David owns a house which is worth $500. He also has $200 in cash (his total wealth is then $700).

The probability that his house will be destroyed in a natural disaster is 0.05.

He can buy home insurance with a sum insured of $500 (meaning that if his house is destroyed, he gets back the full value of $500 from the insurer).

Suppose that his utility function is u(w) = 10 * w 0.0025 * (w^2)

David, can partially insure his house. He can buy a policy which will pay $S if his house is destroyed (where S < 500). If the premium rate is 0.06*S for a sum insured of S, what amount of insurance should David buy? (That is, what is the value of S that would maximise his expected utility?).