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In this paper , we show that it is possible to estimate discount rate or cost of capital in the calculation of customer lifetime value ( <i> CLV</i> ) for a company using numerical methods instead of the traditional financial approaches. We propose an estimation formula for retention period that the company should keep the customers in order to gain certain benefit from them. We also apply our theoretical approaches to the Mongolian mobile service at company level using statistical data from 2005 to 2016. Some numerical results are included.

Customers are one of the main assets of a company that should be assessed and regulated. Customer lifetime value (CLV) is gaining increasing importance as a marketing metric in both academia and practice [

Customer lifetime value models mainly focus on the estimation of customer lifetime value and customer relationship management. However, it is very important to consider ways of finding the discount rate and retention period which are main components of the models. So we address these two issues in our paper. From a mathematical perspective, the discount rate is a solution of nonlinear equation and the retention period can be calculated from the general model. From a managerial perspective, the discount rate and retention period are important components of the models and decision making process. It is possible to find the discount rate using numerical methods.

The paper is organized as follows. In Section 2, we consider a general CLV model and Newton’s method for estimating the discount rate and generate a formula for retention period. In Section 3, we briefly introduce about Mongolian mobile service and show the numerical results. And we summarize our findings.

Customer lifetime value (CLV) is generally defined as the present value of all future profits obtained form a customer over his or her life of relationship with a firm . CLV is similar to the discounted cash flow approach used in finance [

C L V = ∑ t = 0 n ( p t − c t ) r t ( 1 + i ) t − A C (1)

where p t is the price paid by a customer and c t is the direct cost of servicing the customer at time t, i is the discount rate or cost of capital for the firm, r t is the probability or customer repeat buying at time t, A C is the acquisition cost, and n is the time horizon for estimating CLV.

For simplicity, if we assume that a customer generates margin m t for each period t, the retention rate is 100% and discount rate or cost of capital for a firm is i, then the customer lifetime value is calculated by the following formula:

C L V = ∑ t = 0 n m t ( 1 + i ) t (2)

where n is time horizon for estimating CLV. A key decision is what time horizon to use for estimating CLV. Typically, three to five year is reasonable. With this information and estimated other variables, we can calculate CLV using spreadsheet analysis.

Discount rate or the cost of capital is used to find the present profits from a customer. In most cases, discount rate is considered as a constant about 8 - 16 percent in customer lifetime value calculations [

In our work we use a numerical method which is called the Newton’s method to estimate the discount rate when margins and customer lifetime value are known. Newton’s method is one of the most powerful and well-known numerical methods for solving a root-finding problem [

φ t ( i ) = m t ( 1 + i ) t (3)

Then we can compute i from the following equation

C L V = ∑ t = 0 n φ t ( i ) (4)

for given CLV and m t . Equation (4) has a unique solution [

F ( i ) = ∑ t = 0 n φ t ( i ) − C L V = 0 (5)

So the unique solution can be estimated by the following stage of Newton’s iteration method which starts with an initial approximation i_{0} [

i k = i k − 1 − F ( i k − 1 ) F ′ ( i k − 1 ) , k = 1 , 2 , 3 , ⋯ (6)

Retention period is a time period that a customer stays with the company with certainly. For simplicity, if we assume that the average margin each customer per year is constant, that is m t = m , ( t = 0 , 1 , 2 , ⋯ , n ) ), then the formula (1) has the form

C L V = m ( 1 + 1 i [ 1 − 1 ( 1 + i ) n ] ) (7)

So we can find n, retention period, from the equation as:

n = ln ( 1 − [ C L V m − 1 ] i ) ln ( 1 1 + i ) (8)

for given m and CLV.

It has been more than 20 years since Mobicom corporation first introduced mobile service in Mongolia based on GSM in 1996. As a result of market competition, today there are four major mobile operators such as Mobicom corporation, Unitel, Gmobile and Skytel companies.

In our work, we used the statistical data such as number of subscribers, total income and investment of mobile service from 2005 to 2016 obtained by the annual reports of Communications Regulatory Commission of Mongolia to estimate the main indicators of mobile service operators. According to the annual reports, the number of subscribers is increasing year by years. From the

As cellular telephone companies updated equipment and technology from year to year, the number of subscribers and types of service have increased and tariffs have decreased to some extent to compete with the ever-increasing market. Cellular is presently focused on prepaid and postpaid 2 types of product on the market telecommunication markets. Most of subscribers, 87.72 percent, are prepaid users.

We showed the total revenue of Mobile service and total found raising to the state budget from Mobile service in the

In our work, we assume that the total profit generated by mobile operators is 20 percent of its total income, since we are lack of information about the total cost of mobile companies. So we can find the average margin per customer for a mobile service company by dividing the total profit to the number of active subscribers.

years, this amount has dropped to 29.2 thousand MNT in 2016.

To find discount rate numerically, we solved the following problem by Newton’s method starting with the initial approximation i 0 = 0.001 using C++ program:

C L V = ∑ T = 0 5 m t ( 1 + i ) t

where m t and CLV are given. Results are shown in

According to the calculation, if the customer life time value is 130 thousand MNT, discount rate should be 18 percent, however if it is 170 thousand MNT then discount rate must be smaller, that is 4.6 percent. We also showed the iteration numbers which are required to achieve within 0.0001 accuracy.

Since there was tendency to stabilize the average margin per customer of a mobile service company of Mongolia, we assumed that the average margin for

Customer lifetime value, /thous. MNT/ | Discount rate, % | Iteration number | |
---|---|---|---|

1 | 110 | 29 | 8 |

2 | 130 | 18 | 6 |

3 | 150 | 11 | 5 |

4 | 170 | 4.6 | 4 |

each customer is constant, that, about 32 thousand MNT and calculated how long the company should retain the customers using the Equation (9) to gain certain profit from the customers. Numerical results are given by the following table.

From

1) We showed that it is possible to find the discount rate using the numerical methods. We used the Newton’s method to solve the problem.

2) We proposed a formula for calculating retention period of CLV model for a constant margin and any customer lifetime values.

3) We applied the proposed approach to the Mongolian mobile service at company level and estimated the average margin per customer and CLV of a mobile service company using annual data by the Communications Regulatory Commission of Mongolia.

Discount rate, % | CLV (thous. MNT) | Retention period | |
---|---|---|---|

1 | 8 | 200 | 7 |

2 | 8 | 150 | 5 |

3 | 12 | 200 | 9 |

4 | 12 | 150 | 5 |

5 | 16 | 200 | 11 |

6 | 16 | 150 | 6 |

4) We hope that our approach will provide the tools for estimating some of main components of CLV model but also give a theoretical contribution in the field.

Bayanjargal, D., Davaasuren, B. and Enkhbat, R. (2018) A Numerical Approach to the Customer Lifetime Value. iBusiness, 10, 85-91. https://doi.org/10.4236/ib.2018.102005