How the Three Statements Link Together

"Depreciation goes up by $10. Walk me through the three statements." It is the oldest question on any finance desk, and it is asked for one reason: you cannot answer it without knowing how the statements are wired to each other. Somebody who has memorised what each statement contains will get lost in the second sentence. Somebody who knows the wiring answers in thirty seconds and never gets it wrong.

Articulation is the accountant's word for that wiring. It means the three statements are mathematically forced to agree, because they are three views of the same set of transactions. Every dollar shows up in all three places, in different clothing.

The two identities behind it

The balance sheet is a photograph: assets equal liabilities plus equity, on one date. The other two statements are films — they explain how you got from last year's photograph to this year's.

Equity moves by one relation, the clean surplus:

$$\text{RE}_{\text{end}} = \text{RE}_{\text{begin}} + \text{net income} - \text{dividends}$$

In words: retained earnings is everything the company has ever earned and not paid out, so this year's profit adds to it and this year's dividend subtracts.

Cash moves by the other:

$$\text{cash}_{\text{end}} = \text{cash}_{\text{begin}} + \text{CFO} + \text{CFI} + \text{CFF}$$

In words: the bank balance changed by exactly the sum of the three sections of the cash flow statement — nothing is unexplained and nothing is left over.

Worked example: the $10 of depreciation

Assume a 25 percent tax rate and that the tax authority allows the same deduction, so the tax saving is cash this year.

Income statement. Depreciation rises $10, so pre-tax income falls $10. Tax falls by $10 \times 0.25 = 2.5$. Net income falls $7.50.

Cash flow statement. Start at net income, now $7.50 lower. Add back the depreciation, which is $10 higher. Operating cash flow rises by $-7.5 + 10 = 2.5$. Investing and financing are untouched, so cash rises $2.50.

Balance sheet. Cash is up $2.50 and net property, plant and equipment is down $10, so total assets fall $7.50. On the other side, retained earnings falls $7.50 with net income. Both sides move by the same amount, so it balances.

Read the punchline carefully: the company reported $7.50 less profit and ended the year with $2.50 more cash. Depreciation itself moved no money — the money left when the asset was bought. The only genuine cash consequence is the tax shield, and it is exactly $2.50.

Worked example: a transaction that never touches profit

Articulation is easiest to see on a transaction the income statement ignores entirely. A retailer buys $100 of inventory on 30-day credit.

Period 1. No sale has happened, so revenue and cost of goods sold are both zero and net income is unchanged. On the balance sheet, inventory rises $100 and accounts payable rises $100. On the cash flow statement, net income is zero, the inventory build is $-100$ and the payables build is $+100$, so operating cash flow is zero. Cash is unchanged. All three agree that nothing economic has happened yet.

Period 2. The goods sell for $150 cash and the supplier is paid. Revenue $150, cost of goods sold $100, pre-tax profit $50, tax $12.50, net income $37.50. On the cash flow statement: net income $37.50, inventory releases $+100$, payables unwind $-100$, so operating cash flow is $37.50. On the balance sheet cash rises by $150 - 100 - 12.5 = 37.5$, inventory falls $100 and payables fall $100. Assets fall $62.50, liabilities fall $100, equity rises $37.50 — and $-62.5 = -100 + 37.5$. It balances.

The three checks

When a model is wired correctly, three tests pass every period, and if any fails the error is always in the wiring rather than the assumptions.

1. Assets equal liabilities plus equity, every column, to the cent.
2. Ending cash on the balance sheet equals beginning cash plus the total of the cash flow statement.
3. The retained earnings roll-forward ties: opening balance, plus net income, minus dividends, equals the closing balance.

Where the wiring leaks

Articulation is exact for a single entity that does nothing unusual. Two things routinely break the naive version. First, acquisitions: buying a company makes receivables and inventory jump on the balance sheet with no operating cash flow, which is why filings label the working-capital lines "net of effects of acquisitions". Second, foreign currency translation, which moves balance sheet accounts through equity rather than through profit and gets its own line at the bottom of the cash flow statement. When a balance sheet delta refuses to match a cash flow line, one of those two is almost always the reason.

Key terms

• Articulation — the property that the three statements are forced by construction to agree.
• Clean surplus — closing retained earnings equals opening plus net income minus dividends.
• Hinge — the two connection points: net income, and the net change in cash.
• Tax shield — the cash saved because a non-cash charge is deductible.
• Circularity — the interest/debt/cash loop that a three-statement model must resolve iteratively.