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Hazard Functions and Survival Analysis in Marketing 389

A group of customers with different
tenures are stacked on top of each
other. Each bar represents one
customer.
time

At each point in time, the edges
count the number of customers
Customers
Number of

active at that time.

Notice that the sum of all the areas is
the sum of all the customer tenures.

Making the vertical axis a proportion
instead of a count produces a curve
Proportion of

that looks the same. This is a
Customers

retention curve.

The area under the retention curve is
the average customer tenure.
Figure 12.3 Average customer tenure is calculated from the area under the retention curve.

This value, called truncated mean lifetime by statisticians, is very useful. As
shown in Figure 12.4, the better customers have an average 10-year lifetime of
6.1 years; the other group has an average of 3.7 years. If, on average, a cusВ­
tomer is worth, say, \$100 per year, then the premium customers are worth
\$610 вЂ“ \$370 = \$240 more than the regular customers during the 10 years after
they start, or about \$24 per year. This \$24 might represent the return on a retenВ­
tion program designed specifically for the premium customers, or it might
give an upper limit of how much to budget for such retention programs.

Looking at Retention as Decay
Although we donвЂ™t generally advocate comparing customers to radioactive
materials, the comparison is useful for understanding retention. Think of cusВ­
tomers as a lump of uranium that is slowly, radioactively decaying into lead.
Our вЂњgoodвЂќ 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 uraВ­
nium is measured in billions of years, as opposed to smaller time scales.
390 Chapter 12

100%

90%

80%
High End
70%
Percent Survived

Regular
60%

50%
40%
average 10-year tenure high
30% end customers =
average 10-year tenure regular
73 months (6.1 years)
20%
customers
10%
44 months (3.7 years)
0%
0 12 24 36 48 60 72 84 96 108 120
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 knowвЂ”or more preВ­
cisely, scientists have determined how to calculateвЂ”exactly how much uraВ­
nium 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. WouldnвЂ™t it be nice to have such an equaВ­
tion for customer retention?
It would be very nice, but it is unlikely, as shown in the example in the sideВ­
bar вЂњParametric Approaches Do Not Work.вЂќ
To shed some light on the issue, letвЂ™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. ExponenВ­
tial decay would make it easy to forecast the number of customers in the future.
Hazard Functions and Survival Analysis in Marketing 391

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%
90%
80%

70%
Percent Survived

60%
50%
40%
30%

20%
10%
0%
0 1 2 3 4 5 6 7 8 9 10 11 12

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 isвЂ”base times heightвЂ”simply the proportion
retained. The sum of all the rectangles, then, is just the sum of all the retention
values in the curveвЂ”an easy calculation in a spreadsheet. VoilГ , 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.
392 Chapter 12

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

Y
the values quite well. The statistical measure of fit is R2, which varies from 0

FL
to 1. Values over 0.9 are quite good, so by standard statistical measures, all
these curves fit very, very well.
AM
100%
y = -0.0709x + 0.9962
90%
R2 = 0.9215
80%
TE

y = 0.0102x 2 - 0.1628x + 1.1493
70%
Percent Survived

R2 = 0.998
60%

50%
40%
30%
y = 1.0404e -0.1019x
20%
R2 = 0.9633
10%
0%
1 2 3 4 5 6 7 8 9 10 11 12 13

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-FlyВ®
Hazard Functions and Survival Analysis in Marketing 393

PARAMETRIC APPROACHES DO NOT WORK (continued)

100%
90%

80%
70%
Percent Survived

60%
50%
40%
30%
20%

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