Geometric LTV vs Heterogeneous-Churn LTV (Fader-Hardie sBG)

Use a geometric model as an explicit baseline or scenario when constant survival and contribution are defensible. A single aggregate revenue-churn value does not establish customer survival. When evidence is weak, show a finite-horizon sensitivity range or decline a decision-grade forecast.

Use sBG when renewal-period cohort survival suggests churn heterogeneity and the model fits the observation process. Fit churn parameters to customer counts, account for censoring, and evaluate held-out periods and cohorts. Cohort duration alone does not establish identification or forecast quality.

With constant contribution c, survival p and zero discount, a geometric model gives c/(1-p) including the initial payment and c×p/(1-p) for future payments only. Use monthly contribution with monthly survival, or annual contribution with annual survival. For churn probability q~Beta(α,β), sBG survival is S(t)=B(α,β+t)/B(α,β), with S(0)=1. The undiscounted expected number of paid periods including the initial payment is (α+β-1)/(α-1) for α>1. Future-only periods equal β/(α-1). The infinite sum diverges for α≤1 when contribution is positive; finite-horizon and discounted values remain distinct calculations. There is no universal 20–60% downward correction. At the same initial churn, heterogeneity leaves a more persistent survivor population and can produce a longer mean lifetime. A late-cohort hazard applied to new customers can instead overstate their value. The direction depends on the comparison, timing, data and model fit. These are survival models, not complete cash-flow models. Conditional contribution, pricing, support, inference, retention and expansion costs can change with tenure. Validate those inputs before using LTV to set an acquisition budget.

LTV is a discounted expected-contribution estimate with explicit customer-survival, timing, horizon and cost assumptions. GRR/NRR do not identify customer survival, and no universal sBG haircut is applied.