Formulas and Maths Of Arbitrage

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 :-

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 instantly 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. 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 required for each runner. Equal liabilities against each runner. Liability on each runner reducing as more runners are Layed. Percentage of your book at all stages. Total of Lay stakes at all stages - - - the "Skinner" amount. 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. Note that this spreadsheet is also available on the Dutching and Laying web pages. After payment via PayPal, select the "Return to merchant" option on the PayPal screen. PayPal should then route you to a download web page where you can obtain the Excel spreadsheet file. Be A Bookie spreadsheet price = £5. Payment is by PayPal, but you don't need a PayPal account to use the payment button below.

Hedge a bookie bet.
The name "Arbing" is also often applied to the strategy of betting with a bookie and laying the same selection on a betting exchange.
This is in fact a Hedge strategy.
This is a very well known method of trading to guaranteed winnings no-matter what the result of a horse race, football match or any other event.
When making these trades, and "Arber" will compare the bet odds offered by a bookie with the lay odds available on a betting exchange such as Betfair.
If the bookie bet odds are bigger than the Betfair lay odds, guaranteed winnings are there to be traded, but commission on the lay side of the trade needs to be taken into consideration.
To make guaranteed winnings from these trades, usually more than one tick difference between bet and lay odds will be required.

An alternative strategy to trading in this way for an equal win or lose profit is to bias the staking to favour a win outcome.
The theory here is that in making these trades, we are continually betting at good value odds with the bookie, so in the long term the profit from hitting winners should outweigh the profit from hitting losers.. . . . That's the basic theory.
To make that pay, the lay stake is adjusted for zero profit or loss on a loser, which gives a bigger profit if we hit a winner.
We then trade a load of races (these trades are usually done in volatile horse race markets), and make a profit from every winner whilst breaking even on all the losers.

The spreadsheet below can be used to make these trades.
A second spreadsheet in this Excel file shows the lay stake required to break even if we hit a loser.
Input the size of your bet stake with the bookie, plus the bet and lay odds and the spreadsheet shows a range of lay odds and stakes for trading to zero loss on losers after commission deductions, and a profit if we hit a winner.
Remember that with commission built into these calculations, to make a profit, you will probably need 2 or 3 ticks or more difference between the bookie bet odds and the Betfair lay odds when you make these trades.

Excel spreadsheets are a quick and reliable way of doing calculations. You can build a spreadsheet to trade the strategy described above, or purchase an excellent spreadsheet here for only a fiver. Hedge Your Bookie Bets spreadsheet. You will need Excel 2000 or a later version to view this spreadsheet. Input the amount of your Bet Stake with the bookie. Input the bookie bet odds and the Betfair lay odds and the spreadsheet shows :- A range of lay odds and lay stakes Equal profit or loss if we then lay at any of those odds using the lay stake shown.. The liability of your lay. An alternative spreadsheet allows us to bias the staking to favour a win outcome and break even on all losers. Commission is built into the lay side of the calculations. If we hit a loser, we would pay commission on our successful lay stake. If we hit a winner, we have bet winnings with the bookie, but lay liabilities with Betfair, so there would be zero Betfair commission to pay. After payment via PayPal, select the "Return to merchant" option on the PayPal screen. PayPal should then route you to a download web page where you can obtain the Excel spreadsheet file. Hedge Your Bookie Bets spreadsheet price = £5. Payment is by PayPal, but you don't need a PayPal account to use the payment button below.

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