An individual is offered the following choice. Invest in a project that pays £900 with probability 0.5 and £400 with probability 0.5. What is the expected value of this project? Suppose that the utility function is given by U(y) = lny. Setting out your workings clearly with a full accompanying explanation, establish whether or not the individual will turn down this project if she is asked to pay £650. Calculate the value of the risk premium associated with this project.
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- An investor with capital x can invest any amount between0 and x; if y is invested then y is eitherwon or lost, with respectiveprobabilities p and 1− p. If p > 1/2, how much should be invested byan investor having a exponential utility function u(x) = 1 − e −bx ,b > 0.Questions 18 through 20 refer to the following information: Shawn's consumption is subject to risk. With probability 0.75 he will enjoy 10000 in consumption, but with probability 0.25 he will have only 3600. His utility function for consumption is given by v(c) = vc. Question 18 What is the expected value of Shawn's consumption? Question 19 What is his expected utility?Your utility function is given by M1/2. You have $100 and are planning to invest in a venture where you can win or lose 50 with equal probability. Will you accept the venture? What is the minimum gain you need to make in the good scenario such that you will invest in the venture?
- Cheryl Druehl Retailers, Inc., must decide whether to build a small or a large facility at a new location in Fairfax. Demand at the location will either be low or high, with probabilities 0.6 and 0.4, respectively. If Cheryl builds a small facility and demand proves to be high, she then has the option of expanding the facility. If a small facility is built and demand proves to be high, and then the retailer expands the facility, the payoff is $230,000. If a small facility is built and demand proves to be high, but Cheryl then decides not to expand the facility, the payoff is $183,000. If a small facility is built and demand proves to be low, then there is no option to expand and the payoff is $250,000. If a large facility is built and demand proves to be low, Cheryl then has the option of stimulating demand through local advertising. If she does not exercise this option, then the payoff is $45,000. If she does exercise the advertising option, then the response to advertising will…Leora has a monthly income of $20,736. Unfortunately, there is a chance that she will have an accident that will result in costs of $10,736. Thus leaving her an income of only $10,000. The probability of an accident is 0.5. Finally assume that her preferences over income can be represented by the utility function u(x) = 2ln(x).a) What is the expected income? What is Leora’s expected utility (you may leave in log form)? b) What is the certainty equivalent to her situation? What is the risk premium associated with her situation?c) What is the maximum that Leora would be willing to pay for a full insurance policy?d) Illustrate her expected utility, expected wealth, certainty equivalent, the risk premium and her willingness to pay for a full insurance policy in a diagram.Suppose that you graduate from college next year and you have two career options: 1) You will start a job in an investment bank paying a $100,000 annual salary. 2) You will start a Ph.D. in economics and, as a student, you will receive a $20,000 salary. You are bad with decisions, so you are letting a friend of yours decide for you by flipping a coin. The probabilities of options 1 and 2 are, therefore, each 50%. a) Illustrate, using indifference curves, your preferences regarding consumption choices in the two different states of the world. Assume that you are risk-averse. [Include also the 45 degrees line in your figure] b) Now show how the indifference curves would change if you were substantially more risk averse than before. Explain. c) Now show the indifference curves if you are risk neutral and if you are risk loving. d) Show your expected utility preferences from point a) mathematically.
- Suppose that you have two opportunities to invest $1M. The first will increase the amount invested by 50% with a probability of 0.6 or decrease it with a probability of 0.4. The second will increase it by 5% for certain. You wish to split the $1M between the two opportunities. Let x be the amount invested in the first opportunity with (1-x) invested in the second. Find the optimal value of x. Using expected value as the criterion (linear utility) Using the flowing utility function: u(x)=2.3 ln〖(1+4.5x)Show that a decision maker who has a linear utilityfunction will rank two lotteries according to their expectedvalue.Problem 3. Carol's risk preference is represented by the following expected utility formula: U(T, C₁; 1 T, C₂) = π √√ √₁+ (17) √√C₂. i) Suppose Carol is indifferent between the following two options: the first option A returns $100 with probability and $X with probability, and the second option B returns $49 for sure. Determine X. ii) Consider the following three lotteries: L₁ = (0.9, $100; 0.1, $49), L2 = (0.7, $225; 0.3, $49), and L3= (0.5, $400; 0.5, $0). What is the ranking of these lotteries for Carol? Calculate the risk premiums of these lotteries for Carol. 1
- Show that a decision maker who has a linear utility function will rank two lotteries according to their expected value.Priyanka has an income of £90,000 and is a von Neumann-Morgenstern expected utility maximiser with von Neumann-Morgenstern utility index u(x) = √√x. There is a 1 % probability that there is flooding damage at her house. The repair of the damage would cost £80,000 which would reduce the income to £10,000. a) Would Priyanka be willing to spend £500 to purchase an insurance policy that would fully insure her against this loss? Explain. b) What would be the highest price (premium) that she would be willing to pay for an insurance policy that fully insures her against the flooding damage?A manager is deciding whether to build a small or a large facility. Much depends on the future demand that thefacility must serve, and demand may be small or large. The manager knows with certainty the payoffs that willresult under each alternative, shown in the following payoff table. The payoffs (in $000) are the present values offuture revenues minus costs for each alternative in each event.What is the best choice if future demand will be low?