customer lifetime value in service contracts (christoph heitz)

Customer Lifetime Value in servicecontracts

The world is not Markovian!

Christoph Heitz, Andreas Ruckstuhl, Marcel Dettling

Zurich University of Applied Sciences

Swiss Institute of Service Science

Frontiers in Service Conference, Karlstad, June 10-13, 2010

Content

Customer lifetime value (CLV)– What is CLV?– Contractual vs noncontractual settings– Classical models for calculating CLV

CLV in contractual settings– Modeling customer dynamics: Why the Markov

assumption does not hold, and why this mattersSemi-markov modelApplication: Swiss newspaper subscription


Measuring customer value

Concept of customer lifetime value (CLV)– sum of future revenue– discounting net present value– well known concept in marketing

CLV depends on what the customer will do in the future: ck(t)=?Needed: Modeling of future customer behavior

∑∞

=

=1

)(t

tkk tcCLV α

Future revenue - stochastic process


Contractual vs noncontractual settings

Main question: What will customer do?Non-contractual setting

– Start business– Stop vs. continue business– Increase business

Contractual setting– Subscribe new contract– Keep contract vs. cancel– Change contract (e.g. upgrade)

acquisition

retention

Customer development

acquisition

retention

Customer development


Modeling customer dynamics

Model 2: Always-a-share– multi-state model– More complete dynamics (includes Lost-

for-good dynamics)– Modeling issues: describe state changes– Classical model: Markov Chains

(Pfeiffer/Carraway (2000), Piersma/Jonker(2000), Tirenni (2005))

– Basic assumption: the probability of a statechange („hazard rate“) does not depend on the past, in particular not on the sojourntime!

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Model 1: Lost-for-good (Dwyer 1989)– Two-state model: customer / no customer– Customer who has left never returns– Modeling issue: lifetime analysis


Specifics of contractual settings

observabilityContract impacts behavior of customer

– e.g. minimum duration: customer might want to cancelbut is not allowed to!

– Fixed renewal periods allow cancelling only at specifictimes

– Contradiction to Markov assumption!Contract design is an important driver for customerlifetime valueIs it important to account for „contract mechanics“when determining CLV??


Typical hazard functions for contractual settings

h(t)

t

h(t)

t

h(t)

t

h(t)

t

Markovian dynamics

Minimum contract duration withoutcancelling

Long-time customers are more loyal

Contract cancellation after minimumcontract duration

h(t)

t

Periodic withdrawal dates


Empirical example: contract durations fornewspaper subscription

0 50 100 150

010

0020

0030

0040

00

Lebensdauer in Wochen

Häu

figke

it

Lebensdauer Festabo in Wochen


A simple example

Contract with minimum duration periodAssumed customer behavior:

– 50% cancellation after one year, expected lifetime if notcancelled: additional 5 years

– This results in average lifetime of 3 years– Constant revenue stream during contract duration

Calculation of CLV with– Markov model (reflecting correct avg. fifetime)– Correct formula

h(t)

t


h(t)

t

CLV under non-markovian dynamics

CLV(t)

t

CLV calculated with best Markov model

True CLV

40% difference

Markov model results in wrong CLV at any given time!Deviation can be substantialTaking contract into consideration can be crucial forany marketing decision

1 yr


Modeling of dynamics with Semi-Markov models

Semi-Markov models aregeneralization of Markov models

– Dynamics consist of two steps• Sojourn in a state• Jump to another state

– Lifetime in state may be arbitrarily distributed• Hazard rate: Rate of leaving state• Hazard rate may depend on sojourn time

– Jump to another state may depend on sojourn time as wellModeling elements:

– Hazard function for each state: hi(t) = probability of leaving state i at sojourn time t

– Matrix of jump probabilities pij(t)

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Applying SMM to customer dynamics

Semi-Markov models allow incorporating many importantcontract rules, e.g.

– Minimum contract duration– Specific renewal dates– Upgrading possible at each time, but downgrading restricted

At the same time, Semi-markov models allow modeling knowneffects such increasing loyalty of customers

– Churn rate tends to decrease with contract durationAdditional modeling elements:

– hazard functions hi(t) for each state– Jump probabilities pij(t)

Integrating in CLV calculation framework– CLV can be calculated analytically with simple operations

∑∞

=

=1

)(t

tkk tcCLV α


Analytical calculation of CLV, discrete time version

( ) ( )α−

⋅−Ι−Ι= −

11 cxyJ

rr

∑

∑∞

=

∞

=

=

=

1

1

)()(

)(

T

Tiijij

T

Tii

TfTpy

Tfx

α

α

( )0( ) 11

ii i ij j

j

cCLV T x y Jα

= ⋅ − + ⋅− ∑% %

Monthly discount factor

Discrete lifetime distribution, calculated from hazard function

Jump matrix elements

Monthly revenue in states

Current sojourn time in state i


Estimation of model parameters

data

hazard function hi(t) of leaving state i at sojourn time t

Individual matrix of jump probabilities pij

CLV

Individual jump probabilities pk,ij: – Estimated by (multinomial) logistic regression models based on the

recent past Individual hazard function hk(t) :

– Estimated by forward continuation ratio model with proportional hazard properties (discretized version of proportional hazard model)


Application

Subscription of national newspaper of SwitzerlandData: Contract history of 450k customers in 2002-2008Modelling with SMM, and estimation of CLV for each customer

Aktionsabo

Kein Abo

Probeabo

Festabo

evtl.


Average empirical hazard function for standardsubscription

0 50 100 150 200 250 300

0.00

50.

010

0.01

5

Wochen

Haz

ard

Rat

e pr

o W

oche

Empirical Hazard Festabo


Results of case study

Clear non-markovian dynamics in nearly all states– Validated with empirical data

Parameters of Semi-markov model could be estimated on individual customer basis with high accuracy

– Validation with repeatedly simulated data for 450k customers– Average statistical error in individual CLV estimate less than 1%

Approach seems viable for marketing optimization, in particularfor direct marketingSAS and R/MATLAB implementations available (idp, SAS Switzerland)


Conclusion

Markov chain models not suited for many contractualsettings

– Risk of substantially wrong CLVs for individualcustomers

Framework for Semi-Markov modeling developed– parameter estimation on individual customer level– Formulas for CLV calculation, given model parameters

Use of model: – Operational marketing planning: Optimum selection of

customers for marketing campaigns– Strategic and tactical marketing planning


Thank you for yourattention!

customer lifetime value in service contracts (christoph heitz)

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