There are a number of potential arbitrage deals. Below is an explanation of some of them including formulas and risks associated with these arbitrage deals. The table below introduces a number of variables that will be used to formalise the arbitrage models.
Variable Explanation
s1 Stake in outcome 1
s2 Stake in outcome 2
o1 Odds for outcome 1
o2 Odds for outcome 2
r1 Return if outcome 1 occurs
r2 Return if outcome 2 occurs
Bookmaker only Arbitrage
This type of arbitrage takes advantage of different odds offered by different bookmakers. Assume the following situation:
We consider an event with 2 possible outcomes (e.g. a tennis match - either Federer wins or Nadal wins), the idea can be generalized to events with more outcomes, but we use this as an example.
The 2 bookmakers have differing ideas of who have the best chances of winning, they offer the following Fixed-odds gambling on the outcomes of the event:
Bookmaker 1 Bookmaker2
Outcome 1 1.25 1.43
Outcome 2 3.9 2.85
For an individual bookmaker, the sum of the inverse of all outcomes of an event will always be greater than 1.
As they are in this case: 1.25−1 + 3.9−1 = 1.056 and 1.43−1 + 2.85−1 = 1.051
The fraction above 1, is the bookmakers return rate, the amount the bookmaker earns on offering bets at some event. Bookmaker 1 will in this example expect to earn 5,6 % on bets on the tennis game. Usually these gaps will be in the order 8 - 12%.
The idea is to find odds at different bookmakers, where the sum of the inverse of all the outcomes are below 1. Meaning that the bookmakers disagree on the chances of the outcomes. This discrepancy can be used to obtain a profit.
For instance if you bet at outcome 1 at bookmaker 2 and outcome 2 at bookmaker 1:
1.43−1 + 3.9−1 = 0.956
Placing a bet of 100 on outcome 1 with bookmaker 2 and a bet of 100 * 1.43 / 3.9 = 36.67 on outcome 2 at bookmaker 1 would ensure the punter a profit.
In case outcome 1 comes out, you could collect r1 = 100 * 1.43 = 143 from bookmaker 2. In case outcome 2 comes out, you could collect r2 = 36.67 * 3.9 = 143 from bookmaker 1. You would have invested 136.67, but have collected 143, a profit of 6.33 (4.6%) no matter the outcome of the event.
So for 2 odds o1 and o2, where o1−1 + o2-1 < 1. You wish to place stake s1 at outcome 1, then you should place s2 = s1 * o1 / o2 at outcome 2, to even out the odds, and receive the same return no matter the outcome of the event.
Or in other words, if you have two outcomes a 2/1 and a 3/1, by covering the 2/1 with 500 and the 3/1 with 333, you are guaranteed to win 1000 at a cost of 833, giving a 16% profit. More often profits exists around the 4% mark or less.
Quick version of calculating the percentages
Fractional odds:
- When using fractional UK-style odds (e.g. 5/6), divide the right hand number by the sum of both numbers and times by 100.
- Chelsea at 5/6 calculates as 6 / (5 + 6) x 100 = 54.54%
- Liverpool at 7/5 calculates as 5 / (7 + 5) x 100 = 41.66%
Decimal odds:
- When using decimal odds (e.g. 1.83), divide 100 by the odds.
- Chelsea at 1.83 calculates as 100 / 1.833333 = 54.54%
- Liverpool at 2.4 calculates as 100 / 2.40 = 41.66%
These percentages represent the cover of the event that the bookmakers have. Anything under 100% is an under-round and means that it is an arbitrage opportunity. So adding these percentages comes to 96.20% Subtract this from 100% and we have an arbitrage opportunity of 3.80%.
Quick version stakes calculation
Calculating how much money to stake on each selection is a vital part of the sports arbitrage process. If you don't stake your money proportionately you wont guarantee an even return no matter which selection wins. You must back each selection to a stake that is proportionate to their percentages that we calculated above. So in our case, stakes of £54.54 on Chelsea at 5/6 and £41.66 on Liverpool at 7/5 both return £100 no matter which selection wins and you only invested £96.20!
So, now you know what is possible with sports-arbitrage, let's take a look what is involved...