Jen is choosing a portfolio. For this choice, she is an expected utility maximizer. We fix the following preference representation for Jen: if she earns w dollars with robability 1, her utility is √w. There are two stocks she can buy, A or B. She will hoose one. Stock A will be worth 1000 with probability 1/2, and it will be worth 000 with probability 1/2. Stock B will be worth 250 with probability 2/3 and 5000 with probability 1/3. Which does she choose?
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- You are considering a $500,000 investment in the fast-food industry and have narrowed your choice to either a McDonald’s or a Penn Station East Coast Subs franchise. McDonald’s indicates that, based on the location where you are proposing to open a new restaurant, there is a 25 percent probability that aggregate 10-year profits (net of the initial investment) will be $16 million, a 50 percent probability that profits will be $8 million, and a 25 percent probability that profits will be −$1.6 million. The aggregate 10-year profit projections (net of the initial investment) for a Penn Station East Coast Subs franchise is $48 million with a 2.5 percent probability, $8 million with a 95 percent probability, and −$48 million with a 2.5 percent probability. Considering both the risk and expected profitability of these two investment opportunities, which is the better investment? Explain carefully.Let U(x)= x^(beta/2) denote an agent's utility function, where Beta > 0 is a parameter that defines the agent's attitude towards risk. Consider a gamble that pays a prize X = 10 with probability 0.2, a price X = 50 with probability 0.4 and a price X = 100 with probability 0.4. Compute the agentís expected utility for such gamble and find the value of Beta such that the agentis risk neutral? Suppose B= 1, what is the certainty equivalent of the gamble described above? What is the Arrow-Pratt measure of absolute risk aversion?Consider the following prospects – A: (0.5, 0, 0.5: $100, $60, $10) B: (0, 0.9, 0.1: $100, $60, $10) C: (0.2, 0.5, 0.3: $100, $60, $10) D: (0.4, 0.2, 0.4: $100, $60, $10) Show that D>A>B>C is consistent with expected utility theory and that this preference ordering implies “risk-loving” preferences. Show that C>B>D>A is consistent with the expected utility theory.
- 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.Gary likes to gamble. Donna offers to bet him $31 on the outcome of a boat race. If Gary’s boat wins, Donna would give him $31. If Gary’s boat does not win, Gary would give her $31. Gary’s utility function is p1x^21+p2x^22, where p1 and p2 are the probabilities of events 1 and 2 and where x1 and x2 are his wealth if events 1 and 2 occur respectively. Gary’s total wealth is currently only $80 and he believes that the probability that he will win the race is 0.3. Which of the following is correct? (please submit the number corresponding to the correct answer). Taking the bet would reduce his expected utility. Taking the bet would leave his expected utility unchanged. Taking the bet would increase his expected utility. There is not enough information to determine whether taking the bet would increase or decrease his expected utility. The information given in the problem is self-contradictory.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.
- Obi-Wan is considering whether to buy a lightsaber. With probability 0.50 he will value the lightsaber at $4,000, and with probability 0.50 he will value it at $1,000. If new lightsabers sell for $2,500, then buying a new lightsaber is a: Multiple Choice fair gamble. better-than-fair gamble. less-than-fair gamble. less-than-fair gamble if Obi-Wan risk neutral.A risk-averse expected-utility maximizer has initial wealth w0 and utility function u. She facesa risk of a financial loss of L dollars, which occurs with probability π. An insurance companyoffers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in theevent of a loss). Denote by x the number of dollars of coverage.(a) Give the formula for her expected utility V (x) as a function of x.(b) Suppose that u(z) = −e−zλ, π = 1/4, L = 100 and p = 1/3. Write V (x)using these values. There should be three variables, x, λ and w. Find the optimal value of x,as a function of λ and w, by solving the first-order condition (set the derivative of the expectedutility with respect to x equal to zero). (The second-order condition for this problem holds butyou do not need to check it.) Does the optimal amount of coverage increase or decrease in λ,where λ > 0?(c) Repeat exercise (b), but with p = 1/6.(d) You should find that for either (b) or (c), the optimal coverage…A woman with current wealth X has the opportunity to bet an amount on the occurrence of an event that she knows will occur with probability P. If she wagers W, she will received 2W, if the event occur and if it does not. Assume that the Bernoulli utility function takes the form u(x) = with r > 0. How much should she wager? Does her utility function exhibit CARA, DARA, IARA? Alex plays football for a local club in Kumasi. If he does not suffer any injury by the end of the season, he will get a professional contract with Kotoko, which is worth $10,000. If he is injured though, he will get a contract as a fitness coach worth $100. The probability of the injury is 10%. Describe the lottery What is the expected value of this lottery? What is the expected utility of this lottery if u(x) = Assume he could buy insurance at price P that could pay $9,900 in case of injury. What is the highest value of P that makes it worthwhile for Alex to purchase insurance? What is the certainty…
- Suppose that Mira has a utility function given by U=2I+10√I. She is considering two job opportunities. The first job pays a salary of $40,000 for sure. The second job pays a base salary of $20,000 but offers the possibility of a $40,000 bonus on top of your base salary. She believes that there is a probability of p=0.50 that she will earn the bonus. What is the expected salary of the second job? Which offer gives Mira a higher expected utility? Based on this information, is Mira risk adverse, risk neutral, or risk-loving?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?]An investor is considering three strategies for a $1,000 investment. The probable returns are estimated as follows: • Strategy 1: A profit of $10,000 with probability 0.15 and a loss of $1,000 with probability 0.85 • Strategy 2: A profit of $1,000 with probability 0.50, a profit of $500 with probability 0.30, and a loss of $500 with probability 0.20 • Strategy 3: A certain profit of $400 Which strategy has the highest expected profit? Explain why you would or would not advise the investor to adopt this strategy.