HO25-PracticeFinal1

MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002

Final Examination 2001-2002: Part I

Please read the following instructions carefully!

1. This exam is closed book and closed notes. You may use one double-sided 8½’’x11’’ sheet of paper with anything you wish. Three sheets of scratch paper are provided at the back of this exam. You may also use a calculator and a foreign language dictionary if you wish. Please sit in alternate seats.

2. The answer sheet is stapled to the back of the exam. Please tear it off, print your name and SUID number, and sign the Honor Code and exam agreement. Your exam will not be graded unless you have printed your name and signed this statement.

3. Part I contains 60 points. All exam questions in this part carry the same weight. Do not spend too much time on any one question. Read each question carefully. You may not need all the information given in a particular problem.

4. At the end of this portion of exam, please make sure your probability assignments are written clearly in the appropriate boxes on the answer sheet. This portion of the exam will be graded solely on the basis of your probability assignments. Turn in the single answer sheet.

5. This part of the exam will end at 8:00pm. You will have five minutes to turn in your answer sheet to your TA and pick up Part II. In fairness, we will not accept Part I after 8:05 pm.

6. You will receive your answer sheet back when you turn in Part II. Please submit Part I answers on the web by 11:59am (just before NOON) on Friday, December 14, 2001.

7. In the spirit of the Honor Code, the Teaching Team will remain outside the classroom during the exam. If you have questions please ask us. Clarifications will be announced to the class.

8. 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.

9. If your probabilities do not add up to one the teaching team will normalize them. If you leave boxes blank and your probabilities add up to less than one we will distribute the remaining probability equally among the empty answer boxes. If your probabilities add up to more than one and you have left boxes blank we will normalize the non-blank boxes and place zeros in the empty boxes. For example,

Question Number a b c d i 0.3 0.5 0.3 0.3 j 0.7 k 0.9 0.2

Would become:

Question Number a b c d i 0.214 0.357 0.214 0.214 j 0.1 0.1 0.7 0.1 k 0.82 0.18 0 0

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 10. You be will scored according to the following scoring function:

Score = )4lnpln

1(*1060

+ , where p is the probability you assign to the correct answer.

11. The graph and table below show the number of points you will receive per question

depending on the probability you assign to the correct answer.

(20.00)

(18.00)

(16.00)

(14.00)

(12.00)

(10.00)

(8.00)

(6.00)

(4.00)

(2.00)

-

2.00

4.00

6.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Probability

Score

Prob. Score0.00 -∞0.01 (13.93)0.02 (10.93)0.03 (9.18)0.04 (7.93)0.05 (6.97)0.06 (6.18)0.07 (5.51)0.08 (4.93)0.09 (4.42)0.10 (3.97)0.15 (2.21)0.20 (0.97)0.25 -0.30 0.790.35 1.460.40 2.030.45 2.540.50 3.000.55 3.410.60 3.790.65 4.140.70 4.460.75 4.750.80 5.030.85 5.300.90 5.540.95 5.781.00 6.00

12. Relax and make good decisions!

DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO!

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 1. The following conversation is taken from the book Alice in Wonderland by Lewis

Carroll. Alice is walking on a path in the woods. She comes to a crossroads that divides the path into a north path and south path, and meets the Cheshire Cat.

Alice: Which way shall I go? Cat: It depends a great deal on where you want to go. Alice: I do not really care where I go. Cat: Then it does not matter where you go.

Alice: As long as I go somewhere from here.

Cat: You are sure to do that as long as you keep going.

How many of the Five Rules of Actional Thought has Alice violated in this conversation? a) 0 b) 1 c) 2 d) 3

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 2. Sebastian loves pizza. You have already assessed his preference probabilities for the

following pizza toppings: Pepperoni 1.00 Mushrooms 0.77 Olives 0.46 Zucchini 0.32 Squid 0.04 Anchovies 0.00

You now are interested in his preference probability for sausage pizza. Sebastian says that sausage is his favorite topping of all. You assess the following preference probability p = 0.88:

Anchovies

Sausagep

1−p

Pepperoni ~

Anchovies

Sausagep

1−p

Pepperoni ~

Sausagep

1−p

Pepperoni ~

Which of the following is closest to the preference probability of olive pizza, now that you have included sausage pizza in the analysis?

a) 0.32 b) 0.40 c) 0.46 d) 0.52

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 3. Nina is risk-neutral and owns the following deal for which her certain equivalent is $34:

She has the chance to pick one ball out of a bucket containing blue balls and red balls. If she picks a red one, she will win $100. If she picks a blue one, she will win nothing.

What is the value of clairvoyance on the color of the ball Nina will pick?

a) $0 b) $4 c) $60 d) Cannot be determined from the information given

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 4. Bruce’s u-curve is given by: u(x)=sqrt(x), and he is offered the following game:

Outcome 1

p= 0.5

Outcome 2

p= 0.5

$16

$36

If Bruce’s current wealth is $100, what is the most he should pay to play this game?

a) $22.25 b) $25.75 c) $25.66 d) Cannot be determined from the information given

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 5. Ralph has an unknown risk attitude and faces the following decision situation:

Take Sales Job

Graduate

0.9

Don’t Graduate

0.1

Bad Job

0.3

Great Job

0.7

Night School

Keep Current Job

Succeed

0.6

Fail

0.4

$90,000 − X

$30,000 − X

$20,000 − X

$80,000

$30,000

$50,000

Take Sales Job

Graduate

0.9

Don’t Graduate

0.1

The cost of night school, X, is not yet known. Which of the following MUST be true? I Ralph’s optimal decision depends on his risk tolerance

II “Night School” is optimal if Ralph is a delta-person and the cost of night school, X < $4,800

III Ralph’s decision is insensitive to the probability of graduating

a) I only b) II only c) I and II d) I and III

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 6. Doris has consulted you for investment advice. Together you draw this decision diagram:

Invest

BondPrice

$

StockPrice

Invest

Bond Price

Return$

Stock Price

Which of the following MUST be true? I Bond Price is relevant to Stock Price, given & II The Value Node is a deterministic function of all the other nodes III The return Doris will get is uncertain IV At least one arrow indicates “influence” a) I and II only b) II and III only c) III and IV only d) I, II, and III

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 7. Ling has a disease that can be cured with a very painful treatment. His doctors offer him

two alternatives, as described by the diagram below.

Cure

p = 1/2,500

1− p

Cure

Death

− $30,000

Cure

Pain + Cure

p = 1/2,500

1− p

Cure

Death

− $30,000

Ling is has no preference among the three alternatives. According to the readings assigned for this class, What is Ling’s micromort value?

a) $0.03 b) $4 c) $75 d) $400

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 8. Jason is deciding if he should spend the weekend skiing in Lake Tahoe, 180 miles from

Palo Alto. He has two ways to get there, by car or by plane. Because of the travel time he will only be able to ski one day of the weekend if he drives, but can ski both days if he flies. A round-trip plane ticket costs $180, and he estimates the costs associated with driving to be $55 (including gas, tolls, and car depreciation) each way.

Jason is aware of the risks involved with these alternatives. He knows that for this

particular one-way flight, the chance of dying is 3 micromorts. He also knows that in the US, the chance of dying in the road is to 7 micromorts per 1,000,000,000 passenger miles. He also knows that the chance of dying during a single day of skiing at Lake Tahoe is 4 micromorts.

Jason enjoys skiing, and so believes that the total benefit (net of rentals, lift ticket and other costs) he would get from this trip is $550 if he skis both days, and $450 if he skis one day.

Assuming that Jason is risk neutral and has a micromort value of $48, what should he do?

a) Drive to Lake Tahoe and ski one day b) Fly to Lake Tahoe and ski one day c) Fly to Lake Tahoe and ski both days d) Do not go to Lake Tahoe at all

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 9. Which of the diagrams below accurately represents this decision tree?

Uncertainty W Uncertainty X Uncertainty Y Uncertainty Z

W1

X1

W2

X2

Y1

Y2

Z1

Z2

X1

X2

Y1

Y2

Y1

Y2

Y1

Y2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

0.2

0.6

0.4

0.6

0.25

0.4

0.8

0.75

0.2

0.8

0.8

0.2

0.8

0.2

0.1

0.9

0.5

0.5

0.2

0.8

0.1

0.9

0.5

0.5

0.5

0.5

0.5

0.5

0.2

0.8

Uncertainty W Uncertainty X Uncertainty Y Uncertainty Z

W1

X1

W2

X2

Y1

Y2

Z1

Z2

X1

X2

Y1

Y2

Y1

Y2

Y1

Y2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

Z1

Z2

0.2

0.6

0.4

0.6

0.25

0.4

0.8

0.75

0.2

0.8

0.8

0.2

0.8

0.2

0.1

0.9

0.5

0.5

0.2

0.8

0.1

0.9

0.5

0.5

0.5

0.5

0.5

0.5

0.2

0.8

a) b) c) d)

W

ZX

YW

ZX

Y W

ZX

YW

ZX

YW

ZX

YW

ZX

Y W

ZX

YW

ZX

Y

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 10. Pedro, who is risk-neutral, faces a decision situation with three alternatives: X, Y and Z.

After careful analysis, you have produced the following graph which plots Pedro’s certain equivalent (CE) against the probability p of a particular event E occurring or not.

0.0 0.2 0.4 0.6 0.8 1.0

$80

$40

CE

$0 0.44 0.65

Y

X

Z

$100 $90

$50

Z

X Y

0.0 0.2 0.4 0.6 0.8 1.0

$80

$40

CE

$0 0.44 0.65

Y

X

Z

$100 $90

$50

Probability of event E

Z

X Y

Pedro believes the probability of event E is 60%. A $9 test is available to predict the outcome of the event. The test, which is symmetric, has 60% accuracy. What should Pedro do? a) Pedro should not purchase the test because it is observable but not relevant. b) Pedro should not purchase the test because it is observable and relevant but not

material. c) Pedro should not purchase the test because it is observable, relevant, and material

but not economic. d) Pedro should purchase the test.

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MS&E 252 FINAL EXAM Part I: ANSWER SHEET

LAST Name (print): __________________________________________

FIRST Name (print): __________________________________________

SUID: Section (circle one): Ali John Andy Mathias Armando In full recognition of 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. I also understand that obtaining a score of negative infinity on this exam will result in failing the course. I hereby attest that if I assign a zero probability to any answer, it is done consciously and against the advice of the teaching team. Signature: ________________________________________________

Question Number a b c d

1

2

3

4

5

6

7

8

9

10

PLEASE submit your answers, as stated above, on the course webpage (www.stanford.edu/class/msande252) by11:59 am Friday, December 14, 2001.

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http://www.stanford.edu/class/msande252


Final Examination 2001-2002 Part II: Quantitative Problems

MS&E 252 FINAL EXAM Part II: ANSWER SHEET

LAST Name (print): __________________________________________

FIRST Name (print): __________________________________________

SUID: Section (circle one): Ali John Andy Mathias Armando In full recognition of 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. Signed: ___________________________________________________________ 1) This exam is closed book and closed notes. You may use one double-sided 8½’’x11’’

sheet of paper with anything you wish. 2) Several sheets of scratch paper are provided at the back of this exam. You may use a

calculator and a foreign language dictionary if you wish. Please sit in alternate seats. 3) Use the blank spaces after every problem to write your answers. 4) Show all your calculations to earn partial credit. Wrong answers without calculations

cannot earn partial credit. 5) This section of the exam is worth a total of 120 points. Point values for each question are

indicated on each page. Budget your time so that you do not spend too much time on any one question.

6) Please turn in this entire section at the end of the examination. 7) 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. 8) You may begin work on this part of the exam as soon as you receive it.

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 1) Rain Bets and the Party Problem: (15 points) You learned in DA1 that you could construct a deal if people differ in probability assignments about the same uncertainty. Kim believes there is a 60% chance of rain on the day of the Party. However, Jane believes there is a 75% chance of rain. a) Construct a rain-bet deal for Kim and Jane such that the sum of absolute pay-offs is $100 and

they both find the deals equally attractive. (Assume Risk neutrality). (10 points) b) What is the maximum amount of money you can make while acting as a broker between Kim

and Jane? (5 points)

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2) Tornado Diagrams: (15 points) You have just joined a management consulting firm. You are sent out to help the Free-Mark Abbey winery with a decision situation. This time they are risk averse, follow the delta property and have a risk tolerance of $333,000. They are interested in Profit = (Amount Sold) * (Selling price / unit – Cost / unit) They give you 3 fractiles (Hi-Base-Low) for several uncertainties: Uncertainty Low Base High Amount Sold 10,000 100,000 2,000,000 Acidity Level 15 % 20% 25% Next Year’s Growth 7% 9% 11% Selling Price/ unit 10 15 20 Cost/ Unit 5 10 12 1) Sketch the tornado diagram for NPV on profit. (10 points).

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2) In your analysis, you decide to keep only the uncertainties that capture up to 90% of the total

variation. (Max swing squared). Which uncertainties, if any, would you remove from your

analysis? (5 points)

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3) Buying Stocks: (45 points) Bill is an investor who has $2,000 to invest in stocks. There are two stocks, “Imperium” and “Technomania”. Each costs $1,000/share. Bill believes that: Technomania will either go to $2,000/share or $1,000/share with equal odds Imperium will either go to $3,000 / share or $250/share with equal odds. The movements of the two stocks are irrelevant to each other. Bill is risk neutral. Assume that he must invest all of his funds and can only buy whole shares. So he has $2,000 and must divide between two stocks leaving him with 3 alternatives (2-Imperium and 0-Technomania, 1-Imperium and 1-Technomania, 0-Imperium and 2-Technomania).

A) What is the best investment strategy for Bill? (5 points)

B) What is Bill’s Certain Equivalent for the best strategy? (5 points)

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C) Assume the probability that Technomania will go up is p. Bill has not yet made his investment decision. Plot Bill’s Value of Clairvoyance as a function of the probability that Technomania will go up. (10 points)

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D) Ted is a delta person with risk tolerance of $10,000. He has the same beliefs as Bill on the movement of the stocks. What is his Certain Equivalent for his best investment strategy? (10 points)

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 Now both Bill and Ted believe that the two investments are relevant to each other and both stocks will either go up or down. They believe, however that the marginals of the two stocks are the same.

E) In this case, will the CE for Bill move up, down, stay the same, or is it uncertain (relative to the case when the investments were irrelevant)? (5 points)

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F) In this case, will the CE for Ted move up, down, stay the same, or is it uncertain (relative to the case when the investments were irrelevant)? (5 points)

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G) Bill has a stock analyst friend who can give him imperfect information about the movement of “Technomania” stock for a certain fee. Draw the decision diagram for Bill’s situation when he believes there is relevance in the movement of the two stocks.

(5 points )

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4) FLLOYD INSURANCE (45 points)

On the island of Diatia, all people follow the delta property, and are risk-averse and with a risk tolerance ρ=500. Note that if you use the u-function u (x)=1-exp (-x/ρ), the CE of a deal with u-value, U, is given by: CE(U) = -ln (1-U)*ρ. There are two types of drivers in the island: good and bad drivers. Good drivers have a probability of accident of 10% per year, whereas bad drivers have a probability of 20%. For simplicity, assume that each driver can have one accident at most each year. When an accident occurs, for good or bad drivers, it results in damages that cost $200 to the driver. Flloyd unlimited is a risk-neutral insurance business, and wants to sell insurance contracts to drivers of the Diatia Island. An insurance contract has the following form:

• The buyer pays the insurer a premium in the beginning of the year,

• If an accident occurs during the year, the insurer will pay the buyer a coverage amount.

For the buyer, the decision of buying an insurance contract results in the following tree:

{Accident|&} : ACCIDENT - damageDON'T BUY

1- {Accident|&} : NO ACCIDENT 0

BUY INSURANCE?{Accident|&} : ACCIDENT - damage - premium + coverage

BUY1- {Accident|&} : NO ACCIDENT - premium

For the insurance company, the decision of selling a contract results in the following tree:

DO NOT SELL 0

SELL? {Accident|&} : ACCIDENT +premium-coverageSELL CONTRACT

1- {Accident|&} : NO ACCIDENT +premium

You can assume that drivers know if they are good or bad drivers, but that Flloyd cannot know this information. You can assume that, if it is in a person’s interest to say he/she is a good/bad driver, then he/she will say that.

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(1) Total coverage for good drivers. Flloyd Inc. wants to design a contract for good drivers. The contract offers total coverage of the damage; that is, the driver receives $200 in case of accident. What is the maximum premium Flloyd could price the contract, so that it remains attractive for good drivers? Would selling this contract (with maximum premium) to good drivers be interesting for Flloyd? (9 points)

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(2) Flloyd decides to price the “Total coverage for good drivers” contract at a premium of $23.50. How would bad drivers like this contract? If a bad driver lied and bought such a contract from Flloyd pretending he is a good driver, what would be the result for Flloyd? (9 points)

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 (3) To avoid this problem, Flloyd thinks of designing a total coverage contract for bad drivers. What would be the minimum premium Flloyd should charge to be satisfied? How would good drivers like this contract? How would bad drivers like it? (9 points)

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 (4) Finally, Flloyd decides to launch two contracts: “Total coverage” and “Minimal coverage”. The characteristics of both contracts are the following:

Total coverage

Minimal coverage

Coverage $200 $25 Premium $43 $3

If you were to compute the CE of these contracts for good and bad drivers, you would get the following results:

Minimal coverage

Total coverage

CE for good driver -$23.5 -$43.0 CE for bad driver -$43.2 -$43.0

Which contract would a good driver prefer? Which contract should a bad driver prefer? What would be the result for Flloyd? (9 points)

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 (5) Flloyd now hesitates between two policies.

• (a) They could decide to sell the combination “Total Coverage” and “Minimal Coverage”, as described in question (4). In this case, there is no risk, because every contract they sell has a positive CE.

• (b) They could sell only “Total coverage for good drivers”, as described in question (2),

and take the risk to loose money if bad drivers buy the contract. Depending on the proportion p of good drivers in the population, what should be Flloyd’s policy? (Assume that a driver who comes to buy insurance from Flloyd is a good driver with probability p, and a bad driver with probability (1-p). (9 points)

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MS&E 252 HO31 Final Examination: Part I Decision Analysis I December 3rd, 2002 (This page is left intentionally blank).

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HO25-PracticeFinal1

Documents

foreign language dictionary

earn partial credit

ho31 final examination

left intentionally blank

sided 8x11 sheet

give unpermitted aid

final examination 2001

answer sheet