1 a stock market investor has $500 to spend and is considering purchasing an option contract on 1000...

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A stock market investor has $500 to spend and is considering purchasing an option contract on 1000 shares of Apricot Computer. The shares themselves are currently selling for $28.50 per share. Apricot is involved in a lawsuit, the outcome of which will be known within a month. If the outcome is in Apricot’s favor, analysts expect Apricot’s stock price to increase by $5 per share. If the outcome is unfavorable, then the price is expected to drop by $2.75 per share. The option costs $500, and owning the option would allow the investor to purchase 1000 shares of Apricot stock for $30 per share. Thus, the investor buys the option and Apricot prevails in the lawsuit, the investor would make an immediate profit. Asides from purchasing the option, the investor could (1) do nothing and earn about 8% on his money, or (2) purchase $500 worth of Apricot shares.

Problem 4.15

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a. Construct cumulative risk profiles for the three alternatives, assuming Apricot has a 25% chance of winning the lawsuit. Can you draw any conclusions?

To construct risk profiles (cumulative or not cumulative), we have to first draw the decision tree

Purchase Option

Favorable Unfavorable

$3,000

-$500

-$500

$40 Do Nothing

Buy Stock


$86.24

1000*($33.50-$30.00)=$3,500 $0

$500*8%=$40

-17*$28.5= -$484.5

17*$33.50 + $15.5*8%=$570.74 17*$25.75 + $15.5*8%=$438.99

-$45.51

(0.25)

(0.75)

(0.25)

(0.75)

Lawsuit Outcome

Lawsuit Outcome

Assumptions: 1) 8% is the monthly interest rate; 2) the investor can only purchase an integer number of shares and put the remaining money to savings

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Decision Strategies:

1) Purchase option

2) Do nothing

3) Buy stock


$3,000

-$500

(0.25)

(0.75)

Purchase Option Payoffs-$500$3,000

Probabilities0.750.25

$40 Do Nothing Payoffs

$40Probabilities

1

Buy Stock


$86.24

-$45.51

(0.25)

(0.75)

Payoffs-$45.51$86.24

Probabilities0.750.75

4

40-45.51

86.24

Return on Investment

5

40-45.51

86.24

Therefore, no immediate conclusions can be drawn since no one alternative dominates another

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b. If the investor believes that Apricot stands a 25% chance of winning the lawsuit, should he purchase the option? What if he believes the chance is only 10%? How large does the probability have to be for the option to be worthwhile?

Assuming 8% is the monthly interest rate and let p be the probability that Apricot will win the lawsuit

1) The expected monetary value associated with purchasing the option is:EMV(Purchase Option) = 3,000p – 500(1 – p) = 3,500p – 500

2) The expected monetary value associated with doing nothing is:EMV(Do Nothing) = 40

3) The expected monetary value associated with purchasing the stock is:EMV(Buy Stock) = 86.24p – 45.51(1 – p) = 131.75p – 45.51.

When p=0.25, EMV(Purchase Option) = $375, EMV(Do Nothing)=$40, EMV(Purchase Stock) = $-12.57

When p=0.1, EMV(Purchase Option) = -$150, EMV(Do Nothing)=$40, EMV(Purchase Stock) = $-32.33

EMV(Purchase Option) > EMV(Do Nothing) 3500p-500>40 p>0.154

EMV(Purchase Option) > EMV(Buy Stock) 3500p-500>131.75p-45.51 p>0.135 p>0.154

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Job Offers

Robin Pinelli is considering three jobs. In trying to decide which to accept, Robin has concluded that three objectives are important to this decision. First, of course, is to maximize disposable income – the amount left after paying for housing, utilities, taxes, and other necessities. Second, Robin likes cold weather and enjoys winter sports. The third objective relates to the quality of the community. Being single, Robin would like to live in a city with a lot of activities and a large population of single professionals.

Developing attributes for these three objectives turn out to be relatively straightforward. Disposable income can be measured directly by calculating monthly take-home pay minus average monthly rent (being careful to include utilities) for appropriate apartment. The second attribute is annual snowfall. For the third attribute, Robin has located a magazine survey of large cities that scores those cities as places for single professionals to live. Although the survey is not perfect from Robin’s point of view, it does capture the main elements of her concern about the quality of the singles community and available activities. Also, all three of the cities under consideration are included in the survey.

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Income

Rating

Snowfall

Rating

Magazine

Rating

Madison Publishing

$1,500

100(0.15)

MPR Manufacturing

0

(0.6)

Disposable

Income

200(0.70) 400(0.15) 100(0.15) 200(0.70) 400(0.15)

$1,300 (0.4)

Snowfall

75

7575

2525

100 150(0.15) 230(0.70) 320(0.15)

25

100100

Pandemonium Pizza0

25

10050

2550

37.5

100

57.580

100

56

5656

5656

056

00

$1,600

$1,200

0

Magazine

50

75

95

* The gray numbers are not in the original decision tree shown in the textbook

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1. Verify the ratings in the consequence matrix are proportional scores

To do a tradeoff analysis, we have to first make sure different attributes have comparable measuresConvert the measures of three attributes – income, snowfall, and magazine score – to the scale of 0-100.

Income: Set $1600 = 100, $1200 = 0.

For an intermediate value x , its converted score = (x-min)/(max-min) = (x-1200)/(1600-1200)When x =$1300, (1300-1200)/(1600-1200)=25%, so its converted score is 25. When x =$1500, (1500-1200)/(1600-1200)=75%, so its converted score is 75.

Snowfall: set 400 =100, 0=0.

For an intermediate value x , its converted score = (x-0)/(400-0) When x = 100, (100-0)/(400-0)=25%, so its converted score is 25.When x = 150, (150-0)/(400-0)=37.5%, so its converted score is 37.5.When x = 200, (200-0)/(400-0)=50%, so its converted score is 50.When x = 230, (230-0)/(400-0)=57.5%, so its converted score is 57.5.When x = 320, (320-0)/(400-0)=80%, so its converted score is 80.

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Magazine Score: Set 95=100, 50=0For an intermediate value x , its converted score = (x-50)/(95-50) When x = 75, (75-50)/(95-50)≈56%, so its converted score is about 56.

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3. After considering the situation, Robin concludes that the quality of he city is most important, the amount of snow is next, and third is income. Furthermore, Robin concludes that the weight for the magazine rating in consequence matrix should be 1.5 times the weight for the snowfall rating and three times as much as the weight for the income rating. Use this information to calculate the weight for the three attributes and do calculate overall scores for all of the end of branches in the decision tree.

Denote the weights of income, snowfall or magazine as Ki , Ks, and Km, respectively.

Km = 1.5Ks, Km = 3Ki, and Km+ Ks + Ki = 1.

Solving the equations, we can getKm = 1/2, Ks = 1/3, and Ki = 1/6

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Income Snowfall Magazine Overall

Weights: 1/6≈0.17 1/3≈0.33 ½=0.50 Score

Ratings: 75 25 56 49

75 50 56 57

75 100 56 74

Madison 25 25 56 41

25 50 56 49

25 100 56 66

100 37.5 0 29

MPR 100 57.5 0 36

100 80 0 43

Pandemonium 0 0 100 50

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4. Analyze the decision tree using expected values. Calculate expected values for the three measures as well as for the overall score

There is an expected value (EV) for each attribute in each job

Income:

• For Madison

Madison Publishing

$1,500

MPR Manufacturing

(0.6)

$1,300 (0.4)

Pandemonium Pizza

$1,600

$1,200

0

75

100

25

Income

Rating

Original

Income

EV(Income) = $1500*0.6+$1300*0.4=$1,420

Or at the converted scale,

EV(Income) = 75*0.6+25*0.4=55

• For MPR

EV(Income) = $1,600 (Constant)


EV(Income) = 100

• For Pandemonium

EV(Income) = $1,200 (Constant)


EV(Income) = 0

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Madison Publishing

100(0.15)

MPR Manufacturing

(0.6)

200(0.70) 400(0.15) 100(0.15) 200(0.70) 400(0.15)

(0.4)

Snowfall

150(0.15) 230(0.70) 320(0.15)

Pandemonium Pizza 0

Snowfall

Rating

0

25

100

50

2550

37.5

100

57.580

Snowfall:

• For Madison EV(Snowfall) = (100*0.15+200*0.7+400*0.15)*0.6+ (100*0.15+200*0.7+400*0.15)*0.4 =215


• For MPR

EV(Snowfall) =150*0.15+230*0.7+320*0.15=231.5Or at the converted scale,

• For Pandemonium

EV(Snowfall) = 0 (Constant) Or at the converted scale, EV(Snowfall) = 0

EV(Snowfall) = (25*0.15+50*0.7+100*0.15)*0.6+ (25*0.15+50*0.7+100*0.15)*0.4 =53.75

EV(Snowfall) =37.5*0.15+57.5*0.7+80*0.15=57.875

100*0.15+200*0.7+400*0.15U1

E(U1)=

U2

100*0.15+200*0.7+400*0.15E(U2)=

UE(U)=0.6*E(U1)+0.4*E(U2)

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Magazine Score:

• For Madison EV(magazine) = 50 (Constant) Or at the converted scale, EV(magazine) = 56

• For MPR EV(magazine) = 75 (Constant) Or at the converted scale, EV(magazine) = 0

• For Pandemonium EV(magazine) = 95 (Constant) Or at the converted scale, EV(magazine) =100

Overall Score:

Madison Publishing

MPR Manufacturing

(0.6)

(0.15)

(0.15)

(0.4) (0.

15)(0.70)(0.15)

Pandemonium Pizza

(0.15)(0.70)

(0.15)(0.70)

Overall

Score

50

49

7457

4149

2966

3643

U1

49*0.15+57*0.7+74*0.15E(U1)=

U2

41*0.15+49*0.7+66*0.15E(U2)=

UE(U)=0.6*E(U1)+0.4*E(U2)

• For Madison

EV(Overall) = (49*0.15+57*0.7+74*0.15)*0.6+ (41*0.15+49*0.7+66*0.15)*0.4 =55

• For MPR

EV(Overall) =29*0.15+36*0.7+43*0.15=36

• For Pandemonium

EV(Overall) = 50 (Constant)

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5. Do a risk-profile analysis of the three cities. Create risk profiles for each of three attributes as well as the overall score. Does any additional insight arise from this analysis?


1) Madison Publishing

2) MPR

3) Pandemonium

Income:

Madison Publishing

$1,500 (0.6)

$1,300 (0.4)

Income$1,300$1,500

Probabilities0.40.6

Income$1,600

Probabilities1

Income$1,200

Probabilities1

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0

0.2

0.4

0.6

0.8

1

1000 1200 1400 1600 1800

Madison Publishing MPR Pandemonium

Risk Profiles of Income

18

0

0.2

0.4

0.6

0.8

1

1000 1200 1400 1600 1800

Cumulative Risk Profiles of Income


MPR stochastically dominates Madison which stochastically dominates Pandemonium

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2) MPR

3) Pandemonium

Snowfall:

Madison Publishing

100(0.15)

(0.6)

200(0.70) 400(0.15) 100(0.15) 200(0.70) 400(0.15)

(0.4)

MPR Manufacturing 150(0.1

5) 230(0.70) 320(0.15)

Snowfall100200400

Probabilities0.6*0.15+0.4*0.15=0.150.6*0.70+0.4*0.70=0.700.6*0.15+0.4*0.15=0.15

Snowfall150230320

Probabilities0.150.700.15

Snowfall0

Probabilities1

20

Madison Publishing

0

0.2

0.4

0.6

0.8

1

-100 0 100 200 300 400 500

MPR Pandemonium

Risk Profiles of Snowfall

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Cumulative Risk Profiles of Snowfall

Madison Publishing

0

0.2

0.4

0.6

0.8

1

-100 0 100 200 300 400 500

MPR Pandemonium

Both MPR and Madison stochastically dominates Pandemonium but no domination relation between MPR and Madison

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2) MPR

3) Pandemonium

Magazine Score:

Magazine50

Probabilities1

Magazine75

Probabilities1

Magazine95

Probabilities1

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0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Risk Profiles of Magazine Score


24

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Cumulative Risk Profiles of Magazine Score

Pandemonium stochastically dominates MPR which stochastically dominates Madison


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2) MPR

3) Pandemonium

Overall Score:

Madison Publishing (0.6)

(0.15)

(0.15)

(0.4)

(0.15)(0.70)

(0.15)(0.70)

49

7457

414966

MPR Manufacturing (0.15)(0.70)(0.15)

293643

Overall4149576674

Probabilities0.4*0.15=0.06

0.6*0.15+0.4*0.70=0.370.6*0.7=0.42

0.4*0.15=0.060.6*0.15=0.09

Overall293643

Probabilities0.150.700.15

Magazine50

Probabilities1

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0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Risk Profiles of Overall Score


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0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Cumulative Risk Profiles of Overall Score

Madison and Pandemonium stochastically dominates MPR, but there is obvious domination relationship between Madison and Pandemonium


1 a stock market investor has $500 to spend and is considering purchasing an option contract on 1000...

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