5. (Rain insurance) Gavin Jones's friend is planning to invest $1 million in a rock concert to be held 1 year from now The friend figures that he will obtain $3 million revenue from his $1 million investment-unless, my goodness, it rains. If it rains, he will lose his entire investment. There is a 50% chance that it will rain the day of the concert Gavin suggests that he buy rain insurance He can buy one unit of insurance for $.50, and this unit pays $1 if it rains and nothing if it does not He may purchase as many units as he wishes, up to $3 million.

(a) What is the expected rate of return on his investment if he buys a units of insurance? (The cost of insurance is in addition to his $1 million investment.)

(b) What number of units will minimize the variance of his return? What is this minimum value? And what is the corresponding expected rate of return? [Hint Before calculating a general expression for variance, think about a simple answer.]