# Flows: customers and cash

> One rule for every stock, end = start + in − out: MRR, churn, NRR and GRR; profit vs cash, working capital, DSO and runway; CAC payback and incrementality against a holdout.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Unit economics, the bridge
- Lesson 32/36, ~39 min
- Source: https://productovi.com/courses/math-basics/flows-customers-and-cash

## One rule for every stock

**Alex Tutor:** Last bridge, still at Klasna. Many business numbers are **stocks**: subscribers, monthly recurring revenue, cash. They change only through **flows** in and out, like water in a bathtub.

$$
\text{end} = \text{start} + \text{in} - \text{out} \qquad MRR_{end} = MRR_{start} + new + expansion - contraction - churned
$$

*MRR: monthly recurring revenue. Expansion: existing members paying more; contraction: paying less; churned: revenue from members who left.*

**Sofia:** Klasna Club started March with 300,000 of MRR: 45,000 new, 15,000 expansion, 6,000 contraction, 24,000 churned. So 300 + 45 + 15 − 6 − 24 = 330,000.

**Alex Tutor:** Now three ratios, all divided by the **start**. Revenue churn = 24 ÷ 300 = 8%. **NRR**, what last month's members pay now: (300 + 15 − 6 − 24) ÷ 300 = 95%. **GRR** ignores expansion: (300 − 6 − 24) ÷ 300 = 90%.

![MRR goes from 300,000 to 330,000: +45,000 new, +15,000 expansion, −6,000 contraction, −24,000 churned; NRR is 95% and GRR 90%.](https://productovi.com/courses/math-basics/diagrams/m8-l2-mrr-bridge.png)

*New revenue sits outside NRR: it measures only last month's members.*

**Sofia:** So with NRR below 100%, last month's members shrink as a group, even while total MRR grows on new members.

**Alex Tutor:** First the bridge itself, on April's numbers.

> **Exercise:** Klasna Club starts April with 330,000 UAH of MRR. During April: new 50,000, expansion 12,000, contraction 9,000, churned 27,000. What is MRR at the end of April, in UAH?

> **Exercise:** MRR at the start of April is 330,000 UAH. Expansion 12,000, contraction 9,000, churned 27,000, new 50,000. What is NRR, in percent? Two decimals.

**Alex Tutor:** One more ratio on the same base.

> **Exercise:** April at Klasna Club: start 330,000 UAH of MRR, expansion 12,000, contraction 9,000, churned 27,000. Its GRR is higher than its NRR.

## Cash is a stock too

**Alex Tutor:** Cash follows the same rule, but its flows aren't the profit lines. When Klasna for Teams invoices 300,000 on 60-day terms, profit counts it now; cash arrives later. **Receivables** (AR) grow instead.

**Alex Tutor:** Two flows go the other way: unpaid bills to suppliers, **payables** (AP), and prepaid lessons not yet taught, **deferred revenue**. Both hold cash that profit hasn't counted yet.

$$
\Delta \text{cash} \approx \text{profit} - \Delta AR + \Delta AP + \Delta \text{deferred} \qquad WC = AR - AP - \text{deferred}
$$

*Δ: the change over the period; WC: working capital, the money tied up in day-to-day operations.*

**Sofia:** So a month can be profitable and still lose cash if receivables grow faster than profit.

**Alex Tutor:** Prepaid packages are the friendly case: a family pays in September for lessons taught until December. The cash is in the bank before the profit appears.

> **Exercise:** A month: operating profit 200,000 UAH; receivables up 150,000; payables up 30,000; deferred revenue up 80,000. By how much does cash change, in UAH?

## How long and how fast

**Alex Tutor:** Two ratios turn these stocks into time. **Runway** = cash ÷ monthly net burn: 2,400,000 ÷ 300,000 = 8 months. **DSO**, days sales outstanding, = AR ÷ yearly credit sales × 365: how many days customers take to pay.

**Sofia:** And the cash conversion cycle: days we wait to be paid minus days we wait to pay. With no inventory, DSO − DPO.

**Alex Tutor:** Careful with **burn**: it's cash out minus cash in per month, the change in the cash stock, not the loss on the P&L.

**Sofia:** So runway uses the cash formula from before, not profit. A profitable company can still have a short runway.

> **Exercise:** Klasna for Teams has 450,000 UAH of receivables and 2,190,000 UAH of credit sales a year. What is its DSO, in days?

**Alex Tutor:** Now runway, with burn measured the right way.

> **Exercise:** Klasna has 2,400,000 UAH in the bank. Its profit statement shows a loss of 180,000 UAH a month, but each month 950,000 of cash goes out and 650,000 comes in. What is the runway, in months?

## CAC, payback and what was really added

**Alex Tutor:** On the customer side: **CAC** = spend ÷ new customers. **Payback** is the first month k where the cumulative margin reaches CAC.

**Sofia:** For a Club member at 200 a month with 5% churn and CAC 1,500: month 9 gives 1,479, month 10 gives 1,605. Payback in month 10, and LTV ÷ CAC = 4,000 ÷ 1,500 ≈ 2.7.

**Alex Tutor:** Payback in numbers, before we ask what the spend really bought.

> **Exercise:** A new Klasna Club member cost 1,200 UAH to acquire and leaves 150 UAH of margin a month; 4% of members leave each month. The expected margin in month k is 150 × 0.96^(k − 1), so the cumulative margin after k months is 3,750 × (1 − 0.96^k). In which month does the member pay back the CAC?

**Alex Tutor:** But was every new member caused by the spend? Klasna ran ads only in Lviv. Lviv had 2,000 sign-ups before and 2,300 during the campaign. The rest of the country, the **holdout**, grew 5% with no ads.

**Sofia:** So Lviv would have had 2,000 × 1.05 = 2,100 anyway. Only 200 are **incremental**. With 300,000 spent, that's 1,500 each, not 300,000 ÷ 300 = 1,000.

![Lviv grew from 2,000 to 2,300 sign-ups during the ads; at the holdout's 5% growth it would have reached 2,100 anyway, so only 200 are incremental and incremental CAC is 1,500, not 1,000.](https://productovi.com/courses/math-basics/diagrams/m8-l2-incrementality.png)

*Expected = before × the holdout's growth; incremental = actual − expected.*

![A cat sits on the rim of a bathtub that fills from a tap while water runs out through the drain.](https://productovi.com/courses/math-basics/inline/m8-l2-i1.webp)

*A stock changes only through what flows in and out.*

> **Exercise:** Odesa had 800 sign-ups before a campaign and 980 during it; the holdout grew 10% over the same weeks. The campaign cost 140,000 UAH. What is the incremental CAC, in UAH?

## Your turn: profitable but short of cash

**Alex Tutor:** Hanna's question: in May, Klasna for Teams showed 200,000 of operating profit, but its cash fell by 100,000. Leo thinks something is wrong in the books.

**Sofia:** In May it signed two big clients and invoiced them on 60-day terms. Receivables went up 300,000.

**Alex Tutor:** Then explain it to Leo with the cash formula, and say what would bring the cash back.

> **Exercise:** Explain to Leo, Klasna's CEO, in 3–4 sentences how 200,000 UAH of profit became −100,000 of cash in May (receivables +300,000, other balances unchanged), and what would bring the cash back.

## Case: the Club's first quarter

**Alex Tutor:** The last task of the course's bridge. Hanna wants one page on Klasna Club's first quarter: revenue that stayed, cash left, and what a Lviv campaign really bought.

**Sofia:** MRR bridge and the two retention ratios, then runway, then expected versus actual for Lviv.

**Alex Tutor:** And keep every ratio on its right base. That's where most mistakes hide.

> **Exercise:** Compute the Club's quarter and write Hanna, Klasna's head of finance, a short note.

## Where you will meet this

> **In short**
>
> - **Stock end = start + in − out.** MRR_end = start + new + expansion − contraction − churned.
> - Divide by the **start**: churn = lost ÷ start; **NRR** = (start + expansion − contraction − churned) ÷ start; **GRR** leaves out expansion.
> - **Δcash ≈ profit − ΔAR + ΔAP + Δdeferred**; **WC = AR − AP − deferred**. **Runway** = cash ÷ net burn; **DSO** = AR ÷ credit sales × 365.
> - **CAC** = spend ÷ new; **payback** = first month with cumulative margin ≥ CAC. **Incremental** = actual − before × holdout growth; incremental CAC = spend ÷ incremental.

> **These lessons use it**
>
> - [MRR and churn](/courses/bootstrapped-product/mrr-and-churn): the MRR bridge and churn on the right base.
> - [Profit is not cash](/courses/finance-for-product/profit-is-not-cash): why a profitable month can lose cash.
> - [Burn, runway and staying alive](/courses/finance-for-product/burn-and-runway): runway = cash ÷ average net burn ahead.
> - [Working capital: who pays whom first](/courses/finance-for-product/working-capital): AR − AP − deferred revenue.
> - [Receivables and payment terms](/courses/finance-for-product/receivables-and-terms): what slow payers cost.
> - [Discounted LTV and CAC payback in cash](/courses/finance-for-product/ltv-and-cac-payback-in-cash): payback as the first month with ΣCM ≥ CAC.
> - [CAC, payback and ROI](/courses/product-analytics/cac-payback-roi): CAC and payback for ad cohorts.
> - [Channels and marketing attribution](/courses/product-analytics/channels): what ads really add.

**Alex Tutor:** That's the bridge done. The business side of all this lives in the finance and product-analytics courses; you now have the math to read every formula there.

- Previous lesson: [Contribution margin and break-even](https://productovi.com/courses/math-basics/contribution-and-break-even.md)
- Next lesson: [Simple interest, days and annualizing](https://productovi.com/courses/math-basics/simple-interest-and-day-counts.md)

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