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Decision Models & Optimization Term II, 2007

Sample Final Examination Page 1 of 5

Indian School of Business Hyderabad, India Decision Models & Optimization Term II, 2004 Professors Anjani Jain and Ziv Katalan Department of Operations and Information Management The Wharton School University of Pennsylvania Instructions This examination is open book and open notes. Calculators are allowed, but not laptop or handheld computers capable of running spreadsheets. The examination will last exactly two hours; feel free to ask questions during the examination if there is any confusion. Write all your answers in the space provided within this exam booklet. Please show all work — partial credit will be given for all questions except the multiple-choice ones.

Final Examination Good luck! Last Name:

_____________________

First Name:

_____________________

Section:

_____________________

Question ……. 3

Part

a b c d e f g total

points

out of

6 6 4 6 6 6 6 40

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 2 of 5

3 GlobTech Expansion Options (40 points) Julie Dewent, the CFO of GlobTech, is considering building a new plant, which will start generating revenues in one year. The stream of revenues is uncertain, and GlobTech believes that the mean of its value one year from now will be $108,329,000. At a discount rate of r = 8%, this is equivalent to $100,000,000 = e-0.08 $108,329,000 today. During the coming year, GlobTech’s revenue expectations may change. More specifically, GlobTech believes that one year from now, the expected value V (in $millions) of the revenues is lognormally distributed with σ = 60% volatility, V = 100 e(r – σ^2/2)+ σ *Z = 100 e(0.08 – 0.6^2/2)+0.6*Z where Z is a standard normal random variable with mean value of 0 and standard deviation of 1. GlobTech is considering the following options:

Base Case Build the plant now If the plant is built now, its cost will be a fixed $104 million. Because the expected present value of the revenue today is only $100 million, simply building the plant now does not appear to be worthwhile. That is, Expected NPV(Build Now) = -104 + e-0.08 108.329 = -4.

Option 1: Postponement Option GlobTech waits one year, at which time V is known, before making the decision of whether or not to invest in this project. The delay will decrease the project revenues by 25% and construction costs will grow at a rate of 8%. In this case, NPV(Postponement Option) = e-0.08 MAX[0.75V – e0.08(104), 0].

Option 2: Expansion Option GlobTech invests $104 now and has the option to spend more in one year. By that time, the company will know the value of V, and it can choose to spend $50 million at that time on a plant expansion that will increase the project revenues by 50%. Alternatively, GlobTech can just accept V. In this case, NPV(Expansion Option) = e-0.08 MAX[(1.5V – 50), V] –104.

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 3 of 5

Note: For all of the questions that follow, simply check the correct answer. No partial credit will be given. GlobTech has simulated the performance of Options 1 and 2. The results of a 900 trial Monte Carlo simulation are summarized in the tables below: Statistics Trials Mean Median Mode Standard Deviation Variance Skewness Kurtosis Coeff. of Variability Range Minimum Range Maximum Range Width Mean Std. Error Percentiles 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

NPV (Postponement) 900 9.71 0.00 0.00 33.11 1,096.00 5.85 48.61 3.41 0.00 413.44 413.44 1.10 NPV (Postponement) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.98 26.27 413.44

NPV (Expansion) 900 11.07 -19.91 --95.64 9147.60 3.02 18.21 8.64 -91.69 884.72 976.41 3.19 NPV (Expansion) -91.69 -64.25 -54.15 -42.87 -33.15 -19.91 1.44 28.07 59.80 110.38 884.72

a) (6 points) If the option to postpone the project by one year requires an additional payment of $1 million now, what is the approximate probability that the NPV of the project will be positive? ( ) 100% ( ) 80% ( ) 50% ( ) 20% ( )

0%

( ) It cannot be determined from the simulation

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 4 of 5

b) (6 points) What would be the approximate width of the 95% confidence interval for the true expected NPV of the postponement option if the sample size were 1800 instead of 900? ( ) $8.8 million ( ) $4.4 million ( ) $3.1 million ( ) $2.2 million c) (4 points) Which of the following statements is true? ( ) The true expected NPV of the postponement option is higher than the true expected NPV of the expansion option. ( ) The true expected NPV of the expansion option is higher than the true expected NPV of the postponement option. ( ) It cannot be determined from the simulation results which of the true expected NPVs is higher. d) (6 points) Suppose the annual volatility of the project revenues increases to 80%. Without running a simulation, which of the following statements would you expect to be true? ( ) The expected NPV of the postponement option will be higher when the volatility increases while the expected NPV of the expansion option will stay the same. ( ) The expected NPV of the postponement option will stay the same when the volatility increases while the expected NPV of the expansion option will be higher. ( ) The expected NPV of both options will be lower when the volatility increases. ( ) The expected NPV of both options will stay the same when the volatility increases. ( ) The expected NPV of both options will be higher when the volatility increases. e) (6 points) Looking at the distribution of percentiles, the observed chance that the NPV of Expansion will be between -$54,150,000 and $59,800,000 is about 60%. Suppose the number of trials were increased to 3600. What is the best estimate of the chance that the NPV of Expansion will be between -$54,150,000 and $59,800,000 in these new results? ( ) about 90% ( ) about 80% ( ) about 60% ( ) about 30% ( ) about 15%

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 5 of 5

Contraction Option Instead of paying the entire $104 million plant cost now, GlobTech can pay ONLY $50 million now. Then, one year from now, when V is known, the company can choose to pursue either of two options. •

Pay the remaining $54 million cost, plus interest, and obtain the full project value. Here, total costs and revenues will be the same as they would have been in the base case.

•

Contract the scale of the project by paying only $25 million next year. Here, the present value of the revenues will be worth only 50% of what they would have originally been worth.

f)

(6 points) Which of the following expressions will evaluate correctly the NPV of the contraction option? ( ) e-0.08 MAX[(V – e0.08104), 0.5V - 25] ( ) e-0.08 MAX[(V – e0.0854), 0.5V - 25] – 50 ( ) e-0.08 MAX[(0.5V – 25), 0] - 50 ( ) e-0.08 MAX[(0.5V – 25), V] - 54

The results of a 100-trial Monte Carlo simulation of the contraction option are summarized in the following table: Statistics Trials Mean Median Mode Standard Deviation Variance Range Minimum Range Maximum Range Width Mean Std. Error

NPV (Contraction) 100 -6.86 -27.13 --63.89 4,081.62 -62.72 301.24 363.96 6.39

g) (6 points) How does the true expected NPV of the contraction option compare with that of the base case, in which GlobTech simply invests $104 million today to build the plant? ( ) The true expected NPV of the contraction option is definitely no lower than that of the base case. ( ) The true expected NPV of the contraction option is definitely no higher than that of the base case. ( ) Given the simulation results, there is a significant probability that the true expected NPV of the contraction option is either higher or lower than that of the base case. To precisely determine which, one must run the simulation with a much larger number of trials. ( ) It cannot be determined whether or not the true expected NPV of the contraction option is higher or lower than that of the base case.

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Sample Final Examination Page 1 of 5

Indian School of Business Hyderabad, India Decision Models & Optimization Term II, 2004 Professors Anjani Jain and Ziv Katalan Department of Operations and Information Management The Wharton School University of Pennsylvania Instructions This examination is open book and open notes. Calculators are allowed, but not laptop or handheld computers capable of running spreadsheets. The examination will last exactly two hours; feel free to ask questions during the examination if there is any confusion. Write all your answers in the space provided within this exam booklet. Please show all work — partial credit will be given for all questions except the multiple-choice ones.

Final Examination Good luck! Last Name:

_____________________

First Name:

_____________________

Section:

_____________________

Question ……. 3

Part

a b c d e f g total

points

out of

6 6 4 6 6 6 6 40

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 2 of 5

3 GlobTech Expansion Options (40 points) Julie Dewent, the CFO of GlobTech, is considering building a new plant, which will start generating revenues in one year. The stream of revenues is uncertain, and GlobTech believes that the mean of its value one year from now will be $108,329,000. At a discount rate of r = 8%, this is equivalent to $100,000,000 = e-0.08 $108,329,000 today. During the coming year, GlobTech’s revenue expectations may change. More specifically, GlobTech believes that one year from now, the expected value V (in $millions) of the revenues is lognormally distributed with σ = 60% volatility, V = 100 e(r – σ^2/2)+ σ *Z = 100 e(0.08 – 0.6^2/2)+0.6*Z where Z is a standard normal random variable with mean value of 0 and standard deviation of 1. GlobTech is considering the following options:

Base Case Build the plant now If the plant is built now, its cost will be a fixed $104 million. Because the expected present value of the revenue today is only $100 million, simply building the plant now does not appear to be worthwhile. That is, Expected NPV(Build Now) = -104 + e-0.08 108.329 = -4.

Option 1: Postponement Option GlobTech waits one year, at which time V is known, before making the decision of whether or not to invest in this project. The delay will decrease the project revenues by 25% and construction costs will grow at a rate of 8%. In this case, NPV(Postponement Option) = e-0.08 MAX[0.75V – e0.08(104), 0].

Option 2: Expansion Option GlobTech invests $104 now and has the option to spend more in one year. By that time, the company will know the value of V, and it can choose to spend $50 million at that time on a plant expansion that will increase the project revenues by 50%. Alternatively, GlobTech can just accept V. In this case, NPV(Expansion Option) = e-0.08 MAX[(1.5V – 50), V] –104.

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 3 of 5

Note: For all of the questions that follow, simply check the correct answer. No partial credit will be given. GlobTech has simulated the performance of Options 1 and 2. The results of a 900 trial Monte Carlo simulation are summarized in the tables below: Statistics Trials Mean Median Mode Standard Deviation Variance Skewness Kurtosis Coeff. of Variability Range Minimum Range Maximum Range Width Mean Std. Error Percentiles 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

NPV (Postponement) 900 9.71 0.00 0.00 33.11 1,096.00 5.85 48.61 3.41 0.00 413.44 413.44 1.10 NPV (Postponement) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.98 26.27 413.44

NPV (Expansion) 900 11.07 -19.91 --95.64 9147.60 3.02 18.21 8.64 -91.69 884.72 976.41 3.19 NPV (Expansion) -91.69 -64.25 -54.15 -42.87 -33.15 -19.91 1.44 28.07 59.80 110.38 884.72

a) (6 points) If the option to postpone the project by one year requires an additional payment of $1 million now, what is the approximate probability that the NPV of the project will be positive? ( ) 100% ( ) 80% ( ) 50% ( ) 20% ( )

0%

( ) It cannot be determined from the simulation

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 4 of 5

b) (6 points) What would be the approximate width of the 95% confidence interval for the true expected NPV of the postponement option if the sample size were 1800 instead of 900? ( ) $8.8 million ( ) $4.4 million ( ) $3.1 million ( ) $2.2 million c) (4 points) Which of the following statements is true? ( ) The true expected NPV of the postponement option is higher than the true expected NPV of the expansion option. ( ) The true expected NPV of the expansion option is higher than the true expected NPV of the postponement option. ( ) It cannot be determined from the simulation results which of the true expected NPVs is higher. d) (6 points) Suppose the annual volatility of the project revenues increases to 80%. Without running a simulation, which of the following statements would you expect to be true? ( ) The expected NPV of the postponement option will be higher when the volatility increases while the expected NPV of the expansion option will stay the same. ( ) The expected NPV of the postponement option will stay the same when the volatility increases while the expected NPV of the expansion option will be higher. ( ) The expected NPV of both options will be lower when the volatility increases. ( ) The expected NPV of both options will stay the same when the volatility increases. ( ) The expected NPV of both options will be higher when the volatility increases. e) (6 points) Looking at the distribution of percentiles, the observed chance that the NPV of Expansion will be between -$54,150,000 and $59,800,000 is about 60%. Suppose the number of trials were increased to 3600. What is the best estimate of the chance that the NPV of Expansion will be between -$54,150,000 and $59,800,000 in these new results? ( ) about 90% ( ) about 80% ( ) about 60% ( ) about 30% ( ) about 15%

Decision Models & Optimization Term II, 2007

Sample Final Examination Page 5 of 5

Contraction Option Instead of paying the entire $104 million plant cost now, GlobTech can pay ONLY $50 million now. Then, one year from now, when V is known, the company can choose to pursue either of two options. •

Pay the remaining $54 million cost, plus interest, and obtain the full project value. Here, total costs and revenues will be the same as they would have been in the base case.

•

Contract the scale of the project by paying only $25 million next year. Here, the present value of the revenues will be worth only 50% of what they would have originally been worth.

f)

(6 points) Which of the following expressions will evaluate correctly the NPV of the contraction option? ( ) e-0.08 MAX[(V – e0.08104), 0.5V - 25] ( ) e-0.08 MAX[(V – e0.0854), 0.5V - 25] – 50 ( ) e-0.08 MAX[(0.5V – 25), 0] - 50 ( ) e-0.08 MAX[(0.5V – 25), V] - 54

The results of a 100-trial Monte Carlo simulation of the contraction option are summarized in the following table: Statistics Trials Mean Median Mode Standard Deviation Variance Range Minimum Range Maximum Range Width Mean Std. Error

NPV (Contraction) 100 -6.86 -27.13 --63.89 4,081.62 -62.72 301.24 363.96 6.39

g) (6 points) How does the true expected NPV of the contraction option compare with that of the base case, in which GlobTech simply invests $104 million today to build the plant? ( ) The true expected NPV of the contraction option is definitely no lower than that of the base case. ( ) The true expected NPV of the contraction option is definitely no higher than that of the base case. ( ) Given the simulation results, there is a significant probability that the true expected NPV of the contraction option is either higher or lower than that of the base case. To precisely determine which, one must run the simulation with a much larger number of trials. ( ) It cannot be determined whether or not the true expected NPV of the contraction option is higher or lower than that of the base case.

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