We explain how to calculate arbitrage bets to give you an understanding of the maths behind arbitrage betting, and why it’s a low-risk betting strategy. Learn how to calculate arbitrage bets between bookmakers and an exchange to maximise your potential arbitrage betting profit.

If you’re new to arbitrage betting we have already explained in detail what it is, but to provide context for this article, it is a trading technique which benefits from price differentials between two or more bookmakers or betting exchanges to guarantee a profit regardless of the outcome.

Arbitrage sports betting is the method of placing bets on all outcomes of an event at odds that guarantee a profit regardless of the eventual result.

Let’s explain this briefly. A fair market would be priced at 100% based on the likelihood of an event occurring, however, bookmakers will price their odds to go above a 100% probability, therefore giving them an edge.

Arbitrage opportunities are the reverse of this, whereby an arber will bet on all eventualities across a number of betting providers, giving them the opportunity to take advantage of discrepancies in price so the probability of the odds they have bet on is lower than 100% -  therefore in their favour.  

It is fundamental when making an arbitrage bet to understand the maths behind arbitrage betting,  however if you need to quickly calculate an arbitrage bet our arbitrage calculator is free and easy to use.

Arbitrage calculator

Using the Smarkets arbitrage calculator is simple. Just follow the below instructions:

1. Input the total stake you want to risk.
   
   
2. Input the odds for outcome 1 and the commission if on a betting exchange (if you are arbing between bookmakers set the commission value to 0).
   
   
3. Repeat for the other outcomes you would like to arb (add new outcome if you require more than 4 outcomes on a market).
   
   
4. The arb calculator will then automatically inform you how much you need to stake on each outcome and your profit.

How to calculate arbitrage betting between a bookmaker and exchange

The most common arbitrage bet is made by taking positions in the market across a bookmaker and a betting exchange - backing at the bookmaker and then laying the same outcome on the betting exchange.  

As an example let’s say you want to bet on a tennis match between Player A and Player B. The odds on the bookmaker and the market implied probabilities in brackets - learn how to calculate betting margins - are displayed below:

	Player A to win	Player B to win	Market margin
Odds	2.20 (45.45%)	1.68 (59.52%)	105%

You now look at the lay price on Player A to win - betting that he won’t win - which is 1.98 with the Smarkets exchange. The table below highlights this is an arbitrage opportunity as the combined implied probability for bets is 95.96%. 

	Player A to win at bookmaker	Player A not to win at Smarkets	Combined market margin
Odds	2.20 (45.45%)	1.98 (50.51%)	95.96%

Calculate your lay stake for arbitrage bets on a betting exchange

Now you stake £200 on Player A to win on the bookmaker with odds of 2.2. To calculate your lay stake on the Smarkets exchange you simply use the following calculation:

(back price x back stake) / (current lay odds - exchange commission)

You would then make a £224.49 lay bet on Player A to win at odds of 1.98 on the betting exchange.

Calculate your profit if you win at the bookmaker

Calculating your profit if your bet wins with the bookmaker is simple:

Profit = (back odds - 1) * back stake - (lay odds -1) * lay stake

Calculate profit if the lay bet wins at the exchange:

Next, you calculate your profit if your lay bet with the exchange is successful:

Profit = (lay stake * (1 - commission)) - bookmaker stake

Overall arbitrage profit

Your overall arbitrage profit regardless of the outcome is displayed below :

Outcome	Bookmaker	Smarkets exchange	Profit/loss
Player A wins	+£240	-£220	+£20
Player A loses	-£200	+ £220 (includes 2% commission)	+£20

So if you placed this arbitrage bet, you would guarantee a profit of £20 irrespective of Player A winning or not.

How to calculate arbitrage between two bookmakers

Before betting exchanges arbitrage bets were made between two or more bookmakers - these are becoming less prevalent because most bookmakers have very similar odds. Having said that they do still occur.

For an arb to occur between bookmakers you need to identify the two bookmakers offering the highest odds on each outcome. The table below can help you quickly recognise if there is a potential arb across bookmakers. If you find odds similar to outcome 1, then odds which are greater than what is identified in outcome 2 would present an arbitrage opportunity:

Outcome 1 odds	Outcome 2 odds
1.2	>6
1.3	>4.33
1.4	>3.5
1.5	>3
1.6	> 2.66
1.7	> 2.42
1.8	> 2.25
1.9	>2.11
2.0	> 2.0

As an example let’s say you want to bet on a darts match between Player A and Player B. The odds on both bookmakers and the market implied probabilities are displayed below:

	Player A to win	Player B to win	Market margin
Bookmaker A odds	1.30 (76.92%)	3.70 (27%)	104%
Bookmaker B odds	1.40 (71.43%)	2.98 (33.56%)	105%

This clearly shows an arbitrage opportunity between bookmaker A and bookmaker B. The table below combines the odds and implied probabilities and shows they are under 100% - or in other words, in your favour - learn how to convert odds in to implied probability.

	Player A to win at bookmaker B	Player B to win at bookmaker A	Combined market margin
Odds	1.40 (71.43%)	3.70 (27%)	98.43%

By cross-matching the odds for each outcome across the two bookmakers the combined market margin is now in the bettor's favour providing a guaranteed return of 1.57%.

Calculating how much to stake for arbitrage bets

Now you must calculate how much to stake with each bookmaker to guarantee an equal return without consideration for the result. Let’s say you are willing to stake £100 overall.

Stake at each bookmaker = (Overall stake * Bookmaker implied probability)/Combined market margin

Overall arbitrage profit

Your overall arbitrage profit regardless of the outcome is displayed below :

	Bookmaker A	Bookmaker B	Profit/loss
Player A wins	-£27.43	+£29.02	+£1.59
Player B wins	+£74.06	-£72.57	+£1.46

How to calculate arbitrage on three-way betting markets

Now you know how to calculate arbitrage opportunities for a simple two-way arbitrage bet, you can look at three-way or 1X2 arbs. Three-way arbs are more prevalent in football where there is a possibility of a draw.

When arbing with a betting exchange there is no need for three-way arbs as you could back Manchester United to win on the bookmaker against Liverpool and then lay them to win on the exchange - which covers anything other than a Liverpool win.

So this example is for a three-way arb between three separate bookmakers. Let’s say you find the following odds on a game between Manchester United and Liverpool: 

	Man United to win	Draw	Liverpool to win	Market margin
Odds	Bookmaker A 2.12  (47.17%)	Bookmaker B 3.55 (28.17%)	Bookmaker C 5.40 (18.52%)	93.86%

This highlights an arbitrage opportunity between the three bookmakers as the table above combines the odds and implied probabilities and shows the market margin is under 100%. 

Calculating how much to stake for arbitrage bets

Now you need to calculate how much to stake with each bookmaker to guarantee an equal return depending on which bookmaker you win with. Let’s say you want to stake £1500 overall.

Stake at each bookmaker = (Overall stake * Bookmaker implied probability)/Combined market margin

Overall arbitrage profit 

Your overall arbitrage profit regardless of the outcome is displayed below :

Outcome	Bookmaker A	Bookmaker B	Bookmaker C	Profit/loss
Man United win	+£844.30	-£450.19	-£295.97	£98.14
Draw	-£753.84	+1147.98	-£295.97	£98.17
Liverpool win	-£753.84	-£450.19	+1302.27	£98.24

So for this example, if you placed £1500 stake across the different bookmakers as suggested you would guarantee a profit of around £98.

Apply this to betting

Now you know how to calculate arbitrage bets you are in the best position to take advantage of price discrepancies across bookmakers and betting exchanges. Remember Smarkets accept arbitrage bettors and also offer the best odds online - with our industry-low 2% commission.

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