Formulas and Maths Of Arbitrage

Arbitrage. An Arbitrage or Arb, is a transaction that takes advantage of price differences offered by different bookies and betting exchanges for the same event. The event may be a tennis game, football match or even a television talent show, but whatever the event, we only need prices offered by bookies to differ slightly in order to place a profitable Arb. No-matter what the subject of the Arb,  we need to find prices offered which allow us to Dutch the market of an under-round book, in order to make a guaranteed profit.

On this web page Over-round, under-round Making a book Arb a Win market spreadsheet New Arb Each Way spreadsheet

The Basics.  Odds as a percentage.
To recognize an Arb, we need to understand odds expressed as a percentage.

To calculate the percentage of your odds, divide the decimal "Betfair" odds into 100.
For example, we know that even money is a 50 - 50 chance, so evens should equal 50%.
Decimal "Betfair" odds for even money is 2.0.
100 divided by 2 = 50%

To calculate the percentage of fractional odds, add 1 to your odds and divide into 100.
For example, evens (1/1) + 1 = 2.0
100 divided by 2 = 50%

Another example, 3/1 + 1 = 4.
100 divided by 4 = 25%

Over-round and under-round of a book.
An Arb situation exists when the odds available from a selection of bookies allow us to place bets covering all outcomes of an event, and the odds obtained for our bets give us an under-round book.

Over-round or under-round is calculated by expressing all the odds of an event as a percentage, and adding them together.
If the total is more than 100%, the book is over-round, if less than 100%, the book is under-round.

The simplest Arb is an event with just 2 possible outcomes, such as a tennis match, where a draw is virtually an impossibility - either one player or the other will be declared the winner.
Other Arbs may involve 3 outcomes such as a boxing or football match which involve 2 wins and a draw, or an event with many outcomes such as a television talent show.

Making a book.
For example, a horse race has runners priced at evens, 3/1, 4/1 and 9/1.
1/1 is 2.0 in decimal odds, so 100 / 2 = 50%
3/1 is 4.0 in decimal odds, so 100 / 4 = 25%
4/1 is 5.0 in decimal odds, so 100 / 5 = 20%
9/1 is 10.0 decimal odds, so 100 / 10 = 10%
Adding all those percentages together gives 105%.  Our book is 5% over-round.
If we only had evens, 3/1, and 4/1, our book would be 95%, so would be 5% under-round.

Arb example.
Suppose we were interested in a tennis match.
We could look at a selection of bookmakers and compare the odds on offer :-

Bookmakers	Player A	Player B	Book %
Lads	Evens (2.0)	4/5 (1.8)	50 + 55.56 = 105.56%	Over-round
Hills	Evens (2.0)	4/5 (1.8)	50 + 55.56 = 105.56%	Over-round
Tote	Evens (2.0)	4/5 (1.8)	50 + 55.56 = 105.56%	Over-round
Coral	6/4      (2.5)	8/13 (1.61)	40 + 62.11 = 106.56%	Over-round

In this fictitious event, Coral take a slightly different view of the outcome.
Although each bookie has an over-round book of 105% or 106%, we can bet UNDER-round by betting with different bookies and obtaining odds of 6/4 (2.5) and 4/5 (1.8).
Those odds give a percentage of 40 + 55.56 = 95.56 which is under-round by 4.44%.

We can eliminate gambling and make a guaranteed profit on this match by Dutching both outcomes for an equal profit, no-matter what the outcome.
Whether Player A wins or Player B wins in irrelevant.
Get this right, and you cannot lose.
This is the equivalent of an "All Green Screen" on Betfair.

There is no need to learn any complicated math's in order to Dutch or Arb.
An Excel spreadsheet can easily work out stakes as fast as you can type, and will also eliminate mathematical errors.

Note that this spreadsheet limits your Total Stakes to a set £ amount.
In order to keep the Total Stake at a set amount input by the user, this spreadsheet does not use the percentage calculations described above.

Excel spreadsheets are a quick and reliable way of doing calculations. You can build a spreadsheet using the percentage formulas above, or purchase an excellent spreadsheet here for only a fiver. Be A Bookie spreadsheet. You will need Excel 2000 or a later version to view this spreadsheet.  Watch an on-line demo of the Be_A_Bookie spreadsheet This Be A Bookie spreadsheet calculates the lay stakes required to Lay up to 25 selections to an equal liability. You could of course use this spreadsheet for any event other than horse racing. A 2nd spreadsheet within this Excel file calculates the bet stakes required to bet up to 25 selections to an equal profit. Input the amount of your Total Payout, and as you input the odds of your selections, the spreadsheet shows :- Total payout on any winner remaining constant at your original input. The Lay stake or bet stake required for each runner. Equal liabilities or profit against each runner. Liability on each runner reducing as more runners are Layed. Profit on each runner that you have Bet. Percentage of your book at all stages. Total of Lay stakes at all stages - - - the "Skinner" amount. OR, total of bet stakes if you are betting. The more runners you Lay, the less your liabilities become. The more runners you Lay, the bigger the payout on a "Skinner" - a horse you haven't layed. The Betting spreadsheet can be used for Arbitrage of any event. Bet all runners for a book percentage that is less than 100% and you have guaranteed winnings with zero risk. These zero risk trades are available daily.