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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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??
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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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Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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)
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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.
Frontiers in Service Conference, Karlstad, June 10-13, 2010
Average empirical hazard function for standardsubscription
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0.00
50.
010
0.01
5
Wochen
Haz
ard
Rat
e pr
o W
oche
Empirical Hazard Festabo
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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)
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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
Frontiers in Service Conference, Karlstad, June 10-13, 2010
Thank you for yourattention!