Being able to calculate betting margins is useful as it tells you whether or not there is value in a bet.
When arbitrage betting, if the combined implied probability is below 100%, you have found an arb. Whereas if the combined implied probability is greater than 100%, the bookmaker has a margin and there is no value in the market at the current odds.
what are betting margins?
Every outcome of a bet has a true probability, and in a fair market, the combined probability of all outcomes equals 100%.
With the coin toss, there are two possible outcomes - Heads or Tails. Each of these outcomes has the same chance of occurring and so they each have a true probability of 50%.
Bookmakers do not price markets fairly. This is how they make a profit. So, if they were taking bets on a coin toss, they wouldn't give each outcome a 50% chance of occurring. Instead, they may give each outcome a 52% chance which would give a combined probability of 104%.
- The probability that the bookmaker sets is the implied probability.
- The actual chance of a particular outcome is the true probability.
In this example, the implied probability of either heads or tails is 52% and the true probability is 50%.
The difference between the implied probability and the true probability is the bookmakers margin, which in this case, is 4% (104% - 100%).
In betting, we're not used to seeing probability displayed in percentages. Instead betting sites use odds and so it can sometimes be difficult to know whether there is value in a market or not.
By converting the probability of all outcomes from odds to a percentage, we're able to determine whether there is value and whether or not we can make a profit by covering all outcomes.
When we do this, there are two outcomes:
- The combined probability is greater than 100%
- The combined probability is less than 100%
Should the result be outcome 2 then we have found an arb and it is us that has the edge rather than the bookmaker.
We can use the following formulas to determine the margins on markets.
calculating margins with two possible outcomes
Markets which have two possible outcomes include tennis matches or other sports with two players where a draw isn't possible.
We can calculate the margin using the following formula:
Combined Margin = (1 ÷ odds of outcome 1) + 100 + (1 ÷ odds of outcome 2) x 100
For example, in a tennis match where Player 1 has odds of 1.5 with Bookmaker A and Player 2 has odds of 3.2 with Bookmaker B, the margin would be:
Combined Margin = (1 ÷ 1.5) x 100 + (1 ÷ 3.2) x 100
Combined Margin = 97.91%
As the combined margin is less than 100%, this tells us that this is an arb and we can make a profit from betting on all outcomes. Take a look at our Arbitrage betting calculations & formulas article to learn how to calculate your stakes to ensure you return a profit or simply use an arbitrage calculator.
In this example, the margin is 2.09% (100% - 97.91%) which is the profit we can expect to make based on our total stake betting on all outcomes.
calculating margins with three possible outcomes
The most common market with three possible outcomes is the match result market of a football match where a home win, draw and away win is possible.
We can calculate the margin for three-way markets using the following formula:
Combined Margin =
(1 ÷ odds of outcome 1) + 100 + (1 ÷ odds of outcome 2) x 100 + (1 ÷ odds of outcome 3)
Let's apply this to a football match example with the following odds:
- Home win odds: 1.8
- Draw odds: 3.8
- Away win odds: 6.1
Combined Margin =
(1 ÷ 1.8) + 100 + (1 ÷ 3.8) x 100 + (1 ÷ 6.1)
Combined Margin = 98.26%
Again, the combined margin is less than 100% and so this is a profitable arb.
In this example, the margin is 1.74% (100% - 98.26%) which is again the profit we can expect to make based on our total stake.
If a market has more than three possible outcomes, you can simply add more to the formula.
Exchange commissions & margins
Arbitrage betting can be done either by using bookmakers or betting exchanges for your bets. Bookmakers do not charge commission whereas betting exchanges do.
You can calculate betting margins by factoring in exchange commissions using the formula below which will determine the odds for each outcome with the exchange commission applied.
We will use the tennis example above and an exchange commission of 5%.
1 + ((1 – (exchange commission / 100)) x (odds – 1))
Player 1 odds with exchange commission factored in:
1 + ((1 – (5 / 100)) x (1.5 – 1))
Player 1 odds with exchange commission applied = 1.475
Player 2 odds with exchange commission factored in:
1 + ((1 – (5 / 100)) x (3.2 – 1))
Player 2 odds with exchange commission applied = 3.09
As you can see, the odds of Player 1 have shortened from 1.5 to 1.475 and the odds of Player 2 have shortened from 3.2 to 3.09.
To determine whether or not we have a margin and if this is still an arb with the exchange commission factored in, we would use these newly calculated odds in the formula mentioned earlier.