Dead Reckoning: Counting Your Customers Who Never Say Goodbye

Abstract

Firms in non-contractual commerce face the challenge of knowing how many customers they actually have because customers can stop buying without ever saying they have left. Buy-Till-You-Die models address this by estimating each customer's probability of being alive, a quantity called P(alive) and used in every major software tool for dashboards, churn, customer equity, and enterprise valuation. We show this practice confounds two distinct quantities. Within the beta-geometric family, P(alive) is the infinite-horizon limit of an observable family of finite-horizon repeat-purchase probabilities. Every finite-horizon estimate, such as the probability of repeat purchase within 12 months, is a forecast of a verifiable event. The infinite-time limit can only be reached by extrapolation, a customer count obtained by dead reckoning. The implied count is therefore only partially identified: realized returners are the lower bound, and estimation conventions determine the reported point estimate above that. On a seven-year panel of 31,683 customers, specifications with nearly identical observable forecasts estimate the number of alive customers anywhere from 3,654 to 27,734, a factor of 7.6; a default software weighting parameter alone swings the count 42 percent; and five years of later purchases falsify the maximum-likelihood count from below. The patterns replicate on the CDNOW benchmark, with a 2.4x spread. Most of what practice calls miscalibration is instead a category error: summed P(alive) overshoots realized eighteen-month returners by 2.25x, while the same model's own eighteen-month forecast errs by just 1.18x. The remedy is to report an auditable horizon count, estimate return probabilities at a stated horizon, audit them across scoring dates and horizons, recalibrate as cohorts drift, and report the total count, if at all, as an interval rather than a point.