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VEE Accounting and Finance · Project Comparison and Uncertainty · Tutorial 493 of 996

Accounting Break-Even Versus NPV Break-Even

Learn to calculate and interpret three project break-even measures, including how taxes, depreciation, and discounted cash flows separate them.

Advanced 12 min read

What You'll Learn

  • Define accounting break-even as the output level that produces zero accounting earnings.
  • Distinguish zero operating cash flow from cash-cost break-even before taxes.
  • Calculate break-even output when depreciation and tax effects are included.
  • Derive the annual cash flow required for zero NPV.
  • Explain why zero accounting profit does not imply zero NPV.
  • Check break-even calculations against project cash flows and integer output constraints.

Why Three Break-Even Points Differ

A project can report zero accounting income and still fail to recover the cash invested in it. Accounting income includes depreciation, which is not a current cash payment, while net present value (NPV) also recognizes the initial investment and the time value of money. The phrase “break-even” therefore needs a precise definition: the output that makes accounting income zero need not make operating cash flow zero or NPV zero.

The calculations below use a deterministic operating model: the project sells \(Q\) units each year at price \(P\), incurs variable cash cost \(v\) per unit and fixed cash operating costs \(F\), and depreciates its investment by \(D\) per year. Assume no interest expense, constant annual output and costs, and a tax rate \(T\). Where stated, tax benefits from operating losses can be used immediately against other taxable income. There is no probability distribution or mortality assumption in this model; project cash flows are treated as known. The discount rate \(i\) is the effective annual required return.

Definition: Accounting break-even is the output at which accounting earnings are zero. Cash-flow break-even is the output at which operating cash flow (OCF) is zero under a stated tax convention. NPV break-even is the output at which the present value of project cash flows equals the initial investment, so NPV is zero.

Accounting Income and Operating Cash Flow

Let the contribution margin per unit be \(P-v\). Earnings before interest and taxes (EBIT) are revenue less variable costs, fixed cash operating costs, and depreciation:

$$ \mathrm{EBIT}=(P-v)Q-F-D. $$

With no interest expense and with tax effects available immediately, net income is \(\mathrm{EBIT}(1-T)\). Operating cash flow adds back depreciation because it reduced accounting income but did not require a cash payment in the year:

$$ \mathrm{OCF}=\mathrm{EBIT}(1-T)+D =\big((P-v)Q-F-D\big)(1-T)+D. $$

Setting EBIT equal to zero gives the accounting break-even quantity. Setting OCF equal to zero gives the cash-flow break-even quantity under the stated tax assumption. These formulas assume \(P>v\); if \(P\leq v\), increasing output cannot cover positive fixed costs.

$$ Q_{\mathrm{accounting}}=\frac{F+D}{P-v}, \qquad Q_{\mathrm{cash\ flow}}= \frac{F-\dfrac{DT}{1-T}}{P-v}. $$

The second expression follows by expanding OCF as \((P-v)Q(1-T)-F(1-T)+DT\), setting it to zero, and dividing by \(1-T\). It depends on the assumption that a tax benefit from a loss is usable now. If losses cannot generate an immediate tax benefit, the cash-flow calculation below the accounting break-even point changes. In the simplest pre-tax cash-cost convention, cash break-even instead solves \((P-v)Q-F=0\), or \(Q=F/(P-v)\). State which convention is being used rather than treating these quantities as interchangeable.

Worked Example: Comparing Accounting and Cash-Flow Break-Even

A hypothetical equipment project sells output for $50 per unit and has a variable cash cost of $30 per unit. Annual fixed cash operating costs are $120,000. The project costs $100,000, is depreciated straight-line to zero over five years, and has no salvage value. The tax rate is 25%, and operating losses can be used immediately against other taxable income. Find its accounting and cash-flow break-even quantities, then evaluate NPV at accounting break-even. Assume annual output is constant, the required return is 10%, and there are no other cash flows.

Step 1: Find contribution margin and depreciation. Contribution margin is \(P-v=\$50-\$30=\$20\) per unit. Straight-line annual depreciation is \(D=\$100{,}000/5=\$20{,}000\).

Step 2: Find accounting break-even. EBIT is zero when contribution covers both fixed cash costs and depreciation:

$$ Q_{\mathrm{accounting}} =\frac{\$120{,}000+\$20{,}000}{\$20} =7{,}000\text{ units}. $$

At 7,000 units, revenue less variable cost is \(7{,}000(\$20)=\$140{,}000\), so EBIT is \(\$140{,}000-\$120{,}000-\$20{,}000=\$0\). Net income is also zero.

Step 3: Find cash-flow break-even. Using the stated immediate tax-benefit assumption:

$$ Q_{\mathrm{cash\ flow}} =\frac{\$120{,}000-\dfrac{\$20{,}000(0.25)}{0.75}}{\$20} =\frac{\$113{,}333.33}{\$20} =5{,}666.67\text{ units}. $$

At this output, contribution is \(\$113{,}333.33\), EBIT is \(\$113{,}333.33-\$120{,}000-\$20{,}000=-\$26{,}666.67\), and net income is \(-\$26{,}666.67(0.75)=-\$20{,}000\). Adding back depreciation gives OCF of \(-\$20{,}000+\$20{,}000=\$0\), as required. For comparison, the pre-tax cash-cost break-even quantity is \(\$120{,}000/\$20=6{,}000\) units; that convention does not include the immediate tax benefit from losses.

Step 4: Check NPV at accounting break-even. At 7,000 units, EBIT and taxes are both zero, so OCF is the depreciation add-back of $20,000 per year. The five-year annuity factor at 10% is \(a_{\overline{5}|}=3.790787\). Thus:

$$ \mathrm{NPV} =-\$100{,}000+\$20{,}000(3.790787) =-\$24{,}184.26. $$

The project has zero accounting income at this output but a negative NPV. The calculation is checked by noting that the present value of five annual $20,000 cash flows is only $75,815.74, less than the $100,000 initial cost. The common mistake is to treat accounting break-even as investment break-even; depreciation in the income statement does not itself ensure recovery of the initial investment.

Why NPV Break-Even Requires More Cash Flow

For a project requiring initial investment \(I\), with equal OCF at each year-end for \(n\) years and no salvage value, working-capital recovery, or other cash flows, NPV is:

$$ \mathrm{NPV}=-I+\mathrm{OCF}\,a_{\overline{n}|}, \qquad a_{\overline{n}|}=\frac{1-(1+i)^{-n}}{i}. $$

The annuity factor is the sum of the discount factors for the \(n\) year-end cash flows. Indeed, discounting each OCF gives \(\mathrm{OCF}(v+v^2+\cdots+v^n)\), where \(v=1/(1+i)\); summing this geometric series yields the stated annuity factor. Setting NPV to zero and solving for OCF gives the required annual cash flow:

$$ \mathrm{OCF}_{\mathrm{NPV\ break-even}}=\frac{I}{a_{\overline{n}|}}. $$

This formula applies only to equal annual cash flows at the stated period ends and no additional cash flows. If there is a terminal value, working-capital recovery, or an uneven cash-flow pattern, include its discounted value in NPV before solving. A positive accounting profit is not enough to establish positive NPV: the project must generate cash flows sufficient to repay the investment and earn the required return.

Worked Example: Zero Accounting Income but Negative NPV

A hypothetical three-year project requires $90,000 for equipment, depreciated straight-line to zero over three years. Annual fixed cash operating costs are $60,000, price is $120 per unit, and variable cost is $70 per unit. The tax rate is 20%, and tax benefits from losses are immediately usable. There is no salvage value or other cash flow. At what output are accounting income and OCF each zero? At accounting break-even, calculate NPV using an 8% required return.

Step 1: Establish the annual inputs. Contribution margin is \(\$120-\$70=\$50\) per unit, and depreciation is \(\$90{,}000/3=\$30{,}000\) per year.

Step 2: Calculate both break-even quantities. Accounting income is zero when:

$$ Q_{\mathrm{accounting}} =\frac{\$60{,}000+\$30{,}000}{\$50} =1{,}800\text{ units}. $$

The OCF equation is \(\mathrm{OCF}=(\$50Q-\$60{,}000-\$30{,}000)(0.80)+\$30{,}000\), which simplifies to \(\mathrm{OCF}=\$40Q-\$42{,}000\). Therefore, cash-flow break-even is \(Q=\$42{,}000/\$40=1{,}050\) units. Under the pre-tax cash-cost convention, the corresponding quantity would be \(\$60{,}000/\$50=1{,}200\) units.

Step 3: Evaluate the project at accounting break-even. At 1,800 units, contribution is \(1{,}800(\$50)=\$90{,}000\), EBIT is \(\$90{,}000-\$60{,}000-\$30{,}000=\$0\), and OCF is $30,000 after the depreciation add-back. The three-year annuity factor at 8% is \(a_{\overline{3}|}=2.577097\), so:

$$ \mathrm{NPV} =-\$90{,}000+\$30{,}000(2.577097) =-\$12{,}687.09. $$

The discounted cash flows total $77,312.91, which is $12,687.09 below the investment. This verifies the negative NPV despite zero accounting income. The method avoids a common trap: comparing annual accounting income with the initial investment, rather than discounting the project’s cash flows.

Worked Example: Finding the Minimum Output for NPV Break-Even

A hypothetical project requires $60,000 at the start and runs for three years. Its asset is depreciated straight-line to zero over those three years. Annual fixed cash costs are $35,000, price is $75 per unit, variable cash cost is $45 per unit, and the tax rate is 25%. Assume constant annual output, immediate use of any tax benefit, no salvage value, and no other cash flows. The required return is 10%. Find the continuous NPV break-even output and the minimum whole-unit output with nonnegative NPV.

Step 1: Write OCF as a function of output. Depreciation is \(\$60{,}000/3=\$20{,}000\), and contribution margin is \(\$75-\$45=\$30\). Hence:

$$ \mathrm{OCF} =(\$30Q-\$35{,}000-\$20{,}000)(0.75)+\$20{,}000 =\$22.50Q-\$21{,}250. $$

Step 2: Find the OCF needed for zero NPV. At 10% for three years, \(a_{\overline{3}|}=2.486852\). Therefore, the required OCF is \(\$60{,}000/2.486852=\$24{,}126.89\), rounded to the nearest cent.

Step 3: Solve for output. Set the OCF expression equal to the required OCF:

$$ Q_{\mathrm{NPV}} =\frac{\$24{,}126.89+\$21{,}250}{\$22.50} =2{,}016.75\text{ units, approximately}. $$

Since output must be a whole number, check 2,016 and 2,017 units. At 2,016 units, OCF is \(\$22.50(2{,}016)-\$21{,}250=\$24{,}110\), giving NPV \(-\$60{,}000+\$24{,}110(2.486852)=-\$42.00\), rounded to the nearest cent. At 2,017 units, OCF is \(\$22.50(2{,}017)-\$21{,}250=\$24{,}132.50\), giving NPV \(-\$60{,}000+\$24{,}132.50(2.486852)=+\$13.96\). Thus 2,017 units is the minimum whole-unit output for nonnegative NPV. Rounding the continuous threshold down would give the wrong decision.

How to Use the Measures Correctly

The three calculations answer different questions, so report both the definition and the assumptions used. Accounting break-even asks when reported operating income reaches zero. Cash-flow break-even asks when operations provide zero cash under a specified tax convention. NPV break-even asks when discounted project cash flows exactly compensate for the initial investment at the required return.

A frequent exam trap is to confuse cash-cost break-even with after-tax OCF break-even. Before taxes, the cash-cost threshold is \(F/(P-v)\). With depreciation and immediately usable tax benefits, the OCF equation also includes the tax effect of depreciation and of operating losses. Another trap is to ignore the initial investment: OCF of zero means no operating cash is generated, not that the project has recovered its cost. At NPV break-even, by contrast, annual operating cash flow is generally positive and must be large enough to cover the investment after discounting.

When the result is a fractional number of units, retain it for the mathematical threshold, then round in the direction required by the decision. For a minimum whole-unit production level, round up and verify NPV at both the adjacent lower and higher quantities. These distinctions are aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, state the break-even convention and assumptions you use. Round dollar amounts to the nearest cent when needed.

  1. A product sells for $64 per unit, has variable cost of $40 per unit, fixed cash costs of $72,000 per year, and annual depreciation of $24,000. Find accounting break-even output. (Answer: 4,000 units.)
  2. A project has fixed cash operating costs of $50,000 and contribution margin of $20 per unit. Ignoring taxes, find the pre-tax cash-cost break-even output. (Answer: 2,500 units.)
  3. A project has contribution margin of $40 per unit, fixed cash costs of $80,000, depreciation of $20,000, and a 30% tax rate. Assume loss tax benefits are immediately usable. Find the continuous output at which OCF is zero. (Answer: 1,785.71 units, approximately.)
  4. A project has an initial investment of $60,000 and equal year-end OCF for three years. At a 10% required return, use \(a_{\overline{3}|}=2.486852\) to find the annual OCF required for NPV break-even. (Answer: $24,126.89 per year.)
  5. A project costs $60,000 initially, lasts three years, and has annual fixed cash costs of $45,000 and depreciation of $15,000. Price is $52, variable cost is $32, and the tax rate is 25%. Assume annual output is constant, tax benefits are immediately usable, there are no other cash flows, and the required return is 10%. Using \(a_{\overline{3}|}=2.486852\), find the minimum whole-unit output for nonnegative NPV. (Answer: 3,609 units.)

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