# Market prediction, utility and information

A country is home to several million citizens; all citizens are risk averse, and each citizen is exposed to an independent risky loss each year of 1,000 dollars with a probability of $$\frac{1}{40}$$.

The country's Financial Regulator licences the first insurance company and allows it to write up to $$10 000$$ policies. The insurance company charges a premium of 35 dollars for full insurance.

However, a study of the population shows that the risk of loss is in fact just $$\frac{1}{60}$$ for the three-quarters of the population with Blood Group ‘O’, and $$\frac{1}{20}$$ for the other Blood Groups. Citizens are easily able to check their blood group, but regulations do not permit insurers to request or to use medical information. Assuming all people have the utility function $$u(w) = 1 - e^{-0.001w}$$, predict what will happen to the country's insurance market.

So it's easy to check that only the "other Blood groups" (probability $$\frac{1}{20}$$-ones) would prefer to buy the insurance (by comparing expected utilities with and without it). How to proceed after that?

• How did you incorporate the utility function? Note that the people decide by regarding their utilities. – callculus Apr 30 at 0:43

Once you've done that, you should compare the premium income from the insurer, which comes from $$\frac{1}{4}$$ of the population, i.e. $$35\frac{P}{4}$$ where $$P$$ is the number of policies written (max $$10,000)$$, with the expected outgo from paying claims, which will be $$1000\frac{P}{4}\frac{1}{20}$$. This is the profit which the insurance company makes. As you can tell, the insurance company is going to go bust pretty quickly unless it allows patients to use their medical information.