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- 1 Question 2. Suppose that there is one risk free asset with return rf and one risky asset with normally distributed returns, r ∼ N(µ, σ2). The investor has an expected utility maximizer with the CARA utility u(r) = −e −Ar. Write down the investor’s maximization problem of choosing α fraction of his wealth will be invested in the risky asset Find the optimal fraction of wealth that the investor will invest in the risky asset α∗Hint: Use the fact that if a random variable x is distributed normally with mean µx and variance σ2x , then for any constant α, What happens to the optimal fraction of wealth that the investor will invest in the risky asset as the risk aversion A increases? Explain the intuition behind your result.Anticipated consumer demand in a restaurant for free-range steaks next month can be modeled by a normal random variable with mean 1,200 pounds and standard deviation 100 pounds. a. What is the probability that demand will be between 1,100 and 1,300 pounds? Calculate in 4 decimal place. b. The probability is 0.10 that demand will be more than how many pounds?In any year, the weather can inflict storm damage to a home. From year to year, thedamage is random. Let Y denote the dollar value of damage in any given year.Suppose that in 95% of the years Y = 0, but in 5% of the years Y = 20,000.a. What is the mean of the damage in any year?b. What is the standard deviation of the damage in any year?c. Consider an “insurance pool” of 100 people who homes are sufficientlydispersed so that, in any year, the damage to different homes can be viewed asindependently distributed random variables. What is the probability that ?exceeds $2000?
- Please do not give solution in image format thanku Two Manufacturers supply food to a large cafeteria. Manufacturer A supplies 40% of the soup served in the cafeteria, while Manufacturer B supplies 60% of the soup that is served. 3% of the soup cans provided by Manufacturer A are found to be dented, while 1% of the cans provided by Manufacturer B are found to be dented. Given that a can of soup is dented, find the probability that it came from Manufacturer B.Suppose that there are two types of workers: high and low. Employers cannot distinguish between different types during an interview. Employers value high type at $200,000 and low type at $100,000. Employers are in a competitive market (i.e. zero profit applies). High type workers have a reservation wage of 140,000 and low type workers have a reservation wage of 80,000. Suppose that 50% of all workers are high type. The productivities, reservation wages, and the probabilities are common knowledge). What wage would the employers offer? Please explain the solution!A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond, and the third is a T-bill money market fund that yields a rate of 8%. The mean and the standard deviation of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12% 15% The correlation between the fund returns is 0.10. Your client’s degree of risk aversion is A = 3.5. Given the utility function: U = E(r) - 1/2 A Sigma^2 What proportion, y, of the total investment should be invested in the tangency portfolio so that your client can maximize his/her expected utility? What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio?
- Suppose that a particular large hotel has 790 rooms. Furthermore, suppose that for weekend nights the mean demand for rooms is 730 rooms with a standard deviation of 35 rooms. What is the probability that over a sample of 35 weekend nights, the hotel will sell at least an average of 740 rooms over this sample? (please round your answer to 4 decimal places).Suppose there are n assets which are uncorrelated. (They mightbe n different “wild cat” oil well prospects.) You may invest in any one, or in any combination of them. The mean rate of return r¯ is the same for each asset, but the variances are different. The return on asset i has a variance of σ2i for i = 1, 2, . . . , n.(a) Show the situation on an r¯-σ diagram. Describe the efficient set.(b) Find the minimum-variance point. Express your result in terms of σ¯2 =!"ni=11σ2i#−1.Exercise 3: Risky Investment Charlie has von Neumann-Morgenstern utility function u(x) = ln x and has wealth W = 250, 000. She is offered the opportunity to purchase a risky project for price P = 160, 000. 1 1 With probability p = 2 the project will be a success and return V > 160, 000. With probability 1 −p = 2 the project will fail and be worthless (i.e. it returns 0). For simplicity assume there is no interest between the time of the investment and the time of its return, that is r = 0 . How large must V be in order for Charlie to want to purchase the risky project? [Hint: What is Charlie’s expected utility is she does not purchase the project? What is Charlie’s expected utility is she purchases the project?]
- The accompanying table gives the outcomes and probability distribution of the number of times a student checks her e-mail daily: Outcome (X) (number of email checks) Probability Distribution f(x) 0 0.05 1 0.15 2 0.30 3 0.25 4 0.15 5 0.08 6 0.02 Calculate the expected value and the variance.Using the random variables X and Y from Table 2.2, consider two new random variables W = 4 + 8X and V = 11 - 2Y. Compute (a) E(W) and E(V); (b) J2W and J2V; and (c) JWV and corr(W, V).If patient insurance inquiries arrive at Blue Choice insurance with a mean rate of 3.6 calls per minute, the exponential probability of waiting more than 0.5 minutes to get the next inquiry call is _____. Question 6Select one: a. 0.2407 b. 0.1222 c. 0.5000 d. 0.1653