The Customer Lifetime Value Timeline

Walk a customer's relationship through time and watch each piece of value light up: the acquisition cost paid up front, the Value to Date already earned, the Residual Lifetime Value still to come, and how they sum to E(CLV) — the post-acquisition value of a just-acquired customer. Drag Today to split past from future; change the economics to reshape the whole picture.

Business setting
Whose value?
Population

Definitions used here
Acquisition cost (CAC)
What you pay up front to acquire the customer. Sits below zero at t=0. Acquire only when E(CLV) > CAC.
Value to Date
Past, backward-looking: the discounted contribution earned from acquisition through Today. Already in the business — not part of forward-looking CLV.
Residual Lifetime Value (RLV)
Forward-looking, from Today onward. For a still-active customer, E(RLV) = (1−θ) × E(CLV) — and because churn θ is stationary, it doesn't depend on how long they've been active.
Post-Acquisition Value — E(CLV)
The whole forward value of a just-acquired customer: the present value of all future expected cash flows. E(CLV) = m × (1+d)/(d+θ). The first payment is certain, so E(CLV) can't be zero.
Margin multiple
(1+d)/(d+θ) — the number of "years of margin" a customer is worth. Multiply by per-period margin m to get E(CLV).
AOV & AOF
Average Order Value and Average Order Frequency. Revenue per active customer = AOV × AOF; contribution margin m is the net-of-cost slice of that.
% active / retention
Survival S(t) = (1−θ)t in the homogeneous case. In a heterogeneous cohort the observed retention rate rises over time — the "shakeout": those who stay are the champions.
Heterogeneous world (BG)
Churn propensity varies across customers, drawn from a Beta(a,b) distribution — the Beta-Geometric model. Each customer's own churn stays constant; the cohort's retention rises only because high-churn customers leave first. Polarization φ = 1/(a+b+1) sets the spread. Because of the fat right tail of survivors, E(CLV) is higher than a homogeneous model with the same average churn — ignoring heterogeneity always undervalues the base — and Residual Lifetime Value grows with tenure.

Teaching illustration. Churn is modeled as geometric (homogeneous world) or Beta-Geometric (heterogeneous world), with annuity-due timing — the first payment arrives at acquisition. Population constructs E(CLV) and E(RLV) are computed as discounted infinite-horizon sums over the survival curve. Numbers are illustrative, not a forecast of any specific customer base.