My book’s name, Fine Margins, comes from the following. Every odds quote from a bookmaker combines two things: the bookmaker’s assessment of a bet’s chance of winning and a built-in profit margin. Understanding those margins is the key to seeing precisely how sports betting works.
Fair Odds
To see how this works, consider a bet with decimal odds of D. That means that a €1 bet returns €D if it wins (including the original bet). Now assume the bookmaker offers fair odds. This means that on average a €1 bet gets back €1. Sometimes you win, sometimes you lose, but on average you break even.
Suppose they think the bet has probability p of winning. Then fair odds would be
D = \frac{1}{p}A fraction p of the time, you get back D and the rest of the time you get nothing. So you hand over €1 and, on average, you get €1 back.
Note that you can tell the probability that the bookmaker used to set the odds. It is just the reciprocal of the decimal odds.
Introducing the Margin
Alas, the world is not fair – neither in general, nor in the sense that bookmakers decline to take a profit – and this makes mapping odds into probabilities trickier.
Every odds quote you see from a bookmaker has been set as
D = \frac{1-m}{p}where m is a positive number between zero and one.
This means m represents the expected profit margin per unit staked for the bookmaker and the expected loss rate for the bettors. Over time, the bettor will win a fraction p of the time getting an average return of
p D = 1 - m
on one unit bets. In other words, over time, bettors accepting this bet will lose a fraction m of their money and the bookmaker will win a fraction ,
In practice, this m will vary across events and sports. In the book, I present lots of evidence on this.
Calculating the Margin
Generally, just staring at the one number that is your odds quote, you can’t figure out its two components – the probability assumed by the bookmaker and their profit margin.
However, as I explained in my research with Tadgh Hegarty (an experienced bookmaker), there is one simple condition under which you can calculate the bookmaker’s profit margin just from the odds. If the odds are strongly efficient (as defined by Nobel Prize winner Richard Thaler and William Ziemba), meaning the same margin has been applied to each possible outcome of an event, then you can calculate the margin directly from the odds.
Let’s see how that works. Suppose the odds are set so that for all of the potential outcomes of an event, the built-in profit margin is the same. And assume there are N possible outcomes each with its own odds quote Di (so the odds on the first outcome are D1, the odds on the second outcome are D2 and so on.) So all the odds are set as
D_i = \frac{1-m}{P_i}where the Pi are the bookmaker’s different probability estimates for each outcome.
I know most people don’t like equations but this next step isn’t too hard to see. If all the odds have the same margin, then the probabilities the bookmaker used for each of them can be written as
P_i = \frac{1-m}{D_i}where the Pi are the bookmaker’s different probability estimates for each outcome.
I know most people don’t like equations but this next step isn’t too hard to see. If all the odds have the same margin, then the probabilities the bookmaker used for each of them can be written as
Overround = \frac{1} {D_1} + \frac{1}{D_2} + …. + \frac{1}{ D_N} = \frac {1}{1-m}Specifically, the reciprocal of the overround tells us what the margin is
1 - m = \frac{1}{Overround}That number, 1 – m, tells us the average amount of money the bettor gets back. For example if the overround is 1.1 then that is 0.909. For every €1 they stake, bettors get €90.9 cents back – meaning an average loss rate for the bettor of 9.1%. And an average profit rate for the bookmaker of about 9.1%.
A Common Error
There are plenty of online discussions of this topic that tell you the bookmaker’s margin can be calculated by subtracting one from the overround. For example, they tell you that when the overround is 1.1, the margin is 10%. There are plenty of formal-looking “calculators” that use this method.
But you can see that is not correct – in this case, it is actually 9.1%. This error is small enough when the margin is low but margins are not always low. For example, an overround of 1.5 implies a 33% margin, not a 50% one. Either way, you are better off using the correct formula.
Calculating Probabilities
Now, note something else. Since the probability used by the bookmaker is the ratio of 1 – m to Di and you know what the odds are, this means that once you know the margin, you also know the probability. Plugging in the margin into our last formula above for Pi, you get
P_i = \frac {\frac {1}{D_i}} {\frac{1} {D_1} + \frac{1}{D_2} + …. + \frac{1}{ D_N} }This looks ugly, but it is a calculations millions of bettors do every day. Start with the raw estimate of the probability implied by fair odds. Sum them up. Then divide each raw probability by the sum. These are known as normalised probabilities. Generally, they are seen as a sensible but slightly ad hoc solution. You have some raw estimates of probabilities that you know are not correct because they sum to more than one. So “normalise” them by dividing them by the sum.
What we have seen here is that strong efficiency means this is not an ad hoc procedure. If the odds are strongly efficient then you can calculate the bookmaker’s margin and also the normalised probabilities are equal to the probabilities used by the bookmaker. Without the bookmaker telling you what they were thinking, these odds reveal everything that went into setting them.
Two Free Spreadsheet Calculators
So it’s not hard to work out estimates of the bookmaker’s margins or probabilities. But not everyone wants to be working through the calculations above. For this reason, I have put together two simple spreadsheets that can do these calculations for you, starting from whatever the odds are on the event you’re interested in.
First, a simple calculator that allows you to enter either fractional, decimal or American odds and two, three, four or five possible outcomes. It tells you the bookmaker’s margin and their probabilities, under the assumption that the odds are strongly efficient.
Second, a more general calculator that allows you to enter as many different odds as you want, e.g. for 20 horses in a race.
An Important Caveat
This calculations are perfect if betting markets are strong efficient. Alas, they are typically not and so these calculations, while generally a useful guide, can sometimes be misleading. Fine Margins contains plenty of discussion of this issue and what it means for the true probability your bet has of winning and what the likely true bookmaker’s margin is. There will be more posts on this soon.