On June 24, 2016 at 00:19 GMT, 1% of the votes of the Referendum of the United Kingdom’s membership on the European Union, also known as the Brexit, were revealed. With 320,280 votes counted, votes for leaving took a slight 51.1% to 48.9% lead. Although only a small portion of votes were revealed at this time, GBP tumbled 3% against USD to 1.45 after the count announcement (Figure 1).

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**Bloomberg Tradebook**
June 27, 2016

**Bloomberg Tradebook**June 27, 2016

*Figure 1. Percentage of votes for leaving EU vs GBPUSD on June 24, 2016. Source: Bloomberg BRXBIN Index, BRXBOUT Index, GBPUSD Curncy.*

Was the market over reacting to the small portion of counted votes while there was only a marginal difference in the results? In hindsight, the early anticipation of a leave vote foreshadowed its inevitable triumph. Votes for leaving eventually took a 51.9% to 48.1% victory, which stunned the market. GBPUSD was down 8% to 1.37 after all votes were counted.

In this article, we want to examine whether the success of the early anticipation of the Brexit vote count is statistically sounded. Notice that the FX market had been reacting to the vote counts during counting process. However, percentage of vote counts did not incorporate some important elements on predicting the final outcome. First, the more votes counted should have revealed more information on voters’ general intention. Second, the amount of remaining uncounted votes would have indicated the level of uncertainty. One example is that GBPUSD depreciated by 1.2% to 1.42 between 03:00 GMT and 03:15 GMT but the percentage of leaving votes only increased by 0.6% to 50.9%. A statistical model would be needed to incorporate such features that go beyond the simple vote counts.

Our goal was to estimate the probability of the final result being to leave EU. We first assumed that each voter would vote for leaving with an unknown probability. Let’s call it the **intention probability**. Before any votes were counted, we assumed no knowledge on the intention probability in a sense that it can be anything between 0 and 1 with equal probability. When more voting results were revealed, we would gain more information on the voters’ intention.

Next we assumed that each voter was independent. This allowed us to estimate the intention probability by the **Bayesian approach**. Without going into the technical details, the intention probability will follow a so-called Beta distribution after each update by observing the latest vote counts. More realized vote counts will lead to smaller estimation error and the estimated intention probability is equal to the realized percentage of votes for leaving. In order to incorporate part of the sampling bias in the vote counting process (e.g. Scotland comes first, and London comes later), we consider realization of 0.1% of the total available votes as one independent sample when updating our probabilistic model. Figure 2 illustrates how the distribution of the intention probability changed at different points in time of the vote counting.

*Figure 2. Probability distribution that a voter will vote for leaving (Brexit). *

To estimate the final result of the referendum, we considered the percentage of votes for leaving as a composition of counted votes (known) and uncounted votes (unknown). The counted votes will be taken from the time series data and the uncounted votes will then be estimated by the Bayesian approach above.

So how confident are we statistically that the UK would vote for leaving the EU when only part of the votes were counted? To answer this question, we calculated the probability that the percentage of votes for leaving was larger than 50%, or the **probability of leaving **for short. Figure 3 compares the probability of leaving to the percentage of leaving votes. At 00:18 GMT when the percentage of leaving votes went beyond 50% for the first time, the probability of leaving increased from 40% to 51%. That is also the time that GBPUSD first experienced a sharp depreciation after the vote count had been started. When the percentage of leaving votes increased gradually from 48.9% to 51.6% from 02:40 to 03:40 GMT, the probability of leaving increased from 38% to 89%. The increase in probability is much more significant than the percentage of votes because the uncertainty was reduced as more information (votes) was revealed.

*Figure 3. Probability of leaving from the model vs. percentage of votes for leaving (Brexit).*

Figure 4 compares the probability of leaving to GBPUSD movement. We can see their correlation as a reflection of each other. GBPUSD depreciated in a magnitude that was more comparable to the increase in probability of leaving than the percentage of votes for leaving. One interesting observation is that while the probability of leaving reached 90% at 04:01 GMT, GBPUSD was trading at 1.35 which is the same level after all votes were counted. While GBPUSD once dropped to 1.32 at 05:25 GMT, the model had already suggested a 99% probability of leaving over an hour before when GBPUSD is trading at 1.35. These suggest that the market was over reacting while the result was almost certain based on the probabilistic model.

*Figure 4. Probability of leaving from the model vs. GBPUSD.*

By incorporating information processing and statistical uncertainty, the probabilistic approach appears to better explain the early anticipation in the FX market than the simple percentage of counted votes. When a large enough number of votes had been revealed, it was sufficient enough to determine the end result of Brexit with a high level of confidence even several hours before all votes were counted. This also explains the GBPUSD reversion in the final hour of the counting process to the price level that is suggested by the model.