How to Remove the Bookmaker’s Margin to Calculate True Odds
28 January, 2020

For any tipster or bettor signed up to the principles of value betting – and if you’re not, you should give up now – the most important thing to worry about is the true probability of something happening. Without some knowledge of what that is, you will have no way of knowing whether the odds you back at a bookmaker will offer you what is called expected value. If you don’t have expected value, you will end up losing money in the long run.

Expected value measures how much profit as a percentage of your turnover you expect to make over the long run, once good and bad luck have evened up. Suppose you had a biased coin and you tricked people into betting 50-50 propositions, secretly knowing that the coin was weighted 55% heads, 45% tails. After 5 wagers you might have been unlucky and seen 3 tails and 2 heads. But over the long run, you’d expect 55% heads and 45% tails. If every bet was \$1 to win \$1, then you would expect to win \$55 for every \$45 lost, leaving you \$10 profit from the total of \$100 wagered, or 10%. 10% is your expected value.

To calculate the true odds we must remove the margin. To remove it, we first must know how the bookmaker applied it. But bookmakers aren’t in the habit of telling us anything about how they do business.

How do you calculate the true probability of a sporting outcome? One method uses the closing price of a sharp bookmaker like Pinnacle. Because Pinnacle’s model of bookmaking is based on large turnover, it is reasonable to assume, for large betting markets (like the Premier League or Serie A) at least, that the price or odds a team or player settles at just before the match starts – called the closing (market) price – is the best measure of the true probability. Why? Because it contains the most amount of market information, expressed in the form of customers’ opinions via the money that they have staked, about the match in question. For odds quoted in decimal notation, you just invert the figure to give you the outcome probability. Odds of 4.00, for example, imply a 25% probability (1/4.00 = 0.25 or 25%).

But how can a bookmaker’s odds represent the true probability if it contains a margin? A bookmaker’s margin is effectively a commission they earn for the effort to offer the betting market to customers in the first place. Instead of offering fair odds, they reduce them a little bit to make them unfair. If the bookmaker manages to balance their customers’ money in the right way, then regardless of the match outcome, they will make a little profit. The size of their margin dictates the size of their expected profit. You can read more about the bookmaker’s margin in GodsOfOdds article about how the bookmaker makes money.

To calculate the true odds, then, we must remove that margin. To remove it, we first must know how the bookmaker applied it. But bookmakers aren’t in the habit of telling us anything about how they do business, and understandably so. Hence, we must make an educated guess.

The most obvious assumption is that a bookmaker applies their margin equally across all sides in a match. Suppose for a two-player event the true odds were 2.00 and 2.00. With a margin of 2.5% applied equally, their published prices would drop to 1.95 for both sides. But can we be sure that the margin is applied equally to both sides like this? For matches where the possible outcomes are similarly likely, it’s not a bad assumption. But for matches where there is a strong favourite (and hence an underdog too), this is not so accurate.

It is now well documented that in sports where there are many favourites and underdogs, underdogs tend to be priced disproportionately shorter than favourites relative to their true chances of winning. The reasons for this are complex (for example the biased behaviour of bettors and the risk aversion of bookmakers protecting against variance), but it means that when bookmakers apply their margin, they place more weight on the underdogs than the favourites. This is called the favourite–longshot bias.

There are several conceivable methods that a biased margin might be applied, and hence removed. For the purposes of estimating the expected value of our tipsters, GodsOfOdds has chosen the ‘Margin Weights Proportional to the Odds’ method, as published by Football-Data.

This method assumes that the margin weight applied to each outcome is proportional to the size of the betting odds (or inversely proportional to the outcome probability). Hence, for a market with n runners and overall profit margin M, the specific margin weight applied to the fair odds for the ith runner (Oi) will be given by:

Mi = MOi / n

For a tennis match betting market with 2 possible outcomes (player 1 or player 2 win), the margin weights would be:

M1 = MO1 / 2

M2 = MO2 / 2

For example, fair odds of 1.25 and 5.00 and a margin (M) of 0.05 or 5% would have differential margin weights of 0.03125 (3.125%) and 0.125 (12.5%) respectively. To calculate the actual prices one then simply divides the fair price by the margin weight plus 1. In this example these would be 1.25/1.03125 and 5.00/1.125, giving 1.212 and 4.444. By contrast, if the margin had been applied with equal weight across both players, the odds would have been 1.190 and 4.762. You can see from this exercise that a biased weighting of odds in this manner shortens longshots more significantly than favourites.

We can reverse the process to calculate what fair odds the bookmaker will have estimated in the first place, given their margin and applying this model of biased margin weighting. Hence, for the published odds Obookmaker, the fair odds, Ofair, from which they came will be given by:

Ofair = nObookmaker / (n-MObookmaker)

For a 2-outcome market just substitute 2 for n in the equation above. Try putting 1.212 and 4.444 into the equation where M = 0.05 and see what the fair odds are. You should get 1.25 and 5.00. To calculate the margin (M) for any market, simply sum the reciprocals of the bookmaker’s odds and finally subtract 1. In this example, then, M = (1/1.212) + (1/4.444) – 1 = 0.05.

To conclude, following the assumption that Pinnacle’s closing price is the best measure of the true odds, to calculate it simply remove their margin following the methodology above. Dividing the odds you bet by these true odds and subtracting 1 will then give your expected value. If you bet 2.20 and the true price was 2.00, you will have found 10% expected value (2.20/2.00 – 1).

## 1 comment

It's hard to wrap around my head what really happens with this formula, but it is actually really simple.
Let's say the odds are 1.333, 4 and 16.66667.
Convert them to probabilities: 1 / 1.3333 = 0.75 and 1 /4 = 0.25 and 1 / 16.6667 = 0.06
So we add them up 0.75 + 0.25 + 0.06 = 1.06.
So there's 0.06 margin.
Divide that margin by 3, since there 3 options (home / draw / away), which is 0.02 margin per option.
Take the implied probability and subtract that from the implied probability we have, which makes: 0.75 - 0.02 = 0.73, 0.25 - 0.02 = 0.23, 0.06 - 0.04.
So the actual probabilities are: 0.73, 0.23 and 0.04.
If you want we can convert those back to actual odds: 1 / 0.73 = 1.37 and 1/ 0.23 = 4.35 and 1/ 0.04 = 25.
So the method looks really complicated, but it is just: take the margin, divide it by 3 and subtract that in equal parts from the implied probability. That's it.
Btw, this method fails if the implied probability is for example 0.01, and if you would then subtract 0.02 you would get a negative probability.