Measuring Efficiency in the Fixed Odd Football Betting Market: A...
Transcript of Measuring Efficiency in the Fixed Odd Football Betting Market: A...
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University of Maastricht
Faculty of Economics and Business
Administration
Maastricht, 7 December 2007
Sonnenschein, B.P.M.
I162205
Student International Business
Supervisor: Assistant Professor Bodnaruk, A.
Final Thesis
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By
Bart Sonnenschein
At the University of Maastricht
Abstract
Using data of odds placed on matches played in the Premier League for four consecutive seasons. This study
critically investigates the efficiency of the fixed odd football betting market. In contrast to prior studies this
study investigates several specific parts of the market and presents a detailed analysis of these specific markets to
detect market inefficiencies. The data is split into five sub samples that are tested for market inefficiencies.
These market inefficiencies consequently are tested for profitable trading strategies. The study’s results could
also help explain prior literature findings more accurately. In order to test for market inefficiencies the current
study uses three methods that are all based on the spread measure. Spread measures the difference between
realized probabilities of the odds minus implied probabilities of the odds. None of the study’s results are
significant. Profitable trading strategies appear to be minimal, in line with the study’s prior results. Although the
study does not find significant inefficiencies it does offer a nice and chronologic analysis for punters and other
investors, who want to place their bets in the best way possible. Additionally, the study finds quite some
evidence that the value or range of odds influences punters’ returns positively for low odds and negatively for
high odds. The study further shows that punters should definitely not place their money on away win matches
with high odds.
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Table of Contents
1. INTRODUCTION............................................................................................................................. 4
2. MARKET EFFICIENCY ................................................................................................................. 9
3. THE FIXED ODD BETTING SYSTEM....................................................................................... 12
4. THE FIXED ODD BETTING MARKET AND MARKET EFFICIENCY............................... 16
4.1 THEORETICAL EVIDENCE ON MARKET INEFFICIENCY ............................................................ 16
4.2 FURTHER EVIDENCE OF MARKET INEFFICIENCY IN THE FIXED ODD BETTING MARKET ...... 21
5. TESTS OF MARKET EFFICIENCY ........................................................................................... 27
5.1 THE SPREAD BETWEEN IMPLIED PROBABILITIES AND REALIZED PROBABILITIES ................ 27
5.2 Efficiency for whole sample home win odds.................................................................................... 31
5.3 Further tests of market efficiency for the remaining sub samples............................................. 49
6. TRADING STRATEGIES.............................................................................................................. 57
7. CONCLUSION................................................................................................................................ 61
8. LIST OF REFERENCES ............................................................................................................... 66
9. LIST OF FIGURES......................................................................................................................... 69
10. LIST OF TABLES......................................................................................................................... 70
11. APPENDICES ............................................................................................................................... 71
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1. Introduction
Historically there is a consistent interest among scholars concerning the particulars of betting
markets. Two of the most pronounced streams of research within the betting market literature
are the efficiency of betting markets and the possibility to create profitable betting
opportunities based on some specifics of the betting market. Kuypers (2000) and 1×2Betting
(2007) for example both indicate in their research that market inefficiencies exist within the
fixed odd betting market that could offer profitable betting opportunities. Most of these
articles show that theoretically inefficiencies should exists in the betting market because
bookmakers take advantage of punters’ reaction functions. Palomino, Renneboog and Zhang
(2005) investigate whether significant abnormal returns can be generated by testing stock
price reactions of listed soccer clubs to the information embedded in the betting odds placed
on the matches of these soccer clubs. Several other authors more specifically link different
types of news to soccer clubs that are listed on exchanges. Most of these studies investigate
the relationship between soccer game results and stock market price reactions of listed
companies (e.g. Palomino, Renneboog & Zhang, 2005; Ashton, Gerrard & Hudson, 2003).
The study of Palomino, Renneboog and Zhang. (2005), however, specifically links the betting
market to a club’s financial performance measured via the stock returns. Palomino et al.
(2005) further claim that the odds represent experts’ opinions on game outcomes and hence
inform investors on a weekly basis. Furthermore, Palomino et al. (2005) find that odds are
excellent predictors of game outcomes and therefore should quite naturally influence a club’s
stock prices. Game-outcome related information, e.g. the information embedded in odds,
should have a direct relation to a club’s stock prices. This relation between the financial
performance of a club or its stock returns and the game results or game-outcome related
information is evident. One could think of the proceeds reaped from national TV deals, which
are distributed in England according to a performance-based scheme (Falconieri, Palomino &
Sakovics, 2004). One could think of promotion to the Premier League or playing in the
Champions League, which bring about more revenues. Additionally, good game results may
increase ticket sales, merchandise or sponsor deals (Palomino & Sakovics, 2004). For all
these reasons one would expect investors to perceive the game-outcome related information
embedded in odds as stock price information.
Palomino, Renneboog and Zhang (2005), however do not find a significant relationship
between odd information and stock price reaction. Palomino et al. find that stock markets
react strongly to news about game results. They however do not find a significant
relationship, neither in share prices nor in trading volumes, to the release of betting odds.
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Palomino et al. find this surprising as betting odds are excellent predictors of game outcomes.
They explain this result by indicating that odd prices form a non-salience type of information,
which is therefore not reflected in a club’s share prices. Furthermore the authors argue that
game results receive very high media coverage, whereas betting odds come less under the
attention of the audience. Palomino et al. further conclude that non-salient information is
neglected by investors. This implies that due to the absence of investors incorporating the
news into the stock prices, the information embedded in the odds can be used to predict short-
run market returns. This paper however argues that the absence of a market reaction to the
disclosure of betting odds, may be due to the fact that Palomino et al. do not focus on specific
inefficiencies in the betting market that were not modeled in their paper.
The current finance study finds the research by Palomino et al. extremely interesting and
believes that the non-significant relationship maybe due to specific inefficiencies that exist
within the betting market that have not been modeled in the relationship between the
information embedded in odds and a club’s stock price information. The current study,
however, finds the salience explanation weak and rather argues that inefficiencies in the
betting market may explain the insignificant result. Furthermore, the market may be aware of
the fact that bookmakers are able to set inefficient odds. The inefficient odds are consequently
not reflected in the soccer clubs’ share prices. Alternatively, one may argue that the market or
investors are simply reluctant to incorporate the odds due to a well-known phenomena of
human beings’ need for wealth maximization and greed. Instead of offering valuable
information, the market may perceive part of the odds as bookmakers’ personal means of
gaining wealth. Not only would a critical and more specific assessment of the efficiencies or
inefficiencies of the betting market help explain findings of prior research better, e.g. the one
by Palomino et al., a more specific and detailed analysis of inefficiencies that exists in
particulars segments of the betting market could help identify better profitable trading
strategies for punters as well. Most prior studies indicate that the betting markets are
inefficient but then lack the detailed breakdown of the specific betting market inefficiencies
that could lead to more profitable trading strategies for punters and the like (e.g. Kuypers,
2000).
The whole discussion relates to the literature on market efficiency mostly set out by Fama
(1970). Fama defines an efficient market as a market whose prices fully reflect all the
available information. This implies that market inefficiency would result in profitable betting
opportunities. Furthermore, as will become clear in a subsequent part of this paper, this means
that bookmakers can increase their expected profits by setting market inefficient odds. Prior
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literature has debated quite extensively the efficiency of information markets. Information
markets can either be financial markets as well as betting markets. This paper investigates a
specific betting market, namely the fixed odds betting market, i.e. bets placed on soccer
matches played in the English leagues. Only few authors have researched the fixed odd
betting market (e.g. Kuypers, 2000), which makes the topic at hand even more fascinating.
Many authors have also rejected the efficient market hypothesis in favor of the inefficient
market hypothesis (see Figlewski and Wachtel, 1981). Many of these authors argue that
inefficient markets are due to agents that employ information in an inefficient way.
The current study thus focuses on the efficiency of the betting market. More specifically it
focuses on a more critical and specific assessment of the efficiency of the betting market,
something prior research neglected somewhat. Based upon this information the paper reveals
whether investors or punters can create profitable trading strategies out of this information.
Additionally, some of the specific results could be used to explain prior study’s findings. In
order to test these specific elements of the betting market. The fixed-odd betting system of
football in Great-Britain is used, which uses bookmaker experts to generate betting odds for
the games to be played in the next few days. The fixed odd betting market is scarcely
researched by scholars but nevertheless provides several characteristics that make it an
interesting market for empirical investigation. First, these markets give detailed price and
outcome information on regular time intervals and second the odds are fixed in advance and
do not move in response to betting before the event (Kuypers, 2000).
Information concerning the odds is collected for Premier League matches for the seasons:
2002-2003; 2003-2004; 2004-2005 and 2005-2006. To more critically assess possible
inefficiencies that exist within the fixed odd football betting market. The collected sample is
further sorted into several sub samples that may reveal specific inefficiencies in the fixed odd
football betting market that may later be tested in other betting markets as well and yield
some profitable trading strategies for punters or investors. The sub samples analyze the whole
sample of odds, the odds placed on the big five teams in the League, the odds placed on newly
promoted teams to the League, the odds placed on teams with large followings and lastly the
odds placed on team with obscure followings. These sub samples are then further divided into
home and away win odds and into lower and higher value odds.
The current paper’s main idea to more critically assess specific inefficiencies in precise parts
of the fixed odd betting market, may help explain why bookmakers set market inefficient odds
in specific areas and why not in others. This kind of information, however, can only be
revealed once significant market inefficiencies are found. Another explanation of inefficient
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odds that may be tested once inefficiencies are found deals with cash flows or dollar volumes
placed on specific bets in specific parts of the fixed odd football betting market. Furthermore,
betting odds set up by bookies do not reflect the true or unbiased probabilities of game
outcomes, because bookies take into account not only the probabilities of game outcomes, but
also expectations about the dollar volume put on each outcome. Since many people put their
money not with their brains, but with their hearts dollar volumes do not split according to the
expected efficient probabilities of the game outcomes. Bookies, therefore, adjust their odds to
account for that. Henceforth, in order to extract information from the odds one has to adjust
the implied probabilities derived from betting odds for the dollar volumes (Bodnaruk,
Personal communication, October 2007). It may be interesting to see whether these so-called
cash flows differ between specific segments of the betting market. This would then add
further robustness to punters’ trading strategies, who want to take advantage of inefficiencies
that exist in the betting market due to these cash flows. This in turn may be an explanation
why Palomino et al. (2005) do not find a significant relationship between a market reaction
and betting odds. However, in order to research this correctly, the current study must find
some significant inefficiencies in the betting market first.
Overall the study tries to discover whether market inefficiencies exist within precise parts of
the betting market and for specific characteristics of the odds. Based upon this information it
then tries to identify profitable trading strategies for punters. Additionally, some of the
findings could be used to explain prior research findings more accurately and to explain why
odds are priced inefficiently in certain areas of the betting market and why not in other areas,
e.g. the cash flow explanation. In order to research this correctly we have to answer several
more questions. What is the exact meaning of market efficiency in this context? What are the
characteristics of the fixed odd betting market? How does the efficient market hypothesis
relate to the fixed odd betting market and does theory argue that based upon this relationship
profitable betting opportunities can be created? Are the possible fixed odd betting market
inefficiencies significant? Are the inefficiencies significant enough to result in profitable
trading strategies?
To investigate the problem statement and the related sub questions the thesis outline is as
follows. Chapter 2 briefly discusses the efficient market hypothesis to properly define the
meaning of market efficiency used in this paper and relates it to the efficiency of the betting
market. Chapter 3 explains the principles behind the fixed odd betting system of the among
others English football matches. Chapter 4 relates the fixed odd betting market to market
efficiency and indicates how profitable betting opportunities may be exploited. Chapter 5 and
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6 test all the current study’s sub samples for market inefficiencies. Finally, chapter 7
concludes, links the findings to the problem statement, limitations of the research
methodology are addressed and suggestions for future research are given. Furthermore, a
tentative answer to the propositions will be given, based upon the study’s findings
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2. Market efficiency
In one of the most influential papers of the last decades, Fama (1970), presents a coherent
picture of the main issues on efficient markets. In general terms the efficient market
hypothesis investigates whether prices at any point in time reflect all available information.
Fama conducts three types of tests of the efficient market model. The first test is titled the
weak form and tests whether prices, e.g. security prices, reflect historical prices or return
sequences. The second test is titled the semi-strong form and tests whether prices are assumed
to fully reflect all obviously publicly available information. Fama finds significant evidence
that both support the weak and semi-strong form of market efficiency. The last form of
market efficiency is titled strong-form market efficiency. Evidence in favor of strong-form
markets would mean that prices reflect all available information. This implies that specialists
or insiders could not use any monopolistic access to information and use this information to
generate trading profits because prices would already reflect this type of information. Fama
uses this test of market efficiency as a benchmark against which deviations from market
efficiency can be judged.
This latter definition of market efficiency is especially interesting for our purposes as it would
mean that none of the players involved in the betting market could make any additional profits
due to some kind of monopolistic access to information. It would also mean that clubs´ share
prices fully reflect all the information embedded in odds. For matters of convenience this
paper does not refer constantly to the different types of market efficiency, but simply labels an
efficient market as a market that fully reflects all available information. Nevertheless it is
important to keep in mind the different types of market efficiencies for the rest of this study
and they will briefly be explained with respect to the betting market as well later on.
Another important subject that Fama (1970) discusses in his paper on market efficiency, are
the market conditions consistent with efficiency. Furthermore as this study later discusses the
specifics of the betting market, i.e. the fixed odd betting system, it is important to recognize
whether all the conditions are available for an efficient market to exist. Fama stresses several
market conditions that help or hinder efficient adjustments to prices. According to Fama
security prices reflect all available information when there are no transaction costs, all
available information is costless and available to all market participants and when all the
market participants agree on the implications of current information for the current price and
distributions of future prices for each security. Fama argues that if all these conditions are
present it would quite naturally bring about market efficiency, but the author further argues
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that not all of them have to be present for an efficient market to exist. As an example one
could mention transaction costs that inhibit the flow of transactions. Furthermore although
there may be high transaction costs in a particular market this does not necessarily mean that
the prices will not fully reflect all available information. Similarly, Fama indicates that market
efficiency can exist even if information is not freely available to all investors or if there exists
some disagreement about the implications of some kind of information. For the current
study’s discussion on market efficiency in combination with the betting market, or more
specifically the pricing of odds, the discussion on transaction costs is an important item to
keep in mind for the remainder of this study. Although factors such as transaction costs may
not necessarily inhibit a market from pricing efficiently, they are potentially sources of market
inefficiency.
This study focuses on the efficiency of betting markets and in particular the fixed odd betting
market. The subsequent part of the paper will dedicate a discussion on the specifics of the
fixed odd betting system but for now it is important to understand what efficiency or
inefficiency exactly means in the fixed odd betting market. Kuypers (2000) investigates in his
paper the efficiency of this fixed odd betting market and tests how market participants utilize
the available information. Kuypers finds that a profit maximizing bookmaker may set market
inefficient odds . The market inefficiency may subsequently lead to profitable betting
opportunities. Similarly, this paper tries to identify, based upon a dataset consisting of bets
placed on predominantly football matches in the Premier League, whether odds are set
efficiently. This is an important first step to realize before investigating whether profitable
trading strategies can be realized, as much of the prior literature on betting market efficiency
leads to inconclusive results. This is shown in table 1, which summarizes the investigation of
betting market efficiency. The table further shows that the results are mixed.
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The table also clearly portrays that the different studies utilize different types of tests of
market efficiency. Kuypers (2000) explains what these different forms of market efficiency
exactly imply in a betting market. Weak form efficiency in a betting market implies that it is
impossible to obtain abnormal returns by using just price information, i.e. the odds. This holds
for both the punter as well as for the bookmaker. For matters of completeness it is important
to mention that Kuypers defines abnormal returns as returns different from the bookmaker’s
take. The next chapter will further clarify the odd price setting system by introducing a
numerical example. According to Kuypers semi-strong efficiency implies that no abnormal
returns can be achieved with the usage of publicly available information for both the punter
and bookmaker. More specifically it means that incorporating publicly available information
does not improve the accuracy of outcome predictions based on odds. Strong form efficiency
in a betting market context implies that no group in society can make abnormal returns. The
information content encompassed in private information would thus not help in reaching more
accurate outcome predictions based on these odds. The subsequent chapter introduces the
specifics of the betting market or system, which is necessary to comprehend, for
understanding the tests thereafter that deal with market efficiency and the fixed odd betting
system.
Table 1. Betting market efficiency in the literature
Table 1. Betting market efficiency. From: Information and efficiency: an empirical study of a
fixed odds betting market (p. 1354), by T. Kuypers, 2000, London: Routledge.
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3. The fixed odd betting system
Betting in England is huge and everyone must have seen the commercials for betting on the
billboards next to the football fields in especially the English premier league. The spurs these
betting companies give, to induce customers to start betting is enormous. Although the betting
companies and culture are salient within the English society the principles behind the fixed
odd betting system are less clear cut. In order to fully understand the remainder of this study it
is therefore wise to briefly discuss the essentials of the fixed odd betting system.
The system uses the expertise of bookmakers to come up with game outcomes in the English
and Scottish leagues a couple of days before the matches. Since the betting system is a fixed
odd betting system, the odds are fixed several days prior to the match and it is extremely
unlikely that the odds will change during these days. This betting system is therefore different
from other betting systems in which the odds are not fixed but react to the amount of money
bet on each outcome up to the start of the respective match or any other event, bets could be
placed on (Palomino et al., 2005). An example of such a different betting system is the one
used in the U.S. that is often called a pari-mutuel system. Furthermore, in the U.S. pari mutual
markets have been the principal means of wagering on horse races due to the fact that state
prohibitions on bookmaking were passed in the beginning of the twentieth century (Sauer,
1998). Another common type of betting in the U.S. is point spread betting. In such a system
the payoff depends on the difference in points scored by the two opposing teams. It is beyond
the scope of this study to scrutinize in detail the different types of betting systems and we will
therefore focus our attention solely on the fixed odd betting system and only compare it to
other betting systems when useful.
It is important to know that betting markets fulfill two functions. It can namely be looked at as
an information market and as a service market. The information market can simply be
compared to the markets in stocks and shares. The betting market also fulfills the function of a
service market because it gives punters the opportunity to bet (Kuypers, 2000). The study
previously mentioned the bookmakers’ take, which is simply the price customers or punters
pay for the facilities bookmakers create to bet. The prices in the information market are the
relative odds. The revenue bookmakers receive for creating the betting possibilities is again
unique to the fixed odd betting system. Furthermore, in pari-mutuel betting, bookmakers
receive a predetermined percentage of the whole betting pool to cover the bookmakers’ costs.
The residue is given to winning bettors in proportion to their bet stakes (Sauer, 1998). The
revenue bookmakers make, the bookmakers’ take, in fixed odd betting is measured by the
over-roundness of the book. The over-roundness represents the bookmaker’s gross margin
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(Palomino et al., 2005). To further clarify the principles of the fixed-odd betting system the
following numerical example in which a model of bookmakers’ odds setting decision is
replicated is given.
To understand the fixed odd betting system we can use the study’s dataset. The study uses
odds, of the company Bet365, placed on fourteen football clubs active in predominantly the
Premier League. Most of the terms used can best be illustrated by an example. On December
3rd
, 2005, Manchester United played Portsmouth. The bet365 odds placed on this match are
1,2 home win, 5,5 draw and 17 away win. Thus one pound placed on a home win would result
in a 1,2 pound return if the bet proved correctly. Notice that these odds are notated in
European format1. Obviously a home win means that Manchester United wins and an away
win means that Portsmouth wins. Subsequently the percentages can be calculated from these
odds:
Home win (100/ 1,2) * 100 = 83,3 %
Draw (100/ 5,5) * 100 = 18,2 %
Away win (100/ 17) * 100 = 5,9%
Total probabilities: 83,3% + 18,2% + 5,9% = 107,4%
The sum of the three probabilities is larger than 100%, which is due to the over-roundness or
the bookmaker’s gross margin. True or correct probabilities can be calculated by dividing
each probability by the sum of all three the probabilities, which in this case equals 107,4%.
This leads to the following true probabilities:
Home win 83,3% / 107,4% = 77,6%
Draw 18,2% / 107,4% = 16,9%
Away win 5,9% / 107,4% = 5,5%
Total probabilities: 77,6% + 16,9% + 5,5% = 100%
The correct probabilities naturally lead to 100%. For matters of calculus convenience the
current study uses a slightly different approach in calculating the true probabilities, which is
1 One can use the following formula to switch from odds notated in English format to odds notated in European
format: 1+ English format odd. An 5/6 odd thus results in an 1,833 European format odd.
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stipulated below. Keep in mind that literature often refers to these true or correct probabilities
as implied probabilities. The remainder of this study also uses the term implied probabilities
to indicate the odds true or correct probabilities.
The over-roundness of this match can be calculated by adding the percentages and subtracting
it by 100. This leads to an over-roundness of approximately 7,4%. A balanced book means
that the bookmaker takes stakes on the three outcomes in the proportion 83,3; 18,2 and 5,9
(Kuypers, 2000). In this manner the bookmaker will keep 7,4% and will be guaranteed of a
return of 7,4/ 107,4 = 6,9% of the total stake (Kuypers, 2000). Prior literature has claimed
that the average over-roundness of football fixed odds is remarkably constant at around 11,5%
(Kuypers, 2000). The over-roundness of this match seems remarkably small. Overall the
average over-roundness in the study’s sample of 1848 games is 9,69 % with a standard
deviation of only 1,7%. The over-roundness of the current study’s data is therefore lower
than what is indicated by prior literature sources, this may be due to differences between the
betting companies used in the different studies or due to the recent explosion in gambling,
which increased the competition on the bookmakers. Nevertheless the current study calculates
the implied probabilities by assuming that the book is fixed and is 9,69%. In order to calculate
the implied probabilities the formula below can be used (Kuypers, 2000). Note that this
formula translates the prior odds to sum to 100 % and thus presents the true game outcome
expectancies.
( )oddsyprobabilitimplied
0969,1
1=
Plugging in the odds of the Manchester United versus Portsmouth game results in the
following implied probabilities 75,97%; 16,58% and 5,36% for respectively a home win,
draw and away win. Ideally these implied probabilities should sum to 100% and the over-
roundness should be zero. Summing the percentages however leads to an overall percentage
or total implied probability for the match of approximately 98%. Total implied probabilities
of less than 100% are counterbalanced by total implied probabilities of more than 100%,
which is obviously due to the fact that the study uses the average over-roundness of the
sample of 9,69% in the denominator of the implied probability formula. Furthermore
recalculating the total implied probabilities by using the implied probability formula above
results in a total implied probability of 100%, i.e. an over-roundness of 0, and a standard
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deviation of 1,58%. Naturally this standard deviation explains the total implied probabilities
of matches that do not sum to 100%.
In the previous chapter we defined abnormal returns as returns different from the
bookmaker’s take. More specifically abnormal returns can now be specified as returns better
than the bookmaker’s take. The bookmaker’s take for our dataset is 9,69%. This implies that
abnormal returns are returns better than 9,69%. The following chapter further specifies the
exact relationship between the fixed odd betting system and efficient markets.
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4. The fixed odd betting market and market efficiency
This chapter interrelates the fixed odd betting market and market efficiency. The first part of
the chapter provides theoretical evidence why bookmakers may set inefficient odds. The
second part of the chapter combines theory with a more pragmatic view of market
inefficiency in the betting market.
4.1 Theoretical evidence on market inefficiency
In his work on information and efficiency, Kuypers (2000) presents a model based on the UK
football betting market. The model provides theoretical evidence that bookmakers can set
odds inefficiently to increase their expected profit. The model consists of three decision
points. These three points are the bookmakers who decide to quote odds, the punters who
have to decide on which odds to bet and finally the outcome of the game. The model assumes
that the market is semi-strong efficient, more specifically the model assumes that bookmakers
have no private information but can evaluate publicly available information. Below the main
points of Kuypers’ model are replicated and applied to the current study’s example given in
the previous chapter. Kuypers’ model incorporates the reaction functions for punters’ decision
on which outcome to bet. Similar to the example above, punters can bet on three outcomes.
These outcomes are a home win, draw and an away win. These outcomes are respectively
denoted by the subscripts 1, 2 and 3. The bookmaker’s return or handle is denoted by H and
the amount bet on each of the possible game outcomes as h1, h2 and h3. In order to calculate
the bookmaker’s expected profit, Kuypers introduces the following variable that represents
the share of the handle on each game outcome:
H
hs 11 =
H
hs 22 =
H
hs 33 =
Subsequently the bookmaker’s subjective probabilities of the possible game outcomes are
presented by b1, b2 and b3. The sum of these subjective probabilities naturally is 1. Obviously
the model also introduces the bookmaker’s posted odds, which are indicated by o1, o2 and o3.
In the previous chapter we saw that the over-roundness of the study’s whole sample is 9,69%,
which is calculated by using the formula below.
0969,1111
321
=++ooo
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Subsequently the implied probabilities from the odds are necessary for Kuypers’ model and
were calculated in the previous chapter by applying the formulas below. Again, summing the
implied probabilities should lead to 1, i.e. for the whole sample the average of this sum is 1.
)(0969,1
1
1
1o
d = )(0969,1
1
2
2o
d = )(0969,1
1
3
3o
d =
Before the expected profit function is shown it is important to know that Kuypers (2000)
assumes that punters accept the over-roundness set by the bookmaker and that the punter’s
reaction functions, how they spread their bets, are only used in the model to determine the
share of the handle on each outcome. Additionally, Kuypers assumes that bookmakers
understand punters’ reaction function and that the bookmakers are risk neutral and want to
maximize expected profits. The expected profit function for the bookmaker is:
[ ] [ ] [ ]333322221111)( hohbhohbhohbH +−+−+−=ΠΕ
The terms between brackets indicate that punters receive the amount bet on each outcome
multiplied by the odd and their original stake. Overall the formula indicates that bookmakers
receive the handle less their subjective probabilities of each possible game result times the
accompanying payout for each possible game result. Kuypers (2000) then continues by
rewriting the bookmaker’s expected profit function. Thereby taking into account the
following equations: hi = Hsi and 10969,1
1−=
i
id
o .
+
−−
+
−−
+
−−=ΠΕ 3
3
332
2
221
1
11 10969,1
11
0969,1
11
0969,1
1)( Hs
dHsbHs
dHsbHs
dHsbH
Kuypers (2000) uses one more equation to arrive at his ‘main’ expected bookmakers’ profit
function. The study already indicated that the sum of the implied probabilities should lead to
1. Kuypers therefore argues that d3 = 1- d1- d2. Incorporating this into the above bookmaker’s
expected profit function leads to the following final profit function:
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( )( )21
33
2
22
1
11
10969,10969,10969,1 dd
Hsb
d
Hsb
d
HsbH
−−−−−=ΠΕ
The expected profit function indicates that bookmakers try to maximize their profit via the
punters’ reaction function. The formula further indicates that bookmakers try to maximize
profits by setting implied probabilities, which are the bookmakers’ decision variables.
Kuypers (2000) then argues that the share bet is a function of the implied probabilities and the
distribution of punters’ subjective probabilities over the possible game results. Kuypers uses
this to further rewrite the bookmaker’s profit function to indicate that in order for the market
to be efficient the implied probabilities from the odds should be equal to the bookmakers’
subjective probabilities. More concrete this means that di equals bi. The next few equations in
Kuypers work show that the market need not be efficient. A small numerical example, based
upon the current study’s previous example, further clarifies that expected profit maximizing
implied probabilities need not be equal to the subjective probabilities of bookmakers.
In the previous chapter there was a numerical example given based upon the match
Manchester United versus Portsmouth. The odds were 1,2 home win; 5,5 draw and 17 away
win. This led to the following implied probabilities of respectively a home win, draw and
away win: 75,97%; 16,58% and 5,36%. These implied probabilities are based upon the
average over-roundness of the current study’s dataset of 9,69%. For ease of calculation these
implied probabilities are slightly modified so as to sum to 100%. Furthermore summing the
implied probabilities should by definition lead to 1 or 100%. The implied probabilities in the
remainder of this example are therefore modified into 76%; 17% and 7% for respectively a
home win, draw and away win. The numerical example assumes that there are ten punters, six
Portsmouth fans and four neutrals. The example further assumes that the Portsmouth fans are
slightly biased and ascribe better changes to a draw or Portsmouth win than the implied
probabilities would suggest. Similar to Kuypers (2000) model punters follow the following
betting rule:
idppi
idpdpi
iii
iiii
∀==
∀≠−=
)max(arg
)max(arg
The first betting rule implies that punters try to maximize the difference between their
subjective probabilities (pi) and the implied probabilities (di). The second betting rule implies
that punters will bet on the most likely event in case their subjective probabilities equal the
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implied probabilities. The subscript i, again represents the possible game outcomes, i.e. 1 =
home win, 2 = draw and 3 = away win. In contrast to Kuypers (2000), the current example
changes the probabilities for all game results and is therefore slightly more realistic.
The six Portsmouth punter fans believe that Portsmouth has a better change to win or play a
draw than the implied probabilities set by the bookmakers would suggest. Furthermore the
Portsmouth fans have subjective probabilities of p1port = 0,68; p2port = 0,21 and p3port = 0,11.
The neutral fans share the same thoughts as the bookmakers and therefore follow the
subjective probabilities of the bookmakers, i.e. b1 = p1neut = 0,76; b2 = p2neut = 0,17 and b3 =
p3neut = 0,07. If the bookmaker would set the market efficient level of odds this entails that
bookmakers’ subjective probabilities are equal to implied probabilities, i.e. bi = di. The
implied probabilities are therefore d1 = 0,76; d2 = 0,17 and d3 = 0,07. The following table
nicely tabulates all the probabilities mentioned thus far and the direct consequences of the
probabilities based upon the betting rules given above.
Table 2. Finding the punters’ betting shares
Neutral punters’
subjective
probability equals
bookmakers’
subjective
probability
Market Efficiency:
Implied
probabilities equal
bookmakers’
subjective
probabilities
Portsmouth punters’
subjective probability
Portsmouth
punters’ share bet.
Based upon
decision rule:
i = arg max(pi-di)
Neutral punters’
share bet.
Based upon
decision rule:
i = arg max(pi)
p1neut = b1 = 0,76 d1 = b1 = 0,76 p1port = 0,68 0,68-0,76 = -0,08
s1 = 0
s1 = 0,4
p2neut = b2 = 0,17 d2 = b2 = 0,17 p2port = 0,21 0,21-0,17 = 0,04
s2 = 0,3
s2 = 0
p3neut = b3 = 0,07 d3 = b3 = 0,07 p3port = 0,11 0,11-0,07 = 0,04
s3 = 0,3
s3 = 0
The fourth column shows Portsmouth punters’ share betting decisions. These punters try to
bet on the outcome that maximizes the difference between their subjective probabilities and
their implied probabilities. The outcome of their subjective probability of a draw minus the
implied probability of a draw is equal to the outcome of their subjective probability of an
away win minus the implied probability of an away win. It is therefore assumed that
Portsmouth punters equally divide their bets among a draw and an away win, which is
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indicated by s2 = 0,3 and s3 = 0,3. Remaining are the four neutral punters in column five. The
table clearly indicates in the first column that neutral punters’ subjective probabilities are
equal to bookmakers’ subjective probabilities. Consequently, neutral punters cannot maximize
the difference between their subjective probabilities and implied probabilities and therefore
bet on the most likely event, which is indicated by s1 = 0,4.
Remember that the bookmaker’s expected profit function is given by the following formula:
( )( )21
33
2
22
1
11
10969,10969,10969,1 dd
Hsb
d
Hsb
d
HsbH
−−−−−=ΠΕ
The only unknown variable in this function is the handle of the bookmaker, which is the
bookmaker’s return of the total stake. The current study’s over-roundness of the sample is
9,69%. The handle of the bookmaker therefore becomes 9,69/ 109,69 = 8,83%. Plugging in
the numbers in the formula above leads to the following bookmaker’s expected profit:
( ) 78,007,00969,1
3,083,807,0
17,00969,1
3,083,817,0
76,00969,1
4,083,876,083,8 =
×
××−
×
××−
×
××−=ΠΕ
The bookmaker, however can also choose to set odds that are not the market efficient level of
odds. The bookmaker can set odds that take into account the bias among Portsmouth punters,
who believe that Portsmouth has better changes to play a draw or win than the implied
probabilities suggest. The bookmaker could for example set odds in such a manner that the
following implied probabilities would result: d1 = 0,72; d2 = 0,19 and d3 = 0,09. These implied
probabilities are incorporated in table 2. The resulting table 3 is shown below, which indicates
how the punters would bet with these new odds or new implied probabilities.
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Table 3. Finding punters’ betting shares using new implied probabilities
Neutral punters’
subjective
probabilities
Portsmouth punters
subjective
probabilities
New implied
probabilities
Portsmouth punters’
share bet.
Based upon
decision rule:
i = arg max(pi-di)
Neutral punters’
share bet.
Based upon
decision rule:
i = arg max(pi-di)
p1neut = 0,76 p1port = 0,68 d1new = 0,72 0,68-0,72 = -0,04
s1 = 0
0,76-0,72 = 0,04
s1 = 0,4
p2neut = 0,17 p2port = 0,21 d2new = 0,19 0,21-0,19 = 0,02
s2 = 0,3
0,17-0,19 = -0,02
s2 = 0
p3neut = 0,07 p3port = 0,11 d3new = 0,09 0,11-0,09 = 0,02
s3 = 0,3
0,07-0,11 = -0,04
s3 = 0
The table clearly indicates that with the new odds the punters’ share bets are identical to when
the bookmaker chooses the efficient level of odds. The bookmaker’s expected profit function
becomes:
( ) 39,109,00969,1
3,083,807,0
19,00969,1
3,083,817,0
72,00969,1
4,083,876,083,8 =
×
××−
×
××−
×
××−=ΠΕ
The difference between the bookmaker’s expected profit when setting the market efficient
level of odds and the bookmaker’s expected profit when setting market inefficient odds is
1,39-078 = 0,61. Simply by using the punter reaction function the bookmaker is better of by
setting market inefficient odds. The model above, therefore, offers theoretical prove that odds
may be set inefficiently in practice.
4.2 Further evidence of market inefficiency in the fixed odd betting market
Further evidence of market inefficiency comes from a 1×2betting company paid system
document, titled 1×2Betting’s Value Hot Favourites Betting System, which describes a
method of making small but regular profits over the long term by making use of market
inefficiencies in match betting odds (1×2Betting’s Value, n.d.). The report sets of by
indicating that there is a general belief among punters and many betting experts that betting
on underdogs will result in greater returns in the long run than betting on favorites. These
betting experts and punters claim that if one has the patience to wait for the surprising results
to occur, betting on these underdogs will pay of in the long run, because odds from underdogs
return much more to the punter if he or she is correct due to the higher quotient on these odds.
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They back their beliefs by claiming that the majority of punters bet on favorites and that
therefore the bookmaker has to lower the odds on these favorites, which makes the underdog
a value bet. More specifically they assume that both the favorite and the underdog are priced
with the same measure of bookmaker’s profit margin built into them. If consequently the
favorite becomes underpriced the underdog must become overpriced. According to the
1×2Betting company document’s findings, however, lower odds for the favorite are
unrealistic (1×2Betting’s Value, n.d.).
Before we run into calculus to explain the reasoning above, a small anecdote based on horse
racing may explain why the intuition of many punters and betting experts that betting on
underdogs in the long run is a value bet, is unrealistic. Furthermore, why it is unrealistic that
bookies lower the odds on favorites and thereby overprice the odds on underdogs. Let´s
suppose a punter can bet on two different horses with the following odds: o1 = 3 and o2 = 32.
The latter horse, the underdog, is often called a longshot (1×2Betting’s Value, n.d.). In horse
racing bookmakers are often exposed to inside information, which is an added liability to the
bookmakers. This is further proved by Schnytzer and Shilony (1995), who test for inside
information in the Australian horse betting market and find that even exposure to ‘second
hand‘ inside information leads to changes in behavior and more significantly leads to rises in
punters’ payoffs and adds power to the prediction of game results. This kind of inside
information is especially risky with respect to longshots. Furthermore, if punters have some
kind of inside information, which according to Schnytzer and Shilony adds power to the game
prediction capabilities of punters, they could draw on this information to bet on longshots.
Inside information on a longshot exposes the bookmaker to enormous potential losses, i.e.
even higher losses than inside information utilized on favorites’ odds.
In the current example one can clearly observe the bookmaker’s risk exposure discrepancy
between the odds on the favorite and the odds on the longshot or underdog. In order to reduce
this added liability the bookmaker therefore most naturally reduces the longshot odds.
Reducing the longshot odds is exactly the opposite of what many of the so-called betting
experts claimed, who indicated that the majority of punters like to back the favorite and that
consequently the bookmaker must lower the odds on the favorite to handle the added liability
thereby making the underdog a value bet, i.e. with the same measure of bookmaker’s profit
margin built into both. Although inside information probably plays a lesser role in the fixed
odd football betting market, bookmakers operate in a similar manner to reduce their risk
exposure. As a result one may expect that fixed odd football bookmakers set odds similarly
and underprice the underdog and thereby overprice the favorite, i.e. set odds inefficiently. The
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remainder of this part gives theoretical prove and presents results that are in line with the
longshot bias pricing.
To prove why bookmakers fundamentally overprice favorites, we once more turn to our
Manchester united versus Portsmouth game. The example again deviates from the original
document’s example, because it considers all possible game outcomes, whereas the original
example uses a cup final match and therefore solely focuses on a ‘home’ win and ‘away’ win.
Let’s assume that the true expectancy that Manchester will win is 80%. For Portsmouth the
true expectancy of a win is 10% and the chance that the match will result in a draw is thus
10%. This leads to the following odds for respectively a home win, draw and away win: 1,25;
10 and 10. For ease of calculation and interpretation it is assumed that the full over-round is
10%, i.e. the true expectancy plus the bookmaker’s expected profit margin. Based on this
information the table below is created, which shows the influence of pricing biases on a
bookmaker’s return.
Table 4. The influence of pricing biasing on a bookmaker’s returns
Overpriced Underdog No pricing bias Overpriced Favorite
Bookmaker’s expectancy
for Manchester victory
91% 88% 85%
Bookmaker’s expectancy
for draw
11% 11% 11%
Bookmaker’s expectancy
for Portsmouth victory
8% 11% 14%
Over-round 110 110 110
Bookmaker’s expectancy
divided by true
expectancy for
Manchester victory
1,138 1,10 1,063
Bookmaker’s expectancy
divided by true
expectancy for
Portsmouth victory
0,8 1,10 1,40
Odds Manchester victory 1,10 1,14 1,18
Odds draw 9,1 9,1 9,1
Odds away win 12,5 9,1 7,1
The table shows that it is assumed that the bookmaker’s expectancy for a draw remains
constant under the different scenarios. Remember that the true expectancies for respectively a
home win, draw and away win are 80%, 10% and 10%. The bookmaker’s expectancies
divided by these true expectancies reveal the bookmaker’s expected profit margin under each
scenario. The table shows that overpricing the underdog is not a wise thing to do. Furthermore
overpricing the underdog by lowering the result expectancies from 11% to 8%, can lead to an
expected bookmaker’s profit margin drop from 10% to -20%. Similarly, overpricing the
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favorite by lowering the result expectancies from 88% to 85% for a Manchester victory can
lead to an expected bookmaker’s profit margin drop from 10% to 6,3%.
The 1×2Betting article further proves that overpricing the underdog is never a good risk
management strategy. This can be seen in the following tables. The two tables show that in
reality bookmakers can make an error of judgment concerning the true result expectancies.
Let us now assume that the true expectancies for a Manchester victory are 76% under the first
scenario and 84% under the second scenario. Because the example assumes that the
bookmaker’s true expectancies for a draw remain 10% under each scenario, this leads to the
following percentages for a Portsmouth victory under respectively scenario one and two:
14% and 6%. How these errors of judgment affect the bookmaker’s return is illustrated in the
two tables.
Table 5. Bookmaker’s expected profit margin under different errors of judgment with respect to a Manchester
victory
Overpriced underdog
No pricing bias Overpriced
favorite
Expectancies 91 88 85
Expectancies Odds 1,10 1,14 1,18
Fair 76% 1,32 19,7% 15,8% 11,8%
80% 1,25 13,8% 10% 6,3%
84% 1,19 8,33% 4,8% 1,2%
Table 6. Bookmaker’s expected profit margin under different errors of judgment with respect to a Portsmouth
victory
Overpriced underdog
No pricing bias Overpriced
favorite
Expectancies 8% 11% 14%
Expectancies Odds 12,5 9,1 7,1
Fair 14% 7,14 -42,9% -21,4% 0%
10% 10 -20% 10% 40%
6% 16,67 33,3% 83,3% 130,3%
The tables clearly indicate that overpriced underdogs can lead to substantial bookmaker
losses. These losses can amount to -42,9% in the present example. Bookmakers therefore try
to avoid offering any value to punters by not overpricing the underdog in their risk
management strategies. On the contrary, if bookmakers overprice favorites they seem to avoid
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offering any value to punters on all possible outcomes. The tables even show that overpricing
the favorite is a more suitable bookmaker risk management strategy than introducing no
pricing bias. Furthermore, table 6 specifies a potential bookmaker loss of -21,4% if the true
expectancy of a Portsmouth victory turns out to be 14%. Bookmakers are notoriously risk-
averse and therefore avoid offering value to punters (1×2Betting’s Value, n.d.). The most
logical risk management strategy for bookmakers therefore is to overprice the favorite.
The evidence confirms that bookmakers overprice the favorite. Furthermore 1×2Betting
conducted a detailed analysis of over 20,000 football matches from 19 European divisions for
3 consecutive seasons starting in 2000/01. The outcomes of that research indicate that backing
all home and away prices with odds higher than 3.00 would return £0.78 for every unit stake.
In contrast betting on all games with odds lower than 1.50 would return £0.96 (1×2Betting’s
Value, n.d.). These outcomes, however, may be slightly biased as 1×2Betting is not a not for
profit organization. Furthermore the afore-mentioned theory and results are based upon a
document called ‘1×2Betting’s Value Hot Favourites Betting System’ for which one has to
pay. Additionally 1×2Betting has many links on their website to most of the leading
bookmaker companies and 1×2Betting offers services for which punters have to register at the
company (1×2betting, 2007). Despite these caveats the document offers more interesting
results. One of the findings indicates that punters would not make any loss if they would back
all European league games at average prices with odds lower than 1.25 during the three
seasons. The average price is the fair or true price (1×2Betting’s Value, n.d.). The punter’s
return on investment at different odds is portrayed in the figure below.
100,595 92,5 90,6 87,9
82,4
67,8
0
20
40
60
80
100
120
5.00
ROI (%)
Figure 1. Return on investment: blind level stakes betting at different price
ranges (European league games 2001-2003). From: 1×2Betting’s Value Hot
Favourites Betting system (p. 4).
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The above graph portrays average prices. Punters, however, may also compare online
bookmakers and select the best odd prices available, which may be a few per cent higher than
the average price. Based upon these best prices the document finds for the same odds data that
backing all selections with average prices less than 1.25 would result in a profit turnover of a
little over 2,5% (1×2Betting’s Value, n.d.). Punters could further increase their profit
turnovers by comparing as many online bookmakers as possible, selectively studying the
specifics of each match and by using not only the closing prices of the odds. 1×2Betting’s
research further indicates that betting on favorites imposes a lower risk of bankruptcy.
Favorites have shorter prices, which implies that a punter’s bankroll size should fluctuate less
(1×2Betting’s Value, n.d.).
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5. Tests of market efficiency
The previous chapter gave several reasons why odds are most likely to be priced inefficiently
in the market. Furthermore Kuypers’ model (2000) indicated that bookmakers can set odds
inefficiently and the second part of the chapter indicated that punters can create abnormal
returns if they consistently bet on the favorite. The sections of this chapter investigate whether
these abnormal returns can be created in the study’s current sample and thereby thus further
investigates whether market inefficiencies exist in the betting market, by performing several
tests.. The following section focuses on the spread between implied probabilities and realized
probabilities, which is one of the current study’s tests for identifying market inefficiencies.
Thereafter independent sample t-tests and regressions will further test for inefficiencies.
5.1 The spread between implied probabilities and realized probabilities
According to Kuypers bookmakers are inclined to set inefficient odds because they want to
take advantage of punters’ reaction functions to increase their expected profits. In Kuypers’
model the punters accept the over-roundness in the market and that is something the current
paper assumes as well for testing market efficiency. The expected profit function of Kuypers
uses the punters’ reaction function to maximize bookmakers’ expected profits. More
specifically this means that bookmakers try to maximize expected profits by setting the
implied probabilities, which are the bookmakers’ decision variables. Kuypers then argues
that in order for the market to be efficient the implied probabilities from the odds should be
equal to the bookmakers’ subjective probabilities. Because it is practically impossible to
obtain the bookmakers’ subjective probabilities the current study tests for market efficiency
differently. Similar to Kuypers approach, in which the implied probabilities form a major
input for testing efficiency, the current paper uses the implied probabilities in its analysis for
testing market efficiency. Furthermore the following basic equation is used for testing market
efficiency:
yprobabilitimpliedyprobabilitrealizedSpread −=
The realized probability indicates the number of times the odd really occurred. More
specifically it is calculated as the percentage number of times that the odd really occurred.
This is done for all the different odds that occurred in the sample and for different sub
samples as will be explained later. The implied probability is simply the ‘true’ probability of
the odd to occur. Naturally the sum of the home, draw and away odd is 100%, as was
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explained in chapter three. For matters of completeness the formula for calculating the
implied probabilities is showed once more below.
( )oddsyprobabilitimplied
0969,1
1=
The 1.0969 refers to the average over-roundness in the current sample. To reach the ‘true’
expected probabilities of a game result outcome it is necessary to tackle the bookmaker’s take,
i.e. the over-roundness. This formula thus estimates the game result probabilities thereby
taking into account the bookmaker’s take. To make it a bit more pragmatic a concrete
example is given below which stipulates the steps for reaching the spread of a specific odd.
In the whole sample the B365 home win odd 1,25 occurs 22 times. The number of times that
the home team really won with the odd is 15 times. The realized probability can now simply
be calculated as follows:
%18,6822
15==yprobabilitrealized
The implied probability for this B365 home win odd is:
( )%93,72
25,10969,1
1==yprobabilitimplied
The spread is simply the realized probability minus the implied probability, which equals in
this example -4,75%. The implication of this negative spread is not beneficial for punters.
Furthermore it means that if punters would bet on all the 22 B365 home win odds of 1,25 they
would loose money. Furthermore the percentage of times that the odd really occurs is lower
than the percentage implied from the odd. Consequently if punters would bet 1 unit on the 22
B365 home win odds of 1,25 they would have the following winnings or earnings:
25,3
00,22122:
75,1825,115:
−=
−=×
=×
Winnings
Outflow
Inflow
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Positive spreads are of course beneficial to punters. Notice that this example does not take
into account any transaction costs. These kind of issues will be dealt with in the following
chapter. The following chapter investigates any inefficiencies by exploring betting strategies
that may result in abnormal returns for punters.
Based on the previous chapter one expects substantial differences between the realized and
implied probabilities, i.e. one expect substantial spreads, in the current sample. Furthermore
the previous chapter indicated that bookmakers try to avoid offering any value to punters. The
best manner to do this is to overprice the favorite. This way bookmakers avoid offering any
value to punters on all possible outcomes. There was even indicated that overpricing the
favorite is a more suitable bookmaker risk management strategy than introducing no pricing
bias. The consequences for our spread measure would be positive spread percentages for the
lower odds, i.e. the odds placed on the favorite teams. Furthermore, according to the previous
chapter, bookmakers overprice the favorite, which results in lower implied probabilities and if
the realized probabilities stay constant the spreads would naturally turn positive. Based upon
the previous chapter it is therefore expected that especially in the lower odd ranges the spread
would be substantial positive. If one assumes that both the favorite and the underdog are
priced with the same measure of bookmaker’s profit margin built into them, the spread would
gradually decline and become negative for the higher odd ranges. Furthermore if the favorite
is overpriced the underdog must be underpriced.
The spread is a perfect measure for testing efficiency. Remember from chapter 2 that in
general terms the efficient market hypothesis investigates whether prices at any point in time
reflect all available information. This hypothesis can be used as a benchmark against which
deviation from market efficiency can be judged. Evidence in favor of strong form efficiency
would mean that none of the players involved in the betting market, or any other group in
society, could make any additional profits due to some kind of monopolistic access to
information. This form of efficiency may be difficult to test, it does however seem possible to
test whether the current sample odds are semi-strong efficient. This would imply that no
abnormal returns can be achieved with the usage of publicly available information for both
the punter and bookmaker. More specifically it means that incorporating publicly available
information does not improve the accuracy of outcome predictions based on odds. This is
something that the study can test with the help of the spread measure. Furthermore, large
positive spreads for some odd ranges would result in positive trading strategies for the punter.
Additionally, one could say that all the information necessary for calculating the spread is
publicly available. Based on the theory described earlier one would thus expect positive
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trading strategies that would result in additional profits for the punter in the lower odd ranges.
This would then prove that the betting market based on the study’s sample is not semi-strong
efficient.
The spread measure is also interesting as most other betting studies use different methods to
test for market inefficiencies. Additionally the spread measure can nicely be graphed against
the different odd ranges, which results in comprehensible graphs. Furthermore if our
assumptions and interpretations are true, the ‘spread graphs’ should show a downward sloping
pattern. For the low odd ranges the graphs should show a high and positive spread and for the
lower odd ranges the graphs should show a lower and probably negative spread. Again, this is
in line with chapter four, which argues that it is most likely that bookmakers will overprice
the favorites.
This chapter will offer the different ‘spread’ graphs and the ‘spread’ statistics for the sub
samples that are created during the research. The current study’s sample consists of the B365
odds placed on the Premier League games of the seasons 2002-2003, 2003-2004, 2004-2005
and 2005-2006. The outcomes of the whole sample are discussed first. Thereafter the study
discusses the results of the odds placed on the big five teams in the Premier League, the
promoted teams per season to the Premier League, the teams with large followings and finally
the teams with obscure followings in the Premier League. These sub samples are further
divided in B365 home wins and B365 away wins. The spread for B365 home win odds is thus
calculated by calculating the implied probabilities from the B365 home win odds and
subtracting them from the realized probabilities, which are simply the percentage number of
times that the home team really won with the specific B365 home win odd. The spreads for
the B365 away wins are calculated by calculating the implied probabilities from the B365
away win odds and subtracting them from the realized probabilities, which are the percentage
number of times the away team really won with the specific B365 away win odd.
Dividing the sample in different sub samples is unique in the fixed odd football efficiency
literature and may result in some interesting results. Furthermore, there are three specific
reasons why the current study uses these sub samples. First, dividing the sample in sub
samples centers the focus of attention on outcomes we are most likely to find. Although an
analysis of the whole sample probably results in findings in line with theory, dividing it into
sub samples may result in some further confirmation of theory that was specified in the
previous chapter. Second, the sub samples may impound some interesting trading strategies
for punters that become present after investigating the results. These possible trading
strategies will then be further highlighted in the next section. The results may be used to
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explain prior study’s findings more accurately, e.g. the study by Palomino, Renneboog and
Zhang (2005)
5.2 Efficiency for whole sample home win odds
First we investigate the B365 home win odds for the whole sample. Keep in mind that the
spread is measured as the realized probabilities minus the implied probabilities of these home
win odds. The graph of this spread is portrayed below.
Figure 2. B365 home win spread for whole sample.
Spread
-100,00%
-80,00%
-60,00%
-40,00%
-20,00%
0,00%
20,00%
40,00%
60,00%
80,00%
100,00%
1,1
1,2
1,33
1,5
1,66
1,83
2,25
2,6
2,87
3,5
4,75 8
Odds
Percentage
Spread
The figure above depicts the spread of 365 home win odds portrayed against their respective
odds. The first point of the graph shows the spread of the B365 home win odd of 1,1. This is
the odd given to a Manchester United home win over Sunderland played on the 14th
of April
2006. This home win odd of 1,1 occurred only once in the sample. As the match between
Manchester United and Sunderland ended in a draw the realized probability for this odd is
0%. The implied probability of this odd is approximately 82,88%. This consequently results
in a negative spread of -82,88%, which can be seen in the graph by the first dot. Similarly all
the other dots in the graph are created. Due to space limitations it is impossible to portray all
the accompanying odds on the X-axis. Similar to the example of the 1,1 odd stipulated above,
many of the calculated spread values in the graph are based on too little observations and may
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therefore form not a good indication of the efficiency of the odds. The current study, for that
reason, bundles several odds into odd ranges, which consequently leads to more observations
per ´dotspread´ and thus leads to more trustworthy spreads. The bundles are created in such a
manner that there are sufficient observations per range and the ranges do not become too
width. Especially in the lower ranges bundles are kept deliberately small. This approach is
further used throughout the remainder of the sub samples. To give an indication of the
bundling process, table 7 is depicted below. Table 7 portrays the spreads of bundles or ranges
of odds. The spread table of the whole sample with spreads calculated per odd separately is
portrayed in appendix A. This table is thus used to create figure 2 above.
Table 7. Home win spreads calculated per bundle for the whole sample
B365H Himpl # of times odd Really won Realized Spread
from-until Prob occurred in sample with the odd Prob
1,1-1,143 80,62% 6 5 83,33% 2,71%
1,16-1,2 76,57% 19 16 84,21% 7,64%
1,22-1,25 73,48% 32 24 75,00% 1,52%
1,28-1,3 70,66% 41 39 95,12% 24,46%
1,33-1,364 67,74% 28 22 78,57% 10,83%
1,4-1,444 64,24% 54 38 70,37% 6,13%
1,5-1,533 60,34% 70 44 62,86% 2,52%
1,57-1,571 58,06% 51 37 72,55% 14,49%
1,61-1,615 56,57% 63 34 53,97% -2,60%
1,66-1,67 54,83% 74 46 62,16% 7,33%
1,72-1,75 52,89% 81 50 61,73% 8,84%
1,8-1,833 50,28% 146 73 50,00% -0,28%
1,9-1,909 47,92% 95 51 53,68% 5,76%
2 45,58% 101 44 43,56% -2,02%
2,1 43,41% 107 47 43,93% 0,51%
2,2-2,25 40,97% 169 76 44,97% 4,00%
2,3-2,38 39,01% 154 59 38,31% -0,70%
2,4-2,5 36,92% 146 55 37,67% 0,75%
2,6-2,63 34,88% 87 26 29,89% -4,99%
2,7-2,75 33,33% 64 19 29,69% -3,64%
2,8-2,875 32,13% 35 10 28,57% -3,56%
3-3,25 29,34% 60 16 26,67% -2,67%
3,4-3,75 25,51% 30 7 23,33% -2,18%
4-4,75 21,23% 53 17 32,08% 10,85%
5-6,00 17,05% 48 4 8,33% -8,72%
6,5-17 12,13% 34 5 14,71% 2,58%
The home implied probabilities in this table are calculated as the sum of the individual
implied probabilities multiplied by the number of times each of these implied probabilities
occurred in the bundle divided by the total number of times the odds occurred for that specific
odd bundle. This leads to a new spread graph, which is shown below.
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Figure 3. Spread for bundles of the B365 home win whole sample odds.
Spread
-15,00%
-10,00%
-5,00%
0,00%
5,00%
10,00%
15,00%
20,00%
25,00%
30,00%1,1-1,143
1,22-1,25
1,33-1,364
1,5-1,533
1,61-1,615
1,72-1,75
1,9-1,909
2,1
2,3-2,38
2,6-2,63
2,8-2,875
3,4-3,75
5-6,00
Odds
Percentage
Spread
Although there are some outliers, the graph shows a downward trend. To further investigate
the meaning of this graph, statistics are used to investigate the significance of the spread. If
appropriate, the statistical tests of all the studies set the significance level at 5%. First an
independent sample t-test is used. An independent sample t-test investigates how the mean of
a quantitative variable differs between two populations or two subpopulations. The current
study prefers this test over a paired sample t-test because we have one quantitative variable,
which is in this case the percentage change of a home win or the probability of a home win,
and we have two sub populations. Furthermore, the essence of the spread measure is to
investigate whether the probabilities or percentage change of home wins differ between the
probabilities given by the implied probabilities and the probabilities given by the realized
probabilities. A paired sample t-test would in this case assume that the realized and implied
probabilities are two quantitative variables, which we think is less logical as treating it as two
subpopulations. The basic null hypothesis and alternative hypothesis of this test are:
Null hypothesis: )0..(: 0210 == DeiH µµ
Alternative hypothesis: )(: 21 sidedtwoH a −≠ µµ
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Where µ1 is the population mean of the realized probability and µ2 is the population mean of
the implied probabilities. The hypotheses thus test whether in the population, the average of
realized probabilities and implied probabilities is the same, versus whether the average of
realized probabilities differs from the average of implied probabilities. In order to test this we
let Excel and Spss run the following test statistic:
21
021 )(
xxS
Dxxt
−
−−= with
2
2
2
1
2
1
21 n
s
n
sS
xx+=−
Where x is a point estimate of the mean of the realized and implied probabilities based on the
sample used. Similarly S is an estimated standard deviation and the n is simply the number of
the samples used. This test assumes that we use independent samples from both
subpopulations and that the variables are normally distributed or that both samples are large,
i.e. n1, n2 > 30. This is in line with the central limit theorem, which states that if the sample
size n is sufficiently large, then the population of all possible sample means is approximately
normally distributed, with mean µ x = µ and standard deviation σ x = σ/ n , no matter what
probability distribution describes the sampled population. Although the central limit theory
ideally argues that the samples should be equal or larger than 30, Bowerman, O’Connell and
Hand (2001) argue that these formulas hold exactly if the sampled population is infinite and
hold approximately if the sampled populat