Every Insurance Company Can Save You Money!!!

One of the most entertainingly useless statistics touted by insurance companies is the average amount of money that customers save by switching to their insurance.

The first hint that this information is useless is that every company seems to mention it. Now, they can't all be cheaper than each other, although that's certainly the impression that their ads convey. At a minimum, you might think there's some subset of non-advertising ripoff insurance companies that have only imbeciles as customers, and this is the segment that's being targeted in the ads.

The reality is different. It's obvious that customers who actually switch will save money, because nobody switches to a more expensive insurance company. The relevant statistic is what percentage of customers who actually get a quote from you end up choosing your insurance. Because that tells you whether it's worth spending the 15 minutes typing in information to find out if they're actually cheaper. Of course, they never tell you that, because the number is probably pretty low, and probably wouldn't actually motivate you to visit their website. It's much better to just tout the savings of the guys who switch, hoping that idiots won't realise that it's useless information.

Let's assume that all insurance companies are ex-ante identical, and that when a customer comes along, they form an estimate of the premium required for that customer that is drawn from the customer-specific random variable.- e.g., each insurance company quote for you is drawn from N(Shylock mean, Shylock Std Dev). A customer gets their first quote, then goes to the second company for a second draw from the distribution and chooses company #2 if it's cheaper. They repeat this process until they estimate that the time cost of drawing another quote is greater than the expected saving.

Now, in this model all insurance companies are equally good ex-ante. But there'll be some people who start with company A (who had an abnormally high quote), then get a quote from company B that's lower and switch. This will be offset on average by the same number of people whose first quote from B was at the high end, then got a draw from company A that was cheap and switched.

In other words, every company will be able to tout the same pointless statistic about how much their customers saved, even though it actually tells you nothing about the average cost of insurance from that company.

All it tells you is that insurance companies have different premiums for the same customer. Which, when you think about it, is a very interesting point. But somehow I don't think they're spending all this advertising money to raise interesting questions about microeconomics.

Update: The Greek notifies me that he in fact was able to write down a proper version of the model sketched above which generate the conclusions listed. Now that's intuition you can take to the bank!