One of the most persistent — and pernicious — myths about insurance, health insurance in particular, is the notion that insurance works by sharing risk: that any combination of proposed benefits is perfectly reasonable since everyone is sharing the risk and that’s all happy-happy Kumbaya.

This is one of those myths that’s almost true. It’s subtly wrong, but all the more pernicious because of that.

To explain this, we’re going to have to appeal to the risk equation R=P×H again. I’ve hit this a lot, so I won’t explain it in detail (you can go here, here, and here for more details). The basic idea is that if you’re afraid of an event that costs H dollars, but is unlikely, instead of saving H dollars against that unlikely rainy day, you can make a bet with another party against the chance that unlikely rainy day will happen. To make that bet, you put up a little more than R dollars as your side of the bet. If the rainy day does happen, you “win” the bet, and the other side pays you H dollars.

For a concrete example, let’s say you’re worried that a Bad Thing will happen in the course of a year, and if that Bad Thing happens it’s going to cost you \$100,000.

The Bad Thing can be anything — your house burns down, you have a heart attack, the Rockies win the World Series. So you go to a special kind of bookie called an insurance company, and ask them to take a bet with you: you get paid \$100,000 if the Bad Thing happens over the course of a year. Based on their knowledge of Bad Things, the insurance company determines that the probability PBad Things = 0.001. The hazard H is \$100,000, so the product PBad Things×H of course is \$100. That’s the risk. So the insurance company asks you to put up a little bit more than \$100, say \$125, to take the bet. That \$25 is to pay their administrative costs, to protect them against a different Bad Thing happening — more on that in a bit — and hopefully to provide a little profit.

Notice, now, that there are only two parties to this deal: you and the insurance company. If you “win” the bet and your house does burn down over the course of a year, you’re supposed to get \$100,000. Period. And you don’t care if Mary down the street makes the same bet on her house; you just care that the insurance company has the assets to pay up. Of course, if you “lose” the bet and your house doesn’t burn down, you’re out \$125, but that’s okay — you paid \$125 to sleep well knowing that you won’t be out on the street.

The insurance company cares, however. (I’ve beaten the thing to death now, I’m sure you’ll remember.) The insurance company now is exposed to a possible loss of \$100,000 and if their accountants are honest, the insurance company needs to put \$100,000 aside to the end of the year. So they send their Bad Things Insurance salespeople out, selling Bad Things insurance to everyone they can find. And here’s the great thing: they know that the chances are that not everyone with Bad Things Insurance will have the Bad Thing happen — not everyone’s house will burn down — and so for every time they have to pay off the bet, there will be many many times they get to laugh and keep the \$125.

The insurance company has it’s own problem, though.

Let’s say they sell a million policies. They take in \$125 million. Over a long enough time, with enough years, they expect to have to pay out 1000 claims on average, or \$100 million, leaving them with \$25 million left over. This is called the expected result, and in fact the risk equation is also called the expected value equation.

The problem is, though, a run of bad luck can mean they might have 2000 claims, and have to pay out \$200 million. We can actually estimate the chances of this, and they’re pretty small. Explaining why would mean going into Gaussian distributions and the Central Limit Theorem and this article is already too long.

So what does the insurance company do? They go to another insurance company, and make their own bet for Different Bad Thing Insurance — specifically that there will be more than 1000 claims in a year. This is called, reasonably, re-insurance.

But notice, no one is sharing risk here.

Each customer is paying the insurance company to accept a risk, and it doesn’t matter to the customer if the insurance company sells one policy or a million of them.

What makes this whole notion pernicious is that when people start talking about “shared risk,” the underlying notion is that somehow that shared risk means that things can be added without limit — that somehow, by magic, no matter what gets paid out, there will always be money to do so. But whether you’re making the bet with Bennie the Bookie, the Upstanding Mutual Assurance Company, or the government, the math always applies: it just doesn’t work that way.