MMRCDISCUSSION PAPER SERIES
No. 468

Deriving Customer Lifetime Value from RFM Measures:
Insights into Customer Retention and Acquisition
Makoto Abe
Graduate School of Economics, The University of Tokyo
March 2015

東京大学ものづくり経営研究センター
Manufacturing Management Research Center (MMRC)
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Deriving Customer Lifetime Value from RFM Measures:
Insights into Customer Retention and Acquisition

Makoto Abe

Graduate School of Economics
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033

JAPAN

TEL & FAX: +81-3-5841-5646
E-mail: abe@e.u-tokyo.ac.jp

January 24, 2015

Deriving Customer Lifetime Value from RFM Measures:
Insights into Customer Retention and Acquisition

ABSTRACTThe wide use of RFM analysis in CRM suggests that these measures contain rather richinformation about customer purchase behavior. This research, using the RFM measures of acustomer, develops an individual-level CLV model that identifies the underlying behavior
traits of purchase rate, lifetime and spending, which are then linked to CLV. In the application
to two datasets, frequent shoppers program data from a department store and a CD chain, themodel produces customer-specific metrics that are useful for identifying preferred customers
and taking marketing actions targeted at the individual level in CRM. The paper then presents
a retention program for existing customers that is most effective in terms of Marketing ROI,
such as what action needs to be taken to which customers at which timing. For prospective
customers without RFM measures, by relating the demographic characteristics to behavioral
traits, insight into acquisition strategy is obtained.

Keywords: Marketing ROI, scoring models, Pareto/NBD, hierarchical Bayes, MCMC

1

1. INTRODUCTION
Three customer measures—recency (time since most recent purchase), frequency(number of prior purchases), and monetary value (average purchase amount per transaction),or RFM—have been used extensively in CRM with great success. At the heart of CRM is theconcept of customer lifetime value (CLV), a long-term approach to identifying profitablecustomers and cultivating relations. The three customer measures of RFM serve the purpose
well by capturing past buying behaviors parsimoniously without burdening a firm’s data
storage limits. Although not all firms maintain each customer’s purchase history, most still
accumulate customer RFM data (Hughes 2000). This suggests that rich, long-term customer
information is condensed into these three measures (Buckinx and Van den Poel 2005).
Despite the importance of CLV and the usefulness of RFM measures, however, theliterature provides conflicting findings on their relationship. In their predictive analysis,
Malthouse and Blattberg (2005) generally found that frequency and monetary measures werepositively correlated with CLV while recency was negatively correlated with CLV. Blattberg,
Malthouse, and Neslin (2009), in their empirical generalization article on CLV, further statethat practitioners share a similar sentiment. However, they also point out at least fiveacademic studies that have conflicting results on the association of RFM with CLV, leadingthem to conclude that there is a need for further research on the issue.
Here, one must be careful to note that RFM themselves are not behavioral constructs butmere statistical measures that are observed as a result of the underlying customer behavior. In
particular, recency is strongly affected by the analyst’s selected time frame. For the same
recency, frequent customers are more likely to be inactive than infrequent ones, thereby
resulting in different CLVs. Frequency is different from purchase rate. Purchase rate considersa customer’s purchases only when that customer is active. In contrast, frequency counts
purchases during the entire observation period that can include a customer’s inactive duration.

2

Hence, observed RFM measures are interrelated in a complex manner, as demonstrated
by the iso-value contour plot (Fader, Hardie, and Lee 2005). A more rigorous approach toinvestigating the relationship between RFM and CLV is to infer the underlying behavior
processes from RFM measures and then seek their association with CLV. In a non-contractual
setting, appropriate behavior traits that can be derived from RFM measures as sufficient
statistics are “purchase rate”, “lifetime”, and “spending per transaction”, which will bedenoted as PLS throughout the paper (Fader et al. 2005).
Such model-based studies of customer behavior with PLS, however, still generatecontradictory findings. Table 1 shows a partial list of such research. For instance, Reinartz andKumar (2003) find that monetary value (monthly spending) is positively related to lifetime
duration. Borle, Singh, and Jain (2008), in a discrete-time contractual setting, found a positive
relationship between purchase rate and lifetime, a negative relation between purchase rate andspending per purchase, and no correlation between lifetime and spending. Singh, Borle, andJain (2009) show that purchase rate is negatively associated with lifetime and spending perpurchase, whereas lifetime and spending have a positive relationship, although theconsequence of investigating only one of the two gamma parameters (the shape instead of theusual scale parameter) is unknown. Fader et al. (2005) did not find any relationship between
purchase rate and average spending in their database, although they noted that its empiricalgeneralization also requires more studies.
-------------------------------Table 1-------------------------------So why is understanding the relationships among purchase rate, lifetime, and spending
important? Borle et al. (2008), for example, found that frequent shoppers spend less perpurchase. If that is the case, an obvious question is whether recruiting frequent shoppers leads
to increased or decreased CLV. The answer has a potentially severe consequence for3managerial decision making in CRM, where incorrect decisions could be detrimental to thefirm. To assess the net influence on CLV accurately, one must tradeoff the relative magnitudes
of increase in purchase rate with the decrease in spending, as well as account for any
associated change in lifetime.
The objectives of this research are threefold. The first is, from the observed RFM
measures of a customer, to develop an individual-level CLV model that identifies theunderlying customer traits of PLS, which are then linked to CLV. By evaluating the individual
difference in the intensity of PLS, one can obtain insights into their correlation and, thus,
accurately assess their net impacts on CLV. The model accounts for both observed(characteristics) and unobserved customer heterogeneity. Figure 1 illustrates the framework of
our approach.
-------------------------------Figure 1-------------------------------The second objective is, using this model, to obtain normative implications formarketing programs that maximize the CLV of existing customers. In particular, weinvestigate implementable programs for customer retention that is most effective in terms ofMarketing ROI, such as what action needs to be taken to which customers at which timing.
The third objective is, by applying our CLV model to prospective customers, to gain
insight into acquisition strategy. Without transaction data, RFM measures of prospective
customers do not exist and, therefore, we cannot calculate their CLV. By examining therelationship between PLS and the demographic characteristics of existing customers, we caninfer the PLS of prospective customers from their demographic information alone and, hence,
their CLV.
The paper is organized as follows. Section 2 discusses the general approach of ourmodel, followed by a detailed specification that relates RFM measures to PLS, which are then4linked to CLV. Then customer-specific metrics that are useful for CRM, such as expected
lifetime, 1-year survival rate, etc. are derived from the model. Section 3 describes the model
estimation. Section 4 presents the empirical analysis with actual customer data. Finally,
Section 5 concludes the paper with summary, limitations, and future directions.

2. MODEL
2.1. The General Approach
Our approach is to construct a Pareto/NBD-based stochastic model of buyer behavior(Schmittlein, Morrison, and Colombo 1987, hereafter referred to as SMC) for estimating CLV
in a “non-contractual” setting. In this case, a customer being “alive” or “dead” is inferredfrom recency-frequency data through simple assumptions regarding purchase behavior. The
CLV research using such a Pareto/NBD model includes Fader, Hardie and Lee (2005;
hereafter referred to as FHL), Reinartz and Kumar (2003), and Schimittlein and Peterson(1994; hereafter referred to as SP). Furthermore, for disaggregate modeling, we adapt thehierarchical Bayes extension of Pareto/NBD proposed by Abe (2009) to manage“heterogeneity.”-------------------------------Table 2-------------------------------Table 2 highlights our methodology in comparison with that of SP and FHL. For
recency-frequency data, both SP and FHL adopt a Pareto/NBD model that presumes a Poissonpurchase and an exponential lifetime processes whose parameters are independently
distributed as gamma. For monetary data, SP posit a normal-normal model, whereby purchaseamounts on different occasions within a customer are normally distributed with the meanfollowing a normal distribution in order to capture customer heterogeneity. FHL use agamma-gamma model, whereby the normal distributions within and across customers in SP5are replaced by gamma distributions. Both methodologies can be characterized as anindividual-level behavior model whose parameters are compounded with a mixture
distribution to capture customer heterogeneity. This in turn is estimated by an empirical Bayes
method.
The proposed methodology posits the same behavioral assumptions as SP and FHL, but
captures customer heterogeneity through a more general mixture distribution to account for
the interdependence among the three behavior processes of PLS using the hierarchical Bayes(HB) framework.
Before describing the model in detail, let us summarize the reasons for our methodology.
(1) Assumptions on customer behavior are minimal and posited in various past studies.
(2) Aggregation can be managed easily without resorting to integration, and individual-levelstatistics can be obtained as a byproduct of MCMC estimation.
(3) It produces correct estimates of standard errors, thereby permitting accurate statisticalinference even when the asymptotic assumption does not apply.
(4) It is easy to incorporate covariates.
2.2. Model Assumptions
This section describes the assumptions of the proposed HB model.
Individual Customer
Assumption 1: Poisson purchases. While active, each customer makes purchases according toa Poisson process with rate .
Assumption 2: Exponential lifetime. Each customer remains active for a lifetime, which has
an exponentially distributed duration with dropout rate .
Assumption 3: Lognormal spending. Within each customer, the amounts of spending onpurchase occasions are distributed as lognormal with location parameter .

6Assumptions 1 and 2 are identical to the behavioral assumptions of a Pareto/NBD model.
Because their validity has been studied by other researchers (FHL; Reinartz and Kumar 2000,
2003; SMC; SP), the justification is not provided here. Assumption 3 is specified because (1)the domain of spending is positive and (2) inspection of the distributions of spending amounts
within customers reveals a skewed shape resembling lognormal. As described previously, SP
and FHL assume normal and gamma, respectively, to characterize the distribution of spending
amounts within a customer.
Heterogeneity across Customers
Assumption 4: Individuals’ purchase rates , dropout rates , and spending parameters follow a multivariate lognormal distribution.

Assumption 4 permits correlation among purchase rate, lifetime, and spending parameters.
Because Assumption 4 implies that log(), log(), and log() follow a multivariate normal,
estimation of the variance-covariance matrix is tractable using a standard Bayesian method. A
Pareto/NBD combined with either a normal-normal (SP) or gamma-gamma (FHL) spending
model posits independence among the three behavioral processes. We will assess the impactof this independence assumption through comparison between multivariate and independent
lognormal heterogeneity. The impact of assuming a different heterogeneity shape (lognormal
rather than gamma) is an empirical issue, which will be evaluated by comparing independent
lognormals of the HB model with independent gammas of Pareto/NBD.
2.3. Mathematical Notations
For recency and frequency data, we will follow the standard notations {x, tx, T} used by
SMC and FHL. Lifetime starts at time 0 (when the first transaction occurs and/or themembership starts) and customer transactions are monitored until time T. x is the number of
repeat transactions observed in the time period (0, T], with the last purchase (xth repeat)7

occurring at tx. Hence, recency is defined as T-tx.  is an unobserved customer lifetime. For
spending, sn denotes the amount of spending on the nth purchase occasion of the customer
under consideration. Using these mathematical notations, the previous assumptions can be
expressed as follows:

(1)

x


(
T
)

e T
 x!
P[ x |  ]  x ( )   
 x! eif   Tx  0 ,1,2 ,..

if   T(2)

f ( )   e   (3)log(sn ) ~ N (log( ),  2 )(4)  2  
 log( ) log(  ) ~ MVN    ,   0
   0  



 
 
 log( ) 
 



 0
sn  0     
 2       2  

where N and MVN denote univariate and multivariate normal distributions, respectively. 2 isthe variance of logarithmic spending amounts within a customer.
2.4. Expressions for Transactions, Sales, and CLV

Given the individual level parameters for (, ), the expected number of transactions inarbitrary time duration w, E[X(w)|, ], is shown by evaluating E[] as(5)

E[ X ( w) |  ,  ]  E[ ] 
1  e  w where  min( , w) .

The expected sales during this period w is simply the product of the expected number oftransactions shown in Equation (5) and the expected spending E[sn|] as(6)

E[ sales( w) |  ,  , ,  ]  E[ sn |  ,  ]E[ X ( w) |  ,  ]  e / 2
2

8



1  e  w .
For CLV, we define “value” to be synonymous with “revenue” because margin and cost
information is unknown in this study. The general formula of CLV for an individual customer
under a continuous time framework, as appropriate for a Pareto/NBD model, is expressed asCLV   V (t ) R(t ) D(t )dt ,0

where V(t) is the customer’s value (expected revenue) at time t, R(t) is the survival function(the probability that a customer remains active until at least t), and D(t) is a discount factor
reflecting the present value of money received at time t (FHL; Rosset et al. 2003). Translating
to Assumptions 1-3, they are V (t )  E[ sn ] where E[S n ]   exp( 2 / 2) from thedefinition of lognormal, and R (t )  exp( t ) . With continuously compounded discounting ofan annual interest rate d, D (t )  exp(t ) , where   log(1  d ) with the time unit being ayear. Therefore, our CLV reduces to the following simple expression.
(7)





0

0

CLV   V (t ) R (t ) D(t )dt   ee / 2e e dt  2 2 / 2  t tHence, if we can estimate , , , and  for each customer from RFM data, we can compute
CLV as in Equation (7).
2.5. Incorporating Customer Characteristics

To control extraneous factors and gain insight into acquisition, we would like to relate
customer characteristic variables for customer i, di (a K × 1 vector) to customer specificparameters i , i , and i . A straightforward extension of Assumption 4 expressed in equation(4) results in a multivariate regression specification as follows:(8)  2  
 log(i ) log(  ) ~ MVN   Bd ,   
2
0
i 
i
i
       log(i )        2  

where B is a 3  K matrix of coefficients. When di contains a single element of 1 (i.e., no9

characteristic variables), the common mean, 0 = i for all customers i, is estimated.
2.6. Elasticities

Useful implications can be obtained from computing elasticities of CLV with respect to(a) , , and , and (b) characteristic variables di. From equation (7),ECLV (9)

CLV / CLV 1,
 / 

ECLV  


 

,

ECLV  1 ,implying that a one percent increase in the purchase rate or spending parameter causes a one
percent increase in CLV, whereas a one percent decrease in the dropout rate leads to less thana one percent increase in CLV (with the magnitude depending on the discount rate . Under ahigh interest rate, the impact of prolonging lifetime on CLV is not as rewarding because future
customer value would be discounted heavily.
The effect of customer characteristics on CLV can be decomposed into purchase rate,
lifetime, and spending processes to provide further managerial insight. Defining dik as the kth(continuous) characteristic of customer i, from Equations (8) and (9), the elasticity becomes

EdCLV
ik

CLVi / CLVi
d ik / d ik CLVi i CLVi i CLVi i  d iki d ik
i d ik  CLVi
 i d ik

(10)
b 


 bk  k i  bk  d ik
i  
 EpdCLV EldCLV
 EsdCLV
ik
ik
ik

where blk (l  { ,  ,}, k  1,..., K ) denotes (l, k)th element of matrix B. Applying the chain
rule, the derivative with respect to dik through i , i , and i , results in the sum of three, EldCLV, and EsdCLV
due, respectively, to purchase rate, lifetime, andelasticities, EpdCLV
ik
ik
ik
spending.
10

3. ESTIMATIONIn the previous section, simple expressions for the customer processes of purchase rate,
lifetime, and spending (and thus CLV) are derived from the basic behavioral Assumptions 1, 2,
and 3. To account for customer heterogeneity, the HB approach is adopted to bypass thecomplex aggregate expressions of the compounding mixture distribution.
The purchase rate and lifetime parts adopt the HB extension of the Pareto/NBD model
proposed by Abe (2009). It is estimated by MCMC simulation through a data augmentation
method. Because information about a customer being active (z = 1) or not at time T is
unknown, and if not active, the dropout time (y < T) is also unknown, z and y are considered
as latent variables.. Both z and y are randomly drawn from their posterior distributions.
The RFM data of a customer are denoted as x, tx, T, and as, where x, tx, and T are defined
as in SMC, and as represents the average spending per purchase occasion. Without theknowledge of spending variation within a customer from one purchase to another, however,there is no means to infer the variance of logarithmic spending 2, specified in equation (3),from RFM data alone. As the RFM provides cross-sectional measures, it contains information
only on spending variation between customers. Since it is easy to obtain 2 given the panel
data, here we assume that 2 is common across customers and estimated from historical data.
Assumption 3 permits standard normal conjugate updating in Bayesian estimation,
whereby the posterior mean is a precision-weighted average of the sample and the priormeans. For this method to work, however, we need the mean of log(spending) (or equivalently,
the logarithm of the geometric mean of spending amounts) from each customer, whereas the
M part of RFM data provides only the arithmetic mean of spending, as.

11





Following Equation (3), log(sn ) ~ N (log( ),  2 ) implies Esn   exp log( )   2 / 2 .

Replacing the expectation E[sn] and log() by their respective sample means,

1 x
 sn andx n 11 x
 log(sn ) , the following approximation can be obtained.
x n 1(11)

1 x
1log(sn )  log(as )   2

x n 1
2

When we evaluated the accuracy of this approximation with the department store FSP data of
400 customers used in Section 4, the correlation between the actual and the approximation
was 0.927, and the mean absolute percentage error was 6.8%.
3.2. Prior Specification

Reinstating customer index i, let us denote the customer specific parameters as

i  log(i ), log( i ), log(i ) , which is normally distributed with mean  i  Bd i andvariance-covariance matrix 0 as in Equation (8). Our objective is to estimate parametersi , yi , zi , i; B, 0  from the observed RFM data xi , t xi , Ti , asi ; i , where the index forcustomer i is made explicit. In the HB framework, the prior of individual-level parameter icorresponds to the population distribution MVN(Bdi, 0). The priors for the hyperparameters B
and 0 are chosen to be multivariate normal and inverse Wishart, respectively:
vec( B) ~ MVN b00 ,  00  ,0 ~ IW  00 , 00 

These distributions are standard conjugate priors for multivariate regression models.
Constants b00 ,  00 , 00 , 00  are chosen to provide very diffuse priors for the hyperparameters.
3.3. MCMC Procedure

We are now in a position to estimate parameters i , yi , zi , i; B, 0  using an MCMC
method. To estimate the joint density, we sequentially generate each parameter, given theremaining parameters, from its conditional density until convergence is achieved. Because

12these conditional densities are not standard distributions, the independent MH-algorithm isused.

4. EMPIRICAL ANALYSIS
4.1. Frequent Shoppers Program Data for a Department Store

We now apply the proposed model to real data. This dataset contains shopping records
from 400 members of a frequent shoppers program (FSP) at a large department store in
Japan.1 These members had joined the FSP during the month of July 2000 and theirtransactions were recorded for 52 weeks. The first 26 weeks of data were used for model
calibration and the second 26 weeks were used for validation. The available customer
characteristic variables were (a) Age, (b) Gender, and (c) Food, the fraction of store visits on
which food items were purchased (which is a proxy for store accessibility). The same data
were also used by Abe (2009), whose descriptive statistics are reported in Table 3. The
variance of log(spending) within a customer, 2, was estimated to be 0.895 from panel data,
as discussed in Section 3.1.
-------------------------------Table 3-------------------------------4.2. Model Validation

The MCMC steps were put through 15,000 iterations, with the last 5,000 used toconstruct the posterior distribution of parameters.
-------------------------------Table 4--------------------------------

1

In the HB estimation, data on sample households are utilized only to construct a prior for individual
customer-specific parameters. For this reason, the estimation result would be rather insensitive to the sample size,
as long as it is sufficiently large (i.e., 400). One will not gain much, for example, by using a sample of 10,000
customers. Hence, the scalability is not an issue in our approach.

13

Table 4 shows the result of three nested HB models: Independent (variance-covariancematrix 0 is diagonal without covariates), Correlated (general 0 without covariates), and Full(general 0 with all covariates). Correlated models with subsets of covariates are not reported
here because the Full model had the best marginal log-likelihood. The performance of the Full
HB model was evaluated with respect to the number of transactions and spending, obtained
from Equations (5) and (6), respectively, in comparison to the benchmark Pareto/NBD-basedmodel. The expected number of transactions, predicted by the Pareto/NBD, was multiplied byaverage spending asi to come up with the customer i’s spending.
-------------------------------Figure 2-------------------------------Figure 2 shows the aggregate cumulative purchases over time. Both models provide good
fit in calibration and good forecast in validation, which are separated by the vertical dashedline. With respect to the mean absolute percent errors (MAPE) between predicted andobserved weekly cumulative purchases, the HB performed better for validation (1.3% vs.
1.9%) and equal to the benchmark Pareto/NBD-based approach for calibration (2.5% forboth).
-------------------------------Figures 3 and 4-------------------------------Fit statistics at the disaggregate level provide more stringent performance measures.
Figure 3 shows the predicted number of transactions during the validation period, averaged
across individuals and conditional on the number of purchases made during the calibration
period. Figure 4 compares the predicted total spending during the validation period in asimilar manner. Both figures visually demonstrate the superiority of the HB over the
Pareto/NBD-based model.
-------------------------------Table 514

-------------------------------Table 5 compares the correlation and mean squared error (MSE) between prediction
and observation with respect to the number of transactions and total spending at the individual
customer level during the calibration and validation periods. The difference between
Pareto/NBD and Independent models, aside from Empirical Bayes (EB) versus HB, can be
attributed to the difference in the assumption of the heterogeneity distributions for  andwhether they are independent gammas (EB) or independent lognormals (HB). The slightadvantage of the Independent model over Pareto/NBD in predicting spending seems to justify
the lognormal heterogeneity for this dataset. The effect of relaxing the independentassumption and incorporating the covariates is reflected, respectively, by the difference
between Independent and Correlated and between Correlated and Full. Because all HBmodels perform similarly, the improvement in fit from accommodating correlation andcovariates is minor.
In sum, the Full HB model seems to fit and predict well in comparison to thePareto/NBD-based model, in terms of the number of transactions and spending at both theaggregate and disaggregate levels. However, the difference is minor. Considering that bothmodels use a Bayesian method (HB vs. EB) but assume a different prior, the result seems tosuggest that the estimated posterior distribution is driven mainly by data. The real advantageof the HB approach is in interpretation rather than prediction, as will be shown in thesubsequent sections.
4.3. Insights into Existing Customers
4.3.1. Interpretation of the Model Estimation

Having established the validity of the HB model, let us now examine Table 4 to interpretthe estimation result. Food, the fraction of store visits on which food items were purchased
and a proxy for store accessibility, is the most important covariate with a significant positive15

coefficient for purchasing (log()) and a significant negative coefficient for spending (log()).
Managerially, food buyers tend to shop more often but spend a smaller amount on eachpurchase. This finding is consistent with the story told by a store manager in that, althoughfood buyers spend a smaller amount on each shopping trip, they visit the store often enough to
be considered as vital. Another significant covariate for log() is Age, signifying that older
customers tend to spend more at each purchase.
Let us now turn our attention to the relationships among the purchase rate (, dropout
rate (, and spending (parameters. To check whether the independence assumption of thePareto/NBD is satisfied, the correlation of 0 must be tested on the intercept-only model(Correlated) but not the covariate model (Full). The reason is that if covariates explain thecorrelation among ,  and completely, then no correlation remains in the error term as
captured by 0. First, Table 4 indicated that the correlation between log( and log() is not
significantly different from zero, implying that the assumption of Pareto/NBD holds here.
Second, the correlation between log( and log(is significantly negative (-0.28), consistent
with the Food variable having opposite signs on log( and log() in the Full model. Figure 5
presents the scatter plot for the posterior means of the individual i and i (i  1,..,400) . One
can visually observe the correlation.
-------------------------------Figure 5-------------------------------Hence, the assumption of the independence between transaction and spending
components in the Pareto/NBD-based model (SP and FHL) does not hold in this dataset. For
researchers using the SP and FHL models, this finding emphasizes the importance ofverifying the independence assumption (as was done in SP and FHL). Managerially, thisnegative correlation implies that a frequent shopper tends to spend a smaller amount on eachpurchase. Furthermore, this correlation remains even after accounting for differences in16

customer characteristics, specifically Food (store accessibility), Age, and Gender, as seenfrom the correlation for the Full model. No correlation was found between purchase rate andlifetime or between lifetime and spending per purchase.
4.3.2. Customer-specific Metrics and CLV

Table 6 presents nine customer-specific metrics for the top and bottom 10 customers interms of CLV, along with the average, minimum, and maximum for the entire sample of 400
customers: posterior means of i, i,, and i; expected lifetime; survival rate after one year;
probability of being active at the end of the calibration period; an expected number oftransactions (using equation (5)); expected total spending during the validation period (using
equation (6)); and CLV (using equation (7)). In computing CLV, an annual interest rate of
15% (= 0.0027 per week) was assumed as in FHL.
-------------------------------Table 6-------------------------------There exists much heterogeneity across customers despite the use of the Bayesianshrinkage estimation. The mean expected lifetime is 10.0 years with the maximum andminimum of 24.7 and 1.3 years, respectively. The probability of being active at the end of thecalibration period ranges from 0.18 to 1.0, with the average being 0.93. Over the validation
period of 26 weeks, the expected number of transactions was 16.0 times with the total amountof 74,000 yen on average (divide by 100 to convert to the approximate US dollars). CLVranges from 40,000 yen to 10.2 million yen, with an approximate average of 0.69 million yen.
-------------------------------Figure 6-------------------------------Figure 6 shows a gain chart (solid line) in which customers are sorted according to thedecreasing order of CLV, and the cumulative CLV (y-axis, where the total CLV is normalizedto 1) is plotted against the number of customers (x-axis). In addition, two gain charts are17

plotted. The dash-dotted line is based solely on recency criterion, whereby customers are
sorted in the order of increasing recency (from most recent to least recent). The dotted line isa gain chart based on customers being ordered according to the sum of the three rankings ofrecency, frequency, and monetary value. The 45 degree dashed line corresponds to the
cumulative CLV for randomly ordered customers. This figure implies that recency criterion
alone is not sufficient to identify good customers, although many companies use this criterion.
On the other hand, combined use of the three measures (recency, frequency, and monetary
value), even with the naïve equal weighting scheme, seems to provide a rather accurateordering of CLV. This finding strongly supports the wide use of RFM analysis andregression-type scoring models among practitioners for identifying good customers(Malthouse and Blattberg 2005). While RFM measures can produce the relative ranking ofcustomers well, without the proposed model, the absolute CLV figure itself cannot be
obtained.
4.3.3. Customer Base and Customer Equity

We can compute the expected number of active customers (customer base) at the end of
the calibration period, that is January 1, 2001, from Table 6. Customer base is the sum ofactive probabilities (column 7) for all customers. Although the dataset contains 400 customers,
the effective number of active customers on January 1, 2001 is only 371.6 ( = 400×0.929 ).
When a customer turnover is high, customer base becomes much smaller than the registerednumber of customers in the dataset.
Customer equity is the expected CLV generated by all active customers at that time. By
aggregating the products of a CLV and the active probability on January 1, 2001, for all 400
customers, it becomes 273 million yen. Customer base and customer equity are firm’svaluable long-term indicators, neither of which are provided by the accounting statement.
4.3.4. Retention Program for Existing Customers

18Let us assume that, if a customer were indeed inactive, an investment level of c per
customer would win her back with probability r(c). Such a response function r(c), as shown in
Figure 7, can be constructed by decision calculus (Blattberg and Deighton 1996; Little 1971).
The figure implies that, by sending a coupon valued at 1,000 yen (i.e., investment c=1,000),
there is a 50% chance that inactive customers can be brought back. However, no matter how
large the coupon’s value is, the firm cannot bring back inactive customers with a probabilityof more than 0.8. The effect of such a coupon on the change in CLV is derived in the appendix.
We now consider two examples of customer-specific coupon mailing.
-------------------------------Figure 7-------------------------------Customer Retention Example 1: Given that a coupon with a different value is sent toeach customer on January 1, 2001, what level of c* maximizes the increase of her CLV?

The second column of Table 7 presents c* for the same 20 customers as in Table 6 (i.e.,
top 10 and bottom 10 in terms of CLV). The optimum coupon value varied by customers
between 0 yen and 1,500 yen, and the average was 500 yen. For 74 out of 400 customers,
c*=0. This implies that, for these customers, a coupon increases their CLV only by a small
amount such that the investment cannot be recouped. The optimal formula for c*, (A3) in the
appendix, indicates that a higher coupon value is more effective for customers with a higher
CLV and a lower active probability, which is consistent with the general pattern of Table 7.
-------------------------------Table 7-------------------------------Customer Retention Example 2: To increase CLV by 10,000 yen, how many
non-purchase days should a firm wait before mailing a coupon with a face value of 500
yen?

19

Here, the logic is as follows. If an active probability is sufficiently high, the investment
would be wasted (as in (A3)). An active probability decreases as time passes (as in (A1)). But
if the firm waited too long, an investment of 500 yen would not be sufficient to induce anincrease of CLV by 10,000 yen. Therefore, there is an optimum timing to step in with thecoupon. The third column of Table 7 shows this timing (i.e., non-purchase days or recency),
which varies by customers from 3.5 days to 120.4 days. In general, an earlier action is betterfor customers with a higher CLV. The pattern is consistent with the formula (A4) in the
appendix.
4.4. Insights into Prospective Customers

We now focus our attention to prospective customers for the purpose of acquisition. In
particular, to assess the influence of demographic covariates on CLV, we examine theelasticity of these covariates on PLS and CLV.
4.4.1. Elasticity of Covariates on CLV

Table 8 reports the decomposition of the elasticity of CLV with respect to each covariate
into purchase rate, lifetime, and spending components, as shown in Equation (10). To accountfor parameter uncertainty, elasticity is computed for each set of the 5000 MCMC draws of blk
and i according to Equation (10). This is then averaged over the 5000 draws and 400
customers. When the posterior mean of blk and i is directly substituted into Equation (10)(bottom table) instead of averaging over MCMC draws (top table), elasticity with respect tothe lifetime component is overestimated by about 50% (because of nonlinearity in i). Thisoverestimation occurs even when customer heterogeneity is accounted for.
One of the advantages of our Bayesian approach is that parameter uncertainty can beevaluated easily with the sampling estimation method. Ignoring their uncertainty andcomputing various statistics from their point-estimates, say MLE, as if parameters are20

deterministic, could produce a biased result and lead to incorrect managerial decisions (Gupta
1988).
-------------------------------Table 8-------------------------------4.4.2. Implication for New Customer Acquisition

To clarify the impact of covariates, the solid line in Figure 8 plots the value of log(CLV)for different values of a covariate when the remaining two covariates are fixed at their meanvalues. These graphs are computed using the mean estimate of the coefficients of the Fullmodel shown in Table 4, assuming that all covariates are continuous. For the Female covariate,
therefore, it should be interpreted as how log(CLV) varies when the gender mixture ischanged from the current level of 93.3% female, while keeping the other two covariates
unchanged. The dotted vertical line indicates the mean value of the covariate underconsideration. Both Food and Age have strong influences on log(CLV), whereas Female
exerts a very weak influence, consistent with the significance of these covariates.
-------------------------------Figure 8-------------------------------Figure 8 also attempts to decompose the influence of covariates on log(CLV) into threecomponents: purchase rate, lifetime, and spending. Taking the logarithm of the basic formulaof CLV in Equation (7) results in the following summation expression:log(CLV )   log(   )  log( )  log( )   2 / 2 Lifetime  component  Purchase rate  component  Spending  component  constant

The graph can be interpreted as stacking the lifetime, purchase rate, and spending
components from top to bottom, thus constituting the overall log(CLV). To account for the
scale differences among these components, each was adjusted to become 1 at the mean valueof the covariate. Therefore, log(CLV) = 3 at the dotted vertical line.
21

The direction and magnitude of the effect of each covariate on the three components are
consistent with the signs of the posterior means blk (l  { ,  ,}, k  1,..., K ) . Increasing thefraction of food buyers improves lifetime and purchase rate, but decreases spending perpurchase, yielding a net increase in the overall CLV. Increasing the fraction of elderly people
increases the spending without much influence on lifetime and purchase rate, thereby
resulting in a net increase in the overall CLV. Increasing the fraction of females leads to little
improvement in all three components and, hence, a negligible increase in the overall CLV.
Elasticity decomposition provides managers with useful insights into acquisition. An
effort to manipulate certain customer characteristics might impact lifetime, purchase rate, andspending components in opposite directions, thereby canceling each other to produce acollectively reduced effect as the total on CLV. For example, much of the improvement inpurchase rate (from increasing the fraction of food buyers) is negated by the decline inspending per purchase. In addition, only the lifetime improvement provides the net
contribution to CLV, as can be seen from Table 8 and the near flat dashed line of Figure 8. In
contrast, an effort to increase the proportion of elderly people is met with a boost in CLV, due
to increased spending per purchase with only a small negative influence on purchase rate.
To build an effective acquisition strategy from these results, managers must strike a finebalance between desired customer characteristics (i.e., demographics), desired behavioral
profiles (i.e., purchase rate, lifetime, and spending), responsiveness (elasticity) of thecharacteristic covariates on CLV, and the acquisition cost of the desired target customers.
4.5. Second Dataset: Retail FSP Data for a Music CD Chain

The second dataset, which was also analyzed in FHL, contains the record of 500
customers in an FSP of a large music CD store chain. The period covers 52 weeks beginning
September 2003. The following customer characteristics were used as covariates: the amountof the initial purchase, age, and gender.

22

-------------------------------Table 9-------------------------------Table 9 reports the model estimation. Let us first examine the Full HB model, which
results in the highest marginal log-likelihood, for significant explanatory variables. First, theamount of an initial purchase is positively significant on log() and log(), implying that
customers with a larger initial purchase tend to buy more often and spend more per purchasein subsequent purchases. Second, older customers appear to spend more per purchase.
Next, we turn our attention to the Correlated model for the relationship among , , and .
First, we see that the correlation between log() and log() is not significantly different from0, implying that the assumption of Pareto/NBD holds here. Second, the correlation between
log( and log(is significantly positive (0.14), consistent with the initial purchase covariate
having the same signs on log( and log(. Once again, the independence assumption of thetransaction and spending components does not hold here. This time, however, the sign is in
the opposite direction, implying that the correlation between purchase rate and spending per
occasion is data dependent. Managerially, the correlation implies that frequent buyers spend
more per shopping occasion. Also, note that when covariates are included (Full model), thecorrelation is no longer significant. This fact indicates that the difference in initial amount andage can explain the correlation between purchase rate and spending.
-------------------------------Table 10 and Figure 9-------------------------------Table 10 shows the elasticity decomposition of CLV into purchase rate, lifetime, andspending components. When parameter uncertainty is not accounted for, the lifetime
component is overestimated by approximately 20%, as was the case for the department store
data. The elasticity decomposition of log(CLV) into the three components for varying levelsof the three covariates is presented in Figure 9. A higher initial purchase amount is related to a23higher CLV by increasing the purchase rate and spending with almost no change in lifetime.
Older customers are associated with a lower purchase rate, longer lifetime, and higher
spending per purchase with a positive net contribution to CLV. Female customers areassociated with a lower purchase rate, shorter lifetime, and less spending with a negative net
contribution to CLV.

5. CONCLUSIONS

The wide use of RFM analysis in CRM suggests that these measures contain rather richinformation about customer purchase behavior. However, the existing literature provides
conflicting findings on the relation between RFM and CLV, and several authors haveadvocated the need for further studies to provide empirical generalization (Blattberg,
Malthouse, and Neslin 2009). The present research sought to clarify the issue through theidentification of the underlying customer traits characterized by the interrelated behavior
process of purchase rate, lifetime, and spending from statistical RFM measures. The PLS
process posited the same behavioral assumptions as the established Pareto/NBD-based modelstudied by other researchers. The hierarchical Bayesian extension for constructing theindividual-level CLV model permitted the application of accurate statistical inference on
correlations, while controlling covariates and avoiding complex aggregation. The model also
related customer characteristics to the buyer behaviors of purchase rate, lifetime, andspending, which were, in turn, linked to CLV to provide useful insights into retention ofexisting customers as well as acquisition of new customers.
Two FSP datasets, one from a department store and another from a CD chain, were
investigated in the empirical analysis. The proposed CLV model provided ninecustomer-specific metrics: posterior means of i, i,, and i; expected lifetime; survival rate
after one year; probability of being active at the end of the calibration period; an expected24

number of transactions; expected total spending during the validation period; and CLV. These
metrics are especially useful for identifying preferred customers and taking marketing actions
targeted at the individual level in CRM. By maximizing Marketing ROI, we illustrated two
examples of retention couponing that specifies what value needs to be mailed to whichcustomers at which timing.
It was also found that recency criterion alone was not sufficient to identify good
customers, although many companies use this criterion. On the other hand, combined use of
the three measures (recency, frequency, and monetary value), even with the naïve equal
weighting scheme, seemed to provide a rather accurate ordering of CLV. This finding strongly
supports the wide use of RFM analysis and regression-type scoring models among
practitioners in identifying good customers.
Finally, by relating the behavioral traits of PLS with demographic characteristics, we
obtained insights into acquisition strategy for prospective customers with a high CLV. For
example, the first dataset exhibited a statistically significant negative correlation between
purchase rate and spending (-0.28). In such a case, recruiting food buyers for the purpose of
improving purchase rate would be negated by the decline in spending per purchase and, as aresult, only an improvement in lifetime contributed to the net increase in CLV. Note that acorrelation between purchase rate and spending is data dependent. Specifically, one of ourdatasets exhibited a negative relation, whereas the other dataset showed a positive relation.
For a negative relationship, their relative magnitudes must be evaluated to assess the net
impact on CLV --- an insight that is crucial, especially in the context of customer acquisition.
The current study is only the beginning of a stream of research addressing customer
behavior in a “non-contractual” setting. Possible extensions are synonymous with limitationsof the proposed method. From the consideration of RFM measures as sufficient statistics, thecurrent model posited the behavioral assumptions of a Poisson purchase, an exponential

25

lifetime and lognormal spending. With a customer’s complete transaction history, however,more elaborate behavioral phenomena can be modeled.

ACKNOWLEDGMENTS

I would like to express my sincere thanks to Prof. John D. C. Little, who supported me
throughout my career since I was a student at MIT. I am also deeply grateful to Glen Urban
and John Hauser, who served my doctoral dissertation committee, for editing this book. This
research was supported by grant #19530383 from the Japan Society for the Promotion of
Science.

26

APPENDIX: Derivation for the Effect of a Retention Program on CLV

Following Blattberg and Deighton (1996), if a customer were indeed inactive, aninvestment level of c per customer would win her back with probability r(c), wherer (c)  R 1  e  kc

where c  0 .

Here R and k are parameters, which can be estimated by decision calculus. Because whether acustomer is active is unknown, we use its stochastic metric, the predicted probability of being
active p (Table 6), derived from the proposed model, as shown below (Abe 2009).
(A1)

1

p
1

e



(    )( recency )1

Let us denote the probability of being active with and without the retention program,respectively, as pa and p. Then, by assumption, pa  p  (1  p )r (c) . Therefore, an increase in
CLV as a result of the program, , is expressed as Equation (A2).
(A2)

   paV  c   pV
 (1  p)r (c)V  cwhere V  CLV

Customer Retention Example 1: Given that a coupon with a different value is sent toeach customer on January 1, 2001, what level of c* maximizes the increases of her CLV?

The optimum c* is obtained by maximizing  with respect to c by solving the first-ordercondition of (A2) as
(A3)

c* 

1
ln(1  p)kRV .
kCustomer Retention Example 2: To increase CLV by 10,000 yen, how many
non-purchase days should a firm wait before mailing a coupon with a face value of 500
yen?

Substitute =10000 and c=500, and solve (A2) with respect to active probability p*.
p*  1 cr (c)V

Substitute p=p* in (A1) and solve for recency.
(A4)

recency*   

1
c
ln 
 1
     r (c)V  c   

27

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Abe, Makoto (2009), “‘Counting Your Customers’ One By One: A Hierarchical Bayes
Extension to the Pareto/NBD Model,” Marketing Science, 28 (3), 541-553.
Blattberg, Robert C. and John Deighton (1996), “Manage Marketing by the Customer Equity
Test,” Harvard Business Review, (July-August), 36-44.
Blattberg, Robert C., Edward C. Malthouse and Scott Neslin (2009), “Lifetime Value:
Empirical Generalizations and Some Conceptual Questions,” Journal of Interactive
Marketing, 23 (2), 157-168.
Borle, Sharad, Siddharth S. Singh and Dipak C. Jain (2008), “Customer Lifetime Value
Measurement,” Management Science, 54 (1), 100-112.
Buckinx, Wouter and Dirk Van den Poel (2005), “Customer Base Analysis: Partial Defection of
Behaviourally Loyal Clients in a Non-Contractual FMCG Retail Setting,” European
Journal of Operational Research, 164 (1), 252-268.
Fader, Peter S., Bruce G. S. Hardie, and Ka Lok Lee (2005), “RFM and CLV: Using Iso-Value
Curves for Customer Base Analysis,” Journal of Marketing Research, 42 (4), 415-430.
Gupta, Sunil (1988), “Impact of Sales Promotions on When, What, and How Much to Buy,”
Journal of Marketing Research, 25 (4), 342-355.
Hughes, Arthur (2000), Strategic Database Marketing (2nd Ed.), New York: McGraw-Hill.
Little, John D.C. (1971), “Models and Managers: The Concept of a Decision Calculus,”
Management Science, 16 (8) (1971), pp. B466–B485
Malthouse, Edward C. and Robert C. Blattberg (2005), “Can We Predict Customer Lifetime
Value?” Journal of Interactive Marketing, 19 (1), 2-15.
Reinartz, Werner J. and V. Kumar (2000), “On the Profitability of Long-Life Customers in a
Noncontractual Setting: An Empirical Investigation and Implications for Marketing,”
28

Journal of Marketing, 64 (4), 17-35.
----- and ----- (2003), “The Impact of Customer Relationship Characteristics on Profitable
Lifetime Duration,” Journal of Marketing, 67 (1), 77-99.
-----, J. S. Thomas and V. Kumar (2005), “Balancing Acquisition and Retention Resources to
Maximize Customer Profitability,” Journal of Marketing, 69 (1), 63-79.
Rosset, Saharon, Einat Neumann, Uri Eick, and Nurit Vatnik (2003), “Customer Lifetime Value
Models for Decision Support,” Data Mining and Knowledge Discovery, 7 (July), 321-339.
Schmittlein, David C., Donald G. Morrison, and Richard Colombo (1987), “Counting yourcustomers: Who are they and what will they do next?” Management Science, 33 (1), 1-24.
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Quantitative Marketing and Economics, 7 (2), 181-205.

29

Table 1. Conflicting Findings on the Correlation among Purchase Rate, Lifetime, andSpending

Schmittlein & Peterson (1994)

Purchase Rateand Lifetime
N/A

Purchase Rateand Spending0

Lifetime
and Spending+

Reinartz & Kumar (2003)

N/A

N/A

+

Reinartz et al. (2005)

+

N/A

N/A

Fader et al. (2005)

0

0

N/A

Borle et al. (2008)

+

-

0

Singh et al. (2009)

-

-

+

Table 2. Comparison with Existing Methods
Empirical Bayes Model
Data

Model

Pareto/NBD
RF(recency-frequency) (SMC 1987)
M(monetary)normal-normal(SP 1994)
gamma-gamma(FHL 2005)

Individual Behavior
Poisson purchase ()Memoryless dropout ()Normal spending (mean )
Gamma spending (scale )

Heterogeneity
Distribution
~ Gamma
~ Gamma and  independent ~ Normal independent
~ Gamma independent

Proposed Hierarchical Bayes Model
Data

Model

Poisson/exponential
RF(Abe 2009)(recency-frequency)
lognormal-lognormalM(monetary)

Individual Behavior
Poisson purchase ()Memoryless dropout ()Lognormal spending(location )

30

Heterogeneity
Distribution~ MVL correlated

Table 3. Descriptive Statistics for the Department Store Data

Number of repeat purchases
Length of observation T (days)
Recency (T-t) (days)
Average purchase amount (×105 yen)
Food
Age
Femalemean
16.02
171.24
24.94
0.067
0.79
52.7
0.93

std. dev.
16.79
8.81
42.82
0.120
0.273
14.6
0.25

minimum maximum
0
101
151
181
0
181
0.0022
1.830
0
1
22
87
0
1

Table 4. Estimation Results of HB Models (department store)(Figures in parentheses indicate the 2.5 and 97.5 percentiles)* indicates significance at the 5% level

Purchaserate
log()

DropoutRatelog()

Spending
Parameterlog()

Independent

Correlated

Full

Food

-0.82
(-0.93, -0.72)
---

-0.81
(-0.92, -0.71)
---

Age

---

---

Female(male = 0)
Intercept

---

---

Food

-6.24
(-7.03, -5.52)
---

-6.13
(-7.10, -5.56)
---

Age

---

---

Female(male = 0)
Intercept

---

---

Food

-3.59
(-3.67, -3.51)
---

-3.57
(-3.64, -3.49)
---

Age

---

---

Female(male = 0)

---

---

---

-0.33
(-0.59, 0.01)-0.28*(-0.39, -0.17)-0.01(-0.31, 0.27)
-2105

-2.03
(-2.52, -1.51)
1.50*(1.11, 1.89)-0.21(-0.84, 0.40)
0.15(-0.20, 0.48)-5.03(-6.49, -3.57)-1.09(-2.66, 0.26)-0.34(-2.35, 1.49)
0.01(-1.20, 1.38)-3.23(-3.61, -2.86)-1.34*(-1.62, -1.06)
1.18*(0.71, 1.65)
0.11(-0.15, 0.39)-0.24(-0.51, 0.09)-0.14*(-0.25, -0.01)-0.07(-0.35, 0.24)-2078Intercept

correlation( log(), log() )correlation( log(), log() )correlation( log(), log() )marginal log-likelihood-----2111

31

Table 5. Disaggregate Fit of Pareto/NBD and HB Models (department store)

Pareto/NBD Independent
Spending
Correlation
MSE

Transactions
Correlation
MSE

Correlated

Fullvalidation
calibration
validation
calibration

0.80
0.99
0.39
0.02

0.83
0.99
0.35
0.06

0.83
0.99
0.35
0.06

0.83
0.99
0.35
0.06

validation
calibration
validation
calibration

0.90
1.00
57.7
1.22

0.90
1.00
57.1
4.61

0.90
1.00
57.0
4.06

0.90
1.00
56.5
3.92

32

Table 6. Customer-Specific Metrics for Top and Bottom 10 Customers (department store)IDmean

mean

mean

Mean

1 yearProbability

Expected

Expected

CLV

Expected

Survivalof being

number of

total

(x105

lifetime

rate

yen)

(years)active attransactions

spending

the end ofinin val.

calibration

validation

periodperiod

(x10

5

yen)

1
2
3
4
5
6
7
8
9
10
…
391
392
393
394
395
396
397
398
399
400

3.20
1.98
1.15
2.55
1.11
2.23
2.86
1.10
2.19
0.87
…
0.29
0.15
0.29
0.10
0.38
0.10
0.24
0.20
0.10
0.14

0.001650.00173
0.00205
0.00188
0.003380.00191
0.00206
0.002020.00206
0.00273…
0.01218
0.00750
0.011510.039150.02974
0.043070.00586
0.006990.047130.01709

0.075
0.056
0.097
0.036
0.088
0.034
0.027
0.067
0.034
0.090
…
0.011
0.016
0.011
0.049
0.009
0.044
0.008
0.009
0.034
0.016

24.7
21.5
19.9
22.1
11.3
22.4
18.6
19.3
19.7
13.8
…
3.3
5.7
3.4
1.4
2.1
1.4
6.5
6.1
1.3
2.2

0.926
0.922
0.910
0.918
0.862
0.916
0.910
0.910
0.909
0.886
…
0.665
0.754
0.666
0.463
0.555
0.480
0.786
0.762
0.454
0.581

1.000
1.000
0.999
1.000
0.997
1.000
0.999
0.996
1.000
0.999
…
0.379
0.803
0.381
0.436
0.182
0.450
0.951
0.862
0.420
0.601

ave
min
max

0.66
0.07
3.78

0.005640.001650.04713

0.038
0.007
0.207

10.0
1.3
24.7

0.823
0.454
0.926

0.929
0.182
1.000

33

81.5
50.3
29.2
64.7
27.7
56.7
72.5
27.8
55.5
21.9
1.7
2.7
1.8
0.7
0.5
0.7
5.5
4.1
0.7
1.6

9.61
4.38
4.43
3.60
3.80
3.01
3.01
2.93
2.91
3.09
…
0.03
0.07
0.03
0.05
0.01
0.05
0.07
0.06
0.04
0.04

102.0
45.4
45.1
37.2
33.9
31.0
30.3
29.6
29.3
29.0
…
0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4

16.0
0.5
96.1

0.74
0.01
9.61

6.9
0.4
102.0

…

Table 7. Customer-Specific Retention Action for Top and Bottom 10 Customers(department store)

ID

CouponValue(yen)

Recency to Probability
CLV
Waitof being(x105 yen)(days)
active atthe end of
calibration

1
2
3
4
5
6
7
8
9
10

407
235
510
221
623
0
385
661
186
372

3.5
6.5
7.8
5.9
7.0
6.9
5.8
10.1
6.9
9.6

1.000
1.000
0.999
1.000
0.997
1.000
0.999
0.996
1.000
0.999

102.0
45.4
45.1
37.2
33.9
31.0
30.3
29.6
29.3
29.0

391
392
393
394
395
396
397
398
399
400

916
684
911
887
949
864
372
561
822
713

60.5
110.8
62.6
54.0
37.8
55.2
96.3
106.7
66.2
120.4

0.379
0.803
0.381
0.436
0.182
0.450
0.951
0.862
0.420
0.601

0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4

ave
min
max

500
0
1500

32.6
3.5
120.4

0.929
0.182
1.000

6.9
0.4
102.0

34

Table 8. Decomposition of CLV Elasticity into Three Components (department store)

Accounting for Parameter Uncertainty
FOOD

AGE

FEMALE

Total

0.53

0.63

0.21

purchase rate: pCLV

1.17

-0.11

0.14

lCLV

0.41

0.12

-0.04

spending: sCLV

-1.05

0.63

0.10

FOOD

AGE

FEMALE

Total

0.76

0.69

0.18

purchase rate: pCLV

1.17

-0.11

0.14

lifetime: lCLV

0.63

0.17

-0.06

spending: sCLV

-1.05

0.63

0.10

lifetime:

Ignoring Parameter Uncertainty* Note that elasticity for lifetime but neither purchase frequency nor spending is different
when uncertainty is ignored. This is because, as shown in equation (10), only  enters theelasticity formula in a nonlinear fashion.

35

Table 9. Estimation Results of Various Models (music CD chain)

Purchaserate
log()

DropoutRatelog()

Spending
Parameterlog()

Intercept
InitialamountAge
Female(male=0)Intercept
InitialamountAge
Female(male=0)Intercept
InitialamountAge

Female(male=0)
correlation

Independent

Correlated

Full

-2.11
(-2.19, -2.03)
---

-2.11
(-2.19, -2.03)
---

---

---

---

---

-5.18
(-5.63, -4.74)
---

-5.14
(-5.64, -4.72)
---

---

---

---

---

-1.18
(-1.22, -1.13)
---

-1.18
(-1.22, -1.13)
---

---

---

---

---

---

0.20
(-0.02, 0.43)-2.10(-2.34, -1.85)
0.37*(0.11, 0.63)-0.26(-0.87, 0.34)-0.13(-0.29, 0.03)-5.06(-5.89, -4.34)
0.02(-1.09, 0.94)-0.15(-1.84, 1.39)
0.05(-0.60, 0.64)-1.49(-1.63, -1.35)
0.50*(0.36, 0.65)
0.47*(0.12, 0.82)-0.03(-0.10, 0.05)
0.19(-0.04, 0.42)

---

0.14*
(0.01, 0.27)

0.10(-0.05, 0.24)

---

0.01
(-0.22, 0.24)

0.01(-0.20, 0.22)

-2908

-2906

-2889

( log(), log() )correlation( log(), log() )correlation( log(), log() )marginal
log-likelihood* indicates significance at the 5% level36

Table 10. Decomposition of CLV Elasticity into Three Components (music CD chain)

AGE

FEMALE

Total

InitialAmount
0.31

0.12

-0.10

purchase rate: pCLV

0.13

-0.09

-0.06

lCLV

0.00

0.06

-0.02

spending: sCLV

0.18

0.15

-0.01

lifetime:

37

Figure 1. Framework of Our Approach

Underlying customer traits are derived from the observed RFM measures, which are in turnlinked to CLV. The model accounts for both observed (characteristics) and unobserved
customer heterogeneity.
Behavior Traits

Observed Measures
R(recency)P(purchase rate)

Model
L

F
(frequency)(lifetime)

CLV

S

M
(monetary value)(spending)

Customer Characteristics
Figure 2. Weekly Cumulative Repeat Transaction Plot

The vertical dashed line separate the calibration period and forecast period. Both proposedmodel (HB) and Pareto/NBD models provide good fit on the aggregate cumulative purchases.

38

Figure 3. Conditional Expectation of Future Transactions

The proposed HB model shows a better forecast than Pareto/NBD model on the predicted
number of transactions during the validation period, averaged across individuals andconditional on the number of purchases made during the calibration period.

Figure 4. Conditional Expectation of Future Spending

The proposed HB model shows a better forecast than Pareto/NBD model on the predictedamount of spending during the validation period, averaged across individuals and conditionalon the number of purchases made during the calibration period.

39

Figure 5. Scatter Plots of Posterior Means  and 

Each point corresponds to one of the 400 households estimated. Note the negative correlation
between purchase rate (lamda) and spending (eta).

Figure 6. Gain Chart for CLV based on HB model and Simple Recency

RFM scoring, whereby three measures (RFM) are combined with equal weights, seems toprovide a rather accurate ordering of CLV. However, customer ranking using of just one
measure of recency is not that accurate.
Lift Curve

1
0.9
0.8

Cumulative CLV

0.7
0.6
0.5
0.4
0.3
0.2
HB
Recency only
RFM scoring
random

0.1
0

0

50

100

150
200
250
No. of Customers

40

300

350

400

Figure 7. Retention Response Function

Investment level of c per customer would bring him/her to become active with probability r(c).
The function can be approximated by an exponential curve.

0.8 =
0.5

1000 yen41

Figure 8. Impact of Covariates on CLV Decomposed into Three Components(Department store)
The dotted vertical line indicates the mean value of the covariate under consideration.

lifetimepurchase rate

spending
Covariate = Food

Increasing the fraction of food buyers improves lifetime and purchase rate, but decreases
spending per purchase, yielding a net increase in the overall CLV.

lifetimepurchase rate

spending
Covariate = Age

Increasing the fraction of elderly people increases the spending without much influence on
lifetime and purchase rate, thereby resulting in a net increase in the overall CLV.
42

lifetimepurchase rate

spending
Covariate = Female

Increasing the fraction of females leads to little improvement in all three components and,hence, a negligible increase in the overall CLV.

Figure 9. Impact of Covariates on CLV Decomposed into Three Components (CD chain)

lifetimepurchase rate

spending
Covariate =Amount of Initial Purchase

A higher initial purchase amount is related to higher CLV by increasing the purchase rate andspending with almost no change in lifetime.
43

lifetimepurchase rate

spending
Covariate = Age

Older customers are associated with a lower purchase rate, longer lifetime, and higher
spending per purchase with a positive net contribution to CLV.

lifetimepurchase rate

Covariate = Female

spending

Female customers are associated with a lower purchase rate, shorter lifetime, and lessspending with a negative net contribution to CLV.
44