# Calculate odds for lay markets based on back markets

Some days ago I read an interesting article about how bookies arrange their margin to the possible outcomes. All bookies keep this of course secret as this offers them a specific range, where they can shorten or lengthen the odds depended on the amounts of placed bets. But I need this information, because I simulate my prediction models for back and lay markets, with just the odds of the back markets. This post will explain, how you should calculate the bookie margin, and how you should not do it.  I handled this topic a little bit naively during the development of my Poisson model, which causes some problems.

## Bookie margin

Every bookie uses a different margin for their business. There are some bookies with a lower margin like Pinnacle (~2.4%) and some bookies with a higher margin like Bet365 (~4.3%). This margin differs a little bit for every sport, bet-type and match. To identify the margin for a single bet, you just need to sum the probability of every possible outcome.

As you can see, the margin differs from game to game. It varies between 4% and 6%. On average it should be about 4.3% for Bet365. But it not only differs between matches. It differs also for every different market for one single match. The next example shows the different margin for the back and lay markets for the bookie Tipico. The match Wolfsburg – Cologne had a margin of 4,6% for the 3-way back market. The single 2-way margins vary a little bit.

And this is the part, where I made an error during my simulations of the Poisson model. I tried to identify value bets for the back and the lay markets. But my current data set just contains the prices for the back markets. So I need calculate the odds of the lay markets without knowledge about the 2-way margin.

## Fixed or match margin

The real probability of the lay markets without the margin is easily described. It is the counterevent to the back market. So the probability for the lay market is:

$P_{lay-home} = 1 - P_{back-home}$

Now the margin needs to be added in order to get the bookie odds. At first I just used the average bookie margin. Bet365 has an average margin of 4.3%.

$P_{lay-home} = 1,043 - P_{back-home}$

As already mentioned, the real margin varies around that average. I expected the possible differences between the real match margin and the average margin do not heavily effect the value detection. But I did not think about the possible difference between the calculated odds of the lay market and the corresponding profit for winning lay bets. Using the match margin instead of the average margin results in some bigger differences. The margin for the lay-market is simply the sum of the both corresponding back markets.

$P_{lay-home} = ( P_{back-home} + P_{back-draw} + P_{back-away} ) - P_{back-home}$

Following examples show the profit difference between the fixed margin and the match margin. Both margin options offer value, but the match margin calculations always provides less profit. This is caused by the margin difference between the average bookie margin and the margin for specific football matches.

So simulating multiple season with the additional Lay Draw and Lay Away markets multiplies the error, which causes an absolute different overall profit for the Poisson model. Suddenly the Poisson model does not look so good anymore. Especially because of the huge loss during the current season.

## Conclusion

Back-testing a model with lay and back markets should always be done with the specific match margin, if your  dataset only contains the back market odds. The margin differs too much in comparison to the average bookie margin. Of course it would be perfect, if the real historic market odds are available. But in my case, this is currently not possible.

In the next post I will sum up my experience of 1-year productive usage of the Poisson model. There I will take a more precise look at the loss of the current season.