A risk averse investor with utility function: u(w) = (w)1/2 where w represents wealth, He has $200 (wo) available to invest in the financial market. He has two alternatives. One is a risk-free asset with interest rate ofr 2%; and the other alternative is to invest in a risky asset whose return is represented by a discrete probability distribution: = (1%, 5%; 1/2, 1/2). %3D Consider the following two portfolios (P1 an P2): P1 consists in investing $50 in che risky asset, and P2 in investing $150 in the risky asset. Calculate the expected utility of every portfolio: u(P1) = (use two decimals) (P2) = (use two decimals). %3D
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- Buying and selling prices for risky investments obviously are related to certain equivalents. This problem, however, shows that the prices depend on exactly what is owned in the first place. Suppose that your utility for wealth (A) can be represented by the utility function u(A) = In [(A)] You currently have R1000 in cash. A business deal of interest to you yields a reward of R100 with probability 0,5 and RO with probability 0,5. 2.1 If you own this business deal in addition to the R1000, what is the smallest amount for which you would sell the deal? 2.2 Suppose you do not own the deal. Formulate an appropriate equation and solve with algebra to find the largest amount you would be willing to pay for the deal. 2.3 Explain why the amounts in 2.1 and 2.2 are slightly different.4) Consider investors with preferences represented by the utility function U = E(r) – Ao². (a) Draw the indifference curve representing a utility level of 10% for an in- vestor with a risk aversion parameter A = 3 in expected return-standard deviation space. (b) In the same graph, draw the indifference curve representing a utility level of 15% for an investor with a risk aversion parameter A = 3. (c) In the same graph, draw the indifference curve representing a utility level of 10% for an investor with a risk aversion parameter A = 5.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.
- 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.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…Suppose you visit with a financial adviser, and you are considering investing some of your wealth in one of three investment portfolios stocks, bonds, or commodities. Your financial adviser provides you with the following table, which gives the probabilities of possible returns from each investment To maximize your expected return, you should choose: Stocks Bonds Probability Return Probability Return 0.15 20% 0.15 16.7% 06 10% T 04 7.5% 0.25 8% 0.45 3.3% OA bonds OB stocks OC. commodities OD. All of the portfolios have the same expected return. If you are risk-averse and had to choose between the stock or the bond investments, you would choose OA the stock portfolio because there is less uncertainty over the outcome OB. the bond portfolio because there is less uncertainty over the outcome. OC. the stock portfolio because of greater expected return. OD. the bond portfolio because of greater expected return. Commodities Probability Return 02 20% 0.2 15% 0.2 8% 02 02 5% 0%
- 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.Exercise 3: Risky Investment Charlie has von Neumann-Morgenstern utility function u(x) = In x and has wealth W = 250, 000. She is offered the opportunity to purchase a risky project for price P = 160, 000. 1 the project will be a success and return V > 160, 000. With probability 1-p= 1 With probability p= 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?]Hugo has a concave ubility function of U(W)=√W. His only asset is shares in an Internet start-up company. Tomorrow he will learn the stock's value. He belleves that it is worth $225 with probability 80% and $256 with probability 20%. What is his expected utsty? What risk premium would he pay to avoid bearing this risk? The stock's expected utility (EU) is EU = (Enter a numeric response using a real number rounded to two decimal places.) han fro
- 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. With probability p= 1 the project will be a success and return V > 160,000. With probability 1-p = 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?]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 || S0.5, if a; = $8 |0.5, if x; = $2 P(x:) = Let a random variable îñ be the profit of trading at each time t, t = 1, 2, . ..,T, (b) Consider now that the trader's utility function is described by u(ñ) = µ(7). What is now the optimal level of position and the associated equilibrium profits?3. Let us consider a utility function: U(x) = (V2x -1. (200sxs800) We have LiL2 where L1=(1, Y) and L2=(0.3, 450, 0.7,648). a. Determine the value of Y. b. Determine RP (Risk Premium) of L2.