Necessary Elements Of A Pratic Market!

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1 Explorng decson makers use of prce nformaton n an effcent speculatve market J.E.V. Johnson, O.D. Jones, L. Tang Abstract We explore the extent to whch the decsons of partcpants n a speculatve market effectvely account for nformaton contaned n prces and prce movements. In partcular, we nvestgate how successful they are relatvely at ncorporatng transparent and obscure nformaton. The horserace bettng market s chosen as an deal envronment to explore these ssues. A condtonal logt model s used to determne wnnng probabltes, based on prces and ther movements. The model ncorporates technques for parametersng prce (odds) curves and a novel approach s used to test the model aganst a random alternatve. The results challenge the consensus from prevous studes that the market s weak form effcent. Ths dsparty arses snce prevous studes assess the degree to whch ndvdual varables are ncorporated n fnal prces, whereas the model we develop uses combnatons of varables and ther nteractons. In addton, we conclude that transparent nformaton, partcularly that assocated wth an enhanced prospect of success, s largely accounted for n the decsons of market partcpants, but that more obscure nformaton s not properly accounted for. 1. Introducton There s a large body of evdence whch suggests that the combned judgements of decson makers wthn fnancal markets effectvely ncorporate nformaton concernng hstorcal prces nto current prces. Consequently, markets are regarded as weak form effcent to the extent that abnormal returns cannot be made f buy/sell decsons are made on the bass of hstorcal prces. The horserace bettng market s one form of fnancal market that has receved consderable scrutny n ths regard. An mportant reason for ths focus s that wagerng markets are especally smple fnancal markets, n whch the prcng problem s reduced. As a result,

2 - 2 - wagerng markets can provde a clear vew of prcng ssues whch are complcated elsewhere (Sauer, 1998, p. 2021). In partcular, wthn horserace bettng markets each asset has a defnte termnaton pont at whch ts value becomes clear. In addton, bettng markets share many features wth wder fnancal markets ncludng a large number of partcpants, access to a wealth of nformaton, an uncertan return on nvestment, and the possblty of nsder tradng. Partcpants n bettng markets make decsons concernng the nature and extent of ther nvolvement based on a subjectve evaluaton of the lkely outcome; the outcome n turn affects ther welfare. Consequently, analyss of behavour n these markets can offer mportant nsghts nto nformaton and decson-makng processes n broader envronments assocated wth rsk. The subjectve judgements of decson makers wthn horserace bettng markets lead, va ther bettng (bettors) or odds settng (bookmakers) decsons, to the odds whch preval n the market. The consensus to emerge from prevous bettng market studes across the world s that the resultng odds fully reflect hstorcal odds nformaton. However, a falng of these studes has been ther focus on a sngle aspect of hstorcal prce nformaton. Consequently, the prncple approach of ths paper s to develop a model for wnnng probabltes, employng a range of prce varables and ther nteractons, that enables us to make a postve return; thereby demonstratng that the market s not weak form effcent. Ths wll also enable us to explore bettors relatve success n ncorporatng transparent and more obscure nformaton. The paper proceeds as follows. Secton 2 provdes an overvew of the UK horserace bettng market, and hghlghts the degree to whch ths settng embodes features that are lkely to promote market effcency. Secton 3 offers a bref revew of the lterature that addresses weak form effcency n bettng markets. Secton 4 outlnes the data employed n ths study, provdes a ratonale for the model whch s used to test for market effcency, and descrbes how the data s parametersed. The ftted model s presented n Secton 5 together wth a dscusson of ts sgnfcant components. In Secton 6 the model s appled to test data usng two dfferent bettng

3 - 3 - strateges. We show that t s possble to make a proft wth both strateges, ndcatng that the bettng market s neffcent. In Secton 7 we compare our model wth a random alternatve, to establsh that the results produced usng the model cannot be attrbuted to chance. Fnally n Secton 8 we summarse our conclusons. 2. The UK horserace bettng market There are two dstnct forms of horserace bettng market that operate n parallel at racetracks n the UK; the bookmaker market, whch forms the settng for ths study, and the parmutuel market. The odds n the parmutuel market are determned (as n the USA) solely by the pattern of relatve bettor demand, accordng to a pre-determned formula, and bets are settled at the odds prevalng at the close of the market. In contrast, the odds on offer n the bookmaker market are determned by both the decsons of bettors and bookmakers and bets are settled at the odds avalable n the market at the tme the bet s struck. The more serous bettors and those wth access to nsde nformaton are most lkely to bet n the bookmaker market snce they can secure ther return, wthout the possblty of a bandwagon effect erodng ther gans (whch can happen n the parmutuel market). Independent bookmakers operate at each racetrack and post odds at the commencement of the market before each race. Bettors are then free to bet wth the bookmaker offerng the best odds on ther selecton up to the off-tme. These odds change accordng to the relatve weght of demand, reflectng bettors opnons, and accordng to each partcular bookmaker s subjectve vew of the horses relatve prospects. In makng ther wagerng decsons bettors may use nformaton from a varety of sources, ncludng fundamental analyss based on the horses prevous form, advce from specalst publcatons and nformaton derved from both movements n the bookmaker odds, and the relatve amounts of money wagered on horses n the parmutuel system (dsplayed on computer screens throughout the course of the market). In addton to the on-course market there s a large

4 - 4 - ( 90% of total market) off-course market n the UK. Bettors are permtted to bet n bettng shops away from the track and are provded wth televsed coverage and nformaton on the evolvng odds at the racetrack. The off-course bookmakers manage ther labltes by placng bets wth on-course bookmakers. Consequently, odds n the on-course bookmaker market ncorporate nformaton from a wde varety of sources. The UK bookmaker bettng market embodes a number of factors that are lkely to lead to a close correspondence between subjectve judgements of market decson makers (whch emerge as odds) and the relevant objectve probabltes of the horses success. These nclude expertse, ncentves and feedback (Johnson and Bruce, 2001). Expertse has been shown to ad calbraton and odds are determned by a range of ndvduals wth consderable expertse; these nclude the bookmakers, who devote consderable resources to accurate odds assessment, and nsders, such as owners and traners, who may be n possesson of prvate nformaton concernng, for example, the horse s ftness. In addton, bettors clearly have a fnancal ncentve to make accurate judgements and the bettng task s a repettve one, nvolvng predcton, where the stmul reman relatvely constant and where feedback s mmedate and regular. All these factors have been shown, n other settngs, to mprove calbraton (Ashton, 1992; McClelland and Bolger, 1994; Wrght and Ayton, 1988). In addton, Cec and Lker (1986) demonstrated that expert bettors employ cogntvely complex models nvolvng a range of varables when assessng a horse s prospects. These observatons suggest that decson makers n bettng markets are lkely to employ nformaton effcently, resultng n fnal odds that are a good reflecton of each horse s prospects. 3. Weak form effcency n horserace bettng markets The overwhelmng consensus to emerge from prevous horserace bettng market studes s that the subjectve probabltes, nherent n the odds, closely correspond wth the observed wn probabltes (Hoerl and Falln, 1974). In addton, prevous studes appear to demonstrate that

5 - 5 - hstorcal prce nformaton s effcently ncorporated nto fnal odds, to the extent that wagerng strateges based on hstorcal prces do not generally yeld postve expected returns. Weak form effcency studes n bettng markets fall nto three broad categores: those whch examne the extent to whch horses wth the lowest/hghest odds under/over estmate the horses observed wn probabltes (the favourte-longshot bas ); those whch explore whether the nformaton contaned n odds movements s fully dscounted n the fnal odds and those whch test whether proftable tradng rules can be constructed employng arbtrage between two or more equvalent bets. We brefly dscuss each category n turn. Studes explorng the dstrbuton of odds n horserace bettng markets offer strong evdence for a consstent over/under estmaton of the probablty of longshots/favourtes wnnng (the favourte-longshot bas). These conclusons have been reached for studes wdely dspersed n tme (McGlothln, 1956; Al, 1977; Zemba and Hausch, 1986; Bruce and Johnson, 2000) and across a varety of countres such as USA (Asch, Malkel and Quandt, 1982; Snyder, 1978; Thaler and Zemba, 1988, Zemba and Hausch, 1986), Australa (Brd and McRae, 1994; Tuckwell, 1983) and the UK (Bruce and Johnson, 2000; Dowe, 1976; Henery, 1985; Vaughan- Wllams and Paton, 1997). However, these studes do not generally detect opportuntes for tradng proftably on ths nformaton 1 ; suggestng that hstorcal prces are effcently dscounted n current odds. The second set of studes explores the nformaton content assocated wth changes n odds. Informaton held by those wth prvleged nformaton may be transmtted to the market va ther bettng behavour; ths may be revealed as bookmakers adjust odds to account for ther labltes. In general, prevous studes suggest that bettng strateges based on these odds adjustments do not yeld postve expected returns. Dowe (1976) found, on average, a close correlaton between ntal and fnal odds, suggestng lttle scope for dscernng nformaton from odds movements. 1 The one excepton, a study conducted by Zemba and Hausch (1986) n the USA, dentfes a small expected

6 - 6 - However, hs analyss faled to dscrmnate between horses whose odds fell and those whose odds ncreased throughout the lfe of the market. Crafts (1985) focussed on horses whose fnal and ntal odds dffered sgnfcantly. A strategy based on backng/avodng horses whose odds fell/lengthened sgnfcantly produced a better return than a random bettng strategy, but t dd not produce a postve expected return. Ths ndcates that horses whch ease/frm n the market perform worse/better than the orgnal bookmaker odds would mply and Dowe (2003) estmates that approxmately 25% more nformaton s ncorporated nto the fnal odds (compared wth openng odds). Snce bookmakers have a strong ncentve, and the commercal opportunty, to be fully apprased of all publcly avalable nformaton, these observatons suggest that bettng actvty reveals nformaton that s not readly avalable pror to the formaton of the market. Ths s a vew confrmed by Tuckwell (1983), Asch, Malkel and Quandt (1982) and Schnytzer, Shlony and Thorne (2003), who all demonstrate an ncrease n the flow of useful nformaton as each bettng market evolves. Taken together, studes explorng odds changes suggest that prvately held nformaton s revealed throughout the course of the bettng market, but no study fnds opportuntes for proftable tradng once the nformaton becomes avalable. Sauer (1998, pp ), n an extensve revew of the bettng market lterature, concludes that the evdence suggests that an nformed class of bettors s responsble for alterng prces n these markets... and the opnons of experts appear to be fully dscounted n market prces. The thrd set of studes explores weak form effcency of horserace bettng markets by attemptng to construct arbtrage bettng strateges, employng nformaton revealed wthn one market n a parallel market. For example, Al (1977) compares the odds on horses n the daly double pool (a parmutuel bet payng a return only f a bettor selects the wnners of two races) proft from bettng on a very small number of extreme favourtes (odds < 3/10).

7 - 7 - wth the odds n the wn pools of the two races. Hs analyss confrms that the wn pool odds provde effcent estmates of the daly double pool odds; a result consstent wth the hypothess that the bettng market s weak form effcent. Hausch, Zemba and Rubnsten (1981) use probabltes revealed by wn pool bettng to construct probabltes of horses fnshng second and thrd. They then construct proftable wagerng strateges based on dfferences between these probabltes and the odds avalable n the place and show pools 2. Also, some studes (Hausch and Zemba, 1990; Schnytzer and Shlony, 1995; Leong and Lm, 1994) demonstrate that postve returns are possble by explotng the degree to whch bets placed n ndependent wn pools (e.g. operated n dfferent locatons) on a gven race produce dfferences n odds. However, the number of opportuntes for proftable bettng employng the above strateges are lmted; n some cases the nformaton requred to explot the opportunty s not avalable n real tme, and n others the degree of neffcency s small. In addton, there s evdence that publcty concernng arbtrage opportuntes has lead to a change n bettng behavour whch has removed the neffcency (Rtter,1994). Consequently, ths thrd set of studes offer lttle more than a crease n what s predomnantly a smooth pattern of effcency n the racetrack bettng market (Sauer, 1998). From the precedng dscusson t s clear that horserace bettng appears to conform to the expectatons developed from calbraton studes n other contexts, namely the odds at market close effcently ncorporate nformaton concernng past odds and odds movements. Our concern wth ths concluson s that prevous tests of weak form effcency n horserace bettng markets have been restrcted to the mpact of a sngle varable, such as odds. However, Cec and Lker (1986) demonstrate that expert horserace handcappng requres bettors to combne 2 In the parmutuel market, separate pools are created for wn bets (those whch attempt to select the wnner), place bets (those whch attempt to select the horse to fnsh second) and for show bets (those whch attempt to select horses to fnsh second or thrd). Odds are determned separately n each pool by the relatve amount of money on each horse.

8 - 8 - relevant nformaton n complex nteractve models. We therefore set out to demonstrate that the decsons of partcpants n horserace bettng markets (as revealed by the odds at market close) do not effcently ncorporate nformaton concernng a combnaton of odds varables, ncludng relevant nteractons. 4. Descrpton of the data and type of model There s evdence that those wth access to prvleged nformaton (e.g. traners, owners) chose to bet wth bookmakers rather than n the parmutuel market (Crafts, 1985; Schnytzer and Shlony 1995). Moreover, the evdence suggests that an nformed class of bettors s responsble for alterng prces n these markets (Sauer, 1998, p. 2049). Accordngly, we collected bookmaker odds data from 1,200 races run at 41 dfferent racetracks n the UK over the perod Aprl June Only flat races of less than 2 mles were ncluded. The data were suppled by SIS, an organzaton that relays on-track bookmaker odds to off-track bettng offces. The number of horses n each race vares from 2 to 20, wth a mode of 11. The bettng perod for each race lasts from 2.5 to 30 mnutes, and averages 12.5 mnutes (σ = 4.5 mnutes). The odds for a gven horse n the sample can change from 0 to 10 tmes, wth a mode of 2 and an average of 2.7 changes. The nformaton employed conssts of, for each race and each horse j = 1,..., k() n race, a sequence of tmes and odds {(t,j (1), u,j (1)),..., (t,j (n), u,j (n))}. The fnal par (t,j (n), u,j (n)) s always the off tme and fnal odds. The length of the sequence n = n(,j) vares from horse to horse, and the number of horses k = k() vares from race to race. When we are n the context of a sngle race we wll drop the ndex and when n the context of a sngle horse we wll drop the ndces and j. We scale the tmes so that t(1) = 0 and t(n) = 1. We regard the odds as a functon of tme, pecewse constant wth jumps where the odds change, and call ths functon a prce curve. The left-hand dagram of Fgure 1 dsplays a typcal example of a prce curve: the ponts * are the ponts (t(1), u(1)),..., (t(n), u(n)). Note that the fnal par (t(n), u(n)) are generated by the start of the race and not by a change n odds, so u(n) = u(n-1).

9 - 9 - Takng the prce curves as ts nput, we wsh to buld a model for p = (p (1),..., p (k)), the vector of wnnng probabltes for race, where p (j) = Pr(horse j wns race ). Suppose that for horse j n race we have parametersed the prce curve by x,j = (x,j (1),..., x,j (m)), where m s fxed over all and j. We use a condtonal logt model for the p. That s, for a fxed vector of coeffcents β = (β(1),..., β(m)), we suppose that where < β, x, j > = = β ( h) x j ( h). m h 1, exp( < β, x, j > ) p ( j) =, (1) k ( ) = exp( < β, x l > ) l 1, We justfy ths choce of model by notng that t allows the exponent < β,x, j > to be nterpreted drectly as the ablty of horse j, ndependent of the race. To see ths, suppose that ε (j), j = 1,..., k() are ndependent dentcally dstrbuted random varables wth the double exponental dstrbuton. That s ε (j) has cumulatve dstrbuton functon F ε (x) = exp(-exp(-x)), for - < x <. If we put W ( j) = < β, x, >+ ε ( j) then t can be shown that Pr( W ( j) W ( l), l = 1, K, k( )) p ( j) (Maddala, 1983). We can nterpret W(j) as a = wnnngness ndex. That s, the wnner of race s the horse wth maxmal W (j), and we can nterpret the determnstc component < β,x, j > of W(j) as a drect measure of horse j s ablty. It s possble to use a dfferent dstrbuton for ε (j), the random component of W (j). For example, usng a normal dstrbuton for ε (j) leads to a multnomal generalzaton of the probt model. However, ths model s computatonally more dffcult to work wth. Moreover, the condtonal logt model has been successfully employed by many authors for a range of dscrete choce problems, t has been demonstrated to gve smlar results to probt models (e.g. McFadden, 1974), t allows the number of horses to vary from race to race, t places no mportance on the order of the horses and t provdes a mechansm for expressng the competton between horses. j

10 If we observe N races, and the wnner of race s horse j*, then the jont lkelhood L = L(β) s the probablty of observng ths set of results, assumng the p are as above. That s N N exp( < β, x, j* > ) L( β ) = p ( j*) =. (2) = 1 k ( ) = 1 exp( < β, x > l= 1, l ) We employ maxmum lkelhood estmaton, as mplemented n Lmdep (Green, 1998), to choose β that maxmzes L(β) Parametersng the prce curve To construct a model for p (j) we requre a consstent parametersaton of the prce curve for each horse. In dong so we have two ams: to provde a general summary of the shape and other physcal characterstcs of the prce curve, and to pck out partcular features that have been dentfed n the lterature or by actve gambles as havng an effect on the horse s wnnng probablty. For all the parameters we consder there s an mplct dependence on the race and horse j, though, to smplfy, we wll not make ths explct n the notaton Orthogonal polynomal expanson We are nterested to nclude parameters that reflect the general shape of the prce curve, snce the precse shape of a prce curve that s lkely to sgnal a potental wnner (or loser) s unclear. To provde a general summary of the shape of a prce curve {(t(1),u(1)),..., (t(n),u(n))}, we use an orthogonal polynomal expanson of order 3. Ths allows us to measure the heght of the curve (fnal odds), the lnear trend, the curvature (quadratc component) and change n curvature (cubc component). By usng an orthogonal polynomal expanson, we can measure the sze of each component (constant, lnear, quadratc and cubc) ndependently of the others. Orthogonal polynomals are a classcal statstcal tool; detals of ther constructon can be found for example n Wetherll (1981). We summarse the procedure here and t s llustrated n Fgure 1.

11 Fgure 1. Orthogonal polynomal decomposton of a prce curve. The frst dagram shows the prce curve and ts constant, lnear, quadratc and cubc approxmatons. The second dagram shows the separate constant, lnear, quadratc and cubc components, whch are added to gve the approxmatons n the frst dagram. Gven a set of ponts {(t(1),u(1)),..., (t(n),u(n))}, an orthogonal polynomal bass s a sequence of polynomals f 0, f 1, f 2,... such that f s of order and n l= 1 f ( t( l)) f j ( t( l)) = 0 for all j. (3) It can be shown that an orthogonal polynomal bass always exsts and that there s a unque set of coeffcents a 0, a 1, a 2,... such that = u( l) a f =0 ( t( l)) for all l = 1, K, n. (4) The a can be found by least squares. By restrctng ourselves to an order 3 expanson we get an approxmaton to u. As the f are orthogonal, we can nterpret a as the sze of the order component n the prce curve. In fact, the equatons (3) do not specfy a unque bass, and we can mpose further constrants wthout compromsng orthogonalty. In our case, because of the mportance of the fnal odds u(n), we take f 0 (t(n)) = f 0 (1) = 1 and f (1) = 0 for all 1. The effect of ths s to make the constant component a 0 equal to u(n). We also norm each f so that ts leadng term s smply t. In partcular ths mples that a 1 s the slope of the least squares regresson lne constraned to pass through (t(n),u(n)). We nterpret a 2 as a measure of the curvature of the prce curve and a 3 as a measure of the change n curvature.

12 A potental problem wth polynomal expansons s that they are unstable when only a small number of ponts are used. That s, f n s small, then a small change n one of the (t(),u()) can produce a large change n a 2 and a 3. To mtgate ths, we regularse the procedure by ntroducng a roughness penalty when fttng the a. Let F( t) 3 = = 0 a f ( t), then we choose the a to mnmze n l= 1 ( u( l) F( t( l))) 2 + λg( F) (5) where G(F) s the roughness penalty and λ s some constant of proportonalty. Typcally G(F) s some measure of curvature such as a Sobelov norm (that s, a norm based on frst, second and sometmes hgher order dervatves of F). However, the slope of F at t(n) s of partcular nterest to us as a measure of late prce movement (see below), so we do not want to depress ths unnecessarly. So, nstead of a Sobelov norm, we put G(F) equal to the area of F above u max = max u() and below u mn = mn u(). That s, 1 1 G( F) = max( F( t) u,0) dt + max( u F( t),0 dt. (6) 0 max mn ) Volatlty measures The volatlty or roughness of a curve s one of ts fundamental characterstcs. Two parameters were used to measure the volatlty of the prce curve, the number of prce changes and the absolute varaton. That s, for prce curve {(t(1),u(1)),..., (t(n),u(n))}, we put n b1 = n 2 and b 2 = u( l) u( l 1). (7) l= 2 Clearly there wll be some degree of correlaton between b 1 and b 2. There are three specfc types of behavour that the volatlty can capture, whch would not be revealed by the orthogonal expanson alone. The frst s the bandwagon effect (Schnytzer and Shlony, 2003; Smth, 2003), whereby an odds reducton prompts further bettng, whch trggers a further odds reducton, and

13 so on. The second s the slow drft of odds for horses that are not backed (Schnytzer and Shlony, 2003). Ths occurs because bookmakers ntally depress odds, but then allow them to mprove gradually f a horse s not backed. The thrd relates to the dsclosure of sgnfcant, perhaps contradctory peces of nformaton concernng the horse s prospects as the market evolves (see Secton 5.1) More specfc features The remanng three parameters were ncluded snce prevous lterature suggests that they may have some nfluence on a horse s probablty of success. A number of studes, as ndcated n Secton 3, have demonstrated a close correspondence between probabltes mpled by fnal odds and wnnng probabltes (e.g. Bruce and Johnson, 2000). The fnal odds for horse j n race mply a probablty of wnnng q (j), va the relatonshp u, j 1 q ( j) ( n) =, q ( j) 1 q ( j) =. (8) 1+ u, ( n) j We call ths the track probablty to dstngush t from the model probablty p (j). We already nclude the fnal odds n the model as a 0 = u,j, whch gves p ( j) exp( β ( a0 ) (1 q ( j)) / q ( j)). However, t s plausble that a more drect relatonshp between p(j) and q (j) would result n a better ft. Accordngly, we nclude the parameter c 1 = log β ( c1 ) q (j) = log(1 + u,j (n)), whch gves p ( j) exp( β ( c ) c ) = q ( j). In fact, Chapman (1994) even found that c1 added sgnfcant explanatory power n a sophstcated fundamental handcappng model that ncluded 20 varables assocated wth the horse and ts jockey. Crafts (1985), Tuckwell (1983) and Brd and McCrae (1987), amongst others, have demonstrated that a horse s enhanced prospects of success are revealed by a large reducton n odds from the start to the completon of the market. Consequently, we take c 2 = fnal odds ntal odds = u,j (n) u,j (1), although ths wll be hghly correlated wth the slope a

14 Those wth access to prvleged nformaton have an ncentve to bet late n the market. Ths allows them to captalse on the general drftng out of odds as the market progresses, confrm that ther horse gets to the start wthout mshap and t enables them to become more fully nformed about rval horses prospects, va odds changes (Asch, Malkel and Quandt, 1983; Schnyter, Shlony and Thorne, 2003). Consequently, we seek to nclude nformaton n our model concernng late changes to odds. Let u(t) be the prce curve, and [a, b] be a small subnterval of [0, 1], close to 1. We take as our measure of late change the most extreme slope ( u(1) u( t)) (1 t) for t [ a, b], that s, the slope wth the largest absolute value. We let the late change parameter be c3, and provde two llustratons n Fgure 2,where c 3 s the slope of the lne plotted though (1,u(1)). Odds u(t) Odds u(t) * * * * * * * * * * 0 a b 1 Tme t 0 a b 1 Fgure 2. Late change parameter. Tme t Values of a and b were chosen to make c 3 reasonably robust, so that small changes n u(t) do not produce large changes n c 3, whle mnmsng correlaton wth the overall trend a 1. Takng [a, b] = [0.9, 0.95] gave reasonable results. If the prce curve were smooth, then c 3 would smply be an approxmaton to the dervatve at t = 1. Consder agan our orthogonal polynomal expanson of the prce curve, F( t) = = a ( ). 0 f t F(t) s a smooth approxmaton of the prce curve, so we expect c3 to be hghly correlated wth F (1) = a1 + 2a 2 + 3a Rescalng and splttng 3

15 Two refnements of the parameter set were ncorporated, based on our understandng of how prces behave n practce. Frstly, t s known that on long-odds horses (those wth hgh fnal odds), the odds change by larger amounts than for short-odds horses, and we beleve that the relatve sze s more mportant than the absolute sze of any change. Consequently, before calculatng parameters a 1, a 2, a 3, b 2, c 2 and c 3, the prce curve {(t(1),u(1)),..., (t(n),u(n))} was rescaled by dvdng u() by u(n) for = 1,..., n. Parameters a 0 and c 1, whch are based on the fnal odds, were not rescaled, and b 1 s unaffected by ths rescalng. Secondly, practcng gamblers nterpret the prce curve dfferently when the odds are comng n (decreasng) or gong out (ncreasng). Ths suggests that we should nterpret prce changes dfferently when they are changes down rather than up. Accordngly, all parameters except a 0 and c 1 were splt nto two parts, x + and x, dependng on whether the odds came n or went out. That s, x + = x f u(n) > u(1) or 0 otherwse, and x = x f u(n) u(1) or 0 otherwse. 5. Model fttng In order to ft and test the model gven n Equaton (1) the data set was splt nto two parts. The frst 800 races were used to ft the model, and the remanng 400 used to test t. A stepwse fttng procedure was used to select a set of parameters sgnfcant at the 95% level. Parwse nteractons of all the parameters were also consdered. In the fnal model the parameters a 1 +, b 2, c 1, c 3 and the nteracton a 1 + b 1 were all sgnfcant at the 95% level. The estmated coeffcents β are gven n Table 1. The log lkelhood rato of the model over the constant alternatve s , whch gves us that the model s sgnfcant wth a p-value of Table 1. Estmated coeffcents of the model. Note that a 1 +, b 2, c 3 and a 1 + b 1 0; c 1 < 0. Parameter Descrpton Coeffcent Standard p-value Error + a 1 slope up b 2 absolute varaton down c 1 log(track probablty) c 3 late change down

16 a 1 + b slope up and number of changes nteracton Interpretaton of ftted parameters Frstly we note that, as a 1, b 2 and c 1 are n the model, t s not surprsng that c 2, b 1 and a 0 are not, as we knew these parameters were correlated. We nterpret ths as sayng that a 1, b 2 and c 1, respectvely, capture more relevant nformaton concernng overall change n odds, volatlty and fnal odds than c 2, b 1 and a 0. Secondly, we note that some degree of correlaton between c 3 and a 1 + 2a 2 + 3a 3 was expected, so the presence of c 3 n the model n part explans why a 2 and a 3 do not appear Track probablty and favourte long shot bas The parameter c 1 = log q (j), whch has coeffcent , has a large nfluence. Ths s because of the relatve sze of c 1 compared to the other parameters. Consderng just the effect of c 1 on the model probablty we have p (j) exp( log q (j)) = q (j) We nterpret ths relatonshp as a reflecton of the so-called favourte-long shot bas. That s, for horses wth long odds, the true probablty of wnnng s sgnfcantly less than the mpled track probablty, whle for horses wth short odds, the true probablty of wnnng s close to the mpled track probablty. A plot of log u,j (n) = log (1-q (j))/q (j) aganst log q (j) gves a very close match to the analogous plot gven n Bruce and Johnson (2000), whch was obtaned by modellng the favourte-longshot bas drectly Slope The parameter a + 1, whch has a coeffcent sgnfcantly dfferent to zero n our model, should be consdered n conjuncton wth the nteracton term a + 1 b 1. a + 1 s non-zero and postve when the odds worsen (e. the least squares regresson lne constraned to pass through t(n), u(n) has a postve slope). It s reasonable that a horse whose odds move out should be less lkely to wn snce ths may reflect lack of confdence n the horse s prospects on the part of bettors and/or

17 negatve nformaton concernng the horse s chances beng released to the market (Crafts, 1985). The effect of the a 1 + b 1 nteracton s to reduce ths effect when the odds worsen n a large number of small steps, as opposed to small number of large steps. Agan ths s expected snce such a pattern wll emerge as bookmakers adjust odds outwards n small steps n order to encourage bets on horses where demand for bets s weak. Ths contrasts wth horses whose odds move out n large steps, as partcular peces of adverse nformaton concernng the horse reach the market (e.g. horse sweatng badly n the paddock). It appears that bookmakers ncremental outward adjustments to stmulate demand are less of a negatve sgnal concernng a horse s prospects than sharp outward adjustments n odds, whch may reflect new adverse nformaton. When the odds come n on a horse then a 1 + s zero, so nether a 1 + nor a 1 + b 1 have an effect, but f the odds come n on horse j, then they wll usually go out on other horses; the combned effect wll be to ncrease p (j) Late change down The late change down, c 3 acts as expected. c 3 s always negatve n sgn, so when there s a late change down the effect s to ncrease the probablty of wnnng. Ths agrees wth the belef that a late change down ndcates large bets close to off-tme by those wth access to prvleged nformaton. We note that a late change up has no sgnfcant effect on the probablty of wnnng. Ths s probably because at a late stage n the market a sgnfcant ncrease n odds s only lkely to arse from new but wdely avalable nformaton, and hence s readly dscounted n fnal odds (e.g. horse bolts on way to start) Volatlty Absolute varaton b 2 was used as a volatlty measure. The coeffcent s negatve, ndcatng that hgh volatlty n the prce curve makes a horse less lkely to wn, but only when the odds have come n. That s, nconsstency n the drecton of odds movements for horses whose odds fall overall durng the bettng perod s a negatve sgnal concernng the horse s prospects. Such

18 behavour can arse from contradctory ndcatons of the horse s relatve prospects flterng nto the market at dfferent tmes Transparent and Opaque Informaton The results suggest that transparent nformaton s more effcently dscounted n bettng markets than more opaque nformaton. Each of the parameters wth sgnfcant coeffcents mght be descrbed as opaque compared wth an equvalent, but more transparent parameter excluded from the model; for example, a 1 vs. c 2, slope of the least squares regresson lne through t(n), u(n) vs. fnal - ntal odds; b 2 vs. b 1, the absolute value of (scaled) odds changes vs. number of odds changes; c 1 vs. a 0, log of the probablty mpled by track odds vs. track odds. Whlst the late change parameter c 3, whch appears n the model, has no drectly comparable transparent alternatve, we found that the coeffcent for the scaled verson of c 3 s sgnfcant whereas that for the non-scaled (more transparent) verson of c 3 s not. 6. Model Testng Races 801 to 1200, run durng May/June 1998, were used to test the model. As we do not have repeated observatons (each race s only run once), we must use ndrect methods to measure the accuracy of the model. We consder two bettng strateges, maxmum expected payoff and maxmum expected log payoff. For each strategy we use the model probabltes and fnal odds as nputs, and analyse the returns produced. Gven correct probabltes as nputs (as opposed to estmated probabltes), both strateges gve non-negatve expected returns. Thus, f they gve non-negatve returns usng our model probabltes p (j) as nputs, we take ths as evdence that the p (j) are reasonably accurate. Moreover a postve return ndcates that the market s weak form neffcent; snce abnormal returns can be made by smply employng hstorcal prce nformaton. 6.1 Maxmum expected payoff

19 For race, let r = (r (1),..., r (k)) be the returns, that s r (j) = 1 + u,j (n) = 1/q (j) s the amount returned on a unt bet on horse j f the horse wns. Let b = (b (1),..., b (k)) gve the amount bet on each horse n race. As before, we wll drop the subscrpt when the context s clear. Under the model, the expected payoff on the race s E(b) = k j= 1 p( j) b( j) r( j) k j= 1 b( j) = k j= 1 b( j)( p( j) r( j) 1). (9) If p(j)r(j) > 1 for some j, then E(b) s unbounded unless we bound b. Accordngly, maxmsng E subject to the constrant b( j) 1, b ( j) 0 for all j, gves the followng strategy. Let j* be the value of j that maxmses p(j)r(j), then b(j) = 0 for j j*, and b(j*) = 1 f p(j*)r(j*) > 1 or 0 otherwse. k j = 1 Followng ths strategy from race to race, the total wealth behaves lke a random walk wth a lnear trend (drft). As such, total wealth can grow at most lnearly, and there s a non-zero probablty of run. In practce we modfy the strategy n two ways to reduce ts varance, and thus reduce the chance of run. Frstly, when a bet s ndcated, only an amount b(j*) = 1/r(j*) s bet (bet to wn amount 1). Ths reduces your exposure on horses that are less lkely to wn. Secondly, n addton to requrng p(j)r(j) > 1, we apply a flter rule, further restrctng bets to horses where track probablty q(j) 0.2, that s1/r(j) 0.2. Ths restrcton s chosen snce prevous studes (e.g. Tuckwell, 1983) ndcate that the expected loss to the bettor ncreases sgnfcantly for horses wth track probablty less than 0.2. Ths strategy provdes two complementary nstructons: whch races to bet on, and how much to bet on each horse (at most one per race n ths case). Applyng ths strategy to the test data set, we obtaned the followng results: average proft per race = ; proporton of races bet on = 32%; average proft per race bet on = ;

20 average proft per pound bet = Fgure 3 plots the cumulatve proft under ths strategy. The cumulatve proft over the test perod was (bettng up to 1 unt each tme), though we note that t dd drop to durng the test perod. As expected, ths strategy gves a (postve) lnear growth for total wealth, and compares very favourably wth a random strategy of ether bettng an equal amount on each horse (average return % per race) or bettng an amount proportonal to the track probablty on each horse (average return % per race). Fgure 3. Cumulatve proft usng maxmum expected payoff strategy. On the left we have the proft for races 1 to 800 (those used to ft the model), and on the rght the proft for races 801 to 1200 (the test data set) Kelly strategy: maxmum expected log payoff In ths settng, nstead of bettng an amount b (j) on horse j n race, we bet a fracton f (j) of our current wealth. Let f = (f (1),..., f (k)). As usual, we wll drop the subscrpt when the context makes t unnecessary. Bettng fracton f, f horse x wns then our current wealth wll ncrease by a factor of 1 = k j 1 f ( j) + f ( x) r( x). The Kelly strategy conssts of choosng f to maxmse the expected log payoff, F(f) where k k F(f ) = p( x)log( f ( x) r( x) + 1 f ( j) ). (10) x= 1 j= 1

21 Ths bettng strategy was ntroduced by Kelly (1956). It was later shown to be asymptotcally optmal by Breman (1961), n the sense that t maxmses the asymptotc rate of growth for wealth, wth 0 probablty of run. Usng the Kelly crteron, the total wealth grows at an exponental rate, though the standard devaton remans proportonal to total wealth and thus also grows exponentally. We also note that ths strategy only gves 0 probablty of run f arbtrarly small bets are allowed. In practce ths caveat has lead some authors to consder modfed Kelly strateges (e.g. Benter, 1994; Zemba and Hausch, 1986), whereby some fxed fracton of f s bet. As we are nterested n the theoretcal rather than practcal performance of our model, we restrct ourselves to the usual form. As for the maxmum expected payoff strategy, the Kelly strategy tells us whch races to bet on, as well as how much to bet on each horse. In ths case we can bet on more than one horse n a race, though our bets are restrcted to horses that gve a postve expected return. Applyng ths strategy to the test data set, we obtaned the followng results: average proft per race = * wealth; proporton of races bet on = 42%; average proft per race bet on = * wealth; average proft per pound bet = Fgure 4 plots the natural logarthm of the cumulatve wealth under ths strategy, startng wth ntal wealth 1. Over the out of sample test perod, total wealth ncreased exponentally by a factor of Thus, although the expected proft per race s sgnfcantly less than that obtaned usng the maxmum expected payoff strategy, compared to , n the long run the Kelly strategy wll perform better. Ths s because the Kelly strategy s cumulatve: the sze of bets made s proportonal to the current wealth, whereas the sze of bets made under the maxmum expected payoff strategy s fxed. We also observe that the Kelly crteron was more 3 For the purpose of comparson we take current wealth as 1 when applyng the Kelly strategy.

22 consstent than the maxmum expected payoff strategy over the test data set. We suggest that ths s because the Kelly crteron effectvely spreads the rsk more, by bettng on more than one horse per race. Fgure 4. Log of cumulatve wealth usng the Kelly strategy. On the left we have the wealth for races 1 to 800 (those used to ft the model), and on the rght the wealth for races 801 to 1200 (the test data set). Here wealth s gven as a multple of orgnal wealth. 7. Comparson wth random bettng Both bettng strateges consdered n the prevous secton gave a postve return over the test perod. To be confdent that we can ascrbe ths to the accuracy of our model, rather than just good luck, we consder how our model compares to an unnformed random alternatve. That s, we test the hypothess that our model performs no better than a model that makes no use of the nformaton contaned n the prce curves, other than the fnal odds. For a gven race, we construct a random model q * = (q *(1),..., q *(k)) for the vector of wnnng probabltes as follows. Let r = (r (1),..., r (k)) be the returns and q = (q (1),..., q (k)) the track probabltes, q (j) = exp(-log(r(j))). The random model should be a random perturbaton of q. Accordngly, let E = (E (1),..., E (k)) be a vector of ndependent normal random varables, mean 0 and varance σ 2, and defne the random model by

23 q *( j) exp( log( r ( j)) E ( j)). That s, puttng Z (j) = exp(e (j)) (so Z (j) has a log-normal + dstrbuton), exp( log( r ( j)) + E ( j)) q ( j) Z ( j) q *( j) = =. (11) k q ( l) Z ( l) k exp( log( r + l= ( l)) E ( l)) 1 l= We ntroduce the random perturbatons n ths manner to mmc the structure of the condtonal logt model. That s, we replace <β,x,j > by log r (j) + E (j), notng that log r (j) = c 1 s an mportant component of x,j. Ths means that the perturbaton to q (j) s multplcatve rather than addtve. 1 Fgure 5. Hstogram of p (j) q ( j) (left) and q *(j) q ( j) (rght), usng σ = 1.4. In general k ( ) j= 1 j q ( ) > 1, snce bookmakers add a margn to ther odds. We norm q = (q (1),..., q (k)) so that t sums to 1 and call the normed track probabltes q = ( q (1), K, q ( k)), where = k l = q ( l 1 q ( j) q ( j) / ). To see that q * has the rght dstrbuton, we look at the dstrbuton of p (j) q ( j) over the test data set, and compare ths wth the dstrbuton of q *(j) ( j) choose σ so that the two dstrbutons have the same standard devaton, gvng σ = 1.4. Fgure 5 plots sample hstograms for the two dstrbutons. The dstrbuton of p (j) q ( j) s exact (gven the test data set). The dstrbuton of q *(j) q ( j) s a Monte-Carlo estmate. We see that whle q. We

24 the random model does not gve an exact ft to the desred dstrbuton, t does capture some of the mportant features of the shape. Further support for ths concluson s obtaned from a QQPlot of the two dstrbutons, whch ndcates a remarkably good degree of agreement. 4 We compare the performance of the random model wth the condtonal logt model n the context of our two bettng strateges. In each case we perform a number of Monte-Carlo smulatons to estmate the dstrbuton of the average proft per pound per race Maxmum expected payoff The random model was employed as follows. For each race n the test data set we smulated a set of random model probabltes q *, and then appled the maxmum expected payoff strategy to determne how much to bet on each horse. Ths gves a smulated proft for each race, whch we then average over all 400 races n the test data set to get an estmate of the average proft per race X. Ths procedure s then repeated 1000 tmes to get an estmate of the dstrbuton of X. Fgure 6 gves the emprcal cumulatve dstrbuton functon (emprcal CDF) of X. Fgure 6. Emprcal cumulatve dstrbuton functon of the average proft per race over the test perod for the random model, usng the strategy of Maxmum Expected Return on the left, and the Kelly Strategy on the rght. 4 A QQPlot plots the quantles of one dstrbuton aganst those of another. A straght lne ndcates a good

25 Usng our condtonal logt model, the acheved average proft per race was Usng the emprcal CDF, we estmate Pr( X > ) = That s, under the hypothess that our condtonal logt model performs no better than the random model, the probablty of achevng an average proft per race of or better s In other words, the test has p-value 0.169, whch s not partcularly sgnfcant. Note that we compared the proft per race rather than per race bet on, as the decson whether or not to bet depends on the probabltes used. Usng the random model we bet on 24% of races on average (cf. 32% usng our condtonal logt model). To judge the senstvty of the test to the value of σ, we repeated the test for a number of dfferent values of σ. The results are gven n Table 2. Table 2. Senstvty of p-value to choce of σ, strategy of Maxmum Expected Return. σ E(X) Pr(X > ) We see that the test s nsenstve to the value of σ used. The relatvely hgh probablty that the random model results n a postve proft over the test perod can be attrbuted to the favourte-long shot bas. Our bettng strategy s restrcted to horses wth track probablty q (j) > 0.2 (odds of 4/1 or lower). As has been prevously noted, the track probablty tends to be an underestmate for short odds horses, meanng that one has a greater chance of a postve return by bettng on short odds horses. We also note that 100% of the tme, ths random strategy performed better than the naïve strateges of bettng an equal amount on each horse, or bettng to wn 1 pound on each horse Kelly strategy: maxmum expected log payoff match between the dstrbutons.

26 For each race n the test data set we generated a set of random probabltes q *, to whch we appled the Kelly strategy to determne how much to bet on each horse. Ths was repeated for all 400 races n the test data set to obtan an average proft per pound per race, denoted Y. Ths whole procedure was then repeated 500 tmes to obtan an estmate of the dstrbuton of Y. The emprcal CDF of Y s gven n Fgure 6. Usng our condtonal logt model, the acheved average proft per pound per race was Usng the emprcal CDF we estmate Pr( X > ) = That s, under the hypothess that our condtonal logt model performs no better than the random model, the probablty of achevng an average proft per pound per race of or better s In other words, the test has p-value 0.006, whch s hghly sgnfcant. To judge the senstvty of the test to the value of σ, we repeated the test for a number of dfferent values of σ. The results are gven n Table 3. Table 3. Senstvty of p-value to choce of σ, for Kelly Strategy. σ E(X) Pr(X > ) Agan, the test s nsenstve to the value of σ used. Gven such low p-values, we can be confdent n sayng that the condtonal logt model s not assgnng probabltes at random. The reason that ths s brought out when usng the Kelly strategy as opposed to the maxmum expected payoff strategy, s that the Kelly strategy requres good estmates of the wn probablty for every horse n a race f t s to be successful. In other words, because the Kelly strategy uses p more extensvely than the maxmum expected payoff strategy, t s more senstve to errors n p. Usng the Kelly strategy, the amount bet on horse j depends on p (j) for each j. Usng the maxmum expected payoff strategy, only the p (j) for whch p (j)r (j) s largest s mportant, and p (j) s only used to decde whch horse to bet on, not how much to bet.

27 8. Conclusons In ths paper we set out to explore whether the bookmaker horserace bettng market fully ncorporates a varety of hstorcal prce nformaton varables, ncludng nteracton effects. We conclude that t does not; suggestng that the market s weak form neffcent. Ths concluson s n sharp contrast to the majorty of studes examnng bettng market effcency. We beleve ths s because prevous studes have focussed on assessng the extent to whch ndvdual factors are effcently ncorporated nto prces, rather than lookng at combnatons of a range of varables. In addton, prevous studes have utlsed readly dscernable varables, most of whch sgnal a horse s enhanced prospects of success. Our fndngs also suggest that market partcpants are largely effectve n dscountng transparent nformaton concernng a horse s prospects n ther decsons but they do not appear to ncorporate more obscure nformaton: odds and the manner n whch odds move, are rch but subtle nformaton sources, whch bettors do not fully utlze. In summary, ths paper adds to our knowledge of the degree to whch dfferent types of nformaton are dscounted n decsons made n bettng markets. It also ntroduces technques for parametersng prce curves and a novel approach for testng a model for producng probabltes, by comparson aganst a random alternatve. Future work explorng other fnancal markets, usng the technques ntroduced here, may yeld nterestng conclusons regardng market effcency and the manner n whch nformaton s employed by market partcpants. References Al, M. M Probablty and utlty estmates for racetrack bettors. J. Poltcal Economy Asch, P., B.G. Malkel and R.E. Quandt Racetrack bettng and nformed behavor. J. Fnancal Economcs

28 Ashton, R.H Effects of justfcaton and a mechancal ad on judgement performance. Organzatonal Behavor and Human Decson Processes Benter, W Computer based horse race handcappng and wagerng systems:a report. D.B. Hausch, V.S.Y. Lo, and W.T. Zemba, eds. Effcency of Racetrack Bettng. Academc Press, London, Brd, R. and M. McCrae Tests of the effcency of racetrack bettng usng bookmaker odds. Management Scence Brd, R. and M. McCrae Racetrack bettng n Australa: A study of rsk preferences and market effcency. W.A. Eadngton and J.A. Cornelus,eds. Proc. 7 th Inter. Conf. Gamblng and Rsk Takng. Insttute for the Study of Gamblng and Commercal Gamng, Reno, Breman, L Optmal gamblng systems for favourable games. Proc. 4 th Berkeley Symposum Math. Statst. Probab Bruce A.C. and J.E.V. Johnson Investgatng the roots of the favourte-longshot bas: An analyss of supply and demand sde agents n parallel bettng markets. J. Behavoral Decson Makng Cec, S.R. and J.K. Lker (1986) A day at the races: A study of IQ, expertse, and cogntve complexty. J. Expermental Psychology: General Chapman R.G Stll searchng for postve returns at the track: Emprcal results for 2000 Hong Kong races. D.B. Hausch, V.S.Y. Lo, and W.T. Zemba, eds. Effcency of Racetrack Bettng. Academc Press, London, Crafts, N.F.R Some evdence of nsder knowledge n horse race bettng n Brtan. Economca Dowe, J On the effcency and equty of bettng markets. Economca