# The daily return of a certain stock is normally distributed with a mean of

Show your work concisely. If you utilize Excel functions or online calculators specify the function/calculator and ALL inputs. You may find the reading on probability fundamentals useful in answering c) and d)The daily return of a certain stock is normally distributed with a mean of 0.30% and a standard deviation of 2.50%.
a.     (already solved on the quiz) Find the probability that the return on a randomly selected day is negative. Provide a simple picture of the normal distribution and clearly label what the probability found refers to.
b.    Find the level, L, such that there is a 20% chance that the daily return on a randomly selected day exceedsL. Provide a simple picture of the normal distribution and clearly label the value found on the picture.
c.     Find the probability that the return is negative on each of 4 randomly selected days.
Problem 2
(From Decision Making Under Uncertainty by David E. Bell and Arthur Schleifer, Jr., Course Technology Inc.)
Sarah Chang, owner of a small electronics company, is considering submitting a proposal for an electronic timing system for the 2008 Olympic Games. The deadline for submitting a proposal is six months away. For several years, Chang’s company has been developing a new microprocessor to create a new timing system that would be superior to any product currently on the market. However, progress in research and development has been slow, and Chang is unsure about whether her staff can produce the microprocessor in time. Chang estimates the chance that they will succeed in developing the microprocessor in time to be 40%. If they succeed in developing the microprocessor, there is an excellent chance that Chang’s company will win the \$1 million Olympic contract. Chang estimates this chance to be 90%.
Chang can also submit a proposal based on an alternative, inferior timing system that has already been developed. She estimates the chance of winning the contract with the existing system to be 5%. She is allowed to submit at most one proposal.
If she pursues the new microprocessor, Chang must invest \$200,000 in further research and development. In addition, making a proposal requires developing a prototype timing system at an additional cost of \$50,000 (regardless of whether the technology used is new or old). Finally, if Chang wins the contract, the finished product will cost an additional \$150,000 to produce (regardless of the technology).
a.     Draw (by hand) a decision tree to model Chang’s decision. Include as Exhibit A (pasting a picture from a smartphone or camera into MS Word is fine, or you can use any software you want to lay out the structure).
o   Do not put any numbers (probabilities, costs, etc.) on your drawing.
o   The tree should show the structure and logic of the decision process.
o   Distinguishing between decision and chance nodes, and naming the various nodes and branches, is critical.
o   Create a tree that is as general as possible – do not eliminate certain branches from consideration because of the specific values of the inputs as presented in the case.
o   Recall that the key to success is determining what decision/event happens first, and then building out the tree in sequential order.
b.    Implement your decision tree using PrecisionTree. Include the full tree as Exhibit B.
o   Keep all the default settings in the window titled “PrecisionTree – Model Settings” except the tree name. You can name your tree whatever you wish.
o   If you wish, you can create a single-page printout of your tree as follows: Go to the Page Setup window by either clicking on Office button/Print/Print Preview/Page Setup, or alternatively from the Page Layout tab/Page Setup group. In the Page Setup window, select the Page tab. Under the Scaling section, select Fit to 1 page wide by 1 page tall. A “Landscape” orientation may give a better fit than a “Portrait” orientation.
c.     Using the “Decision Analysis” tools of PrecisionTree, create the optimal tree. Include a figure of the optimal tree as Exhibit C. Describe the optimal decision.
d.    Again using the “Decision Analysis” tool, summarize the possible outcomes and their probabilities. What is the probability that the project will return a profit?