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 $0.50, 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? [Hint: Before calculating a general expression for variance, think about a simple answer.]