Final Exam Winter 12

8/2/2019 Final Exam Winter 12

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MS&E 352 Decision Analysis II____________________________________________________________________________________________

___________________________________________________________________________________________page 1 of 13 Final Exam

MS&E 352 Final Examination (Winter 2012)

Distributed: Thursday, March 15th

, 2012 at 5:00pm

Due: Wednesday, March 21st, 2012 at 5:00pm

Please Note:

Please copy the Honor Code statement below into the document you submit as a

record of your digital signature.

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. We encourage you to usea calculator or a spreadsheet program where needed.

2. Read over each question carefully. You may not need all the information given in a particular

problem.

3. On the title page, please write out your full name under the Honor Code. Your exam will not

be graded unless you have typed your full name and the Honor Code on the title page. Please

submit your final exam solutions via Coursework. Note: If you plan to submit two

separate files (e.g. one .doc and one .ppt) then please make sure that you paste the

signed honor code on the first page of both documents.

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. If you have any questions, please post them to the Coursework Discussion page. Only if you

feel that your question will reveal a substantive part of the solution should you email a TA.

7. Clearly 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, or otherwise highlight the final answer.

8. The exam is worth a total of300 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 ofmy ability, all Honor Code violations observed.

Name (printed): _______________________________________________


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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 asealed-bid, first-price auction for a classy painting. The value of the painting to Bill G. isindependent of the bid amount. After much thoughtful analysis, you have constructed two graphsthat provide insight into Bill G.s bidding situation. Unfortunately, the numbers on the x-axis ofone of the graphs have been erased. All you can be sure of is that you have done your analysiscorrectly. Here are the graphs (please note that the two graphs might have different horizontalscales):

-12

-10

-8

-6

-4

-2

0

2

4

6

8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Cer

tainEq

uiva

len

t($K)

Bill G.'s Bid Amount ($K)

Certain Equivalent of Bill G.'s Bid Deal vs Bid Amount

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

{BillG.

AcquiresPainting|Bid,&

}

Bill G.'s Bid Amount

Probability Bill G. Acquires Painting vs Bid


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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 willacquire the painting?

c) [5 points] For reasons known only to him, Warren B. wants to keep Bill G. from participatingin this auction. He is willing to pay him not to participate. What is the minimum paymentthat 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. acquiresthe painting in the auction. Now Warren B. wants to buy the painting from Bill G. What isthe minimum payment that Bill G. would accept to sell the painting?

PART II A Quick $100 (30 points)

You are about to bid for a certificate which entitles its owner to an uncertain deal. You believethat this uncertain deal is a lucrative opportunity if you acquire the certificate, you will have a80% chance of winning $100, and a 20% 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, andestimate your risk aversion coefficient to be equal to 0.001.

a) [15 points] In this part, assume that there are Npeople bidding against you in this first-price-sealed auction. Each of them will place a bid distributed according to a Beta distribution withparameters r= 6, n = 12 on the range $0-$100 (i.e. the distribution has a mean of $50). Youbelieve that allNbids are irrelevant to one another given &. ForN= 1, N= 2, and N= 3,compute:

The optimal bid amount; Certain Equivalent given the optimal bid;

The probability of acquiring the certificate at the optimal bid amount.

Conclude what do you think will happen asNincreases? How would you explain this?


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b) [15 points] In this part, assume that there is only one person bidding against you. This personwill place a bid distributed according to a Beta distribution with parameters r, n = 2ron therange $0-$100 (i.e. the distribution has a mean ofr/n). Forr= 1, r= 6, and r= 20, compute:

The optimal bid amount;

Certain Equivalent given the optimal bid;

The probability of acquiring the certificate at the optimal bid amount.

Conclude what do you think will happen as r increases? How would you interpret thisresult, and how would you explain it?


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Problem 2 What Can Go Wrong in a Sensitivity Analysis

(50 points)

Crazy Widgets, a risk-neutral company, faces what they believe to be an interesting investment

opportunity.

They can either invest or not invest in a new market. The size of this new market is 10 millionunits, and Crazy Widgets would sell their goods at a unit price of $21. Unfortunately they believetheir market share and unit cost to be uncertain; after a long meeting with some of their keyexecutives, 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 andrelevant 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 ofdollars.

a) [10 points] Draw a Tornado Diagram for the Invest alternative, using the uncertainties OilPrice,Market Share and Unit Cost.

b) [5 points] Comment on this tornado diagram what are its two major flaws in capturingwhat goes on in this decision problem? Suggest ways to fix the diagram and remediate thesetwo flaws (you do not need to act upon your recommendations; just explain what you woulddo to resolve these issues).

c) [10 points]

Draw an open loop closed loop chart for the Oil Price uncertainty. Does thisaddress 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, andanother open loop closed loop chart for the Unit Costuncertainty. Does this address someof the issues raised in part b)? Why or why not?


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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

M arket Share at 20% Unit Cost of $200.6

0.5 0.6

Unit Cost of $18

0.2

Unit Cost of $23


0.25

Unit Cost of $20

0.2

Unit Cost of $23

0.2

M arket Share at 40% Unit Cost of $20

0.25 0.6

Unit Cost of $18

0.2

Unit Cost of $23

0.7

Increase in Oil Price

0.4 M arket Share at 20% Unit Cost of $20

0.5 0.2

Unit Cost of $18

0.1

Unit Cost of $23


0.25

Unit Cost of $20

0.1


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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 sophisticatedresearch instrument. The government of UDC is among those that have purchased several ROE-1s. Two years ago, UDC bought the ROE-1s for $77,000 each and installed them throughoutthe country. Unfortunately, UDC has had difficulties maintaining the proper operation of theROE-1s.

LIMs management has learnt a painful lesson from the large loss associated with the sale ofROE-1s to UDC. As a result, in reviewing the business plan for the ROE-2 (the follow-upversion of the ROE-1), which will be sold to UDC in 2011, LIM is considering several servicingalternatives:

The first alternative is to continue with the present servicing arrangement. This meansthat LIM itself must send trained service personnel to UDC.

The next alternative is to train UDC engineers so they can repair the machines ontheir own. However, since the user cannot repair all breakdowns, LIM would stillhave to send its service personnel to UDC periodically.

LIM can also sign a contract with IPX, an independent service firm. The contractwould cover any type of breakdown that might occur.

Finally, LIM can choose not to sell any ROE-2s to UDC.

LIMs warranty period is limited to five years (covering 2011-2015). Any breakdowns that occurbeyond the warranty period must be repaired at the expense of the purchaser. All ROE-2 sales

would occur in 2011 but not after that, since LIM is planning to introduce the revolutionaryROE-3 in just a year.

Finally, LIMs CEO has instructed you to use a 2.75% discount rate and to assume risk-neutrality (at least in the first pass of your analysis). 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:

LIMs experts have carefully thought about a few uncertain variables: the number of machinesLIM can sell to UDC in 2011; the average service cost per machine per year during the five yearwarranty period which depends on the servicing alternative selected; the training cost permachine sold.

All machine failures are of one of two types: alpha or beta. The percentage of machines per yearthat will experience an alpha or beta failure is uncertain. The average costs for LIM to repairalpha and beta failures are two more uncertainties.If the train for self-maintenance servicing alternative is selected, LIM will only be liable foralpha failures. However, it will have to include a supply of parts and servicing equipment with


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each unit sold and train the users to fix beta failures. The training would be offered to UDCsengineers in the first year (2011). The parts and servicing equipment per machine sold would bebought by UDC in 2011 as well, and would cost $9,000 per unit sold (in 2011 only and paid byLIM).

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. This one-time payment wouldoccur in 2011.

Of course, LIM could also decide not to sell ROE-2s to UDC. Since this would adverselyimpact the sales of LIMs other products, LIM considers the loss associated with this alternativeto be $8,000 (expressed in 2011 dollars).

LIM's experts have given the following rough assessments of the six primary uncertainties:

The end of this handout shows the detailed assessments of these uncertainties which LIMs

experts would give you if asked.

We will not take tax into account for this case study.

Description Low Base High

Repair cost uncertainties

Avr repair cost, alpha failure $ thousand 9.0 10.0 12.0Avr repair cost, beta failure $ thousand 7.1 8.0 9.4Alpha failures per year fract of machines 0.12 0.15 0.19

Beta failures per year fract of machines 0.64 0.70 0.75

Revenue uncertaintiesNumber of machines sold units 25 30 50

Training costs uncertaintiesTraining cost per machine $ thousand 4.0 8.0 12.0


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3) Your Task:

You are to prepare a Powerpoint presentation aimed at a reader who is well-versed in

Decision Analysis, showing your work through one iteration of the Decision Analysis cycle

for LIMs decision problem.

Your report should comprise the following: 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 DecisionAnalysis tools. Please place each concept or tool on a separate slide, and structure yourreport in the order shown on the Decision Analysis Cycle (Formulation Evaluation Appraisal). Please do not use more than 20 slides; if your report exceeds this limityou will be penalized for doing so (the 20 slides limit does not include the title page orthe table of content). You may include a few more slides beyond the 20-slide limit as anappendix, especially slides showing your trees and calculations in a more detailedmanner, but please be aware that they may only influence your grade if we wish toaward partial credit for an analysis in which you made a mistake. Please make sure

that all the necessary inputs of your model (e.g. the discretized distribution of keyuncertainties) are neatly organized in your appendix so that we can easily cross-check your numbers. Please try to include some slides in the appendix to back up yourresults as well (e.g. probabilistic dominance analysis).

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 youto draw a tree with 729 branches. Instead, use tornado diagrams to identify, for eachalternative, the uncertainties which cover 95% of the variance or more.

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 thosedistributions using the equal areas method or the equal areas shortcut method, and showon one of your slides (within the appendix) the end results of those discretizations. Youcan then use those critical uncertainties to build a probabilistic tree which will help youplot the CDF of each alternative. This CDFs plot should have a step shape since itsbased on the discretized distributions. There is no need to smooth out this CDFs plot byinterpolation.

Please note that for each alternative, some of the critical uncertainties might not need toappear in your tree at all, because they do not affect that particular alternative.

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 aDecision Analysis tool, insist on the insights you are able to derive from it never showanalysis just for the sake of analysis.


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Appraisal is Essential -

During the Appraisal phase you should state on which part of the analysis you wouldfocus next. How would you modify your model? What information would yourecommend LIM gather? Which assumptions would you question? What new ideas andalternatives 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 valuemodels. Therefore, we will provide you with a few numbers to allow you to cross-check yourown results as soon as you are finished building your deterministic value model, please go tothe class website, download the available value model template and carefully check yournumbers. 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 expertassessments are normal.

If you have any questions regarding this part of the final exam, please post it on the

coursework forum.

We hope you find this case a good learning experience.Good luck and clear insights!

The DA2 Teaching Team


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Appendix - Cumulative Distributions on the Uncertainties:

Avr repair cost, Alpha failure

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

$8.0 $9.0 $10.0 $11.0 $12.0 $13.0 $14.0

$ thousand

Avr repair cost, Beta failure

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

$6.00 $6.50 $7.00 $7.50 $8.00 $8.50 $9.00 $9.50 $10.00 $10.50 $11.00

$ thousand


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Alpha failures per year

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0%

Fraction of machines

Beta failures per year

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

55.0% 60.0% 65.0% 70.0% 75.0% 80.0% 85.0% 90.0%

Fraction of machines


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Note: Please assume that fraction of machine can be sold

Number of machines sold

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

20 25 30 35 40 45 50 55 60

Units

Training cost per machine

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 2 4 6 8 10 12 14 16

$ thousand

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Final Exam Winter 12