Behavioural biases never walk alone: An empirical analysis of the effect of overconfidence on probabilities.

Transcription

1 Behavioural biases ever walk aloe: A empirical aalysis of the effect of overcofidece o probabilities. Isabel Abizao, Luis Muga ad Rafael Satamaria Uiversidad Publica de Navarra (SPAIN) October 2014 Abstract: This paper presets evidece of the impact of overcofidece bias i asset prices draw from a study based o data from teis bettig exchages. A series of bettig strategies i touramets with a clear-cut favourite are show to yield sigificat ecoomic returs. The impact of overcofidece bias o bettig odds icreases with tradig volume, media coverage, ad levels of disagreemet betwee overcofidet ad Cumulative Prospect Theory bettors. Just as i traditioal fiacial markets, arbitrage limits are show to be a ecessary coditio for the impact of behavioural biases o prices. Keywords: Overcofidece, bettig exchages, aomalies, behavioural fiace JEL code: G02, G14 ACKNOWLEDGEMENTS This paper has received fiacial support from the Spaish Miistry of Ecoomy ad Competitiveess (ECO C02-01). Correspodig author: Luis Muga. Dpto. Gestió de Empresas. Uiversidad Publica de Navarra. Campus de Arrosadía s/ Pamploa (Spai). Ph Fax rafael@uavarra.es 1

2 Behavioural biases ever walk aloe: A empirical aalysis of the effect of overcofidece o probabilities. Abstract: This paper presets evidece of the impact of overcofidece bias i asset prices draw from a study based o data from teis bettig exchages. A series of bettig strategies i touramets with a clear-cut favourite are show to yield sigificat ecoomic returs. The impact of overcofidece bias o bettig odds icreases with tradig volume, media coverage, ad levels of disagreemet betwee overcofidet ad Cumulative Prospect Theory bettors. Just as i traditioal fiacial markets, arbitrage limits are show to be a ecessary coditio for the impact of behavioural biases o prices. 1.- Itroductio I recet years, bettig markets have proved a useful testig groud for the validity of behavioural fiace theories (see Smith et al., 2006 or Hetherigto, 2006). Bettig markets a much more suitable cotext i which to ivestigate potetial mispricig 1, despite the problems typical of iformatio markets where strategies are traded based o the probability of give outcomes, ad the odds reflect the market s estimate of that probability. This leaves the odds vulerable to the various cogitive biases that may appear i puters probability estimates, the commoest beig those described by the Prospect Theory (Kahema ad Tversky, 1979). Uder these coditios, the presece of traders operatig uder bouded ratioality may cause prices to diverge from their fudametals ad thus skew the odds i bettig markets. Accordig to the teets of the Cumulative Prospect Theory (CPT), market agets operate o the basis of probability trasformatios, whereby low probabilities are over-weighted ad high probabilities are uder-weighted. Give that the probability of the favourite wiig each match is a high probability outcome, it will be uderweighted. This should mea that the retur o a strategy based o systematically bettig o the favourite wiig the set of matches that make up the touramet (a composite evet) will be o-egative, or eve positive if the market is ot fully arbitraged. I this CPT cotext, this paper attempts to show empirical evidece of the price impact of overcofidece ad self-attributio biases, described i theoretical models such as that of Daiel et al. (1998), usig data draw from teis bettig markets 2. Accordig to this model a ru of good ews i cojuctio with the presece of overcofidet ivestors results i asset overpricig. Furthermore, overcofidece bias teds to arise i combiatio with self-attributio bias, whereby traders igore ay iformatio cotradictory to their priors while icorporatig 1 Bettig exchages offer the fial value of the uderlyig asset ad have lower limits to arbitrage, due to the absece of short-sales costraits. 2 Teis bettig markets were selected as our cotext of aalysis because they allow gamblers to bet o a player either to wi or to lose the touramet (a simple evet) or all of the rouds (a composite evet). 2

3 ay cofirmatory iformatio ito their decisios. Fudametals, however, evetually prevail ad prices ted towards equilibrium. Usig this argumet ad drawig a parallel with the teis gamblig market, we use the favourite to wi a give touramet as a aalogy for a stock whose price is drive by a ru of good ews. We defie the favourite as the player whose estimated probability of wiig the touramet accordig to bettig market odds is greater tha that of ay other prior to its commecemet 3. If the model predictios are correct, gamblers will over-estimate the favourite s chaces of wiig the touramet prior to its commecemet, thus reducig the odds o that outcome. However, as the real iformatio emerges, roud by roud throughout the touramet, the odds will adjust ad should icrease. The paper is orgaised ito 6 sectios. Sectio 2 cotais a review of the literature ad its relatioship with the mai characteristics of the bettig market aalysed. Sectio 3 presets the theoretical framework ad the hypotheses to be tested. Sectio 4 describes the data base. Sectio 5 shows the results of the various bettig strategies employed. Sectio 6 describes the robustess check; ad the 7th ad fial sectio cotais the mai coclusios. 2.- Behavioural Fiace ad bettig exchages Previous research o sport bettig markets has focused o the oe had o the so called Favourite-log shot bias (FLB) i bookmaker-based markets (see for example Jullie ad Salaié, 2008). This bias basically reflects a tedecy for higher rewards to accrue from bets placed o favourites, versus lower rewards for bets placed o less likely outcomes (log shots), give the odds offered by the bookmakers. Although this pheomeo might be compatible with the predictios of Cumulative Prospect Theory, as show by Sowberg ad Wolfers (2010), it has also bee attributed to causes such as iformatio acquisitio costs ad the presece of uiformed puters (Hurley ad McDoough, 1995), who ted to place too much moey o very ulikely outcomes ad too little moey o more likely oes. This type of behaviour exerts a pressure o bookmakers that evetually skews the odds. Koch ad Shig (2008) show that it is the patter of the odds set by bookmakers i the Uited Kigdom that teds to accetuate the presece of Favourite-log shot bias. Moreover, Lahvicka (2013), shows evidece of FLB i bookmaker teis bettig markets. This result is explaied by bookmakers protectig themselves agaist better iformed bettors. O the other had, some recet papers have tested differet arbitrage strategies i this type of bookmaker bettig markets. Vlastakis et al. (2009) fid simple bettig strategies that provide 3 The requiremet for bettors to cosider a player the favourite is a ru of good ews. I the evet of beig associated with a sufficietly sigificat ru of bad ews, such a player will automatically cease to be cosidered the favourite (their odds will raise i the bettig market). 3

4 sigificat positive returs usig data from Europea Football Bettig Markets or Ashiya (2013), who shows arbitrage opportuities i the Japaese Racetrack Bettig Market. Recetly, the popularity of the Iteret has give rise to aother type of bettig market, where, the odds are set ot by the bookmakers but by the puters supply ad demad, thus followig a process similar to that of a order-drive fiacial market. That is, the ivestors buy ad sell orders build the order book which orgaises the trasactios. Thus, the spread is ot determied by the ask ad bid price set by the bookmaker, but by the best ask ad bid prices placed o the order book at that momet. This type of market is kow as a bettig exchage. This form of market orgaisatio has some advatages whe compared with bookmaker-based markets i terms of price settig. Firstly, Frack, Verbeek ad Nüesch (2010) show that bettig exchages are better predictors of future evets. These same authors (Frack, Verbeek ad Nüesch, 2013) sigal the presece of arbitrage opportuities betwee these two types of markets due to mispricig i the bookmaker-based market. Smith et al. (2006) further report that pricig is less vulerable to FLB i bettig exchages tha i bookmakers markets, a fidig which they attribute to lower trasactio costs. They also fid liquidity to have a major impact, i that, the more liquid the market, the lower the degree of FLB. Bettig exchages led themselves particularly well to the testig of hypotheses about ivestor behaviour, because the practical absece of arbitrage costraits, which are eve lower tha i may fiacial markets, allows arbitrageurs to exploit ay mispricig due to oise tradig. Uder these premises, bettig exchages offer a ideal settig i which to test for behavioural biases amog agets. Takig ivestor behaviour to be adequately described by the assumptios of the Cumulative Prospect Theory (Tversky ad Kahema, 1992), the aim i the case i had is to test for a possible additioal effect of puters overcofidece bias derivig from a ru of good ews about a potetial outcome, (the favourite wiig the touramet). Cofirmatio of such a effect would help to reiforce the existig idirect evidece about the impact of ivestors overcofidece bias o stock prices i fiacial markets. The presece of ivestors tradig uder the effect of this behavioural bias has bee suggested as the source of the mometum effect (Cooper et al., 2004), ad of the presece of speculative bubbles i stock prices (Hog et al., 2006). The pheomeo of overcofidece bias is ot uique to fiacial markets. Various papers have reported its presece i iformatio markets ad bettig markets. Berg ad Rietz (2010) show that overcofidece bias has a impact o log term price formatio i the Iowa Electroic Market for differet evets ad Arkes (2011) fids evidece that over-optimistic puters bias prices i bettig markets for NBA basketball games. As already metioed, our choice of market for the aalysis proposed i this paper is the teis bettig exchage. The choice of this sport stems from the fact that the competitios are orgaized as sigle elimiatio touramets. The market has a sufficiet degree of liquidity, which is 4

5 importat, because Smith et al. (2006) have show liquid markets to be less vulerable to FLB 4, ad Borghesi et al. (2009) have show that behavioural biases have a greater impact o prices i bettig markets with low liquidity. The fact that teis also attracts wide media coverage allows us to assume that puters will be sufficietly well iformed as to avoid beig affected by the aforemetioed FLB, of which media coverage is the mai driver (Hurley ad McDoough, 1995). Fially, the puters are ot as strogly idetified with the players as they ted to be i other sports which have teams represetig a city or a coutry Theoretical framework ad hypotheses The CPT (Tversky ad Kahema, 1992) is a modified versio of the Prospect Theory (Kahema ad Tversky, 1979) which aalyses ucertai as well as risky prospects with ay umber of outcomes, applies the probability weightig fuctio to the cumulative probability distributio ad ot to the probability desity fuctio 6, ad allows differet weightig fuctios for gais ad for losses. Formally, let (x i, p i ) be a risky prospect, (x -m, p -m ; ; x -1, p -1 ; x 0, p 0 ; x 1, p 1 ; ; x, p ) with x i > x j if i>j where x i is the outcome (positive or egative)that results if evet Ai occurs ad p i =A i )is the probability of evet A i occurrig. Uder Cumulative Prospect Theory, the value of the gamble is π iv( x i ) where i = m π π if i > 0 ad + i = i π π if i < 0, with i = i π w p p ) w ( p p ) if 0 i < ; π + = w + p ) if i = (1) i = ( i i+ 1 + π w p p ) w ( p p ) if m < i 0ad π w p ) if i = m. (2) i = ( m i m i 1 ( m = ( m The fuctios + w ad w are strictly icreasig from the uit iterval ito itself, satisfyig w + ( 0) = w (0) = 0 ad w + ( 1) = w (1) = 1. I this modified versio, Tversky ad Kahema (1992) propose specific fuctioal forms for v(x) ad π : v = α ( x) x if 0 x ; v( x) x β = λ ( ) if < 0 x (3) γ δ + p p w = ad w = (4) λ γ 1/ γ δ δ 1 / δ ( p + (1 p) ) ( p + (1 p) ) where α (0,1), β (0,1), γ (0,1), δ (0,1) ad λ > 1. 4 The top seeds i the ATP (Associatio of Teis Profesioals) rakig have to play i the four Grad Slam touramets (the Australia Ope, Rolad Garros, Wimbledo, ad the US Ope) ad the 9 Masters Series Circuit touramets, which, i the sample period, were (Idia Wells, Miami, Rome, Mote Carlo, Hamburg, Caada, Ciciati, Madrid ad Paris Bercy). This choice guaratees a miimum level of market liquidity. 5 Tetlock (2004) claims that oe of the factors motivatig people to take part i this type of market, amely, their idetificatio with oe of the teams or players i the sportig evet, could skew the results ad prove difficult to cotrol. This assumptio of a lower degree of atioal idetificatio does ot apply i other teis evets, such as the Davis Cup, which are ot cosidered i this paper. 6 This esures that Cumulative Prospect Theory does ot violate first-order stochastic domiace, a weakess of the origial Prospect Theory. 5

6 These fuctios capture the key elemets of the theory. Thus, the fuctio v(x) is cocave over gais, covex over losses, ad exhibits a greater sesitivity to losses tha to gais. The degree of sesitivity to losses is determied by λ, the coefficiet of loss aversio. The weightig fuctios, + w ad w, iclude a oliear trasformatio of the probability scale, which over weights small probabilities ad uder weights moderate ad high probabilities. This form provides a good approximatio to both the aggregate ad the idividual data for probabilities i the rage betwee 0.05 ad 0.95 (see Tversky ad Kahema, 1992, pg.310). It is kow that the probability of a composite evet A = A A... A ) ca be obtaied by j ( j, 1 j,2 j, N multiplyig the probabilities of the N simple evets A j, of which it is composed, such that P = N c j = 1 P j, ; with c Pj, = Pj, ( j,1 j,2... j, 1 ) > 1 (5) where P j is the probability of a composite evet A j ad simple evet A j,. c P j, is the coditioal probability of the If the simple evet ad the composite evet ca be perfectly arbitraged 7, assumig the presece of sophisticated ivestors, it is reasoable to suppose that N c w( P j ) = = 1 w( P j, ) (6) I the case i had, the Wi Touramet (WT) outcome is cosidered a simple evet o which it is possible to bet at odds which allow us to ifer the associated probability. If, for the sake of homogeeity, we assume that this probability is estimated at a poit i time t0 immediately prior to the start of the touramet, N c w( WT Ω t )) = = 1 w( W Ωt )) with (7) 0 0 c P W ) W W W... W ); > 1 ad N=rouds of the touramet (8) ( = c W where w( Ω )) is the probability of wiig the th roud, give the kowledge that a t 0 evet W W... W ) has already occurred, as far as ca be iferred from the market odds for ( that bet, coditioal to the iformatio available at time t0, ad w( Ω )) is the probability of wiig the touramet coditioal to the set of iformatio available at time t0, W t t 0 Ω t 0. Note that the Wi Touramet (WT) outcome is cosidered a simple evet ad this outcome ca also be obtaied from the product of the probabilities of the simple evets that make up the composite evet Wi Touramet, that is, Wi all rouds of the touramet. 7 I absece of perfect arbitrage there is o reaso to fulfill this relatioship. Nevertheless, the presece of sophisticated ivestors ad perfect arbitrage betwee both evets (simple evet ad the composite evet) lead to the fulfillmet of this relatioship because, i other case, it is possible to obtai abormal positive returs without assumig ay risk. 6

7 Give the characteristics of the competitio, however, we caot obtai iformatio about Ω ) for ay >1, give that we do ot kow which players will get through from the first to W t 0 the secod roud. Thus, the probability of the composite evet wi all the rouds of the touramet is ot give by the simple evets as i expressio (7) but by the followig expressio: T Ω t N = c w( W )) = 1 w( W Ω )) (9) N t the value of which may be greater or less tha w( W T Ω )) the differece beig radom. The fact that w( W T Ω )) is obtaied o the basis of a more complete set of iformatio tha w( W T Ω )) t N either icreases or decreases its value, sice this will deped o the player s progress, which will icrease or decrease his probability of wiig throughout the cosecutive rouds of the touramet Testable Hypotheses t 0 t 0 Bettor overcofidece bias ca affect the favourite s estimated probability of wiig the touramet or wiig all of the rouds. Prior to the touramet s commecemet, the favourite will be the primary driver of overcofidece. A player who has, at some poit durig the ru-up to the touramet, held the positio of favourite, will very likely have lost that positio by the time the touramet commeces, if he or she is hit by a ru of bad ews. Thus, it takes a ru of good ews for the market to cosider a player the touramet favourite. Daiel et al. (1998) developed a model i which ivestors overcofidece bias is strogly apparet i a market whe there is a ru of good ews about a particular firm. This bias may also arise i combiatio with self-attributio bias, whereby ivestors igore ay iformatio that cotradicts their priors. The combied effect of these two biases leads to a overreactio of stock prices, such that, if the market has a high proportio of this type of ivestor ad there is a ru of good ews, a stock price will appear overvalued. The model also predicts that prices will revert to their fudametals i the log term, sice iformatio evetually prevails. I the same vei, Vergi (2001) shows that ivestors have a stroger tedecy to overreact to good ews tha to bad ews ad therefore the effect will be more otable i the wake of a ru of good ews, as the above argumet suggests. Trasportig this argumet to the cotext of teis touramets, we must assume that, prior to the commecemet of the touramet, give a build-up of good ews about the favourite to wi ad i the presece of the biases described i Daiel et al. (1998), the bettig market s estimate of the favourites probability of wiig will be higher (ad thus the odds lower) tha uder perfect icorporatio of iformatio without mispricig. It should be oted that, accordig to the teets of the CPT, very high objective probability will be associated with lower trasformed probability. Our mai poit, however, is that overcofidece bias will result i a reductio i the 7

8 uderestimatio of objective probability, that is, i relative terms, it will be overestimated with respect to the trasformed probability estimate i a CPT model with o overcofidece bias. Give that, i this case, overcofidece bias rus couter to the probability trasformatio effect described by the CPT, it is ot easy to predict the fial effect o the bettig market odds, sice they will deped o the degree of probability trasformatio which, i tur, depeds o the objective probability estimate ad the level of overcofidece bias amog the puters. The exteded period prior to the touramet for gamblers to bet o the wier of the touramet, especially i the case of Grad Slam 8 evets, allows the build up of good ews about the favourites, which skews the puters estimates of the favourites chaces of wiig. Further, as already metioed, Berg ad Rietz (2010) fid that the presece of overcofidet traders has a greater effect o prices over the log term. Later, as the touramet proceeds, iformatio pertaiig to the favourite s real likelihood of wiig will gradually reveal itself, thus relievig the dowward pressure o the odds o the various matches ad reducig the degree of overcofidece bias i the iformatio icorporated ito the odds. This suggests the followig hypothesis: H1: The probability of the favourite wiig the touramet (a simple evet) is higher tha the product of his probabilities of wiig all rouds i the touramet icludig the fial (a composite evet), or, i mathematical terms: N c w( WT Ω t )) > = 1 w( W Ωt )) (10) 0 Give that the potetial spread betwee these two probabilities could be exploited through a pseudo-arbitrage strategy (pure arbitrage beig ufeasible), the presece of overcofidet puters will icrease arbitrage costraits due to oise trader risk, which will allow the spread to persist for a period before revertig to equilibrium, as predicted by the model preseted i De Log et al. (1990). It seems reasoable to assume that a miimum of media attetio is required for ews to reach the market. Ideed, Tetlock (2004) ad Avery ad Chevalier (1999) report price overreactio to ews due to media coverage. This leads us to the followig hypothesis: H2: The uderestimatio of the probability of the favourite wiig the touramet (a simple evet) w( W T Ω )) will be lower i markets with higher media coverage. t 0 8 Bettig o the wier of a touramet opes i advace of the start of the touramet, the period varyig from 100 days i the case of Grad Slam touramets, to 6 days i that of Masters Series touramets. Bettig o the outcome of the first roud commeces 3 days i advace, after which there are two days to place a bet o a Grad Slam match ad 1 to place a bet o a Masters Series match. 8

9 N This implies that the spread w( WT Ω t )) = 1 w( W Ωt )) will wide as media coverage icreases, 0 because high media attetio coveys the good ews about the favourite to a greater umber of bettors. Thirdly, the literature relatig to the presece i fiacial markets of traders with behavioural biases claims that such biases will be stroger i idividual traders tha i istitutioal traders 9. Give that traditioal bookmakers cover their positios by operatig through bettig exchages, their presece as sophisticated traders ca have a sigificat impact o prices, which eables us to propose the followig hypothesis: H3: The presece of iformed traders i the market will mitigate the reductio i the uderestimatio of the probability of the favourite wiig the touramet [the simple evet w( W T Ω ))]. t 0 c Fially, the fiace literature has foud a positive relatioship betwee mispricig caused by traders behavioural biases ad total volume traded. For predictig this relatioship, Hog ad Stei (2007) preseted a family of models that might be termed Disagreemet models. Scheikma ad Xiog (2003) develop a model which relates overcofidece bias i stock market ivestors to the emergece of speculative bubbles associated with high tradig volume; ad Hog, Scheikma ad Xiog (2006) demostrate this effect empirically. The disagreemet i the case that cocers us is betwee gamblers with high vs. low levels of overcofidece ad selfattributio bias: i this cotext, the greater the level of volume traded i the market the greater the mispricig of the odds. This leads us to fourth ad last hypothesis: H4: The reductio i the uderestimatio of the probability of the favourite wiig the touramet [the simple evet w( W T Ω ))] will be greater i markets with higher tradig volume. t The database We aalyse data from the Betfair 10 bettig exchage for the period ruig from May 2004 to December The reaso for this choice of sample period is that it leds itself particularly well to the testig of the above hypotheses o ivestor overcofidece because it featured the domiace o the ATP 11 circuit of two outstadig favourites: Roger Federer (o grass ad hard courts) ad Rafael Nadal (o clay). Sice the, with the emergece of Novak Djokowic ad Ady Murray, amog others, the umber of potetial favourites to wi touramets has icreased, 9 Frazzii (2006), for example shows that the dispositio effect is stroger i idividual ivestors tha i mutual fud maagers. 10 Betfair is the largest olie bettig compay i the UK ad the largest bettig exchage i the world with over four millio subscribers. 11 Some examples of rus of good ews about favorites durig this period ( ) iclude: Roger Federer havig wo the highest umber of cosecutive matches i teis history o hard courts ad grass (56 o hard court, 65 o grass, ad 26 agaist oe of the top 10 seeds) ad Rafael Nadal havig achieved the highest umber of cosecutive matches wo (81) o clay. 9

10 thus reducig each player s idividual chace of wiig. Durig the sample period, the wome s circuit (WTA TOUR) does ot fulfil the ecessary coditios for testig the impact of the presece of overcofidet bettors o the odds, give the lack of ay clear favourite 12. The aalysis uses data o ATP touramets i which the top-seeded players must participate ad caot withdraw except i the evet of ijury 13. This guaratees a miimum level of market liquidity (bid-ask spread ad depth). For each evet, the Betfair database icludes both pre-evet (PE) ad i-play (IP) bettig data. We have t however used the odds o the outcome of the touramet oce the evet has started or the odds o idividual matches after their commecemet (IP), because these would capture ot oly gamblers prior beliefs but also iformatio regardig the results of matches ad the progress of the touramet. Furthermore, Docherty ad Easto (2012) usig golf touramet bettig data draw from the Betfair compay s database, show that oce the competitio has started, the odds uder react to iformatio emergig from the proceedigs. Williams (2010) reach similar coclusios usig NBA bettig data. Table 1 gives a summary of each of the touramets cosidered i this paper, icludig the umber of etries per touramet i the database. After cleaig ad filterig, we are left with a dataset that icludes 281,952 bets o the favourite to wi Grad Slam evets, 110,852 bets i the alterative market for these evets, ad 155,521 bets o Masters Series evets placed prior to their commecemet (PE) 14. From this database, we extracted the bettig odds, the umber of bets placed at those odds, the bettig volume i pouds sterlig, the first ad last time each odds were take, evet idetifiers ad the directio of the bet. These data eable the costructio of odds variables such as the volume-weighted average, the closig odds before the start of each match ad the touramet, the odds at which most of the moey was bet, the stadard deviatio, the skewess coefficiet, the average bet size, the total bets take per evet, ad the total duratio of the bettig period for a give outcome (i umber of days). Give the aims of the paper, the said data refer oly to the favourite, that is, the player with the lowest odds before the touramet starts. As idicated i the precedig sectios, this player must, by variable costructio, be the oe that geerates the accumulatio of bettors actig 12 Focusig o the mai evets of the WTA circuit betwee the 2004 ad 2008 seasos, we ca see that up to 9 differet players succeed i wiig at least oe Grad Slam title. 13 As oted, these are Grad Slam ad Masters Series touramets. The aalysis does ot iclude the Hamburg (2006), Madrid (2004) or Paris (2004) touramets, where the average weighted odds o the favourite to wi were higher tha 8. Paris (2006) was also excluded because the favourite, Roger Federer, withdrew before the touramet started. 14 I oly 9 of the 18 wome s Grad Slam touramets icluded i our database it is possible to implemet the strategy proposed i the paper to detect the presece of overcofidet bettors. Proof of this comes from the fact that i 3 cases out of 9, i which the strategy ca be implemeted, the favourite chaged over the course of the touramet. 10

11 uder overcofidece bias. First, we extracted data pertaiig to the market for bets o the favourite to wi i each touramet, the the data pertaiig to the market for bets o the favourite to wi each of the rouds of each touramet. For every match, the Betfair Compay will, i additio, take bets o either the favourite or his rival wiig (or losig). This alterative market proves particularly useful for testig whether markets with the same iformatio but differet types of gamblers geerate the same odds. 5.- Results Returs of the pseudo-arbitrage strategies The probability of the favourite wiig each match is a high probability outcome. I lie with CPT teets, this probability will be uderweighted 15. As i Adrikogiaopoulou ad Papakostatiou (2013) 16, i this aalysis, we assume that idividuals evaluate bets usig the probabilities implied by their odds (Qj) 17. The iformatio to be draw from the weighted average odds shows that, overall, strategies based o bettig o the favourite wiig every match leadig up to ad icludig the fial yield positive, albeit ot sigificat, returs (26.41%, p=0.18). Similar fidigs are obtaied for bettig o the favourite wiig the touramet (15.1% p=0.35). These fidigs provide evidece of the effect of probability trasformatio i the case of high probabilities. Nevertheless, the presece of sophisticated bettors should prevet simple probability trasformatio from makig the probability of the simple evet sigificatly differet from that of the composite evet. The presece of bettors operatig uder overcofidece ad self-attributio biases, however, might explai why the probability of the simple evet is less uderestimated tha that of the composite evet. The presece of behaviourally-biased ivestors i a market should icrease oise trader risk ad tur iformed traders away, thus icreasig arbitrage costraits ad leavig the way ope for temporary mispricig, as postulated by De Log et al. (1990). We explore the possibility of mispricig by selectig a ex-ate ivestmet strategy. It is based o bettig that the favourite will lose the touramet 18, assumig the effect that the presece of overcofidet bettors will have o the estimated probability of this occurrig, ad stakig the potetial wiigs of this bet o the composite evet, that the favourite will wi each of the rouds. This strategy ca lead to three possible outcomes. If the player loses the touramet, the bettor will wi the bet placed o that evet ad lose the bet placed o the match i which the 15 I the 2006 Australia Ope, for istace, Roger Federer s average estimated probability of wiig a match was (implied probability) ad his probability of wiig the touramet was These authors motivate the assumptio that subjective beliefs ca be approximated with the probabilities implied by the prices due to the bettig markets are quite efficiet ad several olie tools help idividuals covert the quoted prices ito the implied probabilities. They also show that there are small but sigificat deviatios of the subjective from implied probabilities, but their results are ot affected. 17 We aalyse data from bettig exchages where there is o bookmaker, so implied probabilities do ot have to take ito accout the broker commissio. IPj=1/ Q j, where IPj is the implied probability o outcome j ad Qj is the resultig market odds o outcome j. 18 As metioed earlier, oe of the advatages of bettig exchages is that they offer both odds i favour ad odds agaist a give outcome. 11

12 player is kocked out of the competitio, leavig the bettor s wiigs at 0. If the player wis the Ω N c T t = t, 0 touramet ad the assumptio of overcofidece bias is fulfilled w( W )) > 1 w( W Ω )) the bettor will receive a positive retur. Coversely, if the above assumptio is ot fulfilled, the returs will be egative or ull. I mathematical terms, this strategy takes the followig form: Rk, N N = 1Q( W Ωt ) Q( WT Ωt ) 0 = (11) Q( WT Ωt ) 0 where R k, N is the retur to strategy k, assumig that it ivolves N simple evets ad that that the strategy ca be fully implemeted. Q( Ω ) is the observed market odds o wiig the W T t 0 touramet coditioal to the set of iformatio available at time t0, Ω ad Q( Ω ) the t 0 W t observed market odds o wiig the th roud coditioal to the set of iformatio available at time t, Ω t. Table 2 gives the results for all those touramets - Grad Slam (GS) ad Masters Series (MS) - i which it was possible to implemet the above strategy after trasactio costs 19. The strategy was implemeted at three differet odds. The first was the volume-weighted average odds for each evet. The secod was the odds with the highest bettig volume ad the third was the closig odds prior to each evet start, assumig these last to capture all the pre-evet iformatio 20. Note, however, that, i a study of Italia football bettig markets, Ioceti et al. (2012) reported gamblers displayig various behavioural biases arrivig o the bettig scee shortly before match commecemet. Usig volume-weighted average odds, we foud a o-sigificative average retur across these strategies of 3.58% for each evet. The retur was 6.80% for the strategy usig the odds with the highest bettig volume ad 10.27% for the oe usig the closig odds which are sigificatly differet from 0 usig the t-statistic. We also ra a o-parametric test (the biomial test). Usig the average weighted odds, the strategy proved fully implemetable i 36 cases, yieldig positive returs i 19 of them ad egative returs i 17 (p=0.3089). The returs usig the odds with the highest bettig volume were positive i 24 cases ad egative i 12 (p=0.0144), ad with the closig odds there were 21 positive ad 10 egative (p=0.0147) 21. The strategy has prove ecoomically sigificat i two cases out of the three cosidered. This fidig is cosistet with the presece of overcofidet bettors, whose ifluece o odds is greater tha that of trasactio costs. I the absece of overcofidet bettors, the differece 19 The costs set by Betfair durig the sample period rage betwee 2% ad 5% of the wiigs per bet placed, depedig o the total bets of the idividual bettor. Give our tradig-itesive strategies, we assume the lowest level of trasactio costs. Betfair does ot apply trasactio costs to losig bets. 20 The raw returs of the strategies are all sigificatly differet from 0 usig the t-statistics. The results are available from the authors upo request. 21 Oly 31 strategies were possible with the closig odds, because of missig values i the data provided by Betfair for

13 betwee the estimated probability of the simple evet (the favourite wiig the touramet) ad that of the composite evet (the favourite wiig all the rouds of the touramet) should differ oly radomly. I short, the results provide evidece to support hypothesis H1. That is, the odds o the favourite wiig the touramet are low i compariso to the odds o his wiig each roud, by which time real iformatio is emergig. The results ca be explaied by the fact that the evets do ot take place simultaeously. Cotrary to the ratioal hypothesis that the absece of behavioural biases that would skew probability estimates is purely due to chace, the overcofidece bias i the case i had is uequally distributed over the two evets. The simple evet shows the effect of a accumulatio of overcofidece i the favourite s chaces of wiig the touramet, which is drive by a fairly log ru of good ews. This probability estimate, which is overestimated with respect to that foud i the absece of such a bias, caot be fully arbitraged because it does ot offer pure arbitrage opportuities. This, together with the impact of oise traders, icreases arbitrage costraits, thus allowig temporary divergece of prices from their equilibrium values. The start of the touramet seds ew iformatio to the market, which gradually adjusts the impact of the aforemetioed bias o the odds ad reduces its effect o the estimated probabilities of the simple evets that make up the composite evet (wi all the rouds of the touramet) Media coverage: Grad Slam Vs Masters Series If overcofidece is the mai explaatio for the fidigs i relatio to H1, this bias should icrease with media coverage of the evet, i lie with the fidigs reported i studies such as those of Tetlock (2004) ad Avery ad Chevalier (1999). We test this hypothesis (H2) by splittig the sample ito two subsamples: bettig markets for GS touramets, which receive greater media coverage, ad MS touramets. Table 2 also gives the results after deductig trasactio costs yielded by implemetig the strategies separately for each subsample. It ca be see that, i the case of GS touramets, both the t statistic ad biomial test values idicate sigificat returs to the described strategies for the three types of odds cosidered. Specifically, strategies based o volume-weighted average odds i GS evets gave estimated average returs of 5.32%, those based o the odds with the highest bettig volume 9.62%, ad those based o the closig odds 17.64%. The same strategy yields lower returs whe used i MS touramets, however. Specifically, average returs were 1.84% (6 evets with positive returs ad 12 with egative returs) for the strategy based o the volume-weighted average odds, 3.98% (8 positive ad 10 egative) for that based o the odds with the highest bettig volume; ad 3.98% (8 positive ad 8 egative) for that based o the closig odds. Neither of the sigificace tests performed showed these returs to be sigificat. 13

14 These fidigs provide evidece to show that the greater media coverage give to GS touramets icreases the imbalace betwee the odds, ad thereby the returs to the strategies, as stated i Hypothesis 2. Testig for differeces i strategy returs betwee touramet types, the t statistics reveal o differeces i terms of average returs, irrespective of the type of odds cosidered ad whether or ot trasactio costs are icluded. Nevertheless, the results of the o-parametric Ma-Whitey U test for the odds with the highest bettig volume ad the closig odds, both before ad after deductig trasactio costs, eable us to reject the hypothesis of equal returs at the 10% level of cofidece. The low umber of strategies ad the lower depedece of the fulfilmet of the hypotheses o the distributio recommed the use of the oparametric test. The results, therefore, provide support for H2, ad additioal evidece to demostrate that the probabilities of the simple evet ( wi the touramet ) ad the composite evet ( wi all the rouds of the touramet ) are ot radom outcomes, are ot uiform across all touramets, ad caot be attributed to the impact of probability overestimatio associated with cojuctive evets. Ideed, the differece appears to be related to variables associated with the level of overcofidece bias amog bettors. To further explore the potetial factors to explai the differece betwee the GS ad MS touramets, Table 3 shows the pricipal characteristics of the bettig market for the wier of the touramet ad the p-values of the t statistic for the differece of meas betwee the two types of touramet. From this table, it ca be see that there is a sigificatly larger umber of differet bettig odds i GS touramets, although the stadard deviatio ad skewess coefficiet for the two types of touramet show o sigificat differeces. The mai differeces are to be foud i the market-volume variables, which show that GS touramets attract a sigificatly higher bettig volume, both i moetary terms ad i umbers of bets, as was expected give the aforemetioed higher media coverage of GS evets. Fially, the average bet size is also show to be sigificatly larger i GS touramets. Sice this variable serves us as a proxy for ivestor type, its higher value suggests a greater presece of istitutioal bettors. Despite the ituitio that greater media coverage should icrease the umber of small bettors i the market, thus reducig the average bet, the results show that this sort of reasoig overlooks a importat additioal factor relatig to bets made by bookmakers i order to hedge the odds offered to their customers. Higher bettig volume resultig from greater media coverage drives bookmakers to make more use of the bettig exchage to exploit its higher liquidity, thus icreasig average bet size. This effect is much greater tha that of the reductio i average bet size resultig from the iflux of idividual bettors, which explais the observed fial result The results of this characterizatio for each roud i the touramets are available from the authors upo request. They show that bettig volume, both i moetary terms ad i umber of bets, the average bet ad the duratio of the bettig period are always greater i Grad Slam touramets, i most cases to a sigificat degree. 14

15 Fially, as oted earlier, the duratio of the bettig period, that is, the umber of days that the market operates prior to the start of the competitio, is cosiderably loger i GS bettig markets. This could facilitate odds mispricig resultig from a ru of good ews about the favourite i combiatio with bettors overcofidece Arbitrage limits: The alterative market The results of the previous sectio have show that market characteristics such as ivestor type ad tradig volume could be havig a impact o fial odds prices. Obviously, the real opportuities for arbitrage betwee bettig markets will also have a importat impact o them. Oe of the issues raised previously is that the strategy for bettig o the simple evet wi touramet is ot strictly equivalet to the product of the simple evets of which it is composed, sice these are ot all available at the startig poit, t0. The strategy does ot, therefore, costitute pure arbitrage, i the strict sese of self-fiaced, risk-free strategies, implemeted simultaeously at the same poit i time, t0. It is, therefore, worth ivestigatig whether the above results ca be attributed to arbitrage limits i the market. Thus, the markets cosidered eable us to obtai a fairly accurate assessmet of the issue raised, sice oe of the features of these markets is that it is possible, i every match, to bet i oe of two alterative markets. That is, the bet ca be placed i the bettig market for evet i agaist j (player i agaist player j), where it is possible to bet o player i wiig (or losig) or, i the bettig market for evet j agaist i (player j agaist player i) where it is possible to bet o player j losig (or wiig). Thus, a bet i the mai market 24 o the favourite wiig a specific match should be equivalet to a bet i the alterative market o his oppoet losig. For the same iformatio to be icorporated ito the odds i both markets, the followig relatioship must be fulfilled 1 QA ( W Ωt ) = + 1 (12) ( Q( W Ω ) 1) t where Q( Ω ) is the odds i ay mai market associated with evet (i the case i had, this W t would be the th match i the touramet, deoted by i agaist j), coditioal to the set of iformatio available at t ad QA( Ω ), the theoretical odds i the alterative bettig market W t for that evet (that is, the odds o the th roud i the touramet, deoted by j agaist i). 23 There are also sigificat characteristic differeces betwee the me s ad the wome s Grad Slam bettig markets. I particular, the average odds i wome s touramets are 2.88 versus 1.92 i me s touramets, ad the average umber of differet odds is versus 34.89, respectively. These data help to illustrate the higher level of ucertaity ad the absece of a clear favourite i the wome s circuit durig the period of iterest The results are available from the authors upo request. 24 The mai market is cosidered to be the oe where player i is the favourite. 15

16 The characteristics of the mai ad alterative markets differ sigificatly for the GS touramets 25. I particular, the umber of differet odds for each roud is greater i the alterative market, except i the case of the fial roud. Bettig volume is sigificatly lower i all rouds of the touramet i the alterative market, where the umber of bets placed is also lower, ad sigificatly so i five of the seve rouds. This effect may be due to bookmakers ad iformed bettors seekig to exploit the more liquid market, with a view to prevetig their trasactios from havig a egative impact o the odds. Fially, the average bet size i all rouds is also sigificatly lower i this market. The fidigs for the above-metioed variables appear to reveal the presece of less iformed bettors i the alterative market, which could lead to some forms of mispricig resultig from their potetial behavioural biases. There are o sigificat differeces betwee the volumeweighted average odds evolvig i the two types of market cosidered. Furthermore, strategies implemeted usig odds draw from the alterative market yield sigificat positive returs for all three types of odds cosidered (see Table 4). Similar fidigs are obtaied for the mai market. I fact, there is o sigificat differece i returs betwee the two markets, either with the t statistic or the Ma-Whitey U-value 26. These fidigs idicate that the presece of behaviourally biased agets is ot sufficiet to ifluece prices i a give market, ad that limits to arbitrage are also ecessary, (Shleifer, 2000). I this case, iformatio o the odds i both markets is simultaeously available to ay aget, providig opportuities for arbitrage, which will equalize prices betwee both markets. The existece of this arbitrage is a guaratee that the market uder cosideratio is efficiet eough to show that the remaider of the fidigs of this paper are ot due to price settig problems. It should be oted, however, that the virtually total lack of arbitrage costraits has ot cacelled out the effect of overcofidece bias or that of the overall uderestimatio of probabilities i the odds i both markets. It has merely equalized these effects by prevetig ecoomic gais from ay iformatioal imbalace betwee these complemetary markets (mai ad alterative). These results suggest that the returs of the iitial strategies are due, ot to pure arbitrage, but to pseudo-arbitrage operatios, which icorporate the oise trader risk effect, whereby prices may diverge temporarily from their fudametals Multivariate Aalysis The results of the precedig sectios have revealed the existece of several explaatory variables for the differece i odds betwee the simple evet wi the touramet ad the composite evet wi all rouds of the touramet. Specifically, they reveal a retur spread betwee strategies 25 Grad Slam touramets were selected because of the statistical sigificace of the strategies ad the fact that they all have the same umber of rouds (seve), thus facilitatig the compariso of the results. The results are available from the authors upo request. 16

17 implemeted i GS vs MS touramets. Market characteristic differeces may, however, be maskig the mai reaso for the retur spread, thus cocealig the explaatory variables for the strategy returs. I additio to whether the touramet is GS or MS, average bet size (as a proxy for the domiat ivestor type i the market) ad bettig volume are both highly importat factors. Likewise, the presece or otherwise of a clear-cut favourite i the touramet could also affect the overestimatio a player s probability of wiig the fial. For a joit aalysis of the explaatory power of these variables, we propose the followig regressio: R k = α + β β β β ) + ε 1. GS + 2Fav+ 3 l( AvgBet) + 4 l( Vol k (13) where Rk is the retur of the strategy k, GS is a dummy variable that takes a value of 1 for a GS touramet, ad 0 otherwise, Fav is aother dummy variable that takes a value of 1 whe the volume-weighted average odds are less tha 2 (i other words, the bettig market gives the favourite a more tha 50% chace of wiig the touramet) ad 0 otherwise, l(avgbet) is the atural logarithm of the average bet size, ad l(vol) is the atural logarithm of the total bettig volume. The results of this aalysis will eable us to determie whether, as the previous results suggest, there is a positive relatioship betwee the volume traded i a market ad tradig strategy returs, which would be cosistet with H4, ad with the family of behavioural fiace models kow as disagreemet models described by Hog ad Stei (2007). They will also eable us to joitly test H2, which postulates a relatioship betwee media coverage ad strategy returs ad H3, which predicts a egative lik betwee the presece of sophisticated traders ad strategy returs. We ra the regressio for the returs of the strategies usig the volume-weighted average odds, the odds with the highest bettig volume, ad the closig odds, raw ad et of trasactio costs. OLS estimatio is used with White s (1980) stadard errors. The results are displayed i Table 5. The coefficiet o the GS variable proves o-sigificat i all cases, while the coefficiet o the FAV variable is positive at levels of sigificace close to 10%, whe the depedet variable is calculated o the basis of the odds with the highest bettig volume ad the volume-weighted average odds, which suggests that the stroger the favourite, the higher the overestimatio of the probability of the simple evet, wi the touramet, due to stroger impact from the overcofidece bias. The coefficiet o the variable l(avgbet) shows a cosistetly egative sig with 10% sigificace whe the depedet variable is costructed based o the closig odds. Give that higher average bet size is a proxy for a higher proportio of sophisticated tradig, these results support H3. 26 The strategy returs et of trasactio costs, which are very similar to those show i Table 5, are available from the authors upo request. 17

18 Fially, the oly variable that proves sigificat i all the estimatios is l(vol). A sigificat positive relatioship ca be observed betwee bettig volume ad strategy returs, as also foud i traditioal fiacial markets ad as modelled i studies such as that of Scheikma ad Xiog (2003), which relate overcofidece bias with the emergece of speculative bubbles. This result provides evidece to support H4. It also suggests that differeces i the results foud betwee GS ad MS touramets are basically due to tradig volume, rather tha the simple effect derivig from the type of touramet. This obviously does ot rule out the potetial effect of media coverage, sice the two variables are closely related; that is, tradig volume icreases with media coverage. I summary, the aalyses performed so far reveal relative overestimatio of the probability of the simple evet, the favourite wis the touramet, with respect to the probability of the composite evet, the favourite wis all rouds, the causal factor beig overcofidece bias i the bettig market, as also predicted by theoretical models of fiacial markets. Overcofidece bias is also closely related to higher tradig volume, sice it creates a divergece of opiio from traders who systematically uderestimate probabilities. Fially, it shows a potetial, albeit weaker, relatioship with the level of media coverage. Somewhat weaker evidece has bee foud for a egative relatioship betwee the presece of sophisticated tradig ad overestimatio of the favourite s probability of wiig the touramet, suggestig that the former teds to reduce the latter 27. Give that, as already oted, the strategy is ot susceptible to pure arbitrage, the activity of oise traders icreases arbitrage limits ad, thereby, the likelihood of positive returs to the described strategy. 6.- Robustess test: Boostrap aalysis We have computed a test that uses the bootstrap method to simulate strategy returs, give the low umber of touramets icluded i the sample, to robust our results. This procedure ivolves extractig, with replacemet, equally sized samples from the iitial sample of touramets i which the strategy is possible. Average strategy returs for all touramets icluded i the bootstrap sample are the calculated. We the repeat the process 1000 times ad calculate the average of the distributio of bootstrap meas, ad the level of sigificace of the ull hypothesis of average returs of the strategy for the whole set of touramets is equal to 0. This procedure is performed for the GS ad MS touramets separately, ad the results are give i Table 6. They show that the average returs to the strategies for the GS touramets are positive ad sigificat, both raw ad et of trasactio costs, whereas for the MS 27 It is importat to ote a additioal explaatio, amely that sophisticated traders may also be affected, albeit less strogly, by overcofidece bias, which would also reduce their arbitrage activity. 18