## Marketing Analytics: Data-Driven Techniques with Microsoft Excel (2014)

### Part VII. Forecasting New Product Sales

### Chapter 28. Using the Copernican Principle to Predict Duration of Future Sales

If you want to determine the future value of a new product, you must have some idea of how long the product will sell. For example, if you want to value the future profits generated by Britney Spears' songs, you need to have some idea of the length of time for which Britney Spears' music will be popular. In this chapter you learn how to use the *Copernican Principle* to model the length of time that a product will sell well. Because the Copernican Principle sheds light on how long a product is likely to sell, it can also aid in the calculations of a customer's lifetime value, discussed in Chapters 19–22.

**Using the Copernican Principle**

The Copernican Principle attempts to estimate the length of time that a product or event remains in existence, for example:

· How long will people listen to Britney Spears' music?

· How long will people listen to Bach's music?

· How long will Stonehenge exist?

To explain the Copernican Principle, you need the following notation:

· *NOW* = Today

· *MIN* = First date a thing came into existence

· *F* = Future lifetime

· *P* = Past lifetime

· *MAX* = Last date a thing is in existence

Nicolaus Copernicus (1473–1543) was a famous Polish Renaissance astronomer. Copernicus contributed greatly to society when he discovered that the Earth was not the center of the universe. Before Copernicus, egotistical earthlings felt that the Earth was the center of the universe and everything revolved around the Earth. Copernicus showed that Earth did not occupy a special place in the universe. In the spirit of Copernicus, assume that the present time (like the Earth in Copernicus's view of the solar system) has no special quality. Therefore assume that *NOW* is equally likely to be anywhere between *MIN* and *MAX*. This implies that (*NOW-MIN*)/(*MAX-MIN*) is equally likely to be any number between 0 and 1.Thus, for example, there is a probability of 38/40 or .95 that __Equation 1__ is true:

__1__

__Equation 1__ may be rewritten as __Equation 2__:

__2__

__Equation 2__ is satisfied if and only if *F* > *P*/39 and *F* < 39*P*. Therefore, there is a 95 percent chance that:

In general there is probability 1 – (1/*N*) that __Equation 3__ is true:

__3__

In __Equations 28.1__ and __28.2__ you used *N* = 20 to get a 95 percent confidence interval for *F/P*. Letting *N* = 2 shows you are 50 percent sure that:

__4__

For example, in 2007 Britney Spears' songs had been listened to for seven years. Therefore when *P* = 7 from __Equation 4__, you are 50 percent sure that:

__5__

After rearranging __Equation 5__, you can find that in 2007 there was a 50 percent chance that Britney Spears' music will be listened to for between 7/3 and 21 more years. In a similar fashion you are 95 percent sure that Britney Spears' songs will be listened to for between 7/39 and 273 more years.

**Simulating Remaining Life of Product**

You can use the Copernican Principle to simulate the remaining lifetime of a product. The Copernican Principle tells you that (*NOW* − *MIN*) / (*MAX* − *MIN*) is equally likely to be any number between 0 and 1. The Excel =RAND() function is equally likely to assume any value between 0 and 1, and therefore in Excel you may model the following equation:

__6__

Now you can solve for *MAX* in __Equation 6__ and find that *MAX* can be modeled as

__7__

For example, suppose in 2007 you wanted to model the last year (highly uncertain) in which Britney Spears' music would be selling. Then *MIN* = 2000 and *NOW* = 2007, so from __Equation 7__ you would model the latest year during which Britney Spears' music would be popular as 2000 + . This modeling of the remaining time for which Britney Spears' music will be popular would be useful in modeling the future value of the royalties earned from Britney Spears' songs.

**Summary**

In this chapter you learned the following:

· You can use the Copernican Principle to model a product's remaining lifetime.

· Underlying the Copernican Principle is the assumption that the current time is equally likely to be any time between the first time the product was introduced and the last time the product will be sold.

· If *F* = Future lifetime of a product and *P* = Past lifetime of a product, then there is probability 1 – (1/*N*) that:

(__28.3__)

· The latest date (*MAX*) at which a product will be sold can be modeled as

(__28.7__)

**Exercises**

**1.** Beatles' music was first played in the United States in 1964. By the Copernican Principle, on average for how many more years will Beatles' music be played in the United States?

**2.** Give a 90 percent confidence interval for the number of future years for which Beatles' music will be played in the United States.

**3.** *The Simpsons* first aired on TV in the United States in 1989. Give a 95 percent confidence interval for the last year in which *The Simpsons* will air.