What does a November 2023 hockey bet have to do with the 2008 financial crisis and the Chunnel connecting England to France? A lot, and the relation is key to understanding financial disasters — not to mention getting paid for longshot sports bets and getting from London to Paris safely.

It’s all about correlations, rare events and how bookies and bankers try to avoid paying.

I ran across the hockey bet in a Washington Post article by Danny Funt, “He Hit Three Monster Bets — And Then the Sportsbook Wouldn’t Pay,” about well-known quantitative sports bettor Christopher Kozak. Kozak bet that in a Nov. 17 hockey game between the Florida Panthers and the Anaheim Ducks, eight specific players would not score goals, and the home team (Anaheim) would score fewer than three goals. He bet $300, and the casino promised him $60,000 if all nine of the events came to pass. Kozak told Funt he thought he had about a 1% chance of winning, so an expected $600 return for his $300 investment.

How would you evaluate this bet either as a bettor or bookmaker?

One starting point is to look at the historical frequencies of the individual events. Using all data from the 2022-2023 NHL season, a player who is one of the top four goal scorers for his team fails to score in 66% of games. The home team fails to score three or more goals 57% of the time. For Kozak’s actual bet, you would look at the probabilities for the specific teams and players — especially which skaters are healthy and expected to play — and other factors that might influence the outcomes, but for our purposes it’s enough to use aggregate figures.

When people try to estimate the probability of a chain of events using intuition, they often wildly overestimate it. In this case a thoughtless person might assume that since each of the nine events is more likely than not, there’s a pretty good chance that all nine will happen.

On the other hand, a person who knows a little statistics might multiply the nine probabilities to get 2%. A person who knows a little more statistics will realize this calculation ignores the correlation among the events. However, adjusting for the correlations does not change the 2% estimate much.

A better approach is what actuaries do; ask how frequent the combined event has been in the past. In the 2022-2023 NHL season, there were 82 games in which the home team scored fewer than three goals and the top four season goal scorers on both teams failed to score. That’s out of 1,392 regular-season and playoff games, or 6% of the time. Unfortunately, the actuarial approach cannot be adjusted easily for the specifics of one particular bet, it can only tell you the average payoff for a class of broadly similar bets in the past.

Before delving into why the historical frequency of the combined event is three times what we would estimate based on historical frequencies of the individual events and their correlations, let’s go back to 2008.

A major contributor to the crisis were super-senior tranches of collateralized debt obligations. These are similar to Kozak’s hockey bet in that they are bets that a chain of events (bond defaults rather than hockey goals) will not happen. The individual probabilities are reasonably well known based on historical frequencies, implied market prices and fundamental analyses. The correlations can also be estimated with fair accuracy. But the probability of the combination of events that would impair the super-senior CDO tranches was much higher than the individual probabilities and correlations implied.

Bookies, whether they work for casinos, Wall Street, insurance companies or elsewhere, sometimes don’t like to pay winners. Kozak had trouble collecting his win because the casino claimed it had gotten the correlations wrong (which was not the problem) and the $60,000 payout was an obvious error.

On Wall Street it’s done differently. You may recall from the movie The Big Short the scene where Michael Burry (played by Christian Bale) is brought to the edge of ruin, despite having won big on his bets against subprime mortgages. The bets had gone in his favor, but the banks were claiming the opposite, and demanding large margin payments. If Burry could not make the payments, the banks could close out his positions at a loss to him.

What about the Chunnel? Its designers commissioned a fire safety report that estimated one serious fire in the tunnel every 850 years. The calculation multiplied the probabilities of a long chain of failures that would have to occur. The first serious fire occurred less than a month after opening the Chunnel to passenger traffic in 1994, and there have been five other serious fires since — one every five years.

In all three cases, the fundamental problem is multiplying probabilities to get the likelihood of a combination of events only works if the events are independent, and almost nothing of practical interest is independent. It is not enough for the events to be uncorrelated, or to estimate and adjust for correlations.

As Nassim Taleb has argued forcefully, it’s impossible to estimate probabilities of extreme events by collecting data on ordinary events. The corollary is it’s impossible to estimate the probability of a long chain of events by knowledge of the individual and pairwise (correlations are pairwise probabilities) events. Many disasters can be traced to ignoring this principle.

The other lesson is that if you bet on rare events, you have to make sure you can get paid if you win, and that’s often harder than finding the attractive bets. If you bet against rare events, like the bookies, banks and Chunnel engineers, make a contingency plan if you lose, because you will likely lose more often than you think.

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