2. Let U(w) denote the utility function for an investor.(a) If the investor is non-satiated, what property must U(w) have?(b) If the investor is risk-neutral, what property must U(w) have?(c) Let X be a fair gamble. Suppose that an investor is risk-seeking. What would theinvestor prefer between X and the empty portfolio (i.e. making no investment)?

a) When we say the investor is not satisfied, it means they always want more wealth. They are never…

Answered: 2. Let U(w) denote the utility function…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter9: Decision Making Under Uncertainty

Section: Chapter Questions

Problem 69P

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A martingale betting strategy works as follows. You begin with a certain amount of money and repeatedly play a game in which you have a 40% chance of winning any bet. In the first game, you bet 1. From then on, every time you win a bet, you bet 1 the next time. Each time you lose, you double your previous bet. Currently you have 63. Assuming you have unlimited credit, so that you can bet more money than you have, use simulation to estimate the profit or loss you will have after playing the game 50 times.
You now have 10,000, all of which is invested in a sports team. Each year there is a 60% chance that the value of the team will increase by 60% and a 40% chance that the value of the team will decrease by 60%. Estimate the mean and median value of your investment after 50 years. Explain the large difference between the estimated mean and median.
In this version of dice blackjack, you toss a single die repeatedly and add up the sum of your dice tosses. Your goal is to come as close as possible to a total of 7 without going over. You may stop at any time. If your total is 8 or more, you lose. If your total is 7 or less, the house then tosses the die repeatedly. The house stops as soon as its total is 4 or more. If the house totals 8 or more, you win. Otherwise, the higher total wins. If there is a tie, the house wins. Consider the following strategies: Keep tossing until your total is 3 or more. Keep tossing until your total is 4 or more. Keep tossing until your total is 5 or more. Keep tossing until your total is 6 or more. Keep tossing until your total is 7 or more. For example, suppose you keep tossing until your total is 4 or more. Here are some examples of how the game might go: You toss a 2 and then a 3 and stop for total of 5. The house tosses a 3 and then a 2. You lose because a tie goes to the house. You toss a 3 and then a 6. You lose. You toss a 6 and stop. The house tosses a 3 and then a 2. You win. You toss a 3 and then a 4 for total of 7. The house tosses a 3 and then a 5. You win. Note that only 4 tosses need to be generated for the house, but more tosses might need to be generated for you, depending on your strategy. Develop a simulation and run it for at least 1000 iterations for each of the strategies listed previously. For each strategy, what are the two values so that you are 95% sure that your probability of winning is between these two values? Which of the five strategies appears to be best?
Based on Marcus (1990). The Balboa mutual fund has beaten the Standard and Poors 500 during 11 of the last 13 years. People use this as an argument that you can beat the market. Here is another way to look at it that shows that Balboas beating the market 11 out of 13 times is not unusual. Consider 50 mutual funds, each of which has a 50% chance of beating the market during a given year. Use simulation to estimate the probability that over a 13-year period the best of the 50 mutual funds will beat the market for at least 11 out of 13 years. This probability turns out to exceed 40%, which means that the best mutual fund beating the market 11 out of 13 years is not an unusual occurrence after all.
You have 5 and your opponent has 10. You flip a fair coin and if heads comes up, your opponent pays you 1. If tails comes up, you pay your opponent 1. The game is finished when one player has all the money or after 100 tosses, whichever comes first. Use simulation to estimate the probability that you end up with all the money and the probability that neither of you goes broke in 100 tosses.
If you own a stock, buying a put option on the stock will greatly reduce your risk. This is the idea behind portfolio insurance. To illustrate, consider a stock that currently sells for 56 and has an annual volatility of 30%. Assume the risk-free rate is 8%, and you estimate that the stocks annual growth rate is 12%. a. Suppose you own 100 shares of this stock. Use simulation to estimate the probability distribution of the percentage return earned on this stock during a one-year period. b. Now suppose you also buy a put option (for 238) on the stock. The option has an exercise price of 50 and an exercise date one year from now. Use simulation to estimate the probability distribution of the percentage return on your portfolio over a one-year period. Can you see why this strategy is called a portfolio insurance strategy? c. Use simulation to show that the put option should, indeed, sell for about 238.
Based on Kelly (1956). You currently have 100. Each week you can invest any amount of money you currently have in a risky investment. With probability 0.4, the amount you invest is tripled (e.g., if you invest 100, you increase your asset position by 300), and, with probability 0.6, the amount you invest is lost. Consider the following investment strategies: Each week, invest 10% of your money. Each week, invest 30% of your money. Each week, invest 50% of your money. Use @RISK to simulate 100 weeks of each strategy 1000 times. Which strategy appears to be best in terms of the maximum growth rate? (In general, if you can multiply your investment by M with probability p and lose your investment with probability q = 1 p, you should invest a fraction [p(M 1) q]/(M 1) of your money each week. This strategy maximizes the expected growth rate of your fortune and is known as the Kelly criterion.) (Hint: If an initial wealth of I dollars grows to F dollars in 100 weeks, the weekly growth rate, labeled r, satisfies F = (I + r)100, so that r = (F/I)1/100 1.)
A project does not necessarily have a unique IRR. (Refer to the previous problem for more information on IRR.) Show that a project with the following cash flows has two IRRs: year 1, 20; year 2, 82; year 3, 60; year 4, 2. (Note: It can be shown that if the cash flow of a project changes sign only once, the project is guaranteed to have a unique IRR.)
In the financial world, there are many types of complex instruments called derivatives that derive their value from the value of an underlying asset. Consider the following simple derivative. A stocks current price is 80 per share. You purchase a derivative whose value to you becomes known a month from now. Specifically, let P be the price of the stock in a month. If P is between 75 and 85, the derivative is worth nothing to you. If P is less than 75, the derivative results in a loss of 100(75-P) dollars to you. (The factor of 100 is because many derivatives involve 100 shares.) If P is greater than 85, the derivative results in a gain of 100(P-85) dollars to you. Assume that the distribution of the change in the stock price from now to a month from now is normally distributed with mean 1 and standard deviation 8. Let EMV be the expected gain/loss from this derivative. It is a weighted average of all the possible losses and gains, weighted by their likelihoods. (Of course, any loss should be expressed as a negative number. For example, a loss of 1500 should be expressed as -1500.) Unfortunately, this is a difficult probability calculation, but EMV can be estimated by an @RISK simulation. Perform this simulation with at least 1000 iterations. What is your best estimate of EMV?
You now have 5000. You will toss a fair coin four times. Before each toss you can bet any amount of your money (including none) on the outcome of the toss. If heads comes up, you win the amount you bet. If tails comes up, you lose the amount you bet. Your goal is to reach 15,000. It turns out that you can maximize your chance of reaching 15,000 by betting either the money you have on hand or 15,000 minus the money you have on hand, whichever is smaller. Use simulation to estimate the probability that you will reach your goal with this betting strategy.
It is January 1 of year 0, and Merck is trying to determine whether to continue development of a new drug. The following information is relevant. You can assume that all cash flows occur at the ends of the respective years. Clinical trials (the trials where the drug is tested on humans) are equally likely to be completed in year 1 or 2. There is an 80% chance that clinical trials will succeed. If these trials fail, the FDA will not allow the drug to be marketed. The cost of clinical trials is assumed to follow a triangular distribution with best case 100 million, most likely case 150 million, and worst case 250 million. Clinical trial costs are incurred at the end of the year clinical trials are completed. If clinical trials succeed, the drug will be sold for five years, earning a profit of 6 per unit sold. If clinical trials succeed, a plant will be built during the same year trials are completed. The cost of the plant is assumed to follow a triangular distribution with best case 1 billion, most likely case 1.5 billion, and worst case 2.5 billion. The plant cost will be depreciated on a straight-line basis during the five years of sales. Sales begin the year after successful clinical trials. Of course, if the clinical trials fail, there are no sales. During the first year of sales, Merck believe sales will be between 100 million and 200 million units. Sales of 140 million units are assumed to be three times as likely as sales of 120 million units, and sales of 160 million units are assumed to be twice as likely as sales of 120 million units. Merck assumes that for years 2 to 5 that the drug is on the market, the growth rate will be the same each year. The annual growth in sales will be between 5% and 15%. There is a 25% chance that the annual growth will be 7% or less, a 50% chance that it will be 9% or less, and a 75% chance that it will be 12% or less. Cash flows are discounted 15% per year, and the tax rate is 40%. Use simulation to model Mercks situation. Based on the simulation output, would you recommend that Merck continue developing? Explain your reasoning. What are the three key drivers of the projects NPV? (Hint: The way the uncertainty about the first year sales is stated suggests using the General distribution, implemented with the RISKGENERAL function. Similarly, the way the uncertainty about the annual growth rate is stated suggests using the Cumul distribution, implemented with the RISKCUMUL function. Look these functions up in @RISKs online help.)

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