# The rational approach to buying insurance

To understand when to buy insurance, you need to understand two things:

• First, you will probably lose money buying insurance.
• Second, sometimes losing money to buy insurance is a good thing

## You will probably lose money buying insurance

Let’s say I’m going to start selling Ed’s Travel Insurance, a super-simple version of travel insurance. Before you take a flight, you pay me an insurance premium. In exchange, if you lose your bags on the flight, I’ll pay to replace it.

To figure out how high the premium for Ed’s Travel Insurance should be, I hire a bunch of very smart people. They report to me that if I sign up 100 customers, I won’t have to pay anything to 85 of them, but 15 will lose their bags, which are on average worth \$100. In other words, I’m going to pay out about \$1,500 for every 100 customers.

This means that I must set my premium to be at least \$15 - otherwise, I’d be paying out more in claims than I’m making in premiums, and I won’t be an insurance company for very long. Indeed, factor in the salaries and offices and so on for all those very smart people, and the fact that I want to turn a profit, and the premium for Ed’s Travel Insurance is probably closer to \$20.1 That way for every 100 customers, I’ll be collecting \$2,000, paying \$1,500 in claims, and have \$500 left over for my overhead and profit.

The more mathematical way of putting it is, the cost of the claims (\$C) multiplied by the probability of those costs occurring (P) is each customer’s expected claim value (\$C x P). And so my insurance company needs to make sure that for each customer, I’m taking in more in premiums than the customer’s expected claim value, or, in other words:

Cost of Premium > (Cost of claim) x (Probability of claim)

Put in these terms, Ed’s Travel Insurance looks like a raw deal for you! You’re paying \$20 so that on average, you can avoid paying \$15. In the long run you are just losing \$5 every time you buy my insurance, and that \$5 goes straight to me to pay for my overhead and profit.

On a fundamental level, this is always true no matter what insurance you buy. Health insurance, car insurance, pet insurance - all of it, by definition, has to be profitable in aggregate for the insurance company, or else they wouldn’t be able to offer it. And that means that also by definition, every customer’s insurance policy is, on average, a money-loser.

## Losing money to buy insurance might be a good thing

So why, then, would you ever buy insurance?

Well, as it turns out, there’s three good rational reasons to buy insurance:

• Reason 1: Your probability of incurring claim costs (P) is higher than average;
• Reason 2: Your claim costs (\$C) are higher than average;
• Reason 3: Financial catastrophe, aka the nonlinear value of money

### Reason 1: Your probability of incurring those costs (P) is higher than average

The first reason to buy insurance is if your probability of incurring claim costs is higher than average. Suppose you come to buy Ed’s Travel Insurance. You know that I assume my customers will lose their bags 15% of the time. But you know something I don’t know - you know you’re flying a special airline, one notorious for losing its customers’ bags. In fact, this airline loses its customers’ bags a shocking 50% of the time. Now buying my insurance will be a great deal for you: you’re paying \$20 to avoid losing, on average, \$50.

In the real world, insurance companies guard against this in a variety of ways. For car insurance, they ask a bunch of questions to try to figure out your exact probability of getting into a crash. For flood insurance, they prepare exhaustive statistical analyses of the likelihood of flood damage. And of course, it’s almost always considered insurance fraud if you intentionally incur claim costs (e.g., burn down your home, intentionally lose your luggage, etc.).

But at the end of the day, an insurance company simply doesn’t have the manpower to ensure a perfectly customized quote for every single customer. It has to make some statistical assumptions about you, and that means that overcharging some customers and undercharging others. You want to be one of the ones undercharged.

### Reason 2: Your claim costs (\$C) are higher than average

The second reason to buy insurance is if your expected claim costs are higher than average. Suppose that you’re again looking to buy Ed’s Travel Insurance, but this time, you’re transporting priceless cargo. There might still only be a 15% chance that you lose your bags, but if you do, now I’m on the hook to pay you \$100,000. Here buying insurance is a total steal for you - you’re paying \$20 to avoid paying, on average, \$15,000.

In the real world, insurance companies guard against this by imposing limits on the maximum claim amounts they’re liable for. But the point remains the same. If you are confident that your costs would be much higher than what the insurance company thinks your costs are, and that it would be covered under your policy, then you should buy the insurance.

### Reason 3: Financial catastrophe, aka the nonlinear value of money

Finally, the main reason people buy insurance is because the value of money isn’t linear. By which I mean, for most of us, owing \$250,000 is more than 1000 times worse than owing \$250.

Health insurance is perhaps the best example of this. Almost all of us pay out more in health insurance premiums than we get back in terms of benefits, because most of the health care we need is cheap. But if something catastrophic happens and you owe \$250,000 in medical bills, well, owing that much money warps your life in unfathomable ways. You have to liquidate your savings and retirement accounts. You have to mortgage your home or sell your car. You have to move. And all of this comes on top of being so sick you need \$250,000 in care in the first place.

In other words, you’re buying insurance against the possibility of a financial burden so awful that it trumps the cold expected value calculations I described above. Sure, buying health insurance, on average, you’ll be losing money. But the downside financial risk is so catastrophic, and affects so much of your life, that you are more than happy to “overpay” to avoid it.

## Real-world examples and recommendations

Applied to real-world examples of insurance, you get my standard recommendations below. Keep in mind that each type of insurance is still very different; these are just general principles.

• Health insurance: Yep, as discussed above, for Reason 3.
• Car insurance: For liability, it’s legally mandated, but would be a good idea for Reason 3 anyway. For collision, it comes down to how much of an impact losing your car would be on your personal finances.
• Home insurance: Yep, for Reason 3, and it’s usually required by your mortgage lender. But you need to carefully consider for yourself the value of earthquake, flood, and hurricane insurance, which, because of Reason 1 / Reason 2 abuse, are now often excluded and/or overpriced.2
• Travel insurance: No. This is a good example of where none of the above considerations apply. Your odds of losing your luggage will not meaningfully differ from the statistical average. Even if you’re carrying super-valuable cargo, the insurance policy will almost certainly be capped. And most of us do not carry luggage of such value that we would be financially devastated by losing it. (Plus most credit cards offer lost luggage protection for free anyway.)
• Pet insurance: If you are the kind of person that would without hesitation drive yourself into deep debt for your pet, then pet insurance makes sense for Reason 3. Also, the industry is far less developed than human health insurance, so there are many more opportunities for arbitraging pet insurance premium rates under Reasons 1 or 2. (For instance, you might not get charged as much as you should for insuring a constantly-sick pet.) Absent one of those considerations, I don’t recommend it.
• Life insurance: If you’re the sole breadwinner for a family, you definitely want it for Reason 3. If your family can get by without your income, it’s probably not. Child life insurance policies are never worth it unless your child is the primary income stream for the family. And whole life insurance is 1% of the time a reasonable and sound component of estate planning, and 99% of the time a scam preying on the elderly.
• Phone insurance (e.g., AppleCare): Probably not. But my wife and I got AppleCare on our new phones because of our kid, who likes to a) play with our phones, and b) break things - a great example of Reason 1.
• Credit insurance: Absolutely not. You shouldn’t be carrying credit card debt; you doubly shouldn’t be carrying so much that you might not pay it back; and you triply shouldn’t be carrying so much that failure to pay it would be financially catastrophic.

1. As it turns out, insurance companies are not particularly profitable for a variety of complicated reasons, though they do have excellent return on invested capital↩︎

2. For example, it’s really hard to accurately estimate the probability of a earthquake and how much expected damage there is, and so insurance companies tend to be overly cautious as a result. ↩︎