Funnels and unit economics · Unit economics

Payback month from a cohort

Write the function that finds the month a cohort's expected margin covers its CAC, then read whether a $500 channel ever pays back.

codeeasyExercise · 4 min · 18 of 60

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A payback month computed by dividing CAC by margin assumes nobody leaves. Finance needs the month in which a cohort's expected margin actually covers what it cost, and one channel in this plan never gets there.

Half the 0.90 cohort has gone by month 8, so each later month adds less margin; the 0.95 row keeps paying long enough to pay back four months sooner.

The situation

A subscription product acquires customers at $300 each. A customer contributes $40 of gross margin in month 1 and survives each later month with probability 0.9, so expected margin in month n is 40 × 0.9^(n−1): $40, then $36, then $32.40. Finance wants three payback months: for this cohort; for a scenario in which a retention project lifts monthly survival to 0.95; and for a new channel whose CAC is $500 at the same $40 and 0.9. The planning horizon is 36 months, and a cohort that has not paid back by then is treated as one that does not.

Payback is the definition from CAC, LTV and payback: the first month in which cumulative expected margin, margin × survival summed month by month, reaches CAC. That lesson's running numbers, a $521 CAC, $40 of margin and 5% monthly churn, pay back in month 21 by hand: a fourth case to agree with.

The question

Does the $500 channel ever pay back, and what is moving retention from 0.9 to 0.95 worth in months? Answer by writing solve(cac, margin, retention, horizon=36), the first month in which cumulative expected margin reaches CAC or -1 if it does not within the horizon, and reading its three results.

def solve(cac, margin, retention, horizon=36):
    """First month in which cumulative expected margin per customer reaches cac.

    cac:        acquisition cost per customer, in dollars
    margin:     gross margin a paying customer contributes in a month
    retention:  probability that a customer paying this month pays next month
    horizon:    months to look ahead
    Expected margin in month n is margin * retention ** (n - 1).
    Returns the month, counting from 1, or -1 if cac is not reached within horizon.
    """

solve(300, 40, 0.9)    ->  14
solve(300, 40, 0.95)   ->  10
solve(500, 40, 0.9)    ->  -1
solve(521, 40, 0.95)   ->  21

A strong answer · Rubric · Where answers go wrong · Solution

Write your answer first, then open

A strong answer

Keep two numbers: the running total of expected margin, and the share of the cohort still paying, which starts at 1.0 because everyone acquired pays in month 1. Each month adds margin × survival to the total and returns the month if the total has reached CAC; otherwise survival is multiplied by retention for the next month. If the loop runs out of horizon, return -1. It never divides CAC by margin, which assumes nobody leaves, and it never loops without a bound, because some CACs are never reached. The tests pin the four printed results and add edges: no churn, which must give the contracted month 14 at $521, and an 18-month horizon, which must turn that case's month 21 into -1.

The three results answer finance. At $300 and 0.9, payback is month 14. At 0.95 it is month 10, so the retention project is worth four months of cash on every cohort, and that is how to price it, rather than by its effect on LTV. The $500 channel returns -1, and not because 36 months is short. Cumulative margin can never exceed margin ÷ (1 − retention) = 40 ÷ 0.1 = $400, so a $500 CAC never pays back at any horizon. The quick ratio, 500 ÷ 40 = 12.5, would have said month 13.

The decision: do not fund the $500 channel at 0.9 retention. It becomes possible only above 0.92, where the ceiling passes $500, and with the retention project it pays back in month 20, still well outside a 12-month target. The evidence that would change the call is this channel's own cohort measured above 0.92 survival, not the company's blended rate.

Rubric

Cumulates expected margin, margin × survival month by month, not contracted margin or revenue
Starts survival at 1.0, so month 1 is paid in full and months count from 1
Bounds the loop with the horizon and explains the -1 with the ceiling margin ÷ (1 − retention) = $400
States all three payback months and prices the retention change as four months of cash
Tests payback, faster payback and never, plus an edge such as a horizon shorter than the payback month

Where answers go wrong

"LTV is $40 ÷ 0.1 = $400 against a $300 CAC, a ratio of 1.3, so the channel pays." It does, eventually, and the ratio hides when. At 100 new customers a month the company lays out $30,000 every month, is about $138,000 down by month 13, and is whole again only in month 36.

"Payback is 300 ÷ 40 = 7.5, so month 8." That is contracted margin, which assumes no churn. It is six months early for the $300 cohort, and it gives month 13 for a $500 channel that never pays back.

"Start survival at 0.9 in month 1." An off-by-one that discounts the first month, which everyone pays: the $300 cohort reports month 18 instead of 14. The 0.95 case still returns 10, which is why one passing test proves little.

Solution

Python 3 · standard library17 lines · 8 of 8 tests passing

"""Payback month from a cohort — cumulate expected margin until it covers CAC."""


def solve(cac, margin, retention, horizon=36):
    # Invariant: after the body for `month`, `total` is the cohort's cumulative
    # expected margin per acquired customer, sum of margin * retention**(m - 1)
    # for m = 1..month, and `survival` is the share still paying next month.
    # Month 1 is paid in full: survival starts at 1.0, before any churn.
    total, survival = 0.0, 1.0
    for month in range(1, horizon + 1):
        total += margin * survival
        if total >= cac:
            return month
        survival *= retention
    # The ceiling is margin / (1 - retention); a CAC above it never pays back,
    # and a CAC below it can still miss a short horizon.
    return -1

The cases that ran

TESTS = [
    ((300, 40, 0.9), 14),
    ((300, 40, 0.95), 10),
    ((500, 40, 0.9), -1),
    # The same $500 channel once the retention project lands: the ceiling
    # rises to 40 / 0.05 = $800, so it pays back after all, in month 20.
    ((500, 40, 0.95), 20),
    # The module's running numbers: CAC $521, $40 margin, 5% monthly churn.
    ((521, 40, 0.95), 21),
    # No churn reduces to contracted payback, 521 / 40 = 13.03, so month 14.
    ((521, 40, 1.0), 14),
    # Pays back in month 21, so an 18-month horizon reports never.
    ((521, 40, 0.95, 18), -1),
    # CAC covered by the first month's margin.
    ((40, 40, 0.9), 1),
]