Looking at Retention as Decay
Although we don’t generally advocate comparing customers to radioactive materials, the comparison is useful for understanding retention. Think of customers as a lump of uranium that is slowly, radioactively decaying into lead.
Our “good” customers are the uranium; the ones who have left are the lead.
Over time, the amount of uranium left in the lump looks something like our retention curves, with the perhaps subtle difference that the timeframe for uranium is measured in billions of years, as opposed to smaller time scales.
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100%
90%
80%
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High End
Regular
60%
vived
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40%
average 10-year tenure high
cent Sur
er
30%
end customers =
P
average 10-year tenure regular
20%
73 months (6.1 years)
customers
10%
44 months (3.7 years)
0%
0
12
24
36
48
60
72
84
96
108
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Tenure (Months after Start)
Figure 12.4 Average customer lifetime for different groups of customers can be compared using the areas under the retention curve.
One very useful characteristic of the uranium is that we know—or more precisely, scientists have determined how to calculate—exactly how much uranium is going to survive after a certain amount of time. They are able to do this because they have built mathematical models that describe radioactive decay, and these have been verified experimentally.
Radioactive materials have a process of decay described as exponential decay. What this means is that the same proportion of uranium turns into lead, regardless of how much time has past. The most common form of uranium, for instance, has a half-life of about 4.5 billion years. So, about half the lump of uranium has turned into lead after this time. After another 4.5 billion years, half the remaining uranium will decay, leaving only a quarter of the original lump as uranium and three-quarters as lead.
WA R N I N G Exponential decay has many useful properties for predicting beyond the range of observations. Unfortunately, customers hardly ever exhibit exponential decay.
What makes exponential decay so nice is that the decay fits a nice simple equation. Using this equation, it is possible to determine how much uranium is around at any given point in time. Wouldn’t it be nice to have such an equation for customer retention?
It would be very nice, but it is unlikely, as shown in the example in the sidebar “Parametric Approaches Do Not Work.”
To shed some light on the issue, let’s imagine a world where customers did exhibit exponential decay. For the purposes of discussion, these customers have a half-life of 1 year. Of 100 customers starting on a particular date, exactly 50 are still active 1 year later. After 2 years, 25 are active and 75 have stopped. Exponential decay would make it easy to forecast the number of customers in the future.
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DETERMINING THE AREA UNDER THE RETENTION CURVE
Finding the area under the retention curve may seem like a daunting mathematical effort. Fortunately, this is not the case at all.
The retention curve consists of a series of points; each point represents the retention after 1 year, 2 years, 3 years, and so on. In this case, retention is measured in years; the units might also be days, weeks, or months.
Each point has a value between 0 and 1, because the points represent a proportion of the customers retained up to that point in time.
The following figure shows the retention curve with a rectangle holding up each point. The base of the rectangle has a length of one (measured in the units of the horizontal axis). The height is the proportion retained. The area under the curve is the sum of the areas of these rectangles.
100%
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Tenure (Years)
Circumscribing each point with a rectangle makes it clear how to calculate the area under the retention curve.
The area of each rectangle is—base times height—simply the proportion retained. The sum of all the rectangles, then, is just the sum of all the retention values in the curve—an easy calculation in a spreadsheet. Voilà, an easy way to calculate the area and quite an interesting observation as well: the sum of the retention values (as percentages) is the average customer lifetime. Notice also that each rectangle has a width of one time unit, in whatever the units are of the horizontal axis. So, the units of the average are also in the units of the horizontal axis.
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PARAMETRIC APPROACHES DO NOT WORK
It is tempting to try to fit some known function to the retention curve. This approach is called parametric statistics, because a few parameters describe the shape of the function. The power of this approach is that we can use it to estimate what happens in the future.
The line is the most common shape for such a function. For a line, there are two parameters, the slope of the line and where it intersects the Y-axis.
Another common shape is a parabola, which has an additional X2 term, so a parabola has three parameters. The exponential that describes radioactive decay actually has only one parameter, the half-life.
The following figure shows part of a retention curve. This retention curve is for the first 7 years of data.
The figure also shows three best-fit curves. Notice that all of these curves fit the values quite well. The statistical measure of fit is R2, which varies from 0
to 1. Values over 0.9 are quite good, so by standard statistical measures, all these curves fit very, very well.
100%
y = -0.0709x + 0.9962
90%
R2 = 0.9215
80%
70%
y = 0.0102×2 – 0.1628x + 1.1493
– 0.1628x + 1.1493
TEAMFLY
60%
R2 = 0.998
= 0.998
vived
50%
40%
cent Sur
30%
er
y = 1.0404e-0.1019x
P
20%
R2 = 0.9633
= 0.9633
10%
0%
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Tenure (Years)
It is easy to fit parametric curves to a retention curve.
The real question, though is not how well these curves fit the data in the range used to define it. We want to know how well these curves work beyond the original 53-week range.
The following figure answers this question. It extrapolates the curves ahead another 5 years. Quickly, the curves diverge from the actual values, and the difference seems to be growing the further out we go.
Team-Fly®
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PARAMETRIC APPROACHES DO NOT WORK (continued)
100%
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Tenure (Years)
The parametric curves that fit a retention curve do not fit well beyond the range where they are defined.
Of course, this illustration does not prove that a parametric approach will not work. Perhaps there is some function out there that, with the right parameters, would fit the observed retention curve very well and continue
working beyond the range used to define the parameters. However, this example does illustrate the challenges of using a parametric approach for approximating survival curves directly, and it is consistent with our experience even when using more data points. Functions that provide a good fit to the retention curve turn out to diverge pretty quickly.
Another way of describing this is that the customers who have been around for 1 year are going to behave just like new customers. Consider a group of 100
customers of various tenures, 50 leave in the following year, regardless of the tenure of the customers at the beginning of the year—exponential decay says that half are going to leave regardless of their initial tenure. That means that customers who have been around for a while are no more loyal then newer customers. However, it is often the case that customers who have been around for a while are actually better customers than new customers. For whatever reason, longer tenured customers have stuck around in the past and are probably a bit less likely than new customers to leave in the future. Exponential decay is a bad situation, because it assumes the opposite: that the tenure of the customer relationship has no effect on the rate that customers are leaving (the worst-case scenario would have longer term customers leaving at consistently higher rates than newer customers, the “familiarity breeds contempt” scenario).
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Hazards
The preceding discussion on retention curves serves to show how useful retention curves are. These curves are quite simple to understand, but only in terms of their data. There is no general shape, no parametric form, no grand theory of customer decay. The data is the message.