Case Study1: The World Cup

Case Study1: The World Cup

Let t=true odds
Let b= Implied Fair odds
Let p=proability of 'implied' fair bet odds
Rank on
6/10/2010
1
2
3
4
5
6
7
8
9
10
11
12
13
13
13
16
16
16
19
19
21
21
23
23
25
26
27
27
29
30
31
31

Team
Spain
Brazil
Argentina
England
Holland
Germany
Italy
France
Portugal
Ivory Coast
Serbia
Chile
Paraguay
Mexico
U.S.A.
Ghana
Cameroon
Uruguay
Denmark
Nigeria
South Africa
Australia
Greece
Switzerland
Slovakia
South Korea
Slovenia
Japan
Algeria
Honduras
New Zealand
North Korea

Odds (b) on
6/10/2010

=>

Wager/Bet

P(win)

P(Lose)

fairbet=>

=>

=>

4
4.5
6.5
7.5
9
14
16
20
25
50
60
70
80
80
80
100
100
100
125
125
150
150
200
200
225
250
300
300
600
1000
2000
2000

=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
1-p

4(p)=1(1-p)
4.5(p)=1(1-p)
6.5(p)=1(1-p)
7.5(p)=1(1-p)
9(p)=1(1-p)
14(p)=1(1-p)
16(p)=1(1-p)
20(p)=1(1-p)
25(p)=1(1-p)
50(p)=1(1-p)
60(p)=1(1-p)
70(p)=1(1-p)
80(p)=1(1-p)
80(p)=1(1-p)
80(p)=1(1-p)
100(p)=1(1-p)
100(p)=1(1-p)
100(p)=1(1-p)
125(p)=1(1-p)
125(p)=1(1-p)
150(p)=1(1-p)
150(p)=1(1-p)
200(p)=1(1-p)
200(p)=1(1-p)
225(p)=1(1-p)
250(p)=1(1-p)
300(p)=1(1-p)
300(p)=1(1-p)
600(p)=1(1-p)
1000(p)=1(1-p)
2000(p)=1(1-p)
2000(p)=1(1-p)

4p=1-p
4.5p=1-p
6.5p=1-p
7.5p=1-p
9p=1-p
14p=1-p
16p=1-p
20p=1-p
25p=1-p
50p=1-p
60p=1-p
70p=1-p
80p=1-p
80p=1-p
80p=1-p
100p=1-p
100p=1-p
100p=1-p
125p=1-p
125p=1-p
150p=1-p
150p=1-p
200p=1-p
200p=1-p
225p=1-p
250p=1-p
300p=1-p
300p=1-p
600p=1-p
1000p=1-p
2000p=1-p
2000p=1-p

5p=1
5.5p=1
7.5p=1
8.5p=1
10p=1
15p=1
17p=1
21p=1
26p=1
51p=1
61p=1
71p=1
81p=1
81p=1
81p=1
101p=1
101p=1
101p=1
126p=1
126p=1
151p=1
151p=1
201p=1
201p=1
226p=1
251p=1
301p=1
301p=1
601p=1
1001p=1
2001p=1
2001p=1

source: paddypower.com

Solving, p= Q1: p(win)=1/(b+1) p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= p= ∑ p(win)
Total

0.2000
0.1818
0.1333
0.1176
0.1000
0.0667
0.0588
0.0476
0.0385
0.0196
0.0164
0.0141
0.0123
0.0123
0.0123
0.0099
0.0099
0.0099
0.0079
0.0079
0.0066
0.0066
0.0050
0.0050
0.0044
0.0040
0.0033
0.0033
0.0017
0.0010
0.0005
0.0005
1.1190
=> ∑ p(win)>1

Executive MBA Program

Date: August 2012

1 of 4

Case Study1: The World Cup

Q1).
Rank on
6/10/2010

The probabilities implied for each team to win at fair bet are as follows:Team
1
2
3
4
5
6
7
8
9
10
11
12
13
13
13
16
16
16
19
19
21
21
23
23
25
26
27
27
29
30
31
31

Odds (b) on 6/10/2010

p=1/ (b+1)

Spain
Brazil
Argentina
England
Holland
Germany
Italy
France
Portugal
Ivory Coast
Serbia
Chile
Paraguay
Mexico
U.S.A.
Ghana
Cameroon
Uruguay
Denmark
Nigeria
South Africa
Australia
Greece
Switzerland
Slovakia
South Korea
Slovenia
Japan
Algeria
Honduras
New Zealand
North Korea

4
4.5
6.5
7.5
9
14
16
20
25
50
60
70
80
80
80
100
100
100
125
125
150
150
200
200
225

0.2000
0.1818
0.1333
0.1176
0.1000
0.0667
0.0588
0.0476
0.0385
0.0196
0.0164
0.0141
0.0123
0.0123
0.0123
0.0099
0.0099
0.0099
0.0079
0.0079
0.0066
0.0066
0.0050
0.0050
0.0044
0.0040
0.0033
0.0033
0.0017
0.0010
0.0005
0.0005
1.1190

250
300
300
600
1000
2000
2000

Total

where

b= odds p=proability of fair odds

Rule1 of Probabilities is TRUE for each of the 32 teams.
The probabilities p range from
'0.0005 to 0.2' and therefore fulfill the condition:- 0 ≤ p ≤ 1
Rule2 of Probabilities is FALSE for this wager \ experiment. Sum of probabilities=
1.119, which is > 1, or ∑p > 1

Source: Statistics for Business and Economics 11th edition, Anderson, Sweeney and Williams, page155

=> ∑ p(win) Total>> = 1.119
Q2

From Q1, we saw that the odds are not fair with an 'overround' of 11.9% for the betting house.
Thus, we could derive the true probabilty by dividing each individual team's probability of winning by
∑ P(win) Total which =
1.1190

Executive MBA Program

Date: August 2012

2 of 4

Case Study1: The World Cup

Let
Let
Let
Let

Rank

1
2
3
4
5
6
7
8
9
10
11
12
13
13
13
16
16
16
19
19
21
21
23
23
25
26
27
27
29
30
31
31

t=true odds b= Implied Fair odds p=proability of 'implied' fair bet odds pt=probabilty of with true bet odds

Implied Fair
Bet
Probability

Team

Odds (b) on 6/10/2010

p=1/ (b+1)

Spain
Brazil
Argentina
England
Holland
Germany
Italy
France
Portugal
Ivory Coast
Serbia
Chile
Paraguay
Mexico
U.S.A.
Ghana
Cameroon
Uruguay
Denmark
Nigeria
South Africa
Australia
Greece
Switzerland
Slovakia
South Korea
Slovenia
Japan
Algeria
Honduras
New Zealand
North Korea

4
4.5
6.5
7.5
9
14
16
20
25
50
60
70
80
80
80
100
100
100
125
125
150
150
200
200
225
250
300
300
600
1000
2000
2000

0.2000
0.1818
0.1333
0.1176
0.1000
0.0667
0.0588
0.0476
0.0385
0.0196
0.0164
0.0141
0.0123
0.0123
0.0123
0.0099
0.0099
0.0099
0.0079
0.0079
0.0066
0.0066
0.0050
0.0050
0.0044
0.0040
0.0033
0.0033
0.0017
0.0010
0.0005
0.0005
1.1190
∑ p(win)

Totals

pt=probabilty of win with true odds

pt = p /∑ p(win)

/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/

1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190
1.1190

= pt

0.1787
0.1625
0.1192
0.1051
0.0894
0.0596
0.0526
0.0426
0.0344
0.0175
0.0147
0.0126
0.0110
0.0110
0.0110
0.0088
0.0088
0.0088
0.0071
0.0071
0.0059
0.0059
0.0044
0.0044
0.0040
0.0036
0.0030
0.0030
0.0015
0.0009
0.0004
0.0004
1.0000

Gain of Implied Fair Bet
Odds vs. True Bet Odds

True Odds
=1/pt - 1
4.59
5.15
7.39
8.51
10.19
15.78
18.02
22.50
28.09
56.07
67.26
78.45
89.64
89.64
89.64
112.02
112.02
112.02
139.99
139.99
167.97
167.97
223.91
223.91
251.89
279.86
335.81
335.81
671.51
1,119.10
2,238.08
2,238.08

Odds
Difference
(True Implied Fair)
0.59
0.65
0.89
1.01
1.19
1.78
2.02
2.50
3.09
6.07
7.26
8.45
9.64
9.64
9.64
12.02
12.02
12.02
14.99
14.99
17.97
17.97
23.91
23.91
26.89
29.86
35.81
35.81
71.51
119.10
238.08
238.08

2b). Accept £32 on Spain, £31 on Brazil descending…to £2 on NZ, £1 on N. Korea

2a). Accept £1 per team

Probability of Loss (i.e. win for Betting house)
= 1-pt
0.8213
0.8375
0.8808
0.8949
0.9106
0.9404
0.9474
0.9574
0.9656
0.9825
0.9853
0.9874
0.9890
0.9890
0.9890
0.9912
0.9912
0.9912
0.9929
0.9929
0.9941
0.9941
0.9956
0.9956
0.9960
0.9964
0.9970
0.9970
0.9985
0.9991
0.9996
0.9996
31.0000

Bet Amount' in £s
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
32

Expected
Profit [pt *
Odds
Difference*
Bet
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
3.40

Income (from
Bets) - PayOut if
Win (£s)
27
26.5
24.5
23.5
22
17
15
11
6
-19
-29
-39
-49
-49
-49
-69
-69
-69
-94
-94
-119
-119
-169
-169
-194
-219
-269
-269
-569
-969
-1969
-1969

Bet in £s

32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
528

Expected Income
(from
Profit [pt
Bets) * Odds
Difference PayOut if
Win (£s)
* Bet
3.40
399.00
3.30
387.50
3.19
332.00
3.08
309.50
2.98
275.00
2.87
149.00
2.76
111.00
2.66
27.00
2.55
-73.00
2.45
-623.00
2.34
-793.00
2.23
-943.00
2.13
-1073.00
2.02
-993.00
1.91
-913.00
1.81
-1173.00
1.70
-1073.00
1.59
-973.00
1.49
-1223.00
1.38
-1098.00
1.28
-1273.00
1.17
-1123.00
1.06
-1473.00
0.96
-1273.00
0.85
-1273.00
0.74
-1223.00
0.64
-1273.00
0.53
-973.00
0.43
-1873.00
0.32
-2473.00
0.21
-3473.00
0.11
-1473.00
56.14

2a) and 2b) , Both provide expected probability for the betting house to make profit.
The expected profit for the betting house per £1 bet on any of the 32 team is consistent at £0.11. However, should Ivory Coast or a lower ranking team win, where the odds are >32, Betting house loses money (Cells: R104:R126).
With 2b). the situation is similar whereby if Portugal or a lower ranking team wins, then Betting house loses money (cells: U103:R126).
Hence, Arbitrage is NOT achieved with 2a) or 2b).

Executive MBA Program

Date: August 2012

3 of 4

Case Study1: The World Cup

ARBITRAGE SYSTEM

Rank on
6/10/2010

Team

Odds (b) on
6/10/2010

Q1: p(win)=1/(b+ 1)

Income (from
Arbitrage
'System' Bets)
- PayOut if
Win (£s)

Arbitrage 'System' Bet Amount in £s

1
2
3
4
5
6
7
8
9
10
11
12
13
13
13
16
16
16
19
19

Spain
Brazil
Argentina
England
Holland
Germany
Italy
France
Portugal
Ivory Coast
Serbia
Chile
Paraguay
Mexico
U.S.A.
Ghana
Cameroon
Uruguay
Denmark
Nigeria

4
4.5
6.5
7.5
9
14
16
20
25
50
60
70
80
80
80
100
100
100
125
125

0.2000
0.1818
0.1333
0.1176
0.1000
0.0667
0.0588
0.0476
0.0385
0.0196
0.0164
0.0141
0.0123
0.0123
0.0123
0.0099
0.0099
0.0099
0.0079
0.0079

=p(win Spain)/p(win North Korea)
=p(win Brazil)/p(win North Korea)
=p(win Argentina)/p(win North Korea)
=p(win England)/p(win North Korea)
=p(win Holland)/p(win North Korea)
=p(win Germany)/p(win North Korea)
=p(win Italy)/p(win North Korea)
=p(win France)/p(win North Korea)
=p(win Portugal)/p(win North Korea)
=p(win Ivory Coast)/p(win North Korea)
=p(win Serbia)/p(win North Korea)
=p(win Chile)/p(win North Korea)
=p(win Paraguay)/p(win North Korea)
=p(win Mexico)/p(win North Korea)
=p(win U.S.A.)/p(win North Korea)
=p(win Ghana)/p(win North Korea)
=p(win Cameroon)/p(win North Korea)
=p(win Uruguay)/p(win North Korea)
=p(win Denmark)/p(win North Korea)
=p(win Nigeria)/p(win North Korea)

400
364
267
235
200
133
118
95
77
39
33
28
25
25
25
20
20
20
16
16

21

South Africa

150

0.0066

=p(win South Africa)/p(win North Korea)

13

21
23
23
25
26
27
27
29
30

Australia
Greece
Switzerland
Slovakia
South Korea
Slovenia
Japan
Algeria
Honduras

150
200
200
225
250
300
300
600
1000

0.0066
0.0050
0.0050
0.0044
0.0040
0.0033
0.0033
0.0017
0.0010

=p(win Australia)/p(win North Korea)
=p(win Greece)/p(win North Korea)
=p(win Switzerland)/p(win North Korea)
=p(win Slovakia)/p(win North Korea)
=p(win South Korea)/p(win North Korea)
=p(win Slovenia)/p(win North Korea)
=p(win Japan)/p(win North Korea)
=p(win Algeria)/p(win North Korea)
=p(win Honduras)/p(win North Korea)

13
10
10
9
8
7
7
3
2

31

New Zealand

2000

0.0005

=p(win New Zealand)/p(win North Korea)

1

31

North Korea

2000

0.0005
1.1190

=p(win North Korea)/p(win North Korea)

1
2,239

637.28
600.90
503.88
472.49
437.18
370.48
354.78
332.36
314.04
276.31
269.88
265.26
261.78
261.78
261.78
256.89
256.89
256.89
252.96
252.96
250.33
250.33
247.03
247.03
245.93
245.05
243.73
243.73
240.41
239.08

To Achieve arbitrage, the betting house needs to devise a system of Bets that are proportional to the probabilities (Implied Fair Bet odds) from 'Q1'.
By dividing all the probabilities of the top ranked 31 (of the 32 teams) by the 32nd team (with implied fair probability 0.0005) will give us the bet amounts that would provide arbitrage Shown in Yellow shading
(cells:i140:.171). (Total Bets = £2,239)
Thus, for example. The betting house should accept £400 for Spain, £364 for Brazil.. all the way to £1 for North Korea. Sum of all the bets will come to £2,239. Obviously, betting amounts in the same ratio as this would also work to provide arbitrage e.g. £1,200 for Spain, £1,092 for Brazil...£3 for North Korea

The adjacent column shown in green shading (cells:j140:.171) show verification that for each individual country were to win, the net income for the betting

238.08
238.08

source: paddypower.com

Executive MBA Program

Date: August 2012

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