LTV: Lifetime Value, what one user is worth over their whole life

The question LTV answers. How much money does one user bring in from signup to churn, over their whole lifetime? That number sets the ceiling on acquisition cost: customer acquisition cost (CAC) must stay below LTV, or every acquired user loses money.

The derivation starts from the retention curve. Let monthly retention be r: the share of this month's paying users who still pay next month. The share still present after t months is r to the power of t. Expected paying months are the sum of those shares: 1 + r + r² + r³ + …, a geometric series summing to 1/(1-r). So:

LTV = ARPU \times margin \times \frac{1}{1-r}

ARPU is average revenue per user, margin is the share left after direct costs, and 1/(1-r) is the expected lifetime.

Lucerna numbers, with assumptions labelled. Prices are settled: ¥18 per month, ¥180 per year. Retention and margin are assumptions, not measured data, used only to walk through the formula. Monthly plan at 90% monthly retention: expected lifetime 1/(1-0.9) = 10 months, gross LTV 18 × 10 = ¥180, and at a 75% margin about ¥135. The ¥180 per year is the annual plan's first-year payment, not a lifetime total: at 60% annual retention the expected lifetime is 1/(1-0.6) = 2.5 years, giving about 180 × 2.5 × 75% ≈ ¥337.

Two refinements that belong in the full version.

Conversion: the formula above is payer LTV, which assumes the user already pays. What CAC must be compared against is acquisition LTV, the expected value per acquired user:

acquisition\ LTV = conversion\ rate \times payer\ LTV

At a 10% paid conversion and a ¥135 payer LTV, each acquired user is worth ¥13.5 on expectation. With a two-stage funnel (visitor to signup to paid), multiply both conversion rates.

Discounting: future money is worth less than present money, so the strict version discounts each future period by (1+d)^t, where d is the discount rate:

discounted\ LTV = \sum \frac{retention_t \times ARPU \times margin}{(1+d)^t}

The monthly example undiscounted is ¥180 gross; at a 1% monthly discount rate it is about ¥165. The gap grows with lifetime.

Sensitivity tables (gross, before margin). Monthly plan: 80% retention → 5 months → ¥90; 85% → 6.7 months → ¥120; 90% → 10 months → ¥180; 95% → 20 months → ¥360. Annual plan: 50% annual retention → 2 years → ¥360; 60% → 2.5 years → ¥450; 70% → 3.33 years → ¥600; 80% → 5 years → ¥900. Multiply by margin for net LTV.

Four details that matter in practice. First, a real retention curve is not one exponential: a steep early drop, a flat middle, and a long tail of core users, so a fixed r misleads and cohorts must be read separately (paid-acquired cohorts retain far worse than organic ones). Second, margin is contribution margin: payment fees, servers, model costs, refunds and bad debt are deducted, fixed costs are not. Expansion revenue (upgrades, family plans) pushes LTV above the base case; net revenue retention above 100% means negative churn. Third, payback period can kill before LTV matters: a high LTV with an 18-month payback still breaks cash flow, and the annual plan's upfront ¥180 is what makes payback immediate. Fourth, retention is the biggest LTV lever because 1/(1-r) is convex: monthly retention from 90% to 95% doubles lifetime from 10 to 20 months, while a 5% price rise adds only 5%.

What is still missing. Retention rate, exact per-user margin, and CAC have no real data yet: the product has not run paid cohorts. Until then the sensitivity tables frame the range; once paid cohorts run, replace the fixed-r formula with month-by-month summation over the measured cohort curve, sourced from the standalone store and its shadow ledger.