1.
Customer Lifetime Value in service
contracts
The world is not Markovian!
Christoph Heitz, Andreas Ruckstuhl, Marcel Dettling
Zurich University of Applied Sciences
Swiss Institute of Service Science
2.
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 matters
Semi-markov model
Application: Swiss newspaper subscription
Frontiers in Service Conference, Karlstad, June 10-13, 2010
3.
Measuring customer value
Concept of customer lifetime value (CLV)
– sum of future revenue
– discounting net present value
– well known concept in marketing
∞
CLVk = ∑ ck (t ) α t
t =1
Future revenue - stochastic process
CLV depends on what the customer will do in the
future: ck(t)=?
Needed: Modeling of future customer behavior
Frontiers in Service Conference, Karlstad, June 10-13, 2010
4.
Contractual vs noncontractual settings
Main question: What will customer do?
Non-contractual setting
acquisition
– Start business
– Stop vs. continue business retention
– Increase business Customer development
Contractual setting
– Subscribe new contract acquisition
– Keep contract vs. cancel retention
– Change contract (e.g. upgrade) Customer development
Frontiers in Service Conference, Karlstad, June 10-13, 2010
5.
Modeling customer dynamics
Model 1: Lost-for-good (Dwyer 1989)
– Two-state model: customer / no customer
– Customer who has left never returns
– Modeling issue: lifetime analysis
5 5 5 5 5
Model 2: Always-a-share
– multi-state model 4
4 4 4 4
– More complete dynamics (includes Lost-
for-good dynamics) 3 3 3 3 3
– Modeling issues: describe state changes
– Classical model: Markov Chains 2 2 2 2 2
(Pfeiffer/Carraway (2000), Piersma/Jonker
(2000), Tirenni (2005)) 1 1 1 1 1
– Basic assumption: the probability of a state
change („hazard rate“) does not depend on
the past, in particular not on the sojourn
time!
Frontiers in Service Conference, Karlstad, June 10-13, 2010
6.
Specifics of contractual settings
observability
Contract impacts behavior of customer
– e.g. minimum duration: customer might want to cancel
but is not allowed to!
– Fixed renewal periods allow cancelling only at specific
times
– Contradiction to Markov assumption!
Contract design is an important driver for customer
lifetime value
Is it important to account for „contract mechanics“
when determining CLV??
Frontiers in Service Conference, Karlstad, June 10-13, 2010
7.
Typical hazard functions for contractual settings
h(t) h(t) Contract cancellation after minimum
Markovian dynamics contract duration
t t
h(t) Minimum contract duration without h(t)
cancelling Periodic withdrawal dates
t t
h(t)
Long-time customers are more loyal
t
Frontiers in Service Conference, Karlstad, June 10-13, 2010
8.
Empirical example: contract durations for
newspaper subscription
Lebensdauer Festabo in Wochen
4000
3000
Häufigkeit
2000
1000
0
0 50 100 150
Lebensdauer in Wochen
Frontiers in Service Conference, Karlstad, June 10-13, 2010
9.
A simple example
Contract with minimum duration period
Assumed customer behavior:
– 50% cancellation after one year, expected lifetime if not
cancelled: additional 5 years
– This results in average lifetime of 3 years
– Constant revenue stream during contract duration
h(t)
t
Calculation of CLV with
– Markov model (reflecting correct avg. fifetime)
– Correct formula
Frontiers in Service Conference, Karlstad, June 10-13, 2010
10.
CLV under non-markovian dynamics
h(t)
CLV(t)
True CLV
t
40% difference
CLV calculated with best
Markov model
1 yr
t
Markov model results in wrong CLV at any given time!
Deviation can be substantial
Taking contract into consideration can be crucial for
any marketing decision
Frontiers in Service Conference, Karlstad, June 10-13, 2010
11.
Modeling of dynamics with Semi-Markov models
Semi-Markov models are 5 5 5 5 5
generalization of Markov models
4 4 4 4 4
– Dynamics consist of two steps
• Sojourn in a state 3 3 3 3 3
• Jump to another state 2
2 2 2 2
– Lifetime in state may be arbitrarily distributed
1 1 1 1 1
• Hazard rate: Rate of leaving state
• Hazard rate may depend on sojourn time
– Jump to another state may depend on sojourn time as well
Modeling elements:
– Hazard function for each state: hi(t) = probability of leaving state i
at sojourn time t
– Matrix of jump probabilities pij(t)
Frontiers in Service Conference, Karlstad, June 10-13, 2010
12.
Applying SMM to customer dynamics
Semi-Markov models allow incorporating many important
contract 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 known
effects such increasing loyalty of customers
– Churn rate tends to decrease with contract duration
Additional 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
∞
CLVk = ∑ ck (t ) α t
t =1
Frontiers in Service Conference, Karlstad, June 10-13, 2010
13.
Analytical calculation of CLV, discrete time version
Discrete lifetime distribution, calculated from hazard function
∞
xi = ∑ f i (T ) α T
T =1
Monthly discount factor
∞
y ij = ∑ pij (T ) f i (T ) α T
T =1 Jump matrix elements
( )
r r
J = Ι− y
−1
(Ι − x )⋅ 1 − α
c Monthly revenue in states
Current sojourn time
ci in state i
CLVi (T0 ) = ⋅ (1 − xi ) + ∑ yij ⋅ J j
% %
1−α j
Frontiers in Service Conference, Karlstad, June 10-13, 2010
14.
Estimation of model parameters
hazard function hi(t) of
leaving state i at
sojourn time t
data
CLV
Individual matrix of
jump probabilities pij
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
15.
Application
Subscription of national newspaper of Switzerland
Data: Contract history of 450k customers in 2002-2008
Modelling with SMM, and estimation of CLV for each customer
Probeabo
Aktionsabo
evtl.
Kein Abo Festabo
Frontiers in Service Conference, Karlstad, June 10-13, 2010
16.
Average empirical hazard function for standard
subscription
Empirical Hazard Festabo
0.015
Hazard Rate pro Woche
0.010
0.005
0 50 100 150 200 250 300
Wochen
Frontiers in Service Conference, Karlstad, June 10-13, 2010
17.
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 particular
for direct marketing
SAS and R/MATLAB implementations available (idp, SAS
Switzerland)
Frontiers in Service Conference, Karlstad, June 10-13, 2010
18.
Conclusion
Markov chain models not suited for many contractual
settings
– Risk of substantially wrong CLVs for individual
customers
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
19.
Thank you for your
attention!
Frontiers in Service Conference, Karlstad, June 10-13, 2010
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