(Mt) – UCI Expected Value Worksheet

SOLUTION AT Academic Writers Bay

Homework 2 1. Most of us have experienced trouble trying to find parking on campus even when we have a parking permit. Having just recently taken the project management class, we want to explore whether running a valet parking service would be profitable. Our plan, like most naïve plans, is simple: Step1: We will lease land for parking about three miles away (east of I-805). Step2: Employ students to drive the cars from UCSD campus to our off-site parking lot, and back when requested. Step3: profit!! a) At first glance, does this plan pass the “smell test”. That is, does the plan make economic sense? Why or why not? [argue in favor of your position using approximate (order-of-magnitude) estimates why the plan does or does not make sense] b) Since the service is superior to that offered by the UCSD parking services, we may be able to charge a higher price. How much is the minimum you need to charge each car per hour to break-even? Assume it takes 20 minutes for a student employee to take the car to our off-site parking lot. So a student can take no more than 3 cars per hour. Even if they are getting minimum wage $14 per hour, this means that the cost per student employee per car is $14/3 = $5. Note that this does not account for the “real estate” cost of parking, or for the insurance you need, or for other overhead costs (like costs connected to coordinating). Given that UCSD has daily parking for about $5 per day, it is doubtful that our plan can succeed. 2. A project requires $10 million dollars in initial investment. The projected revenue is $3 million dollars per year for the next 5 years. If we apply a discount factor of 5%, what is a) the break-even period? b) the discounted cash-flow and the NPV of the proposed project? c) the IRR of the project? We shall assume that revenue is accrued at the end of the year. Specifically, $3 million at end of year 1, year 2, year 3, … It will take 10/3 = 3.33 years to breakeven DCF = 3/(1+0.05) + 3/(1+0.05)2 + 3/(1+0.05)3 + 3/(1+0.05)4 + 3/(1+0.05)5 which works out to 12.98 million NPV = -10 + DCF = 2.98 million IRR is the discount rate that makes NPV equal to 0. If we use discount rate of 15.2%, DCF will be equal to $10, and NPV will be equal to 0. So IRR is 15.2% 3. You are managing a lab which tests prototypes for compliance with safety regulations. A project manager has given you two prototypes to test, A and B, for his project, with the goal to identify at least one that meets the safety regulations. You calculate that prototype A has 30% likelihood of meeting the regulations and B has 40% chance of meeting the regulations. The profits (value) if we meet the regulations using any prototype is $100, and the value if we do not meet the regulation is $0. Suppose the cost of testing each prototype is $20. [Note that we only want to identify one prototype, and there is no additional value in identifying two prototypes that comply with regulations] a) If we could test only one prototype, which prototype would we test? What is the expected value of following this strategy (of testing only one prototype)? Value from testing A: 0.3×100 – 20 = $10 Value from testing B: 0.4×100 – 20 = $20 So we should test B (if we can test only one prototype) and obtain expected value $20. b) Suppose we decide to test both prototypes simultaneously, and then choose the prototype that complies with the safety regulation. What is the expected value of following this strategy? If neither A nor B meets the regulation, then only do we get the value $0. In all other cases, we get the value $100. The former event happens with probability (10.3)x(1-0.4) = 0.42, and hence the latter event happens with probability 1-0.42 = 0.58 So expected value from testing both is 0.42×0 + 0.58×100 – 20 – 20 = $18 Suppose we build in enough flexibility in our resource scheduling and follow the following sequential strategy: we can test one prototype and then continue onto test the next one only if the first one didn’t meet the regulations. What is the value of following this sequential strategy if c) We test A first. And if A doesn’t meet the guidelines, we test B After testing A, if it didn’t meet regulations, then we test B and get an expected value $20 (see part a) So before testing A, when we look forward, we will see value of $100 if it succeeds and value of $20 if it fails (because then we can go ahead and test B). So expected value of testing A : 0.3×100 + 0.7×20 – 20 = $24 d) We test B first, and if B doesn’t meet the guidelines, we test A. After testing B, if it didn’t meet regulations, then we test A and get an expected value $10 (see part a) So before testing B, when we look forward, we will see value of $100 if it succeeds and value of $10 if it fails (because then we can go ahead and test A). So expected value of testing A : 0.4×100 + 0.6×10 – 20 = $26 [Note that to undertake this sequential strategy we need to be able to tentatively schedule resources for the second test, with the understanding that there is a non-zero chance that the resource would not be utilized and hence would have to re-allocated. Hence the strategy requires flexibility in resource scheduling] 4. As a project manager, you have to select one of four possible designs for a phone that your firm is creating. The first two designs D1 and D2 are similar to previously used designs and you believe will have values that are normally distributed with mean 80 and standard deviation 5. The third and fourth designs D3 and D4 are both novel and you believe they will both have value that is normally distributed with mean 70 and standard deviation 30. a) If you were allowed to test only one design, which design will you test? What is the expected value (not counting the costs) of your design? With exactly one design, you should test the one which has higher mean, which in this case is D1 (or D2). b) If you were allowed to test only two designs in parallel, with the goal of finally choosing one of these tested designs as the final one, which two designs will you test? What is the expected value (not counting the costs) of your chosen set? If we test D1,D2: the expected value of the maximum of two designs is 82.81 If we test D1,D3: the expected value of the maximum of two designs is 87.77 If we test D3,D4: the expected value of the maximum of two designs is 86.92 So it is best to choose to test D1,D3 (or D1, D4; or D2,D3; or D2, D4). c) If you were allowed to test only three designs in parallel, with the goal of finally choosing one of these tested designs as the final one, which three designs will you test? What is the expected value (not counting the costs) of your chosen set? If we test D1,D2,D3: the expected value is 89.54 If we test D1,D3,D4: the expected value is 93.83 So it is best to test D1,D3,D4 (or D2,D3,D4) d) Explain the intuition behind your chosen design set [Specifically, are all the tested designs high risk, or are all of them lower risk. And what is the intuitive purpose of each of the designs you choose to test] It appears that it is best to test higher variance/risk designs as long as you have at least one low variance design. The low variance design provides us with a lower bound on the value, whereas the high variance design allows us to get the possible upside. Use the simulation hosted at https://sanjiverat.com/ptesting to answer the above questions 5. [BONUS QUESTION] You have signed a contract that gives you a payment for 100,000 for a project. Suppose this project can be broken down into two serial activities – A and B. The time taken by activity A is uniformly distributed between 10 days and 60 days. Activity B takes exactly 50 days to complete if worked on by 2 people. But you may choose to have more people work on activity B and reduce the time it takes to complete. Specifically, you calculate that it would take 40 days to complete if worked on by 3 people. The extra cost of this additional person for working on activity B is $1500 The contract you signed with your client is structured so that a quadratic penalty gets deducted based on how many days you take to complete the project. Specifically, if your project gets completed in 70 days, your net payment would be $100,000 minus (70)2 = 100,000 – 4900 = $95,100 Similarly, if your project gets completed in 80 days, your net payment would be $100,000 minus (80)2 = 100,000 – 6400 = $93,600 a) What is the expected value you receive if you choose to use only 2 people on activity B? The project can take 10+50=60 days to 60+50=110 days to complete. The penalty if it takes d days is d2. So average penalty can be calculated as follows: (602 + 612 + 622 + … + 1102)/51 = 7441 So expected value is 100,000 – 7441 = $92,559 b) What is the expected value you receive if you choose to use 3 people on activity B? The project can take 10+40=50 days to 60+40=100 days to complete. The penalty if it takes d days is d2. So average penalty can be calculated as follows: (502 + 512 + 522 + … + 1002)/51 = 5842 So expected value is 100,000 – 5842 = $94,158 Note that this extra person costs $1500. So the net gain is $94,158-1500 = $92,658 You decide to run the project as follows: you will defer making the choice of whether or not have 3 people work on activity B. Specifically, you decide that if activity A takes less than 20 days to complete, then you will use only 2 people to work on activity B. if it takes more than 20 days to complete, then you will use 3 people to work on activity B. c) What is the expected value of following the above strategy? If A takes 10 days to complete, then project takes 10+50=60 days .. If A takes 20 days to complete, then project takes 20+50=70 days If A takes 21 days to complete, then project takes 21+40=61 days [because we increased the resources for B!]. note that in this case, the resource costs also went up by $1500. As before, if we compute the average penalty we will see that it works out to (602 + 612 +… 702 + 612 … + 1002)/51 = 6100 Moreover, the average resource costs will be (0 + 0 + … 0 + 1500 + .. + 1500)/51 = 1176 [Note that the resource costs are incurred only when A takes more than 20 days] So the net value will become 100,000 – 6100 – 1176 = 92,724 Change the number of days you defer the choice from 20 days (as give above) to other values. For instance, try deferring the choice for 25 days, 30 days, 35 days, …. d) From the above, how many days should we defer the choice? Repeating the calculations above should convince you that deferring by 30 maximizes the value to $92739. The key insight to understand is that by deferring the choice, we save on the resource cost, but give up on penalty. e) Finally, will the optimal length of time you delay making the choice increase or decrease if the cost of the additional person is $1800 (instead of $1500 that you used for your prior questions). With the above intuition, you should be able to see that deferring the choice should be more valuable if the resources are more expensive (since we can save more on resource costs) Unnecessarily Advanced Technical Bonus: Note: the above calculations assumed that the duration is always a whole number (i.e., time taken by A is 10, 11, 12, … For a continuous distribution, the calculations below are more appropriate. Suppose we wait till day d of activity A, and then only crash B (if needed). That is, if A takes less than d days, then we don’t crash B. If A takes longer than d days, then we crash B. Then the time taken by the project is (E[A|Ad) [Note that the first term is the time taken by A + non-crashed time for B; and the second term is time taken by A + crashed time for B] Which can be simplified as ((10+d)/2 + 50) ( d – 10)/50 + (d+60)/2 + 40) ( 60-d)/50, which after some tedious algebra is equal to 73 + d/5 The expected (quadratic) penalty can be calculated as The first term corresponds to the penalty when A takes less than d days, and second term corresponds to penalty when A takes more than d days. The above can be simplified to The expected resource cost is given by Which simplifies to 1800 – 30d So the total of expected penalty + expected resource cost is given by Minimizing the above, we can see that the expected penalty + resource cost is least when d=30 Similarly, if we assume that c is the cost of the extra resources, then the only part which changes is that the expected resource cost becomes Hence, the total expected penalty + expected resource cost is given by minimizing the above, we can see that the expected penalty + resource cost is least when d=c/20 – 45. Stated differently, we should defer the decision to crash till later on when the cost of crashing is greater.

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