# Why Bookmakers Didn’t Find a Brexit More Likely – and why everyone thought they did

Bookies made a lot of money on the Brexit

As anyone knows by now, the British people voted for a Brexit, meaning that the United Kingdom will leave the European Union. The stock market didn’t particularly like this news, as indicated by the fact that the German DAX index opened 1.000 points, or nearly 10%, lower on the day the results became known.

It appears the majority of people expected the UK to stay in the EU. This expectation was likely fuelled by the fact that news agencies worldwide said that bookmakers (companies making their money by accepting bets) found it highly likely that Britain would stay in the EU. They pointed at the odds, or pay-offs, the bookies gave. The day before the referendum, the odds Betfair (a big bookmaker) gave were as follows: 1.16 for Bremain, 8 for Brexit. Meaning: if you would bet 1\$ on a Brexit, and it happened, you would get 8\$. For a Bremain you would get only 1.16\$.

Such a big difference in pay-off would only occur if Betfair was very sure that Britain would stay in the EU, right? Otherwise they would lose a lot of money, right?

Wrong. This is nonsense. Betfair didn’t find it more likely that Britain would stay in the EU. It probably didn’t even care, because they would profit regardless of the outcome. Let me explain this via some easy calculations.

First I show the reasoning used by the public and news agencies, such as CNBC (which said that bookies found a Bremain at least twice as likely as a Brexit, and that this was because “voting for the unknown takes higher conviction than voting for the status quo”). Suppose odds of 1.16:8 as given by Betfair. Now suppose a person bets 1\$ on Bremain. In case Bremain happens, Betfair has to pay him 1.16\$. If not, they earn his 1\$ bet. The same reasoning goes for a Brexit: the person earn 8\$ in case of Brexit, Betfair earns 1\$ in case of Bremain.

Now, what can Betfair expect to earn from this 1\$ bet? Either way, it will get the 1\$ bet. Now take p to be probability of Bremain. Then Betfair will have expected earnings of 1\$ – 1.16\$p in case of Bremain, and 1 – 8\$(1-p) in case of Brexit. The bookie of course doesn’t want to lose money. Now the only value of p at which it doesn’t expect to lose money, regardless of a Bremain or Brexit occurring, is 0.87 (calculate this if you don’t believe me). For any other probability, the person betting the 1\$ can expect to make money by betting on either Bremain or Brexit. In case of p = 0.6, for example, Betfair expects to lose 1 – 8\$(0.6) = 3.8\$ in case of a Brexit. Given that Betfair is not stupid, and doesn’t want to risk losing a lot of money, they must be damn sure of this 0.87 probability of a Bremain. Or so it would appear…

### But this is not how it works.

You can also use the expected value calculation in another way, the way used by bookies. Contrary to my example, there is more than 1 person making a bet. You can you use this information. Let’s say that 100 people make a bet of 1\$ each. Let’s say the bookie sets a pair of random odds of 4 to 9. Now assume 87 people bet for Bremain, and 13 for Brexit.

Suppose there is a Bremain. Then the bookie will earn 100\$ (100 times the 1\$ bets he receives) – 87 * 4\$(pay-out per person) = -248\$. In case of a Brexit they will earn 100\$ – 13 * 9\$ = -17\$. Something is going wrong here: the bookie will lose money in case of either a Bremain or Brexit. Apparently the odds don’t match the proportion of people voting for Bremain and Brexit.  In case the bookie changes the odds to 4:6, he will make money only in case of a Brexit, but not in case of a Bremain. Now, the only odds that will make sure the bookie never loses or earns any money, given the 87 people betting Bremain and 13 Brexit, is 1.16 to 8. This is the break-even set of odds.

Proof: in case of Bremain, the bookie will earn 100\$ – 1.16\$*87 = + – 0\$, and in case of Brexit 100\$ – 13*8\$ = +- 0\$. So irrespective of what will happen, the bookie will never lose money.  Now, by only charging a fee for using its service, or a percentage of the amount bet, the bookie will always make money. The bookie can also change the odds slightly, so that – given the same 87 people voting for Bremain and 13 for Brexit – he can make money without charging fees: if he would change the odds to 1.12:8, for example, he is still sure not to lose money in case of a Brexit, but he will actually make money in case of a Bremain.

You might think: these odds work only in case 87 of 100 people vote for Bremain, and 13 for Brexit. In case the ratio changes, so will the bookie’s earnings. That’s true. Maybe more people will be lured into betting Brexit by the 8\$ pay-off. You might get 20 of the next 100 people betting on Brexit, giving the bookie a negative pay-off of 100\$ – 20*8\$ = -60\$ in case of Brexit. What to do then? Well, you just lower the pay-off: in this case a pay-off of 5\$ for Brexit will do the trick. You can even leave the pay-off of Bremain at 1.16\$, implying that you will make money in case of Bremain and not lose anything in case of Brexit. The point being: the odds are constantly adjusted, reflecting the ratio of bets at the time, so that the bookie is sure never to lose money. He is likely to use some margin of error in the odds so that, until a more accurate set of odds is reached, he will still not lose any money.

So we cannot say that the bookies thought a Bremain more likely than Brexit. Looking at the odds, we can only infer that much more money was bet on a Bremain than a Brexit, as shown by the implied probability of 0.87.

But what was the real probability that British people would vote for a Brexit? Until two days before the Brexit, polls were published, which showed the referendum to become a very close call. Some polls showed Leave to be in front with 52% to 48%, others (such as the Financial Times poll) showed Stay to be in front. Either way you look at it, judging by the polls, there was no reason to assume that a Bremain was much more likely than a Brexit. Given an expected value calculation, and assuming probabilities of 55% for Stay and 45% for Leave, betting for a Brexit would give you an expected: -1 + 0.45*8 = 3.60\$ while betting on a Bremain would lose you an expected: -1 + 0.55*1.16 = 0.36\$. Hence you should have always bet for a Brexit. Not only in retrospect (I didn’t do this; read the sidenote).

This is one of the clearest examples of a positive expected value calculation I have ever seen in practice. There is pretty much a 50-50 probability of something happening, and choice A provides you with a pay-off of 1.16, while choice B gives you 8. Then what do you choose?

Sidenote:
I must admit that I expected the British people to stay in the EU. I didn’t think it likely that the average British person would vote for an option that, to me so, obviously seems to decrease their wealth, and therefore their well-being. Leaving the EU forces the UK to start a lengthy process of negotiating new trade contracts with the EU. While the EU has many parties it can trade with, since it already has many contracts in place, the same will not hold for the UK once it is out of the EU. This implies that it has a bad bargaining position, since the need for the UK to trade with the EU is much larger than the other way around. This will likely lead to suboptimal trade contracts for the UK. Furthermore, given the slowdown in economic growth and downturn in the financial markets that were to be expected, the average Brit could have expected a decrease in his material well-being, either through a decrease in job certainty or a decrease of his pension money. Not to speak of the 15% more expensive holidays and imported computers. It was a valuable lesson to me: do not fare on possibly wrong assumptions while the data is so clear.