*As a Sum of Cohort Retention, not a Proof but an Experiment*

### Inspiration

This is a short-read follow-up post to the one I wrote on determining the average lifetime value of a cohort using retention. I have suggested a way how you can extrapolate customer retention on newer cohorts to get the lifetime, and here, I would like to explain why it makes sense to add up cohort retention to get an average lifetime of a customer.

**The Setup**

Now, imagine we’re looking at one cohort only, and it has 12 people in it. Let’s use one month as a base period to measure retention on. We’ll say Tenure = 1 when we mean that this has been 1 month since the customer has joined. For simplicity, we’ll also assume that we’re running a subscription-based service, and churned customers have churned for good. This might not hold true for real life, but you’d find a way to account for it, i.e. by creating a new customer_id for resubscriptions and tracking those in another cohort, or using a unique identifier, such as an email, and estimating the probability of a customer re-subscribing. That said, I can show how the sum of cohort retention will reflect the average lifetime.

**Tenure (maturity) = 1**

The period we acquire them, let’s consider it as period 1, retention will be 12/12 = 100%, as all of the 12 customers placed at least 1, or *their first* order. Hence, the average lifetime of a cohort is 1.0, or 100% expressed as a percentage.

**Tenure (maturity) = 2**

Now we’re in the second month following the acquisition. 6 customers out of the original 12 customers re-subscribed, hence, cohort retention is 6/12 = 50%. What about lifetime of an original cohort? Well, we have 6 customers, or 50% of the cohort, that purchased in the second month, or demonstrated the lifetime of 2 months. The rest 6 customers stuck to a 1-month subscription. This gives us a cohort *weighted average lifetime* of 2 (months) * 50% (of a cohort) + 1 (month) * 50% = 150%, or 1.5 as an integer.

As you might have noticed, this 1.5 corresponds to a sum of retention for months 1 and 2, which is 100% and 50%.

**Tenure (maturity) = 3**

Let’s say in month 3, only 3 customers re-subscribed, which gives us cohort retention of 3/12 = 25%. So, 3 out of 6 customers from month 2 have not re-subscribed. This way, 25% of the cohort have a lifetime of 3 months, 25% remained at lifetime of 2 months, and 50% will have a lifetime of 1 month. Notice that the weights add up to 100%. Given that, the weighted average lifetime for the cohort will be 3 (months) * 25% + 2 (months) * 25% + 1 (month) * 50% = 175%, or 1.75 as an integer.

The figure below summarizes the above text: “customers” in red will have the tenure (maturity) that you’ll see on the left by the arrow, and the ones in green will be re-subscribing the next month.

Figure 1: Legend for Figure 2.

Figure 2: Retained vs not retained customers over 3 months.

Same thing, if you add up cohort retention over 3 months, you get 100% + 25% + 25% = 175%, or 1.75.

**Summing it up**

I hope this small experiment has served its purpose and made you witness that the sum of cohort retention over time gives us a weighted average lifetime of customers acquired at tenure 1. To me, it looks like a very simple and elegant way of getting to an average lifetime, which can be used on actual and extrapolated data.

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