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Show, using equations or a diagram, that an expected utility maximizer requires a higher return for a riskier asset
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- 5. Find the expected value assuming the risk factor is 30 % and the interest rate is 12% , if you will receive $20,000 one year from today.3. Sarah's current disposable income is £90,000. Suppose there's a 1% chance that Sarah's house may be flooded, and if it is, the cost of repairing it will be £80,000, reducing her disposable income to £10,000. Suppose also that her utility function of income M is: U = VM (a)Calculate Sarah's expected income and expected utility given the risk of flooding. (b)For her to take an insurance that fully insures her in the event of house flooding, Sarah would have to pay a price for such an insurance, which would reduce her disposable income. What would be the minimum certain disposable income required for Sarah to take an insurance that fully insures her in the event of house flooding? Explain your answer.10. Which one of the following measures may be used to measure the risk of an investment on its own? a) Expected return of the investment. b) Expected utility of the investment for an investor. c) Standard deviation of the possible outcomes of the investment. d) The Bernoullian utility function's value of a good investment outcome.
- 2. Consider an individual with a current wealth of $100,000 who faces the prospect of a 25% chance of losing $20,000 through theft of her car during the next year. If the person’s utility function is U(X) = ln(X), where X is wealth: a. calculate expected utility without insurance, b. calculate the actuarially fair premium for full insurance, c. calculate expected utility with full insurance at the actuarially fair premium d. calculate the maximum amount the individual would pay for full insurance.1. Tanner is choosing between two investment options. He can invest $500 now and get (guaranteed) $550 in one year, or invest $500 now and get (guaranteed) $531.40 back later today. The risk-free rate is 3.5%. Which investment should Tanner prefer? A) $531.40 later today, since $1 today is worth more than $1 in one year. B) $550 in one year, since it is $50 more than he invested rather than $31.40 more than he invested. C) Neither - both investments have a negative NPV. D) Tanner should be indifferent between the two investments, since both are equivalent to the same amount of cash today.4) Luke is planning an around-the-world trip on which he plans to spend $10,000. The utility from the trip is a function of how much she spends on it (Y ), given by U(Y) = InY a). If there is a 25 percent probability that Luke will lose $1000 of his cash on the trip, what is the trip's expected utility. b). Suppose that Luke can buy insurance to fully against losing the $1,000 with a actuarially fair insurance. What is his expected utility if he purchase this insurance. Will he purchase the insurance? c). Now suppose utility function is U(Y) = Y/1000 What is his expected utility if he purchase the insurance in b). Will he purchase the insurance?
- 3) A risk-loving individual has $1000 to invest. The individual maximizes his/her expected utility and has a monotonic utility function. Show that he/she will never choose a diversified portfolio - that is, show that he/she will either keep the entire $1000 in a safe, or invest the entire $1000 in a risky assesst, for which each $1 invested yields $] with probability p, and SB with probability (1-p), where $B<$1<$J.6. Expected Utility. Consider a Princeton student who feels that a "normal" year on campus is like gaining $A, compared to taking the year off, which gives utility u(wof). But she fears Princeton will renege on its promise of a normal year, and move everything online; she feels that this is like losing $L (compared to a normal on-campus year). If she expects this to happen with probability p, (a) Write out the expected utility from enrolling. (b) Write out the expected utility from taking the year off. (c) Show: if the student chooses to take the year off, and is risk-loving, it must be that L> A/p.4) You are a financial professional working in a corporate loan department. A company named Mitch Hedberg Inc. (MH) comes to you for a loan. MH has debt from a previous loan (given by a different firm than yours) of 200. Your company analysts say that MH is likely to earn either 180, 240, or 300 this year - each with a probability of 1/3. MH wants you to lend them 100. MH could use this borrowed 100 to do either project X or project Y. Project X has a guaranteed return of 125 if the 100 is put there. Project Y may return either 0 or 210; each has probability of 1/2 and also costs 100 to do. a) Which project, X or Y, has the larger expected value? b) If you lend MH the 100, what will they do with the money? Why? Show your math. c) Should you lend MH the money or not? Show your math. d) Why did I choose the letters "MH" for this problem? What financial economic concept with initials "MH" is important in this problem?
- 3. Suppose that Jon Snow's utility function is given by U(I)=501 where I represents annual income in thousands of dollars. a. Is Jon risk loving, risk neutral, or risk averse? Explain b. Suppose that Jon is currently earning an income of $1000 and can earn that income next year with certainty. He is offered a chance to take a new night watch job that offers a 0.25 probability of earning $2000 and a 0.75 probability of earning $500. Should he take the new night watch job?2. Ronald has $18,000. But he is forced to bet it on the flip of a fair coin. If he wins he has $36,000. If he loses he has nothing. Ronald's expected utility function is 0.5x0.5 + 0.5y0.5, where x is his wealth if heads comes up and y is his wealth if tails comes up. What safe income would make him exactly as well off as this bet?2. Two individuals have the same income ($100,000), but different potential healthcare expenses. Person A's probability of having $80,000 in healthcare expenses is 0.5 percent. Person B's probability of having $800 in healthcare expenses is 50 percent. Assume your utility is U = VI where I is your income. Calculate each person's expected income and expected utility. Calculate each person's certainty equivalent. What does value of the certainty equivalent tell you about how much each person would be willing to insure against their loss?