What if I want to calculate exactly what insurance is worth to me, so I can decide if I'm getting a good deal or not?
Let's take our lightning-strike example from before and apply it to a single person making the decision to buy insurance for medical costs resulting from being struck. This person is 20 years old, expects to live at least 50 more years, and has a 1 in 1000 chance of being struck by lightning each year, costing him $20000 in medical expenses.
The odds of him being struck at least once in those 50 years are very close to 5%. (He actually has a 4.76% chance of being struck once and a 0.12% chance of being struck twice. The odds of being struck three times or more are negligible.[1])
Anyway, he has a 4.76% chance of paying out $20,000 in fifty years, and a 0.12% chance of paying out $40,000. Taken together, his expected lifetime medical expenses from lightning strikes is almost exactly $1000.
This works out to an annual cost of $20.
If he can buy insurance for less than that, the odds are very much in favor of him making money. If he is a risk-averse person, he might even be willing to pay slightly more than that for the feeling of security, or if the consequences of him having to pay a $20,000 medical bill all at once would severely impact his life.
It should be noted, however, that if this man is willing to keep aside $20,000 to cover possible medical costs, buying lightning insurance for anything over $20 per year is a bad deal for him.
In real life, of course, this calculation is complicated by the fact that we don't know exactly how long we will live, we don't know exactly how likely it is that we personally will be struck by lightning, etc.
Insurance companies, however, do know all of these things, and this is one way that they make money. If our man pays the insurance company $20 a year for lightning insurance, by his calculations he is breaking even. The insurance company may know from extensive data and research that this man also has a 1% chance of getting hit by a car over those 50 years or dying of natural causes, in which case they wouldn't be paying his lighting-strike medical bills. More specifically, they won't know that for a fact about that particular man, but they'll know that on average over a large population, these things will be true, and they can still make a profit by charging what the man calculates to be a break-even price.
In effect, by being very good at statistics and by spreading out the cost over an entire population, the insurance company profits by mitigating risk. The insurance company has very little financial risk because over a large population, they can calculate extremely accurately their costs. An individual, being only one person, cannot calculate his costs with as high a certainty. In effect, we buy this certainty from them.
Let's use another example. I may be able to say with a very high degree of confidence that given 1000 New Yorkers, half of them will be Yankees fans. However, given a single person, even knowing that half of New Yorkers are Yankees fans, I have at best a 50/50 chance of guessing correctly whether that person likes the Yankees.
At this point, buying insurance may seem like a fool's deal, but that depends on how risk-averse you are. If you will spend more than your expected return for the peace of mind that if something bad happens to you you won't lose your life or your house, then insurance has value to you.
However, if you are able to afford any such expenses either through savings or other means, you have essentially no reason to buy insurance to cover those costs, because that will likely end up costing you more money than you would otherwise spend.
In short, you should only ever buy insurance to cover possible costs that are so large that you could not otherwise pay them. Otherwise, you are losing money to insurance company overhead.
Since most people don't realize this, insurance companies actually can make quite a bit of money by covering expenses that people can easily pay for out-of-pocket. The premiums will be much higher, but when people don't actually pay or even see the cost of the premium (my company has great insurance!) they never have to make this decision.
The sad part is that we could significantly reduce health care costs simply by teaching people statistics.
- For those of you wondering where these numbers are coming from, the Binomial Distribution accurately calculates the probability of k number of events occuring within n trials.
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