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and keep him. This amount is calledcustomer lifetime value (LTV).

Customer value calculations helpenterprises solve various fundamentalproblems like budgeting customeracquisition expenses, selection ofrecruiting media (the LTV is generallydifferent according to the media used),or of types of offer or distributing effortsbetween prospecting and preservingcustomers. Well-conducted LTV analysiscan also help build a competitivestrategic advantage.1

The modelling of customer valuedepends on the context and on thecustomer’s relationship behaviour(whether the situation is contractual ornot). A brief analysis shows that

INTRODUCTIONThe development of ‘customer oriented’interactive database marketing brings tothe fore, in most contexts, the need forcustomer value and customer portfolioassessment methods. These methods haveeither been fine-tuned in directmarketing and catalogue sales ordeveloped in sales force effort allocationmodels. There are many ways in whichthey could be improved but they requiresystematisation and unification first.

From a customer attraction andretention perspective, a profitablecustomer is a customer whose income,generated during the commercialrelationship, exceeds with an acceptableamount costs supported to attract, satisfy

124 Journal of Targeting, Measurement and Analysis for Marketing Vol. 11, 2, 124–147 � Henry Stewart Publications 1479-1862 (2002)

Papers

Customer value modelling:Synthesis and extension proposalsReceived (in revised form): 14th August, 2002

Mihai Calciuis Associate Professor of Marketing at the Institute of Business Administration (IAE) — European Institute of Direct Marketing(IEMD), University of Sciences and Technologies, Lille, France. His preferred subjects are marketing decision support modelsand systems, interactive marketing, e-commerce and customer relationship management.

Francis Salernois Professor of Marketing and Head of the Marketing Department in the Institute of Business Administration (IAE) — EuropeanInstitute of Direct Marketing (IEMD), University of Sciences and Technologies, Lille, France. He specialises in strategicmarketing, interactive marketing and customer relationship management.

Abstract Customer lifetime value is a key concept in relationship marketing. Thispaper develops, synthesises and organises formulae for computing this lifetime value ina progressive manner and continues a research direction started some years ago.Calculations are organised using a double taxonomy: customer relationship behaviourmodels (retention and migration models) and algebraic and matrix computing methods.The formulae suggested allow for the explicit representation of recency-censoredcustomer migration processes, the flexible integration of desynchronised financial flowsand extend customer value optimisation procedures from the retention to the migrationmodel.

Mihai Calciu andFrancis SalernoInstitute of BusinessAdministration — EuropeanInstitute of DirectMarketing, University ofSciences and Technologies,104 Avenue du PeupleBelge, F- 59043 Lille cedex,France.

Tel: �33 3 2012 3409;Fax: �33 3 2012 3448;e-mail:[email protected]

The algebraic approach and the matrixapproach are treated apart. Formulationsprogressively integrate transactional flowsexpressing probabilities to pass orders,financial flows composed of gains andexpenses and customer and prospectvalue optimisation procedures based uponlong-term calculations.

The use of certain representationartefacts (migration trees and diagrams,transition probability matrixes) facilitatesthe development of some recursivealgebraic formulae for LTV calculations.

Several additional aspects concerningdynamics of customer migration, likeright and left censored migrationprocesses or mechanics of purging thecustomer list, are treated explicitly.

EXISTING MODELS AND NATUREOF MODELLING APPROACHES

Retention, migration and mathematicalapproaches of customer value

Dwyer2 presents examples of LTVcalculations drawn from the two majorcontexts that were distinguished by

modelling efforts are based on thedistinction between customer retentionand migration.

This study of existing models showsalso that, in a first set of contributions,the mathematical developments arealgebraic and focus on retention modelswhile, in another set, they are based onmatrix approaches and applied mainly tothe migration model.

The need to desynchronise financialflows (expenses and gains) in thecustomer relationship, is also put forwardby these contributions.

Based upon results emerging from abroad examination of these models, thispaper guides towards systematisation andunification. It finds elements that arecommon and identifies those that arenot. It adds components that seem absentfrom the other approaches. In this way, aset of formulae that complete existingsolutions is developed and adapted.

These, existing or new, formulae aregrouped in a progressive customer valuecalculation framework. It is presentedschematically in Figure 1 and in detail inAppendix 1.

� Henry Stewart Publications 1479-1862 (2002) Vol. 11, 2, 124-147 Journal of Targeting, Measurement and Analysis for Marketing 125

Customer value modelling: Synthesis and extension proposals

Figure 1 Modelling customer value, a research framework

Approach Customer relationship behaviour

modelsStages and formulae in customer

value calculations

Algebraic

Matrix

Retention

model

Migration

model

Migration and

retention model

Transaction flows:

• Buying probabilities

Financial flows:

Customer value•

Prospect value

Optimisation:

•

Customer value•

RFM (recency, frequency and monetary)modelling framework that dominatesindustry practice in direct marketing andcatalogue sales. Some developments thatare more specific to these sectors can alsobe found in Birtran and Mondschein.11,12

A comparative analysis of thesecontributions brings several points to theforefront, which underline the need of asystematisation and unification effort:

— Berger and Nasr insist on thedesynchronisation of financial flows(expenses and gains) during acustomer relationship. They adopt analgebraic approach that they applyessentially to the retention model andneglect the migration model to acertain extent

— Blattberg and Deighton’sdevelopments are also algebraic. Theyare limited to the retention model,insist on customer value optimisationin a long-term perspective and focuson prospects

— the matrix solutions of Pfeifer andCarraway are applied primarily to themigration model. They adapt somelong-term actual value calculationsfrom Blattberg and Deighton to thematrix approach, but the optimisationprocedures adopted by them aredifferent (specific to Markov decisionsprocesses).

The nature and mechanics of thecustomer’s individual response

The two types of temporal behaviour ofcustomers in a relationship with acompany (that have been nameddifferently according to authors: ‘lost forgood’/‘always has share’ by Jackson,‘retention model’/‘migration model’ byDwyer or ‘contractual’/‘non-contractual’by Reinartz and Kumar),13 displaydifferent migration patterns.

The retention model considers that a

Jackson3 in the industrial buyingenvironment: buyers of type ‘always ashare’ and buyers of type ‘lost for good’.

On this occasion Dwyer4 reveals themore general character of this typology,extends it to the consumer market, andshows that the ‘always a share’behaviour, that he associates with the‘migration model’, is generallyrepresentative for catalogue buying; whilethe ‘lost for good’ behaviour, that heassociates with the ‘retention model’, isrepresentative for financial services, presssubscriptions, etc.

Berger and Nasr5 use this distinctionbetween customer retention andmigration and present mathematical LTVcalculation formulae for four situationsimplying retention models and for onecontext requiring a migration model.

Blattberg and Deighton6 build a modelthat computes ‘customer equity’ andfinds the optimal balance betweencustomer acquisition (investments toconvert prospects into customers) andcustomer retention (investments toconvince active customers to continue tobuy) expenses. This model uses aninfinite time horizon (long term) andLTV formulas are therefore simple, easyto use and avoid all summing operationsthat limited time horizon (life time)models suppose. The concept ofcustomer equity is further developed inBlattberg, Getz and Thomas.7

Building upon these works of Bergerand Nasr8 and of Blattberg andDeighton,9 Pfeifer and Carraway10 applyMarkov chains to customer relationshipmodelling and LTV calculations. Theyinsist on the flexibility of models basedon Markov chains and show that allsituations modelled by the previousauthors can be solved with Markov chainmodels. While the previous contributionsare limited to recency-based customervalue models, these authors illustrate thelink between Markov chains and the


Calciu and Salerno

Distinction between transaction andfinancial flows

In order to encourage a progressiveassessment of customers’ potential andto facilitate the resolution of problemswith increasing complexity it isconvenient to distinguish transactionaland financial flows that were generatedby customers during the number ofperiods considered. The first will helpestimate the number of transactionsincurred by a customer during therelationship with the company and theothers will facilitate profitabilitycalculations for the customerrelationship.

The status of a customer is given byan initial transaction. In order to assesstransactional flows it is appropriate tocount the transactions generated duringseveral periods or business cycles by anumber of initial transactions t0 orcustomers. Let us note tj the number oftransactions generated by t0 customers orinitial transactions after j periods. Thecustomer’s response probability after jperiods is given by the number oftransactions generated after j periodsdivided by the number of initialtransactions or tj when t0 � 1. Thetransaction potential of a customer willthen be given by the expected numberof transactions generated during thelifetime of a customer. This correspondsat each period to the sum of probabilitiesto generate a transaction:

Tj �j�

k=0

tk.

THE RETENTION MODELThe ‘lost for good’ type of behaviourimplies that the customer remains activeuntil the moment they leave thecompany. Simplified, this means that ateach business cycle those who remaincustomers are going to generate atransaction.

person or a company remains a customeras long as they generate transactions.This means that if at some givenmoment customers do not renew theircontracts or do not generate anytransaction they can be considered as‘lost for good’ or as ‘ex-customers’. Italso means that if an ex-customer buysagain they are considered as a newcustomer and one deals with anacquisition rather than a customerretention issue. The evaluation ofcustomer potential in a relation of thiskind will only take into account thecustomer’s probability to remain activefrom one period to another or whatBlattberg and et al. call the customer’ssurvival probability. The customerlifetime corresponds to the number ofsuccessive periods during which thecustomer is and remains active.

The migration model considers thatcustomers can reappear (turn up again)after some periods during which theydid not make transactions and tracestheir probability of ‘reactivating’. Theevaluation of the customer’s potential insuch a relationship relies on the jointprobabilities of remaining and becomingagain an active customer after a fixednumber of periods or, in other words,on the ‘survival’ and ‘reactivation’probabilities. The lifetime of thecustomer corresponds to the number ofsuccessive periods during which thecustomer, according to the company’sestimations, either remains active orreactivates. For a customer to preservethis status after several periods ofinactivity their probability ofreactivating, as estimated by thecompany, must be superior to theprospects’ response probability.Alternatively, the customer is ticked offthe customer database and they becomean ex-customer or a ‘purge’ if oneadopts a certain terminology used bythe direct marketing profession.



by retained customer are $28 and thediscount rate is 20 per cent.

Formalisation of calculations

The survival probability at each periodj is

j�k=0

pk

the product of probabilities to remain acustomer during all those periods. Itindicates the expected number oftransactions per customer on each period(tj).

The cumulated expected number oftransactions per customer during theanalysed periods is the sum of survivalprobabilities:

Tj �j�

k=0�

k�l=0

pl� .

In order to facilitate customer valuecalculation the notion of discountedtransactions14 is introduced. The expectedand discounted number of transactionsduring a customer lifetime is given bythe discounted sum of the customer’ssurvival probabilities:

Tja �

j�k=0

�k�

l=0�pl/(1 � a)k

where a is the discount rate and j is thelifetime of a customer expressed number

General case (retention probabilityvariable with time)

The management of subscribers to aconsumer magazine represents thissituation well and raises a genericproblem of customer retention andattrition. The customer potential and thecustomer value come from individualswith whom the relation persists. In anexample drawn from the consumermagazine subscription business, Dwyerpresents a situation in which theprobability to remain a customer byrenewing the subscription increasesslightly with time and becomes stableafter some periods.

Numeric example

Dwyer gives a numeric example that isrepeated by many of the other authorsmentioned before. This example issummarised and adapted in order toreveal essential mechanisms of theretention model and to allow forevaluation of similar situations in otherindustries by decision calculus. It dealswith a company in the magazinesubscription business which evaluates thetransactional potential and the value of itscustomers for an estimated lifetime offive periods. The margin by retainedcustomer is $40 of which $20 areincome from subscriptions and $20 areadvertising revenues. The marketing costs


Calciu and Salerno

Table 1: The magazine subscriptions case (growing retention rate)

Periodsfromacquisition

Retentionrate

Survivalrate(II 2)

Expectednumberof activecustomers(n*3)

Profit percustomerand period(m � c) ($)

Discountedprofitper customerand period(5/(1 � d)^1) ($)

Totaldiscountedprofit perperiod(4*6) ($)

101234Lifetime

210.710.790.830.84Transactions

310.7080.5610.4630.3893.121

41000.000

707.701560.602463.325389.395

3121.000

51212121212

61210876Value

712000707744853243233629141

The sum of survival probabilities givesthe transaction potential of a customer

j�k=0

pk

which is herej�

k=0

0.8k � 3.36.

The discounted transaction potential isthen

j�k=0

[p/(1 � a)]k

that is4�

k=0

[0.8/( � 0.2)]k � 2.6.

The net present value of a customerresults from the profit of a transactionmultiplied by the discounted transactionpotential:

(m � c)j�

k=0

[p/(1 � a)]k

that is ($40 � $28) 2.6 � $31.2

‘Long-term’ customer value

When purchase probabilities remainconstant from one period to the other,it is convenient to compute the

of periods. In the example, the expectednumber of transactions during thelifetime of a customer is 3.12 (see Table1) and its discounted version is 2.43.The expected value of the customers (Vj)is the profit of a transaction multipliedby the discounted expected number oftransactions:

Vj � (m � c)j�

k=0�

k�l=0

pl/(1 � a)k� .

The case where retention probabilitiesremain constant

In mature market situations one cansuppose that for a constant marketingeffort purchase probabilities remainconstant from one period to another.Under these circumstances formulae aresimplified and make it easier to estimatelong-term evolutions. As a result survivalprobabilities can be expressed as powersof the constant retention probability andno more as a product of variableprobabilities.

Numeric example

Dwyer’s example adapted to a situationwith constant retention probability leadsto the following calculations (see Table 2).

Formalisation of calculations

The survival probability after j periods ispj.



Table 2: The magazine subscriptions case (constant retention rate)

Periodsfromacquisition

Retentionrate

Survivalrate(2..)

Expectednumberof activecustomers(n*3)

Profit percustomerand period(m � c) ($)

Discountedprofitper customerand period(5/(1 � d)^1) ($)

Totaldiscountedprofit perperiod(4*6) ($)

101234

LifetimeLong-term

210.80.80.80.8

310.80.640.5120.41

Transactions

41000

800640512409.6

33625000

51212121212

61210876

Value

7120008000512035842458

3116236000

Optimisation of the long-term presentcustomer value

Up to this point the calculations haveused given retention rates. It is yetjustified to consider that the companycan influence retention rates throughmarketing actions oriented towards itscustomers. This supposes to express thecustomer retention probability as afunction of the marketing effort (budget)directed towards the customers duringeach period. The retention probabilityexpressed as a function of the retentionbudget increases with a growth rate thatdecreases progressively towards the endwhen the retention probability reaches itsceiling (see Figure 2). A simpleexponential function like

pr � ceiling*(1 � exp(b*R))

can be estimated by decision calculususing subjective estimations frommanagers.

The optimum retention budget for acustomer R can be found by maximisingthe present long-term customer value,that is

max(m � R/pr) (1 � a)/(1 � a � pr)

where pr � f (R).In the example with a $23 retention

budget R by customer the companyachieves a 0.8 retention rate, thiscorresponds to $28 by retained customerand gives a long-term present value ofthe customer of (40 � 28)*(1 � 0.2)/(1 �0.2 � 0.8) � $36. If the response to theretention budget is expressed by thefunction pr � 0.81(1 � exp(�0.16*R)),one can see that this result is not anoptimum one. In such a situation, wherethe retention rate is limited upwards to0.81 and marks an accelerated growth(coefficient of elasticity 0.16) with aneffort of only $7 per customer aretention rate of 0.785 is obtained. This

long-term present value of a customer.The long term indicates a hypotheticalsituation where the customer’s lifetimewould be endless. Computing thelong-term transaction potential andcustomer value is not of great interestin itself. It helps optimise marketingefforts: avoiding spending too much incustomer retention programmes with nobudget constraints or finding theoptimum retention budget when budgetconstraints exist.

It can be shown that the long-termtransaction potential

lim→n=�

n�j=0

pj is 1/(1 � p).

In the long term, the discountedtransaction potential will then be

lim→n=�

n�j=0

[p/(1 � a)]j � (1 � a)/(1 � a � p).

For example, if the (constant) retentionprobability is 0.8 then the long-termtransaction potential of a customer is1/(1 � 0.8) � 5. This means that if thelifetime of a customer lasts forever thatcustomer would generate fivetransactions. The discounted transactionpotential is (1 � 0.2)/(1 � 0.2 � 0.8) � 3and the customer’s present value willbe (m � c)(1 � a)/(1 � a � p) or($40 � $28)*3 � $36. If oneremembers that the discountedtransaction potential of a customer was2.6 transactions and the customer’spresent value was $31.2, it becomesclear that these long-term measuresmust be seen as limits towards whichtend the cumulated transactions and thecustomer value.

These limits offer a fast computinginstrument that avoids fastidioussummations and produces someindicative values of the transactionpotential and of the customer.


Calciu and Salerno

expenses by prospect divided by theacquisition probability pa, which itself is afunction of these expenses.

Insuring customer relationshipsprofitability reduces to successiveoptimisation of the customer acquisitionand retention value.

Max[pam � A � pa(mpr � R)/(1 � a � pr)]

where pa � f (A) p and pr � f (R).The optimisation procedure, that hasbeen adapted from Blattberg andDeighton, finds first prospect expensesthat maximise the acquisition value(pam � A) of a customer and then usesthe resulting acquisition rate (pa�) inorder to compute the retention value(pa�(m pr � R)/(1 � a � pr)). Themaximum customer value is the sum ofthe optimum acquisition and retentionvalues calculated in this way. Due to itssimplicity and conciseness this model iswell adapted to decision calculus.

In order to set the Blattberg andDeighton problem in the dualperspective of customer acquisition andretention, it is first necessary to expressthe acquisition probability as a functionof acquisition expenses by prospect, for

results in $8.9 per retained customer.The long-term value becomes then(40 � 8.9)*(1 � 0.2)/(1 � 0.2 � 0.785) �$90 which is optimum.

Finding the balance between customeracquisition and retention

In order to have the complete picture ofcustomer relationship profitability, onecannot ignore the customer acquisitionstage that transforms prospects intocustomers. Blattberg and Deighton15 callthe measure of customer profitability‘customer equity’. By putting theproblem in a prospect’s16 perspective theysuggest a model that makes it possible tooptimise customer acquisition andretention.

CEj � pa �m � ca � (m � cr)j�

k=1

[pr/(1 � a)]k� and lim CEj→n=�

� pa[m � ca � (m � cr)*pr/

(1 � a � pr)]

Following the reasoning previouslyapplied to retention costs, customeracquisition costs ca can be expressed as



Figure 2 Maximisation of the long-term customer value (without the acquisition stage)

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30

Retention cost per customer (R)

Customerlong-term

value

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

value can be easily derived here bydividing the prospect value by theacquisition rate (probability)$11.16/0.16 � $68.32. The result iscoherent with the $90 customer valueobtained when acquisition costs wereignored.

Desynchronisation of the financial flows inthe retention model

In their survey of retention modelsBerger and Nasr17 pay special attentionto the distinction between the momentswhen input (gains) and output (expenses)flows intervene. In the previous examplesit was considered that gains (m) andpromotional expenses (c) intervened atthe same moment during a business cyclewhere from the formula

Vj � (m � c)j�

k=0

[p/(1 � a)]k.

example pa � 0.4(1 � exp( � 0.1*A)).The ceiling acquisition rate (probability)and its elasticity should be lower thantheir corresponding parameters in theretention rate function. The optimisedvalues can be found graphically (seeFigure 3).

With a $5 budget per prospect a 0.16response probability (acquisition rate) isreached. It gives the best acquisitionvalue for a prospect 0.16*40 � 5 � $1.53and can be read on the negative scale inFigure 2. This acquisition rate enters thecalculation of the retention value whichbecomes maximum when retentionexpenses by customer are $7 and resultin a retention rate of 0.78. In this way,the maximum retention value that canbe achieved per prospect is $9.62. Thetotal value of a prospect is the sum oftheir acquisition and retention values$1.53 � $9.62 � $11.16.* The customer


Calciu and Salerno

Figure 3 Maximisation of the customer long-term value (with the acquisition stage)

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

0 2 4 6 8 10 12 14 16 18 20

Acquisition (A) and retention (R) costs per prospect

(2) retention rate(pr)

(4) retention value (vr=pa'*(m*pr-R)/(1+a-pr))

(5) customer value (va + vr)

(1) acquisition rate (pa)

(3) acquisition value (va = m*pa - A)

THE MIGRATION MODELThe ‘always a share’ type of behavioursupposes that the customer does not haveto be active during all the analysedperiod and that the customer canreactivate with a probability thatincreases with recency of the lasttransaction.20 In order to get a betterunderstanding and use ofretention/reactivation mechanisms and ofstochastic processes that characterise thiskind of behaviour, several artefacts likedecision trees, migration trees, diagramsand matrixes of transition probabilitiesare used. They facilitate the developmentor adaptation of a series of algebraic andmatrix formulae.

Algebraic approach

With time, customers have purchasepatterns that alternate active and inactiveperiods. A prolonged inactivity can leadto elimination from the customer list.The progressive shaping of purchaseprofiles is seen in Figure 4.

By regrouping customer purchasepatterns at each stage according torecency criteria, by first separating theactive customers from those that are notactive and by segmenting afterwards theinactive ones according to the recency oftheir last purchase a customer migrationtree is obtained. It is a key representation

A more attentive analysis of thealternation of gains and expenses of acompany during the life of a customer,indicates the following timing. Ifacquisition costs that occur in theprevious cycle are ignored, the customerbrings gains at the beginning of the cycleand the company makes its promotionalexpenses at the middle of the cycle inorder to assure repeat purchase. Cyclelengths can be equal, lower or superiorto one year. There can be differentcycles for gains and for expenses.

The following formula generalises andregroups several formulas suggested byBerger and Nasr,18 in order to deal withvarious situations:

Vj � mj*nm�k=0

[( p)k/(1 � a)k/cm]

� cj*nm�k=1

[( p)(k–1)*nm/nc/(1 � a)(k–0,5)/nc ]

where cm and cc are the number ofcycles per year for gains and forexpenses.

Berger and Nasr19 give detailedcalculations for the three casessummarised here and for two other cases,presenting situations where profits perretained customer are not constant butvariable with time (Case 3) and situationswhere financial flows are not discrete butcontinuous (Case 4).



Table 3: Cases that illustrate commercial cycle particularities and the desynchronisation between the financialinput (gains) and output (expenses) flows in different industries

Case Insurance (1) Health club (2a) Car leasing(2b)

Gains (m)Number of gain cycles per year (nm)Expenses (c)Number of expense cycles per year (nc)Retention probability (p)Number of years ( j)Discount rate (a)Customer value (V )

2601

5010.75

100.2

569

1252

2520.840.2

355

70000.33

9510.3

120.2

8273

a recurrent expression that has as astarting point the first transaction, theone that turns a prospect into acustomer:

tj �min(j,R)�

k=1� tj–kpk

k–1�l=1

(1 � pt)�where t0 � 1.

This formulation21 is more generaland more concise than the formulasgiven by Berger and Nasr22 andDwyer.23 It integrates a recency limit,R, that enables right censoring of themigration process and makes thisexpression and the other ones that

form for understanding transitionmechanisms in this model.

The migration tree shows particularlywell (see Figure 5) the recurrentcharacter of the transition process towhich the customer is submitted. Thisfacilitates the development of powerfuland flexible algebraic formulae forcomputing the transactional andfinancial flows associated to themigration model.

The expected number of transactionsgenerated by a customer after j periodsis the sum of reactivation probabilitiesof transactions initiated by the samecustomer in the previous periods. It is


Calciu and Salerno

Figure 4 Stochastic decision tree and purchase patterns

Figure 5 Customer migration tree

Recency \Periods 0 1 2 3 4 Grouped profiles

1 1000.0 300.0 217.4 166.6 130.7 Σ (*1)

t-1 recency 1 300.0 90.0 65.2 50.0 Σ (*11)

t-1 recency 2 127.4 38.2 27.7 Σ (*101)

t-1 recency 3 63.2 19.0 Σ (*1001)

34.1 Σ (*10001)

2 700.0 210.0 152.2 116.6 Σ (*10)

3 572.6 171.8 124.5 Σ (*100)

4 509.4 152.8 Σ (*1000)

lost for good 475.3 Σ (*10000)

Period 1decision

ProbabilityPeriod 2decision



Probability Buyingprofile

Finalprobabilityof path

buy 0.3 buy 0.3 buy 0.3 buy 0.3 1111 0.0080.0190.0110.0520.0110.0270.0190.1530.0110.0270.0160.0730.0190.0440.0340.475

4no buy 0.7 1110 3

no buy 0.7 buy 0.182 1101 3no buy 0.818 1100 2

no buy 0.7 buy 0.182 buy 0.3 1011 3no buy 0.7 1010 2

no buy 0.818 buy 0.11 1001 2no buy 0.89 1000 1

no buy 0.7 buy 0.182 buy 0.3 buy 0.3 0111 3no buy 0.7 0110 2

no buy 0.7 buy 0.182 0101 2no buy 0.818 0100 1

no buy 0.818 buy 0.11 buy 0.3 0011 2no buy 0.7 0010 1

no buy 0.89 buy 0.067 0001 1no buy 0.993 0000 0

of buyingdecision

of buyingdecision

of buyingdecision

of buyingdecision

Numberofpurchases

Profilepresentvalue11.95311.77611.30311.0059.4129.1618.3487.7241.8461.5950.9260.504

–2.278–2.693–4.511–6.02

expresses reactivation of customers thatwere inactive during the first period. Inthe fifth period the whole process restsessentially on the reactivation process, asthe survival probability diminishes(0.002); at the same time intervenes thecensorship of customers who passed therecency limit (R) because they have notreactivated for five periods (theirreactivation probability becomes toosmall).

The notation tj expresses a customermigration process that starts with anumber of manifest customers, that iscustomers that have just ordered or inother words customers for which therecency of the last order is one. Thestarting number of customers or thenumber of generating transactions can benoted tr,j. In a customer database largenumbers of latent customers ortemporarily inactive customers are keptwhose recency of the last order is greaterthan one. The transaction potential ofthese customers needs to be explicitlyconsidered in certain situation andjustifies extending lifetime valuecalculations to customer recency greaterthan one.

tr,j �min(j,R–r+1)�

k=1�tj–kpr+k–1

r+k–2�l=1

(1 � pl)�where t1,j � tj, tr,0 � 0

In this formula transactional flows withrecency greater than one (tr,j, r � 1) areexpressed as a function of recency onetransactional flows. Notations tj and t1,j

are equivalent. It is a generalisation ofthe previous formula that accepts leftcensored migration processes.

A customer with recency two, forexample, will have a zero purchaseprobability at the beginning (column 5in Table 4), in period two probability toreactivate is 0.18, in period three thepurchase probability of 0.145, will becomposed of the survival (remain active)

have been derived therefromcomparable to the matrix formulationssuggested by Pfeifer and Caraway.24

Censorship according to recency islargely used by the industry. When asegment or a customer crosses thisrecency limit, the customer’s responseprobability becomes lower or equal tothe prospect’s. Consequently, there are nomore reasons to consider that individualas a customer and by convention thecustomer’s reactivation probability (PR) isfixed to zero.

Table 4 illustrates these aspects.Calculations are inspired from Dwyer’sexample, which has been slightlymodified in order to facilitate estimatingthe reduction of response probabilitywith recency by decision calculus.25

A company is forecasting thebehaviour of 1,000 customers, knowingtheir purchase probability according tothe recency of the last purchase. Themarketing margin m is $40 by transactionand promotion expenses are $4 bycustomer. For simplicity these values arekept constant.

The Table shows that with a migrationmodel the retention probability isgenerally low and that companies relyless on their survival probability of acustomer than on the customer’sreactivation probability in order to boostsales and to increase the customer value.

The calculation of the purchaseprobability in column 5 uses theabove-mentioned formula in order toillustrate this recurrent retention andreactivation process: t1 � [1*0.3] � 0.300.t2 � [t1*0.3 � t0*(1 � 0.3)*0.182] � 0.217etc. This means that an initial transactiont0 � 1 generates 0.300 transactions afterone period and 0.217 transactions aftertwo periods. In the second period the0.217 transactions come from the twoevoked sources: t1*p1 � 0.9 expresses theretention or the survival and t0*(1 � p1)*p2 � 1*(1 � 0.3)*0.182 � 0.127




Calciu and Salerno

Tab

le4:

The

cata

logu

esa

les

case

Per

iod

sfr

om

acq

uisi

tio

n

Res

po

nse

pro

bab

ility

and

rece

ncy

Sur

viva

lp

rob

abili

ty(2

..)

Rea

ctiv

atio

np

rob

abili

ty(5

-3)

Pur

chas

ep

rob

abili

ty

Num

ber

of

rem

aini

ngcu

sto

mer

s(n

-7)

Num

ber

of

pur

ged

cust

om

ers

Exp

ecte

dnu

mb

ero

fac

tive

cust

om

ers

(5*n

)

Pro

fitp

ercu

sto

mer

and

per

iod

(m*5

-c(1

-7)

($)

Dis

coun

ted

pro

fitp

ercu

sto

mer

and

per

iod

(9/1

�d

)^1)

($)

Tota

ld

isco

unte

dp

rofit

per

per

iod

(6*1

0)($

)

1 10 1 2 3 4 Life

time

2 1 0.3

0.18

20.

110.

067

1.65

9

3 1 0.3

0.09

0.02

70.

008

1.42

5

4 0 0 0.12

70.

140.

123

0.39

5 1 0.3

0.21

70.

167

0.13

11.

815

610

0010

0010

0010

0052

4.67

4525

7 0 0 0 047

5.33

457.

33

810

0030

021

716

713

118

15

9 36 8 4.69

2.66

3.13

54.5

10 36 6.66

73.

261.

542

1.51

48.9

8

1136

000

6667

3260

1542 79

248

979

they were not active in the period thatprecedes these actions. The marketingmargin comes only from the activecustomers while promotional costs areincurred by default with all eligiblecustomers and not only with the activeones as seems to be suggested by theformula developed by Berger andNasr.26 By eliminating thedesynchronisation aspect in the timingof gains and expenses, their formula canbe reduced to

V1,j �j�

k=0

{(m � c)t1,k/(1 � a)k}.

It reflects a very particular case wherepromotion costs apply only to the activecustomers, which is a reduction,compared to the calculations presentedby Dwyer, which the authors havefollowed.

For reasons of clarity the discount rateis set to zero. The formula becomes then

(m � c)j�

k=0

t1,k

or again (m � c)Tr,k. The customer valuethat results ($40 � $4)*1.815 � $65.34, islargely greater than the value in thegeneral case that is presented in Table 4.As might be observed, the Berger andNasr formulation is limited to a veryspecific situation. Rare are the industriesthat can count on the spontaneousreactivation of customers without havingto make the marketing efforts tostimulate this process.

Companies spend money in order tomaintain the retention and reactivationprocess described by the migration tree.The simplified formula becomes then

j�k=0

(mtk � c) or mTj � cj.

By applying it to the data given in Table4, the customer value after five periodsbecomes $40*1.815 � $4*5 � $52.4.

probability of 0.055 � 0.18*0.3 and theprobability to reactivate with recencythree 0.09 � 0.92*0.11. After fiveperiods (0 to 4) the purchase probabilityor the expected number of generatedtransactions t2,4 becomes 0.062 which isnaturally less than t1,4 � 0.131 for acustomer whose migration process startedwith the active state (recency 1). Thetotal number of generated transactionsT2,4 during the five periods becomes 0.5as compared to T1,4 � 1.815. Thus, acustomer of recency two can still bring atransaction out of two during the fivefollowing periods, while an activecustomer can bring two transactions.

This algebraic formulation oftransactional flows in a migration modelis equivalent with the matrix formulassuggested by Pfeifer and Carraway andmakes a direct link to the matrixapproach presented in the next section.

Financial flows and customer valuecalculations

From the number of transactionsincurred during each period followingthe launching of a relationship marketingprogramme, it is possible to calculate thecustomer’s profitability for the estimatedduration of the relationship that is thelifetime value. The separation of input(gains) and output (promotion expenses)financial flows in the migration model isneeded not only in order to mark thedesynchronisation between the two flows(central topic in Berger and Nasr’s work)but also in order to reflect the differenttransactional basis that applies to the twoflows.

By definition, the migration modelconsiders as a customer every individualhaving ordered already and not havingyet exceeded the pre-established recencylimit. This implies that all customersdefined in this way are eligible forpromotional activities conducted by thecompany during several periods even if



generated by a customer of recency r inperiod j; m � net margin on atransaction; c � mailing cost;pl � response probability of a segment ofrecency l; R � recency limit beyondwhich a customer is ticked off thedatabase; a � discount rate;qk � cumulated probability for acustomer to be eliminated (purged) afterk periods.

This calculation cumulates the presentgains produced by transactions that weregenerated during the projection periodand systematically deducts mailing costs.It supposes that mailings are sent tocustomers only if they are maintained inthe database, which is a standardmarketing policy in a migration model.

The formula also integrates themodelling of the elimination process forcustomers who exceed the recency limit.Elimination intervenes when there is asuccession of non-response (no-purchase)periods equal to the recency limit.

For an active customer (recency 1) theprobability to be eliminated intervenesafter R periods and corresponds at eachperiod k to the probability to generate atransaction after k � R � 1 periodsmultiplied with the probability of thenon-response succession. In the examplein Table 4, an active customer can beeliminated starting with period 4 (therecency limit), with a probability of0.475, that is the product of theprobability to generate a transactiont1,4 � 4 � 1 and the probability of the no-answer succession (1 � 0.3)(1 � 0.18)(1 � 0.11)(1 � 0.06) � 0.475.

For an inactive customer (r � 1) theprobability to be eliminated intervenes intwo stages. The first stage intervenesafter R � r � 1 periods. For a customerof recency 3 the first eliminationintervenes after two periods (4 � 3 � 1).The second stage begins after R � 1periods (here 4 � 1 � 5). The first stageeliminates customers who do not

Censorship is needed for economicreasons because, if one continues toengage in promotional actions towards allmembers of a cohort, costs by activecustomer increase up to a point wherethe margin, even if it remains constant, isno longer enough to cover these costs(see column 9).

With censorship included, thecustomer value formula is slightlymodified to become

j�k=0

(mtk � c(1 � qk))] or mTk � cR

�j�

k+R

[c(1 � qk)],

because when the period j arrives to therecency limit R � 4 customers that didnot reactivate are purged (excluded). Thecustomer value becomes $40*1.815 �4$*4 � $4*(1 � 0.475) � $54.5.

The customer value formula suggestedhere allows for residual value calculationsfor customers that are inactive at theassessment time (left bound censorship),takes into account the promotional coststhat are maintained by default during thelifetime of a customer and integratescustomer elimination.

Vr,j �j�

k=0

{[mtr,k � c (1 � Qk)]/(1 � a)k},

where Qk �k�

l=0

qk

and

qk ��t1,k–R

R�l=1,k�R

(1 � pl), r � 1

or, else

R�l=1,k=R–r+1

(1 � pl) � tr,k–R

R�l=1,k�R+1

(1 � pl),r > 1

where: Vr,j � value of a customer withrecency r; tr,j � expected transactions


Calciu and Salerno

possibilities from a given recency stateare limited to two states: recency oneand the given recency incremented byone, after two periods possible transitionsinclude three states and after four periodsall the five states become accessible. Thesum of the P matrixes to powersincremented from 0 to 4 (Table 5c)contains the cumulated probabilitiesduring five periods. It should be noticedthat the starting period is consideredequal to zero and that matrix P to thepower zero corresponds to the identitymatrix. It can be shown that in the longterm the sum of the probability matrixesP at incremented powers tends towardsthe inverse matrix of the differencebetween the identity matrix and P (Table5d).

In order to obtain a strict equivalencewith the algebraic formulae presentedbefore and summarised in Table 6, thematrixes in Table 5 are multiplied withthe vector noted 11, that has thefollowing shape [1, 0. 0. . .]. In this waythe expected number of transactionsvector is extracted. It consists of firstcolumn elements of the transitionprobabilities matrix and of itstransformations.

The vector t1 expresses the expectednumber of transactions that have beengenerated after one period by customersof different recency while the vector t4

expresses transactions generated after 4periods. In the given example a customerof recency two can generate 0.182transactions after one period and 0.117after four periods. The transactionpotential after four periods is containedin the vector T4 and the long-termtransaction potential is contained in thevector Tn. In the given example acustomer of recency two can generate0.507 transactions during the fourfollowing periods and 0.674 transactionsin the long term. The sum of transitionprobabilities during several periods can

reactivate any more with a probabilitythat corresponds to the non-responsesuccession between the recency of thecustomer and the recency limit. Acustomer of recency 3 has a probabilityto be eliminated at the first stage of(1 � 0.11)(1 � 0.06) � 0.83. The secondstage begins with the reactivation ofcustomers from a cohort with givenrecency and continues up to theelimination of all customers from thiscohort.

The probability to be eliminated ateach period k is the product of theprobability to generate a transaction afterk � R � 1 periods that multiplies theprobability of the non-responsesuccession. A customer of recency 3 willhave in period 5 a probability to beeliminated of 0.0524 � 0.11*0.475. It isthe product of the probability togenerate a transaction t3,5 � 4 � 0.11 andof the probability of the non-responsesuccession 0.475.

Matrix approach

The matrix approach suggested by Pfeiferand Carraway27 suits both retention andmigration models. While for theretention models, the algebraicformulations are simple andstraightforward enough, for the migrationmodel the matrix approach represents aflexible and elegant alternative to thealgebraic formulae.

The transition probabilities matrix P inTable 5a summarises the migrationprocess. It expresses transitionprobabilities, between two periods, froma given recency state either towardsrecency one (buying customer) ortowards a recency state incremented byone (customer who does not buy).According to Markov Chain theory thesame matrix P to the power 4 (Table 5b)indicates transition probabilities after fourperiods. If after one period transition



The value for the anticipated lifetimeof a customer can be easily calculatedby multiplying the discounted sum ofprobability matrixes with the rewardsvector R. By considering gains andpromotional expenses constant, thevector R1 looks like this:[m � c, � c, � c,. . .], meaning thatcustomers of recency one bring gainsin addition to promotional costsbecause they buy, while for customers

be interpreted also as the expectednumber of periods spent in a state beforetransiting to the following state, the laststate being the absorbing state. The linetotal of the long-term sum ofprobabilities matrix is the expectedabsorption time for the state representedby this line. In the example a customerin recency state one has an expectedabsorption time in the ‘ex-customer’ stateof 5.85 periods.


Calciu and Salerno

Table 5: Transition probabilities matrix and its transformations

a) After one period b) After 4 periods

r1r2r3r4r5

r10.30.1820.110.0670

r20.70000

r300.818000

r4000.8900

r50000.9331

r1r2r3r4r5

r10.1310.0650.0310.0110

r20.1170.0810.0290.010

r30.1240.0830.0530.0110

r40.1530.0930.0560.0340

r50.4750.6790.830.9331

P P4

c) Cumulated: periods 0 to 4 d) Cumulated: long-term

r1r2r3r4r5

r11.8150.5070.2760.1130

r21.1791.310.1710.0710

r30.8691.0051.1160.050

r40.6620.820.9461.0340

r50.4751.3582.493.7325

r1r2r3r4

r12.1030.6750.3570.141

r21.4721.4720.2500.099

r31.2041.2041.2040.081

r41.0721.0721.0721.072

4

�k=0

pk (I � P)–1

Table 6: The vector of the expected number of transactions in the transition probabilities matrix and itsderivatives

a) After one period b) After 4 periods

r1t1,1

0.300t2,1

0.182t3,1

0.110t4,1

0.067 r1t1,4

0.131t2,4

0.117t3,4

0.124t4,4

0.153

t1 � P11 t4 � P411

c) Cumulated: periods 0 to 4 d) Cumulated: long-term

r1T1,4

1.815T2,4

0.507T3,4

0.276T4,4

0.113 r1T1,n

2.103T2,n

0.674T3,n

0.357T4,n

0.141

T4 �4

�k=0

Pk11

limTn � (I � P)–111→n=�

budget is $4 the ceiling is 0.3 as inDwyer’s example, when the budget is $1the ceiling becomes 0.1 and when thebudget increases to $10 the ceilingapproaches the limit of 0.5. In thesecircumstances, finding the optimum costper customer consists of maximising thelong-term present value of a recency onecustomer:

max V1 � [I�P/(1 � a)]�1R 11

where

R� � [(m � C/pp), � C/pp, � C/pp,� C/pp, 0]

and

p11 � pp � f (C),

and

P � f (pp, recency).

The marketing expenses c are C/pp; pp isa function of expenses per customer Cand the purchase probability depends onthe ceiling pp and the recency.

By applying this optimisation procedureto the given example it is optimum tospend $3 per customer and not $4. Thegraphical solution is presented in Figure 6.Details of the calculations are given inAppendix 2.

It is easy to extend this reasoning to theacquisition of new customers and to findan optimum balance between customeracquisition and retention budgets. In thisway the Blattberg and Deighton28

optimisation procedure is adapted tocustomer migration situations.

This logic seeks the best trade-offbetween long-term efforts and gains. Forthe migration model, other logics withshorter temporal horizons can also beenvisaged. Pfeifer and Carraway29 presentoptimisation procedures extracted fromthe rich literature on Markov decision

of superior recency there are onlycosts.

Vj �j�

k=0

[Pk/(1 � a)k]R

↓V1,4

48.974

V2,4

2.524

V3,4

�0.714

V4,4

�1.350

By applying the customer valuecalculation formula to Dwyer’s exampleusing a five periods time horizon (k � 0to 4) that seems to be this industry’sestimated customer lifetime, a customerof recency one has a value of $48.97, acustomer of recency two has a value of$2.52 and customers with a recencysuperior to two have negative values.The long-term calculations confirm theseresults.

lim vj � [I � P/(1 � a)]–1R→j=� ↓V1,4

50.765

V2,4

3.556

V3,4

�0.211

V4,4

�1.166

Optimisation of the customer value

By applying the logic used with retentionmodels it is possible to represent thepurchase probability with recency as afunction of the marketing effort. Toreduce complexity, let us suppose that bycontrolling the number of mailings sent tocustomers it is possible to modify theceiling purchase probability leaving theother parameters unaltered.

In the example the function ofpurchase probability with recency was 0.3exp(�0.5*recency � 1), where 0.3 is theceiling (pp) the function reaches when therecency is equal to one and the mailingcosts per customer are $4. If one spendsmore for mailings this ceiling can grow toa limit of 0.5 and if one spends less it canfall to 0. The ceiling purchase probabilityexpressed as a customer cost function ispp � 0.3 (1 � exp( � 0.5*C)). When the



customer lifetime value that have beendeveloped in this paper are organised insystematic and progressive way accordingto a double taxonomy: the one ofcustomer relationship behaviour modelsand the one of calculation methods. Theretention/migration model dichotomy isbased on strong theoretical foundationsin consumer behaviour that have beenunderlined by the previous studies.

Besides the evoked behaviouralcharacteristics, the comparative analysis ofthe two categories of customerrelationship models reveals their economicsubstrata. In a migration model theretention probability is usually relativelylow and companies rely less on thecustomer’s survival probability than on thecustomer’s reactivation probability inorder to increase sales and customer value.

The two calculation methods,algebraic and matrix based, are stepwiseand progressively applied to bothrelationship behaviour models. On thatoccasion several stages are distinguished

— a physical, quantitative level oftransaction flows that are stochasticallytreated

processes. Birtran and Mondschein30

formalise the direct marketing problem(that integrates a particular migrationmodel) as a dynamic programmingproblem and develop optimisationheuristics adapted to this operationsresearch problem. These approaches useRFM stratification and vary coverage andintensity of mailing campaigns in order tomaximise the value of customers.

It can be shown that without budgetaryconstraints the optimal policy is to sendmailings to all customer segments withpositive lifetime value. As formonotonously decreasing responseprobabilities the customer value is alsodecreasing, when margins and mailingcosts are constant, a breakeven responserate can be calculated for which thelifetime value is equal to zero. Using thisproperty and the breakeven response rate,it is possible to calculate a profitabilityindex from the response rate of eachlayer/segment.

CONCLUSIONSThe formulae for computing theeconomic value of a customer or the


Calciu and Salerno

Figure 6: Maximisation of the long-term customer value for the migration model

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6 7

Mailing costs per customer

Longterm

customervalue

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Maxim

um

responseprobability

customer-centred marketing, situationsfor which models of LTV are applicablebecome dominant. This incites pursuingthe customer evaluation and dynamicmanagement models’ systematisation andunification efforts.

Note* Please note that where formulae contain currency

values, these have been rounded to two decimalplaces.

References1 Jackson, D. R. (1996) ‘Achieving strategic

competitive advantage through the application of thelong term customer value concept’, Journal ofDatabase Marketing, Vol. 4, No. 2, pp. 174– 186.

2 Dwyer, F. R. (1989) ‘Customer lifetime valuation tosupport marketing decision making’, Journal of DirectMarketing, Vol. 9, No. 1, pp. 79–84.

3 Jackson, B. B. (1985) ‘Winning and keepingindustrial customers: The dynamics of customerrelationships’, Lexington Books, Lexington, MA.

4 Dwyer (1989) op. cit.5 Berger, P. D. and Nasr, N. I. (1998) ‘Customer

lifetime value: Marketing models and applications’,Journal of Interactive Marketing, Vol. 12, No. 1(Winter), pp. 17–30.

6 Blattberg, R. C. and Deighton, J. (1996) ‘Managemarketing by the customer equity test’, HarvardBusiness Review, Vol. 74, No. 4 (July–August), pp.136–144.

7 Blattberg R., Getz, G. and Thomas, J. (2000)‘Customer equity: Building and managingrelationships as valuable assets’, Harvard BusinessSchool Press, Boston.

8 Berger and Nasr (1998) op. cit.9 Blattberg and Deighton (1996) op. cit.

10 Pfeifer, P. E. and Carraway, R. L. (2000) ‘Modelingcustomer relationships as Markov Chains’, Journal ofInteractive Marketing, Vol. 14, No. 2, pp. 43–55.

11 Birtran, S, and Mondschein, S. (1996) ‘Mailingdecision in the catalogue industry’, ManagementScience, Vol. 42, No. 9 (September), pp. 1364–1381.

12 Birtran, S. and Mondschein S. (1997) ‘Acomparative analysis of decision making proceduresin the catalog sales industry,’ European ManagementJournal, Vol. 15, No. 2 (April), pp. 105–116.

13 Reinartz, W. J. and Kumar, V. (2000) ‘On theprofitability of long-life customers in anoncontractual setting: An empirical investigationand implications for marketing’, Journal of Marketing,Vol. 64, October, pp. 17–35.

14 As suggested by Pfeifer and Carraway (2000) op. cit.in their matrix solutions.

15 Blattberg and Deighton (1996) op. cit.16 The Blattberg and Deighton model applies only to

situations where acquisition of a new customer

— a monetary value level of financialflows, the temporal effects of whichare studied

— a decisions level, the one ofoptimisation calculations.

In this way calculations become easier tocompare and their formalisation in orderto solve problems of increasingcomplexity becomes easier.

The algebraic formulae developed forthe migration model represent analternative to the matrix formulations.They make it possible to approach in anexplicit manner the desynchronisation offinancial flows and the censorship ofmigration processes on recency.Calculations to estimate eliminationsfrom the customer list formalise rightcensorship in a migration process;computing the probabilities to generatetransactions for inactive customersformalises left censorship. These aspectsare only implicitly treated in the matrixapproach of Pfeifer and Carraway31 andare absent from the algebraic approachsuggested by Berger and Nasr.32

The optimisation procedure developedfor retention models as well as theBlattberg and Deighton procedure thatoptimally balances marketing effortsbetween customer acquisition andretention have also been adapted to themigration model using the properties oflong-term transition matrixes.

As shown by Mülhern33 customer LTVmodels cannot be applied to allsituations. They require customers tohave some persistent relations withenterprises and that financial flows (gainsand expenses) can be forecast with acertain precision at an individual level.When these conditions are not satisfied,the historic analysis of profitability canreplace LTV calculations. Yet, under thecombined and interdependent impulsesfrom information technology progressand from the adoption of a



recency of the last transaction is formalised as thefollowing exponential function p � min �

(max � min)*exp(�b*recency), if recency � � 1.Recency zero means active customer thus buyingprobability � 1.

26 Formula (11) of the customer value Berger and Nasr(1998) op. cit. p. 26. (see equation below.)GC � Gross marketing contribution per customer,M � promotional costs per customer, Ci � no.customers in period i, d � discount rate, CLV �

Customer Lifetime Value.27 Pfeifer and Carraway (2000) op. cit.28 Blattberg and Deighton (1996) op. cit.29 Pfeifer and Carraway (2000) op. cit.30 Birtran and Mondschein (1996) op. cit.31 Pfeifer and Carraway (2000) op. cit.32 Berger and Nasr (1998) op. cit.33 Mulhern, F. J. (1999) ‘Customer profitability analysis:

Measurement, concentration, and researchdirections’, Journal of Interactive Marketing, Vol. 13,No. 1, pp. 25–40.

(see Ref. 26 above).

remains profitable, which is a rather special situation.In many industries, like direct marketing and mailorder, acquiring new customers induces a cost thatmust be covered by income from existing customers.

17 Berger and Nasr (1998) op. cit.18 Ibid.19 Ibid.20 As for recency a reverse notation is conventionally

used, that is, the last business cycle has recency one,the last-but-one cycle has recency two and so on.We can say that there is a negative correlationbetween the recency and the customer response rate.

21 For a better understanding of the way algebraicformulae concerning the migration model have beenderived, visual support and live examples can befound at http://claree.univ-lille1.fr/cocoon/slides_ltv_uk/slides.xml

22 Berger and Nasr (1998) op. cit., equation 10, p. 25.23 Dwyer (1997) op. cit., p. 12.24 Pfeifer and Carraway (2000) op. cit.25 The relation between the buying probability and the


Calciu and Salerno

CLV � �GC*�C0 �n�

i=1�

i�j=1

�Ci–j*Pi–j*j�

k=1

(1 � Pt–j+k)��(1 � d )i��M*�[C0/(1 � d )0,5] �

n�i=1

�j�

j=1�Ci–j*Pt–j*

j�k=1

(1 � Pt–j+k)��(1 � d )j+0,5��/C0



Ap

pen

dix

1:Fo

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valu

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Ste

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mod

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jp

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5Lo

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cum

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ws

T�

limT j

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(1�

p)

→ j=�

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limT

j�

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1 1→ j=

�

Fina

ncia

lflo

ws:

Cus

tom

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lue

6Va

lue

atp

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dj

7Lo

ng-t

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valu

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�lim

Vj�

(m�

c)(1

�a

�p

)→ j=

�

V�

limV

j�

[I�

P/(

1�

a)]–1

R→ j=

�

Vj�

(m�

c)j � k=0�k � l=

0

pl/(

1�

a)k

�

t j�

j � k=0

pk

Vj�

j � k=0�k � l=

0

Pl/(

1�

a)k �R

Vj�

(m�

c)j � k=0

[t k/(

1�

a)k]

Tj�

j � k=0

Pk 1 1

T j�

j � k=0

pk

T r,j

�j � k=0

t r,k

t j�

min

(j,R

) � k=1

�t j–kp

k

k–1 � l=1

(1�

pl) �,t

0�

1

t r,j�

min

(j,R

–r+

1)

� k=1

�t j–kp

r+k–

1

r+k–

2 � l=1

(1�

pt) �ou

t 1,j

�t j,

t r,0

�0

t j�

j � k=0

Pk,

P0

�I

Vr,j

�j � k=0

[(mt r,

k�

c(1

�Q

k))/(

1�

a)k ],

ouQ

k�

k � l=0

qk

Vj�

j � k =0

[Pk /(

1�

a)k ]R

whe

req

k�

�t 1,k

–R

R �l=

1,k

�R(1

�p

l),r�

1

R �l=

r,k=

R–r

+1(1

�p

l)�t r,

k–R

R �l=

1,k

�R

+1(1

�p

l),r>

1


Calciu and Salerno

Ap

pen

dix

1:co

ntin

ued

Ap

pro

ache

sA

lgeb

raic

form

ulae

Mat

rix

form

ulae

Fina

ncia

lflo

ws:

Pro

spec

tva

lue

(cus

tom

ereq

uity

)

8C

Eaf

ter

jp

erio

ds

(6,3

)w

ithP

and

Rm

odifi

edP

incl

udes

the

pro

spec

tst

ate

and

R�

�[m

�c a

,m

�c r

,�cr

,...

]

9Lo

ngte

rmC

Elim

CE

j�

pa[

m�

c a�

(m�

c r)*

pr/(

1�

a�

pr)]

→ j=�

(7,3

)w

ithP

and

Rm

odifi

edlik

ein

(8,3

)

Op

timis

atio

n:C

usto

mer

/pro

spec

tva

lue

10O

ptim

alre

tent

ion

cost

sR

max

(m�

R/p

r)(1

�a)

/(1

�a

�p

r)w

here

pr�

f(R)

max

(7,3

)w

here

P�

f(R)

and

R�

�[m

�R

/p11,�

R/p

11,.

..]

11O

ptim

alre

tent

ion

(R)

and

acq

uisi

tion

(A)

cost

s

max

[pam

�A

�p

a(m

pr�

R)/

(1�

a�

pr)]

whe

rep

a�

f(A)

etp

r�

f(R)

max

(9,3

)w

here

P�

f(R)

and

R�

�[m

�A

/p11,m

�R

/p21,�

R/p

21..

.]

Lege

nd:

a�

dis

coun

tra

te,

A�

Acq

uisi

tion

cost

sp

erp

rosp

ect,

c�

pro

mot

ion

(mai

ling)

cost

s,c a

�ac

qui

sitio

nco

stfo

ra

new

cust

omer

,c r

�re

tent

ion

cost

for

are

tain

edcu

stom

er,

CE

�cu

stom

ereq

uity

,j�

lifet

ime

valu

eof

acu

stom

er(n

o.p

erio

ds)

,m

�m

arke

ting

mar

gin,

q�

cust

omer

sel

imin

ated

(pur

ged

)fr

omth

elis

t,Q

�cu

mul

ated

pur

ged

cust

omer

s,p

�b

uyin

gp

rob

abili

ty,

pa

�p

rosp

ect’s

acq

uisi

tion

pro

bab

ility

,p

r�

cust

omer

rete

ntio

np

rob

abili

ty,

P�

tran

sitio

n(p

rob

abili

ties)

mat

rix,

r�re

cenc

y,R

�re

tent

ion

cost

per

cust

omer

,R

�ga

ins

vect

or,

t j�

exp

ecte

dtr

ansa

ctio

nsaf

ter

jp

erio

ds,

T j�

cum

ulat

edex

pec

ted

tran

sact

ions

afte

rj

per

iod

s,T

�lo

ng-t

erm

cum

ulat

edex

pec

ted

tran

sact

ions

,V

j�

cust

omer

lifet

ime

valu

e,V

�cu

stom

erlo

ng-t

erm

valu

e,V

j�

cust

omer

lifet

ime

valu

eve

ctor

,V

�cu

stom

erlo

ng-t

erm

valu

e ve

ctor

.

.

CE

j�

pa�(m

�c a

)�j � k=0

[(mt r,

k�

c r(1

�Q

k))/(

1�

a)k ] �

CE

j�

pa�m

�c a

�(m

�c r

)j � k=1

[pr/(

1�

a)]k �



Appendix 2: Matrix calculations for customer long-term valueoptimisation

Cost per customer Transition probabilities Rewards Customer value

1e r1r2r3r4r5

r10.1020.0620.0380.0230.000

r20.8980.0000.0000.0000.000

r30.0000.9380.0000.0000.000

r40.0000.0000.9620.0000.000

r50.0000.0000.0000.9771.000

r1r2r3r4r5

R30

�10�10�10

0

r1r2r3r4r5

r114.872

�22.223�16.915�9.489

0.000

2e r1r2r3r4r5

r10.1840.1110.0680.0410.000

r20.8160.0000.0000.0000.000

r30.0000.8890.0000.0000.000

r30.0000.0000.9320.0000.000

r50.0000.0000.0000.9591.000

r1r2r3r4r5

R29

�11�11�11

0

r1r2r3r4r5

r116.127

�22.722�18.008�10.335

0.000

3e r1r2r3r4r5

r10.2480.1510.0910.0550.000

r20.7520.0000.0000.0000.000

r30.0000.8490.0000.0000.000

r40.0000.0000.9090.0000.000

r50.0000.0000.0000.9451.000

r1r2r3r4r5

R28

�12�12�12

0

r1r2r3r4r5

r116.479

�23.725�19.385�11.313

0.000

4e r1r2r3r4r5

r10.3000.1820.1100.0670.000

r20.7000.0000.0000.0000.000

r30.0000.8180.0000.0000.000

r40.0000.0000.8900.0000.000

r50.0000.0000.0000.9331.000

r1r2r3r4r5

R27

�13�13�13

0

r1r2r3r4r5

r115.853

�25.324�21.110�12.452

0.000

5e r1r2r3r4r5

r10.3410.2070.1250.0760.000

r20.6590.0000.0000.0000.000

r30.0000.7930.0000.0000.000

r40.0000.0000.8750.0000.000

r50.0000.0000.0000.9241.000

r1r2r3r4r5

R25

�15�15�15

0

r1r2r3r4r5

r114.242

�27.557�23.211�13.765

0.000

6e r1r2r3r4r5

r10.3730.2270.1370.0830.000

r20.6270.0000.0000.0000.000

r30.0000.7730.0000.0000.000

r40.0000.0000.8630.0000.000

r50.0000.0000.0000.9171.000

r1r2r3r4r5

R24

�16�16�16

0

r1r2r3r4r5

r111.687

�30.422�25.694�15.255

0.000

7e r1r2r3r4r5

r10.3990.2420.1470.0890.000

r20.6010.0000.0000.0000.000

r30.0000.7580.0000.0000.000

r40.0000.0000.8530.0000.000

r50.0000.0000.0000.9111.000

r1r2r3r4r5

R22

�18�18�18

0

r1r2r3r4r5

r18.260

�33.885�28.542�16.915

0.000

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