How can we determine how valuable our customers are to us? In this article, we will shed light on questions such as what path should we follow if we want to analyze and calculate this mathematically, not based on our feelings. Customer Lifetime Value (CLTV) or if we want to define it in Turkish, Customer lifetime value; is the monetary value that a customer will bring to the company during the relationship communication with a company. Calculating the lifetime value of our customers will help us a lot in making future decisions and if we want to further improve both customer-seller relationships and sales return by examining customer-based sales.

**How is Customer Lifetime Value Calculated?**

CLTV = (Customer Value / Churn Rate )

Profit Margin Customer Lifetime Value = (Customer Value / Lost Customer Rate ) Profit Margin

With the above formulation, we can calculate the Customer Lifetime Value. Well, if this formulation means nothing to us, let’s get to know each variable in more detail:

Customer Value = Average Order Value Purchase Frequency Customer Value = Average Order Value Purchase Frequency

Average Order Value = Total Price / Total Transaction Average Order Value = Total Earnings / Total Orders

Churn Rate = 1-Repeat Rate Lost Customer Rate = 1 -Repeat Rate

Repeat Rate = (Total Transaction > 1 Number of Customers) / All Customers

Profit Margin = Total Price Profit Rate Profit Margin = Total Profit Product Profit Rate

While learning the total values or total numbers in these formulations, we obtain the total values or total numbers by specifying the date on which we want to examine the value from the transaction date to that time.

We can segment our customers according to the values we obtain from here, send these results to the relevant department and take action to activate segment-based customers even more.

Churn Rate and Profit in the formulation above are the values obtained from the mass, that is, they are not customer-based. TotalPrice is the variable that has large values and has a high impact on the results.

Yes, when we apply the process up to this point to our data, we learn the value of our customer until the date we specify. That is, this calculation represents a single time, the time period in which the analysis was made.

As a result of the operations so far, we obtain the Customer Life Time Calculation. We did not use any statistical probability distribution. This is a very important distinction.

So, what do we do when we want to do research for the future with a certain time projection, taking into account the behavior of our customers?

#### Customer Life Time Prediction

With the Customer Lifetime Value value, we calculate the value of the customer to us for the period up to a certain date. But for us, being able to predict the future is extremely important for the institution we work for. We can even target maximum profit by planning the necessary strategies and taking the necessary actions based on this result. Of course, we could have made this prediction with the Machine Learning algorithm, and maybe it would have given more healthy results, but we can make a prediction for the future by including statistical probability distributions by not using ML.

Customer Value = Average Order Value * Purchase Frequency

CLTV = Conditional Expected Number of Transaction * Conditional Expected Average Profit

There was no Conditional Expected statement in the above CLTV formula.

We calculate this by including various statistical probability distributions when making predictions for the future. So what are these probabilistic distributions?

We will use the BG/NBD model in the Conditional Expected Number of Transaction part of the CLTV formula to the right of the equation.

We will use the Gamma Gamma Submodel in the Conditional Expected Average Profit section of the CLTV formula to the right of the equation.

CLTV = BG/NBD Model * Gamma Gamma Submodel In this way, we can note an expression like the one above in order to impose which models will be used more easily.

The Conditional expression takes the Conditional expression, which states that the estimation takes place within the customer information condition since our estimation is calculated based on the customer, that is, considering the individual purchasing behavior of our customers.

When calculating CLTV, after modeling the order behavior of all customers in the Conditional Expected Number of Transaction part, we make our predictions for the future under the condition of the customer’s personal order behavior with the BG/NBD model. And as a result, Conditional Expected Number of Transaction

While calculating CLTV, after the profit rate distribution of all customers is modeled in the Conditional Expected Average Profit section, we make our future prediction using the Gamma Gamma Model under the condition of the customer’s personal profit rate.

**Expected Number of Transaction with BG/NBD (Beta Geometric / Negative Binomial Distribution)**

BG/NBD model aka **Buy Till You Die**.

It is used to estimate how much purchase customers can make in a given time period within the problem.

The BG/NBD model probabilistically models two processes for the Expected Number of Transaction: Transaction Process (Buy) + Dropout Process (Till You Die)

#### Transaction Process (Buy)

As long as a customer is alive (in the buying process), they continue to make random purchases around their transaction rate. And these transaction rates change based on the customer, the gamma is distributed for the entire audience (r, alpha). The gamma here is different from the Gamma Gamma Submodel.

After a customer buys a product, the Transaction Rate is reduced. Because that customer met or partially met their needs from our company. We may not expect to receive products and services from us for a certain period of time. While the Transaction Rate decreases, the Transaction Rate increases in the process when the customer drops. In other words, we can say that Transaction Rate and Transaction Rate are the opposite of each other.

The BG/NBD model will evaluate the Transaction Process for the entire audience. We will then reduce this to the client.

**Dropout Process (Till You Die)**

We can think of it as the reverse of the above process. In the example above, we stated that a customer met their needs after purchasing products and services from us, and assuming that they partially met, there was a drop for a certain period of time, that is, the dropout rate increased. So, what happens when our customer’s needs for the product-service they buy are gradually approaching as long as their needs increase? Yes, Transaction Rate increases, Dropout Rate decreases. Dropout Rates also vary for each customer and beta is distributed over the entire audience (a, b).

If we summarize the event briefly; We specified CLTV = Conditional Expected Number of Transaction * Conditional Expected Average Profit.

We have stated that we will calculate the Conditional Expected Number of the Transaction value with the BG/NBD model. And we mentioned that another name for this BG/NBD model is Buy Till You Die.

The Buy Till You Die process has two stages; Transaction Process (Buy) + Dropout Process (Till You Die)

Transaction Process; It was the part where the transaction rate of the customer increased and modeled with the gamma distribution. (r, alpha). Of course, as it gets closer to the Dropout Process, the transaction rate decreases.

Dropout Process; It was the part where the dropout rate of the customer increased and was modeled with the beta distribution. (a,b)

In the light of the information we have learned so far, we can calculate the Conditional Expected Number of Transaction values in the BG/NBD model.

If we examine the left part of the equation in detail;

· E: Comes from Expected

· X: Customer

· x: number of repeat sales (frequency) for customers who have made at least a second purchase (customer-specific)

· tx: the difference between the customer’s first purchase and the last purchase (recency) (customer-specific)

· T: The time elapsed since the customer first contacted our institution (Teneur) (customer-specific)

· r, alpha: comes from a gamma distribution (Buy) (Audience Feature)

· a, b: income from beta distribution (Till You Die) (Audience Feature)

Under the condition of the customer’s information, we obtain the expected number of transactions specific to a particular customer over the characteristics of the entire audience to us in a certain period of time.

**Expected Average Profit (Gamma Gamma SubModel)**

It is used to estimate how much profit a customer can generate on average per trade. The monetary value of a customer’s transactions is randomly distributed around the average of their transaction values. The average transaction value may change between users over time, but not for a single user. The average transaction value is gamma distributed among all customers. There is a gamma distribution for all customers and this distribution model the average transaction values. Later, we will reduce this model to individuals by conditioning it among all customers.

If we examine the left part of the equation;

· x: number of repeat sales (frequency)

· MX: observed transaction value (monetary)

· p, q, y: variables belonging to the gamma model

We can perform Customer Lifetime Value Prediction by processing the values we obtained using BG/NBD ( Buy Till You Die ) and Gamma Gamma Submodel.