If you are exposed to a 50/50 chance of gaining or losing $1000 and insurance that removes the risk costs $500, at what level of wealth will you be indifferent relative to taking the gamble or paying the insurance? That is, what is your certainty equivalent wealth?

There is a level of wealth when you will be indifferent between the gamble or the insurance, provided that you are risk-averse.

If a person is risk-averse, then the person will always prefer a certain outcome over lottery that provides the same expected payout. In this question, the gamble has an average payout = 0.5 * 0 + 0.5 * (- 1000) = - 500. That is, you are expected to lose $500. The alternative is to lose $500 with certainty by buying the insurance. For a risk-averse individual, the second option is always preferred because the expected payouts are the same in both cases, but the second option provides certainty. Hence, at any given level of wealth, you will always prefer insurance.