Measuring Efficiency in the Fixed Odd Football Betting Market: A …ebettingsystems.com/Files/MeasuringEfficiency.pdf · 2014. 2. 19. · Measuring Efficiency in the Fixed Odd Football

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


13

(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.


14

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


15

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.


16

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


17

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:


18

( )( )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


19

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


20

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.


21

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.


22

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


23

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%


for draw

11% 11% 11%


for Portsmouth victory

8% 11% 14%

Over-round 110 110 110


divided by true

expectancy for

Manchester victory

1,138 1,10 1,063


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


24

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


25

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).


26

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.).


27

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


28

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


29

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


30

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


31

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


32

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.


33

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


34

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

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