,107 0.92 2.12 3,559 1.09 1.69 2,173 0.92 1.86 3,625 1.08 1.79 2,239 0.92 1.58 3,691 1.36 2.03 2,305 0.92 1.60 3,757 1.14 2.09 2,371 0.93 1.51 3,823 1.28 2.05 2,437 0.94 1.63 3,889 1.21 1.87 2,503 0.93 1.53 3,955 1.04 2.01 2,569 0.95 1.52 4,021 1.13 1.82 2,635 0.90 1.52 4,087 1.36 1.65 2,701 0.98 1.44 4,153 0.86 1.87 4,219 0.77 1.86 4,285 0.77 1.71 model of marke prices (Equaio 4) q i = m + se i. (16) Here q i = l(s i ), where S i is give by Equaio 3, ad i =, -1, -2, º, -( d -1). 7 The quaiy m correspods o he equilibrium price, ad e i is he whie oise causig he flucuaios. The quaiy m is esimaed by m = Â i = -( -1) d d q i. (17) If i is assumed ha o ay give day i he prices durig ha day may be approximaed by a radom walk abou is mea, he s i may be esimaed usig he resul of Parkiso (Parkiso 1980), 1 2 p p s = 2 Ê ˆ Er Ë 8 = 8 [ ] Ê Ë ˆ i i i 1 2 r (18) Here r i is give i Equaio 2, E[ ] is expeced value, ad E[r i ] has bee approximaed by is bes esimae r i. The quaiy s is esimaed by averagig over all s i 7 I should be oed ha he logarihm of price, q i, is used raher ha price, S i.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 71 s = Â i = -( -1) d d s i. (19) The momeum p i is give by Equaio 5. A proxy for he momeum, which is dimesioless, is used i he subseque aalysis. I is give by q - m i p = p s =. i i s (20) The momeum-based compoe of he radig sysem ca ow be defied. There are five adjusable parameers: d, l, s, b l, b s, subjec o he cosrai s l < d. There are wo average momeum variables, ad l p = p s = Â Â i = -( -1) l i = l -( s -1) s p i p i (21) (22) The purpose of p l is o measure log-erm momeum ad p s o measure shor-erm momeum. The uderlyig philosophy of he radig sysem is ha, wheever boh shor-erm ad log-erm momea become large i he posiive or egaive sese, he a posiio is ake i he direcio opposie o he momeum; i.e., he momeum idicaors are used as oscillaors. The momeumbased radig rules are as follows. A he close of day 1. if o posiio is held i he marke, ad a. b. if p l < - b l ad p s < - b s, buy oe corac, or if p l > + b l ad p s > + b s, sell oe corac; c. oherwise, do ohig. 2. if a log posiio is held i he marke, ad a. b. if eiher p l > + b l or p s > + b s, close ou he posiio, or if p l > + b l ad p s > + b s, close ou he posiio ad sell oe corac; c. oherwise, do ohig. 3. if a shor posiio is held i he marke, ad a. b. if eiher p l < - b l or p s < - b s, close ou he posiio, or if p l < - b l ad p s < - b s, close ou he posiio ad buy oe corac; c. oherwise, do ohig. The radig sysem is applied o a sample of daa begiig o day for a umber of days ou, ad is profiabiliy is evaluaed. The five adjusable parameers of he radig sysem are deermied by firs opimizig he profis geeraed by he sysem i a sample of daa of size i
72 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 edig o day -1. 8 The procedure is he re-applied wih day beig replaced by day + ou. The procedure is followed uil he radig sysem has bee applied o he eire daa se uder cosideraio. A he ed of each radig period, all rades are closed ou. This isures ha profis geeraed i differe periods are idepede of each oher. Oe beefi of his mehodology is ha profiabiliy of he radig sysem is assessed for a umber of differe, o-overlappig samples, each wih is ow opimized parameers. This should help i avoidig he chace selecio of a profiable se of adjusable parameers. Aoher beefi is ha he adjusable parameers are allowed o evolve i ime, reflecig possible chages i marke codiios. Because of pracical cosideraios, some of he parameers of he radig sysem have bee assiged i a ad hoc maer. Firs, he size of each i-sample, i, used for opimizaio has bee se o 250 days (approximaely oe year). The size of each ou-of-sample, ou, used for evaluaio has bee se o 66 days (approximaely oe quarer). For he firs subse of daa, his resuls i 22 idepede samples for evaluaig profiabiliy. These samples cosis of pois [316, 381], [382, 447], [448, 513],..., [1,702, 1,767]. Pois [1, 66] are used for iiializig average momea, 9 ad pois [66, 315] are he firs se of pois used for opimizig he adjusable parameers which are applied i he firs ou-of-sample [316, 381]. For he secod subse his resuls i 24 idepede samples for evaluaig profiabiliy. These samples are [1,768, 1,833], [1,834, 1,899], [1,900, 1,966],..., [3,286, 3,352]. The calculaios for performig opimizaios are compuaioally iesive. To make he compuaios maageable, he search over values of he adjusable parameers which opimize he profis of he radig sysem have bee resriced. 10 The adjusable parameers ad he resricios placed o heir search values are lised i Table 3. If, i a give search, several differe ses of parameers yield he same opimum profis, he he se of parameers o be used for ou-of-sample esig is seleced by applyig each of he followig crieria, successively, uil oly oe se of parameers remais: selec he se wih he smalles value of (1) d, (2) l, (3) s, (4) b l, (5) b s. The radig sysem has bee applied o he wo subses of daa, ad is profis, alog wih he profis resulig from a BAH sraegy, are preseed i Tables 3.2 ad 3.2. 11 I addiio, for he firs subse he radig profis from he red-followig radig sysem of he sudy of EAO are also preseed. Cocerig he firs subse i is appare ha he profis of he C2M radig sysem are greaer ha hose of he red-followig sysem. However, i appears ha he C2M radig sysem performs slighly worse ha BAH. However, he C2M radig sysem holds posiios i he marke for oly 788 of he 1,451 radig days durig his period. A more appropriae way o coras his radig sysem o BAH is o compare profis oly durig hose periods whe he sysem holds posiios i he marke. The resuls of his compariso are preseed i Table 3.2. Noe i he able ha shor rades, overall, are profiable i spie of he fac ha he geeral red of he marke is up. This suggess ha he C2M radig sysem may be effecive i selecig 8 A sufficie umber of pois, he maximum of d (see Table 3), from he begiig of he series are excluded from he aalysis ad used for obaiig iiial values of average momea. 9 I addiio, 1,000 daa pois have bee prepeded o he daa se o calculae a iiial value of he rescaled rage (see Persise vs. Ai-Persise Behavior ad he Modified R/S Tes ). 10 Approximaely oe day of compuig ime has bee required o perform he opimizaios o he wo samples. Compuaios have bee performed by a dedicaed HP 735/125 worksaio operaig a approximaely 55 Mflops. 11 Profis are preseed i boh S&P 500 pois ad dollars. Noe ha prior o November 3, 1997, oe S&P 500 poi was valued a $500. O ha day he value of oe poi became $250. The umber of posiios carried hrough November 3 was doubled, i.e., ope ieres doubled, o compesae for he reducio i value of he corac. All aalyses i his sudy are doe assumig ha iiially oe corac is raded. O November 3 ad subseque days, he umber of coracs raded is wo. Thus, for purposes here, oe S&P 500 poi coiues o be equivale o $500.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 73 TABLE 3. PARAMETERS TO BE OPTIMIZED AND RESTRICTIONS ON THEIR SEARCH VALUES. Parameer Resricios d (Eq. 16) 22, 33, 44, 55, 66 l (Eq. 21) 1, 2, 3,..., 20 s (Eq. 22) 1, 2, 3,..., 20 b l (See radig rules.) 0, 1 2.88, 2 2.88, 3 2.88,... 19 2.88 b s (See radig rules,) 0, 1 2.88, 2 2.88, 3 2.88,... 19 2.88 profiable rades. Uforuaely, he variabiliy of he profis amog rades makes he average profi per rade o saisically differe from zero. For he secod subse of daa, he C2M radig sysem appears o be iferior o BAH. Buyad-hold is superior eve whe comparig profis oly whe he C2M sysem holds a posiio i he marke (see Table 7.). 12 I is appare from ispecig Figure 2 ha i would be difficul for a radig sysem o ouperform BAH durig his period. I ca be argued ha, for he firs subse of daa, he C2M sysem is as good as or perhaps eve slighly beer ha BAH. This is rue eve whe slippage ad commissios are ake io accou, for which a fair value would be abou $50. 13 Eve BAH will icur commissios sice coracs will eed o be rolled over a heir expiraio. For he secod subse of daa, he C2M sysem is clearly ouperformed by BAH. I he ex secio he radig sysem is modified, improvig profis i he secod subse of daa. Modified Momeum-Based Tradig Sysem As Murphy discusses (Murphy 1986, p. 276), a oscillaor should be used oly as a secodary idicaor. Murphy pois ou ha red aalysis is of primary imporace. Cosequely, i may be possible o improve he momeum-based radig sysem by icludig a TABLE 4. TRADING PROFITS FOR THE FIRST SUBSET OF DATA Corac Profi Profi/Trade Max. Drawdow No. of Trades C2M +127.30 +2.27-34.05 56 +$63,650 +$1137 -$17,025 Tred (EAO) -67.10-1.46-150.25 46 -$33,550 -$730 -$75,125 BAH +63.45 - - - +$31,725 Noes: Profis are give i boh S&P 500 pois ad dollars. The calculaio of maximum drawdow is based o closed rades. 12 As i he case of he firs subse, he average profi (loss) per rade resulig from shor rades is o saisically differe from zero. 13 This value for rasacio coss does o iclude he fac ha some of he rades would, i pracice, eed o be rolled over o avoid radig i he expiraio moh. Thus, a more realisic esimae of rasacio coss would be $60, sice, o average, every rade acually correspods o abou 1.2 rades whe rollovers are icluded.
74 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 TABLE 5. TRADING PROFITS FOR THE SECOND SUBSET OF DATA Corac Profi Profi/Trade Max. Drawdow No. of Trades C2M -60.10 -.90-350.40 67 -$30,050 -$449 -$175,200 BAH +708.50 - - - +$354,250 Noes: Profis are give i boh S&P 500 pois ad dollars. The calculaio of maximum drawdow is based o closed rades. TABLE 6. TRADING PROFITS FOR THE FIRST SUBSET OF DATA Profis Durig Periods of Shor Trades Log Trades Toal Profis No. of Tradig Days C2M +42.85 (31) +84.45 (25) +127.30 788 +$21,425 +$42,225 +$63,650 BAH -42.85 +84.45 +41.60 788 -$21,425 +$42,225 +$20,800 Noes: Comparisos are made oly durig periods whe he radig sysem holds a posiio i he marke. Of paricular ieres is he compariso o BAH whe he radig sysem is shor. The values i pareheses are umbers of rades. The oal umber of possible radig days is 1,451. TABLE 7. TRADING PROFITS FOR THE SECOND SUBSET OF DATA Profis Durig Periods of Shor Trades Log Trades Toal Profis No. of Tradig Days C2M -104.60 (35) +54.50 (32) -60.10 864 -$52,300 +$27,250 -$30,050 BAH +108.00 +67.80 +175.80 864 +$52,300 +$27,250 +$74,550 Noes: Comparisos are made oly durig periods whe he radig sysem holds a posiio i he marke. Of paricular ieres is he compariso o BAH whe he radig sysem is shor. The values i pareheses are umbers of rades. The oal umber of possible radig days is 1,583. compoe ha predics he direcio of he red. The hedge saisic, I (Equaio 15), is used for his purpose. Specifically, if I > 0, he red is prediced o be up; if I < 0, he red is prediced o be dow; if I = 0, o direcio is prediced for he red. The hedge saisic is he used o filer rades; i.e., o log posiios are iiiaed wheever I 0, ad o shor posiios wheever I 0. All posiios are exied as described i he previous secio. I Tables 8, 9, 10, ad 11, he modified radig sysem is compared o BAH. The modified radig sysem appears o perform beer ha he radig sysem wihou a red compoe. For example, i he firs subse of daa, he profi per rade has icreased from +2.27 S&P 500 pois o +2.89. I he secod subse i has icreased from -.90 o +5.46. More impora, he modified radig sysem performs, overall, a leas as well as BAH. This is evideced by he fac ha i boh subses of daa he modified
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 75 radig sysem shows a e profi for shor rades. This is especially oeworhy i he secod subse, where he red has bee decidedly up. I boh he firs ad secod subses, he overwhelmig majoriy of he rades have bee log. Wih hidsigh his is expeced sice, i boh subses, he red has bee up. Aoher poi is ha he modified radig sysem does o hold a posiio i he marke mos of he ime. I fac, for he firs subse he sysem is radig for abou 18 perce of he oal possible radig days ad garers 146 perce of he profis of BAH. For he secod subse i rades abou 20 perce of he ime, showig profis of abou 25 perce of BAH. I coclusio, he resuls sugges ha he modified radig sysem is a leas as good as BAH ad perhaps beer. This coclusio is discussed i furher deail i he ex secio. A fial poi is wheher he radig sysem could have bee implemeed i pracice, yieldig comparable radig profis. The aswer is yes. Sice his radig sysem is similar i some respecs o ha preseed i he sudy of EAO, he argumes preseed here apply here. The pricipal egaive aspec of his radig sysem is ha i may o maage risk a a level accepable o mos raders, as is evideced i he drawdows give i Tables 8 ad 9. Commes abou Marke Timig ad Ivesme Performace The modified C2M radig sysem has bee preseed as a aleraive o BAH. The basic ee is ha performace ca be improved by judiciously selecig whe o buy or sell he marke. Marke imig sraegies have bee discussed exesively. See, for example, he wo-par aricle of Mero ad Heriksso (Mero 1981; Heriksso ad Mero 1981). I par oe (Mero 1981), Mero iroduces he heoreical framework of a geeral marke imig model ad shows how imig sraegies subsumed by he model are equivale o cerai opio sraegies. The model icludes a broad base of radig sraegies desiged o ouperform BAH by ideifyig whe equiies are uder- or overvalued wih respec o fixed icome securiies. Excluded from cosideraio, however, are imig sraegies ha uilize shor-sellig. I par wo (Heriksso ad Mero 1981), Heriksso ad Mero develop saisical procedures for esig he performace of radig sraegies wihi he coex of heir model. Alhough here are similariies bewee he modified C2M radig sysem ad hose imig sraegies icluded i Mero's model, here is a impora differece: he modified C2M sysem uilizes shor sellig o ouperform BAH, raher ha ivesig i fixed icome securiies. Thus, wheher he modified C2M sysem ouperforms BAH depeds primarily o wheher he shor rades of he C2M sysem sigificaly improve is performace over BAH. This cao be direcly resolved usig he saisical procedures of Heriksso ad Mero, ad ideed, resolvig his uambiguously is beyod he scope of his paper. However, he saisical ess proposed by Heriksso ad Mero sugges a mehod of parially resolvig his. Durig he ime frame cosidered i his sudy, he modified C2M radig sysem has iiiaed oly ie shor rades, hree wihi he firs subse of daa ad six wihi he secod. The hree wihi he firs subse are profiable, ad hree of six wihi he secod subse are profiable. Three ou of hree profiable shor rades may sugges profiabiliy, bu hree ou of six profiable rades seems o be o beer ha radom chace. However, because of he exreme upred wihi he secod subse i appears ha radom selecio of ery ad exi pois for a shor posiio would be more likely o resul i a losig rade. Thus, hree ou of six profiable rades i he secod subse is beer ha radom chace. How much beer is o sraighforward o deermie, because he appropriae ull hypohesis for compariso is o obvious. Noeheless, assume he ull hypohesis o be ha shor rades are equally probable o be eiher profiable or uprofiable. Therefore, he umber of profiable rades would be biormally disribued wih p = q =.5, where p ad q are he probabiliies of a rade beig profiable ad uprofiable, respecively. If he umber of shor rades of boh subses are combied, here will be six profiable ad hree
76 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 uprofiable. Uder he ull hypohesis he probabiliy, P, of obaiig six or more profiable rades is 9 - P = Â 9! ( 9 ) 130 (. 5) (. 5) = =. 254.!( 9 - )! 512 = 6 (23) Thus, here is a 25 perce chace of his occurig, assumig he ull hypohesis. Give he overly coservaive aure of he ull hypohesis, his resul provides some quaiaive evidece ha he modified C2M radig sysem ouperforms BAH. To obai a more appropriae ull hypohesis, i.e., more accurae values of p ad q, would require Moe Carlo simulaios. TABLE 8. PROFITS OF THE MODIFIED TRADING SYSTEM FOR THE FIRST SUBSET OF DATA Corac Profi Profi/Trade Max. Drawdow No. of Trades C2M +92.60 +2.89-41.25 26 +$46,300 +$1,447 -$20,625 BAH +63.45 - - - +$31,725 Noes: Profis are give i boh S&P 500 pois ad dollars. The calculaio of maximum drawdow is based o closed rades. TABLE 9. PROFITS OF THE MODIFIED TRADING SYSTEM FOR THE SECOND SUBSET OF DATA Corac Profi Profi/Trade Max. Drawdow No. of Trades C2M +174.80 +5.46-67.05 32 +$87,400 +$2,731 -$33,525 BAH +708.50 - - - +$354,250 Noes: Profis are give i boh S&P 500 pois ad dollars. The calculaio of maximum drawdow is based o closed rades. Table 10. Tradig profis of he modified radig sysem for he firs subse of daa. Profis Durig Periods of Shor Trades Log Trades Toal Profis No. of Tradig Days C2M +15.00 (3) +77.60 (23) +92.60 264 +$7,500 +$38,800 +$46,300 BAH -15.00 +77.60 +62.60 264 -$7,500 +$38,800 +$31,300 Noes: Comparisos are made oly durig periods whe he radig sysem holds a posiio i he marke. Of paricular ieres is he compariso o BAH whe he radig sysem is shor. The values i pareheses are umbers of rades. The oal umber of possible radig days is 1,451.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 77 TABLE 11. TRADING PROFITS OF THE MODIFIED TRADING SYSTEM FOR THE SECOND SUBSET OF DATA Profis Durig Periods of Shor Trades Log Trades Toal Profis No. of Tradig Days C2M +.65 (6) +174.15 (26) +174.80 311 +$325 +$87,075 +$87,400 BAH -.65 +174.15 +173.5 311 -$325 +$87,075 +$86,750 Noes: Comparisos are made oly durig periods whe he radig sysem holds a posiio i he marke. Of paricular ieres is he compariso o BAH whe he radig sysem is shor. The values i pareheses are umbers of rades. The oal umber of possible radig days is 1,583. Discussio ad Coclusios Sice he mid 1980s he S&P 500 sock ad fuures idices have icreased i value appreciably, wih some remarkable gais occurrig durig he 90s. The rae of reur o he fuures idex has bee approximaely 10 perce compouded aually. Eve icludig he commissios associaed wih rollig over coracs, may idividuals would be coe wih such gais, which could have bee realized by adopig a buy-ad-hold sraegy. If he S&P 500 idex should sik io a proraced bear marke, buy-ad-hold may o be he sraegy of choice, bu raher some ype of radig sysem may be. I his sudy a radig sysem has bee proposed which seems o perform, a leas, comparably o buy-ad-hold. Furhermore, resuls of his sudy sugges ha he sysem is profiable wih shor as well as log rades, presumably makig he radig sysem appropriae for bear markes. The radig sysem is based o he fac ha he S&P 500 fuures idex exhibis ai-persise behavior. Qualiaively his implies ha large price moves i a give direcio ed o be followed by price moves i he opposie direcio. This is o a commo rai of all fuures markes. I fac, as show i his sudy, he Japaese ye exhibis persise behavior, implyig ha large price moves i a give direcio ed o be followed by price moves i he same direcio. Because of he ai-persise behavior of he S&P 500 fuures idex, he radig sysem has bee desiged o buy o dowurs ad sell o upurs. Rules for eerig or exiig posiios are based o he values of shor- ad log-erm momea, formulae which are derived from Lagevi equaios. The radig sysem has a umber of adjusable parameers, whose values are obaied by opimizig radig profis durig a 250-radig-day period (approximaely oe year) prior o each 66-day period (approximaely oe quarer) i which he sysem has bee applied. I order o improve he profiabiliy, he radig sysem is modified o iclude a compoe for predicig he direcio of he red. The direcio of he red is prediced from he e ope ieres of he commercials (hedgers). Specifically, if commercials are e log, he red is prediced up, ad if commercials are e shor, he red is prediced dow. Profis geeraed by he radig sysem are cosiderably improved by his modificaio, makig he radig sysem, a leas, as good as buyad-hold. A corollary o his resul is ha his mehod of predicig he red may be icosise wih he idea of a posiive risk premium. If oe believes ha hedgers sell heir risk o speculaors,
78 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 ad herefore speculaors should be rewarded for assumig ha risk, he he direcio of he red, o average, should be opposie o he e posiio of he hedgers. 14 Alhough he resuls of his sudy are suggesive, hey are o saisically rigorous. The obvious problem is he limied amou of daa, makig i impossible o replicae he sudy uder ideical marke codiios. Oe way of circumveig his problem is o cosruc a model of he daa ha akes io accou ai-persisece ad depedecies o he e posiio of hedgers. I addiio, he model should iclude effecs of codiioal heerogeeiy, which is well kow o be prese i log series of fiacial daa. The model could he be used o simulae marke daa. The radig sysem could be applied o he simulaios, ad saisically rigorous comparisos could be made o buy-ad-hold. This is lef for fuure ivesigaio. Refereces Adrews, D. 1991. Heeroskedasiciy ad Auocorrelaio Cosise Covaria Marix Esimaio. Ecoomerica 59: 817-858. J. Bera. 1994. Saisics for Log Memory Process. New York: Chapma ad Hall. Heriksso, R. D., ad R. C. Mero. 1981. O Marke Timig ad Ivesme Performace: II. Saisical Procedures for Evaluaig Forecasig Skills. Joural of Busiess 54: 513-533. Hurs, H. E. 1951. The Log-Term Sorage Capaciy of Reservoirs. Trasacios of he America Sociey of Civil Egieers 116: 770-799. Lo, A. W. 1991. Log-Term Memory i Sock Marke Prices. Ecoomerica 59: 1279-1313. Madelbro, B. B., ad J. R. Wallis. 1969. Compuer Experimes wih Fracioal Gaussia Noises: Par 1. Averages ad Variaces. Waer Resources Research 5: 229-241. Madelbro, B. B., ad J. R. Wallis. 1969. Compuer Experimes wih Fracioal Gaussia Noises: Par 1. Mahemaical Appedix. Waer Resources Research 5: 260. Madelbro, B. B., ad J. R. Wallis. 1969. Compuer Experimes wih Fracioal Gaussia Noises: Par 2. Rescaled Rages ad Specra. Waer Resources Research 5: 260-267. Madelbro, B. B., ad J. W. Va Ness. 1968. Fracioal Browia Moios, Fracioal Noises ad Applicaios. SIAM Review 10: 422-437. Mero, R. C. 1981. O Marke Timig ad Ivesme Performace: A Equilibrium Theory of Value for Marke Forecass. Joural of Busiess 54: 363-406. Murphy, J. J. 1986. Techical Aalysis of he Fuures Markes. New York Isiue of Fiace, New York, NY 10270. Olszewski, E. A. 1998. Assessig iefficiecy i he fuures markes. The Joural of Fuures Markes 18(6): 671-704. Parkiso, M. 1980. The exreme value mehod for esimaig he variace of he rae of reur. Joural of Busiess 53(1): 61-65. Peers, E. E. 1994. Fracal Marke Aalysis. New York: Joh Wiley ad Sos, Ic. 14 I a upublished sudy i has bee show ha, i he case of he Japaese ye ad cor fuures markes he direcio of he red is beer prediced as opposie o he e posiio of hedgers. This seems o be more cosise wih he idea of a posiive risk premium.
JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 79 The commime of raders repor. The Commodiy Fuures Tradig Commissio. The mos rece, as well as previous ediios of his repor, are available a hp://www.cfc. gov/dea/co.hml.