Screen shot of BETDAQ (a betting exchange).

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Odds and bookmaker margin

I think most people know betting sites where they can place bets for certain odds on different sports events. For example the bookmaker Pinbet88 (the Asian version of the "famous" Pinnacle betting site) offers odds of 2.00 for Leicester City to win its next Premier League game against Stoke City.

Odds of 2.00 mean that if for example you place 100 euro on Leicester and win you will win altogether 100 euro ∙ 2.00 = 200 euro. That means your net profit would be 200 euro minus the stake is 100 euro. On the other side if you lose the bet your net loss would be 100 euro, too.
These odds would be fair in case the probability that Leicester wins was exactly 50 % (also if you toss a coin and bet on a certain side, fair odds would be 2.00 because here the probability to chose the correct side is indeed 50 %). Unfortunately betting odds of 2.00 of don't mean that the real probability is 50 % because the bookmaker wants to make some profit.
Lets check the Pinbet88 odds for that game more thoroughly:

Team to win:	Leicester City	Draw	Stoke City
PinBet88 Odds:	2.00	3.49	4.11
Fair odds would be:	2.06	3.59	4.23

Compared to many other bookmakers these odds of Pinbet88 are actually rather good, but there is still a bookmaker margin.

How to calculate the bookmaker margin?

You can calculate this margin by summarizing the inverse values of the odds of each possible outcome:

In case of the Leicester game that would be:

That is a margin of about 3 % and means that you have to place altogether 103 euro (shared on all three outcomes in a way to win the same amount whatever the result is) of the event to get 100 euro back. That is the reason why normally in the long run bookmakers make some profit.

How to calculate the "fair" odds?

Lets finally check the formula when the fair odds are inserted:

As you can see, a bookmaker who would offer these odds couldn't make any profit. How to calculate fair odds if you have the bookmaker odds? I will show you how to get the 2.06 for Leicester (the fair odds for the other results you can calculate in the same way):

Arbitrage betting

One option to make some profit even if bookmakers always implement a margin into their odds is to try to exploit the possible different opinions of different bookmakers about a sports event. In these cases one may be able to play an arbitrage bet. I described in a very basic way how to do that already in German. If there is enough interest I could translate that into English.

How to use betting exchanges?

Another possibility is to use a betting exchange. You can place bets there like with a normal bookmaker (then you "back" them), but in addition you can also sell (lay) them. To lay a bet means to act like a bookmaker yourself and offer own odds which you hope to get accepted by any other user of the betting exchange.

If you bet for example at PinBet88 on Leicester, you will get odds of 2.00. If at the same time you see that any user of any betting exchange wants to place a bet on Leicester as well, you can offer him odds of (for example) 1.95 and hope that he accepts them. If accepted you can then play the bet at PinBet88 and make a save profit. Why? Because if Leicester wins you get odds of 2.00 but have only to pay odds of 1.95 at the exchange. If Leicester loses you lose against PinBet88 but win in the Exchange. You only have to balance the stakes in the correct way.

Consider the fees!

Before I explain how to balance the stakes, I have to point out another aspect one has to include in ones calculation: the owners of betting exchanges want to earn something - therefore you have to pay a little fee for every transaction (bet).

Necessary Calculations:

First lets term the parameters:

• W_B = possible winnings against the bookmaker
• O_B = bookmaker odds
• S_B = bookmaker stake
• W_Lay = possible winnings in the exchange
• S_P = stake which you let other players in the exchange place at your odds
• C = commission rate of the exchange in percent

Then the calculation goes like that:

W_B = (O_B - 1) ∙ S_B

W_Lay = S_P ∙ (1 - C / 100)

If you win against the bookmaker you have to subtract the loss in the exchange from the winnings:
W_B = (O_B - 1) ∙ S_B - (O_Lay - 1) ∙ S_P

If you win in the exchange you have to subtract the loss against the bookmaker from your winnings:
W_Lay = S_P ∙ (1 - C / 100) - S_B

To make sure to win the same amount no matter if you win against the bookmaker or in the exchange you must equate both possible winnings:
(O_B - 1) ∙ S_B - (O_Lay - 1) ∙ S_P = S_P ∙ (1 - C / 100) - S_B

Resolved after the stake which you let other users bet who accept your bet at the exchange:
S_P = (O_B ∙ S_B) / (O_Lay - C / 100)

Oh, so much theory :-) ... finally an example:

Lets assume you want place 1000 euro at PinBet88 on Leicester. Which amount should you let other users of the exchange bet on your offered odds of 1.95 on Leicester now? Lets further assume a fee of 3 % of the net winning in the exchange (in case you win there).

The formula was: S_P = (O_B ∙ S_B) / (O_Lay - C / 100)

Odds, stake in PinBet88 and commission fee inserted:

S_P = (2.00 ∙ 1000 euro) / (1.95 - 3 / 100) = 1041.67 euro

That means you let the opposite player on the exchange bet 1041.67 euro on Leicester (odds: 1.95).

Now lets compare the possible winnings:

PinBet88:

The formula was: W_B = (O_B - 1) ∙ S_B - (O_Lay - 1) ∙ S_P

Parameters inserted:

W_PinBet88 = (2.00 - 1) ∙ 1000 euro - (1.95 - 1) ∙ 1041.67 euro = 10.41 euro

Betting exchange:

The formula was: W_Lay = S_P ∙ (1 - C / 100) - S_B

Parameters inserted:

W_Lay = 1041.67 euro ∙ (1 - 3 / 100) - 1000 euro = 10.42 euro

We have calculated the exact stake we have to offer in the betting exchange so that in case of both results we will have exactly the same net profit of 10.42 euro (the difference of 1 cent is because of some inaccurate rounding) in case we place 1000 euro on a win of Leicester in PinBet88.

Test formula to check which odds you can afford to offer

By the way, how can I be sure to win something at all if I offer odds of 1,95 at the exchange in spite of the fees? There is a test formula:

If the result is positive your offered odds at the exchange are low enough. In our example the formula will indicate a profit of 0.52 % if fees are 3 % and one offers odds of 1.95 for Leicester.

Of course I don't calculate all that manually. It is necessary to automate all calculations.
I have to say that I didn't write anything about trading on exchanges here - that's another topic and could be considered in another article.

List of betting exchanges

• BETDAQ
• Matchbook
• Smarkets
• Betfair; market leader but very greedy (concerning commission fees)
• Fairlay; bitcoin betting exchange!