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The Efficiency of Sport Betting Markets: An Analysis Using Arbitrage Trading within Super Rugby.

Introduction

The popularity of the sport betting industry has grown tremendously over the past few decades, with football, basketball, and horseracing being some of the dominant sports in the betting market. The unprecedented growth resulted from extensive deregulations, abolition of national monopolies, and the advent of online gambling (Vlastakis, 2009). Such growth has also led to a vast amount of literature on the topic of sport betting becoming readily available, with the lion's share of the literature focusing on the key issue of efficiency in the betting market.

A market is observed from an informational perspective, and the efficiency of the market, defined by the level of information reflected in the prices (Fama, 1970). In the context of sports betting, weak form efficiency, in an economic sense, implies that by using price information it is impossible for the punter or bookmaker to make abnormal profits (Kuypers, 2000). Such abnormality implies that the bookmaker would make in excess of the commission charged on the stake, often known as the Vigorish (Vig) (Burkey, 2005), or in excess of the long run expected built-in profit for the bookmaker, known as the over-round (Milliner et al., 2009). Abnormal profits for the punter would be returns that are better than the loss made in paying the Vig or over-round. Hence, punters should not be able to systematically generate returns different to the margin. Semi-strong efficiency would suggest that there is no strategy, based on publicly available information, which will allow either the bookmaker or the punter to make abnormal returns. Finally, strong form market efficiency is the extreme case where even a strategy based on "inside information" will not allow punters or bookmakers to make excess returns.

Multiple studies on market efficiency have been conducted on a variety of sports within several betting systems. Some show no signs of inefficiency, or there is insufficient compelling evidence to prove inefficiency, while others have demonstrated that market inefficiencies do exist, both on a weak and semi-strong form level (for reviews see Sauer, 1998; Vaughn Williams, 1999, 2005). In addition, these studies have also proposed many possible reasons for market inefficiencies. The two most common reasons are due to bookmakers maximizing commercial gains and punters behavioral biases.

A logical point of departure in analyzing the efficiency of the betting market, as with financial markets, would be the identification of arbitrage opportunities. A study conducted by Lane and Ziemba (2004) within the sport Jai Alai, originating from Spain, suggests that there are significant opportunities for arbitrage betting. The authors developed strategies for "perfect" and "risk" arbitrage opportunities, (2) and suggested that such arbitrage trading strategies might have similar positive results within other sports. Paton and Vaughan Williams (2005) also identified risky arbitrage opportunities in the UK football spread betting market.

Vlastakis et al. (2009) showed that by making a combined bet where the maximum odds are taken from each bookmaker, it is possible to discover an "under-round" book (see also Ashiya, 2013, for combined bets in horse-racing). In addition, they demonstrate the results through a study conducted on the European football betting market by analyzing the closing odds of five bookmakers. The authors noted that, with the extensive regulatory changes and as the internet became more readily available (causing a surge in online gambling volumes), it was expected that the efficiency of the market would have improved over recent years. However, the information contained in the odds was still exploitable to gain profits through arbitrage and active betting strategies. Arbitrage opportunities, although rare in football betting, were shown to be highly profitable with returns ranging between 12% and 200% per trade.

Franck et al. (2009) looked at the occurrence of sure bets between the bookmaker market and the exchange market. In the exchange market, bets can be backed or laid by the punter, and therefore these bets can be bought in the bookmaker market at favorable odds, and sold for a higher price in the exchange market. The authors examined 5,478 European football matches and found only 0.18% of games created intra-market arbitrage trades that beat the commission charged. However, when considering the inter-market opportunities, an astonishing 26.47% of games were arbitrage trades, with an average return of 1.2%. Using multivariate tests, it is proven that the bookmaker's odds are less efficient than the exchange odds. In addition to the above, Shin (1993) and Marshall (2009) have also discovered arbitrage opportunities between bookmakers.

In comparison to other studies, Levitt (2004) examines a dataset of approximately 20,000 spread betting wagers on the NFL, placed by 285 bettors in a high-stakes online competition. This study has the advantage over previous studies in that the prices (i.e., spread offered), as well as the quantities on each bet, can be observed. It was found that the betting market is inefficient due to the bookmaker balancing the exposure on each side of the wager and attempting to profit from punter behavioral biases. Bookmakers may attempt to benefit from known punter biases, such as the "favorite-longshot bias" (see Woodland and Woodland, 1994). Notably, such an anomaly has also been observed in both financial derivatives and prediction markets (see, for example, Hodges, Tomkins, and Ziemba, 2003, and Wolfers and Zitzewitz, 2004), further emphasising the value of studying the efficiency of sport betting markets as a proxy for financial markets. Connections between the two markets were further explored and bridged by the more recent study of Moskowitz (2015), which analysed the relationship between certain unique features of sport betting markets and the behavioral theories of cross-sectional asset pricing anomalies.

Pope and Peel (1989) conducted a study on market efficiency using football fixed odds betting in the UK. Fixed odds are posted prior to a game by bookmakers and not altered due to market forces, but rather due to bookmaker expectations and balancing the book. Odds therefore differ between bookmakers. The authors found no evidence of a trading rule that produces abnormal profits and hence proved that the market is efficient. It is suggested that there are better forms of pricing by pooling of information to produce superior forecasts, however, this profit is eroded due to tax. There has also been a wide range of more recent studies on the efficiency of the UK football market. Readers are referred to Demir et al. (2012), Graham and Stott (2008), as well as Goddad and Asimakopolous (2004), where the former proposed an interesting stream of study by implementing the Fibonacci strategy to analyse betting market efficiency.

The level of efficiency within the Super Rugby betting market has become a topic of recent interest. While such a betting market has become more popular over the recent years, particularly with the advent of online gambling, there is a shortfall within the current literature to address the efficiency of the market. In this paper, we attempt to fill such a gap in the literature by investigating the efficiency of the Super Rugby betting market. To the best of our knowledge this is the first paper to explore the efficiency of said market through exploiting odds arbitrage (see below). Within the Super Rugby betting market there are various betting categories, such as, among others, Match Outcome, Handicap, and Winning Margins. The first category, which we shall explore in this paper, allows the punter to bet which team will win the match. This is not dissimilar to that of the football betting market, as analyzed by Vlastakis et al. (2009). The second involves bookmakers giving advantage to a particular team with the aim of balancing the game. Lastly, Winning Margins simply allows the punter to predict the margin of a victory. In the sequel, we will first examine whether arbitrage opportunities do in fact exist in the Super Rugby betting market along the lines of Vlastakis et al. (2009). We then show through an empirical experiment, by implementing a straightforward practical combined betting strategy, that such opportunities are indeed exploitable by the layman. As emphasized above, if the market is indeed efficient, then such a strategy should not exist.

The Super Rugby betting market was chosen for a number of key reasons. Super Rugby is considered one of the most widely gambled sports in certain countries (e.g., South Africa), even though the profit margins (or over-round) that bookmakers aim for is low in comparison to other gambled sports. For instance, bookmakers generally have an over-round of between five and seven percent in rugby, while it is not uncommon to see over-rounds of over 30 percent in horse racing. (2) Furthermore, Super Rugby is an international event, which means that bookmakers around the world quote odds on the same match. Hence, dissimilar odds may result due to biased opinions on the outcomes from individual bookmakers. (3) This, together with the findings of Vlastakis et al. (2009) regarding the efficiency of betting markets with the advent of online gambling, suggests that it may be fruitful to explore the efficiency of the Super Rugby betting market as well. Finally, information on Super Rugby is readily available to both bookmakers and the general public, whereby neither party is prejudiced in anyway. Such a betting environment implies that arbitrage opportunities can develop quickly when bookmakers offer slightly different odds. Hence, investigating the efficiency of the Super Rugby betting market should yield interesting results.

The remainder of the paper is laid out as follows. The next section discusses the methodology and framework used in this paper. An investigation for empirical evidence to support the claim of inefficiency in the Super Rugby betting market is presented in the third section. The fourth section focuses on the implementation of a practical betting strategy, and we show through empirical analysis that arbitrage opportunities are indeed exploitable in the Super Rugby betting market.

Methodology and Theoretical Framework

Arbitrage Definitions and Mathematical Framework

Arbitrage trades in the financial market exist due to market inefficiencies, and generally involves the buying and selling of assets to make a riskless profit, based on the difference in prices. Within sports betting, however, bets cannot be sold (unless dealing with a Betting Exchange, such as Betfair, where it is possible to back or lay a bet), and therefore betting only has two types of arbitrage strategies. The first type of arbitrage trade is based on differing handicaps quoted over numerous bookmakers with the same or very similar odds. The second type, which this paper focuses on, is odds arbitrage. In both strategies, more than one bookmaker must be involved in order to benefit from price discrepancies between the bookmakers.

In an odds arbitrage trade, the punter takes what is known as a combined bet, placing money on all outcomes of the event, by choosing the maximum odds per outcome over all the bookmakers. An arbitrage trade therefore exists if the mispricing is such that the amount won on any outcome will cover the loss of the stake on all the remaining outcomes. Mathematically, Vlastakis et al. (2009) defines the expected gain for a bookmaker on an event with n outcomes as

[mathematical expression not reproducible]

where [P.sub.1] is the true probability of the ith outcome, [w.sub.i] the proportion of bets, and [d.sub.t] the quoted odds on the ith outcome. (4) However, since the portion of the bets, [w.sub.i], is not publicly available, only an implied margin, M', can be calculated. If the odds were set based on the true probabilities (i.e., the fair odds), there would be zero expected gain for the bookmaker. Hence, in order to allow for an expected gain for the bookmaker, the odds are smaller than the fair odds, and thus, the implied probabilities are larger than the true probabilities. This results in the expected implied margin being calculated as

[mathematical expression not reproducible] (1)

where is the implied probability of the ith outcome.

In our context of Super Rugby betting, the three possible outcomes for a punter to place bets on a home win, draw, or away win. Hence, Equation (1) implies that should the punter place bets on all outcomes, with one bookmaker, there would be a certain loss equal to that bookmakers' margin or over-round. If the punter selects the maximum odds for each outcome over J bookmakers, then there may be an opportunity for arbitrage, should the margin on the synthesized book be negative

[mathematical expression not reproducible] (2)

where J is the set of all bookmakers' quoted odds and [d.sub.ij] the odds on outcome i quoted by bookmaker j. Should Equation (2) hold then the arbitrage profit is equal to -[??] with the optimal proportion of the stake (Franck et al., 2009) being

[mathematical expression not reproducible] (3)

placed on each ith outcome over all n outcomes.

Empirical Evidence

Data

The investigation conducted in this paper makes use of the closing odds from multiple bookmakers spanning the 2009 to 2012 Super Rugby seasons. A total of 438 games are analyzed over the four years. It is worth mentioning that the format of the Super Rugby competition changed after the 2010 season, with more teams being introduced and the structure of the tournament altered. Consequently, for the 2009 and 2010 seasons there were 94 games in each season, and then it increased to 125 games per season during 2011 and 2012. Odds were obtained from two South African local bookmakers, Mbet and Marshalls World of Sport, with the remaining data for odds offered globally extracted from Oddsportal.com. Hence, our data covers odds offered from bookmakers both locally in South Africa and internationally, with the majority from Europe. In addition, it is also expected that any results that suggest inefficiency could in fact be understated. Since the investigations in this paper are based on closing odds (i.e., the odds that were quoted immediately prior to the game), it would be expected that market forces would have already corrected any mispricing.

Closing Odd Investigations

Similar methodologies to those utilized by Vlastakis et al. (2009) and Franck et al. (2009) on the European football betting market are implemented in our paper on the Super Rugby betting market. In essence, this investigation measures the frequency and size of arbitrage opportunities embedded within Super Rugby betting. This is done in order to establish whether said mispricing exists, and determine if the market is indeed weak form inefficient. More precisely, if the market is efficient, it is not possible for the punter to systematically generate abnormal returns different to the margin. In the section that follows we will further investigate whether these arbitrage opportunities are practically exploitable.

The data supplied by Oddsportal.com indicated changes in the number of bookmakers, which quoted odds on games, between different matches. During the 2009 season, on average 19.2 bookmakers quoted on each game. This steadily increases over the seasons with 39, 8 bookmakers quoting on average during the 2012 season (See Table 1). The highest odds offered on each outcome of the game were considered over all the bookmakers quoting for each match. Using the definition of arbitrage explained in the preceding section, the games either had an arbitrage opportunity or not.

Table 1 illustrates that an arbitrage opportunity exists nearly one in every three trades. This strongly suggests that the Super Rugby betting market is in fact weak form inefficient. We conduct a formal hypothesis test to examine whether the average arbitrage profits are indeed significantly greater from zero (i.e., [H.sub.0]: [micro] = 0 against the alternative [H.sub.1]: [micro] = 0). The resulting t-statistics and the corresponding p-values for each season under evaluation are presented in Table 2. The results indicate that the average arbitrage profits are significantly positive at a 1% level of significance, further supporting the claim that the Super Rugby betting market is at least weak form inefficient across the individual seasons observed.

Interestingly, the total number of arbitrage opportunities does not appear to be directly proportional to the number of quoting bookmakers. A direct correlation would be expected when there are few bookmakers, however, it appears that this effect becomes negligible when a large number of bookmakers are considered. The lack of correlation between the number of arbitrage opportunities and the average number of bookmakers, as shown in our findings, also gives rise to the possibility that majority of mispricing comes from a subset of bookmakers. Hence, we analyse the amount of involvement of each individual bookmaker in the various European arbitrage opportunities discovered. Table 3 presents the amount of arbitrage opportunities offered by each bookmaker as a percentage of the total number of European opportunities discovered per season (we omit any bookmakers that did not offer arbitrage opportunities in the periods investigated). Interestingly, while the majority of arbitrage opportunities tend to involve only a small subset of bookmakers in the early 2009 and 2010 seasons, the spread becomes more diversified in the 2011 and 2012 seasons.

Utilizing equations (1) to (3), we analyze each arbitrage opportunity and the possible profits for the punter and present our findings in Table 4. It can be seen that, by taking combined bets over multiple bookmakers (as discussed previously), the average over-round is very close to zero (see Table 4). Therefore, it only takes a bookmaker with slightly different odds, for whatever reason, to create a situation where the over-round becomes negative, and an arbitrage opportunity is created. Surprisingly, our results also seem contradictory to the findings of Vlastakis et al. (2009) and Franck et al. (2009). Although Vlastakis et al. (2009) found arbitrage opportunities with returns as high as 200%, they were very rare and therefore a sign that they could be due to bookmakers making errors. Franck et al. (2009) found many inter-market (between bookmakers and the exchange) arbitrage opportunities, but very few intra-market arbitrages. In contrast, our results show that more arbitrage opportunities are realized in the Super Rugby betting market, albeit the majority of these arbitrage opportunities exhibited lower returns, with the average just below two percent (return on closing odds). However, it is worthwhile to highlight that the maximum arbitrage return found over the four seasons investigated was 12.62% for a single match. Notably, bookmakers will start to quote odds several days before the matches. Hence, such returns may be over a couple days or, since these are the closing odds, it could in fact be merely over a few hours.

From Table 1, we can also observe the significance of the area factor in creating an arbitrage. As Burkey (2005) suggests, each bookmaker is exposed to varied market forces. Different areas will offer majority support towards different teams, meaning if bookmakers choose to balance their book it may cause arbitrage opportunities to occur. The results from our investigation confirm such a claim. During most years there are more arbitrage opportunities arising from cross betting over continents than there are from betting within the continent. Since Australia and New Zealand have teams competing in the Super Rugby tournament, fans may force bookmakers within their respective countries to alter odds. This gives rise to the suspicion that if arbitrage traders were to also set up additional accounts in either Australia or New Zealand, in order to exploit quoted odds from locally based bookmakers, more opportunities may bediscovered.

In addition to the standard deviation of the returns per season reported in Table 3, we present the quartiles through abox whistar plot m Figurel to analyse the distribution of the arbitrage returns. Apart from the higher mean and greater dispersion, we also observe more extreme arbitrage re turns during the 2012 season in comparison toprior years. Figure 2 presents the box whisker plot to inrestigate the difference in returns between the different regions (i.e., intonationally and locally in South Africa). Our results also suggest that a number of high-return opportunities emerged from cross-continent arbitiage. Interestingly, the returns in the upper quartile resulted from opportunities involving the majoriry of bookmakers listed in Table 3. Finally, we test the hnpothesis that: the average returns between the two regions are different. We obtained a resulting p-value of 0.8068, suggesting no significant differences in the arbitrage returns across the two different regions.

Implementation of Practical Arbitrage Strategy

The Empirical Evidence section looked at 438 Super Rugby games spanning over four years and determined the total number of arbitrage opportunities available. The significant number of arbitrage trades, together with the significant results of our hypothesis testing, strongly suggests that the market is indeed weak form inefficient. In this section, we provide an empirical experiment by implementing a straightforward arbitrage strategy to demonstrate the possibility of exploiting the arbitrage opportunities in a practical trading environment. As discussed previously, if the market is indeed efficient, then such a strategy should not exist.

We demonstrate how the arbitrage opportunities may be exploited under both "perfect world" and "real world" scenarios. The former describes a framework whereby there are no transactional costs and no risk of a trade not being completed. It is noteworthy that these assumptions are not too far from reality (especially for the latter). Although, the biggest assumption made under such a perfect environment is that the accounts can be rebalanced after every trade. This allows a trader to enter every arbitrage trade that becomes available. Assumptions in the latter scenario are believed to be closest to(or includes at least all) actual market characteristics. In particular, we allow for a withdrawal charge of one percent and that the rebalancing of accounts only occurs after every second game. Finally, we also incorporate a 1% probability that a trade will not be secured (see Securing a Bet below). By overstating the conditions, the "replworld" scenario we propose here will encompass at least that of the resl trading environment. For example, the condition of punters having to rebalance each trading account due to limited capital already encompasses the big players in the market that are without such limitations. In addition, the probability of not securing a bet is derived from an overcompensation of the time duration required to place a bet online.

Further conditions and assumptions for our practical trading experiment under the two different scenarios are as follows.

Accounts. The previous study made use of over 30 bookmakers on average per game. In this model the reasonable assumption that an individual could only manage five accounts has been implemented. It should be noted that this is by far an over-estimation of trading restrictions, as an arbitrage trader may easily open more accounts if he/she so chooses in practice (see Securing a Bet below for further discussions). These five accounts were chosen in order to fulfill a few requirements, which were believed to maximize the likelihood of arbitrage occurrences. The first requirement was that the accounts span several geographical regions. As explained in the Empirical Evidence section, this is done in order to ensure that the bookmakers are exposed to varied market forces. This requirement was restricted further to the condition that there is no restriction of opening an account with any of the bookmakers. Secondly, the accounts were chosen under the subjective requirement that they are large and well-known. The larger bookmakers generally offer the most competitive odds with the smallest over-round. Finally, the last requirement is that the history of odds from each bookmaker is available for all four seasons under review. Such conditions led to Marshalls, Mbet, Bet365, Sportingbet, and Betfred being selected. The odds for Sportingbet were not available for all four seasons. Therefore, it is assumed that on March 25, 2012 that the Sportingbet account is closed and reopened with StanJames.

Withdrawal Charge. It is worth mentioning that in South Africa taxes are exempt on sports betting and the majority of bookmakers do not charge any administration fees (applicable for both Marshalls and Mbet). Hence, all the bookmakers chosen for this model do not charge any administration costs, besides Betfred who may charge a two percent withdrawal charge, depending on method of payment. Hence, this assumption is included for all bookmakers in order to over-estimate any expenses that could be incurred (e.g., bank charges). The withdrawal charge is assumed to be a percentage of the amount withdrawn from each account when rebalancing.

Rebalancing of Accounts. The adjustable assumption here is the number of trades that are made before rebalancing the accounts. When an arbitrage trade is made, this generally entails one account winning a bet, and either one or two other accounts losing. In most cases this results in some accounts emptying, as the maximum amount would have been gambled, in order to benefit fully from the risk-free return. Should another arbitrage trade become available before the accounts have been rebalanced there may not be enough funds in an account to benefit from such trade. In many cases, several games happen per day and therefore it would be unrealistic to assume that rebalancing of accounts is possible after every game. Notably, such a restriction is not applicable for large players in the market who are not subject to such capital limits.

Securing a Bet. When an arbitrage trade is discovered, a punt needs to be placed on every possible outcome. In our practical experiment it involves placing a bet on the home win, draw, and away win. If the odds were to be altered after placing an initial bet, but before completing all three bets, then the trade could become a risky gamble. However, with modern internet gambling and the infrequency of bookmakers altering pre-match fixed odds, the likelihood of this occurring is almost negligible. If the odds were to change, such that an under-round was no longer offered, the trader could choose to either still complete the trade, ensuring a small but manageable loss, or cease further action and therefore accept the gamble. Our practical experiment assumes the latter course of action. In order to calculate the probability of not securing the arbitrage trade, the probability of not securing each of the three individual trades in the combined bet is first calculated. To calculate an approximate probability, Marshalls 2012 Super Rugby odds are assessed. Referring to Table 5 below, it can be seen that that if the time required to place a single punt is grossly over-estimated at three minutes, then the estimated probability of not securing that punt is 0.1% (Similar results are obtained when using odds from other bookmakers). In practice, however, the time required to place a single punt for an experienced punter would only require a few seconds. It is worth mentioning that the advancement of super computers increases the possibility of traders handling large number of accounts. In addition, with the advent of algorithmic trading, the efficiency of trades has also improved significantly. These conditions allow traders to minimize the probability of not securing a bet and benefit from more possible arbitrage opportunities.

Choosing an Amount to Bet. When an arbitrage is discovered three punts are made leading to the same profit being realized regardless of the outcome. The stake on the [i.sub.th], outcome, [S.sub.i], is determined by the formula:

[mathematical expression not reproducible] (4)

where B is the total amount bet on the game, and [O.sub.i] the highest odds (stakes included) offered on the ith outcome. The model determines the stakes by increasing the total amount of bet, B, until it becomes unaffordable for an existing account (i.e., the maximum amount is traded on each game under the affordability constraints placed on the accounts).

Practical Trading Results

Notably, with the overstated assumption on number of account restrictions, we can expect less arbitrage opportunities. It is worth noting that under such assumptions, any arbitrage profits may be understated, as punters may easily open more accounts with the advancement of technology and the improved efficiency of the internet. Referring to Table 6, it can be seen that 12.33% of games produced arbitrage returns. This is a significant decrease from the 31.74% of games that produced arbitrage opportunities in the previous section. It should be emphasised that identical periods were examined under both investigations, and therefore the reason for the dissimilarity is due to the limited number of bookmakers being considered (overly reduced to only five in this experiment for robustness). Relaxing this rigorous assumption will result in more arbitrage trades being available.

Our experiment results show that arbitrage returns are over and above the loss made on the long-term over-round, signifying abnormal profits and suggesting market inefficiency. Moreover, it can also be seen that the arbitrage returns (converted into effective annual for ease of comparison) even outperform the benchmark risk-free rate (LIBOR) over the same time horizon. Such comparison with a risk-free investment in the financial market further emphasizes the significant arbitrage opportunities available in the betting market. Our results of the two scenarios tested are presented in Tables 7 and 8 below.

Perfect World. When we consider the setting whereby i) no transactional costs involved; ii) there is no risk that the trade will not be completed; and iii) betting accounts could be rebalanced immediately after each trade, the trader would experience returns as stated in Table 7. When returns are converted to effective annual rates to be compared to the benchmark risk-free rate (LIBOR) it is clear that very high relative returns are achieved. The relatively lower effective annual return earned over the total period of the investigation is due to the consideration of inactive periods between seasons, whereby no trading could take place.

Our results clearly show that arbitrage opportunities are in fact exploitable in the Super Rugby betting market under a perfect environment. However, the assumptions are not true representations of the real world, and therefore, in the sequel, results under more realistic assumptions are observed and analyzed.

Real World. When considering more realistic assumptions, including the overstating of certain conditions to account for limiting cases, the return that would have been earned by the trader is presented in Table 8. These returns, albeit slightly lower (as expected) than those achieved under a perfect environment (see Table 7), are still considered to be significantly higher than the benchmark risk-free rate. Nevertheless, our results clearly indicate that arbitrage opportunities are indeed exploitable in a "real world" setting. It is also worth mentioning that the arbitrage trading strategy greatly outperforms an asset with comparable risk in the financial market.

Over the course of the investigation, the average return on an arbitrage trade was 1.94%. It is important to emphasize that such returns can be made over just a few days, or even just over a few hours, due to closing odds data being used. Such a result suggests that an astonishingly high effective rate of arbitrage returns is exploitable in the Super Rugby betting market. Moreover, while the arbitrage returns are lower than those discovered in the football betting market on average (see Vlastakis et al., 2009), the number of exploitable opportunities is higher in the Super Rugby betting market.

The slight decrease in return compared to the perfect environment is due to the relaxing of rigorous assumptions. Firstly, every time the accounts are rebalanced, a withdrawal expense is charged, leading to lesser profits. Secondly, the over-estimation of rebalancing of accounts after every second game implies that often the accounts will not be set up to fully benefit from all possible arbitrage opportunities. An account may restrict a trade as it may be offering the highest odds on an outcome, and hence, be a critical part of an arbitrage trade, yet may also have reduced or insufficient funds. However, it is also noteworthy that such drawbacks do not affect large players in the market, who will have enough funds to avoid rebalancing of account and are not susceptible to withdrawal charges. Finally, there is a risk that a trade will not be secured. Over the four seasons, seven arbitrage trades (about 13% of available arbitrage opportunities) were missed due to funding restrictions, and two missed due to incomplete trades. Hence, the number of games with exploitable arbitrage opportunities is reduced to 8.33% of all matches played. However, as mentioned in Securing a Bet above, the probability of not securing a trade has been largely overstated to overcompensate, which renders lower arbitrage returns in our experiment as a result. With the advent of online gambling, algorithmic trading and the advancement of technology, such risk becomes negligible even for traders holding a large number of betting accounts.

Results from our empirical experiment clearly indicate the possibility of exploiting the existing arbitrage opportunities in the Super Rugby betting market. Even after allowing for rigorous assumptions to incorporate (or to go beyond) characteristics of the real world scenario, the realized returns were not only over and above the losses of the long-term over-round, but also significantly outshone the comparative risk-free investment in the financial market. We conclude that the Super Rugby betting market is indeed weak form inefficient, and the arbitrage opportunities are exploitable.

Conclusion

In this paper, we analysed the efficiency of the Super Rugby betting market. We extend the existing literature by analyzing the database of odds quoted by multiple bookmakers spanning the 2009 to 2012 Super Rugby seasons, and investigated the efficiency of the Super Rugby betting market. In addition, we conduct an empirical analysis to demonstrate that the arbitrage opportunity in the Super Rugby betting is indeed exploitable.

Our study was carried out in two folds. The first investigation used an average of 36.6 bookmakers over the 2009 to 2012 seasons to identify the existence of arbitrage opportunities in the market. Astonishingly, it was discovered that nearly one in every three games offers an arbitrage opportunity in Super Rugby betting, and that cross-continental forces also contribute greatly to the occurrences of such opportunities. In contrast to the popular European football betting market, which exhibits less arbitrage opportunities with higher returns on average, we discover an exorbitant amount of opportunities in the Super Rugby betting market, albeit having significantly lower returns on average. The empirical analysis, which followed thereafter, applied a straightforward arbitrage-trading model to the available data, where significant positive returns that outperform even the benchmark risk-free rate were achieved under varying assumptions. The results are particularly useful to demonstrate the possibility of exploiting the existing arbitrage for the layman without having to apply a sophisticated model. Prospective research may include exploring the most optimal and effective arbitrage model to implement. We conclude that arbitrage opportunities exist within Super Rugby betting, and are in fact exploitable under normal market conditions, demonstrating that the market is at least weak form inefficient.

Possible future avenues of research may include conducting more tests to identify the reason for the mispricing and the existence of arbitrage opportunities. In particular, identifying the underlying cause as to why market forces are not aligning odds. Some factors to consider may include, but are not limited to, the small bookmakers margin in Super Rugby, misestimation of "home field advantage" in bookmaker odds in favour of the home team, and international timing issues. Having small margins implies that its only takes a small mispricing in order for arbitrage to exist. While an overestimation of home field advantage, coupled with favorite-longshot biases, may lead to reverse "home-underdog" bias, as shown in the European football odds (see Vlastakis et al., 2009).

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Endnotes

(1) The former (perfect arbitrage) occurs in a strategy whereby the net gain of all secured bets is always non-negative and involves no risk of losing. The latter could incur losses, but occurs more frequently and can have a higher average return.

(2) The average over-round offered by Mbet and Marshalls World of Sport over the 2009 to 2012 Super Rugby seasons were 6.58 percent and 5.75 percent respectively (See Table 4).

(3) Such as the commonly known "home-ground advantage" effect on bookmakers.

(4) Notation: European decimal odds (i.e., Stake is included in odds so that even odds = 2). In South Africa the accepted notation is that the decimal odds do not reflect the stake (i.e., even odds = 1).

Matthew Buckle (1) and Chun-Sung Huang (2)

(1) School of Management Studies, University of Cape Town, South Africa

(2) Department of Finance & Tax, University of Cape Town, South Africa

Matthew Buckle is an alumnus of the University of Cape Town and a fellow actuary currently on sabbatical. His research focuses on sport economics.

Chun-Sung Huang is a senior lecturer in the Department of Finance & Tax and an associate of the African Collaboration for Quantitative Finance and Risk Research (ACQuFRR) unit. His research focuses on financial economics, econometrics, and stochastic processes.
Table 1. Statistical Summary of Super Rugby Arbitrage

Seasons                         Total    2012     2011     2010

Established number of games:    438      125      125      94
Total arbitrage opportunities:  139       49       29      35
European opportunities:          46       26       11       5
South African opportunities:     13        1        2       3
Cross-continental                80       22       16      27
opportunities:
Percentage of trades that are    31,74%   39,20%   23,20%  37,23%
arbitrage:
European arbitrage:              10,50%   20,80%    8,80%   5,32%
SA arbitrage:                     2,97%    0,80%    1,60%   3,19%
Cross-continental arbitrage:     18,26%   17,60%   12,80%  28,72%
Average number of                31,60    39,78    36,36   26,65
bookmakers:

Seasons                         2009

Established number of games:    94
Total arbitrage opportunities:  26
European opportunities:          4
South African opportunities:     7
Cross-continental               15
opportunities:
Percentage of trades that are   27,66%
arbitrage:
European arbitrage:              4,26%
SA arbitrage:                    7,45%
Cross-continental arbitrage:    15,96%
Average number of               19,21
bookmakers:

Table 2. Hypothesis Testing Results ([H.sub.0]: [micro] = 0, [H.sub.1]:
[micro] > 0)

Seasons:  2012           2011           2010           2009

t-stats   5,1346         4,0895         4,6375         4,1661
p-values  5,3167 X 10-7  3,8600 X 10-5  5,7496 X 10-6  3,4653 X 10-5

Table 3. Involvement of Individual European Bookmakers in Arbitrage
Opportunities

              Location   2012  2011  2010  2009

188Bet        UK          4%    0%    0%     0%
888Sport      UK         19%    0%    0%     0%
bet365        UK         35%    0%    0%     0%
Betclic       UK          4%   18%   40%     0%
Betfred       UK         27%   27%   40%     0%
Betsson       Malta       4%   18%    0%     0%
BetVictor     Gibraltar   8%    9%    0%     0%
Betway        Malta       4%    9%    0%     0%
bwin          Gibraltar  35%    0%   20%    50%
Interwetten   Malta       4%   18%    0%   100%
Ladbrokes     UK         27%    0%    0%     0%
NordicBet     Estonia    12%    0%    0%     0%
Sportingbert  UK         31%   36%   80%   100%
Tipico        Malta       4%    0%    0%     0%
Titanbet      Malta      19%   18%    0%     0%
Unibet        UK          0%    9%    0%    50%
William Hill  UK         27%   27%   40%     0%

Table 4. Statistical Summary of Super Rugby Arbitrage Returns

                               Total   2012     2011    2010    2009

Average under-round:           0,0172   0,0206  0,0173  0,0135  0,0158
Max under-round:               0,1120   0,1120  0,0732  0,0707  0,0577
Average return:                1,75%    2,10%   1,76%   1,36%   1,60%
Max return:                   12,62%   12,62%   7,89%   7,61%   6,13%
Std. Deviation of returns:     1,90%    2,42%   1,77%   1,39%   1,44%
Average Over-round:            0,0081   0,0024  0,0088  0,0084  0,0137
Average Marshalls Over-round:  0,0575   0,0579  0,5467  0,0630  0,0548
Average Mbet Over-round:       0,0658   0,0589  0,0604  0,0731  0,0739

Table 5. Assessing Probability of Not Securing a Trade

Assumptions
Minutes required to place punt:          3
Output
Number of Games:                      125
Avg minutes market open per game:    4435,2
Avg number of odd changes per game:     0,48
Avg minutes before market change:    2996,8
Probability of not securing a bet:      0,001

Table 6. Practical Model Result Summary

Number of games:             438
Number of arbitrage:          54
Arbitrage trades:          12,33%
Number of days per season
2009:                        106
2010:                        106
2011:                        141
2012:                        162

Table 7. Seasonal Returns of "Perfect World" Scenario

                                        Effective Annual  LIBOR

Total Return (over 4 years):  49,7487%  12,3257%          0,5614%
 2009 Season Return:           7,1645%  26,9046%          0,6884%
 2010 Season Return:           7,9398%  30,0942%          0,5549%
 2011 Season Return:           5,7179%  15,4815%          0,5623%
 2012 Season Return:          22,4565%  57,8449%          0,6532%

Table 8. Seasonal Returns of Best Estimate to "Real World" Scenario

                                         Effective Annual  LIBOR

Total Returns (over 4 years):  44,9508%  11,2777%          0,5614%
2009 Season Return:             6,5057%  24,2384%          0,6884%
2010 Season Return:             6,5546%  24,4347%          0,5549%
2011 Season Return:             4,5036%  12,0789%          0,5623%
2012 Season Return:            22,2206%  57,1606%          0,6532%
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Author:Buckle, Matthew; Huang, Chun-Sung
Publication:International Journal of Sport Finance
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Date:Aug 1, 2018
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