The compounding formula both ways, a cohort of 1,000 customers month by month, and the LTV error that hides in the annual version.
Annual churn is one minus the twelve-month survival rate: 1 − (1 − monthly churn)12. A 2.0 per cent monthly churn is 1 − 0.9812 = 21.5 per cent a year, not the 24.0 per cent you get by multiplying by twelve. In the other direction, 10 per cent a year is 0.874 per cent a month. And once the conversion is right, an annual LTV formula built on it still overstates lifetime value by 11.5 per cent.
Worked in full in The Growth Equity Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Churn arrives in a growth equity data room in whatever unit the company tracks: monthly for a product sold to small businesses, annually for enterprise contracts, sometimes both. Comparing companies, or feeding a monthly figure into an annual model, needs a conversion, and the obvious one is wrong. The error is small at low churn and large exactly where it matters, in the high-churn segments that drive the diligence questions.
| Input | Value |
|---|---|
| Customers at the start | 1,000 |
| Monthly churn (share of customers at the start of each month who leave in it) | 2.0% |
| Average revenue per account, per month | 1,000 |
| Gross margin | 80% |
Annual retention = (1 − monthly churn)12 = 0.9812 = 78.47%
Annual churn = 1 − annual retention = 21.53%
Monthly churn = 1 − (1 − annual churn)1/12, so 10% a year is 0.874% a month
In Excel, with monthly churn in B1: =1-(1-B1)^12, and with annual churn in B2: =1-(1-B2)^(1/12).
Each month's 2.0 per cent applies to the customers still there. Watching the cohort makes the point:
| Month | Customers remaining | Lost to date |
|---|---|---|
| 1 | 980.0 | 20.0 |
| 3 | 941.2 | 58.8 |
| 6 | 885.8 | 114.2 |
| 12 | 784.7 | 215.3 |
Twenty customers leave in month 1, but only 16.0 in month 12, because the base has shrunk. Multiplying by twelve assumes 20 leave every month and reports 240 lost instead of 215.3: the naive figure overstates annual churn by 2.47 points, or 11.5 per cent.
| Monthly churn | Times 12 | Compounded | Gap, points | Annual retention | Average life, months |
|---|---|---|---|---|---|
| 0.5% | 6.0% | 5.8% | 0.2 | 94.2% | 200 |
| 1.0% | 12.0% | 11.4% | 0.6 | 88.6% | 100 |
| 2.0% | 24.0% | 21.5% | 2.5 | 78.5% | 50 |
| 3.0% | 36.0% | 30.6% | 5.4 | 69.4% | 33 |
| 5.0% | 60.0% | 46.0% | 14.0 | 54.0% | 20 |
At 0.5 per cent a month the shortcut is harmless. At 5.0 per cent it reports 60.0 per cent annual churn for a business that keeps 54.0 per cent of its customers, a company that looks broken on one reading and merely leaky on the other. Average customer life is one over the monthly rate: 50 months at 2.0 per cent.
The reverse conversion matters just as much when a company reports an annual figure and the model runs monthly. Dividing an annual 10 per cent by twelve gives 0.833 per cent a month; the compounded figure is 0.874 per cent, and only the second one returns to 10 per cent after twelve months.
The conversion is right, but plugging it into an annual lifetime value formula creates a second error. Gross profit per account is 800 a month, or 9,600 a year.
| Model | Formula | LTV |
|---|---|---|
| Monthly | 800 ÷ 2.0% | 40,000 |
| Annual, churn times 12 | 9,600 ÷ 24.0% | 40,000 |
| Annual, compounded churn | 9,600 ÷ 21.53% | 44,592 |
The monthly model is the one to trust, and the naive annual model matches it only because its two errors offset. The compounded annual version is 11.5 per cent too high because an annual model charges a full year's revenue for every customer alive at the start of the year. In reality customers who leave in March stop paying in March: the expected gross profit in year one from a starting customer is 8,611, not 9,600.
Rule: convert churn by compounding, but build lifetime value at the frequency the customer pays. If the company bills monthly, use the monthly rate; if it bills annually in advance, the annual model is correct, because a customer who leaves at renewal has already paid for the year.
Annual churn is 1 − (1 − monthly)12: 2.0 per cent a month is 21.5 per cent a year. Then build lifetime value at the billing frequency, or it overstates by 11.5 per cent here. When expansion offsets churn, lifetime value needs a different formula, set out in LTV to CAC when net retention exceeds 100 per cent. The Growth Equity Investor covers underwriting growth companies on unit economics like these, and the free workbook for this case works the arithmetic of follow-on reserves and pro rata rights.
Because each month's churn applies to a smaller base: customers who have already left cannot leave again. Of 1,000 customers at 2.0 per cent a month, 20 leave in month 1 but only 16.0 in month 12, and 215.3 have gone by year end. Multiplying by 12 assumes the full 20 leave every month, giving 24.0 per cent instead of 21.5.
Monthly churn = 1 - (1 - annual churn)^(1/12). An annual churn of 10 per cent is 0.874 per cent a month, not 0.833 per cent. In Excel: =1-(1-B1)^(1/12). The difference looks small, but it compounds into every monthly cohort model built from an annual assumption.
It depends on the customer segment, and any benchmark is illustrative. A business selling to small companies often runs at 2 to 5 per cent a month, which compounds to 21.5 to 46.0 per cent a year; enterprise software is usually measured annually and is expected to sit well below that. Compare like-for-like segments and the same definition.
This article is one calculation from The Growth Equity Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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