How to find EMV(), () for CE =15 & CE=20. Suppose that Dons risk attitude can be modeled with an exponential utility function. If his

How to find EMV(μ), (σ) for CE =15 & CE=20. Suppose that Don’s risk attitude can be modeled with an exponential utility function. If his certainty equivalent were 15 billion credits, find his risk tolerance. What would his risk tolerance be if his CE were 20 billion? We have, CE ≈ μ - [(0.5*(σ^2))/R] & exponential utility function U(x) = 1 - [e^(-x/R)]. {Solve to get these values for our problem μ = EMV = 50.28, and σ^2 = 3553.88}

Given that: Don Newcomb was perplexed. It had been five years since he had founded Interplants, Inc., a research and- development firm that developed genetically engineered plants for interplanetary space flight. During that five years, he and his scientists had made dramatic advances in developing special plants that could be used for food and air-purification systems in space stations and transports. In fact, he mused, their scientific success had been far greater than he had ever expected. Five years ago, after the world superpowers had agreed to share space-travel technology, the outlook had been quite rosy. Everyone had been optimistic. Indeed, he was one of many investors who had jumped at the chance to be involved in the development of such technology. But now, after 5 tumultuous years, the prospects were less exciting. First, there had been the disappointing realization that none of the superpowers had made substantial progress on an ion engine to power space vehicles. Such an engine was absolutely crucial to the success of interplanetary space flight, because—theoretically, at least—it would make travel 10 times as fast as conventionally powered ships. When the importance of such an engine became obvious, the superpowers had generously funded a huge multinational research project. The project had made substantial progress, but many hurdles remained. Don’s risk assessors estimated that there was still a 15%chance that the ion engine would prove an infeasible power source. If this were to happen, of course, Don and the many other investors in space travel technology would lose out. Then there was the problem with the settlement policy. The superpowers could not agree on a joint policy for the settlement of interplanetary space, including the deployment of space stations as well as settlements on planets and their satellites. The United American Alliance urged discretion and long-range planning in this matter, suggesting that a multinational commission be established to approve individual settlement projects. Pacificasia and the Allied Slavic Economic Community were demanding that space be divided now. By immediately establishing property rights, they claimed, the superpowers would be able to develop the optimum space economy in which the superpowers could establish their own economic policies within their “colonies” as well as determine trade policies with the other superpowers. Europa favored the idea of a commission, but also was eager to explore other available economic possibilities. The discussion among the superpowers had been going on since long before the founding of Interplants. Five years ago, progress was made, and it appeared that an agreement was imminent. But 18 months ago the process stalled. The participants in the negotiations had established positions from which they would not budge. Don had followed the discussions closely and had even provided expert advice to the negotiators regarding the potential for interplanetary agricultural developments. He guessed that there was only a 68%chance that the superpowers would eventually arrive at an agreement. Naturally, until an agreement was reached, there would be little demand for space-traveling plants. Aside from these external matters, Don still faced the difficult issue of developing a full-scale production process for his plants. He and his engineers had some ideas about the costs of the process, but at this point, all they could do was come up with a probability distribution for the costs. In thinking about the distribution, Don had decided to approximate it with a three-point discrete distribution. Thus, he characterized the three branches as “inexpensive,” “moderate,” and “costly,” with probabilities of 0.185, 0.63, and 0.185, respectively. Of course, his eventual profit (or loss) depended on the costs of the final process. Don also had thought long and hard about the profits that he could anticipate under each of the various scenarios. Essentially, he thought about the uncertainty in two stages. First was the determination of costs, and second was the outcome of the external factors (the ion-engine research and the negotiations regarding settlement policy). If costs turned out to be “inexpensive,” then, in the event that the superpowers agreed and the ion engine was successful, he could expect a profit of 125 billion credits. He would lose 15 billion credits if either the engine or the negotiations failed. Likewise, if costs were “moderate,” he could anticipate either a profit of 100 billion credits if both of the external factors resulted in a positive outcome, or a loss of 18 billion if either of the external factors were negative. Finally, the corresponding figures in the case of a “costly” production process were profits of 75 billion credits or a loss of 23 billion. “This is so confusing,” complained Don to Paul Fiester, his chief engineer. “I really never expected to be in this position. Five years ago none of these risks were apparent to me, and I guess I just don’t tolerate riskwell.” After a pause, Paul quietly suggested, “Well, maybe you should sell the business.” Don considered that. “Well, that’s a possibility. I have no idea how many crazy people out there would want it.” “Some of the other engineers and I might be crazy enough,” Paul replied. “Depending on the price, of course. At least we’d be going in with our eyes open. We know what the business is about and what the risks are.” Don gave the matter a lot of thought that night. “What should I sell the company for? I hate to give up the possibility of earning up to 125 billion credits. But I don’t like the possibility of losing 23 billion either—no one would!” As he lay awake, he finally decided that he would let the business go—with all its risks—for 20 billion credits. If he could get that much for it, he’d sell. If not, he’d just as soon stick with it, in spite of his frustrations with the risks.