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- Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x) = . There is a safe asset (such as a US government bond) that has net real return of zero. There is also a risky asset with a random net return that has only two possible returns, R₁ with probability 1-q and Ro with probability q. We assume R₁ 0. Let A be the amount invested in the risky asset, so that w - A is invested in the safe asset. 1) What are risk preferences of this investor, are they risk-averse, risk neutral or risk-loving?An investor allocates $30,000 and $50,000 to two assets (A1 and A2). These assets generate 5% and -4.5% rate of returns, respectively. She allocates the remaining 50% of her portfolio to an asset (A3), which provides 4.5% rate of return. Calculate the portfolio's rate of return.Let b(p,s,t) be the bet that pays out s with probability p and t with probability 1−p. We make the three following statements: S1: The CME for b is the value m such that u(m)=E[u(b(p,s,t))]. S2: A risk averse attitude corresponds to the case CME smaller than E[b(p,s,t))]. S3: A risk seeking attitude corresponds to a convex utility function. Are these statements true or false?
- 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) (3) Consider two investments X and Y, where X pays $0 and $10 with equal probability and Y pays 0 with probability 0.75 and $20 with probability 0.25. What investment would an investor choose if her utility function is u(x) = x? u(x) = x u(x) = 1-e 10 () (i) (ii)An investor has utility function U = 10 + 5P – 0.02P2. What is the expected utility of the following investment option? Probability Payoff (P) 0.4 10 0.3 20 0.2 30 0.1 40
- Natalie entered a raffle recently and never checked her tickets. She has recently learned the exact number of the other unchecked tickets. Based on this information she knows that there is a 30% chance that she has won the raffle prize of $1,600. If she does not win the raffle her wealth will be zero. Natalie has a von Neumann- Morgenstern utility such that she wants to maximize the expected value of cvc, where cc is total wealth. What is the minimum price for which Natalie would sell her raffle tickets? $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…You plan to invest $1,000 in a corporate bond fund or in a common stock fund. The following table represents the annual return (per $1,000) of each of these investments under various economic conditions and the probability that each of those economic conditions will occur. Compute the expected return for the corporate bond and for the common stock fund. Show your calculations on excel for expected returns. Compute the standard deviation for the corporate bond fund and for the common stock fund. Would you invest in the corporate bond fund or the common stock fund? Explain. If choose to invest in the common stock fund and in (c), what do you think about the possibility of losing $999 of every $1,000 invested if there is depression. Explain.
- Mr Usu has an expected utility function with u(x) = x0.5. He is analyzing an investment opportunity that promises to pay out $1,160 with prob. 0.6 and $2,800 with prob. 0.4. What is the expected utility of this opportunity?Steve has received a stock tip from Monica. Monica has told him that XYZ Corp. will increase in value by 100%. Steve believes that Monica has a 25% chance of being correct. If Monica is incorrect, Steve expects the value of XYZ Corp. will fall by 50%. a. If Steve's utility of income is U(I)=50I. What is Steve's expected utility from buying $1,000 worth of XYZ Corp. stock? b. If Steve's utility of income is U(I)=I0.5. What is Steve's expected utility from buying $1,000 worth of XYZ Corp. stock?Draw a utility function over income u(I) that describes a man who is a risk lover when his income is low but risk averse when his income is high. 1.) Using the 3-point curved line drawing tool, draw the low income portion of his utility function. Label it U₁. 2.) Using the 3-point curved line drawing tool, draw the high income portion of his utility function. Label it UH. Carefully follow the instructions above, and only draw the required objects. C 500- 450- 400- 350- 300- 250- 200- 150- 100- 50- 0 Utility 20,000 40,000 60,000 80,000 100,000 Income
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