# Question

Play Things is developing a new Hannah Montana doll. The company has made the following assumptions:

• The doll will sell for a random number of years from 1 to 10. Each of these 10 possibilities is equally likely.

• At the beginning of year 1, the potential market for the doll is one million. The potential market grows by an average of 5% per year. The company is 95% sure that the growth in the potential market during any year will be between 3% and 7%. It uses a normal distribution to model this.

• The company believes its share of the potential market during year 1 will be at worst 20%, most likely 40%, and at best 50%. It uses a triangular distribution to model this.

• The variable cost of producing a doll during year 1 has a triangular distribution with parameters $8, $10, and $12.

• The current selling price is $20.

• Each year, the variable cost of producing the doll will increase by an amount that is triangularly distributed with parameters 4.5%, 5%, and 6.5%. You can assume that once this change is generated, it will be the same for each year. You can also assume that the company will change its selling price by the same percentage each year.

• The fixed cost of developing the doll (which is incurred right away, at time 0) has a triangular distribution with parameters $4, $6, and $12 million.

• Right now there is one competitor in the market. During each year that begins with four or fewer competitors, there is a 20% chance that a new competitor will enter the market.

• Year t sales (for t > 1) are determined as follows.

Suppose that at the end of year t – 1, n competitors are present (including Play Things). Then during year t, a fraction 0.9 – 0.1n of the company’s loyal customers (last year’s purchasers) will buy a doll from Play Things this year, and a fraction 0.2 – 0.04n of customers currently in the market who did not purchase a doll last year will purchase a doll from Play Things this year. Adding these two provides the mean sales for this year. Then the actual sales this year is normally distributed with this mean and standard deviation equal to 7.5% of the mean.

a. Use @RISK to estimate the expected NPV of this project.

b. Use the percentiles in @RISK’s output to find an interval such that you are 95% certain that the company’s actual NPV will be within this interval.

• The doll will sell for a random number of years from 1 to 10. Each of these 10 possibilities is equally likely.

• At the beginning of year 1, the potential market for the doll is one million. The potential market grows by an average of 5% per year. The company is 95% sure that the growth in the potential market during any year will be between 3% and 7%. It uses a normal distribution to model this.

• The company believes its share of the potential market during year 1 will be at worst 20%, most likely 40%, and at best 50%. It uses a triangular distribution to model this.

• The variable cost of producing a doll during year 1 has a triangular distribution with parameters $8, $10, and $12.

• The current selling price is $20.

• Each year, the variable cost of producing the doll will increase by an amount that is triangularly distributed with parameters 4.5%, 5%, and 6.5%. You can assume that once this change is generated, it will be the same for each year. You can also assume that the company will change its selling price by the same percentage each year.

• The fixed cost of developing the doll (which is incurred right away, at time 0) has a triangular distribution with parameters $4, $6, and $12 million.

• Right now there is one competitor in the market. During each year that begins with four or fewer competitors, there is a 20% chance that a new competitor will enter the market.

• Year t sales (for t > 1) are determined as follows.

Suppose that at the end of year t – 1, n competitors are present (including Play Things). Then during year t, a fraction 0.9 – 0.1n of the company’s loyal customers (last year’s purchasers) will buy a doll from Play Things this year, and a fraction 0.2 – 0.04n of customers currently in the market who did not purchase a doll last year will purchase a doll from Play Things this year. Adding these two provides the mean sales for this year. Then the actual sales this year is normally distributed with this mean and standard deviation equal to 7.5% of the mean.

a. Use @RISK to estimate the expected NPV of this project.

b. Use the percentiles in @RISK’s output to find an interval such that you are 95% certain that the company’s actual NPV will be within this interval.

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