You plan to invest $4 million in the construction of an oil well which has a potential 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 there is a hurricane of category 1 or 2, this will disrupt your production and the well will produce half its capacity, and if there is a category 3 or 4, your well will produce one fifth of its capacity, if there is a category 5 your well will be closed. 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 $3 if the region is hit with a storm of category 1 or 2, $5 if the region is hit with a storm of category 3 or 4, and $7 if the region is hit by a storm of category 5. We know from the weather prediction that there is 20% chance that the region will be hit with a category 1 or 2 storm, 20% for a category 3, or 4, and 20% for a category 5, 40% chance that the region will not be hit.
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?