Johnne Johnson, Owen Jones and Lele Tang Explorng decson-makers use of prce nformaton n a speculatve market 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. The horserace bettng market s chosen as an deal envronment to explore these ssues. A condtonal logt model s constructed to determne wnnng probabltes based on bookmakers closng prces and the tme ndexed movement of prces to the market close. The paper ncorporates a technque for extractng predctors from prce (odds) curves usng orthogonal polynomals. The results ndcate that closng prces do not fully ncorporate market prce nformaton, partcularly nformaton whch s less readly dscernable by market partcpants. 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, 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, wagerng markets can provde a clear vew of prcng ssues whch are complcated elsewhere (Sauer, 1998, p. 2021). Prevous studes explorng the extent to whch horserace bettng markets ncorporate closng odds and the tme ndexed movement of pre-closng odds fall nto three broad categores. One set of studes explores the dstrbuton of closng odds n horserace bettng markets. These offer
- 2 - 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 and across a varety of countres (e.g. McGlothln, 1956; Al, 1977; Tuckwell, 1983; Zemba and Hausch, 1986; Brd and McRae, 1994; Bruce and Johnson, 2000). However, these studes do not generally detect opportuntes for tradng proftably on ths nformaton 1 ; suggestng that hstorcal prces are largely dscounted n closng odds. A second set of studes explores the nformaton content assocated wth changes n odds from the openng of the market to ts close. 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 or parmutuel odds change to reflect relatve volumes of bets. In general, prevous studes suggest that bettng actvty reveals nformaton that s not readly avalable pror to the formaton of the market, but that bettng strateges based on these odds adjustments do not yeld postve expected returns (e.g. Asch, Malkel and Quandt, 1982; Brd and McCrae, 1987; Crafts, 1985; Schnytzer and Shlony, 2002; Tuckwell, 1983). Sauer (1998, pp. 2048-2049), revewng these studes, concludes that the evdence suggests that an nformed class of bettors s responsble for alterng prces n these markets.. (but) the opnons of experts appear to be fully dscounted n market prces. A thrd set of studes attempt to construct arbtrage bettng strateges whch employ nformaton revealed wthn one market n a parallel market. Some of these studes have demonstrated that postve returns are possble by explotng dfferences n pre-closng odds n ndependent wn pools (e.g. Hausch and Zemba, 1990; Schnytzer and Shlony, 1995) or between wn and place or show pools 2 (e.g. Hausch, Zemba and Rubnsten,1981). However, 1 The one excepton, a study conducted by Zemba and Hausch (1986) n the USA, dentfes a small expected proft from bettng on a very small number of extreme favourtes (odds < 3/10). 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 frst or second) and for show bets (those whch attempt to select horses to fnsh frst, second or thrd). Odds are determned separately n each pool by the
- 3 - opportuntes for proftable bettng employng the above strateges are lmted and the degree of neffcency s often small. 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). It s clear from the precedng dscusson that varous aspects of closng odds and odds movements from the openng of the market to ts close, taken alone (e.g. under-bettng of favourtes (Vaughan-Wllams and Paton, 1997); odds assessed by dfferent handcappers (Fglewsk, 1979); dfferences between mornng lne and closng odds (Crafts, 1985); odds at dfferent tme ponts (Lo, 1994)) do contan nformaton, but that ths s dscounted n closng odds, to the extent that proftable wagerng strateges based on ths nformaton cannot be constructed. However, Cec and Lker (1986) demonstrate that expert horse handcappng requres bettors to combne dfferent types of nformaton n complex, nteractve models. We set out, therefore, to develop such a comprehensve model. Ths model combnes closng odds and varables derved from the movement of pre-closng odds to closng odds that capture nformaton whch (a) may not be closely assocated n the publc s mnd wth a horse s success (e.g. the volatlty of odds changes) and (b) are less readly dscernable by bettors (e.g. odds changes scaled by closng odds). In addton, we wll nclude nteracton terms between the varables. Our vew s that bettors may not readly dscount the combned effect of such varables and ther nteractons. Consequently, t may be possble to construct proftable wagerng strateges based on such a model, and hence demonstrate that the horserace bettng market s not weak form effcent. To acheve our am the paper proceeds as follows. Secton 2 outlnes the data employed, provdes a ratonale for the model whch s used to test for market effcency, and dscusses the relatve amount of money on each horse.
- 4 - sgnfcant components of the ftted model. In Secton 3 the model s tested, to explore the extent to whch t demonstrates that nformaton on changes n pre-closng odds s dscounted n closng odds. Fnally, n Secton 4 we summarze our conclusons. 2. Descrpton of the data and the model There are two dstnct forms of horserace bettng market that operate n parallel at racetracks n the UK; the parmutuel and the bookmaker markets. The latter forms the settng for ths study. The odds on offer n the bookmaker market are determned by the decsons of both 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 prvleged nformaton are most lkely to bet n the bookmaker market (Crafts, 1985; Sauer, 1998; Schnytzer and Shlony 1995) snce they can secure ther return, wthout the possblty of a bandwagon effect erodng ther gans (whch can happen n the parmutuel market). Consequently, snce we seek to explot as much nformaton as possble contaned n closng odds and pre-closng odds and ther changes over tme, t seems approprate to use bookmakers odds. Independent bookmakers (generally 10 to 50) operate at each racetrack and post odds at the commencement of the market before each race. 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. Bettors n the UK are also permtted to bet n off-track bettng offces at the pre-closng odds (or the closng odds) prevalng n the on-course market at any gven tme. An ndependent organsaton, SIS, transmts the evolvng racetrack bookmakers odds to off-course bettng offces. SIS employs assessors who use ther judgement to determne a unque value for the odds avalable on each horse at each moment n tme; these are the maxmum bookmaker odds avalable to a substantal wager at the track. Accordngly, we collected pre-closng and closng bookmaker odds data suppled by SIS from 1,200 races run at 41 dfferent racetracks n the UK over the perod Aprl June 1998. Only flat races of less
- 5 - than 2 mles were ncluded. 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))}. Ths sequence s unque for horse j n race, beng the odds transmtted by SIS. The fnal par (t,j (n), u,j (n)) s always the off tme and closng 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). 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 extracted from the prce curve, predctors 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 exp( < β, x, j > ) p = k ( ) = exp( <, (1) β, x ) 1, > l l m, h j h 1, ( ) where< β, x j >= = β ( h) x. 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.
- 6 - 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 (v) = exp(-exp(-v)), for - < v <. If we put W = < β, x, >+ ε then t can be shown that Pr( W W ( l), l = 1, K, k( )) p (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. 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 j N N exp( < β, x, j* > ) L( β ) = p ( j*) =. (2) k = 1 l ( ) = 1 exp( < β, x > = 1, l ) We employ maxmum lkelhood estmaton to choose β that maxmzes L(β). 2.1. Extractng predctors from the prce curve: To construct a model for p (j) we requre a consstent set of predctors, drawn from 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 gamblers as havng an effect on the horse s wnnng probablty. For all the predctors 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. We are nterested to nclude predctors that reflect the general shape of the prce curve; whch, amongst other thngs, wll reflect the tmng of bets by those wth prvleged nformaton. However, the precse shape of a prce curve that s lkely to sgnal a potental wnner (or loser) s unclear. We therefore provde a general summary of the shape of a prce curve {(t(1),u(1)),..., (t(n),u(n))} by usng an orthogonal polynomal expanson of order 3. Ths allows us to measure the heght of the curve (closng odds), the lnear trend, the curvature (quadratc component) and
- 7 - 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. 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 ( t( l)) = 0 for all j. (3) j 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 = =0 u ( l) a f ( t( l)) for all l = 1K,, 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
- 8 - mpose further constrants wthout compromsng orthogonalty. In our case, because of the mportance of the closng 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. 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) = = a 0 mnmze 3 f ( t), then we choose the a to 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) umax,0) dt+ max( u F( t),0) dt. (6) 0 In addton to a general summary of the shape of the prce curve (dscussed above) we also ncluded two predctors to measure the volatlty or roughness 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 0 mn
- 9 - n 1 b 1 = n 2 and b 2 = u( l) u( l 1). (7) l= Clearly there wll be some degree of correlaton between b 1 and b 2. 2 The remanng three predctors n the model were ncluded snce prevous lterature suggests that they may have some nfluence on a horse s probablty of success. Frstly, a number of studes have demonstrated a close correspondence between probabltes mpled by closng odds and wnnng probabltes (e.g. Bruce and Johnson, 2000). The closng odds for horse j n race mply a probablty of wnnng q (j), va the relatonshp u, j 1 q ( n) =, q 1 q =. (8) 1+ u, ( n) j We call ths the track probablty to dstngush t from the model probablty p (j). We already nclude the closng odds n the model as a 0 = u,j, whch gves p exp( β ( a0 ) (1 q ) / q ). 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 predctor c 1 = ln q (j) = ln(1 + u,j (n)), whch gves p β = β ( c1 ) exp( ( c1) c1 ) q. In fact, Chapman (1994) even found that c 1 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 = closng odds ntal odds = u,j (n) u,j (1), although ths wll be hghly correlated wth the slope a 1. Those wth access to prvleged nformaton have an ncentve to bet late n the market (Asch, Malkel and Quandt, 1983; Schnytzer, Shlony and Thorne, 2003). Consequently we use a predctor to capture late changes n the bettng. Let U(t) be the odds at tme t, that s for t ( ) t< t( + 1), U(t) = u(). Let [a, b] be a small subnterval of [0, 1], close to 1. We take as our
- 10 - 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 predctor be c 3, 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 predctor. 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) 3 = = 0 a f ( t). F(t) s a smooth approxmaton of the prce curve, so we expect c 3 to be hghly correlated wth F (1) = a 1 + 2a 2 + 3a 3. Fnally, two refnements of the predctor 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 closng 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 predctors 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. Predctors a 0 and c 1, whch are based on the closng 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
- 11 - changes down rather than up. Accordngly predctors a 1, c 2 and c 3 were splt nto pars x + and x, where x + = x f x > 0 or 0 otherwse, and x = x f x < 0 or 0 otherwse. Smlarly, predctors b 1 and b 2 were splt nto two parts, dependng on whether or not a 1 > 0. 2.2. 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 predctors sgnfcant at the 5% level. Parwse nteractons of all the predctors were also consdered. In the fnal model the predctors a + 1, b 2, c 1, c 3 and the nteracton a + 1 b 1 were all sgnfcant at the 5% level. The estmated coeffcents β are gven n table 1. The log-lkelhood rato of the model over the constant alternatve s 857.05, whch gves us that the model s sgnfcant wth a p- value of 0.0000. Table 1. Estmated coeffcents of the model. Note that a 1 +, b 2 and a 1 + b 1 0; c 1 and c 3 0. Predctor Descrpton Coeffcent Standard p-value Error + a 1 slope up -2.0493 0.8110 0.0115 b 2 absolute varaton down -0.4227 0.1907 0.0266 c 1 ln(track probablty) 1.1678 0.0648 0.0000 c 3 late change down -0.0666 0.0331 0.0440 a + 1 b 1 slope up and number of changes nteracton 0.4371 0.1937 0.0240 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 predctors 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 closng odds than c 2, b 1 and a 0. Secondly, 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. The predctor c 1 = ln q (j), has a large nfluence. Consderng just the effect of c 1 on the model probablty we have p (j) exp( 1.1678 ln q (j)) = q (j) 1.1678, whch s smlar to the utlty
- 12 - functon derved n Al (1977). We nterpret ths relatonshp as a reflecton of the so-called favourte-long shot bas. That s, the shorter a horse s closng odds, the nearer the true probablty of wnnng s to the mpled track probablty. A plot of ln u,j (n) = ln (1-q (j))/q (j) aganst ln q (j) 1.1678 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. The sgnfcance of the predctor a + 1 suggests that when odds lengthen (.e. the least squares regresson lne constraned to pass through t(n), u(n) has a postve slope) the horse has a sgnfcantly lower chance of wnnng. 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. The late change down predctor, 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. The coeffcent of b 2 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. Each of the predctors wth sgnfcant coeffcents mght be descrbed as relatvely dffcult to dscern (relatvely opaque) compared wth an equvalent, but more transparent predctor excluded from the model; for example, a 1 vs. c 2, slope of the least squares regresson lne through t(n), u(n) vs. closng - ntal odds; b 2 vs. b 1, the absolute value of (scaled) odds changes vs. number of odds changes; c 1 vs. a 0, natural log of the probablty mpled by closng track odds vs. closng track odds. Whlst the late change predctor 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. Taken together, the results suggest that more readly dscernable (or relatvely transparent) nformaton s more effcently dscounted n bettng markets than more opaque nformaton.
- 13-3. Model testng In order to test the model we: () explore, usng log-lkelhood rato tests, whether the model contans more nformaton than a model based solely on closng odds () test whether the addtonal nformaton contaned n the model can be exploted to make profts () use crossvaldaton and jack-knfng to check that the observed profts do not arse by fortunate selecton of the tranng and valdaton sets; (v) use parametrc bootstrappng (on the tranng set employed n ()) to test whether the profts we observe arse because our model ncorporates more nformaton than that avalable from closng odds alone or whether the profts arse by chance (v) explore the features of wnnng bets suggested by the model n order to dentfy any systematc features of such bets.. 3.1. Log lkelhood tests: In order to explore the jont mportance of the predctors n the model we conduct log-lkelhood rato tests. We compare the log-lkelhood (LL) of the full model gven n table 1 ( LL = -2243.9) wth that of a model smply usng normed fnal track probabltes (LL = - 2262.7). A LL rato test comparng the two models has a p value < 0.0001. Ths test confrms that there s sgnfcant sample evdence to suggest that the full model ncorporates more nformaton than fnal track probabltes alone. In the followng secton we explore the extent to whch ths extra nformaton s substantve, to the extent that we are able to employ t proftably. 3.2. Kelly Bettng: To assess whether the observed neffcency can be exploted suffcently to make profts, races 801 to 1200, run durng May/June 1998, were used to test the predctve ablty of the model. As we do not have repeated observatons (each race s only run once), we must use ndrect methods. We consder a bettng strategy based on maxmum expected log payoff (Kelly strategy). We use the model probabltes and closng odds as nputs, and analyse the returns produced. Gven correct probabltes as nputs (as opposed to estmated probabltes), the strategy gves non-negatve expected returns. Thus, f t gves non-negatve
- 14 - 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 abnormal returns can be made by smply employng hstorcal prce nformaton. Let r (j) = 1 + u j (n) be the return on a bet of 1 pound f horse j wns race. The Kelly strategy requres that n race we bet a fracton f (j) of our current wealth on horse j. 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 + f ( x) r( x). The Kelly strategy conssts of choosng f to maxmse the expected log payoff, F(f) where k x= 1 k ( f ( x) r( x) + 1 f ) F(f ) = p( x) ln. (9) j= 1 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 led 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. The Kelly strategy tells us whch races to bet on, as well as how much to bet on each horse. We can bet on more than one horse n a race, though our bets are restrcted to horses that gve a postve expected return. In ths manner, the Kelly bettng strategy makes use of the whole vector of probabltes provded by the model, not just for the horse most lkely to wn. By bettng on a number of horses n a race the rsk of losng s reduced at the expense of reducng the expected
- 15 - return. None the less, bet szes suggested by the Kelly strategy wll be larger when the probablty of wnnng s greater (for the same expected return) and when the expected return s greater (for the same wnnng probablty). Consequently, a Kelly bettng strategy makes greater use of nformaton provded by the model than a smple strategy of maxmzng the expected return. Fgure 3 plots the natural logarthm of the cumulatve wealth, applyng the Kelly strategy to the test set, startng wth ntal wealth 1. Over the out of sample test perod, total wealth ncreased by a factor of 2.4597. Fgure 3. Ln 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. In testng the sgnfcance of returns from bets t s common to consder the proft per pound bet B. However, f we let X n be the wealth after n races, then we obtan X ) + 1 = X + b X B = (1+ b B X, where b s the amount bet per pound of ntal wealth on race. That s, we cannot express the ncrease n wealth solely n terms of the B. Let w be the proft made per pound of ntal wealth on race, so that B = w /b (defnng B = 0 when b = 0). In the context of Kelly bettng, a more natural object to consder than B s W = 1 + w, whch s the factor by whch wealth has ncreased after race. That s X = W +1 X. Takng
- 16 - logs we turn the multplcatve form of the cumulatve wealth nto an addtve form, to obtan ln( X n ) = ln( X n + = ln(1+ 0 ) w 1 ). Let A = ln(1 + w ), then A s a natural object to average, and X n exhbts long term growth f and only f E(A ) > 0. Note that t s not the case that X n exhbts long term growth f and only f E(B ) > 0, snce there s (typcally) postve correlaton between b and B, n whch case E(1+ b B ) > 1+ E( b )E( B ). From the testng set we estmate µ = E(A ) = 0.00226 and 2 σ = Var(A ) = 0.0477 2. When usng Kelly bettng, µ s the asymptotc growth rate for wealth (per race). From the Central Lmt Theorem our estmators are approxmately Normally dstrbuted, whch allows us to calculate that a test of the hypothess µ = 0 aganst the alternatve µ > 0 s sgnfcant at the 17% level. Usng our estmates for µ and 2 σ we can estmate the number of races that we would need to consder to ensure that we are 95% certan of makng proft (notng that not all n = 1 races are bet on). Ths corresponds to the smallest value of n such that Pr( A > 0) 0. 95. Usng the Central Lmt Theorem to approxmate the sum by a Normal random varable, we fnd that n = 1,653. We conclude from these results that usng the full model together wth the Kelly bettng strategy there s some evdence of makng proft f bettng n a suffcently large number of races. Ths n turn suggests that closng odds do not fully ncorporate actve market odds nformaton; that s the market s not weak form effcent. An mportant operatonal ssue s the effect on closng odds of wagers based on the model predctons. Wth the par-mutuel system, large wagers automatcally reduce the odds at whch a bet s settled. However, our model s developed for a bookmaker market and requres that bets are placed close to the start of a race. Although there may be some feedback to closng odds f a large bet s made ths would not mpact on returns snce these are fxed n a bookmaker market at
- 17 - the tme the wager s made. It s possble that an ndvdual bookmaker may refuse a bet or offer reduced odds on a very large bet. However, the odds employed n ths study ( provded by SIS as ndcated n secton 2) are the odds avalable to a substantal wager at the track. In addton, n the UK there are upwards of 20 ndependent bookmakers at any racetrack as well as thousands of ndependent off-track bookmakers who accept bets at the odds on offer at racetracks at any gven tme. Consequently, a well organsed group of ndvduals could splt a very large wager nto smaller amounts and place these smultaneously at several outlets wthout mpactng the odds at whch the bet would be settled. We note that to make a proft n the racetrack bettng market t s necessary to overcome the bookmakers margn, whch s typcally between 15% and 20%. That s, bettng at random we would expect to see E(W ) 0.85. For our full model, usng the test data set we get a 95% CI for E(W ) of (0.9993, ), sgnfcantly hgher than the 0.85 we would expect from random bettng. 3.3. Cross-valdaton and Jack-knfng: We use cross-valdaton and jack-knfng to explore whether the profts we observe from applcaton of the model arse by fortunate selecton of the tranng and valdaton sets. In partcular we use them to renforce our estmate of µ. For the cross-valdaton we splt the data nto 12 blocks of sze 100. We then consder all possble choces of 8 blocks for the tranng set and 4 blocks for the testng set. For each of the 12 C 8 = 495 possble choces we ft the model to the tranng set and then estmate µ from the testng set. Let µˆ j be the estmate obtaned from the jth tral, then the cross-valdaton estmate 1 495 = of µ s ˆ = 0. 00218 1 495 j µ j. We also obtan that 80% of the cross-valdaton estmates are postve, whch s agan n close agreement wth the analyss above. As theµˆ are all hghly correlated, t s not possble to obtan a confdence nterval for µ drectly from the crossvaldaton estmates. j
- 18 - We also appled a partcular varant of cross-valdaton known as the jack-knfe; race s set asde for testng and the remanng races are used for model fttng. Ths s repeated for each of the 1200 races. Let J A be the observed wealth growth rate obtaned by usng the ftted model to bet on race. Note that the model wll change margnally each tme as the tranng set changes slghtly. Let J µ be the mean of J A. We can vew J µ as a contnuous non-lnear functon of the J estmated model parameters β, whch now also depend on. Thus, n general, µ µ, where µ s the asymptotc growth rate of wealth (per race) for the parameters β ftted usng the orgnal tranng set (races 1 to 800). However, maxmum lkelhood estmaton produces consstent estmators, so our estmates of β wll converge to some lmt as the sze of the tranng set J ncreases. Thus t s reasonable to assume that µ µ Because the models used to produce the None-the-less most of the varaton n the on, and not the model tself. Thus the. J A are hghly dependent, the J A are also dependent. J A comes from the race whch the model was tested J A wll be approxmately ndependent, whch allows us to gve a confdence nterval for µ. The jack-knfe sample had mean 0.00309 and standard devaton 0.07055, gvng an approxmate 95% confdence nterval for µ of 0.00309 ± 0.00399 = (-0.00090, 0.00708). A test of the hypothess µ =0 aganst the alternatve µ >0 s sgnfcant at the 6.5% level. We note that the mean of the J A s hgher than the mean of the A (gven n secton 3.2), possbly due to a better estmate of β resultng from a larger tranng sample. However, the varance of the J A s hgher than the varance of the A, whch s probably due to the varaton n the models used to calculate each J A. That s we are gettng varance from the model and the test sample pont. Overall the results of the jack-knfe procedure provde further evdence that the
- 19 - model can be exploted to earn postve returns; ndcatng that the market s weak form neffcent. 3.4. Parametrc bootstrappng: The lkelhood rato test shows that our full model captures sgnfcantly more of the nformaton n the prce curve than that avalable from the closng odds alone, and the applcaton of the Kelly bettng strategy suggests that there may be enough actonable nformaton n the model to make a proft. However there s stll a queston as to whether the ablty to make a proft depends only on closng odds (plus some good fortune) or can be attrbuted to the extra nformaton ncluded n the model. We explore ths usng a parametrc bootstrappng approach; whch nvolves re-samplng from a model ftted to the observed data. Note that f we apply the Kelly bettng strategy usng a model based on fnal track probabltes, then we never actually place any bets as the model never overcomes the bookmakers margn. Consequently, n ths secton we consder a model based on normed fnal track probabltes whch does allow us to apply the Kelly bettng strategy, and thus provdes a comparson wth the full model on the bass of proft earned rather than lkelhood. We norm the track probabltes q = (q (1),..., q (k)) so that they sum to 1 and call the normed track probabltes q = ( q (1), K, q ( k)), where q = q / q ( l). Under the hypothess H 0 that the model p (j) depends only on closng odds, we have p = q + ε where the ε are ndependent errors, condtonal on j p = 1, that s j ε = 0. We note that log q s just a scaled verson of the predctor c 1. In order to mmc the form of the condtonal logt model, we re-wrte p (j) as exp(log q + ~ ε ) = q exp( ~ ε ) where ~ ε = log(1+ ε / q ) are agan ndependent errors, condtonal on j p = 1. The p (j) are estmated usng the full model as gven n secton 2.2 and ~ ε ( j ) s a measure of the l
- 20 - dscrepancy between p (j) and q. The am of ths analyss s to judge whether or not p (j) provdes more actonable nformaton than q. That s, whether the addtonal nformaton captured by the full model over closng odds alone, s needed to generate a proft. If proftablty depends only on the closng odds then ~ ε ( j ) s effectvely a random adjustment to, whch we can smulate. Consequently, to test whether p (j) tells us more than just q we test whether or not usng the real ~ ε ( j ) s any better than usng a randomly generated ~ ε ( j ). We estmate the dstrbuton of ~ ε ( j ) drectly from the tranng data set. Fgure 4 gves a hstogram of ~ ε log( p / q ) for all horses n races 1 to 800.The sample mean and = standard devaton are 0.1012 and 1.1865. The Kolmogorov-Smrnov test for normalty gves a p-value of 0.1492, ndcatng no sgnfcant cause not to accept ths as a normal dstrbuton (ths justfes the error form used). Thus under the hypothess H 0 that p (j) depends only on closng odds, we obtan q Z p = for Z (j)..d. ln normal(-0.1012, 1.1865 2 ) (10) q ( l) Z ( l) l q Fgure 4. Hstogram of the log error ~ ε ( j ), under the hypothess that p (j) depends only on closng odds
- 21 - We use Equaton 10 to test the hypothess that the ablty of the model to make proft depends only on closng odds. Under H 0 our full model s just as effectve as that gven by Equaton 10. We proceeded as follows. For each race n the test data set (races 801 to 1200) we randomly generated sample p (j) usng Equaton 10. These were then used to calculate the asymptotc growth rate for wealthµ, as n Secton 3.2. We then repeated ths 500 tmes to obtan an emprcal dstrbuton for µ under H 0, From the testng set (n secton 3.2) we estmated µ = 0.00226 and we now estmate that under H 0, P(µ > 0.00226) = 0.006. That s, under H 0 the probablty of observng a value of 0.00226 or hgher (the value acheved by the full model) s less than 0.006. Ths suggests that the proft we observed n secton 3.2 s lkely to be attrbutable to factors n the model other than closng odds. 3.5. Features of wnnng bets: We explore under what crcumstances the model produces successful bets n order to dentfy any systematc features of such bets, and, n partcular, to dentfy what varables/combnaton of varables are/s needed to dstngush the horses we should bet on. To acheve ths, for each race n the holdout sample we determne, for the wnnng horse, ts probablty of wnnng gven by the model, the normed track probabltes, and whether we successfully backed the horse. We plot the results n fgure 5; to spread the ponts we plot ln (model probablty) aganst ln (normed track probablty). Wnnng horses whch we backed are depcted by trangles, horses we faled to back because we backed another horse n the race (thereby losng money) by crosses and n races where no horse was backed, the wnner s depcted by a dot. We see from fgure 5 that we only backed horses when the model probablty was greater then the normed track probablty. The favourte-longshot bas means that ths happens more frequently for short odds horses, and the model s better at pckng short odds wnners than long odds wnners. In fact the results suggest that by restrctng our bets to horses where ln (model
- 22 - probablty) > -2.5 (.e. model probablty > 0.08) profts would mprove. We also note that for wnnng horses the dfferences between the normed track probabltes and the model probabltes were generally not very large, though there are a few exceptons. The relatve mportance to overall proft can be gauged by plottng the proft made on the z-axs. To vsualze ths rather than produce a 3D plot we projected the above ponts onto the man dagonal, and then plotted these aganst the proft made per pound of wealth (fgure 6). From ths we see that proft does not depend on occasonal very large profts on long-odds horses or on bets n a partcular odds range (other than suggestng that bets be restrcted to where the model probablty >0.08). 0 Comparatve probabltes for wnnng horses 0.6 Proft for wnnng horses 0.5 0.5 log normed track probablty 1 1.5 2 2.5 3 3.5 proft per pound of ntal wealth 0.4 0.3 0.2 0.1 0 0.1 4 0.2 4.5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 log model probablty Fgure 5: Analyss of bets on wnnng horses 0.3 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 log probablty Fgure 6: Proft obtaned from wnnng horses To gan further nsght nto how the model works we consder the effects of our predctor varables for all wnnng horses n the holdout sample (set A, 400 horses) and for wnnng horses we successfully bet on (set B, 61 horses). For each set we gve box plots of 2.0493*a + 1 + 0.4371*a + 1 b 1, -0.4227*b 2, 1.1678*c 1 and 0.0666*c 3. The sum of these four gves the wnnngness ndex β > for horse j n race. Box-plots provde a graphcal representaton <,x, j of the dstrbutons of each factor, ncludng the nterquartle range (boundares of the larger box)
- 23 - and the more extreme values of the dstrbuton. From Fgure 7 we see that there s lttle dfference between the factors from set A to set B, other than that wnnng horses whch were wagered upon n the sample start at slghtly lower odds than wnnng horses n general and that there s a preference for selectng lower odds varance horses. However the dfferences are not large enough to conclude that there s any sgnfcant dfference between the four factors from set A to set B, ndcatng that n general the model requres a combnaton of the factors for t to dstngush whch horses to bet on. Fgure 7. Influences of predctor varables for wnnng horses. Col. 1: -2.0493*a + 1 + 0.4371*a + 1 b 1 ; Col. 2: -0.4227*b 2 ; Col. 3: 1.1678*c 1 ; Col. 4: 0.0666*c 3. If we look at wnnng horses for whch the model probablty s much larger than the track probablty (by a factor of 1.1 or more) then there are some common characterstcs. There were 14 such horses, for 13 of them the odds ncreased and for 7 there was a late change down. A late change down ncreases β > and thus the model probablty p (j). However, ncreasng odds <,x, j decrease β >, and have a larger mpact than a late change (resultng from the szes of the <,x, j coeffcent and the varables). We conclude that n these 13 cases the model probablty has only mproved because of the relatve change n β > for the wnnng horse compared to the <,x, j
- 24 - other horses n the race. Agan ths ndcates that nformaton from many sources must be carefully weghed to be able to successfully model the probablty of wnnng. In summary, ths analyss suggests that the profts derved from the model are not dependent on a few long-odds wnners and no one predctor can be used to decde when to bet; the model acheves ts success by combnng a varety of factors. Overall, the results of model testng suggest that the model predctors contan sgnfcantly more nformaton than that contaned n closng odds. In addton, there s somethng n excess of an 80% chance that a bettng strategy based on the model wll produce a long term ncrease n wealth. Whlst ths s not conclusve proof that the horserace bettng market s weak form neffcent to the extent that t can be exploted by an expert to hs/her advantage, the results suggest that ths may be the case. 4. Conclusons In ths paper we set out to explore whether the horserace bettng market fully ncorporates a varety of hstorcal bookmaker prce nformaton varables, ncludng nteracton effects. We conclude that there s valuable nformaton contaned n odds and pre-closng odds movements whch s not fully dscounted n closng odds; suggestng that the market s weak form neffcent. Our attempts to use a bettng strategy to explot ths neffcency are also suggestve, f not conclusve, that our model could be employed to make profts. We beleve ths s because, as Cec and Lker (1986) observe, expert handcappng requres the ablty to combne dfferent types of nformaton n complex, nteractve models. Our results suggest that bettng market partcpants as a whole do not acheve ths to the extent of our model, whch ncorporates a range of varables coverng dfferent aspects of nformaton assocated wth closng odds and the movement of pre-closng odds to the closng odds, ncludng ther nteractons. The results also suggest that market partcpants are largely effectve n dscountng readly dscernable (more transparent) nformaton concernng a horse s enhanced prospects n ther decsons but they do not appear to ncorporate less readly dscernable (more obscure) nformaton. We conclude that
- 25 - closng odds and the manner n whch pre-closng odds move, are rch but subtle nformaton sources, whch bettors do not fully utlze. The model has served our purpose n explorng the weak form effcency of horserace bettng markets. However, we have not been tempted to undertake a real-lfe test of the model, snce ths would nvolve consderable effort n terms of real tme data capture and operaton of the model and moreover, the estmated growth rate for wealth, whle postve, s small and may not warrant the potental rsks. 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 a technque for extractng predctor varables from prce curves usng orthogonal polynomals and a varety of approaches for testng a model that produces probabltes. 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. Acknowledgements We are grateful to the anonymous referees, the Assocate Edtor and the Department Edtor, Detlof von Wnterfeldt for ther nsghtful comments on an earler draft of the paper. References Al, M. M.1977. Probablty and utlty estmates for racetrack bettors. J. Poltcal Economy 82 803-815. Asch, P., B.G. Malkel and R.E. Quandt 1982. Racetrack bettng and nformed behavor. J. Fnancal Economcs 10 187-194. Benter, W. 1994. 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, 183-198. Brd, R. and M. McCrae. 1987. Tests of the effcency of racetrack bettng usng bookmaker odds. Management Scence 33 1552-1562.
- 26 - Brd, R. and M. McCrae. 1994. 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, 228-258. Breman, L. 1961. Optmal gamblng systems for favourable games. Proc. 4 th Berkeley Symposum Math. Statst. Probab. 63-68. Bruce A.C. and J.E.V. Johnson. 2000. Investgatng the roots of the favourte-longshot bas: An analyss of supply and demand sde agents n parallel bettng markets. J. Behavoral Decson Makng 13 413-430. Cec, S.R. and J.K. Lker (1986) A day at the races: A study of IQ, expertse, and cogntve complexty. J. Expermental Psycholny: General 115 255-266. Chapman R.G. 1994. 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, 173-181. Crafts, N.F.R. 1985. Some evdence of nsder knowledge n horse race bettng n Brtan. Economca 52 295-304. Fglewsk, S. 1979. Subjectve nformaton and market effcency n a bettng market. Journal of Poltcal Economy 87 75-88. Hausch, D.B. and W.T. Zemba. 1990. Arbtrage strateges for cross-track bettng on major horseraces. J. Busness 63 61-78. Hausch, D.B., W.T. Zemba and M. Rubnsten. 1981. Effcency of the market for racetrack bettng. Management Scence 27 1435-1452. Kelly, J.L. 1956. A new nterpretaton of nformaton rate. Bell System Techncal J. 35 917-926. Lo, V.S.Y.1994. Applcaton of logt models n racetrack data. D.B. Hausch, V.S.Y. Lo, and W.T. Zemba, eds. Effcency of Racetrack Bettng. Academc Press, London, 173-181.
- 27 - Maddala, G.S.1983. Lmted Dependent and Qualtatve Varables n Econometrcs. CUP, New York, 1983. McGlothln, W.H. 1956. Stablty of choces among uncertan alternatves. Amercan Journal of Psycholny. 69 604-615. Sauer, R.D. 1998. The economcs of wagerng markets. J. Economc Lterature XXXVI 2021-2064. Schnytzer, A. and Y. Shlony. 1995. Insde nformaton n a bettng market. Economc Journal 105 963-971. Schnytzer, A. and Y. Shlony. 2002. On the tmng of nsde trades n a bettng market. The European Journal of Fnance 8 176-186. Schnytzer, A., Y. Shlony and R. Thorne 2003. On the margnal mpact of nformaton and arbtrage. L. Vaughan-Wllams, ed. The Economcs of Gamblng. Routledge, London, 80-94. Shn, H.S. 1992. The prces of state contngent clams wth nsder traders and the favourtelongshot bas. Economc Journal 102 426-435. Tuckwell, R.H 1983. The thoroughbred gamblng market: Effcency, equty and related ssues. Australan Economc Papers 22 106-118. Vaughan-Wllams, L. and D. Paton. 1997. Why s there a favourte-longshot bas n Brtsh racetrack bettng markets? Economc Journal 107 150-158. Wetherll, G. B.1981. Intermedate Statstcal Methods. Chapman and Hall, London. Zemba, W.T. and D.B. Hausch.1986. Bettng at the Racetrack. Dr. Z Investments Inc., L.A.