James W. Taylor February 2012 Imperial College EMBA 2012 Quantitative Methods Individual Assignment
This assignment consists of two parts. Part A is worth 50% of the marks, and Part B is worth the remaining 50%. Your report for the two parts should consist of no more than 1,500 words.

Part A – Blanket Systems Blanket Systems is developing and testing a new computer workstation, OB1, which it will introduce to the market in the next 6 months. OB1 will be sold under a three-year warranty covering parts and labour. The company has decided to subcontract the service support for the warranty and has entered negotiations about the support contract with Fixit Inc. Fixit has proposed two different pricing schemes for the subcontract. The first involves the payment of a fixed fee of $1,000,000 and the second a variable fee of $250 per workstation sold, subject to a minimum fee of $350,000. Under both schemes, the payment will be made one year after the introduction of the workstation to the market at which point the product will be replaced by newer models not covered by the warranty service subcontract. At the moment, there is uncertainty about the sales potential of the new workstation. Sales of OB1 are expected to come from two sources: (i) the successful closure by senior management of a major purchase of 2000 units by a long standing customer, (ii) the efforts of regional sales offices. Given the state of the negotiations with the long-standing customer, the current estimate of the probability of a successful closure of the major purchase is 0.5. Regional sales of OB1 would be boosted by the successful closure, and the management of Blanket Systems has estimated the regional sales potential (in addition to the major purchase), as shown in the table below. If major purchase Probability 0.2 0.3 0.3 0.2 If no major purchase Sales Probability 1000 0.25 2000 0.25 3000 0.25 4000 0.25

Sales 2500 3500 4500 5500

(a) Use a decision tree to establish which of the two pricing schemes minimises the expected value of costs for Blanket Systems? (b) By how much would the probability of a successful closure of the major purchase (currently estimated to be 0.5) have to change before your answer to part (a) changes? (c) Assuming the true value of the probability of a successful closure of the major purchase is 0.5, what is the most that Blanket Systems should pay now for information regarding whether they will achieve a successful closure of the major purchase?

PLEASE TURN OVER FOR PART B

Part B - Quant Investments “It had not been a particularly good year for Quant Investments”, mused Alain over his pate de foie gras and champagne. It was lunchtime on Christmas Eve, and his three colleagues Belén, Carlos and Dawood had already set off on holiday to far-off, exotic locations. The four of them had worked as a close team of analysts for the past 15 months since graduating as MBAs, and Alain realised that their performances had not come up to expectations. Alain feared that all four may lose their jobs shortly. At this moment of low spirits, Xavier, one of the senior partners whom Alain had always found remarkably candid, joined him for lunch. “Well, Alain, I’m sure you have heard rumours about the board meeting this morning. The future of the four of you was discussed and a decision was taken to release one of you. I can’t tell you who - the Chairman is writing to you all immediately. However, I can say that we were unable to make the decision on merit, so we made it at random. The Chairman took one of four cards, each of which had one of your names on the back!” “So, I have a one in four chance of being out of work after Christmas”, said Alain. He felt relieved, as he had thought the odds might be worse. “But of course you know definitely who it is”, he added. Alain felt he could leach rather more out of the partner. “Since we both know that at least two out of Belén, Carlos or Dawood will definitely have a job here next year, if you tell me the name of two of them who will keep their jobs, it will still leave me in the dark as to my own fate. It would be pleasant to contemplate over Christmas the two who will continue in their jobs. Clearly, I’m not going to be able to see any of them before the Chairman’s letters arrive. If all three are to keep their jobs, just choose two names at random to tell me.” Xavier thought about this for a moment and, as he left, he said, “Alright, let me tell you that Carlos and Dawood will keep their jobs. Merry Christmas!” Alain’s immediate reaction was one of further relief. He was pleased that Carlos and Dawood will receive good news on his return. But then his depression deepened. “It’s now between Belén and myself. My chance of being fired seems to have gone up from one in four to one in two. Perhaps I shouldn’t have squeezed that extra information out of Xavier after all!” Discuss whether or not Alain is right in thinking that the probability changed from ¼ to ½?

The essence of the problem in Part B is the probability of the selection of one of four individuals for termination. Essentially, one of four members of an analyst team at Quant Investments was going to be terminated due to poor team performance. The decision on which of the four team members would be terminated was not to be based on individual merit. Rather, the decision was to be based on random selection (wherein each member of the analyst team was equally liable for termination selection.
Probability theory is the science of making decisions based on incomplete information. Thus, probability theory is applied in situations when it not possible to obtain sufficient information on which to base a more certain prediction of an outcome. The probability equation yields a probability level that is an expression of the likelihood of a specified outcome under specified conditions. The probability level is expressed in a range from 0 (no probability) to 1 (certainty). If one possessed sufficient information to arrive at a 0 or a 1, however, there would be no point in performing a probability analysis (Ash, 2008; Gnedenko & Khinchin, 1962).
In the Quant Investments case, the initial selection of any one of the four members of the analyst team was identical. The probability for any one of the four members of the analyst team was one-in-four, which would be expresses as p = 0.25. This probability level expressed the likelihood of an undesirable outcome for any one of the four members of the analyst team.
Following the decision by Quant Investments management of the member of the analyst team selected for termination, but prior to the formal announcement of the decision, one member of the analyst team became privy to additional information. This team member, Alain, learned from a senior manager, Xavier, at Quant Investments the names of two (Carlos and Dawood) of the four members of the analyst team who were not selected for termination. Alain was not identified as one of the two team members who were not selected for termination. This added knowledge told Alain and the fourth team member (Belen), that either he or Belen had been selected for termination.
The question at this point is whether, with respect to Alain being selected for termination, “the probability changes from ¼ to ½”? The correct answer to this question may rest on semantics. The chance of Alain being selected in the first place was 1-in-4. Note that the verb in the preceding sentence was “being”. If that verb were to be changed from “being” to “was”, the answer to the question would vary. Simply because Alain became privy to additional knowledge did not change the probability of his “being” selected for termination. Using the additional information, however, Alain could structure a new probability equation, wherein the conditions reduce to two the number of members of the analyst who could be the person who “was” selected. With the additional information, Alain knows that the probability that he was selected for termination is 1-in-2 or p = .5 and that the probability that Belen “was” selected also is 1-in-4. Alain also knows that the probability that either Carlos or Dawood was selected is 0 (none).

References

Albright, S. C., Winston, W., & Zappe, C. (2010). Data analysis and decision making. (4th ed.). Mason, OH: South-Western College Publishing.
Ash, R. B. (2008). Basic probability theory. Mineola, NY: Dover Publications.
Gnedenko, B. V., & Khinchin, A. Ya. (1962). An elementary introduction to the theory of probability. New York, NY: Dover Publications.
Goodwin, P., & Wright, G. (2010). Decision analysis for management judgment. (4th ed.). New York, NY: John Wiley and Sons, Inc.