## Calculate Bookmaker’s Margin

Betting companies are making a great profit due to their margins, which means that the gambler is expected to lose in a long term an amount equal to the (**Total Betting Amount) X (Margin**). Let’s see how to calculate the bookmaker’s margin by taking as an example the football match **“Man Utd vs Arsenal”** and the odds of Bet365 which is a famous betting company (30 Sep 2019).

As we can see the **odds **are **2.3, 3.4 and 3** for **Man Utd, Draw and Arsenal** respectively. To **calculate bookmakers margins** with decimal odds, all you need to do is divide 1 by the odds for each outcome in the market, and sum together. This sum minus 1 is the bookmaker’s **margin**. Thus, we can easily calculate the margin as follows:

Outcome | Odds | 1/Odds |

Man Utd | 2.3 | 0.434782609 |

Draw | 3.4 | 0.294117647 |

Arsenal | 3 | 0.333333333 |

Total | 1.062233589 |

Hence, the margin is \(1.062233589-1= 0.062233589 \approx 6.22\% \)

## Calculate Bookmaker’s Probabilities

The Bookmaker first estimates the probabilities of each outcome and then adds the margins. The estimated probabilities are the **(1/Odds)/Total**. Finally, the “fair” odds (i.e. without margin) would be the **1/(Estimated Probabilities)**. Let’s calculate those figures:

Outcome | Odds | 1/Odds | Estimated Probs | Fair Odds |

Man Utd | 2.3 | 0.434782609 | 0.409309791 | 2.443137255 |

Draw | 3.4 | 0.294117647 | 0.276886035 | 3.611594203 |

Arsenal | 3 | 0.333333333 | 0.313804173 | 3.186700767 |

Total | 1.062233589 | 1 |

As we can see, the Bookmaker estimates the probability of Man Utd to win to be **40.9%** and the fair odd would be **2.443** times, but he pays back **2.3** times.

## From the Fair Odds to the Market Odds

The process that the Bookmaker follows is, **1) define the margin**, **2) estimate the outcome probabilities**, **3) calculate the fair odds,** and **4) finally, apply the margin to the fair odds in order to get the market odds**. The market odds are calculated as **(fair odds) / (1+margin)**. In our case the (1+margin) is **1.062233589, **so let’s calculate the Market Odds.

Outcome | Odds | 1/Odds | Actual Probs | Fair Odds | Market Odds |

Man Utd | 2.3 | 0.434782609 | 0.409309791 | 2.443137255 | 2.3 |

Draw | 3.4 | 0.294117647 | 0.276886035 | 3.611594203 | 3.4 |

Arsenal | 3 | 0.333333333 | 0.313804173 | 3.186700767 | 3 |

Total | 1.062233589 | 1 |

As we can see the “Market Odds” are the same as the “Odds” as expected. In case you got confused the **Market Odds = 2.3 = 2.443137255 / 1.062233589**.

## Calculate Bookmaker’s Mispricing

The Bookmakers can make a mistake in their estimates, however, they have this margin that we mentioned above which protects them from any “mispricing” in their odds. Let’s say that a gambler, estimates his own probabilities after applying Machine Learning Algorithms or even based on his experience and his gut feeling. Let’s assume that a gambler estimates the following probabilities and its corresponding odds (1/Probs) for the example above.

Outcome | Probs | Estimated Odds |

Man Utd | 0.4 | 2.5 |

Draw | 0.28 | 3.571428571 |

Arsenal | 0.32 | 3.125 |

As we can see, although in all outcome events they did not agree with the Bookmaker, however, there is not any “mispricing” since the gambler’s estimated Odds are higher than the market odds (e.g 2.5>2.3, 3.57>3.4, 3.125>3) which means that the fair price would be let’s say 2.5 but the bookmaker pays 2.3. Let’s see another example where there is a “mispricing”.

Outcome | Probs | Estimated Odds |

Man Utd | 0.38 | 2.631578947 |

Draw | 0.28 | 3.571428571 |

Arsenal | 0.34 | 2.941176471 |

In this example, there is a “mispricing” in Arsenal’s outcome, since the Bookmaker pays **3 times** where the gambler estimates as a fair return to be **2.914** times, so he pays more! Thus, the gambler should go for Arsenal, although he does not believe that this is the more likely event to occur (34%). Again, the “mispricing” is based on the assumption that the “Gambler’s” estimated odds are more accurate than that of the bookmaker.

Note: The Bookmaker’s margin changes when we are dealing with multiple bets.

## Arbitrage Betting

Arbitrage Betting is an example of arbitrage arising in betting markets due to either bookmakers’ differing opinions on event outcomes or errors. When conditions allow, by placing one bet per each outcome with different betting companies, the bettor can make a profit regardless of the outcome. Mathematically arbitrage occurs when there are a set of odds, which represent all mutually exclusive outcomes that cover all state space possibilities (i.e. all outcomes) of an event, **whose implied probabilities add up to less than 1**

This is impossible to happen in one betting company, and if this happened, your bet would be canceled based on “terms and conditions”, but let’s say that you can play on different companies. So, the arbitrage will exist if the sum of 1/odds is less than 1. Let’s consider the following hypothetical odds.

Outcome | Odds | 1/Odds |

Man Utd | 2.6 | 0.384615385 |

Draw | 3.5 | 0.285714286 |

Arsenal | 3.3 | 0.303030303 |

Total | 0.973359973 |

In this case we have an arbitrage of \(arb = 1- 0.973359973 =0.026640027 \approx 2.66\% \). Since we have an arbitrage, what is the best strategy in order to make a guaranteed profit, in other words, what are the **weights **that we should place in every outcome in order to win for every possible outcome?

The optimum weights are the **(1/Odds)/sum**, i.e for Man Utd even is **0.384615385/ 0.973359973 = 0.395141977**. Let’s calculate the “weights” for each outcome.

Outcome | Odds | 1/Odds | Weights |

Man Utd | 2.6 | 0.384615385 | 0.395141977 |

Draw | 3.5 | 0.285714286 | 0.29353404 |

Arsenal | 3.3 | 0.303030303 | 0.311323982 |

Total | 0.973359973 | 1 |

If there is an arbitrage the guaranteed profit that you can make is **1-1/sum(1/Odds)** , which is in our case \(Profit = 1- (1/0.973359973)=0.027369141 \approx 2.7369\% \)

Let’s assume that we invest **1000$** in this game. The amount that we should place for every outcome is as follows (column amount).

Outcome | Odds | 1/Odds | Weights | Amount |

Man Utd | 2.6 | 0.384615385 | 0.395141977 | 395.1419774 |

Draw | 3.5 | 0.285714286 | 0.29353404 | 293.5340404 |

Arsenal | 3.3 | 0.303030303 | 0.311323982 | 311.3239822 |

Total | 0.973359973 | 1 | 1000 |

If the outcome is:

- Man Utd, then we will get 395.1419774 x 2.6 – 1000 =
**27.36914129$** - Draw, then we will get 293.5340404 x 3.5 – 1000 =
**27.36914129$** - Arsenal, then we will get 311.3239822 x 3.3 – 1000 =
**27.36914129$**

Hence, no matter the outcome, we will make a profit of around 2.7369% and this is the definition of the arbitrage.

## Arbitrage in Practice

In practice, it is extremely difficult to find Arbitrage Opportunities, and there is also a risk to be banned from the betting companies. Below, we outline some difficulties.

**Limited Opportunity In Limited Time**

In the Internet age, in most cases, it only takes up to 15 minutes until an arbitrage opportunity disappears. Spatial distance has no meaning on the Internet. This is, of course, an issue if you are hoping to make regular profits from arbitrage betting.

**High turnover, low profits **

Another issue is the diminutive size of typical returns. In almost every arbitrage betting opportunity you will earn a relatively small profit from each transaction, a return of more than 3% on your investment will be a rare event. So while this gain is almost certain, you have to consider the rare nature of arbitrage. This means you need a lot of capital to invest in each opportunity, to justify even the high expenditure of time.

**Be careful with high stakes **

Even if you have a large amount of money to invest, another problem soon arises. Almost all other bookmakers accept large bets only when they are checked and waved through by a manager. So there is a real risk that one of your bets will not be accepted.

## 2 thoughts on “Bookmaker’s Margin and Arbitrage Betting”

Excellent post but I was wondering if you could write a litte more on this subject?

I’d be very thankful if you could elaborate a little bit further.

Thank you!