The insurance game

Let’s play a game. You give me £10, and I’m going to roll a pair of dice. If I get two sixes, I’m going to give you a payout of £350. We can play this game as many times as you like (or at least until one of us runs out of money). Do you want to play?


If we play long enough, you’ll end up losing money. The chance of me rolling a pair of sixes in a game is (1/6)×(1/6) = 1/36. Now imagine we play the game thousands of times. For every 36 games we play you’re losing £10×36 = £360 and you’re only, on average, going to get £350 back. You’re effectively losing about 28p per game.


Our maths only works here because we are imagining thousands of games being played. If you only played the game once, you might walk away after the first game as a winner. Conversely you could play one hundred games without ever winning, and end up being down £1,000. Averaging over thousands of games “smooths out” this randomness and you eventually tend towards always losing.


Rolling the dice

What’s this got to do with insurance? In some sense buying an insurance policy is just like playing an elaborate dice game (even more elaborate than Yahtzee). Let’s look at something like home insurance: the money you pay to play the game is what the insurer charges as a premium. The chance that you’ll have to claim on your policy is equivalent to rolling the dice: there is some probability in the next year that your house might burn down, or that your flat screen TV screen will fall off its wall mount and smash into a thousand pieces. If that happens the insurer will pay out, sometimes more than the money you paid to play the game in the first place (in some cases the payout can be thousands of times higher).


In my dice game I purposely set the entry price to be big enough that across many games I would end up making a small profit. I could do this because I knew the amount I would have to pay out when you won, and I also knew the chance of you winning. Calculating both of these numbers for each customer is one of the most important jobs for the insurance company to make sure they can afford to pay out without going out of business.


Playing fair

But wait a minute: we said before you wouldn’t want to play my dice game because it’s biased in my favour, so why are people happy to play the insurance game? Insurance, of course, is not really a game. You’re not trying to “win” at insurance: payouts are made to cover a large, unexpected loss that might otherwise leave you penniless. As a risk-avoider you’re content to play the game, even if you lose a bit of money, in the hope that when you make a big loss you can offset it with a “win” of a payout.


This means that on an individual level behaviour is very un-smooth: someone might only play the game a handful of times and get one very large payout when their house burns down, whereas you might play for decades and never make a claim. Insurance helps you to ride out these bumps. (This is also why it’s very unwise to be your own insurer, or even for small groups to insure themselves.) To keep things looking smooth and predictable overall the insurance company needs to play the game many times, which means having lots of customers.


But should you play the insurance game at any cost? You might think my dice game was biased, but what if I wanted you to pay £20 per game? Again, in that case you could easily calculate how unfair that game was, but since you yourself can’t work out the probability of having to make an insurance claim it’s very hard to tell how fair the insurer’s price is. Note that “fair” does not imply “cheap”: without knowing more about the payouts and probabilities a cheap insurer could very well be less fair than one that charged more!


Hopefully you have started to see insurance in a slightly different way: it’s not really the absolute price to play that’s important, but how fair that price is compared to the risk you’re taking. To understand this idea of fairness in more detail means diving into insurance pricing science, which is something we’ll look at in future articles.


In the meantime, keep rolling the dice, and best of luck!


This blog was written by Michael, our Product Manager.


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