Finance-Theory · Unit 20 · Video 3 · Interactive Practice

After-Tax Cash Flow and the Depreciation Tax Shield

IKey Formulas

Formula	Name	What you need
$\text{OP} = \text{Revenues} - \text{Opex}_{\text{w/o } D}$	Operating profit	Cash revenues and cash operating expenses
$\text{Taxes} = \tau \cdot \text{OP} - \tau \cdot D$	Project income taxes	Tax rate $\tau$ , depreciation $D$
$\text{CF} = (1 - \tau)\,\text{OP} - \text{CapEx} + \tau D$	After-tax cash flow	All of the above, plus capital expenditures
$D = \dfrac{\text{Cost}}{\text{Life}}$ , shield $= \tau D$	Straight-line depreciation and its tax shield	Asset cost and useful life

Key Insight: Depreciation is not a cash payment; it reaches the cash flow only through the tax it saves, $\tau D$ . In every year of the asset's life, after-tax earnings fall short of after-tax cash flow by exactly $D$ .

IITwo Routes from the Same Machine

Equipment costing $600,000, depreciated over 6 years: the accountant subtracts $D$ , the cash flow subtracts only the tax.

Tax rate τ 25%

IIISame Totals, Different Timing

The same project at $\tau = 25\%$ (cost $600,000, revenues $420,000, operating expenses $180,000): cash flow $205,000 and after-tax earnings $105,000 a year. Over the whole life the two streams add up to the same total, but the cash leaves at date 0.

Date 0

💡 An NPV discounts each date's cash, so the stream you choose matters: the earnings column spreads the $600,000 outlay over six years that the cash column pays at date 0.

IVThe Schedule Moves the Shield, Not Its Total

A faster depreciation schedule changes when the tax shield arrives, never how much it adds up to.

Depreciation schedule

💡 Current US law lets most qualifying equipment acquired after 19 January 2025 be written off in full in its first year; the shields still sum to $\tau \times \text{Cost}$ , they simply arrive sooner.

VQuiz Questions

Problem 1 · Annual After-Tax Cash Flow

Given: equipment costing $600,000, depreciated straight-line over 6 years; revenues $450,000 a year, operating expenses (without depreciation) $190,000 a year, tax rate $\tau = 15\%$ . Find the after-tax cash flow in each of years 1–6.

✅ Correct! The project keeps 85% of its $260,000 operating profit and adds the $15,000 tax shield.

❌ Not quite. That is

(1 - \tau)\,\text{OP}

alone. Depreciation lowers the tax bill, and the saving

\tau D

is cash that stays in the firm.

❌ Not quite. Depreciation is not a cash inflow; only the tax it saves,

\tau D = 0.15 \times 100{,}000

, enters the cash flow, not the full

D

❌ Not quite. That is the after-tax accounting earnings,

(1 - \tau)(\text{OP} - D)

. It subtracts depreciation as if cash left the firm, so it understates the cash by

D

❌ Not quite. Use

\text{CF} = (1 - \tau)\,\text{OP} + \tau D

with no capital expenditure in years 1–6.

Show solution

Straight-line depreciation and operating profit:

D = \frac{600{,}000}{6} = 100{,}000, \qquad \text{OP} = 450{,}000 - 190{,}000 = 260{,}000

There is no capital expenditure in years 1–6, so

\text{CF} = (1 - 0.15)(260{,}000) + 0.15(100{,}000) = 221{,}000 + 15{,}000 = 236{,}000

The after-tax cash flow is $236,000 a year. Check through the tax bill: taxable income is $260{,}000 - 100{,}000 = 160{,}000$ , the tax is $0.15 \times 160{,}000 = 24{,}000$ , and $260{,}000 - 24{,}000 = 236{,}000$ .

Problem 2 · A Faster Schedule

Given: the same project (cost $600,000, revenues $420,000, operating expenses $180,000), now with $\tau = 20\%$ . The firm switches from 6-year to 4-year straight-line depreciation. Find the change in the project's year-1 after-tax cash flow.

✅ Correct! The extra $50,000 of year-1 depreciation saves

0.20 \times 50{,}000

of tax, and that saving is cash.

❌ Close, but… Depreciation itself is non-cash, yet it is deductible: a larger deduction lowers the tax actually paid, and the tax is a cash outflow.

❌ Not quite. That is the change in after-tax earnings,

-(1 - \tau)\,\Delta D

. Earnings fall because the accountant subtracts the larger charge; the cash rises because the tax bill shrinks.

❌ Not quite. The extra $50,000 of depreciation is not cash received; only the tax it saves,

\tau\,\Delta D

, changes the cash flow.

❌ Not quite. Only the shield term

\tau D

depends on the schedule.

Show solution

Year-1 depreciation rises from $600{,}000/6 = 100{,}000$ to $600{,}000/4 = 150{,}000$ , so $\Delta D = 50{,}000$ . In $\text{CF} = (1 - \tau)\,\text{OP} + \tau D$ only the last term changes:

\Delta \text{CF} = \tau\,\Delta D = 0.20 \times 50{,}000 = 10{,}000

Directly: $0.8(240{,}000) + 0.2(150{,}000) = 222{,}000$ versus $0.8(240{,}000) + 0.2(100{,}000) = 212{,}000$ , an increase of $10,000. After-tax earnings move the other way, from $0.8(140{,}000) = 112{,}000$ to $0.8(90{,}000) = 72{,}000$ , a fall of $40,000.

Problem 3 · The Date-by-Date Table

Given: equipment costing $600,000 bought at date 0, depreciated straight-line over 6 years; revenues $400,000 and operating expenses $170,000 in each of years 1–6; $\tau = 35\%$ . Find the annual after-tax cash flow, the tax actually paid each year, and the sum of the after-tax cash flows at dates 0–6.

After-tax cash flow in each of years 1–6?

Tax actually paid each year?

Sum of the after-tax cash flows at dates 0–6?

✅ Correct! The undiscounted cash total equals six years of after-tax earnings,

6 \times 84{,}500 = 507{,}000

: the streams differ only in timing.

❌ Check the cash flow. That is

(1 - \tau)\,\text{OP}

alone. Depreciation lowers the tax bill, and the saving

\tau D = 35{,}000

is cash that stays in the firm.

❌ Check the cash flow. That is the after-tax accounting earnings,

(1 - \tau)(\text{OP} - D)

. It subtracts depreciation as if cash left the firm each year, so it understates the cash by

D

❌ Check the cash flow. That adds the whole

D

back to

(1 - \tau)\,\text{OP}

. Depreciation is not a cash inflow; only the tax it saves,

\tau D

, enters the cash flow.

❌ Check the tax. That is

\tau \times \text{OP}

, the tax with no depreciation deduction. Taxable income is operating profit minus depreciation.

❌ Check the tax. That is the tax shield

\tau D

, the tax the deduction saves, not the tax paid.

❌ Check the tax. That is the taxable income,

\text{OP} - D

. The tax is

\tau

times it.

❌ Check the sum. That adds the six annual cash flows but leaves out date 0, where the $600,000 purchase is a negative cash flow.

❌ Check the sum. Nothing forces the undiscounted total to zero: it is the date-0 outlay plus six positive annual cash flows.

❌ Check the sum. That charges the $600,000 twice: once at date 0 and again through the depreciation inside six years of after-tax earnings. Dates 1–6 carry the cash flow, not the earnings.

❌ Check the cash flow. Keep

1 - \tau

of operating profit and add the shield

\tau D

, not the whole

D

and not nothing.

❌ Check the tax. The tax is

\tau

times taxable income, and taxable income is operating profit minus depreciation.

❌ Check the sum. Date 0 carries the $600,000 purchase as a negative cash flow; dates 1–6 each carry the annual after-tax cash flow.

Show solution

$D = 600{,}000/6 = 100{,}000$ and $\text{OP} = 400{,}000 - 170{,}000 = 230{,}000$ . In each year $t = 1, \dots, 6$ :

\text{CF}_t = (1 - 0.35)(230{,}000) + 0.35(100{,}000) = 149{,}500 + 35{,}000 = 184{,}500

\text{Taxes} = 0.35 \times 230{,}000 - 0.35 \times 100{,}000 = 80{,}500 - 35{,}000 = 45{,}500

Check: $230{,}000 - 45{,}500 = 184{,}500$ . The date-0 cash flow is the capital expenditure, $-600{,}000$ , so

\sum_{t=0}^{6} \text{CF}_t = -600{,}000 + 6 \times 184{,}500 = 507{,}000

Annual cash flow $184,500, tax $45,500, total $507,000. After-tax earnings are $0.65 \times 130{,}000 = 84{,}500$ a year and also sum to $507{,}000$ , because the six depreciation charges add up to the $600,000 outlay.

Problem 4 · Full Write-Off in Year 1

Given: the same equipment (cost $600,000), now with revenues $380,000 and operating expenses $160,000 in each of years 1–6, $\tau = 30\%$ , but the full $600,000 is deducted as depreciation in year 1 and nothing afterwards. The project shows a taxable loss in year 1; the firm's other profits absorb it, so the whole tax shield is realised that year. Find the after-tax cash flow in year 1 and in year 2.

Year-1 after-tax cash flow?

Year-2 after-tax cash flow?

✅ Correct! The whole $180,000 of shield arrives in year 1 instead of $30,000 a year; the cash total over years 1–6 is unchanged at $1,104,000.

❌ Check year 1. That is the 6-year straight-line cash flow, with a $100,000 deduction. Here the whole $600,000 is deducted in year 1, so the shield is

\tau \times 600{,}000

❌ Check year 1. That is

(1 - \tau)\,\text{OP}

with no shield. The $600,000 deduction saves

0.30 \times 600{,}000

of tax in year 1, and that saving is cash.

❌ Check year 1. That is the after-tax accounting result,

(1 - \tau)(\text{OP} - D_1)

, which charges the whole $600,000 as if it were paid in year 1. The cash flow adds the depreciation back: only the tax saved enters.

❌ Check year 2. That still counts a $30,000 shield from 6-year straight-line. After a full write-off,

D_2 = 0

and there is no shield left.

❌ Check year 2. The asset is fully depreciated, but the project still earns operating profit; it keeps

1 - \tau

of it.

❌ Check year 2. That is the operating profit before tax. With

D_2 = 0

the full operating profit is taxed, so the project keeps

(1 - \tau)\,\text{OP}

❌ Check year 1.

\text{CF}_1 = (1 - \tau)\,\text{OP} + \tau D_1

with

D_1 = 600{,}000

. The accounting loss,

(1 - \tau)(\text{OP} - D_1)

, is not the cash flow.

❌ Check year 2. With nothing left to depreciate,

D_2 = 0

: the project keeps

1 - \tau

of its operating profit and has no shield.

Show solution

$\text{OP} = 380{,}000 - 160{,}000 = 220{,}000$ every year. Depreciation is $D_1 = 600{,}000$ and $D_2 = \dots = D_6 = 0$ .

\text{CF}_1 = (1 - 0.30)(220{,}000) + 0.30(600{,}000) = 154{,}000 + 180{,}000 = 334{,}000

\text{CF}_2 = (1 - 0.30)(220{,}000) + 0.30(0) = 154{,}000

Year 1 is $334,000 and years 2–6 are $154,000 each. The project's year-1 tax is $0.30(220{,}000 - 600{,}000) = -114{,}000$ : the firm's other profits absorb the loss, so the project lowers the firm's total tax bill by $114,000. Under 6-year straight-line the cash flow would be $0.7(220{,}000) + 0.3(100{,}000) = 184{,}000$ every year. Both schedules give $334{,}000 + 5 \times 154{,}000 = 6 \times 184{,}000 = 1{,}104{,}000$ over years 1–6; the write-off simply moves $150,000 of shield into year 1, which raises the NPV at any positive discount rate.

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