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Semi-strong inefficiency in the fixed odds betting market: Underestimating the positive impact of head coach replacement in the main European soccer leagues
The Quarterly Review of Economics and Finance 71 (2019) 239–246
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Semi-strong inefficiency in the fixed odds betting market: Underestimating the positive impact of head coach replacement in the main European soccer leagues Giovanni Bernardo a , Massimo Ruberti b , Roberto Verona b a b
Department of Law, University of Palermo, Italy Department of Economics e Management, University of Pisa, Italy
a r t i c l e
i n f o
Article history: Received 17 December 2017 Received in revised form 10 August 2018 Accepted 17 August 2018 Available online 25 August 2018 JEL classification: C12 D43 L83 Keywords: Sports betting market Fixed-odds bets Semi-strong efficiency hypothesis Monte Carlo experiment
a b s t r a c t In this paper we analyse the efficiency of the sports betting market, seeking to ascertain whether the market is efficient in the case of fixed odds provided by bookmakers in the four major European soccer leagues under the semi-strong efficiency hypothesis. By examining the trends of odds in the event of a major change in expectations about team results, i.e. when the head coach of a team is replaced, we attempt to verify the argument that a profitable strategy for the bettor is likely to be possible. In this case, the market under consideration would be inefficient. Analysing the average effect of head coach replacement, we find a positive impact on team performance. Based on this information, we build a betting strategy to find out whether the bookmakers’ odds absorb this change in expectations about the winning probability of involved teams. Comparing our strategy result with a distribution generated in a Monte Carlo experiment, we conclude that the betting market is inefficient in its semi-strong form. © 2018 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.
1. Introduction In the field of sport economics there is a recent area of research which explores the efficiency of regulated gambling markets by applying microeconomic theory and econometric analysis. In this context, our goal was to evaluate the sports betting market with a focus on bookmakers’ odds in the main football leagues in Europe, namely the British Premier League, the Spanish Liga, the Italian Serie A and the German Bundesliga. Drawing a parallel with the concept of information efficiency in financial markets (Fama Eugene, 1970), Vaughan Williams (2005) splits the efficiency of gambling markets into weak, semi-strong and strong forms. In this paper, we only cover semi-strong information efficiency, which occurs when current share prices encompass all publicly-available information. In other words, if a market meets this requirement, no extra return may be achieved on the basis of public information. Semi-strong efficiency means that the return on a bet based on public information must be the same, in terms of cost/risk, as that
on a bet that has not been based on public information (Vaughan Williams, 2005). Starting from the analysis of the trends of odds in the event of some major change in expectations about a team’s results, i.e. when a team’s coach is replaced, we attempt to verify the argument that a profitable betting strategy for the bettor is likely to be possible. First, we seek to establish whether a midseason change in technical staff may improve sporting results. Subsequently, we use such information to gain extra profits by setting a strategy that is consistent with the previous findings. Given the particularities of sports betting odds and the limited time horizon of our analysis, we cannot apply standard methodologies used for testing market efficiency. For this reason, we propose an empirical analysis based on Monte Carlo experiments that help us to accept, or eventually to reject, the semi-strong efficiency hypothesis of the sports betting market. The remainder of the paper is organised as follows. Section 2 seeks to ascertain the impact of the replacement of a head coach on a team’s performance by analysing data from six seasons of Europe’s top soccer leagues. Section 3 discusses methodological aspects of market inefficiency and the main results of our analysis. Finally, section 4 draws some conclusions.
E-mail addresses: [email protected] (G. Bernardo), [email protected] (M. Ruberti), [email protected] (R. Verona). https://doi.org/10.1016/j.qref.2018.08.007 1062-9769/© 2018 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.
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2. Change in expectations: the effect of head coach replacement This project revolves around the idea of a similarity between the betting world and what happens in listed share prices. Consider, for instance, a listed company that announces an unplanned distribution of dividends, a leading company’s interest in a takeover, or a merger with a competitor: the market will respond to such new information by changing its expectations. As expectations change, share prices will increase or decrease depending on whether market players rate the operation as positive or negative. Likewise, sports betting odds may be adjusted based on public information, such as an injury suffered by the best player, a referee’s decision to have the match played behind closed doors, or the replacement of a coach. Rather than being likened to an extra distribution of profits, the type of information we are going to use can be compared to an extra financial operation, since it consists in a structural change in the company’s organisation that may influence future performance. In football clubs, there are two major situations where something similar happens: a change in the players’ squad or in the technical staff. The first case is much more complicated, since one would have to check every match, one by one, to see whether new players were involved in the match, the minute count and physical fitness. For the sake of consistency and objectivity, we decided to take the second instance, i.e. the replacement of technical staff and, in particular, the change in head coach. Part of the sports business literature has sought to ascertain whether replacement of a head coach improves a team’s performance. Fabianic (1994) and McTeer, White, and Persad (1995) found that a change of coach has a favourable impact on a team’s performance, while Brown (1982). Recently, DePaola and Scoppa (2012) looked into the Serie A championship between 1997 and 2009: although the results seemed to corroborate the favourable impact of a change of coach on the team’s performance, they pointed out that comparing the results before and after a change of coach is not methodologically sound. Audas, Goddard, and Rowe (2006) reviewed changes of coaches in the last 25 years in England, finding a negative performance in the following three months and an increased variance in the results. The authors concluded that changes are mainly made to take advantage of such increased variance in performance. In this respect, the team owners would therefore try to gamble by giving such a shock. 2.1. Measurement of the average impact of a change of head coach For the sake of consistency, we will take a stricter definition of ‘change of head coach’. It must meet the following requirements: • The change takes place during a soccer season; • The dismissed coach has trained the team for at least five matches; • The new coach is not just a temporary replacement until a new team manager steps in; • This is the first change of the season. The first requirement is crucial because otherwise our research would be impaired by other factors that could change the team’s structure before the start of the season (transfers, promotion, relegation, etc.). The second and third requirements help understand the structural change properly as well as compare results before and after the change in technical staff. The fourth point makes our sample somewhat more consistent. We do not think that the first change should be compared with subsequent ones, since they tend to occur in very critical circumstances and often result in the return of the first coaches dismissed. Finally, the last requirement lays
Table 1 Number of coach replacement by seasons and leagues. (Source: www.transfermarkt. com).
2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 Tot
Serie A
Liga
P. League
Bundesliga
Tot X Season
6 11 10 8 7 10 52
9 8 7 8 7 3 42
5 5 4 4 5 7 30
4 3 9 7 5 4 32
24 27 30 27 24 24 156
Fig. 1. Teams performance variation after coach replacement – Full sample. (Source: www.football-data.co.uk).
down that the new team manager must have had the chance to train the team for least four matches1 , because otherwise the time horizon would not be wide enough to estimate the impact of the change. Our sample consists of six seasons of European’s leading soccer leagues, namely Italy’s Serie A, the Spanish Liga, the German Bundesliga and the British Premier League, from 2008/2009 to 2013/2014. In these 24 championships, 156 midseason changes of coach fulfilled our requirements. Table 1 lists the distribution of the sample by football season and by league. The Serie A championship is the one in which managers changed most frequently (52 times in six years), followed by La Liga (42). In the Premier League and the Bundesliga, the turnover of coaches was much less frequent: 30 and 32, respectively. The aim at this point is to answer the following question: “On average, does a change of coach have a positive or negative impact on team performance?”. By comparing the aggregate points under the dismissed coaches with those under the new coaches, and then weighting them by the number of matches coached, we found how often a change increased the average value of the team’s score. As shown in Fig. 1, in most cases (73%) a change of coach had a positive impact upon the team’s points. The league with the highest number of wins is La Liga, with 38 changes of coach out of 42 resulting in an improvement in team performance (see Fig. 2). The Premier League is the championship in which a change ended up with the lowest number of wins, albeit still positive in 57% of cases. In all, the teams played 2651 matches under the coach before the change was made, generating 1.042 points out of a potential 3 per match. By contrast, 2652 matches were played under a new manager with an improvement in team performance an average of 1.297 points per match. Table 2 suggests that, both in the four championships and in the six years considered, the team’s scores improved on average after the change. In addition, the team’s per-
1 Changes made after the 30th match by the Bundesliga and after the 34th match by the other leagues were therefore left out of this analysis.
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Fig. 2. Teams performance variation after coach replacement –Sample divided by leagues. (Source: www.football-data.co.uk).
Table 2 Average point variation after coach replacement. (Source: www.transfermarkt.com and www.football-data.co.uk).
2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 Tot
Serie A
Liga
P. League
Bundesliga
Tot
11.00% 33.20% 31.00% 22.00% 18.30% 48.10% 24.50%
23.30% 26.90% 46.80% 39.50% 54.10% 23.80% 35.40%
9.70% 2.20% 22.50% 1.50% 27.70% 23.50% 15.80%
10.50% 83.30% 4.50% 32.80% 45.50% −1.80% 22.90%
9.80% 30% 24.40% 24.20% 35.10% 28.10% 24.50%
formance worsened in just two cases, Serie A in season 2008/09 and La Liga in season 2013/14. To conclude, a change of coach has a positive impact on team performance. Based on such evidence, we can move on to the next logical step of this research, i.e. outlining a strategy to ascertain whether the sports betting market is efficient at absorbing the information of a change ‘on the bench’. The results we provide in this section might present some limitations since we take the average results over all teams who have experienced change of coach. With this respect, DePaola and Scoppa (2012) pointed out that the interpretation of average results can be problematic since not all teams will benefit from changing their coach. Similar results are reported by Constantinou and Fenton (2017) which reinforce these findings demonstrating that team’s performance gets worse when the manager who has been replaced has spent more time with the club. Moreover, this paper argues that other factors, such as transfers and team wages, injuries or EU competition, can have a more considerable impact on team performance than the change of coach. Despite these limitations, we believe that our results are useful to motivate the next step of our analysis. 3. Testing the semi-strong efficiency hypothesis 3.1. Literature review Before moving on to an empirical analysis and description of the results, we first summarise some findings on betting market efficiency. The main part of this research area investigates the weak form of information efficiency which, according to Vaughan Williams (2005), is the “possibility for earning differential (or even abnormal) returns in the future, from betting on the basis of past information about the yield to bets at identified [. . .] odds”. There-
fore, technical analysis should not be able to predict and beat the market using past odds as found elsewhere in the literature (Ali, 1998; Asch, Malkiel, & Quandt, 1984; Swindler & Shaw, 1995). Little evidence of the inefficiency of the pari-mutuel horserace betting market has been found by singling out profitable betting strategies (Bolton & Chapman, 1986; Hausch & Ziemba, 1995; Hausch, Ziemba, & Rubinstein, 1981; and Lo, 1995). Turning our attention to the kind of bets in hand, i.e. fixed-odds bets, Dixon and Coles (1997) found evidence of market inefficiency by applying trading strategies to the 1995/96 British Premier League matches. Similar results were found by Rue and Salvesen (2000), Kuypers (2000), and Dixon and Pope (2004). In addition, Goddard and Asimakopoulos (2004) found that market inefficiency is more evident at the end of the football season than in the early stages of a championship. Forrest, Goddard, and Simmons (2005) showed that market inefficiency increases along a five-year span of time, but it is difficult to find a predictive model which offers positive returns. Marshall (2009) finds that bookmakers’ market removes arbitrage opportunities rapidly, but not immediately. By implementing an ordered probit model, normally used to properly estimate the likelihood of the outcome of a football event, they conclude that no profitable returns may be achieved from bookmakers. Graham and Stott (2008) submitted two predictive models, one based on football results, the other on past odds, to compare rankings based on bookmakers’ opinions with rankings based on sports results. This work found that, albeit affected by systematic errors, bookmakers’ odds cannot be profitable, using the results of predictive models. More recently, Berkowitz, Depken, and Gandar, (2017) show evidences of the existence of reverse favorite-longshot bias in fixed-odds US betting markets in professional baseball and hockey. However, the transaction costs keep these markets efficient. As opposed to the results of such papers, Constantinou, Fenton, and Neil, (2013) built a model of a Bayesian network that can generate profitable strategies from bookmakers’ odds through a combination of market odds, subjective information and historical data. Bernardo et al. (2015), analysing bookmakers’ odds, found a systematic error in the estimate of home advantage. This error could allow bettors to gain extra profit, betting on teams’ home fixtures in the main football leagues in Europe. The most widely accepted definition of semi-strong (and strong) efficiency in betting markets is stated in Vaughan Williams (2005): “the existence of semi-strong efficiency in betting markets [. . .] would imply that the expected returns to any bet, or type of bet,
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Table 3 Simulation of a comparison between a LowOdds and a HighOdds strategy in the case of fixed stake. #Simu-lation
Low Odds Strategy Simulation (Odd = 1.25)
High Odds Strategy Simulation (Odd = 5.00)
Total Stakes
Payoff
Net Payoff
Variance
ROI
Total Stakes
Payoff
Net Payoff
Variance
ROI
1 2 3 4 5 6 7 8 9 10
10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D
9,976 D 10,035 D 9,978 D 9,966 D 10,004 D 9,939 D 10,003 D 10,070 D 10,005 D 9,956 D
−24 D 35 D −23 D −34 D 4D −61 D 3D 70 D 5D −44 D
0.252 0.247 0.252 0.253 0.250 0.255 0.250 0.245 0.250 0.253
−0.2% 0.4% −0.2% −0.3% 0.0% −0.6% 0.0% 0.7% 0.1% −0.4%
10,000 D 10,000 D 10,000 D 10,000 D 10.000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D
10,095 D 9,860 D 10,090 D 10,135 D 9.985 D 10,245 D 9,990 D 9,720 D 9,980 D 10,175 D
95 D −140 D 90 D 135 D −15 D 245 D −10 D −280 D −20 D 175 D
4.029 3.958 4.027 4.041 3.996 4.073 3.997 3.916 3.994 4.053
1.0% −1.4% 0.9% 1.4% −0.2% 2.5% −0.1% −2.8% −0.2% 1.8%
Average
10,000 D
9,993 D
−7 D
0.251
−0.1%
10,000 D
10,028 D
28 D
4.008
0.3%
placed about identical outcomes on the basis of publicly available information, should be identical (subject to identical costs and risks). The same applies with respect to strong efficiency when assessed in respect of all information”. Semi-strong information efficiency is less extensively analysed in the literature and interesting results are less frequent. Most of such papers focus on horserace betting, often with mixed results2 . As to football, one line of research looks into experts’ predictions on the main national media (Andersson, Edman, & Ekman, 2005; Forrest & Simmons, 2000; Song, Boulier, & Stekler, 2007). Above all, Forrest and Simmons (2000) engaged in a review of the suggestions of three professional tipsters’ columns, The Times, the Daily Mail and The Mirror. Despite such predictions being more accurate than those based on a random process, the authors concluded that the experts’ predictive process fails to estimate the publicly-available information properly.
3.2. Betting strategy and sample In order to test the semi-strong efficiency of the sports betting market, we develop a strategy building on the information that “head coach replacement improves teams’ performance”. We carry a short-term analysis focusing on the first few matches following a change in technical staff, such that both bookmakers and bettors have insufficient inputs to properly estimate the teams’ chance of winning. Therefore, we select a time frame of four matches, corresponding to the average number of matches played in one month. In order to determine that bookmakers’ odds regularly underestimate or overestimate the probability of victory in this particular case, we test whether a specific strategy - i.e. betting on teams which change their coach - yields the highest return compared to other scenarios. Conversely, if the market has successfully estimated the probability of winning, there should be no abnormal gain. We derive data from football-data.co.uk, which provides both final results and the odds for Europe’s top four leagues’ football matches. This paper covers six seasons between 2008 and 2014, for a total of 8676 matches. 47.34% of these matches ended with the home side winning, 24.65% in a draw, and the remaining 28.01% with the away side winning. Odds per type of result (i.e. home win, draw or away win) were also derived from football-data.co.uk based on two criteria that is their highest and mean values3 . Odds were taken at a random time before the start of the sports event, and were based on approximately 40 bookmakers per match.
2 See for instance Edelman (2003); Hausch and Ziemba (1990); Smith (2003) and Cain, Law, and Peel (2000)]. 3 Data on football-data.co.uk are in turn derived from betbrain.com, which lists all the odds in the European bookmaking markets.
Differently from previous papers4 we do not assume that, when following any given strategy, gamblers will spend 1D on each bet; rather, in line with the rational expectations theory, we assume that the amount of money placed on each bet will depend on odds’ values. More specifically, we assume that the bettor is willing to pay a smaller amount of money when the odd’s value is high (e.g. 20) since the probability of winning is low. On the contrary, the bettor will pay a higher amount when the odd’s value is low (e.g. 1.25) given that the chance of winning is high. Let us assume that there are 10,000 tennis matches (no draw admitted) in Wimbledon. In each of all 10,000 matches there is a player who is more likely to win (80% probability, oddsfav = 1.255 ), while the other player is the underdog (20%, oddsund = 5.00). If the strategy of the bettors requires a constant stake (e.g. 1D ), they will bet a total of 10,000 D . The bettors who follow a low odds strategy (LO bettors) will bet 10,000 times 1D on the favourite players (oddsfav = 1.25). Those betting on the high odds (HO bettors) would place 1D on each underdogs players, for a total of 10,000D on 5.00 odds. If we assume that the probabilities calculated by bookmakers are correct and that 80% of matches end with the victory of the favourite players, we will have the following results. LO payoff = 80% · 10, 000 · 1.25 · 1D − 10, 000 · 1D = 0
(1)
HO payoff = 20% · 10, 000 · 5.00 · 1D − 10, 000 · 1D = 0
(2)
Since the expected payoffs are exactly the same, the economic agents must take into account the volatility of the investment (Sharpe, 1964), which can be represented by the variance of two betting strategies as shown in Table 3. LO bettors’ variance is approximately 0.25, while HO bettors must take into account a much higher volatility as shown by a variance of 4.00. Given this considerable difference in volatility, we exclude that bettors would be likely to place the same stakes regardless of odds’ magnitude. This is consistent with mainstream financial theory according to which the economic agent invests a lower amount of money on high-risk stocks compared to low-risk ones, when the expected returns of investments are the same. In order to take this into account, we suggest that bettors would place a stake Y on event i according to odds’ magnitude (Oi ). The stake Yi will be just a linear transformation depending on the odds (Oi ). Yi =
4
1D Oi
(3)
See for instance Constantinou and Fenton (2013). In our simulation we assume that there are not bookmakers’ margins in the market. 5
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Table 4 Simulation of a comparison between a LowOdds and a HighOdds strategy in the case of proportional stakes. #Simu-lation
Low Odds Strategy Simulation (Odd = 1.25)
High Odds Strategy Simulation (Odd = 5.00)
Total Stakes
Payoff
Net Payoff
Variance
ROI
Total Stakes
Payoff
Net Payoff
Variance
ROI
1 2 3 4 5 6 7 8 9 10
8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D
7,989 D 8,051 D 8,013 D 8,032 D 7,975 D 8,055 D 7,954 D 8,049 D 7,987 D 7,968 D
−11 D 51 D 13 D 32 D −25 D 55 D −46 D 49 D −13 D −32 D
0.161 0.157 0.159 0.158 0.162 0.157 0.163 0.157 0.161 0.162
−0.1% 0.6% 0.2% 0.4% −0.3% 0.7% −0.6% 0.6% −0.2% −0.4%
2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D
2,011 D 1,949 D 1,987 D 1,968 D 2,025 D 1,945 D 2,046 D 1,951 D 2,013 D 2,032 D
11 D −51 D −13 D −32 D 25 D −55 D 46 D −49 D 13 D 32 D
0.161 0.157 0.159 0.158 0.162 0.157 0.163 0.157 0.161 0.162
0.6% −2.6% −0.7% −1.6% 1.3% −2.8% 2.3% −2.5% 0.7% 1.6%
Average
8,000 D
8,007 D
7D
0,160
0,1%
2,000 D
1,993 D
−7 D
0.160
−0.4%
Table 5 Mean bookmakers market highest odds’ margins (Source: www.football-data.co. uk). Mean Margin
Serie A
Liga
Premier League
Bundesliga
Season Mean
2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 Champ. Mean
0.95% 0.95% 0.96% 0.66% 0.38% −0.10% 0.63%
0.87% 0.81% 0.71% 0.49% 0.21% −0.38% 0.45%
0.67% 0.74% 0.63% 0.57% 0.28% −0.05% 0.47%
0.56% 0.85% 0.58% 0.46% 0.16% −0.11% 0.42%
0.77% 0.84% 0.73% 0.55% 0.26% −0.16% 0.50%
Going back to our Wimbledon example, the stake placed by LO and HO bettors would be respectively: LO stakes : YLOi =
1D = 0.80D 1.25
HO stakes : YHOi =
(4)
1D = 0.20D 5.00
(5)
LO and HO strategies expected values would be:
LO payoff = 80% · 10 000 · 1.25 · 0.8D − 10 000 · 1D = 0
HO payoff = 20% · 10 000 · 5.00 · 0.2D − 10 000 · 1D = 0
(6) (7)
As shown in Table 4, if bettors take into account odds’ magnitude while betting, they will not be subject to volatility asymmetry. In our example, for both LO and HO, the variances of their strategies have the same expected value of 0.160. This means that if the stakes are proportional to the magnitude of the odds, the risk will be constant for any kind of strategy. Moreover, from our scientific perspective, a proportional betting system will not produce false results in the case of exceptionally high odds. From now on, our research will refer to the betting stakes (Yi ) formula mentioned above: Yi =
1D Oi
(3)
Where Oi is the odd of an event’s result i. This method will produce a 1D payoff in case of bet victory, 0D if instead the bet is lost. Based on this setup, if a bettor placed his stakes on the highest odds among the 3 possible outcomes (i.e. 1, X, 2) of the 8676 events in our sample, he would spend D 8719.14. His total payoff would hence be D 8,676, i.e. the number of matches he would have betted on. In this way, the return on their stakes would amount to 99.51%. The fact that the expected value is lower than 100% is due to bookmakers margins. As shown in Table 5, the marginal value of the highest odds on the bookmaking market has considerably decreased over the last six years due to the increased competitiveness of the supply side. Therefore, the expected value of the bettors’ stakes must have been lower in the first seasons than in the last ones. This clearly shows
that the expansion of the betting market, in particular through the Internet, allows people to bet almost “on a par” with bookmakers when the highest odds are selected. The 2013/14 season shows slightly negative marginal values: This is an important sign of betting market expansion and of the resulting growth of in arbitrage opportunity6 . Our analysis focuses not only on the highest market odds. In fact, we acknowledge that results might be affected by a bookmaker’s mistake rather than by market inefficiency per se. Therefore, we also test market average odds. However, it is not possible to compare average and highest odds in our sample given that the margins are excessively high (approximately 7% for average odds versus 0.5% of the highest ones). For this reason, we weigh the average odds on the margin, such that the intrinsic probability of the odds adds up to one. This linear weighting method has been extensively used before7 . First, we calculate the margin () using Eq. (1), with qi being the average odds of each stake: −1 q−1 + q−1 x + q2 = 1 + 1
(8)
Then we recalculate the weighted odds by multiplying the average odds by the margin. If, for instance, we want to calculate the weighted odds of a home win q1 , we obtain:
q1 = q1 ∗
(9)
3.3. Monte Carlo experiment Based on the definition of semi-strong efficiency, a betting strategy that relies on public information must have the same risk/return profile as any other betting strategy that uses no information at all. Therefore, what we are trying to understand here is whether the profit margin of our strategy is significantly higher than that of a random betting choice. In this case, we would contradict such a definition and would confirm this as a case of semi-strong market inefficiency. If the number of head coach replacements is Z, our strategy will consist of 4·Z bets8 . Therefore the change-of-coach betting strategy may be equated to a set of 4·Z bets extracted from the full sample. If the sample consists of Y matches, there will be 3·Y possible bets to be extracted from the sample9 . Therefore, we want to randomly pick some sets of 4·Z bets out of the entire sample and calculate their overall payoff. As in the previous example, a cost of 1/qi will be incurred for each qi 10 , and the return will be either 1 or 0, depending on whether or not the chosen result occurs. If we calculate the ratio
6 7 8 9 10
See for example Constantinou and Fenton, 2013. See, for example Strumbelj and Robnik Sikonja (2010). The number of matches taken into account after the substitution is 4. There are 3 possible outcomes from each match. Where qi is the odds for the i-th result.
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Table 6 Betting strategy results. Seasons 2008-11, highest odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Highest Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2008/09 2009/10 2010/11 Tot Champ
−67.4% 18.7% 23.4% 4.8%
4.1% −14.0% 30.2% 5.7%
55.7% 12.5% 23.8% 32.8%
51.7% 24.9% 3.4% 21.1%
14.2% 8.3% 18.2% 13.9%
of total returns to total costs, we find the payoff of this set of random bets. We then repeat this algorithm a large number of times to calculate the probability distribution of payoffs, for a comparison with the results of our strategy. To sum up the above points as well as our strategy: • We consider the first four matches trained by the new team manager; • We simulate betting on the team that changed its coach to win each match (both home and away); • Stakes are placed on the market’s highest and average odds and will be inversely proportional to such odds; • We compare the results with random betting choices using a Monte Carlo experiment. We follow and test this strategy during the three seasons from 2008 to 2011. If successful, we repeat the same strategy in the following three seasons. If in both intervals we find statistical evidence that our strategy gains some advantage from the information that “head coach replacements improve team performance”, we will conclude that the fixed odds market is not efficient in its semistrong form. 3.4. 2008–2011 Seasons – highest and average odds analysis The first period of our analysis is made up of a sample of the three seasons, from 2008/09 to 2010/11, counting a total of 4338 matches. In this time span, there were 81 head coach replacements that respected the conditions of our definition. Our strategy is applied to the 32311 matches following the substitution. Table 6 shows the profit margin of the above betting strategy. In this case, we selected the highest odds from our dataset. If we look at the season-league pairings, we see that the strategydelivered a positive result in 10 cases out of 12 (83%). Serie A is the championship that delivered the lowest payoff, greatly affected by the 2008/09 season; it is in the Premier League, instead, that we have the highest overall payoff, with a 32.8% profit margin. Overall, our strategy results in a 13.9% positive payoff, thus suggesting that the impact of a new coach on team performance is underestimated. In this first period of analysis, the change-of-coach betting strategy may be equated to a set of bets on 323 results, selected according to specific criteria out of 13,014 results of the entire sample. Therefore, we want to randomly pick some sets comprising 323 bets out of the entire sample of the highest odds and calculate their overall payoff. We then repeat this operation 107 times12 , and build the following probability distribution. The distribution shown in Fig. 3 has a mean value of 0.992276 and a variance of 0.005173. Assuming as a null hypothesis that the value obtained following our strategy (1.139091) belongs to this distribution, we carry out the test on the right-hand column. The
11 Number of replacements (81) multiplied by number of matches taken into account (4). The total (324) is reduced by 1 because once the new manager continued to coach the team for only three matches. 12 We made 10 million reiterations, taking efficiency/time as a principle.
Fig. 3. Betting strategy payoff distribution generated with a Montecarlo simulation. (Source: www.football-data.co.uk).
Table 7 Betting strategy results. Seasons 2008-11, average odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Average Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2008/09 2009/10 2010/11 Tot Champ
−65.4% 18.6% 24.0% 6.1%
−5.0% −13.1% 30.3% 2.5%
57.7% 13.1% 24.7% 33.9%
63.5% 33.9% −6.6% 17.9%
13.9% 9.5% 14.4% 12.8%
result of the z-test is 2.0412 (p-value 0.021), which makes us reject the null hypothesis with a 2.5% significance level. If the mean and highest odds were perfectly aligned, we would expect to have a higher payoff than before. This is because the highest odds that we had used before had an overall margin of approximately 0.78%, while the average odds, which are weighted, do not include any bookmaker’s margin. Table 7 shows that payoffs decreased from before, but are still fully positive: the overall payoff dropped from 13.9% to 12.8%. In addition, the league/season pairings that had a negative payoff were 2 out of 12, instead of 4. Finally, note that all four leagues remained positive over the three-year time span, as did the three seasons over the four leagues. Replaying the Monte Carlo simulation, but using the average odds sample, we obtain a normal distribution with a mean value of 0.999878 and a variance of 0.005267. Testing the value obtained from our strategy (1.127583) we obtain a z-test value of 1.7597 (pvalue 0.039), which makes us reject the null hypothesis with a 5% significance level. Overall, even if the strategy payoff of the average odds is lower than that of the highest odds, this test confirms that, in the short term, bookmakers have underestimated the chance of teams winning after a change of manager. 3.5. 2011–2014 Seasons – highest and average odds analysis After obtaining excellent results in the first three-year period, for a subsequent time span we ascertain whether the bookmakers’ market managed to correct its expectations on the winning probability for a team that has replaced its head coach. The second period of our analysis is made up of a sample of the three seasons, from 2011/12 to 2013/14, counting a total of 4338 matches. In this time span, there were 75 head coach replacements that respected the conditions of our definition. Therefore our strategy will be applied to the 300 matches following the substitution. First, as in the pre-
G. Bernardo et al. / The Quarterly Review of Economics and Finance 71 (2019) 239–246 Table 8 Betting strategy results. Seasons 2011-14, highest odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Highest Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2011/12 2012/13 2013/14 Tot Champ
25.0% 51.2% 34.3% 36.1%
16.4% 9.5% −55.9% −0.5%
−7.0% −27.4% 31.5% 2.8%
−27.4% 41.1% 32.7% 5.9%
1.0% 20.9% 19.3% 13.0%
Table 9 Betting strategy results. Seasons 2011-14, average odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Highest Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2011/12 2012/13 2013/14 Tot Champ
23.2% 49.4% 25.9% 31.4%
16.0% 9.0% −55.9% −0.9%
−6.6% −28.1% 32.3% 2.8%
−35.9% 49.6% 33.5% 1.1%
−3.4% 21.6% 16.5% 10.4%
Table 10 Summary of significance level and betting strategy results (Source: www. transfermarkt.com and www.football-data.co.uk). Payoff & Significance Level
Highest Odds Average Odds
Payoff Significance Level Payoff Significance Level
2008-11
2011-14
13.9% 2.5% 12.8% 5.0%
13.0% 5.0% 10.4% 10.0%
vious interval, we look at the results of our strategy betting on the highest odds, later on average odds. Looking at the season-championship pairings in Table 8, we see that the strategy delivered a positive result in 8 cases out of 12 (67%). Serie A is the league that delivered the highest payoff (51.2%) while in La Liga, instead, we have a slight negative overall payoff with a -0.5% loss. Overall, this strategy results in a 13.0% positive payoff, thus suggesting that the impact of a new coach on team performance is still underestimated. We replay the Monte Carlo experiment on the sample made of the highest odds over the interval 2011-14, to test whether the payoff obtained is still outperforming the expected value of a random strategy. The normal distribution generated has a mean value of 0.997831 and a variance of 0.005556. Assuming as a null hypothesis that the value obtained following our strategy (1.129922) belongs to this distribution, we carry out the test on the right-hand column. The result of the z-test is 1.772146 (pvalue 0.038), which makes us reject the null hypothesis with a 5% significance level. Table 9 shows the results of our strategy applied to the sample of average odds. We can see how payoffs have generally decreased from the previous time interval to a value of 10.4%. This could be a signal that some bookmakers’ estimates have been corrected. Although there are still 4 out of 12 championship/season pairings with a negative payoff, it should be noted that the 2011/12 season yields a negative payoff. Replaying the Monte Carlo simulation, but using the average odds sample, we obtain a normal distribution with a mean value of 1.000003 and a variance of 0.005597. Testing the value obtained from our strategy (1.104345) we obtain a z-test value of 1.3948 (p-value 0.082), which makes us reject the null hypothesis with only a 10% significance level. Finally, Table 10 summarizes the average payoffs of our strategy. It generates positive return for the first four matches after managerial replacement in both the two sub-samples under consideration. Considering the persistence of positive payoffs over time, we can conclude that the betting market is inefficient also in its semistrong form.
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4. Conclusion Since the aim of our paper was to analyse the semi-strong efficiency of the sports betting market, we sought to verify the argument that a profitable betting strategy for the bettor is unlikely through analysis of public available information. In particular, we aimed to ascertain whether the market is efficient in the event of some major change in expectations about results, in our case when a team’s head coach is replaced. The impact of such a replacement on team performance was found to be positive (+24%) in most of our sample (73%). On the basis of this information, we suggested to follow a strategy that makes the bettors place a proportional stake on the victory of the team which changed coach. The results yielded a positive return for the first four matches after managerial replacement. By comparing our results to normal distributions of probability created with a Monte Carlo experiment, we can conclude that: • Betting on highest market odds gives a payoff that outperforms a cost/risk equivalent random strategy. Furthermore this is true in both intervals; • Betting on weighted average market odds returns a lower profit, but still positive and significant. In our analysis, we also used average odds in order to ascertain whether our results are influenced by a few bookmaker errors or by the market’s decision as a whole. Particularly in the second period, some bookmakers seem to have underestimated to a lesser extent the winning probability of the team which changed coach; • We cannot accept the hypothesis of the betting market’s semistrong form of information efficiency. References Ali, M. M. (1998). Probability models on horse-race outcomes. Journal of Applied Statistics, 25(2), 221–229. Andersson, P., Edman, J., & Ekman, M. (2005). Predicting the World Cup 2002 in soccer: Performance and confidence of experts and non-experts. International Journal of Forecasting, 21(3), 565–576. Asch, P., Malkiel, B. G., & Quandt, R. E. (1984). Market efficiency in racetrack betting. Journal of Business, 57(2), 165–175. Audas, R., Goddard, J., & Rowe, G. (2006). Modelling employment durations of NHL head coaches: Turnover and post-succession performance. Managerial and Decision Economics, 27, 293–306. Bernardo G., Ruberti M., Verona R. (2015), L‘efficienza del mercato delle scommesse sportive a quota fissa: un‘analisi empirica, pp. 114–171 - In: Il Gioco d‘azzardo in Italia. Contributi per un approccio interdisciplinare, edited by: Fabio La Rosa, Franco Angeli, Milan, Italy. Berkowitz, J. P., Depken, C. A., & Gandar, J. M. (2017). A favorite-longshot Bias in fixed-odds betting markets: Evidence from college basketball and college football. The Quarterly Review of Economics and Finance, 63(C), 223–239. Bolton, R. N., & Chapman, R. G. (1986). Searching for positive returns at the track: A multinomial logic model for handicapping horse races. Management Science, 32(8), 1040–1060. Brown, M. C. (1982). Administrative succession and organizational performance: The succession effect. Administrative Science Quarterly, 27, 1–16. Cain, M., Law, D., & Peel, D. (2000). The favourite-longshot bias and market efficiency in UK football betting. Scottish Journal of Political Economy, 47, 25–36. Constantinou, A. C., & Fenton, N. (2013). Profiting from arbitrage and odds biases of the European football gambling market. The Journal of Gambling Business and Economics, 7(2), 41–70. Constantinou, A. C., & Fenton, N. (2017). Towards smart-data: Improving predictive accuracy in long-term football team performance. Knowledge-Based Systems, 124, 93–104. Constantinou, A. C., Fenton, N. E., & Neil, M. (2013). Profiting from an inefficient association football market: Prediction, risk and uncertainty using Bayesian network. Gambling Market Knowledge-Based Systems Journal, 50 DePaola, M., & Scoppa, V. (2012). The effects of managerial turnover: Evidence from manager dismissals in italian football teams. Journal of Sports Economics. Dixon, M., & Coles, S. (1997). Modelling association football scores and inefficiencies in the football betting market. Applied Statistics, 46, 265–280. Dixon, M. J., & Pope, P. F. (2004). The value of statistical forecasts in the UK association football betting market. International Journal of Forecasting, 20, 697–711. Edelman, D. (2003). A competitive horse-race handicapping algorithm based on an analysis of covariance. In L. Vaughan Williams (Ed.), The economics of gambling (pp. 106–113). London: Routledge.
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Fabianic, D. (1994). Managerial change and organizational effectiveness in major league baseball: Findings for the eighties. Journal of Sport Behavior, 17, 135–147. Fama Eugene, F. (1970). Efficient capital markets: A review of the theory and empirical work. The Journal of Finance, 25(2), 383–417. Forrest, D., & Simmons, R. (2000). Forecasting sport: The behaviour and performance of football tipsters. International Journal of Forecasting, 16, 317–331. Forrest, D., Goddard, J., & Simmons, R. (2005). Odds-setters as forecasters: The case of English football. International Journal of Forecasting, 21, 551–564. Goddard, J., & Asimakopoulos, I. (2004). Forecasting football results and the efficiency of fixed-odds betting. Journal of Forecasting, 23, 51–66. Graham, I., & Stott, H. (2008). Predicting bookmaker odds and efficiency for UK football. Applied Economics, 40, 99–109. Hausch, D., Ziemba, W., & Rubinstein, M. (1981). Efficiency of the market for racetrack betting. Management Science, 27(12), 1435–1452. Hausch, D. B., & Ziemba, W. T. (1990). Arbitrage strategies for cross-track betting on major horse races. Journal of Business, 63, 61–78. Hausch D. B., Ziemba W. T., Efficiency of sports and lottery betting markets. Handbooks in operations research and management science, 9 (18), 1995. Kuypers. (2000). Information and efficiency: an empirical study of a fixed odds betting market. Applied Economics, 11, 1353–1363. Lo, S. Y. (1995). The application of ranking probability models to racetrack betting. Management Science, 41(6), 1048–1059. Marshall, B. (2009). How quickly is temporary market inefficiency removed? The Quarterly Review of Economics and Finance, 49(3), 917–930. McTeer, W., White, P. G., & Persad, S. (1995). Manager/coach mid-season replacement and team performance in professional team sport. Journal of Sport Behaviour, 18, 58–68.
Rue, H., & Salvesen, O. (2000). Prediction and retrospective analysis of soccer matches in a league. The Statistician, 49(Part 3), 339–418. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425–442. Smith, M. A. (2003). The impact of tipster information on bookmakers, prices in UK horse-Race markets. In L. Vaughan Williams (Ed.), The economics of gambling (pp. 67–79). London: Routledge. Song, C., Boulier, B. L., & Stekler, H. O. (2007). The comparative accuracy of judgmental and model forecasts of American football games. International Journal of Forecasting, 23(3), 405–413. Strumbelj, E., & Robnik Sikonja, M. (2010). Online bookmakers’ odds as forecasts: The case of European soccer leagues. International Journal of Forecasting, 26, 482–488. Swindler, S., & Shaw, R. (1995). Racetrack wagering and the uninformed bettor: A study of market efficiency. The Quarterly Review of Economics and Finance, 35(3), 305–314. Vaughan Williams, L. (2005). In L. Vaughan Williams (Ed.), Information efficiency in financial and betting markets. Cambridge: Cambridge University Press.
Websites www.betbrain.com www.football-data.uk.com www.oddsportal.com www.wikipedia.com www.transfermarkt.com
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The Quarterly Review of Economics and Finance journal homepage: www.elsevier.com/locate/qref
Semi-strong inefficiency in the fixed odds betting market: Underestimating the positive impact of head coach replacement in the main European soccer leagues Giovanni Bernardo a , Massimo Ruberti b , Roberto Verona b a b
Department of Law, University of Palermo, Italy Department of Economics e Management, University of Pisa, Italy
a r t i c l e
i n f o
Article history: Received 17 December 2017 Received in revised form 10 August 2018 Accepted 17 August 2018 Available online 25 August 2018 JEL classification: C12 D43 L83 Keywords: Sports betting market Fixed-odds bets Semi-strong efficiency hypothesis Monte Carlo experiment
a b s t r a c t In this paper we analyse the efficiency of the sports betting market, seeking to ascertain whether the market is efficient in the case of fixed odds provided by bookmakers in the four major European soccer leagues under the semi-strong efficiency hypothesis. By examining the trends of odds in the event of a major change in expectations about team results, i.e. when the head coach of a team is replaced, we attempt to verify the argument that a profitable strategy for the bettor is likely to be possible. In this case, the market under consideration would be inefficient. Analysing the average effect of head coach replacement, we find a positive impact on team performance. Based on this information, we build a betting strategy to find out whether the bookmakers’ odds absorb this change in expectations about the winning probability of involved teams. Comparing our strategy result with a distribution generated in a Monte Carlo experiment, we conclude that the betting market is inefficient in its semi-strong form. © 2018 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.
1. Introduction In the field of sport economics there is a recent area of research which explores the efficiency of regulated gambling markets by applying microeconomic theory and econometric analysis. In this context, our goal was to evaluate the sports betting market with a focus on bookmakers’ odds in the main football leagues in Europe, namely the British Premier League, the Spanish Liga, the Italian Serie A and the German Bundesliga. Drawing a parallel with the concept of information efficiency in financial markets (Fama Eugene, 1970), Vaughan Williams (2005) splits the efficiency of gambling markets into weak, semi-strong and strong forms. In this paper, we only cover semi-strong information efficiency, which occurs when current share prices encompass all publicly-available information. In other words, if a market meets this requirement, no extra return may be achieved on the basis of public information. Semi-strong efficiency means that the return on a bet based on public information must be the same, in terms of cost/risk, as that
on a bet that has not been based on public information (Vaughan Williams, 2005). Starting from the analysis of the trends of odds in the event of some major change in expectations about a team’s results, i.e. when a team’s coach is replaced, we attempt to verify the argument that a profitable betting strategy for the bettor is likely to be possible. First, we seek to establish whether a midseason change in technical staff may improve sporting results. Subsequently, we use such information to gain extra profits by setting a strategy that is consistent with the previous findings. Given the particularities of sports betting odds and the limited time horizon of our analysis, we cannot apply standard methodologies used for testing market efficiency. For this reason, we propose an empirical analysis based on Monte Carlo experiments that help us to accept, or eventually to reject, the semi-strong efficiency hypothesis of the sports betting market. The remainder of the paper is organised as follows. Section 2 seeks to ascertain the impact of the replacement of a head coach on a team’s performance by analysing data from six seasons of Europe’s top soccer leagues. Section 3 discusses methodological aspects of market inefficiency and the main results of our analysis. Finally, section 4 draws some conclusions.
E-mail addresses: [email protected] (G. Bernardo), [email protected] (M. Ruberti), [email protected] (R. Verona). https://doi.org/10.1016/j.qref.2018.08.007 1062-9769/© 2018 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.
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2. Change in expectations: the effect of head coach replacement This project revolves around the idea of a similarity between the betting world and what happens in listed share prices. Consider, for instance, a listed company that announces an unplanned distribution of dividends, a leading company’s interest in a takeover, or a merger with a competitor: the market will respond to such new information by changing its expectations. As expectations change, share prices will increase or decrease depending on whether market players rate the operation as positive or negative. Likewise, sports betting odds may be adjusted based on public information, such as an injury suffered by the best player, a referee’s decision to have the match played behind closed doors, or the replacement of a coach. Rather than being likened to an extra distribution of profits, the type of information we are going to use can be compared to an extra financial operation, since it consists in a structural change in the company’s organisation that may influence future performance. In football clubs, there are two major situations where something similar happens: a change in the players’ squad or in the technical staff. The first case is much more complicated, since one would have to check every match, one by one, to see whether new players were involved in the match, the minute count and physical fitness. For the sake of consistency and objectivity, we decided to take the second instance, i.e. the replacement of technical staff and, in particular, the change in head coach. Part of the sports business literature has sought to ascertain whether replacement of a head coach improves a team’s performance. Fabianic (1994) and McTeer, White, and Persad (1995) found that a change of coach has a favourable impact on a team’s performance, while Brown (1982). Recently, DePaola and Scoppa (2012) looked into the Serie A championship between 1997 and 2009: although the results seemed to corroborate the favourable impact of a change of coach on the team’s performance, they pointed out that comparing the results before and after a change of coach is not methodologically sound. Audas, Goddard, and Rowe (2006) reviewed changes of coaches in the last 25 years in England, finding a negative performance in the following three months and an increased variance in the results. The authors concluded that changes are mainly made to take advantage of such increased variance in performance. In this respect, the team owners would therefore try to gamble by giving such a shock. 2.1. Measurement of the average impact of a change of head coach For the sake of consistency, we will take a stricter definition of ‘change of head coach’. It must meet the following requirements: • The change takes place during a soccer season; • The dismissed coach has trained the team for at least five matches; • The new coach is not just a temporary replacement until a new team manager steps in; • This is the first change of the season. The first requirement is crucial because otherwise our research would be impaired by other factors that could change the team’s structure before the start of the season (transfers, promotion, relegation, etc.). The second and third requirements help understand the structural change properly as well as compare results before and after the change in technical staff. The fourth point makes our sample somewhat more consistent. We do not think that the first change should be compared with subsequent ones, since they tend to occur in very critical circumstances and often result in the return of the first coaches dismissed. Finally, the last requirement lays
Table 1 Number of coach replacement by seasons and leagues. (Source: www.transfermarkt. com).
2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 Tot
Serie A
Liga
P. League
Bundesliga
Tot X Season
6 11 10 8 7 10 52
9 8 7 8 7 3 42
5 5 4 4 5 7 30
4 3 9 7 5 4 32
24 27 30 27 24 24 156
Fig. 1. Teams performance variation after coach replacement – Full sample. (Source: www.football-data.co.uk).
down that the new team manager must have had the chance to train the team for least four matches1 , because otherwise the time horizon would not be wide enough to estimate the impact of the change. Our sample consists of six seasons of European’s leading soccer leagues, namely Italy’s Serie A, the Spanish Liga, the German Bundesliga and the British Premier League, from 2008/2009 to 2013/2014. In these 24 championships, 156 midseason changes of coach fulfilled our requirements. Table 1 lists the distribution of the sample by football season and by league. The Serie A championship is the one in which managers changed most frequently (52 times in six years), followed by La Liga (42). In the Premier League and the Bundesliga, the turnover of coaches was much less frequent: 30 and 32, respectively. The aim at this point is to answer the following question: “On average, does a change of coach have a positive or negative impact on team performance?”. By comparing the aggregate points under the dismissed coaches with those under the new coaches, and then weighting them by the number of matches coached, we found how often a change increased the average value of the team’s score. As shown in Fig. 1, in most cases (73%) a change of coach had a positive impact upon the team’s points. The league with the highest number of wins is La Liga, with 38 changes of coach out of 42 resulting in an improvement in team performance (see Fig. 2). The Premier League is the championship in which a change ended up with the lowest number of wins, albeit still positive in 57% of cases. In all, the teams played 2651 matches under the coach before the change was made, generating 1.042 points out of a potential 3 per match. By contrast, 2652 matches were played under a new manager with an improvement in team performance an average of 1.297 points per match. Table 2 suggests that, both in the four championships and in the six years considered, the team’s scores improved on average after the change. In addition, the team’s per-
1 Changes made after the 30th match by the Bundesliga and after the 34th match by the other leagues were therefore left out of this analysis.
G. Bernardo et al. / The Quarterly Review of Economics and Finance 71 (2019) 239–246
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Fig. 2. Teams performance variation after coach replacement –Sample divided by leagues. (Source: www.football-data.co.uk).
Table 2 Average point variation after coach replacement. (Source: www.transfermarkt.com and www.football-data.co.uk).
2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 Tot
Serie A
Liga
P. League
Bundesliga
Tot
11.00% 33.20% 31.00% 22.00% 18.30% 48.10% 24.50%
23.30% 26.90% 46.80% 39.50% 54.10% 23.80% 35.40%
9.70% 2.20% 22.50% 1.50% 27.70% 23.50% 15.80%
10.50% 83.30% 4.50% 32.80% 45.50% −1.80% 22.90%
9.80% 30% 24.40% 24.20% 35.10% 28.10% 24.50%
formance worsened in just two cases, Serie A in season 2008/09 and La Liga in season 2013/14. To conclude, a change of coach has a positive impact on team performance. Based on such evidence, we can move on to the next logical step of this research, i.e. outlining a strategy to ascertain whether the sports betting market is efficient at absorbing the information of a change ‘on the bench’. The results we provide in this section might present some limitations since we take the average results over all teams who have experienced change of coach. With this respect, DePaola and Scoppa (2012) pointed out that the interpretation of average results can be problematic since not all teams will benefit from changing their coach. Similar results are reported by Constantinou and Fenton (2017) which reinforce these findings demonstrating that team’s performance gets worse when the manager who has been replaced has spent more time with the club. Moreover, this paper argues that other factors, such as transfers and team wages, injuries or EU competition, can have a more considerable impact on team performance than the change of coach. Despite these limitations, we believe that our results are useful to motivate the next step of our analysis. 3. Testing the semi-strong efficiency hypothesis 3.1. Literature review Before moving on to an empirical analysis and description of the results, we first summarise some findings on betting market efficiency. The main part of this research area investigates the weak form of information efficiency which, according to Vaughan Williams (2005), is the “possibility for earning differential (or even abnormal) returns in the future, from betting on the basis of past information about the yield to bets at identified [. . .] odds”. There-
fore, technical analysis should not be able to predict and beat the market using past odds as found elsewhere in the literature (Ali, 1998; Asch, Malkiel, & Quandt, 1984; Swindler & Shaw, 1995). Little evidence of the inefficiency of the pari-mutuel horserace betting market has been found by singling out profitable betting strategies (Bolton & Chapman, 1986; Hausch & Ziemba, 1995; Hausch, Ziemba, & Rubinstein, 1981; and Lo, 1995). Turning our attention to the kind of bets in hand, i.e. fixed-odds bets, Dixon and Coles (1997) found evidence of market inefficiency by applying trading strategies to the 1995/96 British Premier League matches. Similar results were found by Rue and Salvesen (2000), Kuypers (2000), and Dixon and Pope (2004). In addition, Goddard and Asimakopoulos (2004) found that market inefficiency is more evident at the end of the football season than in the early stages of a championship. Forrest, Goddard, and Simmons (2005) showed that market inefficiency increases along a five-year span of time, but it is difficult to find a predictive model which offers positive returns. Marshall (2009) finds that bookmakers’ market removes arbitrage opportunities rapidly, but not immediately. By implementing an ordered probit model, normally used to properly estimate the likelihood of the outcome of a football event, they conclude that no profitable returns may be achieved from bookmakers. Graham and Stott (2008) submitted two predictive models, one based on football results, the other on past odds, to compare rankings based on bookmakers’ opinions with rankings based on sports results. This work found that, albeit affected by systematic errors, bookmakers’ odds cannot be profitable, using the results of predictive models. More recently, Berkowitz, Depken, and Gandar, (2017) show evidences of the existence of reverse favorite-longshot bias in fixed-odds US betting markets in professional baseball and hockey. However, the transaction costs keep these markets efficient. As opposed to the results of such papers, Constantinou, Fenton, and Neil, (2013) built a model of a Bayesian network that can generate profitable strategies from bookmakers’ odds through a combination of market odds, subjective information and historical data. Bernardo et al. (2015), analysing bookmakers’ odds, found a systematic error in the estimate of home advantage. This error could allow bettors to gain extra profit, betting on teams’ home fixtures in the main football leagues in Europe. The most widely accepted definition of semi-strong (and strong) efficiency in betting markets is stated in Vaughan Williams (2005): “the existence of semi-strong efficiency in betting markets [. . .] would imply that the expected returns to any bet, or type of bet,
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Table 3 Simulation of a comparison between a LowOdds and a HighOdds strategy in the case of fixed stake. #Simu-lation
Low Odds Strategy Simulation (Odd = 1.25)
High Odds Strategy Simulation (Odd = 5.00)
Total Stakes
Payoff
Net Payoff
Variance
ROI
Total Stakes
Payoff
Net Payoff
Variance
ROI
1 2 3 4 5 6 7 8 9 10
10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D
9,976 D 10,035 D 9,978 D 9,966 D 10,004 D 9,939 D 10,003 D 10,070 D 10,005 D 9,956 D
−24 D 35 D −23 D −34 D 4D −61 D 3D 70 D 5D −44 D
0.252 0.247 0.252 0.253 0.250 0.255 0.250 0.245 0.250 0.253
−0.2% 0.4% −0.2% −0.3% 0.0% −0.6% 0.0% 0.7% 0.1% −0.4%
10,000 D 10,000 D 10,000 D 10,000 D 10.000 D 10,000 D 10,000 D 10,000 D 10,000 D 10,000 D
10,095 D 9,860 D 10,090 D 10,135 D 9.985 D 10,245 D 9,990 D 9,720 D 9,980 D 10,175 D
95 D −140 D 90 D 135 D −15 D 245 D −10 D −280 D −20 D 175 D
4.029 3.958 4.027 4.041 3.996 4.073 3.997 3.916 3.994 4.053
1.0% −1.4% 0.9% 1.4% −0.2% 2.5% −0.1% −2.8% −0.2% 1.8%
Average
10,000 D
9,993 D
−7 D
0.251
−0.1%
10,000 D
10,028 D
28 D
4.008
0.3%
placed about identical outcomes on the basis of publicly available information, should be identical (subject to identical costs and risks). The same applies with respect to strong efficiency when assessed in respect of all information”. Semi-strong information efficiency is less extensively analysed in the literature and interesting results are less frequent. Most of such papers focus on horserace betting, often with mixed results2 . As to football, one line of research looks into experts’ predictions on the main national media (Andersson, Edman, & Ekman, 2005; Forrest & Simmons, 2000; Song, Boulier, & Stekler, 2007). Above all, Forrest and Simmons (2000) engaged in a review of the suggestions of three professional tipsters’ columns, The Times, the Daily Mail and The Mirror. Despite such predictions being more accurate than those based on a random process, the authors concluded that the experts’ predictive process fails to estimate the publicly-available information properly.
3.2. Betting strategy and sample In order to test the semi-strong efficiency of the sports betting market, we develop a strategy building on the information that “head coach replacement improves teams’ performance”. We carry a short-term analysis focusing on the first few matches following a change in technical staff, such that both bookmakers and bettors have insufficient inputs to properly estimate the teams’ chance of winning. Therefore, we select a time frame of four matches, corresponding to the average number of matches played in one month. In order to determine that bookmakers’ odds regularly underestimate or overestimate the probability of victory in this particular case, we test whether a specific strategy - i.e. betting on teams which change their coach - yields the highest return compared to other scenarios. Conversely, if the market has successfully estimated the probability of winning, there should be no abnormal gain. We derive data from football-data.co.uk, which provides both final results and the odds for Europe’s top four leagues’ football matches. This paper covers six seasons between 2008 and 2014, for a total of 8676 matches. 47.34% of these matches ended with the home side winning, 24.65% in a draw, and the remaining 28.01% with the away side winning. Odds per type of result (i.e. home win, draw or away win) were also derived from football-data.co.uk based on two criteria that is their highest and mean values3 . Odds were taken at a random time before the start of the sports event, and were based on approximately 40 bookmakers per match.
2 See for instance Edelman (2003); Hausch and Ziemba (1990); Smith (2003) and Cain, Law, and Peel (2000)]. 3 Data on football-data.co.uk are in turn derived from betbrain.com, which lists all the odds in the European bookmaking markets.
Differently from previous papers4 we do not assume that, when following any given strategy, gamblers will spend 1D on each bet; rather, in line with the rational expectations theory, we assume that the amount of money placed on each bet will depend on odds’ values. More specifically, we assume that the bettor is willing to pay a smaller amount of money when the odd’s value is high (e.g. 20) since the probability of winning is low. On the contrary, the bettor will pay a higher amount when the odd’s value is low (e.g. 1.25) given that the chance of winning is high. Let us assume that there are 10,000 tennis matches (no draw admitted) in Wimbledon. In each of all 10,000 matches there is a player who is more likely to win (80% probability, oddsfav = 1.255 ), while the other player is the underdog (20%, oddsund = 5.00). If the strategy of the bettors requires a constant stake (e.g. 1D ), they will bet a total of 10,000 D . The bettors who follow a low odds strategy (LO bettors) will bet 10,000 times 1D on the favourite players (oddsfav = 1.25). Those betting on the high odds (HO bettors) would place 1D on each underdogs players, for a total of 10,000D on 5.00 odds. If we assume that the probabilities calculated by bookmakers are correct and that 80% of matches end with the victory of the favourite players, we will have the following results. LO payoff = 80% · 10, 000 · 1.25 · 1D − 10, 000 · 1D = 0
(1)
HO payoff = 20% · 10, 000 · 5.00 · 1D − 10, 000 · 1D = 0
(2)
Since the expected payoffs are exactly the same, the economic agents must take into account the volatility of the investment (Sharpe, 1964), which can be represented by the variance of two betting strategies as shown in Table 3. LO bettors’ variance is approximately 0.25, while HO bettors must take into account a much higher volatility as shown by a variance of 4.00. Given this considerable difference in volatility, we exclude that bettors would be likely to place the same stakes regardless of odds’ magnitude. This is consistent with mainstream financial theory according to which the economic agent invests a lower amount of money on high-risk stocks compared to low-risk ones, when the expected returns of investments are the same. In order to take this into account, we suggest that bettors would place a stake Y on event i according to odds’ magnitude (Oi ). The stake Yi will be just a linear transformation depending on the odds (Oi ). Yi =
4
1D Oi
(3)
See for instance Constantinou and Fenton (2013). In our simulation we assume that there are not bookmakers’ margins in the market. 5
G. Bernardo et al. / The Quarterly Review of Economics and Finance 71 (2019) 239–246
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Table 4 Simulation of a comparison between a LowOdds and a HighOdds strategy in the case of proportional stakes. #Simu-lation
Low Odds Strategy Simulation (Odd = 1.25)
High Odds Strategy Simulation (Odd = 5.00)
Total Stakes
Payoff
Net Payoff
Variance
ROI
Total Stakes
Payoff
Net Payoff
Variance
ROI
1 2 3 4 5 6 7 8 9 10
8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D 8,000 D
7,989 D 8,051 D 8,013 D 8,032 D 7,975 D 8,055 D 7,954 D 8,049 D 7,987 D 7,968 D
−11 D 51 D 13 D 32 D −25 D 55 D −46 D 49 D −13 D −32 D
0.161 0.157 0.159 0.158 0.162 0.157 0.163 0.157 0.161 0.162
−0.1% 0.6% 0.2% 0.4% −0.3% 0.7% −0.6% 0.6% −0.2% −0.4%
2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D 2,000 D
2,011 D 1,949 D 1,987 D 1,968 D 2,025 D 1,945 D 2,046 D 1,951 D 2,013 D 2,032 D
11 D −51 D −13 D −32 D 25 D −55 D 46 D −49 D 13 D 32 D
0.161 0.157 0.159 0.158 0.162 0.157 0.163 0.157 0.161 0.162
0.6% −2.6% −0.7% −1.6% 1.3% −2.8% 2.3% −2.5% 0.7% 1.6%
Average
8,000 D
8,007 D
7D
0,160
0,1%
2,000 D
1,993 D
−7 D
0.160
−0.4%
Table 5 Mean bookmakers market highest odds’ margins (Source: www.football-data.co. uk). Mean Margin
Serie A
Liga
Premier League
Bundesliga
Season Mean
2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 Champ. Mean
0.95% 0.95% 0.96% 0.66% 0.38% −0.10% 0.63%
0.87% 0.81% 0.71% 0.49% 0.21% −0.38% 0.45%
0.67% 0.74% 0.63% 0.57% 0.28% −0.05% 0.47%
0.56% 0.85% 0.58% 0.46% 0.16% −0.11% 0.42%
0.77% 0.84% 0.73% 0.55% 0.26% −0.16% 0.50%
Going back to our Wimbledon example, the stake placed by LO and HO bettors would be respectively: LO stakes : YLOi =
1D = 0.80D 1.25
HO stakes : YHOi =
(4)
1D = 0.20D 5.00
(5)
LO and HO strategies expected values would be:
LO payoff = 80% · 10 000 · 1.25 · 0.8D − 10 000 · 1D = 0
HO payoff = 20% · 10 000 · 5.00 · 0.2D − 10 000 · 1D = 0
(6) (7)
As shown in Table 4, if bettors take into account odds’ magnitude while betting, they will not be subject to volatility asymmetry. In our example, for both LO and HO, the variances of their strategies have the same expected value of 0.160. This means that if the stakes are proportional to the magnitude of the odds, the risk will be constant for any kind of strategy. Moreover, from our scientific perspective, a proportional betting system will not produce false results in the case of exceptionally high odds. From now on, our research will refer to the betting stakes (Yi ) formula mentioned above: Yi =
1D Oi
(3)
Where Oi is the odd of an event’s result i. This method will produce a 1D payoff in case of bet victory, 0D if instead the bet is lost. Based on this setup, if a bettor placed his stakes on the highest odds among the 3 possible outcomes (i.e. 1, X, 2) of the 8676 events in our sample, he would spend D 8719.14. His total payoff would hence be D 8,676, i.e. the number of matches he would have betted on. In this way, the return on their stakes would amount to 99.51%. The fact that the expected value is lower than 100% is due to bookmakers margins. As shown in Table 5, the marginal value of the highest odds on the bookmaking market has considerably decreased over the last six years due to the increased competitiveness of the supply side. Therefore, the expected value of the bettors’ stakes must have been lower in the first seasons than in the last ones. This clearly shows
that the expansion of the betting market, in particular through the Internet, allows people to bet almost “on a par” with bookmakers when the highest odds are selected. The 2013/14 season shows slightly negative marginal values: This is an important sign of betting market expansion and of the resulting growth of in arbitrage opportunity6 . Our analysis focuses not only on the highest market odds. In fact, we acknowledge that results might be affected by a bookmaker’s mistake rather than by market inefficiency per se. Therefore, we also test market average odds. However, it is not possible to compare average and highest odds in our sample given that the margins are excessively high (approximately 7% for average odds versus 0.5% of the highest ones). For this reason, we weigh the average odds on the margin, such that the intrinsic probability of the odds adds up to one. This linear weighting method has been extensively used before7 . First, we calculate the margin () using Eq. (1), with qi being the average odds of each stake: −1 q−1 + q−1 x + q2 = 1 + 1
(8)
Then we recalculate the weighted odds by multiplying the average odds by the margin. If, for instance, we want to calculate the weighted odds of a home win q1 , we obtain:
q1 = q1 ∗
(9)
3.3. Monte Carlo experiment Based on the definition of semi-strong efficiency, a betting strategy that relies on public information must have the same risk/return profile as any other betting strategy that uses no information at all. Therefore, what we are trying to understand here is whether the profit margin of our strategy is significantly higher than that of a random betting choice. In this case, we would contradict such a definition and would confirm this as a case of semi-strong market inefficiency. If the number of head coach replacements is Z, our strategy will consist of 4·Z bets8 . Therefore the change-of-coach betting strategy may be equated to a set of 4·Z bets extracted from the full sample. If the sample consists of Y matches, there will be 3·Y possible bets to be extracted from the sample9 . Therefore, we want to randomly pick some sets of 4·Z bets out of the entire sample and calculate their overall payoff. As in the previous example, a cost of 1/qi will be incurred for each qi 10 , and the return will be either 1 or 0, depending on whether or not the chosen result occurs. If we calculate the ratio
6 7 8 9 10
See for example Constantinou and Fenton, 2013. See, for example Strumbelj and Robnik Sikonja (2010). The number of matches taken into account after the substitution is 4. There are 3 possible outcomes from each match. Where qi is the odds for the i-th result.
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Table 6 Betting strategy results. Seasons 2008-11, highest odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Highest Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2008/09 2009/10 2010/11 Tot Champ
−67.4% 18.7% 23.4% 4.8%
4.1% −14.0% 30.2% 5.7%
55.7% 12.5% 23.8% 32.8%
51.7% 24.9% 3.4% 21.1%
14.2% 8.3% 18.2% 13.9%
of total returns to total costs, we find the payoff of this set of random bets. We then repeat this algorithm a large number of times to calculate the probability distribution of payoffs, for a comparison with the results of our strategy. To sum up the above points as well as our strategy: • We consider the first four matches trained by the new team manager; • We simulate betting on the team that changed its coach to win each match (both home and away); • Stakes are placed on the market’s highest and average odds and will be inversely proportional to such odds; • We compare the results with random betting choices using a Monte Carlo experiment. We follow and test this strategy during the three seasons from 2008 to 2011. If successful, we repeat the same strategy in the following three seasons. If in both intervals we find statistical evidence that our strategy gains some advantage from the information that “head coach replacements improve team performance”, we will conclude that the fixed odds market is not efficient in its semistrong form. 3.4. 2008–2011 Seasons – highest and average odds analysis The first period of our analysis is made up of a sample of the three seasons, from 2008/09 to 2010/11, counting a total of 4338 matches. In this time span, there were 81 head coach replacements that respected the conditions of our definition. Our strategy is applied to the 32311 matches following the substitution. Table 6 shows the profit margin of the above betting strategy. In this case, we selected the highest odds from our dataset. If we look at the season-league pairings, we see that the strategydelivered a positive result in 10 cases out of 12 (83%). Serie A is the championship that delivered the lowest payoff, greatly affected by the 2008/09 season; it is in the Premier League, instead, that we have the highest overall payoff, with a 32.8% profit margin. Overall, our strategy results in a 13.9% positive payoff, thus suggesting that the impact of a new coach on team performance is underestimated. In this first period of analysis, the change-of-coach betting strategy may be equated to a set of bets on 323 results, selected according to specific criteria out of 13,014 results of the entire sample. Therefore, we want to randomly pick some sets comprising 323 bets out of the entire sample of the highest odds and calculate their overall payoff. We then repeat this operation 107 times12 , and build the following probability distribution. The distribution shown in Fig. 3 has a mean value of 0.992276 and a variance of 0.005173. Assuming as a null hypothesis that the value obtained following our strategy (1.139091) belongs to this distribution, we carry out the test on the right-hand column. The
11 Number of replacements (81) multiplied by number of matches taken into account (4). The total (324) is reduced by 1 because once the new manager continued to coach the team for only three matches. 12 We made 10 million reiterations, taking efficiency/time as a principle.
Fig. 3. Betting strategy payoff distribution generated with a Montecarlo simulation. (Source: www.football-data.co.uk).
Table 7 Betting strategy results. Seasons 2008-11, average odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Average Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2008/09 2009/10 2010/11 Tot Champ
−65.4% 18.6% 24.0% 6.1%
−5.0% −13.1% 30.3% 2.5%
57.7% 13.1% 24.7% 33.9%
63.5% 33.9% −6.6% 17.9%
13.9% 9.5% 14.4% 12.8%
result of the z-test is 2.0412 (p-value 0.021), which makes us reject the null hypothesis with a 2.5% significance level. If the mean and highest odds were perfectly aligned, we would expect to have a higher payoff than before. This is because the highest odds that we had used before had an overall margin of approximately 0.78%, while the average odds, which are weighted, do not include any bookmaker’s margin. Table 7 shows that payoffs decreased from before, but are still fully positive: the overall payoff dropped from 13.9% to 12.8%. In addition, the league/season pairings that had a negative payoff were 2 out of 12, instead of 4. Finally, note that all four leagues remained positive over the three-year time span, as did the three seasons over the four leagues. Replaying the Monte Carlo simulation, but using the average odds sample, we obtain a normal distribution with a mean value of 0.999878 and a variance of 0.005267. Testing the value obtained from our strategy (1.127583) we obtain a z-test value of 1.7597 (pvalue 0.039), which makes us reject the null hypothesis with a 5% significance level. Overall, even if the strategy payoff of the average odds is lower than that of the highest odds, this test confirms that, in the short term, bookmakers have underestimated the chance of teams winning after a change of manager. 3.5. 2011–2014 Seasons – highest and average odds analysis After obtaining excellent results in the first three-year period, for a subsequent time span we ascertain whether the bookmakers’ market managed to correct its expectations on the winning probability for a team that has replaced its head coach. The second period of our analysis is made up of a sample of the three seasons, from 2011/12 to 2013/14, counting a total of 4338 matches. In this time span, there were 75 head coach replacements that respected the conditions of our definition. Therefore our strategy will be applied to the 300 matches following the substitution. First, as in the pre-
G. Bernardo et al. / The Quarterly Review of Economics and Finance 71 (2019) 239–246 Table 8 Betting strategy results. Seasons 2011-14, highest odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Highest Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2011/12 2012/13 2013/14 Tot Champ
25.0% 51.2% 34.3% 36.1%
16.4% 9.5% −55.9% −0.5%
−7.0% −27.4% 31.5% 2.8%
−27.4% 41.1% 32.7% 5.9%
1.0% 20.9% 19.3% 13.0%
Table 9 Betting strategy results. Seasons 2011-14, average odds case. (Source: www. transfermarkt.com and www.football-data.co.uk). Highest Odds
Serie A
Liga
Premier League
Bundesliga
Season’s Tot
2011/12 2012/13 2013/14 Tot Champ
23.2% 49.4% 25.9% 31.4%
16.0% 9.0% −55.9% −0.9%
−6.6% −28.1% 32.3% 2.8%
−35.9% 49.6% 33.5% 1.1%
−3.4% 21.6% 16.5% 10.4%
Table 10 Summary of significance level and betting strategy results (Source: www. transfermarkt.com and www.football-data.co.uk). Payoff & Significance Level
Highest Odds Average Odds
Payoff Significance Level Payoff Significance Level
2008-11
2011-14
13.9% 2.5% 12.8% 5.0%
13.0% 5.0% 10.4% 10.0%
vious interval, we look at the results of our strategy betting on the highest odds, later on average odds. Looking at the season-championship pairings in Table 8, we see that the strategy delivered a positive result in 8 cases out of 12 (67%). Serie A is the league that delivered the highest payoff (51.2%) while in La Liga, instead, we have a slight negative overall payoff with a -0.5% loss. Overall, this strategy results in a 13.0% positive payoff, thus suggesting that the impact of a new coach on team performance is still underestimated. We replay the Monte Carlo experiment on the sample made of the highest odds over the interval 2011-14, to test whether the payoff obtained is still outperforming the expected value of a random strategy. The normal distribution generated has a mean value of 0.997831 and a variance of 0.005556. Assuming as a null hypothesis that the value obtained following our strategy (1.129922) belongs to this distribution, we carry out the test on the right-hand column. The result of the z-test is 1.772146 (pvalue 0.038), which makes us reject the null hypothesis with a 5% significance level. Table 9 shows the results of our strategy applied to the sample of average odds. We can see how payoffs have generally decreased from the previous time interval to a value of 10.4%. This could be a signal that some bookmakers’ estimates have been corrected. Although there are still 4 out of 12 championship/season pairings with a negative payoff, it should be noted that the 2011/12 season yields a negative payoff. Replaying the Monte Carlo simulation, but using the average odds sample, we obtain a normal distribution with a mean value of 1.000003 and a variance of 0.005597. Testing the value obtained from our strategy (1.104345) we obtain a z-test value of 1.3948 (p-value 0.082), which makes us reject the null hypothesis with only a 10% significance level. Finally, Table 10 summarizes the average payoffs of our strategy. It generates positive return for the first four matches after managerial replacement in both the two sub-samples under consideration. Considering the persistence of positive payoffs over time, we can conclude that the betting market is inefficient also in its semistrong form.
245
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Websites www.betbrain.com www.football-data.uk.com www.oddsportal.com www.wikipedia.com www.transfermarkt.com