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.

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*$33.50 + $15.5*8% =$570.74 Buy Stock -17*$28.5= -$484.5 Lawsuit Outcome Unfavorable (0.75) -$45.51 17*$25.75 + $15.5*8% =$438.99

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

40 -45.51 86.24 Return on Investment

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

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

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.

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

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 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.

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.

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

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

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

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+200*0.7+400*0.15)*0.6 + (100*0.15+200*0.7+400*0.15)*0.4 =215 E(U1)= 100*0.15+200*0.7+400*0.15 U1 E(U)=0.6*E(U1)+0.4*E(U2) Or at the converted scale, U 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 U2 100*0.15+200*0.7+400*0.15 E(U2)= For MPR EV(Snowfall) =150*0.15+230*0.7+320*0.15=231.5 Or at the converted scale, EV(Snowfall) =37.5*0.15+57.5*0.7+80*0.15=57.875 For Pandemonium EV(Snowfall) = 0 (Constant) Or at the converted scale, EV(Snowfall) = 0

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*0.15+57*0.7+74*0.15 E(U1)= 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 U1 E(U)=0.6*E(U1)+0.4*E(U2) U For MPR U2 EV(Overall) =29*0.15+36*0.7+43*0.15=36 41*0.15+49*0.7+66*0.15 E(U2)= For Pandemonium EV(Overall) = 50 (Constant)

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

Risk Profiles of Income Madison Publishing MPR Pandemonium

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

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+0.4*0.15=0.15 0.6*0.70+0.4*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

Risk Profiles of Snowfall Madison Publishing MPR Pandemonium

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

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

Risk Profiles of Magazine Score Madison Publishing MPR Pandemonium

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

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*0.15+0.4*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

Risk Profiles of Overall Score Madison Publishing MPR Pandemonium

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