Craig George Leslie Hopf and Gurudeo Anand Tularam. Science Environment Engineering and Technology (ENV),

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1 Joual of Mathematcs ad Statstcs 0 (3): ISSN: Hof ad Tulaam Ths oe access atcle s dstbuted ude a Ceatve Commos Attbuto (CC-BY) 3.0 lcese do:0.3844/jmss Publshed Ole 0 (3) 204 (htt:// A MATHEMATICAL ANALYSIS OF THE INCLUSION OF INSTITUTIONAL BETTING FUNDS INTO STOCK MARKET: THE CASE OF TECHNICAL AND FUNDAMENTAL PAYOFF MODELS IN HORSERACE BETTING Cag Geoge Lesle Hof ad Guudeo Aad Tulaam Scece Evomet Egeeg ad Techology (ENV) Evomet Futue Cete Gffth Uvesty Austala Receved ; Revsed ; Acceted ABSTRACT Hoseace w ad lace etus have yet to be cosdeed seously as otfolo uts facal tadg makets. Howeve thee exst techcal ad fudametal stochastc models of aametc ad oaametc dstbuto that aea to otmze exected etus fom w ad lace vestg the hoseace wageg maket. I ths study a comlex umbe otmzato techque s toduced ad aled to develo a detemstc bettg model that calculates actual etus fom the w ad lace bettg. Usg classes of models metoed feld w bet ayoff esults wee geeated fo a samle of successve global galloe aces fom Austalasa Asa ad the Uted Kgdom. It was oted double dgt etus exceedg 0% wee cosstetly acheved wth mutes of hoseace vestg wth abtage ootuty locked to eace fom the detemstc model. The esults fom ths study ovde evdece fo seous vestgato to the cluso ad beeft of Isttutoal Bettg Fuds to the local stock maket. Keywods: Models o Hoseace Outcomes Isttutoal Bettg Fuds Stock Maket Hoseace Bettg.INTRODUCTION Facal makets ae tycally defed by havg tasaet cg basc egulatos o tadg costs ad fees ad maket foces detemg the ces of secutes that tade (Ivestoeda 203). The hoseace wageg maket hets uceta vestmet etus may atcats ad suly exasve fomato coceg atcats ad oducts tycal of facal makets (Al 998). Bettg Maket elemets clude sttutoal egulatos (Iteatoal Hoseacg Fedeato) the atcats (beede tae owe jockey ad betto) ad the bettg oducts (w lace quella tfecta). Global hoseace aual wageg tuove s sgfcat; fo examle fo yea 2008 the tuove was AUD$0.25 tllo (AGC 20) whle bllo. The amout ca be comaed to the ASX equty aual tuove of AUD $.6 tllo (ASXG 2008). Global hoseace wageg facltes have evolved fom the tack ad Lcesed Bettg Offce (LBO) outlets to ole tote o fxed odd bettg ad eso-to-eso (2) bettg exchage oeatos (Laffey 2005). A bettg exchage s a etty that ovdes tadg facltes fo etal o bookmake customes to buy o sell cotacts (Kog ad Va Velze 2009). Hoseace wage cotacts ae stuctued as bay otos (tycally Euoea style) whee the ayoff s ethe some fxed amout fom a w o lace bet o othg fom a loss. Oe aty s the laye (acceto) of the bet ad the couteaty ae the bet take. Bettg exchage oducts clude hoseace wageg cotacts facal sead bettg ad cotact fo dffeece facal devatves. The bettg exchages ca be clamed to have bought tasaecy ad tadg theaustala hoseace aual tuove was AUD$2.6 Coesodg Autho: Cag Geoge Lesle Hof Scece Evomet Egeeg ad Techology (ENV) Evomet Futue Cete Gffth Uvesty Austala Scece Publcatos 390

2 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): ovato to hoseace bettg makets ecessay fo sttutoal vestmet (Laffey 2005). Bettg actvty o the exchage may be classfed to seculatve hedgg o abtage tadg (Aold 2002). I 20 the UK-fouded Betfa bettg exchage ecoded a aual sale actvty of 96 mllo bets matched; a aveage of 7 mllo tasactos o the bettg exchage daly. Moeove Betfa ecoded a ew eak load of beats e mute dug the 20 Gad Natoal steelechase evet (Betfa 20). Seveal factos aea to have cotbuted to Betfa s gowth that clude educed ole betto tasacto costs the mlcato fom Mooe s Lawmoe affodable comutg gves fms ad customes access to eomous ocessg owe; ad the adatato fom Metcalfe s Law of Netwoks that as the umbe of customes usg Betfa multles so does the utlty of each custome. Othes such as geate lqudty the maket ad hece geate effcecy also aded Betfa s gowth (Daves 2005). The Betfa model s based uo the oe outcy system wheeby backes ad layes stakes ae ool matched (Daves 2005). Goldma Sach s techologcal dvso ovdes sevces coceg the develomet of both techcal ad fudametal model based algothmc tadg softwae. The models geeate automatc executo stateges that ca be used by the bake s teal equty busesses as well as exteal clets such as fud maages ad hedge fuds (Goldma Sachs 20). Reseach ecet tmes has focused o otmal hoseace bettg models that foecast hoseace outcomes based o both techcal ad fudametal aalyss techques (Edelma 2007). The techcal bettg models utlze hoseace maket odds that ae the betto s obabltes wth assumed maket effcecy. Ths model quatfes both hstocal ad cuet hoseace vaable data; whle the fudametal modellg techques utlze selectve eseached vaables fo ace foecast such as a Suot Vecto Mache/Codtoal Logt (SVM/CL) hoseace bettg modelthat ca geeate double dgt etu tal esults (Lessma et al. 2009). The ma am of ths study s to develo ad demostate a ew techcal detemstc bettg model. The aalyss wll show that cosstet oftable tadg at a accetable ewad-to-sk level fo the sttutoal bettg fud ca be acheved. The sgfcace of detemstc models to lock eace guaateed ayoff fo the fud s ctcally examed. I the fst secto a evew of the exstg lteatue o stochastc techcal ad fudametal ak ode hoseace bettg models used to develo tadg algothms s eseted. I the ext secto a multle Scece Publcatos 39 system otmzato theoem ove the comlex umbe feld s eseted ad the detemstc hoseace bettg model develoed fom the theoem s used to geeate etus fom w ad lace bettg. The model testg s the coducted o a adom samle of global hoseaces by smulatg cotuous tadg ad ecodg ayoff esults. The detemstc model ayoff esults ae the comaed wth ayoffs fom selectve techcal stochastc models ad the fal secto esets a dscusso of the alcato of the ew detemstc bettg model sttutoal bettg fud tadg. 2 TECHNICAL AND FUNDAMENTAL STOCHASTIC BETTING MODELS 2.. Race Rak Notato The ak fo the outcome of a evet exessed by a ate s the esult of akg of a adom vaable fom a udelyg oaametc o aametc dstbuto (D Ela 2003). Assgg the obablty to the outcome fom a feld hoseace s equvalet to assgg the obablty fo the ak emutato of the fst aces. I ths study X deotes a deedet o-detcally dstbuted cotuous adom vaable to eeset tme take fo the th ak ace wth obablty dstbuto F(X ; ). The ode of fsh fo a feld sze of aces s eeseted by the emutato = ( 2 ) wheeby eesets the fst laced ace ad s the last laced ace fom the feld. The obablty assged to ak emutato s ( 2 ) ( R R R ) = = X < X <... < X. Ths 2 class of ak models s descbed as ode statstcs models that shae the oety that the ode of ay subset of the tems s deedet to the odeg of ay dsjot subsets (Ctchlow et al. 99; Al 998). Pemutato gou = {} εi eesets the set of comlete ad atal ak ode emutatos fo the ace feld ad () the aametc o oaametc dstbuto o emutato R. The techcal ad fudametal models evsed ths study shae a multstage sequetal ocess to geeate emutato obablty fom betto w odds to calculate exected ayoff. Fo examle w( ) quella ( 2 ) ad tfecta ( 2 3 ) llustate bettg maket oducts. I atcula aametc dstbutos based uo the omal adom vaable f(x ; σ 2 ) the gamma adom vaable f(x ; ) ad the exoetal adom vaable f(x ; = ) have bee oosed fo techcal bettg model alcato. These techcal models shae the L- decomosablty oety that the obablty attached to the akg of a ace s deedet fom the elatve

3 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): odeg of the hghe aked aces (Ctchlow et al. 99). The fudametal bettg models evsed ths study utlze a two stage ocess to combe selectve hoseace vaable data wth the betto odds vaable to geeate w (o lace) obablty. These clude ak ode Multle Lea Regesso/Codtoal Logt (MLR/CL) ad Codtoal Logt/Codtoal Logt (CL/CL) models ad classfe Suot Vecto Mache/Codtoal Logt (SVM/CL) model (Edelma 2007; Lessma et al. 2009) Techcal Bettg Model Otmzato Techques The akg ocesses of the L-Decomosable models of Luce (959; Havlle 973; Hausch ad Zemba 985; Ste 990) all deteme the codtoal oduct of the choce obabltes (.e. efeed ace fom emag aces) acoss the multstage sequece Equato : = ( ) = { }{ }.. { ( -)} = = j jεb () B = {.. } s the set emag at stage ; wth ( ) =. These multstage sequetal ocesses = utlze the betto w odds ( = ) = ( O : w odds) to O calculate w lace ad comoud bettg oducts. The techcal bettg models attemt to acheve cosstet oftable bettg etu whch challeges the sem-stog maket effcet hyothess that hstocal ad ublcly avalable hoseace data has bee coectly factoed to the cuet betto odds. The HZR wageg system calculates otmal lace bets to maxmze exected logathm of fal wealth fom lace wageg (Hausch ad Zemba 985). The model calculates the exected etu fom oe addtoal dolla bet to decde ace selectos agast a beakeve bechmak ad otmal bet szes fo the selected aces ae calculated to maxmze fal wealth. The mode ole bettg otals should oly ehace ths model s efomace fom the esumto that ool wageg occus just o to the close of the bettg eod (Hausch ad Zemba 985). A modfed veso of the HZR model fo algothm develomet usg bothw ad lace totalze betto odds s as follows Equato 2: EX P 2 w q jqk wjqqk wk q jq = + + k= ( q )( q q j ) ( q j )( q j q ) ( qk )( q j qk ) Q P + ( + Pjk ) + INT ( P + ) EX P : exected lace etu ace ' ' w : w odds ace ' '; q : lace odds ace ' ' P EX γ γ : beakeve bechmak aamete Maxmze 2 wq jqk EX ( w f ) = { } l = j k j; k= ( q )( q q j ) Q P + l ( Pjk + jk ) l= j log k w0 l 3 P + Pj + j Pk + k l= 0 ( f ) EX w : exected logathm fal wealth fom lace wageg w : tal wealth; Q : tack ayback P : lace ool P = P + P + P jk j k l : otmal lace bets jk = + j + k (2) Scece Publcatos 392

4 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): The Logstc model geeates emutato obabltes fom the logathm of w obablty atos (Plackett 975). The fst-ode logstc model coesods to the L-Decomosable model.. = 2. A teetato ( )( ( 2)).. of the model s that a ace s ak s deedet of eale selectos accodace wth the L- decomosablty oety. A secod-ode logstc model llustato fo a fou ace feld has emutato obablty (Dase 983) Equato 3: = { < 2} { < } 2 j all j ad ( 2 ) ( 23) = ( 234) = ( 23) + ( 24) The obablty assocato betwee ace as s cosdeed by the secod-ode logstc model. A thd-ode logstc model llustato fo a fve ace ( 23 ) ( 34) feld s =. A model deved fom the ( 34) + ( 35) exteso of the L-Decomosable ad gamma models s the dscout model whch cludes a dscout facto λ k (decease fucto as k ceases ad deedet o shaeaamete ) to dscout dmshed ace efomace wth decease lacg (Lo et al. 995). The log odds ato assumto that ace beats ace j fo k th lace beg a dscouted fucto of ace defeatg ace j fo the w LO( j k) LO( j ) Scece Publcatos (3) λk \ = \. A dscout model tfecta obablty aoxmato s ovded (Lo et al. 995) Equato 4: ( ( j )) λ 2 λ3 ( ( k )) ( jk ) ( ) (4) λ2 λ3 ( ) ( ) s t s t j The dscout model s a fucto of the w obabltes of all the aces the feld ulke the L- Decomosable model whch s a fucto of the w obabltes of oly selectve aces. The vese hyegeometc model smlaly ales a sequetal comaso cteo of betto w o lace odds to vestgate akg ocess outcomes (D Ela 2003). 393 The omal ak ode model s a class of ak model that s a fucto of a sgle deedet vaable of aametc dstbuto N(X ; ) wth jot = df ( ) f X ( : aveage tme ace). Pemutato obablty fo the omal ak model s eeseted by the multvaate tegal Equato 5: = ( 2 ) ( R < R < < R ).. X X X dx 2 f ( X ) f X f X (5) 0 X X 2.. dx ( σ = ) = The w ace obablty fo X R (exected w tme) s exessed X R - = ( ; ) = ( ) = j j f X F X j 0 dx. Heey (98) deved the omal ak aoxmate model fom fst degee Taylo exaso to calculate the emutato obablty as µ ; + = ( = )!! 0. The w lace ad tfecta obablty aoxmatos ae llustated by the followg foms Equato 6: µ ; ( = ) + ( ) jµ k; = + ( j k ) 23 ( ) 3 + µ ; = 3 3 ( 3) j ; = µ j j + ( 3) The gamma ak ode model class s a fucto of bvaate deedet vaables wth gamma dstbuto Γ(X ; ) ad jot df X ex X ( : ate; : dstace). = Γ( ) The gamma ak emutato obablty (Ste 990) s Equato 7: th (6)

5 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): ( ) ( ) ( X ) X 2 X ex 3 ( ).. X X 2 X2 ex 2X ex 2 2 X dx dx 0 Γ 0 Γ 0 Γ ( ) (7) Ad the ace obablty to w (Heey 983) s ( = ) = ( ; ) ( ; ) f X F X 0 = j j dx j Scece Publcatos The gamma desty fucto Equato 8: ( ; ) f X ( X ) ex = X { X C ( X ) D( )} ex + + Γ ( ) (8) Fom whch a fst degee Taylo exaso aoxmato fo the model s deved (Heey 983) as follows Equato 9: ' ( µ ; D ) ( ) = 0 = ( = 0 )! =.. l f ( X; ) f ( X; ) dx = = +! { µ ; }{ } = +!! ( D( ) = l ) A gamma ak obablty aoxmato fo the k th laced ace (Heey 983) s { µ k; }{ } ( = k ) +. The exoetal model ( ) wth desty fucto. (9) f X ; = ex( X ) deotes the gamma ak ode model wth shae aamete =. The exoetal model s equvalet to the L- Decomosable model (Ste 990). The mathematcal devato of the L-Decomosable model fom the gamma ak ode model wth shae aamete = usg codtoal obablty fst cles s accodgly Equato 0: 394 ( = mx j ) X... j X j =.. ( = ) je dx2dx2 dx 0 X X ( = mx j ) X = ( ) X P X ; = [ 2.. ] j X X e dx j 0 = j X j ( X = mx j ) e dx 0 ( X = ) = mx j j ( ) = ( 2 ) = ; B = {.. } = j jε B (0) These techcal ak ode models fo hoseace bettg adot emutato codtoal obabltes ad betto odds to deteme exected outcomes fo tycal hoseace bettg oducts such as quella tfecta ad fst fou hoseace wageg. These elemetay obabltes ca be combed to otmze exected etu fom feld bettg Fudametal Bettg Model Otmzato Techques The fudametal model aalyss utlzes elevat ace vaables fo w ad lace foecast based uo ublcly avalable fomato. The edctve models that attemt to acheve a cosstet oftable bettg etu do ot satsfy the codtos of the sem-stog maket effcecy theoy that ublcly avalable ace fomato has bee factoed to betto odds (Lessma et al. 2009). The cluso of selectve fudametal vaables elevat to cuet ad evous ace fomato combed wth cuet maket odds data s oe attemt to develo oftable stochastc models. CL/CL MLR/CL ad SVM/CL models ossess such a two stage desg; the fst stage quatfes ace s ablty based uo evous ad cuet ace data ad the secod stage utlzes the betto

6 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): odds to ovde cosdeato fo wth-ace cometto. The oaametc MLR/CL techque models ace ak as a lea fucto of selectve fudametal multvaable data to oduce a wgess dex foecast (scoe ablty) the fst stage. Wth-ace cometto s excluded fom the fst stage of the MLR/CL model but cluded the secod stage. Stage two develos a obablty foecast fo a ace w o lace estmated cojucto wth comettos (wth-ace cometto) by usg a multomal logt techque whch models a ace as a etty; matag ace elatosh ad factog maket betto odds. The MLR/CL CL/CL ad SVM/CL techques dffe the fst stage. The CL/CL model cosdes wth cometto wth ts modellg of ublcly avalable ace fomato at stage oe. The SVM/CL techque deves a classfcato model to detfy ace wes o loses ad tetoally elmatg elace uo ak ode egesso. The SVM/CL model utlzes a w o ow dcato vaable athe tha a fshg osto stage oe (Lessma et al. 2009). To costuct olea decso sufaces suot vecto mache methodology ma ut fudametal data to a hghdmesoal featue sace usg a mag fucto to mmze tesve calculato the tasfomed featue sace (Lessma et al. 2009). Keel fuctos ca be emloyed to comute the scala oduct of tasfomed vectos the featue sace. The Gaussa Radal Bass Fucto (RBF) keel has bee aled to hoseace modellg wth outut values lyg betwee zeo ad oe (Edelma 2007). The model below s a feld w bet otmzato stategy to deteme otmal feld bets to maxmze exected logathmc etu ad utlzg the MLR/CL two stage techque to foecast ace w obabltes. Stage oe oduces a wgess dex as follows Equato : Y = β + β X + β X + + β X j ε; X.. X : ace vaables fj Yj = 0.5; Yj = 0 Yj [ ] ( + ) j ' j ' evet ' ' Scece Publcatos Y : omalzed fsh wgess dex ace f : ak fsh ode ace ' j ' evet ' '; j : umbe of aces evet ' ' () 395 The secod stage of the MLR/CL model detemes w obablty foecasts fo the dvdual aces Equato 2: L ( δ δ ) N Races = ( 2 ) = ( ˆ δ ˆ δ2 ) = max ( Y + lo 2 ) ( Yjδ + lojδ 2 ) L δ δ : egulazato aamete estmates µ : bay 0 j j j 2 = ex( Yˆ ˆ ˆ jδ + lojδ 2) ex( Yˆ ˆ δ + lo ˆ δ ) ex µ j jδ jδ ex j j 2 : obablty ace ' j ' ws evet ' ' O : betto w odds ace ' j ' ws evet ' ' j Y : wgess dex foecast j (2) The obablty foecasts ae combed wth the betto w odds to calculate otmal feld bets to maxmze exected logathmc etu (Edelma 2007) Equato 3: Maxmze EX ( R ) = { bj } j log bjoj bj EX R : exected logathm etu Oj : betto w odds; bj : otmal bet fo ace ' j ' evet ' ' (3) The stochastc fudametal ad techcal hoseace bettg maket models aalogous to catal maket models otmze exected etu o dvdual o feld w ad lace assets. Futhemoe a oety of amutuel bettg to becosdeed s zeo etu - to - l sk feld bettg s achevable fom the goss betto odds (befoe commsso); subsequetly ostve actual etus ca be acheved fom favouable tadg. 3. COMPLEX NUMBER MULTIPLE SYSTEM OPTIMIZATION A set of algebac elemets ad elatos that defe the set costtute a mathematcal system. Comlex ad

7 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): hyecomlex (quateos octoos) umbe systems ae fte dmesoal vecto saces ove the eal umbes that satsfy may of the eal umbe system axoms. Comlex aalyss exteds mathematcal alcato beyod estctos evdet wth the eal umbe system s caacty to descbe all featues of hyscal scece. Cosdeato s gve to hoseace bettg ayoff eeseted by a comlex umbe to seaate feld ayoff fom dvdual ace ayoff. A detemstc model s oe whch evey set of vaable states s uquely detemed by aametes the model ad by sets of evous states of these vaables (Yag 2008). I fact detemstc modellg of futue evets though kow aametes has sgfcat alcato fo facal maket vestmet ayoff as evdet the catal debt makets. Schochetma ad Smth (998) develos a algothm to geeate aveage otmal soluto detemstc fte hozo. The Multle System Otmzato (MSO) model s develoed ths study.the model otmzes multle comlex system uts Z(C ;) ove a fte hozo. Theoem Multle System Otmzato (MSO) ove a fte sees of comlex systems geeates a costat eal comoet ove each cosecutve system Equato 4:.. (.. ) ' ' ( ) { max / m} Z C ; = Я Z.. C C2.. C : multle system comlex fucto (4) C : comlex vecto ut; : set of oeatos ( ˆ C ) : comlex outut; Re Poof of Theoem Scece Publcatos = Я The oof fo the MSO theoem s by mathematcal ducto. The oof by ducto volves a two stage ocess; fstly the base stage that s followed by the ductve stage. The base stage vefes that the otmzato ove a comlex system whch comses a comlex vecto agumet ad accomayg elatos geeates a Я eal costat value. The ductve stage vefes that fo a fte sees of cosecutve comlex systems the otmal soluto s Я fo the multle system comlex fucto. Maxmzg o mmzg o the sace C of -tules of comlex umbes must satsfy the -dmesoal Cauchy-Rema equatos ode fo the comlex fucto to be comlex dffeetable Equato 5: 396 = ( 2 ) C j = x j y j j { } Z C ; Z C C C ; ( ) dz dz dz dz =.. dc dc dc2 dc dz Z x Z x 2 Z x.. = dc x C x2 C2 x C Z y Z y 2 Z y.. + y C y2 C2 y C dz Z Z Z Z Z Z =.. dc 2 x y x2 y2 x y Z Z Z = =.. = = 0{ Cauchy Rema } x x2 x Re = Я = costat Mathematcal Iducto ste Equato 6: k k (.. ;) = = { ( ;)} { = } k.. k (.. k ;) ( ;) { ( ;)} ( ;) Z C C C Я Z C k Z.. k 2 k.. k+ = Z C C C Z C = Z C Z C 2 (.. k ;){ } k k +.. k+ 2 + = Я Я = Я = Z C C C QED (5) (6) 3.. Detemstc Bettg Model: A Alcato of MSO It would seem that a detemstc model would be elevat fo modellg bettg maket ayoffs. The bettg maket assets (wage) ossess defed ayoffs wth lmted lablty that s cotast to equty ad devatve maket asset foecast ayoff that ae detemed fom the systematc ad dvesfable factos (Wllams 999). A zeo sk hoseace feld bettg ovdes a abtage ootuty fo the tade ad sttutoal bettg fud. Otmzato of the comlex olea ayoff fucto ovded Equato 7 detemes ace feld w ad lace etus fom the avalable betto fxed ad totalze odds: { max} Payoff b O ; = O b b { m} subject to ( b b.. b ) O b = O b j (7) ( ) ( ) : set of oeatos b : bettg amout b C O : w / lace odds O R j j 2 = =

8 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): Table. Tackvest model-w ad lace ayoff ace esults: R(R R 2 R 3)=(234); w ayoff Z(R ) = Я = 30.30%; lace ayoff Z(R R 2 R 3) = Я = 5.9% Race Mkt Net Race Mkt Max-m Payoff (R ) odds Bet W (R ) etu Payoff (R ) odds Bet Place(R jk) etu (max-m) 7.8 $2000 $23600 $48600 (29.45%0) 3.4 $2000 $7400 ($3600-$43600) (6.6% -2.3%) $0000 $ $43000 (26.06%0) $20000 $72000 ($4200-$43000) (6.9% -2%) $45000 $ $55500 (33.64%0) 3.6 $45000 $72000 ($4200-$43000) (6.9% -2%) $3000 $25800 $50800 (30.79%0) $0000 $73000 ($4200-$42000) (6.9% -20.5%) $2000 $ $43800 (26.55%0) $4000 $74200 ($4200-$40800) (6.9% -9.9%) $47000 $ $55900 (33.88%0) 8.9 $37000 $70300 ($2500-$44700) (6.% -2.8%) $0 $0 -$65000 (0% -00%) 9 $20000 $20000 (-$37800-$44700) (-8% -2.8%) $6000 $26000 $5000 (30.9%0) $20000 $70000 ($2200-$44700) (6% -2.8%) $0000 $25000 $50000 (30.30%0) 2 4 $8000 $72000 ($4200-$43000) (6.9% -2%) $65000 $ souce: Utab Rccato ZS (24 Febuay 202) Table 2. Tackvest model-multbet w ayoff ace esults: R (R R 2 R 3 ) = (234) R 2 (R R 2 R 3 ) = (2);w ayoff Z..2 = (R R 2 ) = (+Я) 2 = ( )(+0.09) =.42Я= =9% Race Mkt W Net Mkt Net (R ) Odds Bet (R ) etu Payoff Race(R ) odds Bet W(R ) etu Payoff 7.8 $2000 $23600 $48600 (29.45%0) 4.4 $2000 $52800 $4500 (9%0) $0000 $ $43000 (26.06%0) $45000 $ $55500 (33.64%0) $7500 $50250 $950 (4%0) $3000 $25800 $50800 (30.79%0) $5500 $550 $2850 (6%0) $2000 $ $43800 (26.55%0) $4600 $49220 $920 (2%0) $2000 $52000 $3700 (8%0) $2800 $50400 $200 (4%0) $47000 $ $55900 (33.88%0) $4000 $2800 -$35500 (0% -73%) $0 $0 -$65000 (0% -00%) $3000 $49800 $500 (3%0) $6000 $26000 $5000 (30.9%0) $000 $50600 $2300 (5%0) $2600 $50960 $2660 (6%0) $0000 $25000 $50000 (30.30%0) $3300 $4970 $870 (2%0) $65000 $48300 Souce: Utab-Rccato ZS ad ZS2 (24 Febuay 202) Table ad 2 llustate feld w ad lace ayoff esults geeated fom algothm whch uses ole o fxed hoseace betto odds ad attemts to lock a e-ace ostve otmal ayoff ove the ace feld fo a total mmum vestmet. The comlex ayoff s detemed usg the MSO-the deved algothm dslays a etu-to-sk tade-off that tyfes facal maket vestmets. Ths techcal detemstc bettg model s outut seaates feld ayoff fom dvdual ace ayoff. The etu-to-sk tade-off show Table s to lock a e-ace 30% w ayoff ove the ace feld excludg the favoute fo the shottem vestmet eod (mutes). The sk s a loss fom the favoute ace wg. A otoal feasble tade-off s lockg to the lowe lace ayoff ad lesse absolute maxmum sk. Scece Publcatos 397 A tade could souce the global maket of bettg exchages ad bookmakes fo efeed w o lace odds. Bet takg w maket odds of 4.7 fo the favoute would have acheved abtage ad a 3% ayoff ove the ete feld. Table 2 dslays the etu-to-sk tade-off fom multle bettg ove two cosecutve aces. The hoseace esults show Table 2 geeated a feasble ayoff of 9% fo the fud fom cosecutve ace vestg. 4 DETERMINISTIC VERSUS STOCHASTIC BETTING PAYOFF- ANALYSIS AND RESULTS A samle of ffty cosecutve galloe aces fo Jauay fom global acetacks of Austalasa

9 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): (Austala ad New Zealad) Asa (Sgaoe ad Hog Kog) ad Geat Bta ovded the data (htt:// testg the efomace ad comaso of the techcal detemstc ad stochastc bettg models ths study. The data gatheed smulates assg the bet global - cotuous tadg o bettg maket aces fom the southe to the othe hemshee. Although dated both Lo et al. (995) ad Al (998) tests show that the omal model foecasted hoseace ak obabltes moe accuately tha the gamma ak o L- decomosable models. Based o the data the ace feld w bettg ayoff fom the detemstc model s comaed wth ayoff fom the omal ak aoxmate ad L-decomosable models. The detemstc model otmzes ace feld ayoff (Я)-to-sk tade-off fo mmum bettg amout o cosecutve aces; zeo sk eflects abtage ootuty ad locks a e-ace ostve etu deedet of wg ace outcome. Both the omal aoxmate ad L-decomosable stochastc models ecoded the same exected ayoff esults. The et exected w etu e $ ut bet fom a wg ace fo both the L-decomose ad omal aoxmate models equals the egatve of the tote tack take (t); ad the et exected feld w ayoff fo both models s E(ayoff) = -(+t-); : feld sze t: tack take. Table 3 dslays model s esults fo the Austalasa Asa ad UK samled egos. Idvdual ad accumulatve ace etus geeated fom both the DBM ad the omal aoxmate techque ae dslayed. The DBM ovde two ayoffs: (a) A l tade stategy whee maxmum exosue fom a dvdual ace losg s 00%; ad (b) a case whee the maxmum sk of loss fom a dvdual ace s caed at a maxmum of 0%. Ufom ut bets o the feld fo the omal aoxmate model esulted samle ayoff comaso wth the DBM ayoff esults. Table 3 esults demostate the sueoty of the DBM ove the stochastc model. The DBM cosstetly geeated double dgt betto ace etu. Smla to the stochastc model esults the accumulatve ayoff geeated fom the DBM ove a sees of aces s affected whe a loss occus wth hgh exosue. Iteestgly howeve the DBM otmzg feld w bets wth bettg exchage tade to maxmze exosue to 0% esulted both excetoal ostve accumulatve ayoffs (+Я) ad successve ace etu Я acheved fo the sees of aces all the egos. Table 3. Techcal models-feld w bet ayoff DBM Results (%) Rego ace ace 2 ace 3 ace 4 ace 5 ace 6 ace 7 ace 8 (+Я) Я(%) UK (Chelteham) (a) (a).44 (a) 7.6 (b) -0 (b).83 (b) 2.8 SGP (Kaj) 7.87 (a) (a) (a) -7.5 (b) (b).405 (b) 7 HKD (Sha T) 5.46 (a) (a) (a) - (b) -0 (b).33 (b) 5.9 AUS (Iveell) (a) (a) (a) (a) -26 (b) -0 (b) -0 (b).86 (b) 2.6 AUS (Mogto) (a) (a) (a) (a) 0.02 (a) -39 (b) -0 (b) -0 (b) -0 (b).48 (b) 4.46 AUS (Muay Bdge) (a) (a) (a) 0.84 (a) -9 (b) -0 (b) -0 (b) (b).4 AUS (Sushe Coast) (a) (a) (a) (a) 0.8 (a) -9.3 (b) -0 (b) -0 (b) -0 (b).34 (b) 3.73 NZD (Wakouat) (a) (a) (a) 0.9 (a) (b) -0 (b) -0 (b).033 (b).09 Nomal (aoxmate) Model Results (%) UK (Chelteham) SGP (Kaj) HKD (Sha T) AUS (Iveell) AUS (Mogto) AUS (Muay Bdge) AUS (Sushe Coast) NZD (Wakouat) Souce: NSW TAB (0 Jauay 202) (a) max sk 00% (b) max sk 0% Scece Publcatos 398

10 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): IMPLICATIONS FOR THE INSTITUTIONAL BETTING FUND The esults show that the DBM ca be aled to all classes of acg cludg galloe totte ad geyhoud. Clealy the DBM otmzes the e-ace (o wth ace) ayoff-to-sk tade-off fo w o lace feld bettg fo selectve aces o cosecutve ace sees ove the fte tme hozo. The algothms of w ad lace combato ae wothy of cosdeato to ehace ayoff efomace. The bet take ad lay tade eables a sk-fee ayoff; that s the bettg exchages tade betto odds that offset the tack commsso to lock- e-ace detemstc etu. The cosstetly geeated double dgt etus fom the DBM fo the global galloe samle ovdes evdece fo futhe ad moe deth vestgato of sttutoal bettg fud atcato ad lqudty cotbuto to the facal makets; cludg futue cosdeato fo otfolo vestmet cluso. Scece Publcatos 6. CONCLUSION The fdgs of ths study show that detemstc ad stochastc facal maket models may be used to deteme the ewad-to-sk tade-off fo a vestmet. The stochastc techcal ad fudametal bettg models ovde ace w ad lace obabltes ad exected ayoffs fom hstocal ad cuet ace fomato. The aalyss shows that the detemstc modellg of a lmted ayoff ad lmtedlablty vestmet geeates a fxed o a ceta outcome. Moeove the DBM s geeate a actual etu fom w ad lace wageg. The MSO geeated a costat eal comoet ove cosecutve systems. The DBM develoed based o ths techque otmzes feld w ad lace bettg ad seaates the otmal feld ayoff fom the dvdual ace ayoff. The dvdual ace exosue s elated to the tack take whch educes the tue betto odds. Both the DBM ad tadg the bettg maket ca offset the tack commsso to acheve ace etu feld coveage at a accetable sk; theefoe clealy abtage s acheveable. The elmay fdgs fom DBM testg of a global hoseace samle ths study has geeated sgfcat ayoff at accetable sk fo the Isttutoal Bettg Fud; as ecuso to attag esults fom lage ace samlg eods to challege the exstg sem-stog effcet maket hyothess towad hoseace bettg REFERENCES AGC 20. A database o austala s gamblg dusty. Austalasa Gamg Coucl. Al M.M Pobablty models o hose-ace outcomes. J. Aled Stat. 25: DOI: 0.080/ Aold G Cooate Facal Maagemet. 2d Ed. Facal Tmes/Petce Hall Halow Eglad. ASXG ASX Ltd Aual Reot. Austala. Betfa 20. Betfa Aual Reot. Lodo. Ctchlow D.E. M.A. Flge ad J.S. Veducc 99. Pobablty models o akgs. J. Math. Psychol. 35: DOI: 0.06/ (9) Dase B.R A ote o emutato obabltes. J. Royal Stat. Socety 45: Daves M. L. Ptt D. Shao ad R. Watso Betfa.com: Fve techology foces evolutoze woldwde wageg. Eu. Maage. J. 23: DOI: 0.06/j.emj D Ela A Modellg aks usg the vese hyegeometc dstbuto. Stat. Modell. 3: DOI: 0.9/47082X03st047oa Edelma D Adatg suot vecto mache methods fo hoseace odds edcto. A. Oeat. Res. 5: DOI: 0.007/s Goldma Sachs 20. Goldma sachs aual eot. Uted States. Havlle D.A Assgg obabltes to the outcomes of mult-ety comettos. J. Am. Stat. Assoc. 68: DOI: 0.080/ Hausch D.B. ad W.T. Zemba985. Tasactos costs extet of effceces etes ad multle wages a acetack bettg model. Maage. Sc. 3: DOI: 0.287/msc Heey R.J. 98. Pemutato obabltes as models fo hose aces. J. Royal Stat. Socety 43: Heey R.J Pemutato obabltes fo gamma adom vaables. J. Aled Pobab. 20: DOI: / Ivestoeda 203. Facal maket. Uted States. Kog R.H. ad B. Va Velze Bettg exchages: The futue of sots bettg? It. J. Sot Face 4:

11 C.G.L. Hof ad G.A. Tulaam / Joual of Mathematcs ad Statstcs 0 (3): Laffey D Eteeeush ad Iovato the UK Bettg Idusty: The se of eso-to-eso bettg. Eu. Maage. J. 23: DOI: 0.06/j.emj Lessma S. M. Sug ad J.E. Johso Idetfyg wes of comettve evets: A SVMbased classfcato model fo hoseace edcto. Eu. J. Oeat. Res. 96: DOI: 0.06/j.ejo Lo V.S.Y. J. Baco-Shoe ad K. Busche 995. The alcato of akg obablty models to acetack bettg. Maage. Sc. 4: DOI: 0.287/msc Luce R.D Idvdual Choce Behavo: A Theoetcal Aalyss. st Ed. Wley New Yok : 53. Plackett R.L The aalyss of emutatos. J. Royal Stat. Socety 24: Schochetma I.E. ad R.L Smth 998. Exstece ad dscovey of aveage otmal solutos detemstc fte hozo otmzato. Math. Oeat. Res. 23: DOI: 0.287/moo Ste H Models fo dstbutos o emutatos. J. Am. Stat. Assoc. 85; DOI: 0.080/ Wllams L.V Ifomato effcecy bettg makets: A suvey. Bull. Ecoom. Res. 5: -39. DOI: 0./ Yag X.S Mathematcal Modellg fo Eath Sceces. st Ed. Dued Academc Pess UK. Scece Publcatos 400