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 $u$ 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?