MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ MS&E 352 Final Examination (Winter 2009) Distributed: Thursday, March 12th, 2009 at 1:00pm Due: Thursday, March 19th, 2009 at 1:00 pm Please Note: Please keep this sheet as the first page of the work you turn in. No exams will be accepted after the due date and time. Late submission of the final exam will result in a grade of 0/300. 1. This exam is a take-home exam. You may use the reference material listed in the course guide as well as any work that you have done during the quarter. Please do not refer to any work done by students of previous years. This is meant to be an individual effort; you should not discuss this exam with anyone except the teaching team. You may (and are encouraged to) use a calculator or a spreadsheet program where needed. 2. Please print your name, and sign the Honor Code. Your exam will not be graded unless you have printed your name and signed the Honor Code. Please keep this sheet as the first page of the work that you turn in. 3. Read over each question carefully. You may not need all the information given in a particular problem. 4. Unless stated otherwise, the characters in the exam prefer more money to less and follow the Five Rules of Actional Thought. Do not make any additional assumptions. 5. Please include a general table of contents for the work that you turn in. 6. Please write clearly and show your work. You will not receive credit for illegible work or if you do not show your work. Partial credit will be given where appropriate. For problems with numerical answers, please draw a box around the final answer. 7. If you have any questions, please post them on coursework. Important clarifications (if necessary) will be e-mailed to the entire class and posted on the announcements page on coursework. 8. The exam is worth a total of 300 points. The points are broken down by question. In recognition and in the spirit of the Honor Code, I certify that I will neither receive nor give unpermitted aid on this exam and that I will report, to the best of my ability, all Honor Code violations observed. Name (printed): _______________________________________________ Name (signed): _______________________________________________ ___________________________________________________________________________________________ Final Exam 03/12/09 page 1 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Problem 1 - Auctions (50 points) PART I - Billionaire Art Lovers (20 points) You work for Bill G., an eccentric risk-neutral billionaire art lover, who is considering entering a sealed-bid, first-price auction for a classy painting. The value of the painting to Bill G. is independent of the bid amount. After much thoughtful analysis, you have constructed two graphs that provide insight into Bill G.'s bidding situation. Unfortunately, the numbers on the x-axis of one of the graphs have been erased. All you can be sure of is that you have done your analysis correctly. Here are the graphs: (Please note that the two graphs might have different horizontal scales) Probability Bill G. Acquires Painting vs Bid 1.0 0.9 {Bill G. Acquires Painting|Bid,&} 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Bill G.'s Bid Amount Certain Equivalent of Bill G.'s Bid Deal vs Bid Amount 8 Certain Equivalent ($K) 6 4 2 0 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 -4 -6 -8 -10 -12 Bill G.'s Bid Amount ($K) ___________________________________________________________________________________________ Final Exam 03/12/09 page 2 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Please answer the following questions, providing a one or two sentence explanation for each one. (For full credit, numerical answers should be within $1000 of actual values.) a) [5 points] What is Bill G.'s optimal bid? b) [5 points] What is the minimum that Bill G. must bid to be virtually certain that he will acquire the painting? c) [5 points] For reasons known only to him, Warren B. wants to keep Bill G. from participating in this auction. He is willing to pay him not to participate. What is the minimum payment that Bill G. would accept to refrain from participating in the auction? d) [5 points] Warren B. decides not to make the payment discussed in (c), and Bill G. acquires the painting in the auction. Now Warren B. wants to buy the painting from Bill G. What is the minimum payment that Bill G. would accept to sell the painting? PART II - A Quick $100 (30 points) You are about to bid on a lottery ticket. You believe that this lottery is an unusually lucrative opportunity - if you acquire the ticket, you will have a 54% chance of winning $100, and a 46% of winning $0. Unfortunately you are not sure how many other people will bid for the lottery ticket, and you are also unsure how those people will bid. You wish to follow the delta property in the range of the prospects described in this problem, and estimate your risk aversion coefficient to be equal to 0.001. a) [15 points] In this part assume that there are N people bidding against you. Each of them will place a bid distributed according to a Beta distribution with parameters r = 5, n = 10 on the range $0-$100 (i.e. the distribution has a mean of $50). You believe that all N bids are irrelevant to one another given &. For N = 1, N = 2, and N = 3, compute: The optimal bid amount; The optimal Certain Equivalent; The probability of acquiring the deal at the optimal bid amount. Conclude - what do you think will happen as N increases? How would you explain this? ___________________________________________________________________________________________ Final Exam 03/12/09 page 3 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ b) [15 points] In this part assume that there is only one person bidding against you. This person will place a bid distributed according to a Beta distribution with parameters r, n = 2r on the range $0-$100 (i.e. the distribution has a mean of r/n). For r = 1, r = 5, and r = 20, compute: The optimal bid amount; The optimal Certain Equivalent; The probability of acquiring the deal at the optimal bid amount. Conclude - what do you think will happen as r increases? How would you interpret this result, and how would you explain it? ___________________________________________________________________________________________ Final Exam 03/12/09 page 4 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Problem 2 - What Can Go Wrong in a Sensitivity Analysis (50 points) Crazy Widgets, a risk-neutral company, faces what they believe is an interesting investment opportunity. They can either invest or not invest in a new market. The size of this new market is 10 million units, and Crazy Widgets would sell their goods at a unit price of $21. Unfortunately they believe their market share and unit cost to be uncertain; after a long meeting with some of their key executives, you have been able to assess the probability distribution shown on the next page. Crazy Widgets computes profit as market size * market share * (unit price - unit cost); therefore, their profit is not directly influenced by oil price but they still believe that this is a useful and relevant distinction when thinking about their market share and unit cost. Assume that the value of not investing is $0, and compute all relevant quantities in millions of dollars. a) [10 points] Draw a Tornado Diagram for the Invest alternative, using the uncertainties Oil Price, Market Share and Unit Cost. b) [5 points] Comment on this tornado diagram - what are its two major flaws in capturing what goes on in this decision problem? Suggest ways to fix the diagram and remediate these two flaws. c) [10 points] Draw an open loop - closed loop chart for the Oil Price uncertainty. Does this address some of the issues raised in part b)? Why or why not? d) [25 points] Draw an open loop - closed loop chart for the Market Share uncertainty, and another open loop - closed loop chart for the Unit Cost uncertainty. Does this address some of the issues raised in part b)? Why or why not? ___________________________________________________________________________________________ Final Exam 03/12/09 page 5 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Unit Cost of $20 Market Share at 40% 0.2 0.25 Unit Cost of $18 0.8 Unit Cost of $23 0.2 Stable Oil Price Market Share at 20% Unit Cost of $20 0.6 0.5 0.6 Unit Cost of $18 0.2 Unit Cost of $23 Market Share at 10% 0.8 0.25 Unit Cost of $20 0.2 Unit Cost of $23 0.2 Market Share at 40% 0.25 Unit Cost of $20 0.6 Unit Cost of $18 0.2 Unit Cost of $23 0.7 Increase in Oil Price 0.4 Market Share at 20% 0.5 Unit Cost of $20 0.2 Unit Cost of $18 0.1 Unit Cost of $23 Market Share at 10% 0.9 0.25 Unit Cost of $20 ___________________________________________________________________________________________ 0.1 Final Exam 03/12/09 page 6 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Problem 3: (200 points) Case Study - Large Instrument Manufacturer, Inc. 1) Problem Statement: Large Instrument Manufacturer, Inc. (LIM) is the producer of the ROE-1, a sophisticated research instrument. The government of UDC is among those that have purchased several ROE-1's. Two years ago, UDC bought the ROE-1's for $77,000 each and installed them throughout the country. Unfortunately, UDC has had difficulties maintaining the proper operation of the ROE-1's. LIM's management has learnt a painful lesson from the large loss associated with the sale of ROE-1's to UDC. As a result, in reviewing the business plan for the ROE-2 (the follow-up version of the ROE-1), which will be sold to UDC in 2007, LIM is considering several servicing alternatives: The first alternative is to continue with the present servicing arrangement. This means that LIM itself must send trained service personnel to UDC. The next alternative is to train UDC engineers so they can repair the machines on their own. However, since the user cannot repair all breakdowns, LIM would still have to send its service personnel to UDC periodically. LIM can also sign a contract with IPX, an independent service firm. The contract would cover any type of breakdown that might occur. Finally, LIM can choose not to sell any ROE-2's to UDC. LIM's warranty period is limited to five years (covering 2007-2011). Any breakdowns that occur beyond the warranty period must be repaired at the expense of the purchaser. All ROE-2 sales would occur in 2007 but not after that, since LIM is planning to introduce the revolutionary ROE-3 in just a year. Finally, LIM's CEO has instructed you to use a 3% discount rate and to assume risk-neutrality. The sales price of the ROE-2 is $100,000. The cost of manufacturing, marketing, and installing an ROE-2 is $70,000. 2) Uncertainties and Additional Assumptions: LIM's experts have carefully thought about a few uncertain variables: the number of machines LIM can sell to UDC in 2007; the average service cost per machine per year during the five year warranty period which depends on the servicing alternative selected; the training cost per machine sold. All machine failures are of one of two types: alpha or beta. The percentage of machines per year that will experience an alpha or beta failure is uncertain. The average costs for LIM to repair alpha and beta failures are two more uncertainties. ___________________________________________________________________________________________ Final Exam 03/12/09 page 7 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ If the \"train for self-maintenance\" servicing alternative is selected, LIM will only be liable for alpha failures. However, it will have to include a supply of parts and servicing equipment with each unit sold and train the users to fix beta failures. The training would be offered to UDC's engineers in the first year (2007). The parts and servicing equipment per machine sold would be bought by UDC in 2007 as well, and would cost $9,000 per unit sold. If the service contract is signed, LIM will pay IPX $750,000 or 23% of the sales price for every ROE-2 sold in UDC during the next year, whichever is greater. The payment would occur in 2007. Of course, LIM could also decide not to sell ROE-2's to UDC. Since this would adversely impact the sales of LIM's other products, LIM considers the loss associated with this alternative to be $8,000 (expressed in 2007 dollars). LIM's experts have given the following rough assessments of the six primary uncertainties: Description Repair cost uncertainties Avr repair cost, alpha failure Avr repair cost, beta failure Alpha failures per year Beta failures per year Revenue uncertainties Number of machines sold Training costs uncertainties Training cost per machine Low Base High 9.0 7.1 0.12 0.64 10.0 8.0 0.15 0.70 12.0 9.4 0.19 0.75 units 25 30 50 $ thousand 4.0 8.0 12.0 $ thousand $ thousand fract of machines fract of machines The end of this handout shows the detailed assessments of these uncertainties which LIM's experts would give you if asked. 3) Your Task: You are to prepare a report aimed at a reader who is well-versed in Decision Analysis, showing your work through one iteration of the Decision Analysis cycle for LIM's decision problem. Your report should comprise the following: A one-page executive summary with your recommendations at this point of the analysis; A table of contents / agenda in which you will identify key parts of your analysis; Presentation slides showing the insights you gathered from the use of the Decision Analysis tools. Please place each concept or tool on a separate slide, and structure your report in the order shown on the Decision Analysis Cycle (Formulation - ___________________________________________________________________________________________ Final Exam 03/12/09 page 8 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Evaluation - Appraisal). Please do not use more than 25 slides; if your report exceeds this limit you will be penalized for doing so. Last, here are some important remarks that we hope will guide you in your analysis: Rely on Tornado Diagrams - Six uncertainties is already a respectable amount of complexity, and we do not want you to draw a tree with 729 branches. Instead, use tornado diagrams to identify, for each alternative, the uncertainties which cover 95% of the variance or more, and base your probabilistic analysis on this choice of uncertainties. You may pick different uncertainties for different alternatives. Once you have identified the critical uncertainties, you are allowed to reassess their low - base - high values using the distributions we are providing. Please discretize those distributions using the equal areas method or the equal areas shortcut method, and show on one of your slides the end results of those discretizations. Insist on your Insights - \"Build a model to translate the problem statement into math. Then do the analysis. Then translate the numbers back into English and burn the math...\" Each time you present a Decision Analysis tool, insist on the insights you are able to derive from it - never show analysis just for the sake of analysis. Appraisal is Essential During the Appraisal phase you should state on which part of the analysis you would focus next. How would you modify your model? What information would you recommend LIM gather? Which assumptions would you question? What new ideas and alternatives would you suggest and explore? Etc. 6) Additional Remarks & Contact Information: The objective of this assignment is obviously NOT to test your skills at building financial value models. Therefore, we will provide you with a few numbers to allow you to cross-check your own results - as soon as you are finished building your deterministic value model, please go to the class website, download the available value model template and carefully check your numbers. Please also note that this template consists of not just one worksheet, but two. Finally, be reminded that you should not assume that any of the distributions provided as expert assessments are normal. If you have any questions regarding this part of the final exam, please post on the Coursework Forum titled \"Final Exam.\" We hope you find this case a good learning experience. Good luck and clear insights! The DA2 Teaching Team ___________________________________________________________________________________________ Final Exam 03/12/09 page 9 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Appendix - Cumulative Distributions on the Uncertainties: Avr repair cost, Alpha failure 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% $8.0 $9.0 $10.0 $11.0 $12.0 $13.0 $14.0 $ thousand Avr repair cost, Beta failure 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% $6.00 $6.50 $7.00 $7.50 $8.00 $8.50 $9.00 $9.50 $10.00 $10.50 $11.00 $ thousand ___________________________________________________________________________________________ Final Exam 03/12/09 page 10 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Alpha failures per year 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 80.0% 85.0% 90.0% Fraction of machines Beta failures per year 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 55.0% 60.0% 65.0% 70.0% 75.0% Fraction of machines ___________________________________________________________________________________________ Final Exam 03/12/09 page 11 of 12 MS&E 352 Decision Analysis II 03/12/2009 ____________________________________________________________________________________________ Number of machines sold 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 20 25 30 35 40 45 50 55 60 12 14 16 Units Training cost per machine 100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 0 2 4 6 8 10 $ thousand ___________________________________________________________________________________________ Final Exam 03/12/09 page 12 of 12