# Measuring Customer Equity in Noncontractual Settings Using a Diffusion Model: An Empirical Study of Mobile Payments Aggregator

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Brief Introduction of CE

#### 2.2. CE Measurement Model

- In terms of the relationship between a firm and its customers, the approaches of customer equity measurement can be divided into contractual models and noncontractual models [21,22]. In contractual settings, such as insurance, telecommunications, and magazine subscriptions, firms can accurately understand the cash flow generated by their customers as customers usually sign long-term contracts with the firms. In such cases, the measurement of customer equity should focus on the prediction of whether customers in the following period will continue to subscribe for products or services of the firm (i.e., investigation of the retention rate) [7]. Noncontractual models are used, for example, in the retail industry, where customers and firms are not bound by contracts and customers may shift among different products or service providers. In such cases, the measurement of customer equity involves not only the prediction of the frequency of customers’ future purchases but also the prediction of customers’ future purchase amount [23]. The measurement of customer equity under a noncontractual relationship thus appears to be more complicated.
- In terms of the data source, the measurement of customer equity can be based on firm-level data (such as financial data from annual reports) or individual-level data (such as individual transaction data) [24]. For the former, the average customer lifetime value is calculated through financial data and is multiplied by the number of customers to obtain customer equity. As no individual customer lifetime value is involved in this top-down approach, it is also called an aggregate-level approach [3,9,25,26]. For the latter, the lifetime value of each customer is calculated based on individual transaction data. Customer equity is then calculated by summing up the lifetime value of all customers. This bottom-up approach is also referred to as a disaggregate-level approach [23].
- In terms of the model type, customer equity can be calculated by taking deterministic and stochastic approaches [2,27]. For the former, there are no random components involved in the model. The heterogeneity among individual customers is usually not considered. Customer equity can thus be directly calculated based on a simple formula [7,25]. For the latter, the stochastic factors and the heterogeneity among customers are fully considered. Although the stochastic approach is more complicated, it can make the calculation of customer equity more accurate [11].
- In terms of the specific modeling approach, customer equity can be calculated using probability models, econometric models, persistence models, computer science models, and diffusion/growth models. Probability models are built under certain probability distribution assumptions of customers’ purchasing behaviors. For example, the customers’ future purchasing frequency can be predicted using a Pareto/NBD model [16] and a BG/NBD model [28], the amount of customers’ future purchase can be predicted using the normal distribution [29] or log-gamma distribution [30], and the Conway–Maxwell–Poisson distribution is used to jointly model the purchase frequency and purchase quantity [31]. The central concept of econometric models is to explain different purchasing behaviors of customers through a series of covariates. This approach emphasizes factors that affect customer equity and its components. Commonly used econometric models include the simple regression model, the logit/probit model [32,33] and the survival analysis model [34]. When the time series data is long enough, persistence models, including the unit root test, cointegration model, and VAR model, can be established to predict customer equity, quantifying the long-term dynamic effects of different covariates in customer equity modeling [35,36]. Computer science models introduce machine learning and modern nonparametric statistics to the measurement of customer equity. These models need no theoretical assumptions and generally have stronger predictive ability than parametric models. The main models used are the support vector machine (SVM) [37], random forest [38], Lasso regression, and generalized additive model (GAM) [39]. Diffusion/growth models emphasize the prediction of the number of future customers and thus lay a foundation for customer equity measurement. The diffusion models are discussed in detail below.

#### 2.3. CE Measurement Using a Diffusion Model

## 3. Theoretical Model

#### 3.1. Customer Segmentation Based on a Diffusion Model

#### 3.1.1. Bass Model

_{t}be the cumulative number of customers who adopt new products at time t and m be the total population of potential adopters. The difference equation of the Bass model is

#### 3.1.2. Bass Model and Adopter Categories Proposed by Rogers

_{1}), (T

_{1}–T

_{peak}), (T

_{peak}–T

_{2}), and (T

_{2}–∞) respectively. In the case that$\frac{{d}^{2}f\left(t\right)}{d{t}^{2}}=0$,

#### 3.2. Estimation of CE and CESR

#### 3.2.1. Estimation of CE

_{t}are constants that do not change with time t, that is, ${\overline{C}}_{t}=CF$ and r

_{t}= r, d is referred to as the discount rate. Without considering marketing costs, customer retention costs and customer acquisition costs, the lifetime value of a single customer is $CLV={\displaystyle \sum _{t=0}^{\infty}\frac{CF\times {r}^{t}}{{\left(1+d\right)}^{t}}}=\frac{CF\times {r}^{0}}{{\left(1+d\right)}^{0}}+\frac{CF\times {r}^{1}}{{\left(1+d\right)}^{1}}+\dots $. It is the sum of an infinite geometric sequence in the case that $\left|\frac{r}{1+d}\right|<1$ and may be rewritten as $CLV=CF\times \frac{1+d}{1+d-r}$. Thus, customer equity $CE={N}_{end}\times CF\times \frac{1+d}{1+d-r}$, where ${N}_{end}$ is the number of customers at the end of the period. In contractual settings, the retention rate $r=1-\frac{{N}_{cancel}}{\left({N}_{beg}+{N}_{end}\right)/2}$, where ${N}_{beg}$ is the number of customers at the beginning of the period and ${N}_{cancel}$ is the number of lost customers. Pfeifer [47] further improved the above method, firstly the author believed that new customers were immune to (initial) churn and the retention rate can be adjusted to the ratio of the number of retained customers to the number of customers at the beginning, that is $r=\frac{{N}_{end}-{N}_{add}}{{N}_{beg}}$, here, ${N}_{add}$ is the number of new customers; secondly, the author proposed that the replacement of the beginning of the period with the middle of the period as the time-point when there is cash flow appears more consistent with the actual situation. Then,

#### 3.2.2. Estimation of CESR

#### 3.3. Estimation of Customer Purchase and Retention Rates Using a Pareto/NBD Model

- 1.
- The time when a customer becomes “inactive” is exponentially distributed with churn rate μ. The probability density function is:

- 2.
- There is a difference in the transaction rate of individual customers. The transaction rate λ across customers follows a gamma distribution. The probability density function is:

- 3.
- There is a difference in the churn rates of individual customers. The churn rate μ across customers follows a gamma distribution. The probability density function is:

- 4.
- Transaction rate λ is independent of the death rate μ. The expected number of transactions from the initial transaction period T
_{0}to the observation period T is therefore

## 4. Methodology

#### 4.1. Research Framework

#### 4.2. Data

## 5. Empirical Results and Analysis

#### 5.1. Parameter Estimation for the Bass Diffusion Model

#### 5.2. User Segmentation Based on the Diffusion Model

_{1}and T

_{2}were first calculated to be 15.2 and 30.2 respectively using Equations (5) and (6). These time points respectively differentiate the early adopters from the early majorities and the late majorities from the laggards. The proportions of the five user categories were then calculated according to Equations (7) and (8). Finally, the proportions of innovators and imitators in each category were calculated. Figure 7 shows the user segmentation.

#### 5.3. Modeling Customer Behavior Using a Pareto/NBD Model

_{m}. For all customers making an initial use at least T

_{m}time units ago, we chose $\gamma /\alpha ,s,\beta $ to minimize

_{0}to the current month T.

_{0}to T. Average these estimates to obtain $\widehat{\alpha}$

#### 5.4. CE and CESR Calculation

#### 5.4.1. Calculating Customer Equity of Current Users

#### 5.4.2. Calculating Customer Equity of Potential Users

## 6. Managerial Relevance and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Concept | Definition | Abbreviation |
---|---|---|

Customer lifetime value | The net present value of profits that a customer can bring to a firm during his or her entire life of transactions with the firm | CLV |

Customer equity | The sum of the lifetime value for all customers of the firm | CE |

Customer equity sustainability ratio | The ratio of a customer’s (or all customers’) future CLV (or CE) to the customer’s (or their) total CLV (or CE) | CESR |

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**Figure 1.**(

**a**): Cumulative distribution function curve of the Bass model. (

**b**): Probability density function curve of the Bass model.

**Figure 4.**(

**a**): Distribution of the length of use in the first stage (in months). (

**b**): Distribution of the length of use in the second stage (in months).

**Figure 5.**(

**a**): Distribution of use frequency (first stage). (

**b**): Distribution of use frequency (second stage).

**Figure 6.**Comparison between the actual cumulative number of new users and fitted values of the models.

Reference | Customer-Firm Relationship | Data Source | Model Type | Modeling Approach |
---|---|---|---|---|

[7] | contractual | firm-level | deterministic | econometric |

[25] | contractual/noncontractual | firm-level | deterministic | econometric |

[22] | noncontractual | individual-level | stochastic | probability |

[3] | contractual | firm-level | deterministic | econometric |

[11] | noncontractual | individual-level | stochastic | probability |

[8] | contractual | firm-level | deterministic | diffusion/growth |

[23] | noncontractual | individual-level | stochastic | probability/econometric |

[9] | noncontractual | firm-level | stochastic | econometric |

[42] | contractual | firm-level | deterministic | diffusion/growth |

[35] | contractual | individual-level | stochastic | persistence |

[38] | contractual | individual-level | stochastic | computer science |

[37] | noncontractual | individual-level | stochastic | computer science |

User Categories | Number of Uses per Month | Churn Rate |
---|---|---|

Innovators | 1.28 | 0.011 |

Early adopters | 0.41 | 0.0003 |

Early majorities | 0.52 | 0.00012 |

Late majorities and laggards | 0.56 | 0.028 |

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**MDPI and ACS Style**

Xue, W.; Sun, Y.; Bandyopadhyay, S.; Cheng, D.
Measuring Customer Equity in Noncontractual Settings Using a Diffusion Model: An Empirical Study of Mobile Payments Aggregator. *J. Theor. Appl. Electron. Commer. Res.* **2021**, *16*, 409-431.
https://doi.org/10.3390/jtaer16030026

**AMA Style**

Xue W, Sun Y, Bandyopadhyay S, Cheng D.
Measuring Customer Equity in Noncontractual Settings Using a Diffusion Model: An Empirical Study of Mobile Payments Aggregator. *Journal of Theoretical and Applied Electronic Commerce Research*. 2021; 16(3):409-431.
https://doi.org/10.3390/jtaer16030026

**Chicago/Turabian Style**

Xue, Wei, Yinglu Sun, Subir Bandyopadhyay, and Dong Cheng.
2021. "Measuring Customer Equity in Noncontractual Settings Using a Diffusion Model: An Empirical Study of Mobile Payments Aggregator" *Journal of Theoretical and Applied Electronic Commerce Research* 16, no. 3: 409-431.
https://doi.org/10.3390/jtaer16030026