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 cus

tomer is worth, say, $100 per year, then the premium customers are worth

$610 “ $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 reten

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 don™t generally advocate comparing customers to radioactive

materials, the comparison is useful for understanding retention. Think of cus

tomers 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 ura

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 know”or more pre

cisely, scientists have determined how to calculate”exactly how much ura

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. Wouldn™t it be nice to have such an equa

tion for customer retention?

It would be very nice, but it is unlikely, as shown in the example in the side

bar “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. Exponen

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 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.

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-Fly®

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%