Risk-Adjusted, Ex Ante, Optimal, Technical Trading Rules in Equity Markets. Christopher J. Neely

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1 WORKING PAPER SERIES Risk-Adjused, Ex Ane, Opimal, Technical Trading Rules in Equiy Markes Chrisopher J. Neely Working Paper D hp://research.slouisfed.org/wp/999/ pdf Revised Augus 200 FEDERAL RESERVE BANK OF ST. LOUIS Research Division 4 Locus Sree S. Louis, MO 6302 The views expressed are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis, he Federal Reserve Sysem, or he Board of Governors. Federal Reserve Bank of S. Louis Working Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Federal Reserve Bank of S. Louis Working Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors. Phoo couresy of The Gaeway Arch, S. Louis, MO.

2 Risk-Adjused, Ex Ane, Opimal Technical Trading Rules in Equiy Markes Chrisopher J. Neely Augus 8, 200 Senior Economis, Research Deparmen Federal Reserve Bank of S. Louis S. Louis, MO 630 (34) (o), (34) (f), Primary Subjec Code: G0 - Financial Economics Secondary Subjec Code: G4 - Informaion and Marke Efficiency Keywords: echnical analysis, geneic programming, rading rule, sock price, equiy The views expressed are hose of he auhor and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis, or he Federal Reserve Sysem. The auhor hanks Ken Koch for research and programming assisance, Chuck Whieman, Paul Weller and hree anonymous referees for commens, Ramazan Gençay for privae correspondence on he implemenaion of he X eff measure and Franklin Allen and Riso Karjalainen boh for making heir programs available and for correspondence on he programs. Any errors are my own.

3 Risk-Adjused, Ex Ane, Opimal Technical Trading Rules in Equiy Markes Absrac: This paper uses geneic programming o consruc risk-adjused, ex ane, opimal, rading rules for he S&P 500 Index and hen characerizes he predicive conen of hese rules. These resuls exend previous resuls by using risk-adjusmen selecion crieria o generae ex ane rules wih improved performance. There is, however, no evidence ha he rules significanly ouperform he buy-and-hold sraegy on a risk-adjused basis. Therefore, he resuls are consisen wih marke efficiency. Neverheless, risk-adjusmen echniques should be seriously considered when evaluaing rading sraegies.

4 Risk-Adjused, Ex Ane, Opimal Technical Trading Rules in Equiy Markes The use of echnical rading rules rading rules based on pas price behavior has been common in equiy markes since he urn-of-he-cenury analysis of Wall Sree Journal edior Charles Dow. Because excess reurns generaed from publicly available informaion would seem o conradic he efficien markes hypohesis, a number of auhors have sudied he usefulness of echnical analysis in equiy markes (Brock, Lakonishok and Lebaron (992), Bessembinder and Chan (995, 998), Allen and Karjalainen (999), Lo, Mamaysky and Wang (2000)). Such sudies have generally evaluaed raw excess reurns raher han explicily risk-adjused reurns, leaving unclear he implicaions of heir work for he efficien markes hypohesis. Riskadjusing he reurns is necessary because he echnical rading sraegies spend ime ou of he marke and herefore have less volaile reurns han he buy-and-hold rule. Therefore, simply comparing he reurns o each sraegy is insufficien o compare he usefulness of he rules. An excepion o he failure o adjus for risk is he work of Brown, Goezmann and Kumar (998) ha found value in he risk-adjused reurns generaed by he marke signals of William Peer Hamilon. Anoher common problem in he rading rule lieraure is ha rules are evaluaed precisely because hey are widely used by echnical raders (see Brock, Lakonishok and Lebaron (992)). Ready (998) argues ha esing such rules is a form of daa snooping. This pracice is likely o produce spurious evidence of echnical rading profis; he rules are widely used precisely because hey would have been profiable on pas daa. Anoher excepion is he work of Bessembinder and Chan (998), which uses varying leverage wih is echnical rading rule o produce a rule wih approximaely he same risk as a buy-and-hold sraegy.

5 Raher han evaluaing widely used rules, one migh search for opimal, ex ane rules wih a nonlinear search procedure such as geneic programming (Koza (992)). Allen and Karjalainen (999) hereafer AK used geneic programming o generae opimal, ex ane, echnical rading rules on daily S&P 500 daa over he period 929 hrough They found ha he ransacions cos-adjused reurns o hese rules failed o exceed he reurns o a buy-andhold sraegy despie he exclusion of dividends from he sock reurn and ha he marke was efficien in his sense. 3 There was, however, some evidence of predicabiliy in reurns as he rules ended o be in he marke during periods of high reurns and ou of he marke during periods of low reurns. Alhough AK aribued his predicabiliy o low-order serial correlaion in he sock index, hey speculaed ha he rules migh be useful on a risk-adjused basis despie heir lower reurns. Even hough he rules do no lead o higher absolue reurns han a buyand-hold sraegy, he reduced volailiy migh sill make hem aracive o some invesors on a risk-adjused basis. (Allen and Karjalainen (999), p. 26) AK did no, however, use any riskadjusmen echniques in heir work. This paper exends he lieraure by invesigaing wheher ex ane, opimal echnical rading rules are useful on a risk-adjused basis in equiy markes. I is no sufficien merely o examine he resuls from previous geneic-programming rules wih common mehods of risk adjusmen. To fairly evaluae risk-adjused reurns, new ses of rules ha maximize riskadjused measures like he Sharpe raio (Sharpe (966)), he X* saisic (Sweeney and Lee (990)) and he X eff measure (Dacorogna e al. (200)) are generaed. In addiion, his paper 2 Neely, Weller and Dimar (997) and Neely and Weller (2000, 999) have applied geneic programming o find rading rules in he dollar foreign exchange marke and he European Moneary Sysem. 3 The reurn o a dynamic sraegy moving in and ou of he marke will be reduced less by he exclusion of dividends han will he reurn o a buy-and-hold sraegy. 2

6 more fully characerizes he predicabiliy found by hese rules and conducs formal ess of marke iming (Cumby and Modes (987)). A cenral poin of his sudy is ha risk adjusmen is no a secondary issue; i is absoluely essenial boh for evaluaing he usefulness of rading rules and for measuring he consisency of resuls wih marke efficiency (Sharpe (966), Jensen (968), Kho (996), Brown, Goezmann, and Kumar (998), Ready (998), Dowd (2000)). The rules fail o consisenly and significanly ouperform he buy-and-hold sraegy by any risk-adjused measure. Thus, his exercise exends previous resuls o find ha risk-adjused, ex ane, opimal rule reurns are consisen wih marke efficiency. The facs ha he marke indices used here exclude dividends and ha some predicabiliy may be due o spurious auocorrelaion only reinforce he negaive resuls. 2. METHODOLOGY A. Geneic Programming Geneic programming is a nonlinear search procedure for problems in which he soluion may be represened as a compuer program or decision ree (Koza (992)). Like is cousin, he geneic algorihm (Holland (975)), geneic programming uses he principles of parallel search and naural selecion o search for candidae soluions o problems of ineres. 4 Essenially, a compuer randomly generaes a populaion of candidae soluions expressible as decision rees o a problem of ineres. The rules are required only o be well defined and o produce oupu appropriae o he problem of ineres a buy/sell decision in he presen case. Of course, mos of hese random soluions will be quie poor, bu some, purely by chance, will "fi" he insample daa reasonably well, generaing excess reurns. The compuer hen allows he 3

7 populaion o "evolve" using reproducion and muaion operaors. Reproducion mixes subrees of he populaion while muaion replaces subrees wih new, randomly generaed subrees. Members of he populaion ha are more fi (profiable) have a greaer chance o reproduce whereas less fi members have a greaer chance of being replaced. In his way he geneic program searches promising areas of he soluion space by evolving a populaion of rules ha ends o become more adep a solving he problem in successive generaions. 5 B. Daa Ses Ten overlapping in-sample esimaion periods (929-35, , ) of daily S&P500 daa from 929 o 995 are inpu o he geneic program o consruc en ses of ex ane rading rules. There were en independen rules from each in-sample period. Each insample period of seven years was broken down ino a raining period (five years) and a selecion period (wo years) o alleviae he problem of overfiing he daa. 6 Rules wih posiive excess reurns over he buy-and-hold sraegy in he raining period were saved for ou-of-sample esing over he remainder of he daa (936-95, ). 7 4 Geneic algorihms require he soluion o he problem o be encoded as fixed-lengh characer srings raher han as decision rees or compuer programs as in geneic programming. 5 An alernaive (bu no equivalen) nonparameric procedure would be o use neural neworks o produce eiher forecass (Gençay (998b), Gençay and Sengos (997)) or rading rule signals (Gençay (998a)) on daily equiy marke daa. 6 Overfiing he finding of spurious paerns in he daa ofen resuls from applying a flexible saisical echnique o one sample of daa. Breaking he in-sample period ino wo subsamples helps alleviae his by requiring he paerns o be in boh samples. 7 To check for robusness, all he exercises were repeaed wih 5-year in-sample periods, wih and wihou runcaed ou-of-sample periods. Excep where oherwise noed, all he resuls in his paper proved robus o longer in-sample periods, wih and wihou shorer ou-of-sample periods. Full resuls are available from he auhor. 4

8 C. Trading Procedures and Reurn Calculaions This paper uses modified versions of AK s programs and similar procedures boh because he procedures are sensible and o mainain maximum comparabiliy o he unadjused resuls. 8 One difference beween AK s procedures and hose used here should be noed: Ineres raes are reaed differenly. AK's code aribues one day s (/365) ineres rae o he rules during each business day no calendar day hey are ou of he marke. This pracice undersaes he reurns o he geneic programming rules by 0.5 percen or less. In his paper, rules earn ineres on calendar days no business days hey are ou of he marke. Table summarizes some of he imporan parameers of ineres. AK provide more informaion on he program and parameers. [Place Table abou here.] Each day, he rading rules generaed by he geneic program observe prices and generae a buy or sell signal indicaing he posiion o ake (he same day). The buy and sell signals are used along wih sock prices and 30-day T-bill ineres raes o compue he coninuously compounded excess reurn of he rule over he reurn o a buy-and-hold sraegy in he sock marke. This daily excess reurn ignoring any ransacions coss over he buy-and-hold sraegy a ime is given by: P + xsr = ( z ) ln ln( + i ) () P where z is an indicaor variable aking he value if he rule is in he marke or 0 if he rule is in T-bills, P is he sock index and i is he ineres rae on he 30-day Treasury bill earned from business day o business day +. The cumulaive excess reurn also called he "finess" for 5

9 a rading rule from ime zero o ime T is he sum of he daily excess reurns less a proporional ransacions cos. AK considered ransacions coss of 0. percen, 0.25 percen and 0.5 percen. For breviy s sake, his paper concenraes on ransacions coss of 0.25 percen. Because i is likely ha rading coss have decreased over ime, he use of 25 basis poin ransacions coss migh underesimae he rue value in earlier periods bu overesimae hem recenly. D. Risk Adjusmen Techniques The crierion of judging he rules o be useful only if hey generae a reurn ha exceeds he buy-and-hold reurn is neiher necessary nor sufficien o conclude ha he rules do no violae he efficien markes hypohesis (EMH). 9 The EMH is usually inerpreed o mean ha asse prices reflec informaion o he poin where he poenial risk-adjused excess reurns do no exceed he ransacions coss of acing (rading) on ha informaion (Jensen (978)). Risk adjusmen is poenially imporan because dynamic sraegies, such as hose found by he geneic program, are ofen ou of he marke and herefore may bear much less risk han he buyand-hold sraegy. Alhough here is no universally acceped mehod of adjusing reurns for risk, his paper will employ four echniques: he Sharpe raio, he X* measure, Jensen s α and he X eff measure of Dacorogna e al. (200). 0 8 Geneic programs wrien by oher auhors produced resuls similar o hose generaed by he AK programs, suggesing ha geneic programming is robus o small change in procedures. 9 Brown, Goezmann and Kumar (998) find ha risk adjusmen is crucial in evaluaing Dow Theory recommendaions. 0 There is some danger of daa snooping in looking a muliple measures of risk. Sullivan, Timmerman and Whie (999) propose a mehod o couner daa snooping in he esing of muliple ypes of rules. This procedure is no appropriae for his paper as his paper does no es he bes rule ou of a group, bu raher a porfolio rule. 6

10 The Sharpe raio he expeced excess reurn per uni of risk for a zero-invesmen sraegy (Campbell, Lo and MacKinlay (997)) is usually expressed in annual erms as he annual excess reurn over he riskless rae ne of ransacions coss o a porfolio over ha porfolio s annual sandard deviaion. The daily excess reurn over he riskless rae ignoring ransacions coss o he rules a ime is given by: P + r = z ln ln( + i ), (2) P where he variables are as defined in equaion (). Alhough he rules may have lower reurns han he buy-and-hold sraegy, lower volailiy may permi he reurns o be leveraged up o exceed he buy-and-hold reurn wih similar risk. As an example, assume he excess reurn o he rading rule was only half ha of he buy-and-hold sraegy, bu he rading rule s Sharpe raio was higher. In his case, he rading rule could ake leveraged posiions in he marke buying wih only a 50 percen margin o obain equal reurns wih lower risk. Buying wih a slighly lower margin would enable he rule o obain higher expeced reurns for he same risk. Sweeney and Lee (990) developed anoher risk-adjusmen sraegy, he X* measure, in he conex of he foreign exchange marke ha may be even more appropriae for equiy Because dynamic sraegies are a an inheren disadvanage, as he marke reurn will, on average, exceed he riskless reurn, Bessembinder and Chan (995, 998) pursue anoher sraegy o compare rading rules o a marke reurn. They permi rules o use double leverage during periods in which hey are in he marke. Ready (998) has quesioned wheher he sraegy of leveraging reurns is implemenable, as he invesor would have o know or predic he ex pos momens o compue he proper amoun of leverage. 7

11 8 markes. 2 They show ha, in he presence of a consan risk premium, an equilibrium daily riskadjused reurn o a rading rule would be given by: ( ) ( ) = = = + = ln ln ln 2 ) ln( ln * T T T i T p P P T p c c T n i z P P z T X (3) where z, P, and i are defined as before, T is he number of observaions, n is he number of oneway rades, c is he proporional ransacions cos, p is he proporion of he ime spen in he marke and p 2 is he proporion of he ime spen in T-bills (p + p 2 =). Noe ha he sum of he hird and fourh erms esimaes he expeced reurn o a zero ransacions-cos sraegy ha randomly is in he marke on a fracion p of he days, earning he marke premium, and in T-bills oherwise. The risk-adjused reurn under he null of no iming abiliy is he acual reurn less he expeced reurn. Posiive X* saisics are inerpreed as evidence of superior risk-adjused reurns. The hird risk-adjusmen measure considered is he X eff measures advocaed by Dacorogna e al. (200). X eff measures he uiliy ha he rading sraegy provides o a consan absolue risk averse individual over a weighed average of reurn horizons. The measure is: = = = + = n i i n i i i i T eff w year w c c n r T X 2 ~ ~ 2 ln σ γ, (4) 2 Sweeney (988) uses he X* measure in he equiy marke. Ready (998) consrucs a saisic similar o Sweeney and Lee's (990) X*. In urn, he es saisic of X*, proposed by Sweeney and Lee (990), is virually equivalen o he es saisic of he coefficien β in he Cumby-Modes es of marke iming if ransacions coss are omied from he X* calculaion.

12 where T T n + c r ln 2 = c is he annualized excess reurn o he rule in percenage erms, ne of ransacions coss; r is defined in equaion (2); σ i 2 is he variance of nonoverlapping reurns of lengh i days; ( year/ i days) is he number of reurns of lengh i days in one year; and γ is a risk aversion parameer. Dacorogna e al. (200) recommend values of γ beween 0.08 and 0.5; his paper ses γ equal o 0.2. The sequence of weighs { w ~ i } akes on a maximum value a a holding period of 90 days: w~ i =. (5) 2 2 i + ln 90 days The reurn horizons i are given by a geomeric sequence {,2,4,8, 6, ec}, whose maximum value is less han or equal o one quarer he number of days in he sample. Ineresed readers should consul Dacorogna e al. (200) for addiional deails on his risk measure. Finally, we also consider he performance of he rules according o Jensen s (968) α, he reurn in excess of he riskless rae ha is uncorrelaed wih he excess reurn o he marke: z n + c. (6) 2T c [ ln( P + / P ) ln( + i )] ln = α + β M [ ln( P + / P ) ln( + i )] + ε If he inercep in equaion (6) α is posiive and significan, hen he rading rule produces excess reurns ha canno be explained by correlaion wih he marke. 3. RESULTS A. In-sample Resuls Table 2 shows he in-sample resuls from implemening a uniformly weighed porfolio sraegy using he excess reurn as he finess crieria. The rules are assessed a 0.25 percen 9

13 ransacions cos for changing posiions, and informaion on day is used o rade he same day. As migh be expeced, he rules did very well in he sample on which hey were rained and seleced. The in-sample, annual excess reurn over he buy-and-hold sraegy, averaged over he en samples, was jus over 5 percen. The analogous Sharpe raio was 0.75, he risk-adjused, annualized X* reurn was 4.34 percen and Jensen s α was 7.2 percen. Rules seleced using he oher finess crieria (Sharpe raios, X*, X eff ) also did well in-sample, producing especially good resuls by he meric for which hey were consruced. [Place Table 2 abou here.] I is worh noing ha, alhough hese rules are opimal in he sense of being he bes rules ha could be found wihin he limiaions of he search procedure, hey are no opimal in he sense of being he bes possible rules one could find on he in-sample daa. Wih sufficien compuaional power, a nonparameric search procedure like he geneic program could find an exremely good in-sample fi ha would be unlikely o be informaive abou ou-of-sample performance. Tha is, i would overfi he daa. The use of boh he raining and selecion periods is one mehod o help guard agains such overfiing. B. Comparison wih Previous Resuls Table 3 shows he ou-of-sample resuls from implemening a uniformly weighed porfolio based on all he good rules found in-sample. 3 As in AK (compare wih Table 2, Panel A in AK), he rules generally failed o produce posiive excess reurns over he buy-and-hold sraegy in he sample. Wih he excepion of he period , for which no good in-sample 3 Resuls for median porfolio rules are broadly similar o slighly beer han hose of he uniform porfolio rules. For he sake of breviy, hey will no be repored separaely. The median porfolio rule goes ino he marke if mos of he N rules are in he marke; oherwise i says ou of he marke. 0

14 rules were found, he ou-of-sample performance was similar o ha found by AK. 4 While AK found only one period in which he mean excess reurn over he buy-and-hold sraegy was posiive, he curren exercise found wo such periods. The rules were long in he marke abou 5 percen of he ime and raded 7.3 imes a year, on average, hough he figures varied widely wih he in-sample period. The period produced unineresing rules ha sayed ou of he marke almos all he ime. [Place Table 3 abou here.] Column 5 of Table 3 shows he mean annual reurn o he marke when he rules are in he marke less he mean annual marke reurn when he rules are ou of he marke (r b -r s ). Alhough here is no measure of saisical significance, posiive numbers favor he proposiion ha he rules have some marke iming abiliy. AK found ha rules from seven of en in-sample periods had marke iming abiliy by his measure. The resuls in his paper are similar, showing ha six of nine have posiive r b -r s. Because he rules' buy/sell decisions could be closely replicaed by moving average rules, AK concluded ha he geneic programming rules were aking advanage of low-order serial correlaion. AK speculaed ha he rules migh be of use o a risk-averse speculaor, bu did no seriously explore ha possibiliy. As noed previously, risk adjusmen is necessary o evaluae he consisency of he rules wih he EMH. The nex secion repors he resuls of such procedures. 4 There are wo reasons why he resuls will no exacly replicae hose found by AK: ) Geneic programming is inherenly sochasic, generaing and recombining populaions probabilisically; and 2) ineres rae reurns were reaed differenly in his analysis.

15 C. Risk-Adjused Resuls The average Sharpe raio of he ransacions cos-adjused geneic programming rules is abou 0.02, lower han he average 0.3 Sharpe raio he index doesn include dividends for he buy-and-hold sraegy over he en subsamples. 5 As Dowd (2000) poins ou, he Sharpe raio only provides enough informaion o choose beween wo invesmens ha are uncorrelaed wih he reurn o he insiuion s porfolio. This is clearly no he case wih hese rading rules. Dowd (2000) proposes ha an invesmen should be underaken if i improves he Sharpe raio of he porfolio, aken as a whole. To implemen his sraegy in he presen siuaion, porfolio shares mus be chosen. Three mehods are considered: ) o arbirarily se he porfolio weighs o ½ on he rading rule and ½ on he buyand-hold sraegy; 2) o choose he weighs o maximize he in-sample Sharpe raio; 3) o choose he weighs ex pos o maximize he ou-of-sample Sharpe raio. While he hird mehod clearly consiues daa mining, i provides an upper bound on he benefi from using he rule. Columns 7 hrough 9 of Table 3 show he improvemen of he combined porfolio s ouof-sample Sharpe raio over ha of he buy-and-hold sraegy, using each of he hree sraegies for choosing porfolio weighs. None of he hree porfolio sraegies significanly improves upon he Sharpe raio of he buy-and-hold sraegy. The second risk adjusmen procedure considered is he X* procedure advocaed by Sweeney and Lee (990). Mos of he annualized X* saisics ne of ransacions coss in 5 Jorion and Goezmann (999) esimae ha dividends made up much of he oal reurn o U.S. equiies over he period 92 o 995. The average Sharpe raio for he buy-and-hold sraegy over he en overlapping ou-of-sample subsamples is 0.3; whereas he Sharpe raio from 929 hrough 995 is 0.06 for he index ex dividends. 2

16 Table 3 are negaive, indicaing ha he rules would no have been useful, even by his riskadjused measure. Almos all he X* saisics would have been posiive hough, if ransacions coss had no been need ou. This suppors he evidence of predicabiliy suggesed by he r b -r s saisics. The mean of he hird risk-adjused reurn measure, X eff, is negaive and lower han he X eff generaed by eiher he buy-and-hold sraegy or he riskless sraegy (X eff = 0). Tha is, he geneic programming rules would no help an invesor who has consan risk averse uiliy over he weighed average of reurns a various horizons. Recall he las measure of risk-adjused reurns Jensen s α esimaes he mean reurn ha is uncorrelaed wih he reurn o he marke porfolio. To measure Jensen s α, reurns o he marke and o he rading rules were aggregaed over nonoverlapping 30-day periods and regression (6) was performed by OLS using annualized reurns. These resuls are shown in columns 4 and 5 of Table 3. The alphas are never significanly posiive in any ou-of-sample period, and he mean among all he ou-of-sample periods is negaive. D. Rules o Maximize Risk-Adjused Measures Of course, he rules rained on an excess reurn crierion may no be he bes risk-adjused rules. To deermine wheher echnical rading rules can produce beer risk-adjused reurns han he buy-and-hold sraegy, ideally we mus rain rules using risk-adjusmen crieria. The resuls of such an exercise using he Sharpe raio as he in-sample finess crierion are shown in Table 4. In several cases, he rules rained on Sharpe raios urned ou o be rivial almos always long or shor in he ou-of-sample period. The non-rivial rules failed o produce higher mean ou-of-sample Sharpe raios, bu hey did show marginally greaer 3

17 predicive abiliy by he sandard of he r b -r s and X* saisics. They also spen less ime in he marke (27 percen long). Using he rules rained o maximize he Sharpe raio in a porfolio wih he buy-and-hold sraegy, as suggesed by Dowd (2000), failed o produce significan improvemen over he buy-and-hold Sharpe raio (see columns 7 hrough 9 of Table 4). [Place Table 4 abou here.] Table 5 shows he analogous ou-of-sample resuls from rules rained o maximize X* as he in-sample finess crieria. There are no rivial (non-rading) X* porfolio rules, and he rules are very even handed. Tha is, here are no cases in which he rules are always in or always ou of he marke. The resuls are generally beer o hose of he rules rained on excess reurns. The annualized excess reurn over he buy-and-hold is abou 60 basis poins greaer han in he benchmark case, and he average Sharpe raio is abou he same as he average buy-and-hold Sharpe raio over all sample periods (0. vs. 0.3). The mean annualized X* saisic is abou zero, which is beer han he mean X* saisics from he rules rained wih excess reurns or he Sharpe raio as finess crieria. However, i should be noed ha even posiive X* resuls may be consisen wih he EMH in he presence of a ime-varying risk premium. The rules mean X eff measure is slighly inferior o ha of he buy-and-hold sraegy. Jensen s alpha is never saisically significan in any sample, and he mean alpha is slighly posiive bu saisically and economically insignifican. [Place Table 5 abou here.] Finally, we consider he resuls from raining rules o maximize he X eff saisics. The resuls are very similar o hose from he oher finess crieria in Tables 3 hrough 5. Table 6 shows ha he ex ane uniform porfolio rule chosen o maximize he X eff saisic never 4

18 ouperforms he buy-and-hold sraegy in a saisically and economically significan way by any of he risk-adjused reurn merics. [Place Table 6 abou here.] 4. CHARACTERIZING LOW-ORDER SERIAL CORRELATION Afer finding ha moving average rules could closely approximae he GP rules' buy/sell behavior, AK aribued he predicabiliy found by heir GP rading rules o "low-order serial correlaion" in he reurns (Campbell, Lo and MacKinlay (997)). One migh speculae ha a simple ime series model of reurns could produce beer decisions han he GP. To es his predicion and o aemp o beer characerize he naure of he predicabiliy found by AK, a variey of ARMA models were fi o he in-sample excess reurns and he bes in-sample models and parameers were chosen by he Akaike, Schwarz and excess reurn crieria. The bes models were used o generae rading signals as wih he geneic programs during he ou-of-sample periods. Table 7 shows ha he non-rivial ARIMA models are even less successful han he rules consruced by geneic programming. If low-order serial correlaion generaes he predicabiliy, he geneic rules are apparenly more successful a esimaing i han are sandard ARIMA models. [Place Table 7 abou here.] Finally, Cumby-Modes ess of marke iming abiliy are used o more formally deermine wheher he rules have predicive conen. The saisical significance of he coefficien (β ) in he regression of excess reurns on signals from he rading rule summarizes he rules one-day-ahead iming abiliy: 5

19 P ln + i = β + β z + ε P ln( ) 0. (7) + Table 8 presens srong evidence ha he rules do possess predicive abiliy: 29 of he 32 available β coefficiens are posiive and 20 of hose are significan a he 5 percen level. Of he hree finess crieria, he X* crieria seems o have produced he rules wih he mos predicive conen. These iming ess illusrae he well-known resul ha profiabiliy is no necessary for a rule o have predicive conen. [Place Table 8 abou here.] 5. CONCLUSION This paper has invesigaed wheher ex ane, opimal rading rules creaed by geneic programming are useful on a risk-adjused basis. Alhough risk-adjusmen improves he relaive araciveness of he rules, neiher Sharpe raios nor Sweeney and Lee's X* saisic, nor Gençay s X eff measure, nor Jensen s α provide evidence ha rules developed by geneic programming would have been useful even o risk-averse speculaors, conrary o reasonable speculaion. Rules rained on he X* measure had he bes risk-adjused performance by all he measures, approximaely equaling he buy-and-hold reurn performance. No oo much should be made of his resul, however. Wih 5-year in-sample periods resuls omied for breviy he Sharpe raio became he bes ou-of-sample performer, again wih performance approximaely equal o he buy-and-hold sraegy. Of course, risk is difficul o measure and any risk adjusmen is subjec o criicism. Neverheless, his paper argues ha rading rule resuls mus be carefully inerpreed in ligh of risk adjusmen. 6

20 I is likely ha he inclusion of dividends in he sock index, he removal of spurious auocorrelaion from he index reurns or accouning for price slippage would only srenghen he negaive resuls of his exercise. 7

21 REFERENCES Allen, F., Karjalainen, R., (999) Using Geneic Algorihms o Find Technical Trading Rules, Journal of Financial Economics, 5, Bessembinder, H., Chan, K., (995) The Profiabiliy of echnical Trading rules in he Asian sock markes, Pacific-Basin Finance Journal, 3, Bessembinder, H., Chan, K., (998). Marke Efficiency and he Reurns o Technical Analysis. Financial Managemen, 27, 5-7. Brock, W., Lakonishok, J., LeBaron, B., (992). Simple Technical Trading Rules and he Sochasic Properies of Sock Reurns, Journal of Finance, 47, Brown, S.J., Goezmann, W.N., Kumar, A., (998). The Dow Theory: William Peer Hamilon's Track Record Reconsidered, Journal of Finance, 53, Campbell, J.Y., Lo, A.W., MacKinlay, A.C., (997). The Economerics of Financial Markes. Princeon Universiy Press, Princeon, NJ. Cumby, R.E., Modes, D.M., (987). Tesing for Marke Timing Abiliy: A Framework for Forecas Evaluaion, Journal of Financial Economics, 9, Dacorogna, M.M., R. Gençay, U.A. Müller, O.V. Pice, (200). Effecive Reurn, Risk Aversion and Drawdowns, Physica A, 289, Dowd, K., (2000). Adjusing for Risk: An Improved Sharpe Raio, Inernaional Review of Economics and Finance, 9, Gençay, R., (998a). Opimizaion of Technical Trading Sraegies and The Profiabiliy in Securiy Markes, Economics Leers, 59, Gençay, R., (998b). The Predicabiliy of Securiy Reurns wih Simple Technical Trading Rules, Journal of Empirical Finance, 5,

22 Gençay, R., Sengos, T., (997). Technical Trading Rules and he Size of he Risk Premium in Securiy Reurns, Sudies in Nonlinear Dynamics 2(2), Holland, J., (975). Adapaion in Naural and Arificial Sysems. Universiy of Michigan Press, Ann Arbor, MI. Jensen, M.C., (968). Problems in Selecion of Securiy Porfolios: The Performance of Muual Funds in he Period , Journal of Finance, 23, Jensen, M.C., (978). Some Anomalous Evidence Regarding Marke Efficiency, Journal of Financial Economics, 6, Jorion, P., Goezmann, W.N., (999). Global Sock Markes in he Twenieh Cenury, Journal of Finance, 54, Kho, B.C., (996). Time-Varying Risk Premia, Volailiy, and Technical Trading Rule Profis: Evidence from Foreign Currency Fuures Markes, Journal of Financial Economics, 4, Koza, J.R., (992). Geneic Programming: On he Programming of Compuers by Means of Naural Selecion. MIT Press, Cambridge, MA. Lo, A.W., Mamaysky, H., Wang, J., (2000). Foundaions of Technical Analysis: Compuaional Algorihms, Saisical Inference, and Empirical Implemenaion, The Journal of Finance, 55, Neely, C.J., Weller, P.A., (2000). Technical Analysis and Cenral Bank Inervenion, FRB S. Louis WP C. Forhcoming in he Journal of Inernaional Money and Finance. Neely, C.J., Weller, P.A., (999). Technical Trading Rules in he European Moneary Sysem, Journal of Inernaional Money and Finance, 8,

23 Neely, C.J., Weller, P.A., Dimar, R.D., (997). Is Technical Analysis in he Foreign Exchange Marke Profiable? A Geneic Programming Approach, Journal of Financial and Quaniaive Analysis, 32, Ready, M.J., (998). Profis from Technical Trading Rules. Unpublished manuscrip. Universiy of Wisconsin, Madison. Sharpe, W.F., (966). Muual Fund Performance, Journal of Business, 39, Sullivan, R., Timmermann, A., Whie, H., (999). Daa-Snooping, Technical Trading Rule Performance, and he Boosrap, Journal of Finance, 54, Sweeney, R.J., (988). Some New Filer Tess, Mehods and Resuls. Journal of Financial and Quaniaive Analysis, 23, Sweeney, R.J., Lee, E.J.Q., (990). In: Aggarwal R., and Lee C.F. (Eds.), Inernaional Dimensions of Securiies and Currency Markes. Advances in Financial Planning and Forecasing Series Vol. 4, Par A. JAI Press, Greenwich, CT,

24 Table : Geneic programming parameers of ineres for AK s implemenaion Parameer AK s Choice Size of a generaion 500 Terminaion crierion 50 generaions or no improvemen for 25 generaions Probabiliy of selecion for 2 rank in populaion - reproducion wih rules ranked from (wors) o 500 (bes). ( size of populaion) 2 arihmeic funcions +, -, *, /, norm, consan beween (0,2) Boolean operaors funcions of he daa "if-hen", "and", "or", "<", ">", "no", "rue", "false" "moving average", "local maximum", "local minimum", "lag of sock index", "curren sock index" 2

25 Table 2: In-sample uniform porfolio resuls from he benchmark case # Rules Excess Reurn Rb-Rs Sharpe X* (s.e.) Xeff Xeff B&H alpha (s.e.) Trades per year % long B&H in-samp B&H ou-samp (4.97) (4.2) (5.5) (4.4) (3.34) (3.09) (4.59) (2.45) (3.43) (2.65) (3.77) (.38) (2.60) (2.68) (2.29) (2.39) (3.04) (2.90) (0.90) NA NA (3.4) (2.88) Noes: Columns and 2 provide he firs and las years of he in-sample period for each case. Column 3 provides he number of rules ha were evaluaed in he in-sample period. Column 4 is he annualized ou-of-sample excess reurn, ne of ransacions cos, o he uniform porfolio rule while column 5 is he annualized mean difference beween average marke reurns on days ha he rules were in he marke and he days ha hey were ou of he marke. The porfolio mean reurn over he riskless rae, ne of ransacions cos, divided by he sandard deviaion of he porfolio reurn is in column 6. Columns 7 and 8 show he annualized X* risk-adjused reurn measure, ne of ransacions cos and is sandard error. Columns 9 and 0 show he Xeff measure of risk-adjused reurns o he uniform rule and he buy-and-hold sraegy, respecively. Columns and 2 show Jensen s alpha and is sandard error. To avoid evaluaing rivial rules, Jensen s alpha was no calculaed if he proporion of long posiions was less han one percen or greaer han 99 percen. Such cells are marked NA. The number of rades per year and he proporion of ime spen in he marke are in columns 3 and 4. The annualized buy-and-hold reurns are shown in columns 5 and 6. The final row displays he means of he columns. 22

26 Table 3: Ou-of-sample uniform porfolio resuls from he benchmark case # Rules Excess Reurn Rb-Rs Sharpe W=.5 Weigh ExAne Weigh ExPos X* (s.e.) Xeff Xeff B&H alpha (s.e.) Trades per year (0.42) (0.29) (.5) (0.79) (.2) (.20) (.55) (0.85) NA NA NA NA NA NA NA NA NA NA NA NA NA NA (2.7) (0.35) (2.05) (.22) (.98) (.32) (0.76) (0.72) (0.2) NA NA mean (.26) (0.84) % long B&H in B&H ou Noes: The column headings are as in Table 2 wih he following excepions. Column 3 provides he number of rules (ou of 0 rials) ha had posiive raining period reurns. Columns 7 hrough 9 display he improvemen in he ou-of-sample Sharpe raio, over a buy-and-hold sraegy, of a porfolio consruced from he uniform rading rule and he buy-and-hold sraegy. Column 7 shows he resuls wih equal porfolio weighs, column 8 requires ha he porfolio weighs for he wo sraegies be chosen on in-sample informaion and column 9 shows he maximum improvemen obainable wih perfec foresigh. 23

27 Table 4: Ou-of-sample uniform porfolio resuls generaed using he Sharpe raio as he finess crierion # Rules Excess Reurn Rb-Rs Sharpe W=.5 Weigh ExAne Weigh ExPos X* (s.e.) Xeff Xeff B&H alpha (s.e.) Trades per year (0.04) NA NA (0.30) (0.34) (0.92) (.04) (.68) (.42) NA NA NA NA NA NA NA NA NA NA NA NA NA NA (.20) (.46) (.45) (.34) (.33) (0.89) NA NA NA NA -0.0 (0.3) NA NA NA NA NA NA NA 0.00 (0.08) NA NA mean (0.79) (.08) % long B&H in B&H ou Noes: See he noes o Table 3. 24

28 Table 5: Ou-of-sample uniform porfolio resuls generaed using he X* measure as he finess crierion. # Rules Excess Reurn Rb-Rs Sharpe W=.5 Weigh ExAne Weigh ExPos X* (s.e.) Xeff Xeff B&H alpha (s.e.) Trades per year (0.69) (0.78) (.0) (.7) (.02) (.09) (.44) (.22) (.99) (.09) (.3) (.26) (.80) (.56) (.60) (.08) (0.75) (0.6) (.33) (0.69) mean (.28) (.05) % long B&H in B&H ou Noes: See he noes o Table 3. 25

29 Table 6: Ou-of-sample uniform porfolio resuls generaed using he Xeff measure as he finess crierion. # Rules Excess Reurn Rb-Rs Sharpe W=.5 Weigh ExAne Weigh ExPos X* (s.e.) Xeff Xeff B&H alpha (s.e.) Trades per year (0.37) (0.3) (.05) (.09) (0.96) (.08) (.5) (.06) NA (2.0) NA NA (2.4) (0.37) (2.07) (.4) (.74) (.33) (0.82) (0.84) NA NA NA NA NA 0.00 (0.08) NA NA mean (.29) (0.9) % long B&H in B&H ou Noes: See he noes o Table 3. 26

30 Table 7: Resuls from ARIMA rules In-sample Search AR MA Daily Excess Sharpe X* Trades Period Crierion Order Order Dummy Reurn R b -R s per year % long AIC SC NA NA Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn AIC SC Excess Reurn Noes: Column 2 shows he in-sample model selecion crierion. Columns 3 and 4 show he chosen orders of he auoregressive and moving average componens. Column 5 summarizes he deerminisic componen of he model: indicaes a simple consan, 2 indicaes a weekend dummy on reurns while 3 indicaes ha a full se of day-of-he-week dummies was used. For he oher columns, see he noes o Table 2. 27

31 Table 8: Cumby-Modes ess of marke iming Benchmark Sharpe X* Xeff Bea (s.e.) p-value Bea (s.e.) p-value Bea (s.e.) p-value Bea (s.e.) p-value NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA Noes: The four panels show he resuls of Cumby-Modes ess (see equaion (7)) on he four cases in which he finess crieria were excess reurns, he Sharpe raio, he X* saisic and he Xeff saisic. The columns of each subpanel show he coefficien, is sandard error and is p- value. The final row displays he number of posiive beas and he number of p-values less han To avoid evaluaing rivial rules, he es was no performed if he proporion of long posiions was less han one percen or greaer han 99 percen. Such cells are marked NA. 28

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