Customer Lifetime Value Modeling and Its Use for Customer

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Customer Lifetime Value Modeling and Its Use for
Customer Retention Planning
Saharon Rosset
Einat Neumann
Uri Eick
Nurit Vatnik
Yizhak Idan
Amdocs Ltd.
8 Hapnina St.
Ra’anana 43000, Israel
{saharonr, einatn, urieick, nuritv, yizhaki}@amdocs.com
ABSTRACT
We present and discuss the important business problem of
estimating the effect of retention efforts on the Lifetime Value of
a customer in the Telecommunications industry. We discuss the
components of this problem, in particular customer value and
length of service (or tenure) modeling, and present a novel
segment-based approach, motivated by the segment-level view
marketing analysts usually employ. We then describe how we
build on this approach to estimate the effects of retention on
Lifetime Value. Our solution has been successfully implemented
in Amdocs’ Business Insight (BI) platform, and we illustrate its
usefulness in real-world scenarios.
Keywords
Lifetime Value, Length of Service, Churn Modeling, Retention
Campaign, Incentive Allocation.
1. INTRODUCTION
Customer Lifetime Value is usually defined as the total net
income a company can expect from a customer (Novo 2001). The
exact mathematical definition and its calculation method depend
on many factors, such as whether customers are “subscribers” (as
in most telecommunications products) or “visitors” (as in direct
marketing or e-business).
In this paper we discuss the calculation and business uses of
Customer Lifetime Value (LTV) in the communication industry,
in particular in cellular telephony.
The Business Intelligence unit of the CRM division at Amdocs
tailors analytical solutions to business problems, which are a high
priority of Amdocs’ customers in the communication industry:
Churn and retention analysis, Fraud analysis (Murad and Pinkas
1999, Rosset et al 1999), Campaign management (Rosset et al
2001), Credit and Collection Risk management and more. LTV
plays a major role in several of these applications, in particular
Churn analysis and retention campaign management. In the
context of churn analysis, the LTV of a customer or a segment is
important complementary information to their churn probability,
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SIGKDD ’02, July 23-26, 2002, Edmonton, Alberta, Canada.
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as it gives a sense of how much is really being lost due to churn
and how much effort should be concentrated on this segment. In
the context of retention campaigns, the main business issue is the
relation between the resources invested in retention and the
corresponding change in LTV of the target segments.
In general, an LTV model has three components: customer’s value
over time, customer’s length of service and a discounting factor.
Each component can be calculated or estimated separately or their
modeling can be combined. When modeling LTV in the context
of a retention campaign, there is an additional issue, which is the
need to calculate a customer’s LTV before and after the retention
effort. In other words, we would need to calculate several LTV’s
for each customer or segment, corresponding to each possible
retention campaign we may want to run (i.e. the different
incentives we may want to suggest). Being able to estimate these
different LTV’s is the key to a successful and useful LTV
application.
The structure of this paper is as follows:
In section 2 we introduce the general mathematical
formulation of the LTV calculation problem.
Section 3 discusses practical approaches to LTV
calculations from the literature, and presents our
preferred approach.
The practical implementation of our LTV calculation,
with some examples, is presented in section 4.
In section 5 we turn to the business problem of
estimating LTV given incentives and using these
calculations to guide retention campaigns.
Section 6 presents our LTV-based solution to the
incentive allocation challenge and illustrates its use for
real-life applications.
2. THEORETICAL LTV CALCULATIONS
Given a customer, there are three factors we have to determine in
order to calculate LTV:
1. The customer’s value over time: v(t) for t≥0, where t is
time and t=0 is the present. In practice, the customer’s
future value has to be estimated from current data, using
business knowledge and analytical tools.
2. A length of service (LOS) model, describing the
customer’s churn probability over time. This is usually
described by a “survival” function S(t) for t≥0, which
describes the probability that the customer will still be
active at time t. We can then define f(t) as the
customer’s “instantaneous” probability of churn at time
t: f(t) ≡ -dS/dt The quantity most commonly modeled,
however is the hazard function h(t) = f(t)/S(t). Helsen
and Schmittlein (1993) discuss why h(t) is a more
appropriate quantity to estimate than f(t). The LOS
model has to be estimated from current and historical
data as well.
3. A discounting factor D(t), which describes how much
each $1 gained in some future time t is worth for us
right now. This function is usually given based on
business knowledge. Two popular choices are:
- Exponential decay: D(t)= exp (-αt) for some α≥0
(α=0 means no discounting)
- Threshold function: D(t)= I{t≤T} for some T>0 (where
I is the indicator function).
Given these three components, we can write the explicit formula
for a customer’s LTV as follows:
∞
LTV = ∫ S(t)v(t)D( t)dt
(2.1)
0
In other words, the total value to be gained while the customer is
still active. While this formula is attractive and straight-forward,
the essence of the challenge lies, of course, in estimating the v(t)
and S(t) components in a reasonable way.
We can build models of varying structural and computational
complexity for these two quantities. For example, for LOS we can
use a highly simplistic model assuming constant churn rate – so if
we observe 5% churn rate in the current month, we can set S(t) =
0.95t. This model ignores the different factors that can affect
churn – a customer’s individual characteristics, contracts and
commitments, etc. On the other hand we can build a complex
proportional hazards model, using dozens of customer properties
as predictors. Such a model can turn out to be too complex and
elaborate, either because it is modeling “local” effects relevant for
the present only and not for the future, or because there is not
enough data to estimate it properly. So to build practical and
useful analytical models we have to find the “golden path” which
makes effective and relevant use of the data available to us. We
attempt to answer this challenge in the next sections.
3. PRACTICAL LTV APPROACHES
In this section we review some of the approaches to modeling the
various components of LTV from the literature and present the
segment-based approach, which follows naturally from the way
analyses and campaigns are usually conducted in marketing
departments. The segment-based approach helps in simplifying
calculations and justifies the use of relatively simple methods for
estimating the functions.
To model LTV we would naturally want to make use of the most
recent data available. Therefore let us assume that we are only
going to use churn data from the last available month for
modeling LOS. So for the rest of this paper we assume we have a
set of n customers, with covariates vectors x1,…,xn representing
their “current” state and churn indicators c1,…,cn. The customers’
tenure with the company is an important churn predictor since
LOS frequently shows a strong dependency on customer “age”, in
particular when contracts prevent customers from disconnecting
during a specific period. Let us denote these tenures by t1,…,tn
.Additional covariates are customer details, usage history,
payment history, etc. Some of the covariates may be based on
time-dependent accumulated attributes (e.g. averages over time,
trends).
Our discussion is going to view time as discrete (measured in
months), and thus the ti’s will be integers and f(t) will be a
probability function, rather than a distribution function.
3.1 LOS Modeling Approaches
We now present a brief description of common Survival Analysis
approaches and their possible use in LOS modeling. Detailed
discussion of prevalent Survival Analysis approaches can be
found in the literature, e.g. Venables and Ripley 1999, chapter 12.
Pure parametric approaches assume S(t) has a parametric form
(Exponential, Weibull etc.) with the parameters depending on the
covariates, including t. As Mani et al (1999) mention, such
approaches are generally not appropriate for LTV modeling, since
the survival function tends to be “spiky” and non-smooth, with
spikes at the contract end dates.
Semi-parametric approaches, such as the Cox proportional
hazards (PH) model (Cox 1972), are somewhat more flexible. The
Cox PH model assumes a model for the hazard function h(t) of the
form:
f i (t )
hi (t ) =
S i (t )
= λ (t ) exp(β ' xi )
(3.1)
or alternatively:
log(hi (t )) = log(λ (t )) + β ' xi
(3.2)
So there is a fixed parametric linear effect (in the exponent) for all
covariates, except time, which is accounted for in the timevarying “baseline” risk λ(t). Mani et al (1999) build a Neural
Network semi-parametric model, where each possible tenure t has
its own output node (the tenure is discretized to the monthly
level). They illustrate that the more elaborate NN model performs
better than the PH model on their data.
The data as described above, makes LOS modeling a special case
of survival analysis where each subject is observed only once in
time, and customers who disconnected before this month are “left
censored”. Consequently we can approach it either as a survival
analysis problem or a standard supervised learning problem where
the time (i.e. customer’s tenure with the company) is one of the
predictors and churn is the response. To include a “baseline
hazard” effect, time can be treated as being factorial rather than
numerical, thus allowing a different effect for each tenure value.
In this setting, a log-linear regression model for churn prediction
using left-censored data would be equivalent in representation to a
Cox proportional hazards survival analysis model. To see this
point, consider that a customer’s churn risk is in fact his h(t) value
(since if the customer already left we would not observe him).
Thus a model of the form:
log(P(ci = 1)) = α (t i ) + β ' xi
(3.3)
is obviously equivalent to (3.2).
The Kaplan-Meier estimator (Kaplan and Meier 1958) offers a
fully non-parametric estimate for S(t) by averaging over the data:
S (t ) =
∑ I (t
i
i
≥ t)
∑ I (t
i
Where:
•
I (t i
i
≥ t ) + Ct
(3.4)
≥ t ) equals 1 if customer i's tenure is at least t months
•
∑ I (t
i
i
≥ t ) is the number of customers whose tenure is at
least t months
Ct is the number of customers who should have been at least
t months old at the current date but have already left.
The data as described above is “left censored” and does not
include Ct. However it can often be calculated based on historical
information found in customer databases, which are typically used
for LTV calculations.
•
3.2 The Segment-Based LOS Approach
When we are considering the use of analytical models for
marketing applications, we should take into account the way they
are going to be used. An important concept in marketing is that of
a “segment”, representing a set of customers who are to be treated
as one unit for the purpose of planning, carrying out and
inspecting the results of marketing campaigns. A segment is
usually implicitly considered to be “homogeneous” in the sense
that the customers in it are “similar”, at least for the property
examined (e.g. propensity to churn) or the campaign planned.
Amdocs Business Insight tools assist marketing experts in
automatically discovering, examining, manually defining and
manipulating segments for specific business problems. We
assume in our LTV implementation that:
the marketing analyst is interested in examining
segments, not individual customers
these segments have been pre-defined using Amdocs
CMS or some other tool
they are “homogeneous” in terms of churn (and hence
LOS) behavior
they are reasonably large
Based upon these assumptions, estimating LOS for a segment is
reasonable and relatively simple. Under these assumptions we can
dispense completely with the covariate vectors x (since all
customers within the segment are similar) and adopt a nonparametric approach to estimating LOS in the segment by
averaging over customers in the segment.
The Kaplan-Meier approach is reasonable here, but as we
discussed before it requires the use of left-censored data referring
to customers who have churned in the past. While this data is
usually available it refers to churn events from the (potentially
distant) past, and so may not represent the current tendencies in
this segment, which may well be related to recent trends in the
market, offers by competitors etc. So an alternative approach
could be to calculate a non-parametric estimate of the hazards
rate:
h(t ) =
∑ I (t
i
i
= t )I (ci = 1)
∑ I (t
i
i
= t)
(3.5)
Where:
•
I (t i = t ) equals 1 if customer i's current tenure is t months
•
I (ci = 1) equals 1 if customer i churned in the current
month
•
∑ I (t
i
i
= t )I (ci = 1) is the number of customers whose
current tenure is t months and churned in the current month.
This approach relies heavily on having a sufficient number of
examples for each discrete time point t (usually taken in months),
but has the advantage of using only current data to estimate the
function. We can obtain an estimate for S(t) through the simple
calculation:
S (t ) = ∏u <t S (u + 1) / S (u ) =
∏ (S (u) − f (u )) / S (u) = ∏ (1 − h(u ))
u <t
u <t
(3.6)
Where S(0) = 1, of course.
In section 4 we describe Amdocs’ LTV platform, which utilizes
this approach and illustrate it on real data.
3.2.1 Theoretical Discussion of a Segment Approach
When examining the adequacy of a modeling approach, we
generally have to consider two statistical concepts:
Bias / Consistency: if we had infinite data, would our
estimate converge to the correct value? How far would
it end up being?
Variance: how much uncertainty do we have in the
estimates we are calculating for the unknown value?
These concepts have concrete mathematical definitions for the
case of squared error loss regression only (although many
suggestions exist for generalized formulations for other cases –
see, for example - Friedman 97). However the principles they
describe apply to any problem:
The more flexible and/or adequate the model is, the
smaller the bias.
The more data one has, and the more efficiently one
uses it, the smaller the uncertainty.
Under the segment-homogeneity assumption mentioned in the
previous section, the bias of our segment-based approach should
be close to zero. Furthermore, even without this assumption, if we
assume that the marketing expert planning the campaign is only
interested in the segment as a whole, then the quantities we want
to estimate are indeed segment averages and not individual values.
Hence the segment-based estimates are unbiased in this scenario
as well.
As for variance, this is obviously a function of segment size.
Parametric estimators will tend to have smaller variance. It is an
interesting research question to investigate this bias-variance
tradeoff between non-parametric and parametric estimates in this
case. Under the assumption that segments are “large” (as are
indeed the segments in most real-life segments encountered in the
communication industry), and that there is a reasonable amount of
churn in each segment, we can safely assume that the segment
based non-parametric estimates will also have low variance, and
hence that our approach is reasonable.
3.3 Practical Value Calculations
Calculating a customer’s current value is usually a straight
forward calculation based on the customer’s current or recent
information: usage, price plan, payments, collection efforts, call
center contacts, etc. In section 4 we give illustrated examples.
The statistical techniques for modeling customer value along time
include forecasting, trend analysis and time series modeling.
However the complexity of modeling and predicting the various
factors that affect future value: seasonality, business cycles,
economic situation, competitors, personal profiles and more, make
future value prediction a highly complex problem. The solution in
LTV applications is usually to concentrate on modeling LOS,
while either leaving the whole value issue to the experts (Mani et
al 1999), or considering customers’ current value as their future
value (Novo 2001).
Working at the segment level also makes the value calculation
task easier, since it implies we do not need to have an exact
estimate of individual customers’ future value, but can rather
average the estimates over all customers in the segment. This does
not solve the fundamental problem of predicting future value, but
it allows us to get a reliable average current value estimate at the
segment level.
4. LTV WITHIN THE CMS
One of the Amdocs’ BI platform systems is the Churn
Management System (CMS). The key outputs of the system are
churn and loyal segments, as well as scores for each individual in
the target population, which represent the individual’s likelihood
to churn. The first step we take in the process of churn analysis is
defining and creating a customer data mart that provides a single
consolidated view of the customer data to be analyzed. It includes
various attributes that reflect customers’ profile and behavior
changes: customer data, usage summaries, billing data, accounts
receivable information, and social demographic data. Relevant,
trends and moving averages are calculated, to account for timevariability in the data and exploit its predictive power. The
preparation of the data for the exact needs of the data mining
process includes Extracting Transforming and Loading (ETL) the
necessary data.
The churn analysis process within the CMS combines automatic
knowledge discovery and interactive analyst sessions. The
automatic algorithm is a decision tree algorithm followed by a
rule extraction mechanism. The analyst can then view and
manipulate the automatically generated predictive segments (or
patterns), and add to them based on his marketing expertise.
The automatic and interactive tools which the CMS utilizes to
discover and analyse patterns, and to perform predictive
modelling, have proven themselves as highly successful when
compared to the state of the art data-mining techniques (Rosset
and Inger 2000, Inger et al 2000, Neumann et al 2000).
The analysis tool includes an easy-to-use graphical user interface.
Figure 1 is a capture of one of the system analysis tool’s screens
which provides the analyst with insight into various customer
population segments automatically identified by their attributes,
churn likelihood and related value. These segments (or rules) are
characterized by several attributes accompanied by statistical
measures that describe the significance of the segments and their
coverage. Additional graphical capabilities of the CMS include
analyzing the distribution of each variable per churn/loyal groups
or in comparison to the entire population and an interactive visual
data analysis, which provides the ability to further investigate
attributes to provide additional insight and support the design of
retention actions.
The data is extracted on a monthly basis and accordingly the
scoring process is performed once a month. The churn score is
one of the main components of the LOS; thus each customer will
have a new LTV every month.
As shown in Figure 1 the system produces segments that
characterize churn and loyal populations. Thus, the segment level
and not the customer level is the basis for the interaction with the
analyst. That is the level on which retention campaigns are
planned and therefore the level on which the analyst is interested
in viewing LTV.
The LOS solution implemented in Amdocs CMS is the segmentbased calculation described in section 3.2. . For value definition
the CMS allows flexibility and it calculates the value individually
per customer. It can be a constant value, an existing attribute
within the data-mart, or a function of several existing attributes.
An example for the customer value can be ‘The financial value of
a customer to the organization’. This value can be calculated from
‘received payments’ minus the ‘cost of supplying products and
services” to the customer.
To effectively use the data mining algorithms in the CMS, the
input is usually a biased sample of the population. Often the churn
rate in the population is very small but in the sample the two
classes (churn and loyal) are much more balanced. The difference
in churn rate between the sample and the population is accounted
Figure 1. Churn and loyal patterns discovered
automatically by the CMS
for in the LOS model, as we describe below. Rosset et. al. 2001
provide a detailed explanation about the relevant inverse
transformation.
LOS is calculated on the segment level. It is calculated for each
“age” group t within the segment, i.e. for each group of customers
with the same tenure in the segment, there will be the same LOS.
This calculation is based on a large amount of data (customers
with the same tenure in the segment). The base for this value is
the proportion of churners for each age t - pt as defined in the
following formula (this is an extended version of the calculation
from equation (3.5) in section 3.3)
∑ I (t
pt =
i
i
= t )I (c i = 1)
factor × ∑ I (t i = t )I (ci = 0 ) + ∑ I (t i = t )I (c i = 1)
i
•
•
LOSj is the Expected Length of Service for the j
customer
ratio is the population to sample ratio of loyal
customers
(4.1)
i
Where:
•
∑ I (t
i
= t )I (ci = 1) is the number of churners at tenure t
∑ I (t
i
= t )I (ci = 0 ) is the number loyal customers at tenure t
i
•
i
•
factor = (churn to loyal sample ratio) / (churn to loyal
population ratio)
This quantity is calculated for each tenure t in the segment. There
are several assumptions underlying this calculation. First, that the
current churn probabilities for customers at tenure t represent the
future ones at tenure t; second, that the customers come from a
“homogeneous”
population
and
third,
that
both
I (t = t )I (c = 1) and
I (t = t )I (c = 0 ) are large enough to
∑
i
∑
i
i
i
i
i
give reliable estimates of pt.
Now, given a customer who is currently at tenure t0 , we can use
(4.1) to get the ‘Probability of a customer to reach age t’ – S(t)
(
S (t ) = (1 − p t −1 ) × (1 − p t − 2 ) × L × 1 − p t0
)
(4.2)
And then we can get the expected LOS as followsh
LOS = ∑ S (t )
(4.3)
t =0
Where h is the horizon, i.e. the number of months until the end of
the interest period. If we are interested in a horizon of two years
then the sum will be over 24 months. Implementing other
discounting functions, in addition to this threshold approach is
planned for the future, and poses no conceptual problems.
Finally, LTV within a segment will be the following sum over all
customers in the segment is
LTV = ratio × ∑ LOS j v j c j
(4.4)
j
where
•
•
•
j is the index of customers in the segment
vj is the value of the j-th customer
cj is an indication for the j-th customer. 0 if he is a
churner, 1 if he is loyal
Figure 2. LTV calculation CMS screen
Figure 2 is a screen capture of the CMS window for calculating
LTV. It is necessary to select the customer’s age (tenure), enter
the horizon and enter the full population churn rate (the sample
churn rate is already derived). It is also necessary to select/define
the value, which may be one of three options: an equal value for
all customers, a field that was previously selected as the value (in
this case the “average bill” was previously selected), or a new
value function.
The result of the LTV calculation can also be seen in Figure 1.
The statistic measures (including LTV) of the identified segments
are already transformed to the full population. In general, Loyal
segments (“Class: Stay”) have higher LTVs than Churn segments,
since the LOS of churners is 0. The aim is to try to increase the
LTV of relevant segments by proper retention efforts, which aim
mainly at increasing the LOS (a secondary purpose is increasing
the value).
5. ESTIMATING THE EFFECT OF
RETENTION EFFORTS ON LTV
We now turn to the most useful and challenging application of
LTV calculation: modeling and predicting the effects of a
company’s actions on its customer’s LTV.
An example of a desirable scenario for a LTV application would
be:
Company “A” has identified a segment of “City dwelling
professionals”, which is of high value and high churn rate. It
wants to know the effect of each one of five possible incentives
suggestions (e.g. free battery, 200 free night minutes, reduced
price handset upgrade etc.), on the segment’s value over time and
LOS, and hence LTV. Each incentive may have a different cost,
different acceptance rate by customers, and different effect if it
gets accepted. The goal of the LTV application is to supply useful
information about the effects of the different incentives, and help
analysts to choose among them.
From the definition of the problem it is clear that there is some
information about the incentives which we must know (or
estimate) before we can calculate its effect on LTV:
1. The cost of the effort involved in suggesting the
incentive. This figure is usually known and depends for
example on the channel utilized (e.g. proactive phone
contact, letter, comment on written bill). Denote the
suggestion (or contact channel) cost by C.
2. The cost to the company if the customer accepts the
incentive (e.g. the cost of the battery offered). Again
this figure is either known or can be reliably estimated
based on business knowledge. Denote the offer cost by
G
3. The probability that a customer in the approached
segment will agree to accept the incentive (which can be
around 100% if the incentive is completely free, but that
is rarely the case). This is a more problematic quantity
to figure out and it has to be estimated from past
experience, or simply guessed (in which case many
different values for it can be tried, to see how each
would affect the outcome). Denote the acceptance
probability by P
4. Change in the value function if the incentive is
accepted. For example, if the incentive is free voicemail, the customer’s calls to the voice-mail can still
generate additional revenue. Similarly to P, the change
in value has to be assumed or approximated from past
data. Denote the new value function by v(i)(t).
5. The effect on customer’s LOS if the incentive is
accepted. The most obvious way for the incentive to
affect LOS is if it includes a commitment by the
customer (i.e. the customer commits not to leave the
company in the next X months). Denote the new
survival function by S(i)(t).
Given all of the above, calculating the change in LTV of a
customer from a retention campaign, in which a given incentive is
suggested is a straight forward ROI calculation:
LTVnew − LTVold =
∞
∞
0
0
P ⋅ ( ∫ S (i )(t)v (i )(t)D(t)dt − ∫ S(t)v(t)D(t)dt − G) − C (5.1)
As for the basic LTV calculation described in section 2, and even
more so, the main challenge is in obtaining reasonable and usable
estimates for the above quantities, in particular the functions v(i),
S(i).
We now describe two approaches to this problem: one that builds
on our segment-level LTV calculation approach presented above,
and another that makes further simplifying assumptions, negating
the need to predict the future.
5.1 Segment-level calculation
As was mentioned before, working at the segment level allows us
to “average” our information over the whole segment and avoid
parametric assumptions, at the price of assuming that the segment
population is “homogeneous”.
To expand the segment-level approach described in section 3.2 to
estimate the effect of incentives on a segment’s LTV, we need to
describe how we change the LOS model per segment, and how we
adjust customer value for the incentive effects.
We define two possible effects of an incentive on LOS:
commitment and percentage decrease. If an incentive includes a
commitment period of X months (usually with a penalty for
commitment violation that makes it unprofitable to leave during
this period), then obviously any customer who accepts the
incentive will not leave during this period. On the other hand,
incentives that do not include a commitment also cause the churn
probability to decrease. Our model allows a percentage decrease
in the monthly churn rate. This percentage is presumed to be
constant in all months and for all customers within the segment.
Thus, to estimate post-incentive LOS for a specific segment and a
specific incentive, we need to know:
Commitment period included in incentive, denote by
cmt(i)
Reduction in churn probability from incentive, denote
by rc(i)
Which gives us for a specific customer:
S (i) (t ) =
I {t < cmt (i ) } + I {t ≥ cmt (i ) }
t
∏ (1 − c(a + u ) ⋅ rc
u = cmt
(i )
)
(5.2)
(i )
where a is the customer’s current “age”, and c(a+u) is the churn
probability estimate for age a+u as estimated for the whole
segment.
Then a similar calculation to the expected LOS calculation in
equations (4.2) and (4.3) now gives us a post-incentive expected
LOS estimate of:
ELOS(i) =
t
1 n h
cmt (i ) + ∑ ∑ ∏ (1 − c(a j + u ) ⋅ rc (i ) )
n j =1 t =cmt ( i ) u =cmt ( i )
(5.3)
where the index j runs over the customers in the segment.
So we are using the homogeneity assumption to average the effect
of the incentive on LOS over all customers in the segment, and we
are assuming again that the “age” effect is the only differentiating
factor of individual behavior within the sample. We also assume
that the probability of accepting the incentive is constant across
the segment and independent of all customer properties (including
age), not used for the segment definition.
We also assume that once the commitment period is over,
customers will “on average” return to the churn behavior that
would characterize them at their “age” have they not churned for
other reasons (rather than the commitment from the incentive).
The incentive’s effect on customer value is assumed to be as a
percentage change in the customer value. This change should
reflect both the reduced value to the company due to the incentive
cost and the increased value due to the increase in the relevant
customer’s usage. For example, when offering a free voicemail
incentive the reduced value would be the voicemail cost and the
increased valued would be derived from the increase in billed
incoming calls and the increase in outgoing calls due to the
customer’s calls to the voicemail box. Thus, we get that for every
customer: v(i)=v⋅ (1+change(i)), where change(i) is the change in
value due to the incentive, assumed constant for all customers.
We can now combine all of the above into an estimate of the
average change in LTV in the segment due to the incentive:
avLTV (i) − avLTV =
1 n
P ⋅ ( ∑ [ ELOS ( i ) ⋅ v (i ) ( j ) − ELOS ⋅ v( j )] − G ) − C
n j =1
(5.4)
Where avLTV(i) is the estimated average LTV per customer in the
segment after the retention campaign and avLTV is the estimated
current average LTV per customer in the segment. If this
difference is positive it means we expect the retention campaign
to be beneficial to the company.
5.2 Simplified Calculation Based on Constant
Churn Assumptions
Let us now assume the following:
1. D(t) is a threshold function with horizon h.
2. The churn risk is constant for each customer in the
segment for any horizon. This would translate to
assuming that for each customer, S(t) =1- pt, where p is
the churn probability of the customer for the next
month.
3. p is small
4. The incentive includes a commitment for h months at
least.
5. Customer value is constant over time, v(t) = v, and is
not affected by the incentive’s acceptance.
Then we get the following value for customer LTV without
retention:
6. RETENTION LTV IMPLEMENTATION
IN THE CMS
We now illustrate how the concepts of the previous section drive
the incentive LTV implementation within the CMS, by following
the details of the steps in the application and the calculation for a
couple of real-life examples.
Figure 3. Churn segment
Figure 3 demonstrates one of the churn segments selected for a
retention campaign. The segment consists of young customers
who don’t have a caller-id feature, whose handset was not
upgraded in the past year and who have recently changed their
payment method (for example from direct debit to check).
A marketing analyst came up with two possible incentives for this
segment: an upgrade at a discounted price (lets assume for
simplicity that all will be offered the same new handset) or a free
caller-id feature. Both incentives will involve a 12 months
commitment period.
h −1
LTVold = v ⋅ ∑ (1 − p ) t ≅
t =0
h −1
v ⋅ ∑ 1 − pt = vh(1 − p (h − 1))
(5.5)
t =0
where the approximation relies on p and h being reasonably small.
And adding retention we get:
(5.6)
LTVnew = P (hv − G ) +(1 − P )LTVold − C
Since if we succeed in giving the incentive we are guaranteed
loyalty for the full relevant period of h months. So the difference
in LTV due to retention is:
LTVnew - LTVold ≅ P ⋅ h(h − 1) ⋅ v ⋅ p − P ⋅ G − C
(5.7)
which, given P,G and C and ignoring the inaccuracy in our
calculation gives us the elegant result that:
LTVnew - LTVold > 0 ⇔
v ⋅ p > ( P ⋅ G − C ) /( P ⋅ h(h − 1))
(5.8)
In other words, we get the intuitive conclusion, that if we have a
reasonable model for v and p, we should suggest the incentive
only to customers whose value weighted risk v⋅p is big enough.
Figure 4. Incentive definition CMS Screen
The first step is to calculate the current LTV of this segment. As
displayed in section 4 we defined the LTV parameters. Recall,
that the field selected as value was the monthly average bill, the
selected horizon is 12 months and the population churn rate is 5%
(in the sample it’s about 50%). The LTV of this segment, which is
already displayed in Figure 3 is $4,967,202.
The next step is to define the possible incentives. An example of
how this is done in the CMS is illustrated in Figures 4 & 5. In
Figure 4 the incentive is defined and in Figure 5 the incentive is
attached to a specific segment. At this stage it is also possible to
refine the segment definition. Note that the same incentive may be
allocated to different segments.
Suppose we wanted to examine the same two incentives for a
different segment, as shown in Figure 7. This is a segment with
many loyal customers, comprised of older customers with stable
usage and medium bill average amounts.
Figure 7. Loyal segment
Figure 5. Incentive allocation CMS screen
Finally, we compare the change in LTV related to each of the
incentives. The cost of giving a discounted handset upgrade is
much higher than the cost of a free caller id (in this example we
used $100 and $10 correspondingly). On the other hand, the
acceptance rate will be higher since it’s a more attractive offer
(caller-id - 10% of the churners and 20% of the loyals, upgrade 20% of the churners and 30% of the loyals). Actually, churners
often switch providers in order to receive an improved handset
promised by the competitor. So, the result of the upgrade
incentive will be a higher retention rate than the caller-id
incentive. Additionally, a more sophisticated handset will
probably increase the usage and thus the added value, while
adding a caller-id will have very little or no impact on the usage
(the relative value increase for the upgrade is 10% in this example
and none for the caller-id). Note that the added value affects both
potential churners who accept the offer and loyal customer who
will accept the offer. Furthermore, loyal customers will also be
committed to 12 more months, so even though they weren’t about
to churn in the next month the incentive may lengthen their LOS.
The new LTV calculation takes into account all these parameters
and the result as can be seen in Figure 6 is that the estimated
increase in LTV due to offering a discounted upgrade is
$2,413,338 and due to offering a free caller-id is $1,982,294.
In addition to the purpose of retaining the churners in this
segment, offering an incentive to this segment is done also to
increase the usage / value of the loyal customers and lengthen
their LOS. The original LTV as displayed on Figure 6.5 is
$29,091,321. The same cost and acceptance rates were applied for
the caller-id and upgrade incentives. Note that since the
acceptance rate is higher for loyal customers the overall
acceptance rate of this segment will be higher then in the previous
churn segment.
The result was that the increase in value and LOS wasn’t large
enough to cover the high cost of the upgrade offers. Thus, the
estimated change in LTV due to that incentive is negative: $485,450. On the other hand, the caller-id incentive yielded an
estimated LTV increase of $1,422,540 (Figure 8).
Figure 8. Estimated LTV change due to a free caller-id and
due to a discounted upgrade offer
The examples illustrate that different incentives may have
different impacts on LTV of the same segment, and the same
incentive may have different impacts on LTV of different
segments. The calculations involved are complex enough that the
differential effect of different incentives on different segments
cannot be easily guessed even when all the incentive’s parameters
are known. Using the application’s mechanism for estimating that
impact, it is possible to fit the appropriate incentive (out of the
given options) to selected segments.
7. SUMMARY AND FUTURE WORK
Figure 6. Estimated LTV change due to a free caller-id and
due to a discounted upgrade offer
In this paper we have tackled the practical use of analytical
models for estimating the effect of retention measures on
customers’ lifetime value. This issue has been somewhat ignored
in the data mining and marketing literature. We have described
our approach and illustrated its usefulness in practical situations.
The approach presented here to LTV calculation is not necessarily
the best approach. However our emphasis is on practical and
usable solutions, which will enable us to reach our ultimate goal -
to get useful and actionable information about the effects of
different incentives. As our approach is modular, additional LOS
and value models can certainly be integrated into the solution we
presented.
We believe that this problem, like many others that arise from the
interaction between the business community and data miners,
present an important and significant data mining challenge and
deserve more attention than it usually gets in the data mining
community. In this paper we have tried to illustrate the usefulness
of combining business knowledge and analytical expertise to build
practical solutions to practical problems.
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“Statistics and Data Mining Techniques for Lifetime Value
Modeling,” Proceedings of KDD-99, 94-103.
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