Look carefully at the examples given Complete 3B 3C Please make sure your answers are correct (Rain insurance) 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 them? [Hint: Before calculating a general expression for variance, think about a simple answer.] The rate of return is {10000000+0.5u30000001,10000000+0.5uu1,ifitdoesnotrain,ifitrains. Because the probability of raining is 50%, the expected rate of return is 1000000+0.5u1500000+0.5u1=1000000+0.5u500000. Similarly, we can calculate the variance of the rate of return as (1000000+0.5u15000000.5u)2. We try to minimize the variance by choosing the number of insurance unit u[0,3000000]. Because the variance is a decreasing function in u when 0 u3000000, it is minimized at u=3000000. Thus, you need buy 3,000,000 units of insurance to minimize the variance, and the minimum variance is 0 . The expected rate of return is 20%. Without calculating the general expression of variance, one can still find the optimal u in the following way. Note that whether or not it rains, the difference in revenue is $3,000,000. On the other hand, the difference in insurance payment is $1 per unit. Therefore, by purchasing 3000000 units of the insurance, the gap in revenue is fully compensated by the insurance payment, ensuring a certain payoff and minimizing the variance of return. 3. (Rain insurance) 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 $2.5 million revenue from his $1 million investment - unless it rains. If it rains, he will lose his entire investment. There is a 20% chance that it will rain on the day of the concert. Gavin suggests that he buy rain insurance. He can buy one unit of insurance for $0.20, 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 2.5 million units. (a) If he buys u units of insurance, what is the expected rate of return on his investment? (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 of variance? And what is the corresponding expected rate of return? [Hint: Before calculating a general expression for variance, think about an intuition for this.] (c) What number of units will minimize the variance, given the additional constraint that the expected rate of return should be at least 80%