13. Value of a Customer. As the manager of credit cardservices at Bank of Hanover (BOH), you’re aware that theaverage proﬁtability of a credit card customer grows with thenumber of years they have used the credit card. Two probabi-listic factors affect actual proﬁtability. The mean proﬁtabilityfunction is given in the table below, which has been gatheredfrom data on BOH customers. The actual proﬁt in a given yearfollows a normal distribution, with a standard deviation equalto 25 percent of the mean proﬁt.In addition, there is a probability less than one that acustomer will continue to use the card during year t. Thisprobability is sometimes called the retention rate. For instance,an 80 percent retention rate means that, during any year, thereis a 20 percent chance the customer will cancel their credit card.Assume that if a customer cancels during year t, then thecancellation occurs at the end of the year, and BOH stillgets proﬁts from year t. The current retention rate has beenestimated at 80 percent. BOH uses a discount rate of 10 percent for calculating netpresent values. a.When the retention rate is 80 percent, what is the averageNPV from a customer?b. When the retention rate is 80 percent, what is the probabil-ity that the NPV for a given customer will exceed $100?c. Determine the average NPV from a customer when theretention rate is 85 percent, 90 percent and 95 percent. Sketch agraph that describes how the average NPV varies with theretention rate, for retention rates above 75 percent; then,interpret the sketch.

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

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13. Value of a Customer. As the manager of credit cardservices at Bank of Hanover (BOH), you’re aware that theaverage proﬁtability of a credit card customer grows with thenumber of years they have used the credit card. Two probabi-listic factors affect actual proﬁtability. The mean proﬁtabilityfunction is given in the table below, which has been gatheredfrom data on BOH customers. The actual proﬁt in a given yearfollows a normal distribution, with a standard deviation equalto 25 percent of the mean proﬁt.In addition, there is a probability less than one that acustomer will continue to use the card during year t. Thisprobability is sometimes called the retention rate. For instance,an 80 percent retention rate means that, during any year, thereis a 20 percent chance the customer will cancel their credit card.Assume that if a customer cancels during year t, then thecancellation occurs at the end of the year, and BOH stillgets proﬁts from year t. The current retention rate has beenestimated at 80 percent. BOH uses a discount rate of 10 percent for calculating netpresent values.

a.When the retention rate is 80 percent, what is the averageNPV from a customer?b. When the retention rate is 80 percent, what is the probabil-ity that the NPV for a given customer will exceed $100?c. Determine the average NPV from a customer when theretention rate is 85 percent, 90 percent and 95 percent. Sketch agraph that describes how the average NPV varies with theretention rate, for retention rates above 75 percent; then,interpret the sketch.

The table below displays the mean profit over a 20-year period, providing a numerical overview of profit changes year by year.

**Yearly Mean Profit Data:**

| Year | Mean Profit |
|------|-------------|
| 1 | -40 |
| 2 | 66 |
| 3 | 72 |
| 4 | 79 |
| 5 | 87 |
| 6 | 92 |
| 7 | 96 |
| 8 | 99 |
| 9 | 103 |
| 10 | 106 |
| 11 | 111 |
| 12 | 116 |
| 13 | 120 |
| 14 | 124 |
| 15 | 130 |
| 16 | 137 |
| 17 | 142 |
| 18 | 148 |
| 19 | 155 |
| 20 | 161 |

**Observations:**
- The mean profit started at a negative value of -40 in the first year.
- There was a steady increase in mean profit every year.
- By the 20th year, the mean profit reached 161, showing significant growth.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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