Question

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?

Answer 1

	Potential capacity	10	$ Million
	Hurricane Category	Capacity	Insurance Policy Cost	Pay	Net Cash Flow	Total Net Cash Flow	Probability	Expected Return	Deviation	Squared
	1	5U	1	3	2	10U	0.2	7.2 U	2.8U	7.84U2
	2	5U	1	3	2	10U	0.2	7.2 U	2.8U	7.84U2
	3	2U	1	5	4	8U	0.2	7.2 U	.8U	.64U2
	4	2U	1	5	4	8U	0.2	7.2 U	.8U	.64U2
	5	0U	1	7	6	0	0.2	7.2 U	(7.2U)	51.84U2
							1			68.8U2
1	Expected rate of Return on your investment if we buy U units
	Expected Return of Portfolio		2*5U*.2+2*5U*.2+4*2U*.2+4*2U.2+6*0U*.2
			10U*.2+10U*.2+8U*.2+8U*.2+0
			7.2U
2	So the variance is the last coloumn of the table the sum of which is 68.68 U2
3	So the number of units that can minimize variance is in the case of the Category 3 and 4 and the sum of the two varainces is 1.28 U2
	and the corresponding rate of return is 1.6 U +1.6 U =3.2 U