You plan to invest $4 million in the construction of an oil well which has a potential yearly revenue of $10 million. The oil well will be located in the Golf of Mexico. As we all know, this region is constantly hit by hurricanes. Assuming that if during an entire year there is a hurricane, this will disrupt your production and your well will lose 20% its yearly production. And if during a year there are two hurricanes, your well will lose 40% of its yearly production, if there are three hurricane your well will lose 60% of its yearly production, 

if there are four hurricanes your well will lose 80% and if there are five or more hurricanes your well will lose its entire yearly production. According to the weather prediction the yearly number of hurricanes that will hit this region follows a Binomial random variable with parameters n = 5 and p = 1 3 . In order to reduce the risk on your investment you plan to buy an insurance policy. One unit of this policy cost $1 and will pay $2 each time the region is hit by a hurricane.

1. What is the expected rate of return on your investment if you buy u units of this policy?
2. What is the variance of the rate of return on your investment if you buy u units of this policy?
3. What is the number of units that will minimize variance, and what is the corresponding rate of return?