ARPU fell 5% last month — how do you tell whether revenue or the active-user count drove it?

## What ARPU is, structurally ARPU (average revenue per user) is total revenue in a period divided by the count of active users in that period. It is a **ratio metric**: a numerator (money) over a denominator (people). Every movement in a ratio is the joint result of both parts, and quoting the ratio alone throws away the one fact a decision needs — which part moved. The first instinct of a weak candidate is to treat an ARPU drop as a revenue drop. It frequently is not. ## The exact arithmetic Let the baseline be `A0 = R0 / U0` and the current period `A1 = R1 / U1`. Then ``` A1 / A0 = (R1 / R0) / (U1 / U0) ``` Writing growth rates `g_R = R1/R0 - 1` and `g_U = U1/U0 - 1`: ``` 1 + g_A = (1 + g_R) / (1 + g_U) g_A = (g_R - g_U) / (1 + g_U) ``` The shortcut everyone reaches for, `g_A ≈ g_R - g_U`, is just the numerator of that expression. It is accurate when the denominator barely moves and increasingly wrong as it does. Concretely, revenue +2% with users +8% gives an exact ARPU change of `(0.02 - 0.08) / 1.08 = -5.56%`, while the subtraction shortcut says -6%. Fine for a whiteboard, wrong on a slide that claims precision. Logs remove the approximation entirely, because the log of a ratio is a difference: ``` ln(A1) - ln(A0) = [ln(R1) - ln(R0)] - [ln(U1) - ln(U0)] ``` Log-point contributions add up exactly, which is why they are the clean way to attribute a ratio movement to its two parts. In the example above: `ln(1.02) = +0.0198` from revenue, `ln(1.08) = +0.0770` from users, net `-0.0572` — about -5.6%, matching the exact figure. ## The three stories a 5% drop can be 1. **Numerator-driven.** Users flat, revenue down. Something happened to monetisation: pricing, a broken checkout, a discount campaign, a large customer leaving. This is the story people assume by default. 2. **Denominator-driven (dilution).** Revenue flat or up, users up faster. The company acquired users who have not started paying yet. Total money is fine or growing; the per-head average fell because the head count grew. Punishing this is how you talk a growth team out of growing. 3. **Both.** The usual case. State each contribution in points rather than picking whichever supports the narrative you already had. ## Splitting each part further The decomposition does not have to stop at two terms. Because `revenue = payers x revenue per payer`, and `payers = active users x payer rate`, you get ``` ARPU = payer rate x revenue per payer ``` This is exact: `(payers / active users) x (revenue / payers) = revenue / active users`. It separates two very different failures — a smaller share of your users converted to paying, versus the ones who did pay spent less each. They land on different teams. On the denominator side, "active users" can grow from acquisition, from reactivation, or from the same population simply being counted as active more often. Which of those it is changes whether the dilution is expected to reverse. ## Why an ARPU decline is often the healthy outcome Any company acquiring users quickly will see ARPU drift down, because new users start near zero revenue and ramp over their first weeks or months. The aggregate number mixes cohorts of very different tenure, so it moves whenever the tenure composition moves. The standard fix is to look at ARPU **by cohort age**: revenue per user at month 1, month 3, month 6, for each acquisition cohort. If every cohort curve is flat or improving, the aggregate decline is composition, and it is the arithmetic price of growth. ## Traps to avoid - Subtracting the two percent changes and presenting it as exact. - Reporting "ARPU is down" with no statement about which part moved — it is an incomplete answer, not a short one. - Confusing ARPU with ARPPU (average revenue per **paying** user). They have different denominators and can move in opposite directions: if the payer rate falls but the remaining payers are heavy spenders, ARPU falls while ARPPU rises. - Treating an unfavourable direction as automatically a problem before checking which part caused it. ## How to state it One line, both parts, then the reading: "ARPU -5.0%: revenue +2.0%, active users +8.0%. Revenue per payer is flat, so this is dilution from the acquisition push, not monetisation decay." That sentence is what the question is actually asking for.