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(a) For what values of γ is the utility function well behaved? (i.e. U′ > 0 and U′′ < 0)
(b) Derive an expression for the stochastic discount factor mt+1?
(c)What are the prices q0s1and q0s2 of complex securities s1 and s2 at t = 0?
d) What weights create Arrow-Debreu securities? (hint: What are the weights of the complex
securities that create payoffs (1, 0)′ and (0, 1)′?)
(e) What are the state prices q1 and q2? (hint: State prices are a function of prices from part
c) and the the inverse of the payoff matrix from part d).)
(f) What is the price of a risk free bond qb?
(g) What are the risk neutral probabilities π1RN and π2RN ?
(h) Using the risk neutral probabilities solve for qs1and qs2
(i) Is π2RNgreater than or less than the true probability π2? Why?
.
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- Suppose an individual faces uncertainty over the following set of monetary outcomes A = {-10, 4, 8, 13, 22} Suppose the individual abides to the vNM axioms. a) How can you represent her preference relation over gambles over A? b) Is this representation unique? c) Is it the only one satisfying the expected utiltiyproperty? d) Suppose u(-10) = 0, u(22) = 1. Also assume that u(4) = 0.5, u(8) = 0.55 and u(13) =0.65. Can you build a gamble that is indifferent to the degenerate gamble that gives 4 with certainty? What is the utility of the following gambles p = (0.1, 0.2, 0.5, 0.2, 0), q = (0.2, 0.2, 0.2, 0.2, 0.2), r =(0.3, 0, 0.2, 0.1, 0.4). How does the individualrank these three gambles?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?By using the expected utility theory approach with u(x)=x2, choose the optimal decision for three different possible outcomes with probabilities p(ω1)=1/2, p(ω2)=p(ω3)=1/4, rewards R(d1,ω1)=£49,R(d1,ω2)=R(d1,ω3)=£25, R(d2,ω1)=£36,R(d2,ω2)=£100,R(d2,ω3)=£0, R(d3,ω1)=£81,R(d3,ω2)=R(d3,ω3)=£0
- 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…"The non-liner transformation of the vNM expected utility function fails to reflect the underlying preference relation." Explain the above statement.The von-Neumann Morgenstern utility function is of the form u(e) - In(e). There is a lottery over consumption outcomes: with probability 0.3, the consumption will be 1 and with probability 0.7 the consumption will be 3; Compute the risk premium (round to 2 decimals).
- Assumptions of rationality are violated because Choose one: A. both low-probability events and high-probability events are underanticipated. B. low-probability events are overanticipated whereas high-probability events are underanticipated. C. low-probability events are underanticipated whereas high-probability events are overanticipated. D. both low-probability events and high-probability events are overanticipated.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.Consider the following claim: “If a decision maker prefers one given lottery that yields $x with probability 1 over another given lottery whose expected return is $x, then we can fully characterize the agent's risk attitude. That is, this information comparing two given lotteries is enough to determine if the decision maker is risk averse, risk loving or risk neutral.” If this claim is TRUE, then provide a proof. If it is FALSE, then prove your argument by providing an explanation.
- Y5 Alfred is a risk-averse person with $100 in monetary wealth and owns a house worth $300, for total wealth of $400. The probability that his house is destroyed by fire (equivalent to a loss of $300) is pne = 0.5. If he exerts an effort level e = 0.3 to keep his house safe, the probability falls to pe = 0.2. His utility function is: U = w0.5 – e where e is effort level exerted (zero in the case of no effort and 0.3 in the case of effort).a. In the absence of insurance, does Alfred exert effort to lower the probability of fire?HINT: Calculate and compare the expected utility i) with effort, and ii) without effort. If effort is exerted, then the effort cost is paid regardless of whether or not a fire occurs.b. Alfred is considering buying fire insurance. The insurance agent explains that a home owner’s insurance policy would require paying a premium α and would repay the value of the house in the event of fire, minus a deductible “D”. [A deductible is an amount of money that the…In the CRRA utility family u(x) = x^(1−γ). If this individual has autility where γ = 0.01, would he prefer the payment of $320 or the gamble?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.