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

A Bit of 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, which you can find here. There, 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. If you’re a visual person, you might want to have a look at the previous post to see what I mean by “cohort retention”.

I saw someone asking this question in A Spreadsheet for Calculating Subscription Lifetime Value post by Eric Stromberg, which is very comprehensive. I had the same question, and as I managed to answer it for myself, I think it would be valuable to give a cue to those who are wondering.

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.

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, the topic which I talked about here.

Ending this blog post, I would like to mention that I feel inspired to continue with this customer analytics series, so if that’s something of your interest, give me some claps, comment and follow me to stay in touch.