1.45x at One Year, 2.5x Over a Lifetime

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.

See the relationship before changing it

The figure reads from left to right. The blue card is monthly churn. The middle card applies the page rule. The green card is expected customer life. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Monthly churn changes expected customer life An input card leads through the rule life = 100 / monthly churn percent to the expected customer life result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Lower churn stretches the whole survival curve, not just the first-year snapshot.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2 %.

  2. 2

    Name the relationship. life = 100 / monthly churn percent

  3. 3

    Substitute with units. 100 / 2 = 50.00 months

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change monthly churn

Try Predict the direction of life = 100 / monthly churn percent. Test another monthly churn, then compare expected customer life.

2 %
Chapter baseline
Expected customer life

Observe Lower churn stretches the whole survival curve, not just the first-year snapshot. Reset monthly churn to 2 and compare expected customer life.

Explain Lower churn stretches the whole survival curve, not just the first-year snapshot.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only monthly churn moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only monthly churn. It is not a field prediction.

TryBegin with monthly churn rates of 2% and 5%, calculate both 12-month survivor counts, and select Check derivation.
ObserveThe one-year panel shows about 78 versus 54 survivors, but expected lifetimes separate to 50 versus 20 months.
ExplainCompounding magnifies the churn gap because lifetime value sums every surviving month, including the widening tail beyond the 12-month snapshot.

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.

Technical boundaries
The churn model deliberately assumes a constant independent monthly rate and does not simulate cohort aging, seasonality, reacquisition, discounting, margin changes, upgrades, or competing risks.

Work the audit first, then check the displayed derivation.

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