Shall we buy insurance?

When we buy air tickets, we are asked to buy insurance on the tickets as well. Shall we buy insurance?

Suppose an air ticket costs 300 dollars. The insurance on the ticket provides one chance to exchange for a ticket in case you need to change the traveling time. This happens about one out of ten times. The cost of the insurance is 50 dollars.

On average, it costs us 500 dollars to get a 300 dollar coverage. That is how insurance companies make money. The net loss for a customer is 200 dollar. If so, why some people still buy insurance?

If we are well off, it is no big deal to fork out 300 dollar for a new ticket. However, if your bank account has very little left, it might be tough to dip into your savings. That is when insurance can help. The poorer we get, the more insurance we will crave.

In the following, we will give a more quantitative analysis. In the year of 1738, Daniel Bernoulli published a paper on why one would buy insurance. In that paper, he introduce the concept of utility function. His utility function is a logarithm function. We will apply his method to our problem.

Suppose the initial wealth of a person is w. For a person who buys insurance, his final wealth is always w – 50, where 50 is the cost of insurance. For a person who does not buy insurance, his final wealth is w 90% of the time, when he doesn’t change his travelling schedule, and w – 300 10% of the time,  when he changes his travelling schedule and buy an extra ticket.

His utility when he buys insurance is ln (w – 50). His utility when he doesn’t buy insurance is 0.9*ln (w) + 0.1*ln(w – 300). When the initial wealth is 310 or 1000, the utility values can be calculated and listed in the following table

For a person with initial wealth of 310, his utility is improved with the purchasing of the insurance. For a person with initial wealth of 1000, his utility is reduced with the purchasing of the insurance.

In general, over long term, the more insurance we buy, the poorer we become.

This happens not only to personal insurance, but also to social insurance. If we are poor, we are more likely to need social insurance. However, more social insurance comes with more taxes. We might get 300 dollar benefit for every 500 dollar tax. There are a lot of costs associated with any program. Over long term, the more social insurance we have, the poorer we will become.

This is not to suggest we shouldn’t buy any insurance, or our society shouldn’t provide any social insurance. We fall on hardship from time to time. Insurance can be invaluable in genuine difficult times. However, it is a sure way to poor house if we insure each and every trifle. When we spend less to buy insurance, we can save more to build up our own financial cushion.

Genuine insurance in our life is our health, physical and mental health. If we are healthy, we are less likely to fall into hardship. When we do, we are more likely to rebound from it.