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Quantitative Marketing and Economics

Erschienen in:

01.06.2009

A generalized framework for estimating customer lifetime value when customer lifetimes are not observed

verfasst von: Siddharth S. Singh, Sharad Borle, Dipak C. Jain

Erschienen in: Quantitative Marketing and Economics | Ausgabe 2/2009

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Abstract

Measuring customer lifetime value (CLV) in contexts where customer defections are not observed, i.e. noncontractual contexts, has been very challenging for firms. This paper proposes a flexible Markov Chain Monte Carlo (MCMC) based data augmentation framework for forecasting lifetimes and estimating customer lifetime value (CLV) in such contexts. The framework can be used to estimate many different types of CLV models—both existing and new. Models proposed so far for estimating CLV in noncontractual contexts have built-in stringent assumptions with respect to the underlying customer lifetime and purchase behavior. For example, two existing state-of-the-art models for lifetime value estimation in a noncontractual context are the Pareto/NBD and the BG/NBD models. Both of these models are based on fixed underlying assumptions about drivers of CLV that cannot be changed even in situations where the firm believes that these assumptions are violated. The proposed simulation framework—not being a model but an estimation framework—allows the user to use any of the commonly available statistical distributions for the drivers of CLV, and thus the multitude of models that can be estimated using the proposed framework (the Pareto/NBD and the BG/NBD models included) is limited only by the availability of statistical distributions. In addition, the proposed framework allows users to incorporate covariates and correlations across all the drivers of CLV in estimating lifetime values of customers.

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Customer Lifetime Value (CLV) is defined as the net of the revenues obtained from a customer over the lifetime of transactions minus the cost of attracting, selling, and servicing the customer, considering the time value of money (Berger and Nasr 1998).

In the CLV literature, researchers have generally considered the revenue stream from a customer to estimate the lifetime value of the customer while ignoring the costs of servicing the customer. We too take this approach in this paper.

The estimation involves multiple evaluations of the Gauss Hypergeometric Function which in turn may be a source of computational difficulty (Fader et al. 2005).

An Exponential(λ _h) distribution is used to draw the interpurchase times. The very first simulated interpurchase time for customer h, \( IPT_{{h,i = X_h + 1}} \) is drawn such that \( IPT_{{h,i = X_h + 1}} > T_h - t_h \)

This is a natural outcome when we use the proposed framework for model estimation.

Using existing approaches, estimation of the Pareto/NBD model requires repeated evaluation of the Gauss Hyper geometric function which itself is an infinite sum series.

We observe defection for all the customers in our dataset.

The choice of January 1, 1999 is for illustration and is not based on any particular reasoning.

The details of the estimation including the set of prior distributions used, the full conditional distributions and the MCMC steps are included in the Supplemental Note.

However, this does not imply that the NBD parameters across the two models will be the same. Please see Fader et al. 2005 for details.

The estimates from these three models are provided in the Supplemental Note.

For CLV models that characterize the defection process in terms of number of lifetime purchases (for example the BG/NBD model) this is the remaining number of lifetime purchases.

A range of discount rates from 10% to 15% was also used, however the relative performance of the models considered does not change.

In our application, out of the 5000 customers 3950 had defected before the slice date.

It is interesting to note that the use of averages can sometimes be misleading. A case in point is the predicted average lifetime using CMP/NBD model in Table 6 (column V). The CMP/NBD model predicts the average remaining lifetime in the population very well. However looking at the MAD statistic (column VI) which is the average of the absolute deviations of the predicted individual remaining lifetime, we can see that the Gamma/NBD and the Gamma/Gamma models outperform the CMP/NBD.

Recollect that the firm observes (for every customer) a string of purchase amounts and string of interpurchase times (with the last observed time being censored), however, the lifetime of a customer remains unobserved.

Males are coded as 0 and females are coded as 1.

The variables ‘Age’ and ‘Salary’ have been estimated from the ‘Block’ Demographic Trends available for the area of residence of the household.

The details of the estimation including the set of prior distributions used, the full conditional distributions and the MCMC steps are included in the Supplemental Note.

Past research has pointed out the difficulty in estimation using the full model likelihood of the Pareto/NBD model (Fader et al. 2005).

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Titel: A generalized framework for estimating customer lifetime value when customer lifetimes are not observed
verfasst von: Siddharth S. Singh
Sharad Borle
Dipak C. Jain
Publikationsdatum: 01.06.2009
Verlag: Springer US
Erschienen in: Quantitative Marketing and Economics / Ausgabe 2/2009
Print ISSN: 1570-7156
Elektronische ISSN: 1573-711X
DOI: https://doi.org/10.1007/s11129-009-9065-0