MKT 626 | Forecasting churn and CLV: Tool 5

The First Dollar

One customer, four contracts, four values. Everything hinges on the first payment: WHEN it arrives, and WHETHER it arrives.

IF you could see one customer's dialsEverything here imagines you could see an individual customer's own retention propensity r, constant period to period, and their own margin m, also constant. NOT the company's rate, not a cohort's, not any group average.

The three ingredients

one individual customer | constant m and r | annual periods | infinite horizon
This customer's own margin, held constant.
Each year of waiting divides by (1+d).
This customer's own propensity, constant period to period.
...Prepaid: pay now, this payment sure
...Good for the money: signed now, billed later
...Free trial just ended: pay now, if they convert
...Maybe pay, maybe leave: billed later, if they stay
...Spread: best contract vs worst, same customer

Model, conventions, and source notes

The model. One customer, infinite horizon, annual periods. Each payment is worth m, needs one more ×r per survival required, and one more ÷(1+d) per period of waiting. The geometric series always collapses to the shared base m / (1 + d − r); the four contracts differ only in the first payment's timing (now vs a period from now) and certainty (probability 1 vs r). Prepaid m(1+d)/(1+d−r); signed now, billed later m/(1+d−r); trial just ended m·r(1+d)/(1+d−r); maybe pay, maybe leave m·r/(1+d−r). Signature value = the middle column gap = m(1−r)/(1+d−r).

Step 6, one customer vs a group. The collapse to m(1+d)/(1+d−r) needs the same r every period. That holds for one customer's own renew-or-churn coin. A group of customers with different coins has a retention rate r(t) that rises over time (the churn-prone leave first), so the formula is not the value of the group, and plugging in an average or observed retention rate misstates it. The group's value is the average of each customer's own value.

Caveats from class. The whole page is an IF: as if you could see one individual customer's own retention propensity r, constant period to period, and their own constant m. Neither is observable in practice, and neither is a company-wide, cohort, or group-average rate; averaging across customers with different r's is a different (and later) lecture. Strictly, free-to-paid conversion at the end of a trial deserves its own rate (r₀ at the buzzer vs r ongoing); this tool sets r₀ = r, as the class deck does. Source: "The First Dollar" class deck (m = $100/yr, d = 10%, r = 80%).