2. Consider a trader with initial fund given by To = 15, and the transaction cost function of holding q shares of stock i is C(q) = 10 + q². The price (x₂) at which this trader sells its position is stochastically distributed according to the following probability distribution: (0.5, 0.5, if x₂ = $2 if x₁ = $8 Let a random variable ♬ be the profit of trading at each time t, t = 1, 2, …, T, (a) If the trader's utility function is given by 1 u(ñ) = µ(ñ) — -—-0(ñ), P(x₁) = where μ() is the mean and o() is the variance, determine the trader's optimal level of position, and the associated equilibrium profits from liquidate the complete position.
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- The investor is considering how to optimally invest 1000 euros in stocks and bonds. Let's assume that the optimal decision is made based on expected utility. Suppose the investor has a utility function u(x)=ln(1+x), where x is their wealth. Let y be the proportion invested in stocks and 1−y be the proportion invested in bonds. By investing in stocks, the investor earns 1% with a probability of 39.5% and 4% with a probability of 60.5%. By investing in bonds, the investor earns a certain 2.8%. What proportion of the investment will the investor allocate to stocks and what proportion to bonds?Speedy Oil provides a single-channel automobile oil change and lubrication service. Customers provide an arrival rate of 4 cars per hour. The service rate is 5 cars per hour. Assume that arrivals follow a Poisson probability distribution and that service times follow an exponential probability distribution. What is the average number of cars in the system? What is the average time that a car waits for the oil and lubrication service to begin? What is the probability that an arrival has to wait for service?2.4 The opening 2018 World Cup odds against being the winning team specified by espn.com were 9/2 for Germany, 5/1 for Brazil, 11/2 for France, 20/1 for England, and 7/1 for Spain. Find the corresponding prior probabilities of winning for these five teams.
- 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?]a. A company produces lightbulbs whose life follows a normal distribution, with mean 1200 hours and standard deviation 250 hours. If we choose a lightbulb at random, what is the probability that its lifetime will be between 900 and 1300 hours? (answer in three decimal places) b. An instructor has found that the time spent by students on a particular homework assignment follows a normal distribution with mean 150 minutes and standard deviation 40 minutes. The probability is 0.8 that a randomly chosen student spends less than how many minutes on this assignment. (answer in one decimal place),A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows: Stock fund (S) Bond fund (B) Proportion in Stock Fund The correlation between the fund returns is 0.10. Tabulate the investment opportunity set of the two risky funds. (Round your answers to 2 decimal places.) 0.00 % Expected Return 20% 12 20.00 40.00 60.00 80.00 100.00 Proportion in Bond Fund Standard Deviation 30% 15 100.00 % 80.00 60.00 40.00 20.00 0.00 Expected Return % Standard Deviation %
- The closing price of Martin's Sporting Goods Inc. common stock is uniformly distributed between $25 and $36 per share. What is the probability that the stock price will be more than $32?21. The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probabiliiy function specified by the p.d.f. f(x), given by : k. x ={k. (10 – x), for 5Q1) An expected utility maximiser owns a car worth £60000£60000 and has a bank account with £20000£20000. The money in the bank is safe, but there is a 50%50% probability that the car will be stolen. The utility of wealth for the agent is u(y)=ln(y)u(y)=ln(y) and they have no other assets.Q1) An expected utility maximiser owns a car worth £60000£60000 and has a bank account with £20000£20000. The money in the bank is safe, but there is a 50%50% probability that the car will be stolen. The utility of wealth for the agent is u(y)=ln(y)u(y)=ln(y) and they have no other assets. Q2) Consider the setup from Question 1. A risk-neutral insurance company is willing to insure the car at the premium of π=£2/3π=£2/3 for every one pound of coverage. Q3) Consider the setup from Questions 1 and 2. How much profits, in expectation, does the insurance company earn on insuring the individual?A proprietor of a food-stall has invented a new item of food delicacy which he calls WHIM. He has calculated that the cost of manufacture is Re. 1 per piece and that because of its novelty and quality it would be sold for Rs. 3 per piece. It is, however, perishable, and any goods unsold at the end of the day are a dead loss. He expects the demand to be variable and has drawn up the following probability distribution. 14 No. of pieces demanded : Probability (i) Find the value of k. 10 11 12 13 15 k + 0:02 k + 0.05 5k -0.02 7k + 0:03 2k + 0-02 2k (ii) Find an expression for his net profit or loss if he manufactures m pieces and only n are demanded; consider separately the two cases n m. (ii) Assume that he manufactures 12 pieces. using the result in (i) ab ove, find his net profit or loss for ea ch level of demand. (v) Also obtain the net profits for di fferent levels of demand and produ ction, each ranging from 10 to 15. (iv) Using the probability distributio n, calculate his expected…Problem 1. A new edition of a very popular textbook will be published a year from now. The publisher currently has 2000 copies on hand and is deciding whether to do another printing before the new edition comes out. The publisher estimates that demand for the book during the next year is governed by the probability distribution on the image . A production run incurs a fixed cost of $10,000 plus a variable cost of $15 per book printed. Books are sold for $130 per book. Any demand that cannot be met incurs a penalty cost of $20 per book, due to loss of goodwill. Up to 500 of any leftover books can be sold to Barnes & Noble for $35 per book. The publisher is interested in maximizing expected profit. The following print- run sizes are under consideration: 0 (no production run) to 16,000 in increments of 2000. What decision would you recommend? Use simulation with 1000 replications. For your optimal decision, the publisher can be 90% certain that the actual profit associated with…SEE MORE QUESTIONS