Expected value

Question

A collector has a rare $50,000 painting that needs to be insured. If in a given year, there is a 1% chance that the painting will be stolen or destroyed, how much at minimum should the insurance company charge to insure this painting?

Answer 1

If the insurance company was being fair, then they would charge you the exact expected outcome of the painting being destroyed.  That's simply the value of the painting minus the premium the customer pays times the likelihood it gets destroyed, which you can then set equal to the likelihood the painting does not get destroyed times the value of the premium.  

 

($50,000 - x) * .01 = x * .99

500 - .01x = .99x

500 = x

 

In other words, 99% of the time the painting stays safe and the insurance company collects their premium.  The other one percent of the time, they have to pay out the $50,000 it's worth minus the premium they already collected.

 

(Of course, in reality, insurance companies are not in the business of being "fair" to customers, and thus will charge you a premium a lot higher than equilibrium to ensure that they are expected to profit on all of their policies. However, let's just assume for this question that the world is perfect and the insurance companies are just in it for the good of society!)