Calculating the correct stakes when arbitrage betting is essential so that you return the same or similar profit whatever the outcome. There are many arbitrage calculators available online which will do these calculations for you based on the odds of each outcome but it is useful to understand the maths behind the calculations so that you are able to spot any obvious errors which may occur should you enter incorrect odds for example.
Our arbitrage betting article goes into detail about what arbing is and lists some useful resources to get started whereas this article is slightly more advanced and explains the maths behind it.
The maths behind an arb
Arbing consists of betting on all outcomes on an event. By doing so, one of these bets will win. In a fair market (one where neither the bookmaker or the bettor has a margin) the probability of all possible outcomes will add up to 100%.
For example, a tennis match has two possible outcomes.
- Player 1 wins
- Player 2 wins
Should these two players be equally matched, they would each have a 50% chance of winning which gives a total probability of 100%.
If a bookmaker was pricing the outcomes of each player, they would price them slightly higher than 50%. For example, they may give each player a 53% chance of winning. This means that the total probability of all outcomes would be 106%. As there can only be a true combined probability of 100% for all outcomes, that extra 6% is the bookmakers edge.
When an arb occurs, the opposite happens and the total probability of all outcomes is below 100%. For example, if the probability of Player 1 winning with bookmaker A was 45% and the probability of Player 2 winning with Bookmaker B was 48%, the total probability of all outcomes would be 93%. This means that there is an opportunity to make a 7% profit from the arb.
- Odds of all possible outcomes = over 100% probability = The bookmaker has an edge
- Odds of all possible outcomes = under 100% probability = The arber has an edge
There are two main types of arbs.
- Between a bookmaker and a betting exchange (a back and lay bet)
- Between two or more bookmakers (covering all outcomes with multiple back bets)
The process for calculating your stakes is different depending on what type of arb it is so let's take a look at each one separately.
calculating lay stakes for arbitrage between a bookmaker and an exchange
These are the the most common and easiest type of arbs. You simply back an outcome at a bookmaker and lay the same outcome on a betting exchange. By doing so, you have covered all outcomes.
You can calculate your lay stakes using either a matched betting calculator or an arbitrage calculator but the following shows you how to calculate your lay stakes manually.
Let's use a tennis match as an example as it has two possible outcomes.
Header | Player 1 win odds (Bookmaker) | Player 2 lay odds (Betting Exchange) | margin |
---|---|---|---|
odds | 2.15 | 2.0 | Cell |
implied probability | 46.51% | 50% | 96.51% |
Calculating betting margins: Read our 'How to calculate bet ting margins' article for more info.
We now need to calculate our lay stakes.
Assuming that we are using a back stake of £100 with the bookmaker, we can use the following formula to determine our lay stake:
Lay Stake = (back odds x back stake) ÷ (lay odds - exchange commission)
Using the example above, and with an exchange commission of 5% (0.05), we get:
Lay Stake = (2.15 x 100) ÷ (2 - 0.05)
Lay Stake = £110.26
calculating your profit
Depending on the outcome of the match, either our bet will win with the bookmaker or with the exchange. The following formulas will calculate your profit based on each outcome.
your bet wins with the bookmaker
Profit = (back odds - 1) x back stake - (lay odds -1) x lay stake
Again, using the above example, our profit if our bookmaker bet wins would be:
Profit = (2.15 - 1) x 100 - (2 -1) x 110.26
Profit = £4.74
your bet wins with the exchange
Profit = (lay stake x (1 - exchange commission)) - back stake
Profit = (110.26 x (1 - 0.05)) - 100
Profit = £4.74
As you can see, the profit (£4.74) is the same whichever player wins.
calculating stakes for arbitrage between two bookmakers
When arbing between bookmakers, no betting exchanges or lay bets are required. You simply place a bet on every possible outcome with different bookmakers.
By choosing the bookmaker offering the highest odds for each possible outcome, it is sometimes possible to create an arb.
Let's use a tennis match as an example again.
Header | Player 1 win odds | Player 2 win odds | margin |
---|---|---|---|
bookmaker a | 1.6 | 2.4 | Cell |
implied probability | 62.5% | 41.67% | 104.16% |
bookmaker b | 1.4 | 3.0 | Cell |
implied probability | 71.42% | 33.33% | 104.75% |
Cell | Cell | Cell | Cell |
Best Odds | 1.6 | 3.0 | 95.83% |
The table above shows that if we were to cover all outcomes using a single bookmaker, we would make a loss as the combined market margin is above 100% (shown in red), giving the bookmaker an edge. However, if we back each outcome using the best odds from more than one bookmaker, the combined market margin drops to below 100% (shown in green), producing an arb.
In this example, our profit would be 4.17% (100% - 95.83%).
We now need to calculate our stakes for each bet so that we return the same profit regardless of the outcome. We can do this using one formula.
Stake = (total stake x implied probability) ÷ combined market margin
As our stakes will be different for each bet, we must apply this formula to each one.
The following is based on using a total stake of £100 across all bets.
Bookmaker A Stake = (100 x 62.5%) ÷ 95.83%
Bookmaker A Stake = £65.22
Bookmaker B Stake = (100 x 33.33%) ÷ 95.83%
Bookmaker A Stake = £34.78
We have now staked a total of £100. Let's take a look at the possible returns.
Bet wins with Bookmaker A: £65.22 @ 1.6 = £104.35
Total Profit = £4.35
Bet wins with Bookmaker B: £34.78 @ 3.0 = 104.34
Total Profit: £4.34
As you can see, our profit is the same whichever player wins.
calculating stakes for arbitrage on three-way markets
Not all events have only two possible outcomes. For example, in a football match there are three outcomes (Home Win / Draw / Away Win) and for other sports there are many more.
In this example, we'll focus on a football match with the following odds for each outcome at three different bookmakers.
Header | home win | draw | away win | margin |
---|---|---|---|---|
bookmaker a | 2.0 | Cell | Cell | Cell |
implied probability | 50% | Cell | Cell | Cell |
bookmaker b | Cell | 3.6 | Cell | Cell |
implied probability | Cell | 27.78% | Cell | Cell |
bookmaker c | Cell | Cell | 5.2 | Cell |
implied probability | Cell | Cell | 19.23% | Cell |
combined market margin | Cell | Cell | Cell | 97.0% |
As the combined market margin is below 100%, we have found an arb.
We now need to calculate the ideal stakes for each of the three outcomes.
The following formula will allow you to calculate the stake.
Stake = (total stake x implied probability) ÷ combined market margin
You will need to apply this to each of the three outcomes.
We will be using a total combined stake of £100.
Bookmaker A Stake = (100 x 50%) ÷ 97%
Bookmaker A Stake = £51.54
Bookmaker B Stake = (100 x 27.78%) ÷ 97%
Bookmaker A Stake = £28.64
Bookmaker C Stake = (100 x 19.23%) ÷ 97%
Bookmaker A Stake = £19.82
We have now staked a total of £100. Let's take a look at the possible returns.
Bet wins with Bookmaker A: £51.54 @ 2.0 = £103.08
Total Profit = £3.08
Bet wins with Bookmaker B: £28.64 @ 3.6 = 103.10
Total Profit: £3.10
Bet wins with Bookmaker C: £19.82 @ 5.2 = 103.06
Total Profit: £3.06
Our profit is roughly the same no matter what the result in the match is.