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Home » Database Storage & Design

#### Database Storage & Design

**Q:**Until uncertainty about the growth rate is resolved, the volatility of the stock price could well be much higher than 18% - perhaps as high at 25%. What would the value be in that case?

**Q:**The executive thinks the growth rate could shrink to 7% per year if the company has growing pains, but on the other hand it could be as high as 15% per year if the company prospers. What is the expected value of the stock options in those cases?

**Q:**NARRBEGIN: SA_105_108 An executive has been offered a compensation package that includes stock options. The current stock price is $30/share, and she has been offered a call option on 2000 shares, which can be exercised five years from now at a price of $42/share. Therefore, if the market price of the shares in five years is more than $42/share, she can buy 2000 shares at $42/share, and then immediately sell the shares at the market price, earning a riskless profit. If the market price of the shares was less than $42/share, she will obviously choose not to exercise the option, and would have zero profit. Assume the price of the stock can be modeled as exponential growth (compounding), which could be calculated as: where, stock price in next period (i.e., price next year) current stock price annual growth rate of the stock price, which has been 10% annual volatility, which is estimated to be 18% normal random variable with mean of zero and standard deviation of 1 NARREND Simulate the price of the stock in five years by calculating five annual increments (steps) with this model, starting from the current price of $30/share. For each price simulated five years from now, model the exercise decision and calculate the resulting profit, which should then be discounted for five years at the current discount rate (5%) to obtain the present value of the options. What is the expected value of the stock options?

**Q:**NARRBEGIN: SA_103_104 Assume you have $1000, all of which is invested in a basketball team. Each year there is a 60% chance that the value of the team will increase by 60% and a 40% chance that the value of the team will decrease by 60%. NARREND (A) Estimate the mean and median value of your investment after 100 years. (B) Explain the large difference between the estimated mean and median.

**Q:**NARRBEGIN: SA_99_102 We are trying to determine the proper capacity level for a new electric car. A unit of capacity gives us the potential to produce one car per year. It costs $10,000 to build a unit of capacity and the cost is charged equally over the next 5 years. It also costs $400 per year to maintain a unit of capacity (whether or not it is used). Each car sells for $14,000 and incurs a variable production cost of $10,000. The annual demand for the electric car during each of the next 5 years is believed to be normally distributed with mean 500,000 and standard deviation 100,000. The demands during different years are assumed to be independent. Profits are discounted at a 10% annual interest rate. We are working with a 5-year planning horizon. Capacity levels of 300,000, 400,000, 500,000, 600,000, and 700,000 are under consideration. (Assume that no more than the demand is ever produced, so that no ending inventory ever occurs.) NARREND (A) Assuming we are risk neutral, use simulation to find the optimal capacity level. (B) Using the answer to (A), there a 5% chance that the actual discounted profit will exceed what value? (C) Using the answer to (A), there is a 5% chance that the actual discounted profit will be less than what value? (D) If we are risk averse, how might the optimal capacity level change?

**Q:**Now assume that the project has an abandonment option. At the end of each year you can abandon the project for the values given below: For example, suppose that year 1 cash flow is $400. Then at the end of year 1, you expect cash flow for each remaining year to be $400. This has an NPV of less than $6200, so you should abandon the project and collect $6200 at the end of year 1. Estimate the mean and standard deviation of the project with the abandonment option. How much would you pay for the abandonment option? (Hint: You can abandon a project at most once. Thus in year 5, for example, you abandon only if the sum of future expected NPVs is less than the year 5 abandonment value and the project has not yet been abandoned. Also, once you abandon the project, the actual cash flows for future years will 0. So the future cash flows after abandonment should disappear.)

**Q:**NARRBEGIN: SA_97_98You are considering a 10-year investment project. At present, the expected cash flow each year is $1000. Suppose, however, that each year's cash flow is normally distributed with mean equal to last year's actual cash flow and standard deviation $100. For example, suppose that the actual cash flow in year 1 is $1300. Then year 2 cash flow is normal with mean $1300 and standard deviation $100. Also, at the end of year 1, your best guess is that each later year's expected cash flow will be $1300.NARRENDEstimate the mean and standard deviation of the NPV of this project. Assume that cash flows are discounted at a rate of 10% per year.

**Q:**NARRBEGIN: SA_95_96 ABC sells refrigerators. Any refrigerator that fails before it is 3 years old is replaced for free. Of all refrigerators, 2% fail during their first year of operation; 4% of all 1-year-old refrigerators fail during their second year of operation; and 8% of all 2-year-old refrigerators fail during their third year of operation. It costs ABC $500 to replace a refrigerator, and ABC sells 6,000 refrigerators per year. NARREND If the warranty period were reduced to 2 years, how much per year in replacement costs would be saved?

**Q:**NARRBEGIN: SA_95_96 ABC sells refrigerators. Any refrigerator that fails before it is 3 years old is replaced for free. Of all refrigerators, 2% fail during their first year of operation; 4% of all 1-year-old refrigerators fail during their second year of operation; and 8% of all 2-year-old refrigerators fail during their third year of operation. It costs ABC $500 to replace a refrigerator, and ABC sells 6,000 refrigerators per year. NARREND Estimate the fraction of all refrigerators that will have to be replaced.

**Q:**NARRBEGIN: SA_92_93 Suppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are as shown below. The correlations between the annual returns on the four stocks are also shown below. NARREND Consider a device that requires two batteries to function. If either of these batteries dies, the device will not work. Currently there are two brand new batteries in the device, and there are three extra brand new batteries. Each battery, once it is placed in the device, lasts a random amount of time that is triangularly distributed with parameters 15, 18, and 25 (all expressed in hours). When any of the batteries in the device dies, it is immediately replaced by an extra (if an extra is still available). Use @RISK to simulate the time the device can last with the batteries currently available.

**Q:**NARRBEGIN: SA_92_93 Suppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are as shown below. The correlations between the annual returns on the four stocks are also shown below. NARREND What is the probability that your portfolio will lose money during the course of a year?

**Q:**NARRBEGIN: SA_92_93 Suppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are as shown below. The correlations between the annual returns on the four stocks are also shown below. NARREND What is the probability that your portfolio's annual return will exceed 20%?

**Q:**NARRBEGIN: SA_86_91 In this example we are estimating the net present value of introducing a new drug to market. We have the following information about the market: The market size is 1,000,000 and is projected to grow at an average 5%, with a standard deviation of 1%, over the next ten years. The market share captured at entry is projected to be between 20% and 70%, with most likely value 40%. Three competitors may enter the market in the future, with each one having a 40% probability of entry per year. For each new competitor per year, the market share goes down by 20%. The marginal profit per unit is $1.80. We want to evaluate the project over ten years, using a discount rate of 10%. NARREND Suppose this new drug will cost $3 million to develop. What is the chance that we could loose money on this project?

**Q:**NARRBEGIN: SA_86_91 In this example we are estimating the net present value of introducing a new drug to market. We have the following information about the market: The market size is 1,000,000 and is projected to grow at an average 5%, with a standard deviation of 1%, over the next ten years. The market share captured at entry is projected to be between 20% and 70%, with most likely value 40%. Three competitors may enter the market in the future, with each one having a 40% probability of entry per year. For each new competitor per year, the market share goes down by 20%. The marginal profit per unit is $1.80. We want to evaluate the project over ten years, using a discount rate of 10%. NARREND Perform a simulation with this model. What is the expected NPV? What is the standard deviation?

**Q:**NARRBEGIN: SA_86_91 In this example we are estimating the net present value of introducing a new drug to market. We have the following information about the market: The market size is 1,000,000 and is projected to grow at an average 5%, with a standard deviation of 1%, over the next ten years. The market share captured at entry is projected to be between 20% and 70%, with most likely value 40%. Three competitors may enter the market in the future, with each one having a 40% probability of entry per year. For each new competitor per year, the market share goes down by 20%. The marginal profit per unit is $1.80. We want to evaluate the project over ten years, using a discount rate of 10%. NARREND What is the appropriate distribution for the probability of competitor entry?

**Q:**NARRBEGIN: SA_86_91 In this example we are estimating the net present value of introducing a new drug to market. We have the following information about the market: The market size is 1,000,000 and is projected to grow at an average 5%, with a standard deviation of 1%, over the next ten years. The market share captured at entry is projected to be between 20% and 70%, with most likely value 40%. Three competitors may enter the market in the future, with each one having a 40% probability of entry per year. For each new competitor per year, the market share goes down by 20%. The marginal profit per unit is $1.80. We want to evaluate the project over ten years, using a discount rate of 10%. NARREND What is the appropriate distribution for initial market size?

**Q:**NARRBEGIN: SA_86_91 In this example we are estimating the net present value of introducing a new drug to market. We have the following information about the market: The market size is 1,000,000 and is projected to grow at an average 5%, with a standard deviation of 1%, over the next ten years. The market share captured at entry is projected to be between 20% and 70%, with most likely value 40%. Three competitors may enter the market in the future, with each one having a 40% probability of entry per year. For each new competitor per year, the market share goes down by 20%. The marginal profit per unit is $1.80. We want to evaluate the project over ten years, using a discount rate of 10%. NARREND What is the appropriate distribution for the market growth rate?

**Q:**NARRBEGIN: SA_86_91 In this example we are estimating the net present value of introducing a new drug to market. We have the following information about the market: The market size is 1,000,000 and is projected to grow at an average 5%, with a standard deviation of 1%, over the next ten years. The market share captured at entry is projected to be between 20% and 70%, with most likely value 40%. Three competitors may enter the market in the future, with each one having a 40% probability of entry per year. For each new competitor per year, the market share goes down by 20%. The marginal profit per unit is $1.80. We want to evaluate the project over ten years, using a discount rate of 10%. NARREND Develop an @Risk model to estimate the NPV given an assumed capacity. What are the variable inputs and outputs?

**Q:**Briefly explain why designing the plant for the expected capacity is clearly not the optimal solution.

**Q:**Considering your answers for Questions 78 through 83, please state how many units of capacity you think the plant should be built for and explain why.

**Q:**For each simulation what is the probability of exceeding $75,000 in NPV (approximate these numbers as closely as possible from the data given in the above table). Please put your answer in the following table:

**Q:**NARRBEGIN: SA_78_85 Suppose we want to choose capacity for a plant that will produce a new drug. In particular, we want to choose the capacity that maximizes discounted expected profit over the next 10 years. We have the following information: Demand for the drug is expected to be normally distributed ~ Normal (50,000, 12,000). A unit of capacity costs $16 to build. The number of units produced will equal the demand, up to capacity limits. The revenue per unit is $3.70 and the cost per unit is $0.20 (variable cost). The maintenance cost per unit of capacity is $0.40 (fixed cost). The discount rate is 10%. NARREND Are there any simulations which indicated there was a chance of getting negative NPV? Briefly explain in one sentence.

**Q:**NARRBEGIN: SA_78_85 Suppose we want to choose capacity for a plant that will produce a new drug. In particular, we want to choose the capacity that maximizes discounted expected profit over the next 10 years. We have the following information: Demand for the drug is expected to be normally distributed ~ Normal (50,000, 12,000). A unit of capacity costs $16 to build. The number of units produced will equal the demand, up to capacity limits. The revenue per unit is $3.70 and the cost per unit is $0.20 (variable cost). The maintenance cost per unit of capacity is $0.40 (fixed cost). The discount rate is 10%. NARREND Which simulation has the most risk as measured by spread or dispersion in the data? Please state clearly what statistic you used to answer this question.

**Q:**NARRBEGIN: SA_78_85 Suppose we want to choose capacity for a plant that will produce a new drug. In particular, we want to choose the capacity that maximizes discounted expected profit over the next 10 years. We have the following information: Demand for the drug is expected to be normally distributed ~ Normal (50,000, 12,000). A unit of capacity costs $16 to build. The number of units produced will equal the demand, up to capacity limits. The revenue per unit is $3.70 and the cost per unit is $0.20 (variable cost). The maintenance cost per unit of capacity is $0.40 (fixed cost). The discount rate is 10%. NARREND Which simulation yields the largest median NPV?

**Q:**NARRBEGIN: SA_78_85 Suppose we want to choose capacity for a plant that will produce a new drug. In particular, we want to choose the capacity that maximizes discounted expected profit over the next 10 years. We have the following information: Demand for the drug is expected to be normally distributed ~ Normal (50,000, 12,000). A unit of capacity costs $16 to build. The number of units produced will equal the demand, up to capacity limits. The revenue per unit is $3.70 and the cost per unit is $0.20 (variable cost). The maintenance cost per unit of capacity is $0.40 (fixed cost). The discount rate is 10%. NARREND Use RISKSIMTABLE with a range of possible values to help the firm decide what the plant capacity should be.

**Q:**NARRBEGIN: SA_78_85 Suppose we want to choose capacity for a plant that will produce a new drug. In particular, we want to choose the capacity that maximizes discounted expected profit over the next 10 years. We have the following information: Demand for the drug is expected to be normally distributed ~ Normal (50,000, 12,000). A unit of capacity costs $16 to build. The number of units produced will equal the demand, up to capacity limits. The revenue per unit is $3.70 and the cost per unit is $0.20 (variable cost). The maintenance cost per unit of capacity is $0.40 (fixed cost). The discount rate is 10%. NARREND Perform a simulation assuming the plant will be designed to meet the expected demand. What is the NPV in that case?

**Q:**NARRBEGIN: SA_74_77 The "winner's curse" refers to a situation where there are several bidders on the same item. Each participant can make his or her independent estimate for the value of the item. When all participants are equally informed their estimates will be unbiased, but, given the difficulty of estimating the value, the estimates may vary widely. Even though the mean of the estimates may equal the expected value, the winner's bid will likely be more than the value of the item. Consider a case where 3 companies are trying to decide how much to bid for a commercial real estate tract. Assume that each bidder independently estimates the value of the tract. This estimated value is a random variable that for each bidder is drawn from a normal distribution with a mean of $1,000,000 and a standard deviation of $200,000. The actual value is also drawn from the same distribution. NARREND What is the probability of winning for the conservative bidder?

**Q:**NARRBEGIN: SA_74_77 The "winner's curse" refers to a situation where there are several bidders on the same item. Each participant can make his or her independent estimate for the value of the item. When all participants are equally informed their estimates will be unbiased, but, given the difficulty of estimating the value, the estimates may vary widely. Even though the mean of the estimates may equal the expected value, the winner's bid will likely be more than the value of the item. Consider a case where 3 companies are trying to decide how much to bid for a commercial real estate tract. Assume that each bidder independently estimates the value of the tract. This estimated value is a random variable that for each bidder is drawn from a normal distribution with a mean of $1,000,000 and a standard deviation of $200,000. The actual value is also drawn from the same distribution. NARREND Next, assume that one of the bidders bids 20% below his or her estimated value, while the other two bidders follow the same strategy as in Question 74. Using 1000 iterations report the expected profit or loss to the conservative bidder.

**Q:**NARRBEGIN: SA_74_77 The "winner's curse" refers to a situation where there are several bidders on the same item. Each participant can make his or her independent estimate for the value of the item. When all participants are equally informed their estimates will be unbiased, but, given the difficulty of estimating the value, the estimates may vary widely. Even though the mean of the estimates may equal the expected value, the winner's bid will likely be more than the value of the item. Consider a case where 3 companies are trying to decide how much to bid for a commercial real estate tract. Assume that each bidder independently estimates the value of the tract. This estimated value is a random variable that for each bidder is drawn from a normal distribution with a mean of $1,000,000 and a standard deviation of $200,000. The actual value is also drawn from the same distribution. NARREND What is the probability of winning for each bidder in the above scenario?

**Q:**NARRBEGIN: SA_74_77 The "winner's curse" refers to a situation where there are several bidders on the same item. Each participant can make his or her independent estimate for the value of the item. When all participants are equally informed their estimates will be unbiased, but, given the difficulty of estimating the value, the estimates may vary widely. Even though the mean of the estimates may equal the expected value, the winner's bid will likely be more than the value of the item. Consider a case where 3 companies are trying to decide how much to bid for a commercial real estate tract. Assume that each bidder independently estimates the value of the tract. This estimated value is a random variable that for each bidder is drawn from a normal distribution with a mean of $1,000,000 and a standard deviation of $200,000. The actual value is also drawn from the same distribution. NARREND Suppose first that all three bidders are aware of the winner's curse so they have decided (independently) to bid 10% below their estimated values. Using 1000 iterations report the expected profit (or loss) to the winner.

**Q:**NARRBEGIN: SA_68_73 Suppose that GM earns a $4000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GM's cars. The customer is assumed to buy a new car every five years, for a total of 10 cars through her lifetime. The customer will keep buying GM cars so long as they are satisfied with them. The probability that the customer will be satisfied with her GM car is 80%. If she is not satisfied with her GM car, she will buy another brand (we"ll call all other brands cumulatively "Toyota"). The probability that she is satisfied with "Toyota" is 85%. NARREND Does the answer to Question 72 match your intuition? Explain why or why not.

**Q:**NARRBEGIN: SA_68_73 Suppose that GM earns a $4000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GM's cars. The customer is assumed to buy a new car every five years, for a total of 10 cars through her lifetime. The customer will keep buying GM cars so long as they are satisfied with them. The probability that the customer will be satisfied with her GM car is 80%. If she is not satisfied with her GM car, she will buy another brand (we"ll call all other brands cumulatively "Toyota"). The probability that she is satisfied with "Toyota" is 85%. NARREND Suppose that a customer satisfaction firm approaches GM with a proposal to increase satisfaction from the current 80% rate to $85% through a low cost maintenance program that will cost GM $300 per customer. Would the program be worth it?

**Q:**NARRBEGIN: SA_68_73 Suppose that GM earns a $4000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GM's cars. The customer is assumed to buy a new car every five years, for a total of 10 cars through her lifetime. The customer will keep buying GM cars so long as they are satisfied with them. The probability that the customer will be satisfied with her GM car is 80%. If she is not satisfied with her GM car, she will buy another brand (we"ll call all other brands cumulatively "Toyota"). The probability that she is satisfied with "Toyota" is 85%. NARREND Using your answers to Questions 69 and 70, and without simulating the model again, estimate how much an extra 5% customer satisfaction is worth to GM.

**Q:**NARRBEGIN: SA_68_73 Suppose that GM earns a $4000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GM's cars. The customer is assumed to buy a new car every five years, for a total of 10 cars through her lifetime. The customer will keep buying GM cars so long as they are satisfied with them. The probability that the customer will be satisfied with her GM car is 80%. If she is not satisfied with her GM car, she will buy another brand (we"ll call all other brands cumulatively "Toyota"). The probability that she is satisfied with "Toyota" is 85%. NARREND What if the GM satisfaction rate is raised further to 90%. What would the customer NPV be in that case?

**Q:**NARRBEGIN: SA_68_73 Suppose that GM earns a $4000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GM's cars. The customer is assumed to buy a new car every five years, for a total of 10 cars through her lifetime. The customer will keep buying GM cars so long as they are satisfied with them. The probability that the customer will be satisfied with her GM car is 80%. If she is not satisfied with her GM car, she will buy another brand (we"ll call all other brands cumulatively "Toyota"). The probability that she is satisfied with "Toyota" is 85%. NARREND Suppose GM could raise it customer satisfaction to 85%, to match Toyota's. What would the customer NPV be in that case?

**Q:**NARRBEGIN: SA_68_73 Suppose that GM earns a $4000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GM's cars. The customer is assumed to buy a new car every five years, for a total of 10 cars through her lifetime. The customer will keep buying GM cars so long as they are satisfied with them. The probability that the customer will be satisfied with her GM car is 80%. If she is not satisfied with her GM car, she will buy another brand (we"ll call all other brands cumulatively "Toyota"). The probability that she is satisfied with "Toyota" is 85%. NARREND Consider a customer whose first car is GM. If profits are discounted at 10% annually, use simulation to estimate the value of this customer to GM over the customer's lifetime.

**Q:**NARRBEGIN: SA_66_67 Suppose that a recent study shows that each week each of 300 families buys a gallon of apple juice from company A, B, or C. Let denote the probability that a gallon produced by company A is of unsatisfactory quality, and define and similarly for companies B and C. If the last gallon of juice purchased by a family is satisfactory, the next week they will purchase a gallon of juice from the same company. If the last gallon of juice purchased by a family is not satisfactory, then the family will purchase a gallon from a competitor. Consider one week in which A families have purchased juice A, B families have purchased juice B, and C families have purchased juice C. Assume that families that switch brands during a period are allocated to the remaining brands in a manner that is proportional to the current market shares of the other brands. Thus, if a customer switches from brand A, there is probability B/(B + C) that he will switch to brand B and probability C/(B + C) that he will switch to brand C. Suppose that the market is currently divided equally: 100 families for each of the three brands. NARREND Suppose a 1% increase in market share is worth $10,000 per week to company A. Company A believes that for a cost of $1 million per year it can cut the percentage of unsatisfactory juice cartons in half. Is this worthwhile? (Use the same values of , , and as in Question 66.

**Q:**NARRBEGIN: SA_66_67 Suppose that a recent study shows that each week each of 300 families buys a gallon of apple juice from company A, B, or C. Let denote the probability that a gallon produced by company A is of unsatisfactory quality, and define and similarly for companies B and C. If the last gallon of juice purchased by a family is satisfactory, the next week they will purchase a gallon of juice from the same company. If the last gallon of juice purchased by a family is not satisfactory, then the family will purchase a gallon from a competitor. Consider one week in which A families have purchased juice A, B families have purchased juice B, and C families have purchased juice C. Assume that families that switch brands during a period are allocated to the remaining brands in a manner that is proportional to the current market shares of the other brands. Thus, if a customer switches from brand A, there is probability B/(B + C) that he will switch to brand B and probability C/(B + C) that he will switch to brand C. Suppose that the market is currently divided equally: 100 families for each of the three brands. NARREND After a year, what will the market share for each of the three companies be? Assume = 0.10, = 0.15, and = 0.20. (Hint: Use the RISKBINOMIAL function to model how many people switch from A, then how many switch from Ato B and from A to C.)

**Q:**NARRBEGIN: SA_57_64Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%.NARRENDSuppose that Coke and Pepsi are fighting for the cola market. Each week each person in the market buys one case of Coke or Pepsi. If the person's last purchase was Coke, there is a 0.80 probability that this person's next purchase will be Coke; otherwise, it will be Pepsi. (We are considering only two brands in the market.) Similarly, if the person's last purchase was Pepsi, there is a 0.90 probability that this person's next purchase will be Pepsi; otherwise, it will be Coke. Currently half of all people purchase Coke, and the other half purchase Pepsi. Simulate one year of sales in the cola market and estimate each company's average weekly market share. Do this by assuming that the total market size is fixed at 100 customers. (Hint: Use the RISKBINOMIAL function.)

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND Suppose again that Amanda will stop investing in the stock market and transfer all of her retirement into a savings account if and when she reaches $500,000. When can she expect to reach this goal under the more conservative investing strategy?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND What is the probability that Amanda will have less than $500,000 in her retirement account after 30 years under the more conservative investing strategy?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND What is the standard deviation of the ending balance? What does the distribution look like now? What should Amanda infer from this?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND Next, suppose Amanda's broker thinks the stock market may be too risky and has advised her to diversity by investing some of her money in money market funds and bonds. He estimates that this will lower her expected annual return to 10% per year, but will also lower the standard deviation to 10%. What can she expect to have in her account after thirty years under this investing strategy?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND Suppose Amanda will stop investing in the stock market and transfer all of her retirement into a savings account if and when she reaches $500,000. When can she expect to reach this goal?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND What is the probability that Amanda will have less than $500,000 in her retirement account after 30 years?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND What is the standard deviation of the ending balance? What does the distribution look like? What should Amanda infer from this?

**Q:**NARRBEGIN: SA_57_64 Amanda is a recent college graduate, and has just started her first job. She would like to know if she saves $5,000 per year out of her salary over the next 30 years what the distribution of the value of her retirement fund after 30 years. She has decided that she will invest all her money in the stock market that she estimates has a return that is normally distributed with mean 12% per year and standard deviation 25%. NARREND Simulate Amanda's portfolio over the next 30 years and determine how much she can expect to have in her account at the end of that period. At the beginning of each year, compute the beginning balance in Amanda's account. Note that this balance is either 0 (for year 1) or equal to the ending balance of the previous year. The contribution of $5,000 is then added to calculate the new balance. The market return for each year is given by a normal random variable with the parameters above (assume the market returns in each year are independent of the other years). The ending balance for the each year is then equal to the beginning balance, augmented by the contribution, and multiplied by (1+Market return).

**Q:**NARRBEGIN: SA_51_56 A firm is considering investing $0.9M in a typical industrial manufacturing application with a three year production planning cycle under a forecasted market price environment. A simple three-period project pro forma cash flow sheet for this project is shown below: In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows are then be discounted at a rate that is commensurate with the riskiness of the project (here, assumed to be 10%). NARREND Given your answers to Questions 51 through 55, would you invest in this project?

**Q:**NARRBEGIN: SA_51_56 A firm is considering investing $0.9M in a typical industrial manufacturing application with a three year production planning cycle under a forecasted market price environment. A simple three-period project pro forma cash flow sheet for this project is shown below: In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows are then be discounted at a rate that is commensurate with the riskiness of the project (here, assumed to be 10%). NARREND What are the chances the firm could loose money on this project, given the price uncertainty?

**Q:**NARRBEGIN: SA_51_56 A firm is considering investing $0.9M in a typical industrial manufacturing application with a three year production planning cycle under a forecasted market price environment. A simple three-period project pro forma cash flow sheet for this project is shown below: In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows are then be discounted at a rate that is commensurate with the riskiness of the project (here, assumed to be 10%). NARREND What does the distribution of the NPV look like?

**Q:**NARRBEGIN: SA_51_56 A firm is considering investing $0.9M in a typical industrial manufacturing application with a three year production planning cycle under a forecasted market price environment. A simple three-period project pro forma cash flow sheet for this project is shown below: In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows are then be discounted at a rate that is commensurate with the riskiness of the project (here, assumed to be 10%). NARREND What is the standard deviation of the NPV? What does it indicate?

**Q:**NARRBEGIN: SA_51_56 A firm is considering investing $0.9M in a typical industrial manufacturing application with a three year production planning cycle under a forecasted market price environment. A simple three-period project pro forma cash flow sheet for this project is shown below: In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows are then be discounted at a rate that is commensurate with the riskiness of the project (here, assumed to be 10%). NARREND Suppose that the forecasted price levels shown in the pro forma cash flow sheet are not deterministic, but rather are expected to fluctuate due to market forces. The prices are expected to be normally distributed in each year, with the means equal to the expected values shown in the pro forma, but with standard deviations of $5.2, $5.3, and $5.5 in years 1, 2, and 3, respectively. Enter this pro forma in an Excel worksheet, with the appropriate @RISK functions for the random prices, and simulate 1,000 iterations. What is the expected NPV now? Would you recommend investing in this project? Explain.

**Q:**NARRBEGIN: SA_51_56 A firm is considering investing $0.9M in a typical industrial manufacturing application with a three year production planning cycle under a forecasted market price environment. A simple three-period project pro forma cash flow sheet for this project is shown below: In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows are then be discounted at a rate that is commensurate with the riskiness of the project (here, assumed to be 10%). NARREND The Net Present Value (NPV) is the sum of the discounted cash flows. What is the NPV of the project, including the required investment?

**Q:**The @RISK function RISKDUNIFORM in the form = RISKDUNIFORM ({List}) generates a random member of a given list, so that each member of the list has the same chance of being chosen.

**Q:**A @RISK output range allows us to obtain a summary chart that shows the entire simulated range at once.

**Q:**In marketing and sales models, the primary issue is the uncertain amount of sales that can be obtained, given an assumed timing.

**Q:**A marketing simulation model can be used to determine the expected profit under uncertain customer loyalty, and then we can use an optimization model to determine the optimal amount to spend on increasing customer loyalty.

**Q:**In marketing models of customer loyalty, we are typically interested in modeling the rate of customer retention, called churn.

**Q:**In investment models, we typically must simulate the random investment weights

**Q:**A key objective in cash flow models is often to determine the amount of debt that must be taken out to maintain a minimum cash balance.

**Q:**A tornado chart lets us see which random input has the most effect on a specified output in a financial model.

**Q:**In financial simulation models, the value at risk (VAR) is the 5th percentile of an output distribution, and it indicates nearly the worst possible outcome.

**Q:**In financial simulation models, we are typically more interested in the expected NPV of a project than in the extremes of the outcomes.

**Q:**Churn is an example of the type of uncertain variable we deal with in financial models.

**Q:**RISKTARGET is a function that allows us to determine the cumulative probability of a particular value in an output distribution, such as the probability of meeting a due date in manufacturing.

**Q:**Using @RISK summary functions such as RISKMEAN, RISKPERCENTILE, and others allows us to capture simulation results in the same worksheet as the simulation model.

**Q:**RISKMAX and RISKMIN are can be used to find the probability of meeting a given due date in a manufacturing model.

**Q:**Uncertain timing and the events that follow in process modeling can be modeled using IF statements.

**Q:**In a manufacturing setting, a discrete distribution is natural for modeling the number of days to produce a batch, and a continuous distribution is appropriate for modeling the yield from a batch.

**Q:**We can use the RISKSIMTABLE function to summarize the results of a single simulation of product lifetime.

**Q:**A common distribution for modeling product lifetimes is the binomial distribution

**Q:**We can use Excel's RAND function inside an IF function to simulate whether some event occurs or does not occur.

**Q:**In warranty cost models, the key input random variable is product lifetime.

**Q:**In a bidding model, once we have the bidding strategy that maximizes the expected profit, we no longer should consider the bidders risk aversion.

**Q:**Although we can determine the optimal bid and the expected profit from that bid in a bidding simulation, we usually cannot determine the probability of winning.

**Q:**In bidding models, the simulation input variable is the number of competitors who will bid.

**Q:**The primary objective in simulation models of bidding for contracts is to determine the optimal bid.

**Q:**Simulation applications involving games of chance are primarily for learning the background of simulation (e.g., modeling gambling casinos of Monte Carlo), since they are not business applications per se.

**Q:**In a marketing and sales model, which of the following might be a good choice for a discrete distribution to model the random timing of sales? a. RAND() b. Normal distribution c. Binomial distribution d. Exponential distribution e. Poisson distribution

**Q:**The main issue in marketing and sales models is: a. the amount invested in marketing b. the timing of marketing c. the profit from sales d. the timing of sales e. the tradeoff between marketing and sales

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