2. Consider a trader with initial fund given by To holding q shares of stock i is C(q) = 10 + q². The price (x;) at which this trader sells its position is stochastically distributed according to the following probability distribution: 15, and the transaction cost function of J0.5, if a; = $8 0.5, if x; = $2 P(x;) = Let a random variable ñ be the profit of trading at each time t, t = 1, 2, ... ,T, (Ъ) Consider now that the trader's utility function is described by u(ñ) = µ(r). What is now the optimal level of position and the associated equilibrium profits?
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- QUESTION 1 Elizabeth has decided to form a portfolio by putting 30% of her money into stock 1 and 70% into stock 2. She assumes that the expected returns will be 10% and 18%, respectively, and that the standard deviations will be 15% and 24%, respectively. Compute the standard deviation of the returns on the portfolio assuming that the two stocks' returns are uncorrelated. 17.4%. 27.4%. 7.4%. 11.4%. QUESTION 2 Elizabeth has decided to form a portfolio by putting 30% of her money into stock 1 and 70% into stock 2. She assumes that the expected returns will be 10% and 18%, respectively, and that the standard deviations will be 15% and 24%, respectively. Describe what happens to the standard deviation of the portfolio returns when the coefficient of correlation ρ decreases. The standard deviation of the portfolio returns decreases as the coefficient of correlation decreases. The standard deviation of the portfolio returns increases as the coefficient…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?
- 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…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.Find the Pratt - Arrow risk - aversion function for a utility function U(W) = log(0.5-W + 500), where W is the amount of wealth in €. Suppose that an investor's wealth is subject to outcomes -800 €, 500 €, 500 € and 1, 000 € which affect the initial amount of 2,500 € with probabilities of their occurrence 40%, 15%, 15% and 30%, respectively. a) Using the Taylor approximation to certainty equivalent, calculate an approximate expected utility value. b) Calculate the certain equivalent of the investor's uncertain wealth. Interpret.
- 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?]Suppose Investor A has a power utility function with γ = 1, whilst Investor B has a power utility function with γ = 0.5 (i) Which investor is more risk-averse(assuming that w > 0)? (ii) Suppose that Investor B has an initial wealth of 100 and is offered the opportunity to buy Investment X for 100, which offers an equal chance of a payout of 110 or 92. Will she choose to buy Investment X?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.
- Using the Utility Function in Portfolio Management, where the utility function is the constant relative risk aversion utility of wealth function U(W) = W^(gamma)/gamma, set gamma to 0.5 and consider a 50-50 bet on winning 50,000 or getting nothing. What is the certainty equivalent wealth for this bet under these assumptions? Group of answer choices 30,000 10,000 25,000 12,500A moderately risk-averse investor has 50% of her portfolio invested in stocks and 50% in risk-free Treasury bills. Show how each of the following events will affect the investor’s budget line and proportion of stocks in her portfolio: A. The standard deviation of the return on the stock market increases, but the expected return on the stock market remains the same. B. The expected return on the stock market increases, but the standard deviation of the stock market remains the same. C. The return on risk-free Treasury bills increases.The value of Jon’s stock portfolio is given by the function v(t) = 50 + 77t + 3t2, where v is the value of the portfolio in hundreds of dollars and t is the time in months. How much money did Jon start with? (y-intercept) What is the minimum value of Jon’s portfolio? (vertex)