Introduction
Single transactions are the building blocks; real businesses chain them together. A company borrows to buy a machine, depreciates it for several years, pays interest on the loan, sells the machine before the end of its life and repays part of the debt with the proceeds. Each step is simple on its own. Strung across several years, the steps interact: depreciation in year two depends on the purchase in year one, interest in year three depends on how much debt was repaid in year two, and the gain on the sale depends on everything that came before.
Multi-year scenarios are how bankers test whether a model's statements are wired correctly over time, and they appear constantly in technical interviews as "walk me through year one, then year two." The method is the same as for any single transaction, applied once per year, as set out in tracing a transaction through the statements. What changes is the bookkeeping discipline: knowing which figures reset every year and which carry forward, then testing that discipline on four complete scenarios, from a debt-financed machine to a legal accrual that is partly reversed.
The Rules That Carry Across Years
Two kinds of numbers behave differently from one year to the next, and keeping them apart is most of the work.
- Flows reset each year: revenue, expenses, net income and every line of the cash flow statement measure one period only. Year two's depreciation is year two's charge, not the running total.
- Balances carry forward: every balance sheet line ends one year and opens the next at the same figure. Net property, plant and equipment (PP&E), debt, retained earnings and cash accumulate the flows of every year so far.
The link between them is the roll-forward: each year's closing balance equals the opening balance plus that year's flows, as described in PP&E and debt roll-forwards. In a multi-year scenario, the year-one trace produces the opening balance sheet for year two, and so on.
- Cumulative effect
The total change in a balance sheet line from the start of a scenario to a given date, equal to the sum of the flows that affected it in every year so far. Cumulative effects are what the balance sheet shows; annual effects are what the income statement and cash flow statement show.
A useful check follows. Over the full life of a transaction, every timing difference reverses: depreciation, net of any gain or loss on sale, adds up to the cash spent on the asset less what it is sold for, accruals are paid, receivables are collected. So the cumulative change in cash equals cumulative net income adjusted only for the net cash raised from or returned to lenders and shareholders, and the cumulative change in retained earnings equals cumulative net income less dividends. If a multi-year trace does not converge that way, a year has been linked incorrectly.
Scenario One: Borrow, Buy, Depreciate, Sell
A company borrows $1,000 at 10% interest at the start of year one and uses it to buy a machine with a five-year life, depreciated straight line at $200 a year. The tax rate is 25%, and tax depreciation is assumed to equal book depreciation, so no deferred taxes arise. To isolate the transaction, ignore any revenue the machine produces. At the end of year three, the company sells the machine for $500 and uses the proceeds to repay $500 of the debt.
Year One: The Purchase and the First Year's Costs
Year one combines the borrowing, the purchase, a full year of depreciation and a full year of interest:
Pre-tax income falls by $300, the $200 of depreciation and $100 of interest; tax falls by $75, so net income falls by $225. On the cash flow statement, operating cash flow is minus $25: the $100 of interest paid less the $75 of tax saved, since depreciation used no cash. The $1,000 purchase and the $1,000 borrowing cancel. On the balance sheet, net PP&E is $800 after a year's depreciation, debt is $1,000, cash is $25 lower and retained earnings $225 lower: assets up $775, matched by $1,000 of debt less $225 of equity.
Year Two: The Same Flows, Larger Cumulative Effects
Year two has no new transaction, only another year of the same costs. The income statement and cash flow statement repeat year one's operating lines exactly: depreciation $200, interest $100, tax benefit $75, net income minus $225, operating cash flow minus $25. Nothing appears in investing or financing.
The balance sheet is where the years accumulate. Net PP&E falls from $800 to $600, cash is now $50 below its starting level, debt is still $1,000, and retained earnings are $450 lower than before the scenario began. The year-two trace started from year one's closing balances, not from zero, which is exactly the discipline that matters: an answer that reported net PP&E of $800 at the end of year two would have reset a balance that should have carried forward.
Year Three: Selling the Machine and Repaying Debt
At the end of year three, after a third year of depreciation, the machine has a net book value of $400. Selling it for $500 produces a $100 gain, and the proceeds repay $500 of debt:
The gain partly offsets the year's costs: pre-tax income falls by $200 and net income by $150. Operating cash flow is minus $50, the $100 of interest paid less $50 of net tax saved, because the tax on the gain absorbs part of the shield from depreciation and interest. The gain is removed from operating activities, the full $500 of proceeds appears in investing and the $500 repayment in financing. On the balance sheet, net PP&E falls by the remaining $600: $200 of depreciation and the $400 carrying amount of the machine sold.
The Scenario in One View
Laying the three years side by side shows the flows resetting and the balances accumulating:
| Line item | Year 1 | Year 2 | Year 3 |
|---|---|---|---|
| Income statement | |||
| Depreciation | (200) | (200) | (200) |
| Interest expense | (100) | (100) | (100) |
| Gain on sale | 0 | 0 | 100 |
| Tax benefit | 75 | 75 | 50 |
| Net income | (225) | (225) | (150) |
| Cash flow statement | |||
| Cash flow from operations | (25) | (25) | (50) |
| Cash flow from investing | (1,000) | 0 | 500 |
| Cash flow from financing | 1,000 | 0 | (500) |
| Change in cash | (25) | (25) | (50) |
| Balance sheet, cumulative change at year end | |||
| Cash | (25) | (50) | (100) |
| PP&E, net | 800 | 600 | 0 |
| Debt | 1,000 | 1,000 | 500 |
| Retained earnings | (225) | (450) | (600) |
Every year balances: the cumulative change in assets equals the cumulative change in debt plus retained earnings.
The cumulative check works. Over three years, net income totals minus $600, and retained earnings are $600 lower. Cash is $100 lower, which equals the $600 cumulative net loss, plus $600 of cumulative depreciation, less the $100 gain, minus the net $500 invested ($1,000 paid, $500 recovered), plus the net $500 borrowed: −600 + 600 − 100 − 500 + 500 = −100. The economics are also visible: the machine cost $1,000 and returned $500, so it consumed $500 of capital, and the company paid $300 of interest, together $800 of cost before the $200 of total tax shield, the same $600 after-tax cost that net income recorded.
A Variation: Keeping the Machine to the End
Had the company kept the machine for its full five years and never repaid the debt, years three to five would simply repeat year one's operating lines: net income down $225 and operating cash flow down $25 each year. By the end of year five, net PP&E would be back to zero, the full $1,000 having passed through depreciation, debt would still be $1,000, cumulative net income would be minus $1,125 and cash would be $125 lower. The difference between those two figures, $1,000, is exactly the depreciation: the machine's whole cost reached the income statement over five years, while its cash cost was paid on day one with borrowed money. Repaying the loan at the end would then reduce cash and debt by $1,000 with no further income statement effect.
A Variation: Changing the Machine's Life
Suppose instead that at the start of year two, management concludes the machine will last only two more years rather than four. A change in useful life is a change in accounting estimate, applied prospectively from the period of change without restating year one, as PwC's guide to determining useful lives explains. The $800 remaining carrying amount is depreciated over two years, so book depreciation rises to $400 in years two and three. If tax depreciation stays at $200, only book income falls, and the extra $200 a year of book depreciation creates a $50 deferred tax asset each year rather than any cash saving. Year two's net income falls by $375 instead of $225, while operating cash flow is still minus $25: a large earnings change with no cash consequence, the case traced in depreciation on the three statements.
- Change in accounting estimate
A revision of a judgment used in the accounts, such as an asset's useful life, salvage value or an allowance for credit losses, made because of new information. It is applied prospectively, in the current and future periods, without restating prior years.
The variation also changes the sale in year three. With $400 of depreciation in each of years two and three, the machine's net book value at the end of year three is zero, so the $500 sale produces a $500 gain for the books rather than $100. For tax, the machine still has $400 of tax basis, so the taxable gain is still $100, and the $100 deferred tax asset built up in years two and three reverses on the sale. Cash is identical to the original scenario in every year; only the income statement timing and the deferred tax balance differ.
Scenario Two: Inventory Bought, Sold and Collected
Working capital transactions also play out over several years. A distributor buys $300 of inventory on credit in year one. In year two it sells the goods for $500, on credit, and pays its supplier. In year three it collects from the customer. The tax rate is 25%, paid in the year profit is recognized.
| Line item | Year 1 | Year 2 | Year 3 |
|---|---|---|---|
| Income statement | |||
| Revenue | 0 | 500 | 0 |
| Cost of goods sold | 0 | (300) | 0 |
| Income tax expense | 0 | (50) | 0 |
| Net income | 0 | 150 | 0 |
| Cash flow from operations | |||
| Net income | 0 | 150 | 0 |
| Change in inventory | (300) | 300 | 0 |
| Change in accounts payable | 300 | (300) | 0 |
| Change in accounts receivable | 0 | (500) | 500 |
| Cash flow from operations | 0 | (350) | 500 |
| Balance sheet, at year end | |||
| Inventory | 300 | 0 | 0 |
| Accounts payable | 300 | 0 | 0 |
| Accounts receivable | 0 | 500 | 0 |
| Cash, cumulative change | 0 | (350) | 150 |
Cumulative cash from operations over three years equals cumulative net income of 150.
The pattern is the essence of working capital. In year one, the inventory build is financed entirely by the supplier, so neither profit nor cash moves. In year two, the company records all of its profit, $150, yet its cash falls by $350: it pays the supplier $300 and the tax $50 while its $500 of revenue is still owed by the customer. In year three, it records no profit at all, yet collects $500. Over the three years, cash rises by $150, exactly matching net income, because every working capital timing difference has reversed. The single-year mechanics are traced in receivables, payables and accruals on the statements.
Scenario Three: A Customer Pays Three Years in Advance
The reverse timing arises when customers pay before delivery. A software company receives $300 at the start of year one for a three-year subscription and recognizes $100 of revenue each year. Ignoring costs, and with tax at 25% paid as revenue is recognized:
| Year | Revenue | Net income | Change in deferred revenue | Operating cash flow | Deferred revenue at year end |
|---|---|---|---|---|---|
| 1 | $100 | $75 | Up $200 | $275 | $200 |
| 2 | $100 | $75 | Down $100 | $(25) | $100 |
| 3 | $100 | $75 | Down $100 | $(25) | $0 |
| Total | $300 | $225 | Nil | $225 |
Year one shows the cash arriving up front: $300 collected, less $25 of tax, gives $275 of operating cash flow against only $75 of net income. In years two and three, revenue continues while no new cash arrives from this customer, and the tax on the recognized revenue makes operating cash flow negative. Over the three years, cash and net income both total $225.
The tax assumption changes the yearly pattern but not the total. If the company were taxed on the full $300 when received, it would pay $75 of tax in year one, recording a $50 deferred tax asset for the tax paid on revenue not yet recognized. Operating cash flow would be $225 in year one and zero in years two and three, as the deferred tax asset unwinds against each year's tax expense, while net income would still be $75 a year. The mechanics behind each year are in deferred revenue on the three statements.
Scenario Four: An Accrual Booked, Paid and Reversed
Some multi-year chains involve estimates that turn out wrong. A company records a $100 liability in year one for a legal claim it expects to settle. For tax, the cost is deductible only when paid. In year two it pays $60 toward the claim, and in year three the claim is settled with no further payment, so the remaining $40 is reversed. The tax rate is 25%.
| Line item | Year 1 | Year 2 | Year 3 |
|---|---|---|---|
| Income statement | |||
| Legal expense (reversal) | (100) | 0 | 40 |
| Current tax benefit (expense) | 0 | 15 | 0 |
| Deferred tax benefit (expense) | 25 | (15) | (10) |
| Net income | (75) | 0 | 30 |
| Cash flow from operations | |||
| Net income | (75) | 0 | 30 |
| Change in accrued liability | 100 | (60) | (40) |
| Deferred income taxes | (25) | 15 | 10 |
| Cash flow from operations | 0 | (45) | 0 |
| Balance sheet, at year end | |||
| Accrued liability | 100 | 40 | 0 |
| Deferred tax asset | 25 | 10 | 0 |
| Cash, cumulative change | 0 | (45) | (45) |
Cumulative net income of (45) equals the cumulative change in cash: 60 paid less 15 of tax saved.
The year-one charge reduces net income by $75 with no cash effect: the liability and a $25 deferred tax asset both appear. In year two, the $60 payment uses cash and reduces the liability, and the tax deduction finally arrives, saving $15 of cash and using up $15 of the deferred tax asset, with no effect on net income at all. In year three, reversing the unneeded $40 raises net income by $30 after the deferred tax asset is cleared, again with no cash. Over the full life, the company lost $45 in cash and $45 in net income; the three years simply spread the recognition, the cash and the tax differently. A reversal like year three's is also why bankers treat large reserve releases as non-recurring income.
Patterns to Take From the Scenarios
The four scenarios produce a short list of lessons that apply to any multi-year question or model:
- Balances accumulate and flows do not: every year's trace begins from the prior year's closing balance sheet, and every balance sheet answer is cumulative. Most errors in multi-year questions come from resetting a balance, such as net PP&E or retained earnings, that should have carried forward.
- Timing differences reverse over the full life: depreciation, net of any gain or loss on sale, adds up to the asset's cost less the sale proceeds, working capital changes cancel, deferred revenue is earned and accruals are paid or reversed. Cumulative operating cash flow therefore converges on cumulative net income plus cumulative non-cash charges, which is why a persistent gap over many years, rather than in one, signals a genuine issue in a real company's accounts.
- Investment and financing flows explain the rest: over the life of scenario one, cash fell by far less than the after-tax cost of the machine and the loan because half the debt was still outstanding at the end. In a real business, the same logic explains why a company can report years of profits while its cash falls, if it keeps investing, or years of losses while its cash rises, if it keeps borrowing.
- Deferred taxes absorb the gaps between book and tax timing: whenever an item is recognized in one year for the books and another for tax, a deferred tax balance carries the difference across the years until it reverses.
In models, the same discipline applies with more lines and more years. Interest is usually calculated on the opening or average debt balance for each period; using the average makes interest depend on repayments, which depend on cash, which depends on interest, a circularity models resolve with iterative calculation or an opening-balance convention. The modeling treatment is set out in the Valuation guide's article on building the three-statement model. Whatever the tool, the test is the one this article uses: each year must balance, and over the full life of every transaction, profit and cash must tell the same story.
