1. Problem 4.15
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.

2. 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
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
Favorable
(0.25)
$3,000
1000*($33.50-$30.00)
=$3,500
Purchase Option
-$500
Unfavorable
(0.75)
Lawsuit Outcome
-$500 $0
Do Nothing
$40
$500*8%=$40
Favorable
(0.25)
$86.24
17*$ $15.5*8%
=$570.74
Buy Stock
-17*$28.5=
-$484.5
Lawsuit Outcome
Unfavorable
(0.75)
-$45.51
17*$ $15.5*8%
=$438.99

3. Decision Strategies:
1) Purchase option
Favorable
Unfavorable
$3,000
-$500
(0.25)
(0.75)
Purchase Option
Payoffs
-$500
$3,000
Probabilities
0.75
0.25
2) Do nothing
Payoffs
$40
Probabilities 1
$40
Do Nothing
3) Buy stock
Buy Stock
Favorable
Unfavorable
$86.24
-$45.51
(0.25)
(0.75)
Payoffs
-$45.51
$86.24
Probabilities
0.75

4. 40
-45.51
86.24
Return on Investment

5. 86.24
-45.51 40
Therefore, no immediate conclusions can be drawn since no one alternative dominates another

6. 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) = p –
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  p>0.135
p>0.154

7. 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.

8. Income Rating
Snowfall Rating
Magazine Rating
Disposable Income
Snowfall
Magazine
100
(0.15) 75 25 56
$1,500
200
(0.70) 50 75 50 56
(0.6)
400
(0.15) 75
100 56
Madison Publishing
100
(0.15) 25 25 56
$1,300
200
(0.70) 25 50 56
(0.4)
400
(0.15) 25
100 56
150
(0.15)
100
37.5
MPR Manufacturing
$1,600
230
(0.70) 75
100
57.5
320
(0.15)
100 80
Pandemonium Pizza
$1,200 95
100
* The gray numbers are not in the original decision tree shown in the textbook

9. 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 measures
Convert the measures of three attributes – income, snowfall, and magazine score – to the scale of
Income:
Set $1600 = 100, $1200 = 0.
For an intermediate value x , its converted score = (x-min)/(max-min) =
(x-1200)/( )
When x =$1300, ( )/( )=25%, so its converted score is 25.
When x =$1500, ( )/( )=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.

10. Magazine Score:
Set 95=100, 50=0
For 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.

11. 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 get
Km = 1/2, Ks = 1/3, and Ki = 1/6

12. Income
Snowfall
Magazine
Overall
Weights:
1/6≈0.17
1/3≈0.33
½=0.50
Score
Ratings: 75 25 56 49 50 57
100 74
Madison 41 66
37.5 29
MPR
57.5 36 80 43
Pandemonium

13. 4. Analyze the decision tree using expected values
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:
Madison Publishing
$1,500
MPR Manufacturing
(0.6)
$1,300
(0.4)
Pandemonium Pizza
$1,600
$1,200 75
100 25
Income Rating
Original Income
For Madison
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)
Or at the converted scale,
EV(Income) = 100
For Pandemonium
EV(Income) = $1,200 (Constant)
Or at the converted scale,
EV(Income) = 0

14. Madison Publishing 100 (0.15) MPR Manufacturing (0.6) 200 (0.70) 400
Snowfall:
For Madison
Madison Publishing
100
(0.15)
MPR Manufacturing
(0.6)
200
(0.70)
400
(0.4)
Snowfall
150
230
320
Pandemonium Pizza
Snowfall Rating 25 50
37.5
57.5 80
EV(Snowfall) = (100* * *0.15)*0.6
+ (100* * *0.15)*0.4 =215
E(U1)=
100* * *0.15 U1
E(U)=0.6*E(U1)+0.4*E(U2)
Or at the converted scale, U
EV(Snowfall) = (25* * *0.15)*0.6
+ (25* * *0.15)*0.4 =53.75 U2
100* * *0.15
E(U2)=
For MPR
EV(Snowfall) =150* * *0.15=231.5
Or at the converted scale,
EV(Snowfall) =37.5* *0.7+80*0.15=57.875
For Pandemonium
EV(Snowfall) = 0 (Constant)
Or at the converted scale,
EV(Snowfall) = 0

15. Madison Publishing MPR Manufacturing (0.6) (0.15) (0.4) (0.70)
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)
(0.4)
(0.70)
Pandemonium Pizza
Overall Score 50 49 74 57 41 29 66 36 43
For Madison
49* *0.7+74*0.15
E(U1)=
EV(Overall) = (49* *0.7+74*0.15)*0.6
+ (41* *0.7+66*0.15)*0.4 =55 U1
E(U)=0.6*E(U1)+0.4*E(U2) U
For MPR U2
EV(Overall) =29* *0.7+43*0.15=36
41* *0.7+66*0.15
E(U2)=
For Pandemonium
EV(Overall) = 50 (Constant)

16. 5. Do a risk-profile analysis of the three cities
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?
Decision Strategies:
1) Madison Publishing
2) MPR
3) Pandemonium
Income:
Madison
Publishing
$1,500
(0.6)
$1,300
(0.4)
Income
Probabilities
0.4
0.6
$1,600 1
$1,200

17. Risk Profiles of Income
Madison Publishing
MPR
Pandemonium

18. Cumulative Risk Profiles of Income
Madison Publishing
MPR
Pandemonium
MPR stochastically dominates Madison which stochastically dominates Pandemonium

19. Madison Publishing 100 (0.15) (0.6) 200 (0.70) 400 (0.4) Snowfall
Decision Strategies:
1) Madison Publishing
Madison Publishing
100
(0.15)
(0.6)
200
(0.70)
400
(0.4)
Snowfall
100
200
400
Probabilities
0.6* *0.15=0.15
0.6* *0.70=0.70
2) MPR
Snowfall
150
230
320
Probabilities
0.15
0.70
MPR Manufacturing
150
(0.15)
230
(0.70)
320
3) Pandemonium
Snowfall
Probabilities 1

20. Risk Profiles of Snowfall
Madison Publishing
MPR
Pandemonium

21. Cumulative Risk Profiles of Snowfall
Madison Publishing
MPR
Pandemonium
Both MPR and Madison stochastically dominates Pandemonium but no domination relation between MPR and Madison

22. Magazine Score:
Decision Strategies:
1) Madison Publishing
Magazine 50
Probabilities 1
2) MPR
Magazine 75
Probabilities 1
3) Pandemonium
Magazine 95
Probabilities 1

23. Risk Profiles of Magazine Score
Madison Publishing
MPR
Pandemonium

24. Cumulative Risk Profiles of Magazine Score
Madison Publishing
MPR
Pandemonium
Pandemonium stochastically dominates MPR which stochastically dominates Madison

25. Overall Probabilities Overall Probabilities Magazine Probabilities
Overall Score:
Decision Strategies:
1) Madison Publishing
Madison Publishing
(0.6)
(0.15)
(0.4)
(0.70) 49 74 57 41 66
Overall 41 49 57 66 74
Probabilities
0.4*0.15=0.06
0.6* *0.70=0.37
0.6*0.7=0.42
0.6*0.15=0.09
2) MPR
Overall 29 36 43
Probabilities
0.15
0.70
MPR Manufacturing
(0.15)
(0.70) 29 36 43
3) Pandemonium
Magazine 50
Probabilities 1

26. Risk Profiles of Overall Score
Madison Publishing
MPR
Pandemonium

27. Cumulative Risk Profiles of Overall Score
Madison Publishing
MPR
Pandemonium
Madison and Pandemonium stochastically dominates MPR, but there is obvious domination relationship between Madison and Pandemonium