Presentation on theme: "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."— Presentation transcript:
1 Problem 4.15A 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 treeAssumptions: 1) 8% is the monthly interest rate; 2) the investor can only purchase an integer number of shares and put the remaining money to savingsFavorable(0.25)$3,0001000*($33.50-$30.00)=$3,500Purchase Option-$500Unfavorable(0.75)Lawsuit Outcome-$500$0Do Nothing$40$500*8%=$40Favorable(0.25)$86.2417*$ $15.5*8%=$570.74Buy Stock-17*$28.5=-$484.5Lawsuit OutcomeUnfavorable(0.75)-$45.5117*$ $15.5*8%=$438.99
5 86.24-45.5140Therefore, 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 lawsuit1) The expected monetary value associated with purchasing the option is:EMV(Purchase Option) = 3,000p – 500(1 – p) = 3,500p – 5002) The expected monetary value associated with doing nothing is:EMV(Do Nothing) = 403) 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) = $-12.57When p=0.1, EMV(Purchase Option) = -$150, EMV(Do Nothing)=$40, EMV(Purchase Stock) = $-32.33EMV(Purchase Option) > EMV(Do Nothing) 3500p-500>40 p>0.154EMV(Purchase Option) > EMV(Buy Stock) 3500p-500>131.75p p>0.135p>0.154
7 Job OffersRobin 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 RatingSnowfall RatingMagazine RatingDisposable IncomeSnowfallMagazine100(0.15)752556$1,500200(0.70)50755056(0.6)400(0.15)7510056Madison Publishing100(0.15)252556$1,300200(0.70)255056(0.4)400(0.15)2510056150(0.15)10037.5MPR Manufacturing$1,600230(0.70)7510057.5320(0.15)10080Pandemonium Pizza$1,20095100* 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 measuresConvert the measures of three attributes – income, snowfall, and magazine score – to the scale ofIncome: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=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.
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 getKm = 1/2, Ks = 1/3, and Ki = 1/6
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 scoreThere is an expected value (EV) for each attribute in each jobIncome:Madison Publishing$1,500MPR Manufacturing(0.6)$1,300(0.4)Pandemonium Pizza$1,600$1,2007510025Income RatingOriginal IncomeFor MadisonEV(Income) = $1500*0.6+$1300*0.4=$1,420Or at the converted scale,EV(Income) = 75*0.6+25*0.4=55For MPREV(Income) = $1,600 (Constant)Or at the converted scale,EV(Income) = 100For PandemoniumEV(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 MadisonMadison Publishing100(0.15)MPR Manufacturing(0.6)200(0.70)400(0.4)Snowfall150230320Pandemonium PizzaSnowfall Rating255037.557.580EV(Snowfall) = (100* * *0.15)*0.6+ (100* * *0.15)*0.4 =215E(U1)=100* * *0.15U1E(U)=0.6*E(U1)+0.4*E(U2)Or at the converted scale,UEV(Snowfall) = (25* * *0.15)*0.6+ (25* * *0.15)*0.4 =53.75U2100* * *0.15E(U2)=For MPREV(Snowfall) =150* * *0.15=231.5Or at the converted scale,EV(Snowfall) =37.5* *0.7+80*0.15=57.875For PandemoniumEV(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 MadisonEV(magazine) = 50 (Constant)Or at the converted scale, EV(magazine) = 56For MPREV(magazine) = 75 (Constant)Or at the converted scale, EV(magazine) = 0For PandemoniumEV(magazine) = 95 (Constant)Or at the converted scale, EV(magazine) =100Overall Score:Madison PublishingMPR Manufacturing(0.6)(0.15)(0.4)(0.70)Pandemonium PizzaOverall Score504974574129663643For Madison49* *0.7+74*0.15E(U1)=EV(Overall) = (49* *0.7+74*0.15)*0.6+ (41* *0.7+66*0.15)*0.4 =55U1E(U)=0.6*E(U1)+0.4*E(U2)UFor MPRU2EV(Overall) =29* *0.7+43*0.15=3641* *0.7+66*0.15E(U2)=For PandemoniumEV(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 Publishing2) MPR3) PandemoniumIncome:MadisonPublishing$1,500(0.6)$1,300(0.4)IncomeProbabilities0.40.6$1,6001$1,200
17 Risk Profiles of Income Madison PublishingMPRPandemonium
18 Cumulative Risk Profiles of Income Madison PublishingMPRPandemoniumMPR stochastically dominates Madison which stochastically dominates Pandemonium
26 Risk Profiles of Overall Score Madison PublishingMPRPandemonium
27 Cumulative Risk Profiles of Overall Score Madison PublishingMPRPandemoniumMadison and Pandemonium stochastically dominates MPR, but there is obvious domination relationship between Madison and Pandemonium