1.45x at One Year, 2.5x Over a Lifetime

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

1.45x at One Year, 2.5x Over a Lifetime

The chapter presses on a single fact: a 2% monthly churn leaves about 78 of 100 customers after a year, while 5% churn leaves only about 54 — a mere 3-point gap. Yet stretched across a full customer life those same rates imply a 50-month versus 20-month lifetime, a 2.5x spread in lifetime value. So why does the same 3-point churn gap read as a modest one-year snapshot but a 2.5x lifetime spread?

Companion to the chapter Financial Metrics — every number here comes from that chapter.

Ada: This section reports two ratios for the same 3-point churn gap, and a quick reader might expect them to agree. They do not — and the mismatch is exactly what “churn compounds” means. Let me carry both out at full precision and reconcile them.

The 12-month survival snapshot:

  • 2% churn: 10,000 x (0.98)^12 = 10,000 x 0.784717 = 7,847 (the chapter rounds 0.98^12 to 0.785, giving 7,850)
  • 5% churn: 10,000 x (0.95)^12 = 10,000 x 0.540360 = 5,404 (rounded to 0.540, giving 5,400)
  • Gap at one year: 7,850 - 5,400 = 2,450 customers (2,444 at full precision)
  • Snapshot ratio: 7,850 / 5,400 = 1.45x

The lifetime lens:

  • Average lifetime is 1 / churn, so 1 / 0.02 = 50 months versus 1 / 0.05 = 20 months
  • Lifetime ratio: 50 / 20 = 2.5x

So why 1.45x at the one-year mark but 2.5x over a lifetime? Because the 12-month figure is a single snapshot, while the average lifetime is the entire survival curve added up: 1 + (1-c) + (1-c)^2 + ... = 1/c. Summing the 2% curve gives 50 months and the 5% curve gives 20 — both land on the penny. The compounding gap keeps widening past month 12, so integrating over the whole relationship stretches a 1.45x snapshot into a 2.5x lifetime spread.

The design lesson lands squarely on LTV: because lifetime value scales with that summed lifetime rather than a convenient one-year checkpoint, judging churn at 12 months captures only about 58% of its true spread. A launch review that shrugs at “78% versus 54% after a year” is reading the wrong ratio — the number that flows into lifetime value is the 2.5x, and it only appears when churn is measured across the full expected life of the customer.

Every number above is taken from the chapter’s own material and re-derived step by step.