Utility theory and insurance

The insurance industry exists because people are willing to pay a price for being insured. There is an economic theory that explains why insureds are willing to pay a premium larger than the net premium, that is, the mathematical expectation of the insured loss. This theory postulates that a decision maker, generally without being aware of it, attaches a value u(w) to his wealth w instead of just w, where u(·) is called his utility function. To decide between random losses X and Y, he compares E[u(w − X)] with E[u(w − Y)] and chooses the loss with the highest expected utility. With this model, the insured with wealth w is able to determine the maximum premium P⁺ he is prepared to pay for a random loss X. This is done by solving the equilibrium equation E[u(w − X)] = u(w − P). At the equilibrium, he does not care, in terms of utility, if he is insured or not. The model applies to the other party involved as well. The insurer, with his own utility function and perhaps supplementary expenses, will determine a minimum premium P⁻. If the insured’s maximum premium P⁺ is larger than the insurer’s minimum premium P⁻, both parties involved increase their utility if the premium is between P⁻ and P⁺.