Consider the following statements when answering this question: 1. The allocation of a risk-averse investor's portfolio between a risk free asset and a risky asset never changes if the rate of return on both assets increases by the same amount. II. Given the choice between investing in a risk free asset or a risky asset with higher expected returns, the utility maximizing portfolio of a risk neutral or risk loving investor would never include the risk free asset. I is true and II is false. BI is false and II is true. I and II are true. I and II are false.
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- 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.Suppose Caroline is choosing how to allocate her portfolio between two asset classes: risk-free government bonds and a risky group of diversified stocks. The following table shows the risk and return associated with different combinations of stocks and bonds.CombinationFraction of Portfolio in Diversified StocksAverage Annual ReturnStandard Deviation of Portfolio Return (Risk)(Percent)(Percent)(Percent)A 0 1.50 0B 25 3.00 5C 50 4.50 10D 75 6.00 15E 100 7.50 20There is a relationship between the risk of Caroline's portfolio and its average annual return.Suppose Caroline currently allocates 75% of her portfolio to a diversified group of stocks and 25% of her portfolio to risk-free bonds; that is, she chooses combination D. She wants to reduce the level of risk associated with her portfolio from a standard deviation of 15 to a standard deviation of 5. In order to do so, she must do which of the following? Check all that apply. Sell some of her stocks and use the proceeds to purchase…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?
- 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.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.5. An individual has a utility function given by (W) - √W, and initial wealth of $100. If he plays a costless lottery in which he can win or lose $10 at the flip of a coin, compute his expected utility. What is the expected gain? Will such a person be categorized as risk neutral?
- 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?5) Philippina's utility of wealth function is VW. She has stocks, bonds and cash worth $360,000 and a house worth $280,000. There is a 1% chance of her house being completely destroyed by fire. Unfortunately, if she has insurance, she is less responsible and so the risk of her house being burned down rises to 2%. a. What is the most Philippina is willing to pay for insurance?10. Karl's utility function is U(w) = 20 is w. = 300. He considers a gamble in which he could win 200 with probability p or lose 200 with probability 1 described by expected utility theory. He is indifferent between keeping his initial wealth for sure or taking the gamble if the value of p is where w is wealth. His initial wealth %3D w+200 - p. Karl's preferences in the face of risk are (a). 4 (b) .5 (c) .6 (d) .7 (e) .8
- 3. In the second example, we will consider the case where the insurance contract involves a deductible this is an amount which is deducted from the final pay-out of the insurance firm in the case of a loss. In other words, the consumer bears this part of the loss herself. For this problem, assume a risk-averse, expected utility maximizing consumer with initial wealth wo who faces a potential loss of size L which will occur with probability p. Her utility-of-final-wealth function is denoted by u(.). Suppose that the consumer can purchase insurance coverage of C > 0 units of wealth from a perfectly competitive insurance firm at a premium of 7 per unit of coverage, but that the firm charges an additive deductible: if C units of insurance is purchased, the insurance firm pays out (C – d) if the loss occurs, where d 20 is a fixed amount independent of C. (a). For this problem, state the consumer's expected utility function. (b). Set up the consumer's utility maximization problem and find…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?10. BB has an endowment (E) which she can invest. With probability p, the investment works out well and BB ends up with E(1+r) but, with probability 1-p, BB loses all of her money. a) If BB is risk neutral, what is the lowest value of p for which she should invest? b) What is the expected value of perfect information? c) Now assume BB is risk averse with utility of money function U(m) = m1/2. What fraction, x, of her endowment should BB invest?