1 · Who churns?
Every customer renews each period with their own probability. The sBG model says those churn probabilities θ are spread across the cohort as a Beta(a, b) distribution — not one shared rate.
2 · Survival & retention
The cohort's survivor curve S(t) and retention rate r(t). Aggregate retention rises over time even though no individual's churn probability changes — the high-churn customers leave first.
3 · After n renewals
Survivors are a selected group. After n renewals the churn-probability distribution among remaining customers is Beta(a, b+n): same a, more b — shifted toward loyal.
4 · Value & payback
Each possible lifetime T is worth PAV(T) − CAC. The expected cumulative discounted cash flow E(CtV) climbs toward E(CLV) as t grows; payback is where cumulative value covers CAC. Click any CLV(T) point to trace that same lifetime in the two distributions below; click again (or elsewhere) to clear.
5 · Distribution of CLV
CLV is a random variable: each lifetime has a value and a probability. The average E(CLV) is just one summary of this whole distribution.
6 · Distribution of RLV vs. PAV
A customer who has survived n renewals has a residual value distribution shifted right relative to a brand-new customer's PAV — survival is informative.