David Moss on Insurance

I'm reading a great (if not a little bit dry) book by David Moss called "When All Else Fails: Government as the Ultimate Risk Manager". In the book, Moss has a really amazing and elegant explanation of how insurance works that I wanted to repeat here. The following example is Moss's work summarized by me.

Here's a very simple statistics refresher for you (it will be necessary). If you already understand expected value, you can skip ahead to the break:

---[definition: expected value]---

Let's say there's a raffle. This raffle has a payout of $100. There are 1,000 tickets given away for free for this raffle and they are all thrown in a hat.

We know that someone is guaranteed to win, because a ticket is going to be chosen. By getting 1 lottery ticket, you have a 1 in 1,000 chance (.001%) of winning the $100. Obviously, if you get your hands on 5 tickets, you have a 5 in 1,000 chance (0.005%) of winning the $100.

To be clear: If each ticket has a 1 in 1,000 chance of winning, and you participated in this raffle on 1,000 separate instances, you're expected to win once.

In statistics, we use a term called expected value. All that means is that, if you played this game over and over and over again forever, there is a certain amount that each play would be worth (or how much money you expect to make on average from each play). We calculate this by multiplying your odds of winning, by how much money you expect in winning. In this instance, the expected value is:

You can also include the calculation for what happens if you lose. But the payout if you lose is $0, so it just cancels out that whole part of the equation and doesn't change anything.

This means that if you end up with a raffle ticket, the expect payout per ticket is $0.10.

---[break]---


Ok, on to Moss's insurance explanation.

Imagine that there is a boat captain and he ferries shipments from Boston to London. If he makes a successful journey, he will be paid $10,000 for his cargo.




In this world, hundreds of boats attempt this journey every month. And because so many boats attempt the journey, we know that on average about 80% of the boats are successful. The other 20% never make it because of storms or attacks by the sea monster. Every boat that makes it gets $10,000 for its cargo.

Given the odds of making it, and the payout of being successful, we can calculate the expected value of a single journey to be the following:

This means that if the sailor makes the journey thousands of times over his lifetime (and doesn't die in one of the wrecks), his average income per journey will end up being $8,000.

Now the problem with this is that while his expected income is $8,000 per journey, the variation between the amounts he can make is huge. It is either $10,000 or $0. And sometimes people can't deal with such a large variation in their expected income (imagine planning your personal finances around a paycheck like that). One of the terms we use for this variability in statistics is standard deviation. 

There's lots of statistical theory behind this, but we can ignore that. The basic idea is that as a sample size gets larger and larger (more and more boats making the trip), the average real earnings gets closer to the expected value.

Maybe our ship captain can't afford to miss a profitable trip because he has bills to pay. Missing a bill (employee salaries, mortgage payments, etc.) would cause him financial distress. In an extreme case it could trigger bankruptcy putting his business at risk. He'd like to be guarded against this risk, so he decides to buy insurance.

Since everyone in the industry is in the same boat (pun intended), they all get together and decide to start an insurance plan. They all put money into a fund that will be paid out to those captains who have failed expeditions. In order for the insurance plan to function properly though, enough money must be paid in to cover the average losses (otherwise the plan would quickly run out of money).

The reason an insurance plan works in this case is because instead of having the potential for a $10,000 or $0 payout, the boat owner pays $2,000 per trip in insurance premiums and GUARANTEES an $8,000 profit per trip ($10,000 payout minus $2,000 premium). He knows that he will lose every fifth boat on average, but he will get paid $8,000 from everyone else's premiums.

(It's kind of like he set aside and saved $2,000 from each successful trip for the failures. You might be thinkings: "Why doesn't he just do that?" The reason is that everyone in the system is spreading the risk across the entire population to avoid strings of failures. So if our boat captain runs into a time where he has 4 failures in a row, the greater population will still statistically make up for it by using the premiums from higher-than-average successes by other captains. And if statistics holds true, the boat captain - if he operates long enough - will eventually diminish the effects of that string of losses on his average earnings.)



His expected value did not change, it stayed the same. But his variation, or standard deviation was reduced... essentially for free, because he coordinated with his fellow boat captains. Of course, if our boat captain loves risk and wants to play the odds at making $10,000 in a trip (and can afford the risk), he won't buy the insurance. But over the long run, he'll end up with the same average profit, just with greater variation.

The beauty of this is that an efficient insurance program where everyone shares the same average risk will keep the insured very close to his or her expected value in a system. It just reduces the likelihood that a participant will run into financial distress if they have a string of bad luck (e.g. if the ship has 4 failures in a row).

Where it gets complicated is in something like health insurance. Perhaps I am very healthy, I have good genes and good eating habits. My expected medical costs over the course of my life might be $20,000. But there is also someone who is very unhealthy with poor genes and poor eating habits who has an expected medical cost of $200,000 over the course of her life.

The example above would state that my premiums should bring me close to my expected costs over time, but that only works in a system where everyone has the same level of risk (like the boat captains). In a system where individuals have varying levels of risk (like health), the insurance plan brings you close to the average expected cost of the entire group (the two numbers are the same in the ship example).

In the imaginary healthcare example with only two people, the average of $20,000 and $200,000 is $110,000. So both I and the other person must pay a guaranteed $110,000 over the course of our lifetime if the system is to function properly. That's a tough bargain for me*, but while healthcare risks are statistically predictable for an entire population, they are hard to predict for individuals. And when an individual encounters a major health problem, the costs can be astronomical and place that person into financial distress. The insurance program doesn't exist to reduce the cost of healthcare; it exists to eliminate the huge standard deviation that causes financial problems beyond what a person can deal with if they get sick.



*This "tough bargain" leads to the problem of adverse selection which I will write about in a future post.